Forms and Concepts for Lightweight Structures 052143274X, 9780521432740

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Forms and Concepts for Lightweight Structures Covering a wide range of structural concepts and presenting both relevant theories and their applications to actual structures, this book brings together for the first time lightweight structures concepts for many different applications and the relevant scientific literature, thus providing unique insights into a fascinating field of human endeavor. Evolved from a series of graduate courses taught by the authors at the University of Tokyo, the Institute of Space and Astronautical Science, the University of Cambridge, and the California Institute of Technology, this textbook provides both theoretical and practical insights and presents a range of examples that also provide a history of key lightweight structures since the Apollo age. This essential guide will inspire the imagination of engineers and provide an analytical foundation for all readers. Koryo Miura is a prominent inventor in the field of space structures and Professor Emeritus at the University of Tokyo and the Institute of Space and Astronautical Science, JAXA. Sergio Pellegrino is a leading academic researcher in lightweight and deployable structures, Joyce and Kent Kresa Professor of Aerospace and Civil Engineering at the California Institute of Technology, and a senior research scientist at the NASA Jet Propulsion Laboratory.

Forms and Concepts for Lightweight Structures K O RY O M I U R A University of Tokyo

SERGIO PELLEGRINO California Institute of Technology

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9780521432740 DOI: 10.1017/9781139048569 © Cambridge University Press 2020 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2020 Printed in the United Kingdom by TJ International Ltd, Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Miura, Koryo, 1930– author. | Pellegrino, S. (Sergio) author. Title: Forms and concepts for lightweight structures / Koryo Miura and Sergio Pellegrino. Description: First edition. | New York : Cambridge University Press, 2020. | Includes bibliographical references and index. Identifiers: LCCN 2019038141 (print) | LCCN 2019038142 (ebook) | ISBN 9780521432740 (hardback) | ISBN 9781139048569 (epub) Subjects: LCSH: Lightweight construction. | Structural engineering. Classification: LCC TA663 .M58 2020 (print) | LCC TA663 (ebook) | DDC 624.1–dc23 LC record available at https://lccn.loc.gov/2019038141 LC ebook record available at https://lccn.loc.gov/2019038142 ISBN 978-0-521-43274-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface

page ix

1

Truss Structures 1.1 Introduction 1.2 Rigidity Theory 1.3 Rod-Like Trusses 1.4 Examples 1.5 Rigidity Computations

1 1 2 11 13 18

2

Space Frames 2.1 Introduction 2.2 Classification 2.3 Continuum Models for Single-Layer Space Frames 2.4 In-Plane Efficiency of Single-Layer Space Frames 2.5 Continuum Models for Double-Layer Space Frames 2.6 Bending Efficiency of Double-Layer Space Frames 2.7 Continuum Models for Multiple Layer Space Frames 2.8 Applications

28 28 33 38 44 48 51 54 56

3

Tension Structures 3.1 Introduction 3.2 Three Simple Cable Nets 3.3 Soap Films 3.4 Shape of Hanging Cables 3.5 Shape of Zero-Superpressure Balloons 3.6 Pneumatic Domes 3.7 Pressurized Membrane Cylinders 3.8 Computational Form-Finding Methods

59 59 61 70 73 76 82 84 88

4

Tension-Stabilized Structures 4.1 Introduction 4.2 Prestressed Stayed Columns 4.3 Tensegrity Structures

96 96 96 103 v

vi

Contents

4.4 Tensegrity Domes 4.5 Parachutes and Other Decelerators 4.6 Tension Field Beams

112 116 123

5

Shell Structures 5.1 Introduction 5.2 Why Shell Structures? 5.3 Membrane vs. Bending Action 5.4 Shell Morphology 5.5 Polyhedral Shells

128 128 131 135 139 151

6

Sandwich Structures 6.1 Introduction 6.2 Theory of Sandwich Plates 6.3 Shape Efficiency of Core 6.4 Foam Core 6.5 Honeycomb Core 6.6 Corrugated Core and Zeta-Core

158 158 161 168 173 177 181

7

Packaging of Membranes 7.1 Introduction 7.2 Geometric Conditions 7.3 Unfolding of Tree Leaves 7.4 Biaxial Folding: Miura-Ori 7.5 Biaxial Folding: Wrapping 7.6 Foldable Cylinders

189 189 191 196 198 209 214

8

Concepts for Deployable Structures 8.1 Introduction 8.2 Packaging through Elastic Deformation 8.3 Articulated Space Frames 8.4 Packaging through Elastic Creasing 8.5 Concepts for Deployment and Deployment Actuation 8.6 Concepts for Stabilization

224 224 227 235 249 252 263

9

Applications of Deployable Structures 9.1 Introduction 9.2 Deployable Booms and Masts 9.3 Deployable Reflector Antennas 9.4 Tension Truss and AstroMesh Antennas 9.5 Springback Reflector 9.6 Inflatable Antenna Structures 9.7 Deployable Mirrors and Solar Sails

280 280 280 290 293 297 298 303

Contents

10

Adaptive Structures 10.1 Adaptivity 10.2 Actuators 10.3 Structural Adaptivity to Increase Material Utilization 10.4 Concepts for Shape Adaptive Trusses 10.5 Variable Geometry Truss 10.6 Homologous Structures

vii

306 306 311 314 320 325 329

Appendix A Geometric Foundations A.1 Introduction A.2 Polyhedra and Tessellations A.3 Surfaces A.4 Curvature of Surfaces A.5 Introduction to Geometric Modeling

337 337 337 347 351 363

Appendix B Structural Mechanics Foundations B.1 Introduction B.2 Stress and Strain Tensors B.3 Generalized Stresses in Rods, Plates, and Shells B.4 Generalized Strains in Rods, Plates, and Shells B.5 Constitutive Relationships B.6 Equilibrium Equations for Axisymmetric Shells B.7 Structural Theorems B.8 Buckling of Columns B.9 Calculus of Variations

366 366 366 369 372 373 375 376 378 380

Appendix C

381

Essential Bibliography

Appendix D Bibliography Index

382 397

Preface

Our motivation in writing this introduction to the wide range of structural forms and concepts for lightweight structures, a field where people of outstanding creativity have combined shapes and materials in fascinating ways, is to inspire the next generation of students, engineers, and inventors. In structural engineering, the design process can be divided into three main steps. First, the specification of the requirements imposed on the structure: its function, environment, global dimensions, accuracy, endurance, cost, and so on. Second, the proposition of one or more structural concepts to meet these requirements; this is the most creative and potentially ground-breaking step, and is the focus of this book. Third, the assessment of the proposed concept, through analysis and experiment, to establish if it is compatible with the initial requirements, and to determine suitable values for the design variables, to obtain the final design of the structure. Space exploration and the digital revolution in manufacturing have led to major advances in structural concepts and architectures. Gravity loading, the main load for large earthbound structures, becomes tiny in orbit and much smaller than solar pressure. Designers of large spacecraft structures had to search for new concepts, adequate for the requirements and environment of space missions, and we have participated in this effort. In the education of our students, we have found that explaining the process of searching for new structural forms and concepts has filled our students with excitement and enabled them to make their own professional contributions. The book is organized around ten main topics, each starting with a short introduction about its significance and followed by an in-depth treatment of the underlying concepts. Each chapter is in fact the subject of many textbooks, specialized monographs and research papers, all listed in Appendix D. We expect that many readers of this book will have a mechanics background at a graduate level, but we have presented the material in a way that makes it accessible to a much wider readership, by presenting foundational material on relevant geometry and structural mechanics, in two extensive appendixes. Also, Appendix C lists the core textbooks that cover fundamental aspects. We have been helped by many people of great inventiveness and unique insight. We are particularly grateful to Chris Calladine and Martin Mikulas and we would also like to pay tribute to the inspiration of the late John Hedgepeth. The AIAA Spacecraft Structures technical committee and the Structural Morphology working group of the International Association for Shell and Spatial Structures have provided a nurturing ground for the discussion and maturation of many of the ideas presented in this book. ix

x

Preface

The following institutions have generously provided facilities and financial support during the preparation of this book: the California Institute of Technology; Corpus Christi College, Cambridge; the Institute of Space and Astronautical Science, Sagamihara; the Royal Society of London; the University of Cambridge; and the University of Colorado at Boulder. We are grateful to several colleagues, students, and collaborators for helping us at various stages of writing. In particular, Chris Calladine, Simon Guest, Ken Livesley, and Tibor Tarnai provided comments on early drafts, and Charlie Dorn, Harmanj Hasanian, Christophe Leclerc, Yang Li, Michael Marshall, Harsha Reddy, Fabien Royer, Alex Wen, and Hesham Zaini helped us with the final proofreading. We also thank Mariella Soprano for generously hosting our long writing sessions, Tomoko Miura for providing a beautiful drawing of Miura-ori for the book cover, Martin Mikulas and William Weber for taking photos for us that have been included in this book.

1

Truss Structures

1.1

Introduction A truss structure, also called a pin-jointed structure, is a structure consisting of straight, slender members, known as bars, connected by frictionless spherical joints. The joints allow each bar to swivel and twist, unless these motions are restricted by connections to other bars. Joints of this kind are an abstraction, as real connections are usually rigidly jointed, but in fact many rigidly jointed structures can be usefully modeled as pin-jointed. Changing the end conditions of the bars in this way has little effect on the overall response of the structure, provided that the pin-jointed version of the actual structure is kinematically determinate. This concept will be explained in Section 1.2. Consider a structure whose straight members are made from a high-modulus material and are rigidly connected to one another. Imagine releasing all of the rotational degrees of freedom at the connections. This change will have little effect on the overall response of the structure if the stress distribution in the constrained rotation members is mainly axial, and so the bending and shearing stresses are a “secondary,” rapidly decaying effect near the joints. This will generally be the case for a kinematically determinate structure. The stiffness of this structure is mainly derived from its axial mode of action, where its members are either in uniform tension or uniform compression, as the stiffness provided by the bending mode of action is several orders of magnitude smaller. Thus, the global behavior of the structure is fully captured by the pin-jointed model of the structure, whereas its local behavior – which is important to determine local stress concentrations, and hence to check the safety of the structure against fracture and failure – needs a detailed stress analysis of the connections. Figure 1.1 shows three planar structures used for a static equilibrium experiment by the engineering undergraduates at the University of Cambridge: the geometrical layout of the three structures is the same, but their construction is different. The first structure is made of thin-walled, square section steel tubes with welded connections; the second is made of extruded aluminum-alloy profiles bolted to gusset plates; and the third is made of thin-walled carbon-fiber-reinforced-plastic (CFRP) tubes (made by winding a carbon filament coated with epoxy onto a mandrel and curing the epoxy in a furnace) bonded to aluminum-alloy joint fittings that are connected by steel pins. During the experiment, the students measure the axial forces in the members of the trusses by comparing the strain in each member of a truss to the strain in the directlyloaded, vertical link under the truss – of identical construction to the rest of the truss. 1

2

Truss Structures

Figure 1.1 Photographs of three truss models in Structures Laboratory at University of

Cambridge.

Figure 1.2 Idealized, pin-jointed model of truss structures shown in Fig. 1.1.

They find that the axial forces in corresponding members are practically identical.1 They also find that the values of the axial forces can be accurately estimated by analyzing the truss shown in Fig. 1.2.

1.2

Rigidity Theory The key question that we want to deal with is whether or not a given pin-jointed framework is rigid.2 This straightforward question is a surprisingly difficult one to 1 Note that this experiment is concerned only with the linear-elastic behavior of the three structures. If the

loads were increased into the nonlinear range, and up to failure, the behavior of the three structures would no longer be identical. 2 By rigid we mean a structure that does not deform at all if is assumed that the bars are inextensional, i.e., do not change their length.

1.2 Rigidity Theory

3

answer. Indeed, in many cases it can be answered in full only by carrying out a detailed analysis, or by testing a physical model of the structure. In two dimensions, i.e., for the case of structures that lie in a plane, the easiest way of constructing a rigid truss is by arranging the bars to form a sequence of triangles. A triangle consisting of pin-jointed bars is the simplest two-dimensional rigid structure, and two triangles with a side in common also form a rigid structure in two dimensions (but not in three dimensions, as one of the triangles can move out of plane by rotating about the common side). To extend this approach to three dimensions, i.e., to structures in a three-dimensional (Euclidean) space, one can use the simplest three-dimensional structure that is rigid, the tetrahedron (there will be more on this later), or alternatively one can form a closed surface that is completely triangulated. These approaches are often followed in the design of practical structures, but there are also many rigid structures that are not triangulated. Hence, it is of great importance to have a general way of telling whether or not a general three-dimensional structure is rigid. A general method to find the answer computationally will be given in Section 1.5. A simpler question, that can be answered much more directly, is whether a pin-jointed structure contains a sufficient number of members to be rigid. The answer is to count the total number of degrees of freedom of its joints and to subtract the number of degrees of freedom suppressed by applying kinematic constraints to the joints, and by connecting pairs of joints by means of bars. In two dimensions, each joint has two degrees of freedom, corresponding to two independent translation components, and hence for a structure with j joints the total number of degrees of freedom is 2j . Denoting by k the total number of kinematic constraints, where, for example, connecting a joint to a foundation counts as two because it suppresses both translation components, and by b the total number of pin-jointed bars – each bar counts one as it imposes a single “distance” constraint between the joints it connects – we require that 2j − k − b ≤ 0

(1.1)

This is known as Maxwell’s equation (Maxwell, 1864). Consider, for example, the structure shown in Fig. 1.3(a). It consists of four triangles, the first of which is connected to a foundation, and hence it is obviously a rigid structure. Substituting j = 6,k = 4,b = 8 (obviously, there is no need for a bar between the two foundation joints) into Eq. 1.1 we obtain 2×6−4−8=0 Hence, we conclude that this structure has (just) enough bars to be rigid. It is important to realize that a structure that has enough bars to be rigid may not in fact be rigid, as its bars may be “incorrectly” placed. For example, if in Fig. 1.3(a) we re-locate the bar bracing the left-hand square, so that the right-hand square is now doubly-braced, as shown in Fig. 1.3(b), we obtain a structure that still satisfies Eq. 1.1 and yet is clearly not rigid. In this case we have a single-degree-of-freedom mechanism, Fig. 1.4(a). A structure that admits no mechanisms is called kinematically determinate.

4

Truss Structures

(a)

(b)

Figure 1.3 Examples of two-dimensional pin-jointed structures that are (a) fully triangulated and

hence rigid, (b) a mechanism.

+1

+1

−√2

−√2

+1

+1

(a)

(b)

Figure 1.4 Mechanism (exaggerated amplitude of a small-amplitude motion) and state of

self-stress of structure shown in Fig. 1.3(b).

Note that the doubly braced square on the right-hand side of the structure in Fig. 1.3(b) admits a state of self-stress, i.e., there is a set of non-zero bar forces that are in equilibrium with zero external forces, as shown in Fig. 1.4(b). A structure that admits no states of self-stress is called statically determinate. Denoting by m the number of independent mechanisms of a structure, and by s the number of states of independent states of self-stress, for the structure of Fig. 1.3(a) we have s = 0 and m = 0 (statically and kinematically determinate), whereas for the structure of Fig. 1.3(b) we have s = 1 and m = 1 (statically and kinematically indeterminate). Here, by independent we mean that if any mechanism is represented by a vector, whose components correspond to the tangent motions of the joint, and any state of self-stress by a vector whose components correspond to the bar forces, it is not possible to obtain one of the vectors as a linear combination of the others. So, Maxwell’s equation in the form of Eq. 1.1 is only a necessary condition for the kinematic determinacy of pin-jointed structures, but not a sufficient condition. It will be shown in Section 1.5 that the general, and most useful way, of writing Maxwell’s equation is: dj − b − k = m − s

(1.2)

1.2 Rigidity Theory

5

(b) (a)

(c) Figure 1.5 Examples of simple three-dimensional trusses (Pellegrino and Calladine, 1986).

where d = 2, or 3 depending on the dimensions of the (Euclidean) space in which the structure is considered. Consider the three-dimensional structures, d = 3, shown in Fig. 1.5. The tripod structure in Fig. 1.5(a) has a single free joint plus three fully constrained joints; so j = 4 and k = 9. The unconstrained joint is connected by three non-coplanar bars, b = 3, to the foundation joints. It has no states of self-stress, s = 0, as the condition for the joint to be in equilibrium in three different directions without external forces requires that the bar forces be zero. Substituting into Eq. 1.2 gives: 3×4−3−9=0=m−0

(1.3)

from which the number of mechanisms is m = 0. Having established that s = 0 for the structure of Fig. 1.5(a), obviously s will remain unchanged if a bar is removed, Fig. 1.5(b). Hence, for this structure j = 3, k = 6, and b = 2. Substituting into Maxwell’s equation: 3×3−2−6=m−0

(1.4)

which gives m = 1. The mechanism involves a rotation of the two bars about an axis passing through the two foundation joints, as shown in Fig. 1.5(b). By an analogous argument, the structure of Fig. 1.5(c), which is obtained by adding a bar to the structure of Fig. 1.5(a), has m = 0 and, from Maxwell’s equation, s = 1. Figure 1.6 shows two examples of pin-jointed structures that are topologically identical to the structure in Fig. 1.5(a), i.e., they have the same numbers of joints, bars, and constraints; but now the bars are coplanar. These structures admit a state of self-stress, e.g., a tension in the two inclined members equilibrated by a compression in the vertical member.

6

Truss Structures

(a)

(b)

Figure 1.6 Examples of pin-jointed structures that are both statically and kinematically

indeterminate. In (a) the three bars are coplanar and the three foundation joints are collinear (Pellegrino and Calladine, 1986).

Since the left-hand side of Eq. 1.3 is unchanged, but now s = 1, here m = 1. In both structures the mechanism is identical to that shown in Fig. 1.5(b), but whereas Fig. 1.6(a) is a finite mechanism, in Fig. 1.6(b) only a small-amplitude motion of the mechanism is possible. This is because the central foundation joint is aligned with the other two in Fig. 1.6(a) but not in Fig. 1.6(b). The truss structure in Fig. 1.6(b) is a simple example of an infinitesimal mechanism. If a structure of this kind is made with infinitely rigid members and perfectly fitting joints, it would admit only an infinitesimal motion of its mechanism. In practice, of course, its members will be elastic and there will be some tolerance in the joints; hence, the stiffness of the structure will be of a “lower order” than that of a normal, kinematically determinate structure. Note that in the mathematics literature on structural rigidity a rigid structure is any structure that is either kinematically determinate or indeterminate but with mechanisms that are only infinitesimal (Connelly, 1993). Engineers tend to use the definition of rigid structures adopted here, which includes the smaller class that admit no mechanisms at all. The existence of structures with infinitesimal mechanisms was first discovered by J. Clerk Maxwell (1864), but it was only more recently that it was realized that they can be given a first-order (geometric) stiffness through a state of prestress (Calladine, 1986). This property has been successfully exploited in the design of prestressed cable nets, see Section 3.2, and tensegrity structures, see Section 4.3.

1.2.1

Polyhedral Trusses Figure 1.7 shows five trusses based on the five platonic polyhedra, more details of which can be found in Appendix A.2. The simplest of these structures is the tetrahedral truss; from Table A.1 j = 4,b = 6, and k = 0. Hence, Maxwell’s equation gives: 3×4−6−0=6=m−s

(1.5)

Because there are only three noncoplanar bars meeting at each joint, for which three equations of equilibrium can be written, the bar forces have to be equal zero if the

1.2 Rigidity Theory

(a) Tetrahedral

7

(b) Cubic

(c) Octahedral

(e) Icosahedral

(d) Dodecahedral Figure 1.7 Regular polyhedral trusses.

external loads are zero. Therefore s = 0 and so, from Eq. 1.5, m = 6. Because the truss has six rigid-body mechanisms as a free body in three-dimensional space, i.e., three independent translations and three rotations, these are the only mechanisms of the truss. Denoting by m the number of independent internal mechanisms, we have m = 0 for the tetrahedral truss, i.e., it is internally rigid. Next, consider the cubic truss, Fig. 1.7(b). From Table A.1, j = 8,b = 12, and k = 0; Maxwell’s equation gives: 3 × 8 − 12 − 0 = 12 = m − s

(1.6)

Because s = 0, which can be shown by the same argument as for the tetrahedral truss, Eq. 1.6 gives: m = 12

8

Truss Structures

Table 1.1 Static and (internal) kinematic determinacy of polyhedral trusses. Shape

s

m

Tetrahedron Cube Octahedron Dodecahedron Icosahedron

0 0 0 0 0

0 6 0 24 0

of which six are rigid-body motions, as above, and the remaining six are internal mechanisms. For example, six independent mechanisms are obtained by deforming each square of the truss into a rhombus. Repeating the same analysis for the remaining trusses it is found that m − s = 6 for the octahedral and icosahedral trusses, but m − s = 30 for the dodecahedral truss. Then, since it can be shown that s = 0 for all of them – although the proof is not straightforward for the octahedral and icosahedral trusses – it can be concluded that the octahedral and icosahedral trusses are internally rigid, but not the dodecahedral truss. These results are summarized in Table 1.1. Note that the five trusses based on the platonic polyhedra can all be regarded as tessellations of triangles, squares and pentagons on a sphere. Also note that only the tessellations of triangles have turned out to be rigid; the cube and the dodecahedron – consisting of tessellations of squares and pentagons, respectively – have many mechanisms. This result was to be expected, in light of the earlier comment, in Section 1.1, on the rigidity of triangulated surfaces.

1.2.2

Cauchy’s Theorem The rigidity of a truss consisting of a tessellation of triangles that lie on a sphere follows from a theorem proved by Cauchy, together with several other theorems for polygons and polyhedra. Theorem 13 of Cauchy (1813) states that: In a convex polyhedron with invariable faces the angles at the edges are also invariable, so that with the same faces one can build only a polyhedron symmetrical to the first one. Thus, every convex polyhedron with rigid faces will be rigid and, since the simplest way of forming a rigid face with pin-jointed bars is to use a triangle, Cauchy’s theorem can also be stated in the specialized form: Every convex polyhedral surface is rigid if all of its faces are triangles. An example of a truss whose rigidity follows from Cauchy’s theorem is shown in Fig. 1.8. This structure has been obtained by considering arcs of great circles that join the vertices of an icosahedron – which by definition lie on a sphere – and by locating an additional joint at the mid-point of each arc. Then, each joint has been connected

1.2 Rigidity Theory

9

Figure 1.8 Truss structure obtained by adding a series of mid-arc nodes to an icosahedron. The

nodes of the original icosahedron are at the center of the pentagons.

with a bar to all of its neighbors. The resulting truss structure has j = 42, as 12 joints coincide with the vertices of the icosahedron, plus there are 30 joints at the mid-points of the great circle arcs. The number of bars is equal to twice the number of edges of the icosahedron, E, plus three times the number of faces, F , whose values are given in Table A.1. Hence b = 2E + 3F = 120. Maxwell’s equation gives m − s = 6 and, since s = 0, the only mechanisms are the six rigid motions. Despite the restriction in Cauchy’s theorem, that the surface should be convex, mathematicians had conjectured for over 150 years that in fact all surfaces consisting of triangles are rigid, even those surfaces that are not convex3 . This was known as the “rigidity conjecture,” which was finally proven to be wrong by a counter-example devised by Connelly (1978). Since it took so long to find a counter-example, we can safely state that “almost all” triangulated surfaces are rigid. This means that one is very unlikely to ever encounter a simply connected triangulated surface of any shape that is not rigid.

1.2.3

Flexible “Sphere” Several examples of concave triangulated structures that admit an infinitesimal motion were found over the years, but none whose motion was finite. Connelly’s discovery of a counter-example to the “rigidity conjecture,” which he called a flexible sphere, led to the subsequent discovery of several such structures by other authors. One of these examples is shown next. Figure 1.9 shows a model, made from the cutting pattern in Fig. 1.10: the pattern is meant to be scaled up on a photocopying machine so that the edge numbers should be lengths in centimeters. This gives a size that is easy to work with. On the pattern, curved

3 Note that the surfaces that we are considering here are simply connected, i.e., topologically identical to a

sphere. Toroidal surfaces, for example, are excluded.

10

Truss Structures

Motion

Figure 1.9 Perspective view of Connelly–Steffen “flexible sphere” made from cutting pattern in

Fig. 1.10 (Dewdney, 1991).

1 cm

Fold up Fold down Join edges 6

6 8.5

6

6 6

5

2.5

6

5

5

5

5.5

2.5

2.5

5.5

2.5

2.5

2.5 6

2.5

6 5

5

2.5

6 5

6 5

Figure 1.10 Cutting pattern for Connelly–Steffen “flexible sphere” (Dewdney, 1991).

arrows indicate pairs of edges that should be attached, e.g., by leaving a tab on one side and gluing it under the other side. After making your own model, try rotating the upper triangle relative to the lower one (not shown in Fig. 1.9): it will move without any resistance until two internal triangles come into contact. Note that while the structure moves, there is no sign of it stiffening

1.3 Rod-Like Trusses

11

up, as would happen in an infinitesimal mechanism. This indicates that the structure is a finite mechanism.

1.3

Rod-Like Trusses The static and kinematic properties of a truss are very important in the design of highperformance structures. As an illustration, here we will use the ideas of kinematic and static determinacy to guide the design of a rod-like truss; we will return to this topic in Section 10.5 when we discuss adaptive structures. Several different kinds of rod-like trusses can be obtained by interconnecting polyhedral trusses of the type described in Section 1.2.1. The first requirement will be that the structure is rigid (m = 0), as this guarantees that the structure can carry any load in its axial mode (stiff), which will hence lead to a low-mass design. The second requirement will be that the structure is statically determinate (s = 0). There are two main reasons for this, first the force distribution in a statically determinate structure is uniquely determined by equilibrium considerations, and so disturbances such as thermal gradients, or manufacturing imperfections will not lead to overstressing and, second, ease of assembly. Static determinacy guarantees that none of the bars are redundant and so, if the structure is assembled, say, by joining together a series of prefabricated elements, each time we attach a new member the distance between the end joints is not already determined by the preexisting members. Another advantage of statically determinate truss structures is that their shape can be adjusted, and indeed varied by large amounts by modifying the length of one member at a time, without inducing any self-stress in the structure. This feature will be exploited in Section 10.5. Substituting the above values for m and s into Maxwell’s equation, Eq. 1.2, and assuming that k = 6 kinematic constraints will be introduced after the design of the structure has been completed, to locate the structure with respect to a foundation 3j − b − 6 = 0 Rearranging j=

b +2 3

and, assuming j and b to be large j≈

b 3

(1.7)

It follows that we need to consider designs where the ratio between the number of bars and the number of joints is equal to three. Therefore, because each bar is attached to two joints, on average there should be six bars connected to each joint. This result is not restricted to rod-like trusses, but holds for any three-dimensional truss.

12

Truss Structures

(a)

(b)

Figure 1.11 Stacks of tetrahedral and octahedral trusses.

(a)

(b)

Figure 1.12 Rod-like trusses topologically equivalent to trusses in Fig. 1.11.

The third requirement is that the truss should consist of identical modules, because a repetitive design requires a smaller number of different parts and, hence, is simpler and cheaper to produce. Of the three rigid polyhedral trusses that have been identified in Section 1.2.1 only the tetrahedron and the octahedron lend themselves to forming a rod-like structure, and the

1.4 Examples

13

resulting trusses are shown in Fig. 1.11. Both of these structures are rigid, because their “building blocks” are rigid and are connected to each other through a common triangle, which does not allow any relative translation or rotation between the building blocks. Of the two structures, the stack of octahedra, Fig. 1.11(b), is easiest to visualize, because the common triangles lie in parallel planes. It has three-fold rotational symmetry about a central, vertical axis, and it has also mirror symmetry through three vertical planes. The stack of tetrahedra, Fig. 1.11(a), is more difficult to visualize as the triangles that are common to consecutive modules do not lie in parallel planes. The resulting structure, called tetrahelix by Buckminster Fuller (1975), has helical symmetry. In practice, these two “designs” are not used in the present form. Instead, since it is often desirable for structures to have straight, continuous longitudinal members – e.g., to reduce the axial forces in the members due to overall axial or bending loads, and thus increase the stiffness and strength of the structure – the forms that are normally used are the topologically equivalent structures shown in Fig. 1.12. Their members have different geometrical lengths.

1.4

Examples In this section we discuss a series of three-dimensional trusses with interesting properties. These structures illustrate the type of behavior that can be encountered when designing trusses, and also form a useful “catalogue” of unexpected effects that one should watch for. Figure 1.13 shows a simple “space frame,” of which some larger scale examples can be found in Section 2.2, consisting of pin-jointed bars – all of equal length – that form a skew cube with surface diagonals. It has j = 8, b = 18, and k = 0 and, substituting these values into Maxwell’s equation 3 × 8 − 18 − 0 = 6 = m − s

(1.8)

which gives m − s = 6. This truss is rigid, because it consists of an octahedral truss – a rigid structure – rigidly connected to two tetrahedra – also rigid. Therefore, its total

Figure 1.13 Space frame module consisting of an octahedron and two tetrahedra; the outer joints lie on six planar faces.

14

Truss Structures

a

a

(a)

+1

D

–1 A

–2

–1

–1

+1

+1

–1

+1

+1

B –2 –1 +1

+1

C (b)

Figure 1.14 (a) Space frame module consisting of five tetrahedra and four half-octahedra (the six nodes in the upper plane are denoted by hollow circles whereas the six nodes in the bottom plane are denoted by solid circles). (b) Shows the state of self-stress, where members carrying compression are drawn thicker.

number of mechanisms is m = 6; these mechanisms are all rigid-body mechanisms. From Eq. 1.8, s = 0. Figure 1.14 shows a more complex space frame, also consisting of bars of equal length a. As the previous example, this structure is also a module for a commonly-used space frame; more details will be provided in Section 2.2. This truss consists of two unbraced squares, √ plus two identical squares in a plane parallel to the first set, and at a distance a/ 2. The joints of the upper square are connected by diagonal members to the joints of the lower square. For this structure j = 12, b = 30, and k = 0 and, substituting into Maxwell’s equation 3 × 12 − 30 − 0 = 6 = m − s

(1.9)

which gives m − s = 6, as before. This structure, however, is not statically determinate. It has s = 1, and Fig. 1.14(b) shows a self-equilibrated set of bar forces. It can be verified by inspection that these axial forces form a state of self-stress. The eight members on the outer edge of the structure form a continuous “tension loop” carrying a force of one unit; at every kink in the loop equilibrium is ensured by a compression of one unit in the member that is locally coplanar with the tension loop,

1.4 Examples

15

D A

A, E

D, H

C B

(a) H E G

C, G

B, F

F

D

D C E

A

H

B

(b)

A

H

C

E G

G

F

F

B

C

C B

D

E

H

A

(c)

D

B

H

E

G F

G

F A

Figure 1.15 Perspective and top views of (a) cubic truss, and trusses obtained by rotating the upper square through (b) 45◦ and (c) 135◦ .

whereas the fourth member – which is not coplanar with the other three – carries no force. The equilibrium of the innermost four joints, e.g., joint A, is ensured by tensions of one unit in the diagonal members, e.g., AB and AC, and a compression of two units in, e.g., AD.

16

Truss Structures

Having shown that this truss has s = 1, it follows from Eq. 1.9 that m = 7. Hence, in addition to the six rigid-body mechanisms, there is an internal mechanism in which the whole truss twists as the squares distort out of plane. The next example is the cubic truss shown in Fig. 1.15(a), with members of length a lying on√ the edges of a cube plus four diagonal bracing members on the side faces, of length a 2. The four joints at the bottom are fully constrained. This truss has j = 8, b = 12, and k = 12, and substituting these values into Maxwell’s equation gives 3 × 8 − 12 − 12 = 0 = m − s

(1.10)

Having found that m − s = 0, it can then be shown by the matrix method of Section 1.5 that this truss is both statically and kinematically determinate (m = s = 0). Now, consider varying the shape of this truss by rotating the upper square in an anticlockwise sense, without translating. Obviously, the lengths of both the diagonal members and of the members that were originally vertical, e.g., AF and AE respectively, vary during this process. Tarnai (1980) has shown that the coefficient matrix of the system of equilibrium equations – further details on this matrix are given in Section 1.5 – has full rank, equal to 12, normally. However, the matrix becomes rank deficient, with rank of 11, in four special configurations. Of these, the configurations that are of greatest practical interest are those that are obtained for a rotation of the upper square through 45◦ , Fig. 1.15(b), and 135◦ , Fig. 1.15(c). In each of these configurations the static and kinematic properties of the structure change from m = s = 0 to m = s = 1. The states of self-stress and mechanisms for these configurations are shown in Figs. 1.16 and 1.17. In Fig. 1.16(a) note that the bar forces in the top square are alternatively positive and negative as one goes round the square, whereas in Fig. 1.17(a) the bar forces in the top square are all of the same sign. There is an important difference between these two special configurations: in the first one the mechanism allows a finite amplitude distortion of the structure, whereas the mechanism of the second configuration allows only an infinitesimal motion. –0.96 +0.96

+1

–1

–0.96

+0.96 +1

–1

–1 –1 +1

(a)

+1

(b)

Figure 1.16 (a) State of self-stress and (b) inextensional mechanism of first special configuration of cubic truss.

1.4 Examples

17

+1 +1

+1.04

+1

+1

–1.98

+1.04

+1.04

–1.98

–1.98 +1.04

(a)

–1.98

(b)

Figure 1.17 (a) State of self-stress and (b) inextensional mechanism of second special configuration of cubic truss.

In practice, imposing a state of prestress on the first structure, e.g., by varying the length of one of its members with a turn-buckle, is impossible, as the structure will change shape instead of becoming self-stressed4 . On the other hand, the structure of Fig. 1.15(c) has the same type of behavior of the structure in Fig. 1.6(b). It is an example of a tensegrity structure, and will be discussed further in Section 4.3. Tarnai (1980) shows that the existence of special configurations that are both statically and kinematically indeterminate is a general feature of trusses based on two interconnected regular polygons with n-sides (in Fig. 1.15 n = 4). In particular, configurations that admit finite amplitude inextensional mechanisms exist for all trusses with n even and ≥4. However, for n odd there are no such special configurations. Truss structures with a layout similar to Fig. 1.15(b) have been used for several applications, often in preference to the layout in Fig. 1.15(a), because their higher degree of symmetry leads to the expectation of a “more uniform” stiffness distribution. An example is the telescope structure shown in Fig. 1.18, here a truss structure with the same layout as that shown in Fig. 1.15(b) supports the secondary mirror. This is a standard design for Cassegrain configuration telescopes, but requires a stiff ring beam at the top of the truss. As we have seen, a three- or five-sided truss would be mechanismfree and hence would not require such a massive ring beam. In addition to the ring trusses described above, another family of trusses that show a strange pattern of static and kinematic indeterminacy are triangulated hyperbolic paraboloids. Figure 1.19 shows a truss structure whose joints lie on a hyperbolic paraboloid, see Appendix A.3. Its edge joints lie on parabolas and are subject to vertical constraints, as shown in Fig. 1.19(a). Three additional constraints are applied to the whole surface, see Fig. 1.19(b), to prevent it from translating horizontally and rotating about a vertical axis. Pellegrino (1988) has shown that this particular structure has m = s = 4 and, in general, a structure whose joints form a n × n grid has m = s = n − 2. Structures with

4 This was pointed out by Kuznetsov (1991), who called such structures unprestressable.

18

Truss Structures

(b) (a) Figure 1.18 (a) 3D model of William Herschel Telescope, Roque de los Muchachos Observatory, La Palma (4.2 m diameter) and (b) view inside dome (images used with permission of Isaac Newton Group of Telescopes, La Palma, 2019).

(a)

(b)

Figure 1.19 Triangulated hyperbolic paraboloid; (a) perspective view; (b) top view.

different boundary shapes, e.g., straight, or with joints on a rectangular n × n grid also produce variable degrees of static and kinematic indeterminacy (Pellegrino, 1988).

1.5

Rigidity Computations Consider a truss structure with j joints that are subject to k kinematic constraints, and b bars. Let d be the number of dimensions of the space in which the structure is to be analyzed (d = 2 or 3). By setting up a system of nr = dj − k

(1.11)

1.5 Rigidity Computations

19

I

tHJ sin αHJ J y

tHI sin αHI

αHI H

tHJ cos αHJ

pHx

αHJ

pH x

tHI cos αHI

x

pHy

pHy (a)

(b)

Figure 1.20 Equilibrium of a general pin-joint.

linear equations of equilibrium that relate the nc = b

(1.12)

axial forces in the bars, arranged in the vector t, to the external load components, arranged in the vector p, we obtain the equilibrium matrix A for this structure. Hence, At = p

(1.13)

Usually nr = nc , and hence the equilibrium matrix is rectangular. Some details on a simple way of setting up this equilibrium matrix are given next. Consider joint H of a two-dimensional pin-jointed structure, connected by bar HI to joint I and by bar HJ to joint J, as shown in Fig. 1.20(a). Let αH I and αH J be the angles between these bars and the x-axis, respectively. These angles are defined to be positive if anti-clockwise. Figure 1.20(b) shows a free-body diagram for joint H, showing the x- and y-components of the bar forces acting on this joint. The bar forces (tH I in bar HI, etc.) are assumed to be positive if tensile. The external force applied to joint H has components PH x and PHy in the x- and y-directions. For equilibrium in the x-direction, we must have cos αH I tH I + cos αH J tH J + pH x = 0 or, in a form more suitable for Eq. 1.13 − cos αH I tH I − cos αH J tH J = pH x An analogous equilibrium equation in the y-direction can also be written. Hence, the equilibrium equations for joint H are  − cos αH I tH I − cos αH J tH J = pH x (1.14) − sin αH I tH I − sin αH J tH J = pHy

20

Truss Structures

I

αHI

dHx cosαHI H

dHx

Figure 1.21 Relationship between change of length of a bar and one of the four components of joint displacement.

A pair of equilibrium equations like this can be written for each unconstrained joint of a pin-jointed structure and, by grouping together these equations, we obtain a system whose coefficient matrix is the equilibrium matrix A. A set of dual equations to the equilibrium equations is the set of nb = b compatibility equations, that relate the extensions of the bars, arranged in the vector e and assumed to be small, to the displacement components of the joints, arranged in the vector d and also assumed to be small. These two vectors are related by the kinematic matrix B: Bd = e

(1.15)

The compatibility equation for a bar is obtained by considering the changes in bar length that are induced by each of the four displacement components of its joints, one of which is shown in Fig. 1.21. For a general bar HI, the compatibility equation is − cos αH I dH x − sin αH I dHy + cos αH I dI x + sin αH I dIy = eH I

(1.16)

It can be shown, by inspection or by virtual work (Livesley, 1975), that the equilibrium matrix is equal to the transpose of the kinematic matrix, hence Eq. 1.15 can be written in the form: AT d = e

(1.17)

The number of independent inextensional mechanisms, m, and the number of independent states of self-stress s can be calculated, after determining the rank r – recall that the rank of a matrix is equal to the number of linearly independent rows or columns (Strang, 1980) – of the equilibrium matrix, from the equations m = nr − r

(1.18)

s = nc − r

(1.19)

Subtracting Eq. 1.19 from Eq. 1.18 and substituting Eqs 1.11 and 1.12 respectively for nr and nc yields the generalized Maxwell’s equation in a form that coincides exactly with Eq. 1.2. Table 1.2 introduces a general classification scheme for truss structures, which identifies four types. Note that the actual values of m and s are not important: what matters

1.5 Rigidity Computations

21

Table 1.2 Classification of truss structures. Type

Static and kinematic properties

Equilibrium and kinematic equations

I

s=0 m=0

Statically determinate and kinematically determinate

Both Eq. 1.13 and Eq. 1.15 have a unique solution for any p and e.

II

s=0 m>0

Statically determinate and kinematically indeterminate.

Eq. 1.13 has one solution for some particular p, otherwise no solution. Eq. 1.15 has infinitely many solutions for any e.

III

s>0 m=0

Statically indeterminate and kinematically determinate.

Eq. 1.13 has infinitely many solutions for any p. Eq. 1.15 has one solution for some particular e, otherwise no solution.

IV

s>0 m>0

Statically indeterminate and kinematically indeterminate.

Eq. 1.13 has infinitely many solutions for some particular p, otherwise no solution. Eq. 1.15 has infinitely many solutions for some particular e, otherwise no solution.

is if they are zero or not: if m = 0 the truss is kinematically determinate (i.e. rigid if the bars are assumed to have fixed length), and if s = 0 the truss is statically determinate (i.e. the bar forces depend only on the external forces).

1.5.1

SVD of Equilibrium Matrix It is shown in Linear Algebra (e.g., Strang, 1988) that for any matrix A of dimensions nr × nc and rank r, there exist: • • •

a nr × nr orthogonal matrix U = [u1, . . . ,unr ] (by orthogonal matrix we mean that UT U = I, i.e., that the vectors ui are orthonormal); a nc × nc orthogonal matrix W = [w1, . . . ,wnc ]; and a nr × nc matrix V with r positive elements vii (i = 1, . . . r) on the leading diagonal: all other elements are zero;

such that A = UVWT See Fig. 1.22 for a pictorial representation of this decomposition.

(1.20)

22

Truss Structures

r

nr-r=m v11 0

nr

A

Ur

Unr-r

T

Wr

r

vrr 0

0

T

Wnc-r

nc-r=s

nc Figure 1.22 Graphical illustration of SVD of equilibrium matrix A = UVWT (Pellegrino, 1993).

The coefficients vii are the singular values of A and the vectors ui and wi are, respectively, the i-th left singular vector and the i-th right singular vector. This is called the Singular Value Decomposition (SVD) of the matrix A, and generalizes the spectral decomposition of a square matrix. The SVD is described in textbooks on matrix computations, such as Golub and Van Loan (1983) where further details, proofs, etc. can be found. For our purposes, given A, all that is needed is a way of computing its SVD; the easiest way of doing it is to type svd(A) in Matlab (Mathworks, 2018). Once the SVD of the equilibrium matrix has been computed, the value of its rank has to be decided. The problem is that, although the definition of the SVD at the beginning of this section states that V contains only r non-zero elements, in practice one often finds that the leading diagonal of V contains a number of singular values of decreasing magnitude, none of which is actually equal to zero although some may be quite small. For accurate and stable computations a threshold value  has to be set, whose value depends on an acceptable round-off error and on the accuracy to which the nodal coordinates are defined: all singular values smaller than  are treated as zero, and the rank of A is defined accordingly. The value of  can be set after the full set of singular values has been calculated and in fact, in particularly sensitive calculations, its choice can be based upon information provided by the singular values themselves. See Pellegrino (1993).

1.5.2

Physical Interpretation of SVD Because of the correspondence between statics and kinematics described above, the singular value decomposition of the equilibrium matrix can be interpreted in two different, but corresponding ways. Figure 1.23(a) shows that the first r left singular vectors are the load systems in equilibrium with the sets of bar forces in the corresponding right singular vectors, multiplied by the corresponding singular values. The remaining s = nc −r right singular vectors are independent states of self-stress. Similarly, Fig. 1.23(b) shows that the first r left singular vectors are the displacement systems compatible with the sets of bar extensions in the corresponding right singular

1.5 Rigidity Computations

×

in equilibrium with .

.

. . . .

.

.

0

0

a set of loads which cannot be carried

bar forces

. . . .

0

. . . .

p a set of loads

23

σ

a state of self-stress

(a)

×1

compatible with .

.

. . . .

d

.

0 .

0

0

a zero-energy mode a set (mechanism) of joint displacements (b)

bar extensions

. . . . . . . .

ε

a set of incompatible strains

Figure 1.23 The first r left- and right-singular vectors are in one-to-one correspondence both

statically and kinematically (Pellegrino, 1993).

vectors, divided by the corresponding singular values. The remaining m = nr − r left singular vectors are independent inextensional displacement modes, or mechanisms. Furthermore, it can be shown that the last m left singular vectors are load conditions which the truss cannot equilibrate in its current configuration, and also that the last s right singular vectors are orthogonal sets of incompatible strains. Note that the SVD is a way of finding bases5 for the four fundamental subspaces of a matrix – the rowspace, the nullspace, the column space, and the left-nullspace (Strang, 1980). Although computationally more expensive than alternative techniques based on row operations (Pellegrino and Calladine, 1986), two advantages of the approach presented here are that (i) ill-conditioning of the equilibrium matrix poses no problems, and (ii) the one-to-one static and kinematic correspondence between the left singular vectors in Ur and the corresponding right singular vectors in Wr , explained above, is very useful for a variety of structural computations (Pellegrino, 1993). 5 A basis of a vector space is a set of linearly independent vectors whose linear combinations span the space

(Strang, 1980).

24

Truss Structures

1.5.3

Rigid-Body and Internal Mechanisms The m independent inextensional mechanisms of a truss structure may include rigidbody displacement modes, combined with the internal mechanisms. For example, if the structure is not attached to a rigid foundation, there will obviously be three rigid-body translations and three rotations, in addition to any internal inextensional mechanisms which the structure may possess. The decomposition described above does not make any distinction between the two different types of mechanisms, and hence, first we generate an independent set of rigidbody mechanisms and, second we remove the rigid-body components from the set of mechanisms obtained from the decomposition and generate an independent set of internal mechanisms. Consider a truss structure, having a total of k kinematic constraints to a rigid foundation, and let the locations of the joints in the original configuration be described with respect to fixed Cartesian coordinates Oxyz. Any rigid-body displacement of the assembly may be described by translation and rotation vectors: ⎡ ⎤ ⎡ ⎤ rx t0x ⎢ ⎥ ⎢ ⎥ (1.21) t0 = ⎣t0y ⎦ , r = ⎣ry ⎦ t0z

rz

Here t0 represents the translation of a reference point attached to the structure, which lies at the origin of the coordinates in the original configuration; r is the rotation about this point. In such a rigid-body motion the displacement of a point i having position vector q in the original configuration is given by di = t0 + r × q

(1.22)

Now if i is a joint of the assembly which is fully restrained to the foundation, the three kinematic conditions ui = vi = wi = 0

(1.23)

are imposed on any rigid-body motion; and thus Eq. 1.23 gives three scalar constraint equations: +qiz ry

t0x t0y t0z

−qiz rx +qiy rx

−qix ry

−qiy rz

=

0

+qix rz

= =

0 0

(1.24)

If this joint had only two or one degrees of kinematic constraint, then two or one of the above three constraints, respectively, would apply. Each of the external restraints imposes one condition on t0 and r; and hence the k external constraints together give a system of k equations in six unknowns:  t0

= 0 (1.25) C r

1.5 Rigidity Computations

25

C is a k by 6 kinematic matrix. The rank rc of this matrix counts how many of the k external constraints effectively suppress rigid-body degrees of freedom of the assembly. Thus, the number of independent rigid-body mechanisms is: rb = 6 − rc

(1.26)

and any set of independent solutions of Eq. 1.25 defines the translation and rotation vectors from which the corresponding displacement components of each joint of the truss can be calculated, through Eq. 1.22. Having determined the rb rigid-body mechanisms in this way, we then find a basis for the m -dimensional space of internal mechanisms of the assembly: m = m − rb

(1.27)

First, we orthogonalize the m inextensional mechanisms found in Section 1.5.2, say d1, . . . dm , to the rb rigid-body mechanisms, which themselves have also been orthogonalized, say r1, . . . rrb , by use of the recursive formula: d i = di −

ai · rj · rj , ||rj ||2

j = 1, . . . ,rb

(1.28)

This operation transforms the original mechanisms into a set of m internal mechanisms. We know that only m of these are linearly independent, and we can use the SVD to detect which columns of the (3j − k) × m matrix [d 1, . . . d m ] are dependent on the others.

1.5.4

Examples We apply the above computational method to the analysis of some of the threedimensional truss structures discussed in Section 1.2. Consider the truss shown in Fig. 1.24(a). The vector of bar forces contains the axial forces Ti in the four bars, and the load vector contains the load components in the x-, y-, and z-directions at joint A. Equation 1.13 takes the form: ⎡ ⎤ ⎤ tAB ⎡ ⎤ −1 0 1 0 ⎢ px ⎥ 1 ⎢ ⎥ ⎢ tAC ⎥ ⎢ ⎥ √ ⎣ 0 1 0 −1⎦ ⎢ ⎥ = ⎣ py ⎦ ⎣tAD ⎦ 2 1 1 1 1 pz tAE ⎡

The equilibrium matrix A has nr = 3 rows and nc = 4 columns. The vectors of kinematic variables e and d contain the four bar extensions ei and the three displacement components of joint A, respectively.

26

Truss Structures

A

A

1

1 C B

z

y

y

1

x

D

B

z x

D

E

1

1 1 1

(a)

1 (b)

Figure 1.24 Truss structures discussed in Section 1.2; (a) consists of four bars connected to one

joint and four foundation joints, while (b) consists of three bars, with the three foundation joints collinear.

The SVD of A, obtained from Matlab (Mathworks, 2018), is as follows: ⎤⎡ ⎤ ⎡ 1.414 0 −0.000 −0.000 1.000 −1 0 1 0 1 ⎢ ⎥ ⎢ ⎥ ⎢ 0.000 1.000 1.000 0.000⎦ ⎣ 0 √ ⎣ 0 1 0 −1⎦ = ⎣ 2 0 0 1.000 −0.000 0.000 1 1 1 1 ⎡ ⎤ 0.500 0.500 0.500 0.500 ⎢ ⎥ ⎢ 0.000 0.707 −0.000 −0.707⎥ ⎥ ×⎢ ⎢−0.707 0.000 0.707 0.000 ⎥ ⎣ ⎦ 0.500 −0.500 0.500 −0.500 ⎡

0 0 1.000

⎤ 0 0⎥ ⎦ 0

It can be verified by inspection that this decomposition satisfies the conditions stated at the beginning of Section 1.5.1. There are three nonzero singular values, hence r = 3, m = nr − r = 0 and s = nc − r = 1. Therefore, the pin-jointed assembly of Fig. 1.24(a) is of Type III. The structure has a single state of self-stress [set of incompatible extensions] involving axial forces [extensions] of equal magnitude but opposite signs in AB, AD and AC, AE, see the bottom row of the WT matrix. The static and kinematic correspondence between the first three left and right singular vectors can be easily verified. For √ example, the load condition u1 which consists of a unit z-load at joint A, amplified by 2 – which is the singular value v11 – is in equilibrium with axial forces of +0.5 in the four bars, forming the stress state w1 . From a kinematic viewpoint, the displacement mode u1√which involves a unit z-displacement of joint A, divided by the singular value v11 = 2, is compatible with extensions of +0.5 of the four bars, forming the set of extensions w1 .

1.5 Rigidity Computations

27

Next, consider the truss shown in Fig. 1.24(b). The singular value of its equilibrium matrix (obtained by deleting the second and fourth columns from the previous equilibrium matrix) is: ⎤ ⎤⎡ ⎤ ⎡ ⎡ 1.000 0.000 −0.707 −0.707 0.000 −0.707 0.707 ⎥ ⎢ ⎢ ⎢ 0 0.000 −1.000⎥ 0 ⎥ ⎦ ⎣0.000 1.000⎦ ⎦ = ⎣ 0.000 ⎣ 0.000 0.000 0.707 −0.707 0.000 0.707 0.707  1.000 0.000 × 0.000 −1.000 There are two nonzero singular values, hence r = 2, m = nr − r = 1 and s = nc − r = 0. Therefore, the pin-jointed assembly of Fig. 1.24(b) is of Type II. The single mechanism (and corresponding set of loads which cannot be equilibrated) involves the translation of joint A in the y-direction, see the last left singular vector.

2

Space Frames

2.1

Introduction There are many examples of structures, both in nature and in the built environment, where lightness and structural efficiency have been achieved by concentrating the material along narrow lines, rather than spreading it continuously; structures of this kind are called space frames. One of the classical examples of space frames that is found in nature is the metacarpal bone in the vulture’s wing, Fig. 2.1, where regularly spaced diagonal members connect together the upper and lower layers of the structure. The earliest examples of engineered space frames are found in Alexander G. Bell’s studies, in the early 1900s, of structural forms for large kites. Bell (1903) was the first to recognize that a structure formed from a series of interconnected tetrahedra can be made very light and yet will be very stiff; he used tetrahedral cells for the construction of boats and towers, as well as kites. Figure 2.2 shows photographs of the Cygnet and Cygnet II kites, flown between 1907 and 1909. These structures, 13 m wide, 3 m deep, and 3 m high, consisted of over 3,000 cells made of metallic rods and covered in red silk. The front view, in Fig. 2.2(a), shows a continuous arrangement of triangles, only interrupted at the center to leave space for the person that flies on the kite. The end view in Fig. 2.2(b) shows that only parts of the structure have in fact been filled. More examples can be found in Pelham (1979). A similar concept was employed by Fuller (1961) in a proposal for a super-efficient hangar for large aircraft (Fig. 2.3). Fuller envisaged a series of plate-like structures arranged to form a complex three-dimensional shape. Each plate-like structure is in fact a space frame consisting of repeating tetrahedral and octahedral units; this is one of the semi-regular tessellations described in Section A.2. This arrangement was the precursor to the popular double-layer grid structure, which will be discussed in more detail in Section 2.2.

Figure 2.1 Metacarpal bone in a vulture’s wing (Thompson, 1917).

28

2.1 Introduction

29

Figure 2.2 Bell’s tetrahedral cell space frames: (a) front view of Cygnet kite, flown in 1907 (Gilbert H. Grosvenor Collection of Photographs of the Alexander Graham Bell Family, Library of Congress, LC-G9-Z1-139,226-A); (b) end view of Cygnet II, flown in 1909 (Alexander Graham Bell family papers, 1834–1974, Library of Congress, Manuscript Division, www.loc.gov/resource/magbell.13900501).

30

Space Frames

Figure 2.3 Space frame for servicing dock by Buckminster Fuller (Fuller, 1961; Borrego, 1968).

A variety of space frames with different patterns are found in nature. The skeletons of Radiolaria, microscopic organisms that live in water, form tessellations of hexagons, or squares, or triangles and hexagons (see Fig. 2.4(a–d)). An example of a space frame at the molecular level is the carbon-cage molecule, C60 (see Fig. 2.4(e)), also known as Bucky-ball or Fullerene. Section A.2 provides the geometric background to plane or space-filling structures. A great variety is possible, beyond the simple regularity of Bell’s tetrahedral cells, but so far the shapes that have been used for most man-made space frames have tended to use only the simplest tessellations of polyhedra. What exactly do we mean when we say that a structure is a space frame? Definitions of space frame structures have been given by several authors, including Tsuboi (1984), Medwadowski (1983), and Kawaguchi (1989). The definition that will be adopted here is a somewhat broader version of these previous definitions: A space frame is a structure whose overall shape is two-dimensional or threedimensional. It consists of linear elements that form a series of repeating units. The load carrying mode of the structure is three-dimensional. Space frames are often constructed by assembling a set of rods or tubular members using standard joints; a range of regular shapes can be achieved by this approach (Makowski, 1981). If the member lengths and joint angles are allowed to be nonuniform, then there are practically no limits to the shapes that can be achieved (Nooshin, 1984). An alternative fabrication technique, normally of lower cost but suitable mainly for uniform shapes, is to use pre-fabricated sub-units of several members. Models of space frames can be made with the Zometool (2018) kit, consisting of plastic struts and spherical joints with connector holes. A simple example is shown in Fig. 2.5. From a structural mechanics viewpoint, some space frames may be analyzed as trusses, if modeling the connections between the members as frictionless pin joints

2.1 Introduction

(a)

(b)

(c)

(d)

(e) Figure 2.4 Space frames in nature: (a) Conarachnium; (b) Litharachnium; (c and d) sketch and

detailed view of Callimitra (all from: Bach, 1990); (e) Bucky-ball (Tarnai, 1993).

31

32

Space Frames

Figure 2.5 Icosahedral space frame made of plastic rods and joints (Zometool, 2018).

Figure 2.6 Double layer space frames with ETFE cushions attached to the outer surface, at the

Eden Project, Cornwall, UK.

2.2 Classification

33

leads to a kinematically determinate – i.e., rigid, see Section 1.5 – truss structure. In such cases there is an obvious overlap between space frames and truss structures. Some other space frames, such as those in Fig. 2.4(a, b, e), are not rigid when the joints are modeled as pin joints. In such cases, the bending, torsional, etc. contributions to the stiffness of the space frame are important and hence the space frame has to be modeled as a rigid-jointed structure. Also, membrane elements may form an integral part of a space frame as, for example, in the ETFE pressurized cushions of the Eden Project domes in Cornwall, UK (see Fig. 2.6). The outer surface of these structures consists of a patterns of (mainly) hexagons and the inner surface consists of triangles and (mainly) hexagons.

2.2

Classification It is clear from the previous section that space frames come in a variety of shapes and structural forms and hence it is useful to introduce a classification scheme, to bring order to this field. Authors who have dealt with a particular type of space frames, e.g. plate-like frames (Borrego, 1968) have introduced detailed classification schemes that suit that particular type. Here we are interested in the shape of a broad range of space frames, and hence we follow a more general approach that considers both the overall shape of a space frame and the shape of its repeating unit. This approach is consistent with the definition given in Section 2.1 and leads to the simple classification that is introduced in Table 2.1. Figure 2.7 shows several space frames. Their overall shape is clearly threedimensional, but the repeating unit is essentially two-dimensional in all cases. The first example, in Fig. 2.7(a) is the first known geodesic dome, used by W. Bauersfeld as the arrangement for the steel reinforcement bars during construction of the Jena Planetarium, in 1923. This happened long before the “invention” of the geodesic dome by Buckminster Fuller; see Wester (1997) for further details. In this structure all of the joints lie on a spherical surface, i.e., a structure of positive gaussian curvature (the gaussian curvature of a surface is defined in Section A.4.4), and are connected by bars of near-constant length arranged to form a series of interconnected triangles. Here, the repeating unit is a triangle, obviously a two-dimensional object. This structure can be modeled as a pin-jointed truss for the purpose of analysis, because it follows from Cauchy’s theorem – Section 1.2.2 – that any convex, triangulated structure that is connected to a rigid foundation along the edge is rigid. Table 2.1 Classification of space frames.

Lattice/Single layer Double layer Multiple layer

Unit

Arrangement of unit

Overall shape

2D 3D 3D

2D 2D 3D

Surface-like (planar or curved) Surface-like (planar or curved) Solid-like

34

Space Frames

Figure 2.7 Examples of single-layer space frames: (a) reinforcing bars of Jena planetarium,

forming the first geodesic dome, built in 1924 (image downloaded from www.wikiwand.com/en/ Planetarium_Jena); (b) geodesic plate dome (Wester, 1997); (c) interstage structure of M-3 launcher (Onoda, 1986); (d) roof structure of St Mary’s Cathedral in San Francisco, CA, designed by Pier Luigi Nervi and consisting of eight hyperbolic paraboloidal surfaces connected to triangulated space frames (image used with permission of Pier Luigi Nervi Project, 2019); (e) broadcasting tower designed by Vladimir Shukhov in Moscow, Russia and built in 1920–1922. It consists of six vertically stacked hyperboloids of revolution forming a structure that approximates a cone and its total height is 148.5 m (Igor Rozhkov/Shutterstock.com).

2.2 Classification

35

Figure 2.7(b) shows a different kind of space frame whose overall shape is similar to the previous example, but here the repeating unit is a hexagonal plate whose interior has been removed. In the actual structure, timber members connected by plywood joints were used. This space frame concept is inherently much more deformable than that in Fig. 2.7(a); its function is to provide temporary cover for dancing and social gatherings, when some of the hexagons would be covered with a nonstructural membrane. Unlike the geodesic dome in the previous example, a pin-jointed model of this space frame would have many inextensional modes of deformation and, hence, considering the bending rigidity of its joints in the analysis is essential. An appropriate method of analysis would be to model each member of the frame as a beam element and to use a structural analysis computer program to analyze the structure. The remaining examples are a structure with cylindrical shape, i.e., zero gaussian curvature, Fig. 2.7(c), and two doubly curved hyperbolic paraboloids (negative gaussian curvature), Fig. 2.7(d,e). The repeating unit is a triangle in all cases, but note the resemblance between these structures and some of the examples discussed in Section 1.4. When analyzed as pin-jointed structures, these structures turn out to be rigid, or not, depending on their support conditions. Figure 2.8 shows several examples of planar or surface-like space frames. Unlike the two-dimensional repeating units in Fig. 2.7, these space frames are based on threedimensional repeating units. Figure 2.8(a,b) show two plate-like space frames formed by inter-connecting two regular tessellations of squares, Fig. 2.8(a), or triangles, Fig. 2.8(b), which lie in parallel planes and are vertically offset. These structures are often called double-layer flat grids. The square-on-square arrangement is most popular for roof structures, because it is suitable for buildings with square and rectangular plan shapes. These space frames can be mass produced; in the simplest case (considered here) the complete structure can be made from bars of the same length, L, and using identical joints. In both designs, the joints of the upper layer lie directly above the center of the squares/triangles √ forming the bottom layer, while the spacing between the two layers is L/ 2 in the √ square-on-square design, and L 2/3 in the triangle-on-triangle design. More general geometries could be considered, for example the spacing between the two layers could be varied. In the first example, Fig. 2.8(a), the three-dimensional repeating unit consists of a tetrahedron joined with a half-octahedron, while in the second example the repeating unit is a tetrahedron joined to an octahedron (see Fig. 2.8(b)). There is a key difference between these two repeating units: the first has a mode of inextensional deformation while the second is rigid. The behavior of these structures has been explained in Section 1.4. For both structures, when several units are connected together, the resulting space frame is able to carry out-of-plane bending moments and shear forces, as well as in-plane forces, in a way that is similar to the behavior of a continuous plate. However, the square-on-square design cannot carry any twisting moments, due to its internal mechanism. The resulting torsional compliance can be exploited to design roof structures that are able to undergo significant differential foundation settlements.

36

Space Frames

Figure 2.8 Examples of double-layer space frames: (a) square-on-square grid (image used with permission of Delta Structures, Inc., 2019); (b) octet truss (image used with permission of Delta Structures, Inc.); (c) barrel vault (image used with permission of Taiyo Kogyo Corporation, 2019); (d) hyperbolic paraboloid (image used with permission of MERO-TSK, 2019).

2.2 Classification

37

An important difference between surface-like space frames based on threedimensional repeating units, such as those shown in Fig. 2.8(a), and those based on two-dimensional units such as those shown in Fig. 2.7, is that the global degree of static indeterminacy is usually much higher. For example, the average ratio between the number of bars and the number of joints, neglecting boundary joints, is 4 in Fig. 2.8(a), 4.5 in Fig. 2.8(b), but only 3 in Fig. 2.7(a,c–e) and 1.5 in Fig. 2.7(b). The same types of repeating units can be used to form curved surface-like space frames. Two examples are shown in Fig. 2.8(c,d). Note that the bar lengths are no longer all the same, and joints with different angles are also required in these structures. There are many variants to the two simple repeating units that have been discussed here. For example, the addition of a third layer of bars is sometimes used to produce stronger/stiffer space frames. Another variant is space frames where the bars connecting the top and bottom layers are replaced with continuous elements of conical or other shape that form a “stressed skin.” These space frames can be assembled more quickly because a smaller number of parts are required, and also can incorporate roofing/insulating elements within the structure itself. Also, more elaborate arrangements of the bars in each layer are sometimes used, often consisting of two different, tessellations of bars in the two layers (the various types of tessellations are explained in Section A.2). An example is the space frame in Fig. 2.6, whose upper layer consists of hexagons and bottom layer of hexagons and triangles. An interesting feature of this particular layout is that the average ratio between the numbers of bars and the number of joints is 2.5, i.e., much lower than all other surface-like structures formed from a 3D repeating unit shown here. The space frames shown so far have all been plate- or surface-like. Three-dimensional space frames, also called multiple-layer, are more rare. They tend to be used as a replacement for continuous materials, mainly for their lightness. A commonly used example is a structural foam, see Section 6.4; another example is a micro-engineered, multi-functional material such as that proposed by Deshpande et al. (2001) and shown in Fig. 2.9.

Figure 2.9 Micro-engineered material made from a casting aluminum alloy and based on the

octet truss space frame layout. The rod length is 14 mm.

38

Space Frames

2.3

Continuum Models for Single-Layer Space Frames A simple relationship between the form of a space frame and its structural properties has many uses for preliminary design; finding such relationships is the focus of this section. The detailed structural analysis of a particular space frame can be done with standard structural analysis software and has been covered in specialized publications such as Makowski (1981) and Kollar and Hegedus (1985). The approach that will be followed here is to describe the general structural properties of a space frame in terms of an equivalent continuum that simulates the properties of the basic repeating unit of the space frame. The idea of linking the behavior of a space frame to an equivalent continuum was pioneered by Hrennikoff (1941). To simplify the structural analysis of space frames containing hundreds of members, Hrennikoff – followed by others – looked for bar layouts and cross-sectional areas that would match the behavior of a continuum, in order to design space frames whose analysis could be replaced by the analytical solution, or iterative numerical solution for the equivalent continuum. Our aim for exploring this equivalence is different, as today’s tools can readily perform the analysis of any space frame, however large the number of joints. Instead, we are interested in exploring how the stiffness of a continuum changes when it is replaced by a structure that can carry loads only througha finite number of load paths. The results of this investigation will allow us to better understand the overall behavior of space frames and how the anisotropy that is inherent in many space frame architectures can be exploited to achieve certain special kinds of behavior. The space frames shown in Fig. 2.10 will be studied. It will be assumed that each space frame is a square with unit side length and that all bars have equal axial stiffness AE per unit length. The bending stiffness, etc. of the members of the space frame is not considered because its contribution to the overall stiffness is small in comparison with the axial stiffness contribution. The in-plane stress resultants per unit length of the equivalent continuum are defined in analogy with Section B.3.2, but the integrals of the stress over the continuous plate in Eqs. B.22–B.24 correspond to an appropriate summation of bar forces in the space frame. The corresponding deformation variables are the average mid-plane strains, defined in analogy with Section B.4.2. Our first objective is to find the homogenized stiffness matrix, A, such that: ⎡

Nx

⎤

⎡

x

⎤

⎥ ⎢ ⎥ ⎢ ⎣ Ny ⎦ = A ⎣ y ⎦ Nxy

(2.1)

γxy

from which we will later evaluate the stiffness of the space frame in some particular directions.

2.3 Continuum Models for Single-Layer Space Frames

1

1

L

L

1

1

(a)

(b)

1

L

39

1

L

1

1

(c)

(d)

Figure 2.10 Single-layer space frames based on (a) square, (b) right-angle triangle, (c) equilateral triangle, and (d) hexagonal tessellations.

2.3.1

Homogenized Stiffness Matrix for Lattice of Parallel Bars Consider a simple lattice consisting of parallel bars with cross-sectional area A and elastic modulus E at a distance  and in a general direction α, Fig. 2.11(a). Obviously, Nα =

AE α 

(2.2)

where Nα is the sum of the bar axial forces in unit width of space frame and α the normal strain in the bars. This equation can be written in matrix form as:

AE

1 α Nα = 

(2.3)

40

Space Frames

Ny

Nxy 1 Nα

Nx

1

Nxy 1

α

1 (b)

(a)

sin α

1

Nα sin α

cos α

Nx

Nα cos α

Nxy

Nxy Ny 1 (d)

(c)

Figure 2.11 Transformation of bar axial forces to continuum stress resultant components: (a) lattice of parallel bars and stress resultant parallel to bars; (b) stress resultants in x,y coordinate system; (c, d) free body diagrams.

This relationship can be transformed to the x,y coordinate system shown in Fig. 2.11(b) by considering the free body diagrams shown in Fig. 2.11(c,d). Resolving horizontally and vertically, respectively, the forces acting in Fig. 2.11(c) gives: Nx = Nα cos2 α Nxy = Nα sin α cos α and doing the same for Fig. 2.11(d) gives: Nxy = Nα sin α cos α Ny = Nα sin2 α Noting that there is a duplicate equation, the remaining equations can be reorganized to obtain: ⎤ ⎤ ⎡ ⎡ cos2 α Nx ⎥

⎥ ⎢ ⎢ (2.4) ⎣ Ny ⎦ = ⎣ sin2 α ⎦ Nα sin α cos α Nxy

2.3 Continuum Models for Single-Layer Space Frames

41

The strain transformation is a special case of the standard plane strain transformation, Eq. B.11, ⎡ ⎤ x 2

⎢ ⎥ 2 (2.5) α = cos α sin α sin α cos α ⎣ y ⎦ γxy Next, we transform Eq. 2.3 by premultiplying both sides by the matrix ⎡ ⎤ cos2 α ⎢ ⎥ ⎣ sin2 α ⎦ sin α cos α to obtain:

⎡ ⎤ ⎤ cos2 α cos2 α ⎥ ⎢ ⎥ AE

⎢ 1 α ⎣ sin2 α ⎦ Nα = ⎣ sin2 α ⎦  sin α cos α sin α cos α ⎡

(2.6)

Finally, combining Eqs. 2.4, 2.6, and 2.5 gives: ⎡ ⎤ ⎡ ⎤ cos2 α Nx ⎢ ⎥ ⎢ ⎥

⎣ Ny ⎦ = ⎣ sin2 α ⎦ Nα = Nxy

sin α cos α ⎡ ⎤ cos2 α ⎢ ⎥ AE 2 = ⎣ sin2 α ⎦ 1 cos α  sin α cos α

⎤ x

⎢ ⎥ sin α cos α ⎣ y ⎦ γxy ⎡

sin2 α

and doing the multiplications gives the final result: ⎡ ⎡ ⎤ sin2 α cos2 α sin α cos3 α Nx cos4 α AE ⎢ ⎢ 2 ⎥ 2 sin4 α sin3 α cos α ⎣ Ny ⎦ = ⎣sin α cos α  sin α cos3 α sin3 α cos α sin2 α cos2 α Nxy

⎤ x ⎥⎢ ⎥ ⎦ ⎣ y ⎦ γxy

(2.7)

⎤⎡

(2.8)

Equation 2.8 can be used to derive the homogenized stiffness matrix for the singlelayer space frames shown in Fig. 2.10, by superposing the stiffness of each set of parallel bars. In the case of the square lattice in Fig. 2.10(a), we superpose the cases α = 0◦, = L and α = 90◦, = L to obtain: ⎡ ⎤ ⎡ ⎤⎡ ⎤ 1 0 0 Nx x ⎥ AE ⎢ ⎢ ⎥⎢ ⎥ (2.9) ⎣0 1 0⎦ ⎣ y ⎦ ⎣ Ny ⎦ = L Nxy γxy 0 0 0 It can be seen that this matrix has rank = 2, because of the lack of shearing stiffness of this lattice.

42

Space Frames

Next, consider the lattice shown in Fig. 2.10(b) which, in addition to the two sets of orthogonal bars, has a set of parallel diagonal bars. Its homogenized stiffness matrix can be found by adding the contribution of the diagonal √ bars to the matrix of Eq. 2.9. The diagonal bars have α = 45◦ and  = L/ 2. Substituting these values into Eq. 2.8 and adding to the matrix of Eq. 2.9 gives: ⎡ ⎤⎡ ⎤ ⎡ ⎤ 1+a a a x Nx ⎥⎢ ⎥ ⎢ ⎥ AE ⎢ (2.10) 1 + a a ⎦ ⎣ y ⎦ ⎣ a ⎣ Ny ⎦ = L a a a Nxy γxy where 1 a= √ 2 2 This matrix has full rank, indicating that there are no zero-stiffness deformation modes. The homogenized stiffness matrix for the lattice shown in Fig. 2.10(c), which is based on a tessellation of equilateral triangles, can be derived by superposing the matrices obtained from Eq. 2.8 for the three cases α = 0◦,60◦,120◦ with  = L in all cases. Hence, we obtain: ⎡ ⎤ 3b b 0 AE ⎢ ⎥ (2.11) A= ⎣ b 3b 0⎦ L 0 0 b where 3 8 The derivation of the homogenized stiffness matrix for the hexagonal lattice of Fig. 2.10(d) is more complex and will not be presented here. The following expression has been derived by Heki and Saka (1972): ⎡ ⎤ c c 0 AE ⎢ ⎥ A= (2.12) ⎣ c c 0⎦ L 0 0 0 b=

where 1 c= √ 2 3 Note that this matrix has rank = 1. Hence, for any imposed deformation, Nx = Ny and Nxy = 0.

2.3.2

Stiffness in Direction θ From the homogenized stiffness matrices derived in the previous section we will obtain an expression for the stiffness of the space frame in any particular direction θ . The derivation makes use of the same geometric transformation for both stress resultants

2.3 Continuum Models for Single-Layer Space Frames

43

Nθ , εθ Nθ , εθ y’ x’

θ

Figure 2.12 Boundary conditions for calculation of stiffness in direction θ.

and strains. It is also assumed that the deformation of the space frame is constrained as shown schematically in Fig. 2.12, and hence assuming: ⎤ ⎤ ⎡ ⎡ = 0 x  ⎥ ⎥ ⎢ ⎢ (2.13) ⎣ y  ⎦ = ⎣ 0 ⎦ γx  y 

0

Substituting the kinematic conditions in Eq. 2.13 into the strain transformation equation, Eq. B.13, doing the matrix multiplications, and finally replacing x  with θ gives:   1 2θ γ x sin θ cos θ cos xy 2 (2.14) = θ 1 y sin θ cos θ sin2 θ 2 γxy which can be reorganized into: ⎡

⎤ ⎤ ⎡ x cos2 θ ⎥ ⎢ ⎥ ⎢ ⎣ y ⎦ = ⎣ sin2 θ ⎦ θ 2 sin θ cos θ γxy

(2.15)

The transformation of the stress resultants can be obtained from Eq. B.6, simply writing N instead of σ and θ instead of x’. The only component of interest is Nθ , which is given by: Nθ = Nx cos2 θ + Ny sin2 θ + 2Nxy sin θ cos θ

(2.16)

Equations 2.15 and 2.16 can be combined with the stiffness matrices derived in Section 2.3.1 to obtain expressions for the stiffness of the space frames in a general θ direction. The stiffness Nθ /θ will be denoted by Eθ∗ .

44

Space Frames

2.4

In-Plane Efficiency of Single-Layer Space Frames The stiffness derived in the previous section will now be compared to that of a homogeneous, continuous plate of arbitrary thickness t and made from the same material as the bars of our space frames. The Young’s modulus of the homogeneous plate is E, its Poisson’s ratio 1/3 – for simplicity – and its density ρ. We define the in-plane efficiency in the θ-direction of a single-layer space frame as μθ =

Eθ∗ /ρ ∗ Eθ /ρ

(2.17)

where: • •

Eθ∗ is the stiffness of the space frame in the θ -direction (with no transverse or shear deformation allowed); ρ ∗ is the density of the space frame: ρ∗ =

• •

mass of bars in repeating unit ; volume of repeating unit

Eθ is the stiffness of the continuous plate in the θ-direction (with no transverse or shear deformation allowed); and ρ is the density of the continuous plate. It is convenient to introduce the relative density, α, of the space frame α=

ρ∗ ρ

and, noting that ρ ∗ is also given by ρ∗ =

volume of bars in repeating unit × ρ volume of repeating unit

the relative density is given by α=

ρ∗ volume of bars in repeating unit × ρ 1 = ρ volume of repeating unit ρ

(2.18)

= volume fraction where the volume fraction is the ratio between the volume of the bars in the lattice and the volume of material in the continuous plate. Substituting Eq. 2.18 into Eq. 2.17 we obtain μθ =

Eθ∗ αEθ

(2.19)

Note that when defining the density of the space frame we take its “thickness” to be equal to t. Also note that, because the homogeneous plate is isotropic, its stiffness in the θ -direction is equal to the stiffness in the x-direction: Eθ = E0

(2.20)

2.4 In-Plane Efficiency of Single-Layer Space Frames

2.4.1

45

Efficiency of Structures in Fig. 2.10 Consider the lattice shown in Fig. 2.10(a). By substituting Eq. 2.9 into Eq. 2.16 we obtain:  AE 2 cos θ x + sin2 θ y (2.21) Nθ = L Then, substituting Eq. 2.15 we obtain:  AE 4 Nθ = (2.22) cos θ + sin4 θ θ L from which  Nθ AE 4 = (2.23) cos θ + sin4 θ Eθ∗ = θ L To find the corresponding relationship for the continuous plate we set y = γxy = 0 in Eq. B.38 and find 9 (2.24) Etx 8 This relationship is also valid if we replace x with θ because the plate is isotropic. Hence we can find the stiffness in the θ-direction: Nx 9 = Et Eθ = E0 = (2.25) x 8 Nx =

The relative density is equal to the volume fraction, hence considering a unit of size L × L, we have the following: volume of bars = 2AL volume of material in homogeneous plate = L2 t and hence 2A 2AL = Lt L2 t Combining the above results, we find:  E∗ 4 4 cos θ + sin4 θ μθ = θ = αEθ 9 α=

(2.26)

(2.27)

and for the special case θ = 0 Eq. 2.27 gives 4 (2.28) 9 The analysis for the remaining space frames shown in Fig. 2.10 follows similar lines and the final results are as follows. For the single-layer structure with diagonally braced squares in Fig. 2.10(b) we find:   √ √ 8 1 μθ = √ √ + sin4 θ + cos4 θ + 2 sin θ cos θ + 2 sin2 θ cos2 θ 9(2 + 2) 2 2 (2.29) μ0 =

46

Space Frames

Table 2.2 In-plane efficiency of 2D lattices in x-direction (θ = 0◦ ). Shape

E0∗

α

μ0

AE L

A 2 tL

4 9 ≈ 0.444

1 1+ √



2 2

AE L

2+

√  A 2 tL

√ 6−2 2 ≈ 0.352 9

√ 3 3 AE 4 L

√ A 2 3 tL

1 ≈ 0.333 3

1 AE √ 2 3 L

√2 A 3 tL

2 9 ≈ 0.222

For the structure based on a regular tessellation of equilateral triangles (isogrid) in Fig. 2.10(c) we find: μθ =

2 1 2 1 sin θ + cos2 θ = 3 3

(2.30)

For the structure based on a regular tessellation of hexagons in Fig. 2.10(d) we find: μθ =

2 9

(2.31)

All of these results have been plotted in Fig. 2.13 and the efficiency values in the x-direction have also been presented in Table 2.2. A surprising feature of the plot in Fig. 2.13 is that the first and fourth space frames appear to have nonzero efficiency in all directions, despite the fact that they can carry loads only in some particular directions. This apparent anomaly is explained by the fact that, although the homogenized stiffness matrices for these structures are singular, in the derivation of the stiffness Eθ no transverse or shear deformation was allowed. This figure shows clearly the relationship between shape efficiency in any particular direction and the arrangement of the members in the lattice. Thus, the only lattice that is isotropic, i.e., its efficiency is independent of θ , is the lattice that consists of equilateral triangles. All other arrangements are highly directional, i.e., anisotropic. An even more surprising result is that the efficiency of the continuous plate, which has the value μ = 1 in all directions as E ∗ = E and α = 1, is unmatched. The best value of μ that can be achieved by the single-layer space frames, but only in a limited range of directions, is 4/9, while insisting on isotropic behavior further reduces μ to 1/3. How can we explain such a large drop in efficiency when we go from a continuum to a discrete structure? The explanation is very simple in the case of a square lattice. Starting from a thin plate (see Fig. 2.14(a)), imagine cutting the plate into a series of thin, parallel strips, Fig. 2.14(b). The stiffness in the direction of the strips is no longer

2.4 In-Plane Efficiency of Single-Layer Space Frames

47

90 60

120

0.5 0.4

150

30

0.1 180

0

210

330

240

300 270

Figure 2.13 In-plane efficiency μθ .

(a)

(c)

(b)

Figure 2.14 Simple explanation of efficiency loss in single-layer space frames.

computed from the constitutive relation for the continuum plate, instead we have to use Eq. 2.21 with θ = 0, which gives: AE L

(2.32)

E ∗ = Et

(2.33)

E∗ = and since for this set of strips A = tL

The homogeneous plate stiffness is given by Eq. 2.25 and the relative density in this case is obviously α = 1. With this, we can find the efficiency in the direction parallel to the strips: μ =

E∗ 8 8 = Et = αE 9Et 9

(2.34)

48

Space Frames

The stiffness in the direction perpendicular to the strips is zero, and so in this direction μ⊥ = 0

(2.35)

To make μ = μ⊥ we slice the strips through their mid-plane and rotate one set through 90◦ . The resulting lattice is depicted in Fig. 2.14(c) and the corresponding efficiencies are 1 8 4 (2.36) μ = μ⊥ = × = 2 9 9 Finally, it should be noticed that the two-dimensional space frames that have been considered in this section are only a limited sample of what is possible. For example, the cross-sectional area need not be the same for all members. A particular case that has some interesting properties is obtained by adding the other set of diagonal elements to the space frame in Fig. 2.10(b), thus forming another isotropic lattice. If this lattice is stretched, say, in the x-direction we obtain an orthotropic space frame, whose μθ varies between a maximum value in the y-direction, and a minimum value in the x-direction.

2.5

Continuum Models for Double-Layer Space Frames We will consider double-layer space frames based on the square-on-square grid and the octet truss, shown in Fig. 2.8(a–b). For simplicity, it will be assumed that in both cases all bars have identical length L and equal axial stiffness per unit length AE. The theoretical model that we use to study these space frame is the Kirchhoff plate, a thin plate model where the normals to the mid-plane are assumed to remain perpendicular to the deformed mid-surface (Jones, 1999). Note that shear deformation and in-plane bending are excluded. A right-handed x,y,z coordinate system is introduced, whose x,y axes lie in the mid-plane parallel to the sides of the space frame and z is perpendicular to the mid-plane, see Fig. 2.15. There are six stress resultants, three mid-plane forces per unit width and three moments per unit width, defined in analogy with Section B.3.2. The corresponding

z

y d

x Mxy Ny

Mxy Nxy

Nxy My

Mx

Nx

Figure 2.15 Definition of coordinate axes and stress resultants for double-layer space frame.

2.5 Continuum Models for Double-Layer Space Frames

49

deformation variables are the average mid-plane strains x , y ,γxy and out-of-plane κy , κxy , see Section B.4.2. curvatures of the mid-plane of the space frame  κx ,  Next, we will determine the homogenized stiffness matrix, ABD, of the two doublelayer space frames. This matrix is defined such that: ⎡ ⎤ ⎡ ⎤ Nx x ⎢ ⎥ ⎢ ⎥ ⎢ Ny ⎥  ⎢ ⎢ y ⎥ ⎢ ⎥ ⎥ ⎢ Nxy ⎥ ⎢ ⎥ ⎢ ⎥ = A B ⎢γxy ⎥ (2.37) ⎢M ⎥ B D ⎢ κx ⎥ ⎢ x⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ κy ⎦ ⎣ My ⎦ ⎣ Mxy  κxy

2.5.1

Homogenized Stiffness Matrix Consider a general double-layer space frame with identical single-layer structures on its faces. The fact that the single-layer structures are identical implies that • • •

they have equal geometry and the same orientation; are made of the same material; and they are made of bars with equal cross-sections.

It will be assumed that the connection between the two single-layer space frames is effectively rigid in shear but is compliant in bending and stretching. The first assumption is equivalent to saying that the Kirchhoff-Love plate model, see Appendix B.3.2, will be used for the space frame. The second assumption means that, apart from providing a shear constraint as described, the contribution of the connecting elements to the stiffness of the space frame can be neglected. Under in-plane loading, this space frame behaves effectively as two separate structures at a distance d = L. Each structure carries half the load and the geometric offset makes no difference. Hence, the matrix A relating the in-plane stress resultants and the corresponding deformations can be obtained simply by multiplying by 2 the stiffness matrix of the single-layer space frames, i.e., A = 2ASL

(2.38)

here the subscript SL stands for single layer. A more formal approach will be adopted for the calculation of the remaining blocks of the ABD matrix. We consider a general inextensional deformation of the space frame mid-plane, as follows: ⎡ t ⎤ ⎡ b⎤ ⎡ t ⎤ x x x ⎢ t ⎥ ⎢ b⎥ ⎢ t ⎥ arbitrary ⎣ y ⎦ and ⎣ y ⎦ = − ⎣ y ⎦ t b t γxy γxy γxy Here the superscripts t and b denote the average strains in the single-layer space frames on the top and bottom faces, respectively.

50

Space Frames

The corresponding values of the stress resultants for each space frame, obtained from Eq. 2.1 are ⎤ ⎡ t ⎤ x Nxt ⎢ t ⎥ ⎢ t⎥ ⎣ Ny ⎦ = ASL ⎣ y ⎦ t t Nxy γxy ⎡

(2.39)

and ⎡

Nxb

⎤

⎡

xb

⎤

⎡

xt

⎤

⎡

Nxt

⎤

⎢ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ b⎥ ⎣Nxy ⎦ = ASL ⎣ yb ⎦ = −ASL ⎣ yt ⎦ = − ⎣ Nyt ⎦ b t t Nyb γxy γxy Nxy

(2.40)

The mid-plane strains of the space frame are x = y = γxy =

xt + xb 2 yt + yb

=0

2 t b γxy + γxy 2

=0

(2.41)

=0

This verifies that the chosen deformation is inextensional. The bending and twisting deformations of the space frame are  κx =  κy =  κxy =

xt − xb d t y − yb d t − γb γxy xy d

=2 =2 =2

xt d yt d t γxy

(2.42)

d

The corresponding stress resultants per unit length in the double-layer space frame are as follows: ⎡ ⎤ ⎡ t ⎤ ⎡ b⎤ Nx Nx Nx ⎢ ⎥ ⎢ t ⎥ ⎢ b⎥ (2.43) ⎣ Ny ⎦ = ⎣ Ny ⎦ + ⎣ Ny ⎦ = 0 t b N N Nxy xy xy and ⎡ t⎤ ⎡ t⎤ ⎡ b⎤ ⎤ Nx Nx Nx Mx ⎢ t⎥ ⎥ d ⎢ t ⎥ d ⎢ b⎥ ⎢ ⎣ My ⎦ = ⎣ Ny ⎦ − ⎣ Ny ⎦ = d ⎣ Ny ⎦ 2 2 t b t Nxy Nxy Nxy Mxy ⎡

(2.44)

2.6 Bending Efficiency of Double-Layer Space Frames

51

Substituting Eqs. 2.43 and 2.44 into the left-hand-side of Eq. 2.37 and substituting Eqs. 2.41–2.42 into the right-hand-side gives: ⎤ ⎤ ⎡ ⎡ 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 ⎥  ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ B 2A 0 0 SL ⎥= ⎢ ⎡ ⎢ ⎡ ⎤⎥ ⎤ (2.45) T ⎥ ⎢ ⎢ t t D ⎢ B x ⎥ Nx ⎥ ⎥ ⎢ ⎢ ⎢ t ⎥⎥ ⎢ 2 ⎢ t ⎥⎥ ⎣d ⎣ Ny ⎦⎦ ⎣ d ⎣ y ⎦⎦ t t Nxy γxy From the first three equations it follows that B = 0. From the bottom three equations we find: ⎡ t ⎤ ⎡ t⎤ x Nx 2 ⎢ t ⎥ ⎢ t⎥ (2.46) ⎣ Ny ⎦ = 2 D ⎣ y ⎦ d t t Nxy γxy Comparing Eq. 2.46 to Eq. 2.39 we find: 2D = ASL d2

(2.47)

d 2 ASL d 2A = 2 4

(2.48)

from which D=

In conclusion, the ABD matrix of a double-layer space frame with identical top and bottom layers has the following expression:  A 0 ABD = (2.49) 2 0 d 4A where A = 2ASL .

2.6

Bending Efficiency of Double-Layer Space Frames We define the bending efficiency in the θ -direction of a double-layer space frame as in Eq. 2.17, but replacing the stiffness Eθ∗ of the space frame and the stiffness Eθ of the continuous plate with the bending stiffnesses in the θ-direction. We determine the bending stiffness of the space frame under a set of kinematic conditions equivalent to those used for the in-plane stiffness, Eq. 2.13: ⎤ ⎤ ⎡ ⎡ = 0 κx  ⎥ ⎥ ⎢ ⎢ (2.50) ⎣ κy  ⎦ = ⎣ 0 ⎦ κx  y 

0

52

Space Frames

For simplicity, we will show the derivation for the special case θ = 0, i.e., the θ -direction coincides with the x-direction. Substituting Eq. 2.50 into Eq. 2.37 and noting that B = 0 Mx = D11 κx

(2.51)

where D11 is the top-left coefficient of the sub-matrix D. Therefore, we have D0∗ =

Mx = D11 κx

(2.52)

For a continuous plate of Young’s modulus E, Poisson’s ratio 1/3 and thickness t = d, under the same kinematic conditions, one obtains from Eq. B.41 3 Ed 3 κx 32

(2.53)

Mx 3 = Ed 3 κx 32

(2.54)

Mx = and so the bending stiffness is D0 =

The relative density α of this space frame is most easily calculated by considering the volume of material in the repeating unit. Thus, the bending efficiency of the space frame in the x-direction is μ =

2.6.1

D0∗ αD0

(2.55)

Efficiency of Structures in Fig. 2.16 Consider the double-layer space frame shown in Fig. 2.16(a). The matrix D can be readily calculated from Eqs. 2.48 and 2.9: ⎡ ⎤ 1 0 0 2 2 d ASL d AE ⎢ ⎥ D= = (2.56) ⎣0 1 0⎦ 2 2 L 0 0 0 and so, from Eq. 2.52 D0∗ =

AEd 2 2L

(2.57)

The relative density α of this space frame is calculated by considering the volume of material in a square prism whose base coincides with one of the squares in the top or bottom layer. This gives √ A α=8 2 2 L

(2.58)

In conclusion, substituting Eqs. 2.57, 2.52, and 2.58 into Eq. 2.55 the bending efficiency of the square-on-square space frame in the directions of the square grid is

2.6 Bending Efficiency of Double-Layer Space Frames

L

1

L

1

53

y

y

x

x z

z x

x

2 L 3

L/ 2 1

1

(a)

(b)

Figure 2.16 Top and side views of (a) square-on-square and (b) octet truss space frames.

D∗ AEd 2 = 0 = αD0 2L √ and substituting d = L/ 2 μ0

L2 √ 8 2A

μ0 =

√ 2L 32 = 3d 3Ed 3

2 3

(2.59)

(2.60)

Note that this value is different from the in-plane efficiency of the same space frame. The in-plane efficiency is obtained by substituting into Eq. 2.19 the in-plane stiffnesses from Eq. 2.25, multiplying by 2 because there are two layers, and taking the relative density from Eq. 2.58. This gives D0∗ 2AE = αD0 L √ then, substituting t = L/ 2 gives μ0 =

L2 √ 8 2A

μ0 =

2 9

2L 8 = √ 9Et 9 2t

(2.61)

(2.62)

A similar calculation for the triangle-on-triangle space frame, Fig. 2.16(b), shows that its bending efficiency is identical to the square-on-square frame, but its in-plane efficiency is also 2/9. More details are presented in Table 2.3. The analysis that has been presented in this section is based on the simplifying assumption that the members connecting the top and bottom layers of the two space frames that have been considered provide a shear-rigid connection between the two layers. A more refined analysis for members with axial stiffness AE per unit length (Heki and Saka, 1972) leads to the conclusion that the matrix D that has been calculated above is unchanged for both space frames, while the coefficients of the matrix A are changed only by small amounts.

54

Space Frames

Table 2.3 In-plane and bending efficiencies in x-direction of double-layer space frames. E∗ , D∗

α

stretching

2AE L

√ 8 2 A2

bending

AEd 2 2L

Shape

stretching bending

2.7

√ 3 3AE 2L √ 3 3AEd 2 8L

μ, μ

L

2 9 2 3

√ 9 2 A2 L

2 9 2 3

Continuum Models for Multiple Layer Space Frames A frequently used multiple layer space frame is shown in Fig. 2.17; it is based on a semi-regular tessellation of octahedra and tetrahedra. This frame can also be generated by translation of the square-on-square double-layer space frame in the direction perpendicular to the plane of the layers. Defining the three-dimensional stress components and the corresponding strain components in the standard way, see Fig. 2.17, the homogenized stiffness matrix for this space frame is derived by extending the analysis of Section 2.3.1 for each of the six sets of parallel members and then converting the corresponding stiffness matrices to a common coordinate system x,y,z. The outcome (Lake, 1992; Lake and Klang, 1992) is as follows: ⎤⎡ ⎤ ⎡ ⎡ ⎤ 2 1 1 0 0 0 x σx ⎥⎢ ⎥ ⎢ ⎢ ⎥ ⎢1 2 1 0 0 0⎥ ⎢ y ⎥ ⎢ σy ⎥ ⎥⎢ ⎥ ⎢ ⎢ ⎥ ⎢1 1 2 0 0 0⎥ ⎢ z ⎥ ⎢ σz ⎥ EA 1 ⎥⎢ ⎥ ⎢ ⎢ ⎥= √ (2.63) ⎥⎢ ⎥ ⎢σ ⎥ 2 L2 ⎢ ⎢0 0 0 1 0 0⎥ ⎢γxy ⎥ ⎢ xy ⎥ ⎥⎢ ⎥ ⎢ ⎢ ⎥ ⎣0 0 0 0 1 0⎦ ⎣γyz ⎦ ⎣σyz ⎦ 0 0 0 0 0 1 σzx γzx where L is the length and AE the axial stiffness per unit length of the (identical) members of the space frame. The corresponding linear-elastic constitutive relationship for an equivalent continuum with modulus E and Poisson’s ratio ν is (Gere and Timoshenko, 1990) ⎤⎡ ⎤ ⎡ ⎡ ⎤ 1−ν ν ν 0 0 0 σx x ⎥⎢ ⎥ ⎢ ⎢ ⎥ 1−ν ν 0 0 0 ⎥ ⎢ y ⎥ ⎢ ν ⎢ σy ⎥ ⎥⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ν ⎢ σz ⎥ ⎢ ⎥ ν (1 − ν) 0 0 0 ⎥ E ⎥ ⎢ z ⎥ ⎢ ⎢ ⎥= 1−2ν ⎥ ⎥ ⎢σ ⎥ (1 + ν)(1 − 2ν) ⎢ 0 0 0 ⎥⎢ 0 0 ⎢ ⎢ xy ⎥ ⎢γxy ⎥ 2 ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ 1−2ν 0 0 0 0 ⎦ ⎣γyz ⎦ ⎣ 0 ⎣σyz ⎦ 2 1−2ν σzx γzx 0 0 0 0 0 2 (2.64)

2.7 Continuum Models for Multiple Layer Space Frames

55

z zx

zy

xz yz

x

z

x

xy

yx

y

y

Figure 2.17 Space-filling frame.

(a)

(b)

Figure 2.18 Repeating module for isotropic space-filling space frame: (a) members added to space frame in Fig. 2.17; (b) resulting cubic lattice (Lake, 1992).

The first thing to note is that this space frame has a Poisson’s ratio of 1/3. This can be verified by taking the inverse of the homogenized stiffness matrix in Eq. 2.63 and then looking at the ratio between non-diagonal and diagonal terms. Also the continuum is isotropic whereas the space structure is not. Lastly, changing the axial stiffness of the members of the space frame can only uniformly change all coefficients of Eq. 2.64. The relative density is √ 6 2A (2.65) α= L2 Lake (1992) has shown that the space frame of Fig. 2.17 can be made isotropic by adding to each octahedron the three diagonal members and by redistributing one fifth of the original material into these new members. The resulting structure consists of cubic frames with singly braced faces, see Fig. 2.18. The diagonal members, forming

56

Space Frames

the original structure, have √ a cross-sectional area of 4A/5; the edges of the cubes have a cross-sectional area of 2A/5. The homogenized modulus and Poisson’s ratio of this isotropic frame are as follows EA ; E∗ = √ 2L2

ν∗ =

1 4

(2.66)

The stiffness-to-density ratio of this isotropic space frame is E∗ E = ∗ ρ 6ρ

(2.67)

Note that this is a somewhat different measure of efficiency of the space frame than that used in the previous sections, as this space-filling space frame fully matches the behavior of an equivalent continuum and so there is no need to apply any kinematic constraints on the deformation of the space frame when studying its equivalent properties. Furthermore, Lake has shown that the result in Eq. 2.67 is valid for all isotropic three-dimensional space frames. An alternative type of space-filling tessellation is based on the octet truss of Figs. 2.8(b) and 2.9. This case has been analyzed by Deshpande et al. (2001) and Christensen (2004).

2.8

Applications It has been shown in the previous sections that the structural efficiency of a space frame is lower than that of a continuum. This result may seem to be at odds with the widespread use of space frames, both in nature and man-made structures, until one realizes that the range of available materials is rather limited and a key advantage of space frames is that by changing the relative density we can in effect achieve any density we require. So the loss of efficiency associated with slicing a continuum, as explained in Section 2.4.1, is the price to achieve a material with tailorable density.

Figure 2.19 Beijing National Aquatics Center (Nokuro/Shutterstock.com).

2.8 Applications

57

There are many other reasons for using a space frame instead of a continuum. Perhaps the main reason is visual: Figs. 2.19 and 2.20 show the Beijing National Aquatics Center, also known as the Water Cube. This structure is a striking example of a space frame with a less obvious repeating unit cell, derived from the Wearie–Phelan space-filling tessellation shown in Fig. A.8.

Figure 2.20 Internal view of Beijing National Aquatics Center (© Arup).

(a1)

(a2)

(b1)

(b2)

Figure 2.21 Bistable space frame based on cubic repeating module: (a1 and a2) stress-free configuration and (b1 and b2) alternative configuration of stable equilibrium.

58

Space Frames

Figure 2.21 shows a space frame based on a cubic repeating module that is bistable. This structure has two configurations of stable equilibrium and can be flipped from one to the other by pulling two opposite corners of the structure. The first stable configuration is the cylindrical stress-free configuration in which the structure has been constructed and is shown in Fig. 2.21(a1–a2). The second stable configuration is also a cylindrical configuration but the axis of the cylinder has now rotated 90◦ . This configuration has nonzero strain energy but corresponds to a local energy minimum and hence it is stable. This structure was designed by Ye and Pellegrino (2005) by investigating the strain energy surface for all possible cylindrical shapes of the structure.

3

Tension Structures

3.1

Introduction Tension structures are the ultimate lightweight architecture and are used for roofs and canopies, bridges, balloons, sail ships, and large spacecraft. Tension structures are thin and flexible, and hence can easily change their shape in response to the applied loads. They carry loads by pure tension: no compression, out-of-plane bending, or shear can be carried. The simplest example of tension structure is the cable, whose shape is such that “the action of any part of the line upon its neighbor is purely tangential” (J. Bernoulli). Cables are made of twisted steel strands (ropes), of parallel or braided polymer yarns, etc. The definition of a cable naturally extends to that of a membrane, which is a flat or curved structural element in which one length scale (thickness) is very small in comparison with the other two length scales. A membrane carries only tensile forces that lie within the plane tangent to the mid-surface. A simple example of a flat membrane is a sheet of paper. Also, polymer foils made of Mylar are used for packaging food and to make birthday balloons, and aluminum-coated Kapton foils are used to wrap flower bouquets and to form thermal blankets for spacecraft structures. Many cables may be joined together to form a cable net, to which a membrane skin can be attached. Figure 3.1 shows a cable-net-and-membrane skin cooling tower; the tension in the cable net is equilibrated by several rings and a central strut, which are in compression. Nets are also found in nature, and Fig. 3.2 shows examples of almost two-dimensional and three-dimensional spiders’ webs. A different type of tension structure, based on continuous membranes stretched over edge cable supports, is shown in Fig. 3.3. This iconic fabric roof structure, whose shape was inspired by the snow-capped Rocky Mountains, was designed by Horst Berger and completed in 1993. Figure 3.4 shows the inflated membrane sculpture Leviathan by the artist Anish Kapoor. The entire surface was made of a single piece of PVC-coated polyester membrane with a surface area of ∼12,000 m2 . This “walk-in” sculpture, engineered and built by Hightex, was exhibited at the Grand Palais in Paris in 2011. It was 100 m long and 34 m high, weighed 12 tons and was installed in 8 days. Another example of a tension structure, where the shape of the outer membrane is a lightly stressed skin that hides a series of internal load-carrying balloons, is the large airship shown in Fig. 3.5. 59

60

Tension Structures

Figure 3.1 Hyperbolic paraboloid cable net (Kullmann et al., 1975).

Figure 3.2 Spider webs are an example of tension nets in nature (Kullmann et al., 1975).

This chapter investigates the basic load-carrying mechanisms of different tension structures in order to explain the relationship between shape and structural behavior. We compare three simple cable structures of given shape and show how their shapes and structural characteristics affect their response. We begin by considering a two-dimensional problem: finding the shape of an inextensional cable by energy minimization. Then, we tackle the problem of determining the axisymmetric shape of a hot-air or helium balloon, both by extending the energy approach and by an equilibrium approach which enforces direct constraints on the stress distribution in the membrane. We present an analysis of the stiffness of pressurized cylinders. We also introduce

3.2 Three Simple Cable Nets

61

Figure 3.3 Denver International Airport (Arina P Habich/Shutterstock.com).

Figure 3.4 Sculpture Leviathan exhibited at the Grand Palais in Paris in 2011 (Photo © Raphael

GAILLARDE / Gamma-Rapho / Getty Images).

general computational techniques that can be used for any form-finding problem involving cables or membranes.

3.2

Three Simple Cable Nets The structures shown in Fig. 3.6 consist of four cables of equal tensile stiffness per unit length AE that are attached to an edge foundation; the projections of the foundation points onto a horizontal plane lie on the edges of a square of side length 3L.

62

Tension Structures

Figure 3.5 The Schütte-Lanz airship (Schütte, 1926).

The cables intersect at four cross-over points and are joined at these points, forming a horizontal square. It is assumed that transverse forces may be applied only at these four points. Modeling the cables as pin-jointed bars (this assumption will be further discussed later on) allows us to do a quick rigidity analysis of these structures using Maxwell’s

3.2 Three Simple Cable Nets

L

L

B

L L

P

L L

P

P

C P

u v v

A

63

D (a)

B P P P

r

C

P r

A

D

(b)

B

P

P C

P P

A

r (c)

D

Figure 3.6 Simple cable structures (a) flat; (b) saddle; (c) bubble.

rule, Eq. 1.2. We set d = 3 because we are dealing with a three-dimensional structure and we substitute for the number of joints j = 12, for the number of bars b = 12, and for the total number of kinematic constraints k = 24. This gives: 3 × 12 − 12 − 24 = m − s ⇒ m − s = 0

(3.1)

and this result is of course valid for all three structures, although we will see that m = s has different values. The main difference between the three structures is in the height of the supports with respect to the central square. In Fig. 3.6(a) the supports are coplanar and thus we have a simple two-dimensional cable net whose behavior is similar to that of a spider’s web, Fig. 3.2. In Fig. 3.6(b) pairs of supports are at a distance r alternately above and below the central square, thus forming a simple saddle-shaped structure which resembles the cable net of Fig. 3.1 and the membrane of Fig. 3.3. In Fig. 3.6(c) the supports are

64

Tension Structures

coplanar, but the central square is a distance r below the plane of the supports. This last structure is a simple bubble-shaped net that captures the behavior of the pneumatic structures in Figs. 3.4–3.5. We compare the stiffnesses of these three structures against a set of four vertical forces P , starting with the case where the structures are initially unstressed.

3.2.1

Flat Cable Net Consider the structure shown in Fig. 3.6(a). It has four independent states of self-stress (each involving equal non-zero forces in one cable only) and, to satisfy Maxwell’s rule, also four independent inextensional mechanisms, hence m = s = 4. Four equal vertical forces P are applied and, by symmetry, the four joints move down by u and horizontally outwards by v, as shown in Fig. 3.7(a). All cables carry approximately the same tension and hence extend by the same amount e. Therefore, due to symmetry, each end of the central cable segment moves horizontally by half this amount: v = e/2

(3.2)

and so the deformation of a side cable segment is as shown in Fig. 3.7(b). Hence we can write   e 2 e  2 u 2 2 +u =L 1− + (3.3) L− L+e = 2 2L L

L

L P

P

u L+e

v

v=e/2

v

L

(b)

P

P

L

P/2

∼T L

u

L P/2

T0

L

T (c)

P/2 T0 T0 (a)

Figure 3.7 (a) Top view and side view of deformation of flat cable net; (b) extension of side cable;

(c) free body diagram for a joint.

3.2 Three Simple Cable Nets

Expanding the square root as a Taylor’s series gives   e u2 e2 L+e ≈L 1− + + 2L 8L2 2L2

65

(3.4)

Then, neglecting e2 and simplifying gives e u2 e≈− + 2 2L

(3.5)

e ≈ u2 /3L

(3.6)

which can be solved for e to obtain:

The tension T in the cables can be assumed to be uniform because the cables are nearly straight in the deformed configuration. Hence, resolving vertically the forces acting on the free body for one of the joints, Fig. 3.7(c), L PL P × = 2 u 2u The corresponding extension of each cable segment is T =

(3.7)

P L2 L = (3.8) AE 2AEu Equating Eq. 3.8 to Eq. 3.6 and re-arranging, we obtain a cubic load-deflection expression

u 3 2 P = AE (3.9) 3 L Hence, the stiffness of this structure is: e=T ×

dP AEu2 (3.10) =2 3 du L where the subscript 0 denotes the fact that the structure is initially unstressed. By substituting u = 0 into Eq. 3.10, we find k0 = 0. In other words, this structure has zero stiffness in the initial, flat configuration. In conclusion, the structure in Fig. 3.6(a) is a mechanism and has zero stiffness against the particular load that has been selected, unless geometry-change effects are included in the analysis. The above geometrically nonlinear analysis has shown that the stiffness increases with the square of the deflection u. k0 =

3.2.2

Saddle Cable Net To analyze the stiffness of the structure shown in Fig. 3.6(b), we note that only the two cables connecting edge BC to edge AD will be effective; the other two cables go slack as they would be required to carry compressive forces. An analysis of the equilibrium matrix of a pin-jointed model of the structure, as described in Section 1.5, gives m = s = 1, and indeed it can be verified that there is a state of self-stress in which all cables are under tension. However, if the slack cable elements are removed from the

66

Tension Structures

Q

L

L

L

V

α

P

P

r R

S

Figure 3.8 One cable of the saddle net.

pin-jointed model, then we are left with two separate hanging cables. Then, applying Maxwell’s rule to each cable leads to the conclusion that there are three independent mechanisms (one in the plane of the cable and two out of plane) and no states of selfstress for each cable. Therefore, we need to analyze the behavior of each cable separately, as shown in Fig. 3.8. Although this geometry looks similar to that analyzed in Fig. 3.7, in the present case the initial rise r is finite and so can be assumed to be much larger than the deflection due to the applied loads. Hence, there is no need to consider the effects of a (small) deflection on the equilibrium equations. In other words, this problem can be treated as geometrically linear. In Fig. 3.8 we define the angle α = arctan

r L

(3.11)

Then, equilibrium of joint R yields the following tensions for the cable segments TQR = TSV =

P ; sin α

TRS =

P tan α

(3.12)

The corresponding extensions are eQR = eSV =

PL ; AE sin α cos α

eRS =

PL AE tan α

(3.13)

and the vertical component of deflection of R (and S) due to these extensions can be found using the Principle of Virtual Work in Eq. B.46. The static system is as shown in Fig. 3.8, with two virtual external forces of P = 1 at R and S, and the cable tensions calculated from Eq. 3.12 for this particular value of P . The compatible system consists of the unknown vertical displacements u of R and S, and the cable extensions in Eq. 3.13. Substituting into Eq. B.46 and solving for u gives: u=

P L 2 + cos3 α AE 2 sin2 α cos α

(3.14)

Re-arranging Eq. 3.14, the load-deflection relationship has the following linear expression P = 2AE

sin2 α cos α u 2 + cos3 α L

(3.15)

3.2 Three Simple Cable Nets

67

The stiffness of the second structure is obtained by differentiation of Eq. 3.15 k=

2AE sin2 α cos α L 2 + cos3 α

(3.16)

and here it should be noted that, unlike Eq. 3.10, this expression is independent of the deflection u.

3.2.3

Bubble Cable Net The stiffness of the third structure, Fig. 3.6(c), can be obtained from the results for the previous structures. Because the geometry of its cables is identical to Fig. 3.7(b), in the case that P is downwards (P > 0) all four cables are effective and so, for a given value of P , the cable tensions are half of those obtained in Section 3.2.2. Therefore, the extensions given by Eq. 3.13 are halved and therefore the deflection u given by Eq. 3.14 is also halved. Thus, the final outcome is that the load-deflection relationship is obtained by doubling Eq. 3.15: P = 4AE

sin2 α cos α u 2 + cos3 α L

(3.17)

In conclusion, the stiffness of this structure – which also remains approximately constant when a small deflection u occurs – is k=

4AE sin2 α cos α L 2 + cos3 α

(3.18)

If P is upwards (P < 0) then all cables are slack and so P = 0 and k = 0. For very large deflections, |u| > 2r, the net flips from a concave to a convex configuration. In this reversed configuration its stiffness for P < 0 is positive.

3.2.4

Comparison Compare the behavior of the three structures for the case L = 333 mm, r = 100 mm, and AE = 660 kN, corresponding to a square of 1 m2 spanned by steel wires with a cross-sectional diameter of 2 mm. The load-deflection responses of the three structures, given by Eqs. 3.9, 3.15, and 3.17, have been plotted in Fig. 3.9(a). The material behavior has been assumed to be linear-elastic, hence neglecting the yield limit of steel. Note that the deflection of the first structure due to P = 1 kN is well over 40 mm, and is over five times larger than the deflections of the other two structures. If the direction of the load is reversed, then obviously the load-deflection relationships for the first two structures are unchanged, but now the third structure has zero stiffness and P = 0.

3.2.5

Effects of Prestress The behavior of the first two structures can be changed by imposing an initial state of prestress. If all four cables of the structure shown in Fig. 3.6(a) are made slightly shorter

68

Tension Structures

3 (c) bubble

(b) saddle

2 1

(a) flat

P (kN) 0 –1 –2 –3 –50 –40 –30 –20 –10

0 10 u (mm)

20

30

40

50

(a)

3 bubble 2

saddle

1 flat P (kN) 0 –1

flat –2

saddle

–3 –50 –40 –30 –20 –10

0 10 u (mm)

20

30

40

50

(b) Figure 3.9 Load deflection curves for three structures shown in Fig. 3.6; (a) without prestress and (b) with an initial prestress of 1 kN in the horizontal cables and corresponding values in the other cables.

than the distance L between the supports, but all of exactly the same length, they will have to stretch a bit before they can be connected to the supports. Thus, they will now carry an equal pre-tension T0 . The reason why the lengths of all four cables have to be separately controlled is that this structure has s = 4. The structure in Fig. 3.6(b) can also be prestressed. Its single state of self-stress can be computed by applying unit vertical forces – representing actions from the other cables – on one of the four cables. For the case L = 333 mm, r = 100 mm, and for

3.2 Three Simple Cable Nets

P = 1, the cable tensions are [3.48 the cable net is α [3.48

3.33 3.33

69

3.38] and hence the state of self-stress in 3.48

3.48

...]

Note that, unlike the first structure, in this case the whole prestress is controlled by a single parameter because s = 1. The third structure cannot be prestressed by changing the length of the cables because its pin-jointed model is statically determinate. However, its cables could be put in a state of pre-tension by applying a set of initial external forces P0 to the four joints, thus inducing the tensions P0 [3.48 3.33 3.48 3.48 . . . ] 2 This makes it possible for the cables to carry both tensile and compressive changes of axial force, due to extra loads P which are therefore no longer restricted to being only downwards. This is precisely what is done in pneumatic structures, where the initial pre-tension results from the pressure of the gas inside the structure. We will now investigate the effects of applying the states of prestress just described to the three cable nets, to see how their behavior changes. For the first structure, assuming that the out-of-plane displacement is not too large, we can calculate the value of P required to cause a specified displacement u by superposing two separate terms. The first term corresponds to the elastic response of the structure, which has been analyzed in Eq. 3.9. The second term corresponds to the geometric stiffness associated with the geometry-change effect shown in Fig. 3.7(c): assuming that the prestress remains equal to T0 , for equilibrium in the displaced configuration a force u (3.19) P = 2T0 L is required. Hence, adding Eq. 3.9 and Eq. 3.19 we obtain

u 3 u 2 (3.20) P = 2T0 + AE L 3 L Differentiating Eq. 3.20 with respect to u and substituting u = 0, we find the stiffness in the initial configuration: 2T0 (3.21) L For the second structure, the main effect of applying a state of prestress is that all cables are now effective up to the point where two cables becomes slack, at which point the compressive force due to P becomes equal and opposite to the initial pretension. The third structure is similarly unaffected by the state of prestress when a downward load is applied. However, the structure will now retain the same stiffness when the load direction is reversed, up to the point where the cables become slack. Figure 3.9(b) shows the load-deflection curves for the three structures with the specific geometry and properties considered in Section 3.2.4, and for an assumed kT0 =

70

Tension Structures

pre-tension T0 ≈ 1 kN. The pre-tension is exactly 1 kN in all cables of the flat net and also in the horizontal cables of the other two nets, but it is about 4% higher in the inclined cables. A comparison of Fig. 3.9(b) to Fig. 3.9(a) shows that the addition of the linear term to curve (a) has the effect of rigidly rotating this curve about the origin, by an amount proportional to the prestress. Line (b) now coincides with (c), and both lines extend into the negative load range. At P ≈ −0.6 kN two cables of the saddle net become slack and its stiffness for additional loads is halved. At the same load, the bubble net becomes completely slack and loses its stiffness altogether. In conclusion, in the case of the flat cable structure, Fig. 3.6(a), prestress has the effect of imparting a linear stiffness, while in the case of the three-dimensional cable structures, Fig. 3.6(b) and 3.6(c), it enables the cables to carry compressive, as well as tensile force changes. The stiffness of the kinematically determinate configurations is usually much larger than the geometric stiffness that can be provided by prestress. Also, it can be shown that the stiffnesses of the second and third structures increase with r, as increasing the rise has the effect of reducing the cable forces, until an analytical maximum is reached for   d sin2 α cos α =0 (3.22) dα 2 + cos2 α After this point, the effect of increasing the lengths of the cables becomes greater than the effect of reducing the cable forces. In practice, almost all prestressed cable nets are saddle-shaped, but they have a high degree of kinematic indeterminacy because of their large number of joints. Hence, under most load conditions their inextensional mechanisms are excited and so the behavior seen in the flat cable net example is observed. This behavior can be described as softlinear with stiffening non-linearity. However, for some special load conditions – see the geometric loads introduced in Section 4.3.1 – a cable net behaves approximately linearly, like a kinematically determinate structure. This is the behavior that was seen in the saddle net example: by applying a symmetric set of loads, the inextensional mechanism was not excited. More details, including an extension to large cable nets of the stiffness calculations performed in this section, can be found in Pellegrino (1992). Analogous considerations apply to prestressed membrane structures as well. Pneumatic structures behave approximately linearly, and the only difference between the response to a general load and a special load condition of the type mentioned above is in the magnitude of the corresponding stiffness.

3.3

Soap Films In the previous section we have focused on the calculation of the stiffness of tension structures with a given shape, but we found that determining the initial, prestressed shape of a tension structure is an important problem in itself. This problem is called form-finding and will be explained by first considering the shape of structures made from a rather special kind of material: soap film.

3.3 Soap Films

(a)

71

(b)

Figure 3.10 Form-finding of a small tent with models made of (a) soap film (Wendland, 2001) and (b) thin fabric (Gass, 1990).

We can determine the shape of a prestressed membrane with prescribed boundaries by dipping a rod with the same shape as the prescribed boundary, into a bath of soap solution. When the rod is taken out of the water, a soap film will form whose shape is approximately equal to the shape of a membrane stretched over the same boundary. Figure 3.10 compares the two corresponding shapes of a soap film model of a tent structure and a model made with fabric (Wendland, 2001). To understand why the two shapes are rather close, we begin by noting that soap film is a special membrane with the property that the tension per unit length N has the same value at all points of the film and, at any one point, it has the same value in all directions and remains approximately constant when the surface area varies within wide limits (Isenberg, 1992). N is the surface tension of the film. Soap film has the property that its mean curvature H is zero everywhere, where H is defined in Section A.4.3. We will now show a proof of this statement starting, for generality, from the case of a soap film subject to a normal pressure p. Consider a small, general curvilinear rectangular element of soap film, ABCD, of area S, as shown in Fig. 3.11. The sides of this rectangle have lengths x and y. The radii of curvature (not necessarily principal) of the sides are r1 and r2 , see Section A.4.1. Now let the surface element ABCD undergo a virtual displacement, δu, normal to the surface, and let A B  C  D  be the new position of the element. The radii of curvature will increase to r1 + δu and r2 + δu, and the sides of the surface element will increase to x + δx and y + δy. The ‘external’ virtual work, i.e., the work done by the pressure p is δW = pSδu

(3.23)

The internal work is equal to the surface tension multiplied by the change of surface area δF = N δS Expressing Eq. 3.24 in terms of x and y

δF = N (x + δx)(y + δy) − xy

(3.24)

(3.25)

72

Tension Structures

C' D' C

du

B'

A' D

z y x p A

r1

B

r2

O1

O2 Figure 3.11 Infinitesimal element of soap film in original configuration ABCD and displaced configuration A B  C  D  .

Expressing δx and δy in terms of x, y, r1 , r2 , and δu      δu δu δF = N xy 1 + 1+ − xy r1 r2

(3.26)

then, neglecting higher-order terms 

1 1 + δF = N xyδu r1 r2

 (3.27)

Equating the internal virtual work (δW in Eq. 3.23) with the external virtual work (δF in Eq. 3.27) and substituting S = xy gives   1 1 + (3.28) pSδu = N Sδu r1 r2 and simplifying:  p=N

1 1 + r1 r2

 (3.29)

This equation is valid for any orthogonal system of coordinates. In particular, it can be written in terms of the principal radii of curvature R1 and R2 , see Section A.4.1, to obtain   1 1 + (3.30) p=N R1 R2 The case p > 0 corresponds to pressurized soap bubbles, whereas the case p = 0 corresponds to soap films.

3.4 Shape of Hanging Cables

73

Figure 3.12 Soap film forming a catenoid surface (Isenberg, 1992).

Comparing Eq. 3.30 and Eq. A.40 for the case of a soap film (p = 0) we find H =0

(3.31)

This is known as the Laplace–Young equation and states that the mean curvature of a soap film is zero. In other words, for a given set of boundary elements, a soap film forms the minimal area surface that spans the given boundary (Isenberg, 1992). Despite the formal simplicity of the Laplace–Young equation, finding analytical solutions to problems with different boundary shapes is difficult and the number of analytical surfaces that are known is small. A problem which has an analytical solution is the shape of the soap film/minimal area surface joining two parallel circular rings. This surface is the catenoid, i.e., a catenary of revolution, Fig. 3.12, if the distance between the rings is less than 1.056a, where a is the radius of the rings. If the distance between the disks is greater, the minimal area surface consists of two separate, flat films one across each ring.

3.4

Shape of Hanging Cables Consider an inextensional cable of length L and uniform mass m per unit length that hangs between two fixed points, as shown in Fig. 3.13. The cable is subject to a uniform gravitational field and hence takes a configuration such that its potential energy is minimized. Let y be the height of the cable above a datum; we want to determine the function y(x). In other words, we want to determine a function y that minimizes the potential energy of the cable, subject to the constraint that the cable length is L. This is known as an isoperimetric (equal perimeter) problem (Langhaar, 1962). The potential energy of the cable is  s2 mgyds (3.32) V = s1

74

Tension Structures

y y2 y1 g L

x2

x1

x

Figure 3.13 Hanging cable.

and, since ds =

  dx 2 + dy 2 = 1 + (y  )2 dx

where y  = dy/dx, we can write V in the form  x2  V = mgy 1 + (y  )2 dx

(3.33)

(3.34)

x1

The length of the cable in a general configuration is given by  x2   s2 ds = 1 + (y  )2 dx L= s1

(3.35)

x1

and finding the minimum length subject to the isoperimetric condition in Eq. 3.35 is a variational problem, see Section B.9. The solution procedure is only outlined here; further details can be found in Langhaar (1962). V and L are combined into a single, modified potential by introducing the Lagrange multiplier λ V = V + λL   x2   2 mgy 1 + (y ) dx + λ = =

x1  x2

 (mgy + λ) 1 + (y  )2 dx

x2



1 + (y  )2 dx

x1

(3.36)

x1

Hence the functional to be minimized is

 F = (mgy + λ) 1 + (y  )2 dx

(3.37)

which is of the type considered in Section B.9. Hence, Eq. B.61 yields, after some manipulation mgy + λ  =k 1 + (y  )2

(3.38)

3.4 Shape of Hanging Cables

75

rearranging y+

 λ k 1 + (y  )2 = mg mg

(3.39)

The integral of this nonlinear differential equation is y=−

λ x − c2 + c1 cosh mg c1

(3.40)

This solution contains three unknown parameters, c1 , c2 , and λ, which in general can be determined by imposing two boundary conditions, i.e., by requiring that the cable passes through the points (x1,y1 ) and (x2,y2 ), plus the condition that the total length of the cable is L. We will restrict our attention to a symmetrically hanging cable between two points at distance ; hence, the coordinates of the end points are assumed to be (±/2,0). It can be shown that c2 = 0 and then, by imposing the remaining boundary condition, we obtain: c1 cosh

 λ − =0 2c1 mg

(3.41)

The length condition gives /2 



1 + sin2

0

x L dx = c1 2

(3.42)

which can be integrated to obtain c1 sinh

 L = 2c1 2

(3.43)

Thus, given  and L, first we determine c1 by solving Eq. 3.43 numerically, and then we determine the cable shape from the equation    x − cosh (3.44) y = c1 cosh c1 2c1 An alternative derivation of this result can be found in Irvine (1992). Four catenary shapes for cables with L/ = 1.1, 1.5, 2, and 5 have been plotted in Fig. 3.14. The hanging cable problem is often analyzed by assuming that the weight per unit projected length of cable over the x-axis is approximately constant, and thus neglecting the term (y  )2 in the expression for ds, in Eq. 3.33. This assumption results in a considerable simplification of the analysis, and it is found that the shape of the cable is a parabola. It is interesting to compare catenaries of different length with the corresponding parabolic curves, see Fig. 3.14. It can be seen that up to L/ = 1.5, corresponding to a sag/span of ≈0.5, there is hardly any difference between the two solutions.

76

Tension Structures

1 1.1 1.5 2

parabolic shape catenary shape

5

Figure 3.14 Comparison of exact and approximate, i.e., parabolic, shapes of hanging cables of different lengths between level supports at a distance of 1.

3.5

Shape of Zero-Superpressure Balloons Balloons filled with helium are used to lift a variety of scientific instruments into the stratosphere. The shape of these balloons is calculated on the basis of the atmospheric pressure at the operational altitude, which is called the float altitude. There are two types of balloons: zero-superpressure balloons, open at the bottom to equalize the internal and external pressures, are the most widely used, and superpressure balloons, forming a closed envelope and able to fly at almost constant altitude (Fairbrother and Pierce, 2009). Due to the much higher density of the atmosphere near the earth’s surface, the shape of these balloons near the ground is highly elongated. The helium forms a small bubble near the top apex and gradually expands as the balloon rises in the atmosphere; at float altitude the balloon envelope is fully inflated, see Fig. 3.15. Note that this extreme shape change is not seen in hot-air balloons, see Fig. 3.16, which are used at low altitudes. These balloons are also open at the bottom, and hence are of the zero-superpressure type, but their shape at launch practically coincides with the shape during flight. The basic idea of all balloons is that the gas inside the balloon is lighter than the surrounding air and the resulting buoyancy provides a net lift. Also, the pressure difference between the inside and the outside of the balloon puts the balloon envelope in a state of tension and so stabilizes the envelope. In the case of small rubber balloons the envelope stretches until it reaches equilibrium, however the envelope of large balloons is made of a stiffer polymer film shaped in a way that achieves a desirable stress

3.5 Shape of Zero-Superpressure Balloons

(a)

77

(b)

Figure 3.15 Equilibrium shapes of helium balloons (a) below float altitude and (b) at float altitude (Baginski and Ramamurti, 1995).

Figure 3.16 Hot air balloons (image downloaded from Pixabay).

distribution. A form-finding analysis is required to determine the cutting pattern for the gores of these balloons. We will present two different ways of performing form-finding calculations for balloons. The first method aims to identify the optimal profile for a balloon, i.e., the shape that maximizes the gravitational potential of the gas contained in the balloon, without any detailed structural considerations. The second method determines the equilibrium shape of the balloon with a prescribed distribution of stress, and hence can guarantee that the balloon envelope is under biaxial stress everywhere. There are more advanced computational methods currently available (Baginski and Schur, 2003, Deng and Pellegrino, 2009, Pagitz and Pellegrino, 2007, Wakefield, 2007) but the

78

Tension Structures

analytical approaches presented here are simpler and provide greater insight. In both methods, we will not consider the weight of the membrane itself, for simplicity; this effect has been analyzed by Smalley (1965).

3.5.1

Energy Approach To find the optimal shape of a balloon we consider configurations that satisfy two conditions: • •

the enclosed volume V is equal to a prescribed value, which is related to the required buoyancy force; and the perimeter of a meridian has a specified length L.

A particular balloon shape, assumed to be axisymmetric, is defined by the height H and the function r(z) defined in Fig. 3.17. Let ρ be the difference between the density of the atmosphere and the gas inside the balloon; g is the gravitational acceleration measured at the float altitude. The gravitational potential energy of the gas for a given configuration of the balloon, W , is given by  H W = πρg zr 2 dz (3.45) 0

We want to find the function r(z) that maximizes W in order to find the most stable shape of the balloon, subject to the iso-perimetric and iso-volumetric conditions  H L= 1 + (r  )2 dz (3.46) 0



V =π

H

r 2 dz

0

where r  = dr/dz. Note that Eq. 3.46 is almost identical to Eq. 3.35. r z=0 dr dz R2

df

R1

f

H

z

Figure 3.17 Meridional profile of balloon.

(3.47)

3.5 Shape of Zero-Superpressure Balloons

79

As in Section 3.4, this is a constrained optimization problem that can be solved using the calculus of variations. First, we multiply the two constraint functions by the Lagrange multipliers λ and μ, to obtain the modified potential function W = W + λL + μV   H zr 2 dz + λ = πρg 0

0

H

  1 + (r  )2 dz + μπ

H

r 2 dz

(3.48)

0

Then, we apply the standard machinery to transform the functional minimization into a differential equation, Eq. B.61, and obtain  r d  + (μ + ρgz)r = 0 (3.49) λ dz 1 + (r  )2 This is a nonlinear differential equation that can be integrated numerically, starting from the top of the balloon. The initial conditions to start the integration at the top of the balloon are r = 0 and r  → ∞. Figure 3.18 shows a series of shapes obtained by Nishimura et al. (1963) by prescribing different values for ρg/λ. Instead of imposing the constraints in Eq. 3.46 and Eq. 3.47 during the integration, they calculated at the end of each integration the length L and volume V corresponding to each particular shape. Nishimura et al. (1963) show that if the hoop stress is assumed to be zero everywhere, then the two terms in Eq. 3.49 correspond to the terms N/R and p, respectively, in the equilibrium equation for a curved string loaded by a lateral pressure p, with tension N and with local radius of curvature R.

1/20 1/40 1/50

1/100 1/120

1/160

1/170 1/176 Figure 3.18 Equilibrium profiles for different values of ρg/λ (Nishimura et al., 1963).

80

Tension Structures

A general feature of this approach is that there is no guarantee that it will produce balloon designs with a particularly desirable stress distribution.

3.5.2

Direct Equilibrium Approach To consider the stress distribution while carrying out the form-finding analysis, we can follow an alternative approach that was first proposed by Taylor (1963) during a study of parachutes. The idea is to choose a particular functional relationship between the stress resultants in the membrane and to use the equations of equilibrium to determine the shape of the membrane required to equilibrate those particular stresses. This approach has been applied to the form-finding of balloons by Irvine and Montauban (1980) and here we will follow their analysis. The pressure loading on the balloon is purely in the normal direction; its distribution varies hydrostatically in the z-direction and has a value of zero at the bottom of the balloon, hence pφ = 0

pn = ρg(H − z)

(3.50)

The functional relationship between the stress resultants is chosen such as to accomplish three things: first, make Nθ = Nφ at the crown of the balloon, to avoid stress singularities, second keep Nθ positive everywhere (the sign of Nφ is controlled by direct equilibrium considerations and hence remains tensile), and third obtain a balloon shape with a pinched meridional profile at the bottom. Irvine and Montauban (1980) adopted the following relationship:

  (3.51) Nθ = 1 − A r/H Nφ √ where A is an arbitrary constant such that 0 ≤ A r/H ≤ 1. The left and right-hand sides of this inequality ensure that Nθ has the same sign of Nφ and that Nθ is not larger than Nφ , respectively. Substituting Eqs. 3.50 and 3.51 into the equilibrium equation in the meridional direction for an axisymmetric shell, Eq. B.43, and integrating gives: 

 ρgrc H (3.52) exp −2A r/H Nφ = 2 

   ρgrc H Nθ = (3.53) 1 − A r/H exp −2A r/H 2 where rc is the radius of curvature at the crown, which is assumed to be locally spherical. Next, we substitute into the equilibrium equation in the normal direction for an axisymmetric shell, Eq. B.44, the expressions for the principal curvatures, Eqs A.29 and A.36, and also the meridional stress resultant given by Eq. 3.52. Rearranging this expression we obtain the following differential equation for the profile of the balloon √

  H (1 − A r/H ) 2 H (d 2 r/dz2 ) + = (H − z) exp 2A r/H (3.54) − rc [1 + (dr/dz)2 ]3/2 r[1 + (dr/dz)2 ]1/2 This equation can be integrated numerically by a shooting technique, starting from the apex of the balloon.

3.5 Shape of Zero-Superpressure Balloons

81

Table 3.1 Dimensions of hot-air balloon that can lift a payload of 2 to 3 kN (Irvine and Montauban, 1980). 2,000 m3

Volume

793 m2 17.1 m 54.6 m 16.4 m 1.7 m

Surface area Height H Radius at crown rc Maximum diameter Radius at throat

-0.3

0.0

0.3 w H

0.0

0.2

0.4

0.6 0.3 r H

0.2 0.6 0.4

0.6

0.9

0.8 1.2 z H

1.0

1.5 s H

Figure 3.19 Typical meridional profile of a hot air balloon and shape of a gore, obtained for rc = 3.20 and A = 1.44 (Irvine and Montauban, 1980).

As an example, Fig. 3.19 shows the solution corresponding to rc = 3.20 H and A = 1.44. Note that the gore cutting pattern on the right, defined by the arc-length along the meridian, s, and the width of the gore, w, assumes that the balloon will be constructed from eight identical gores. The properties of a hot-air balloon with this profile and capable of lifting a payload of 2 to 3 kN are given in Table 3.1. Finally, it should be noted that the weight of the balloon membrane has not been considered in the form-finding analyses shown in this section. This effect has been

82

Tension Structures

incorporated in the variational method by Nishimura (1993) and in the equilibrium analysis by Smalley (1965).

3.6

Pneumatic Domes A variant of the problem studied in the previous section is encountered if one searches for the best shape for an axisymmetric pneumatic dome with a vertical profile at the base. The simplest shape is the hemisphere but this shape corresponds to a rather tall dome structure. Consider a hemisphere loaded by an internal pressure p. The principal curvatures are R1 = R2 = R and substituting these values into the equation of equilibrium in the normal direction, Eq. B.44, one finds: Nφ = Nθ =

pR 2

(3.55)

The span to rise ratio for this shape is 0.5 but often in pneumatic buildings it is required that the rise for a give span be minimized to reduce the volume of dead space at the top of the dome. So, the problem that we want to solve is finding the shallowest possible shape that avoids the formation of wrinkles. This problem was solved by Kawaguchi (1977) and a summary of his study is contained in the book by Irvine (1992). A shallower alternative to the hemisphere is the ellipsoid of revolution. It can be shown that – although Nφ > 0 for all ellipsoids – Nθ becomes negative near the major axes, and hence that the ellipsoid of revolution will wrinkle in the meridional √ direction, if the rise to span is less than 1/2 2, i.e., about 0.35. This particular geometry defines the critical ellipsoid shown in Fig. 3.20. Note that shapes with non-vertical tangents around the perimeter – see the solution labelled partial sphere in the figure – are excluded, because they create additional dead space around the perimeter.

Hemisphere

Critical ellipsoid Shallowest surface Partial sphere Effective height

Figure 3.20 Different shapes for a pneumatic dome (Kawaguchi, 1977).

3.6 Pneumatic Domes

83

Kawaguchi showed that the shallowest wrinkle free pneumatic dome has a rise to span ratio of less than 0.3 and is not an ellipsoid. Kawaguchi’s solution was as follows. First, it was noted that the stress resultants in a dome of revolution subject to uniform normal pressure p are Nφ = pR2 /2

(3.56)

Nθ = pR2 (1 − R2 /2R1 )

(3.57)

Equation 3.56 can be readily obtained from Eq. B.45. Substitution of Eq. B.44, yields Eq. 3.57. The best possible dome design is “just on the point of wrinkling” and hence has Nθ = 0. Hence, setting Nθ = 0 in Eq. 3.57 gives R1 =

R2 2

(3.58)

The solution of this equation can be written in terms of elliptic integrals or can be found by numerical integration; further details on the solution are given in Section 4.5. It provides a solution, shown in Fig. 3.20, with a rise to span ratio of 0.2996. Figure 3.21 shows two physical models that were constructed by Kawaguchi. The model on the left is based on the solution of Eq. 3.58 and is wrinkle free. The model on the right is based on an ellipsoid of revolution with equal semi-axes and is visibly wrinkled.

(a)

(b )

Figure 3.21 Photographs of inflated models (plan and side views) of (a) wrinkle-free shallowest surface and (b) wrinkled ellipsoid of revolution (from Kawaguchi, 1977).

84

Tension Structures

3.7

Pressurized Membrane Cylinders Lightweight inflatable structures depend on internal pressure for much of their loadcarrying ability. They have had a variety of applications in the past and are currently receiving considerable attention for “gossamer” spacecraft structures. Examples are the Echo balloons described in Section 9.6 and the inflatable antenna structure shown in Fig. 3.22, developed by L’Garde for NASA. A commonly used tensioned membrane structural element is the pressurized cylinder. This structure behaves much like a normal beam if the membrane is everywhere in a state of tension, that is if the compressive longitudinal stresses due to the applied loads are smaller than the prestress due to the pressure. Stein and Hedgepeth (1961) studied a cylinder subject to a combination of axial load P and bending moment M. They showed that the initial appearance of wrinkles does not correspond to the maximum load that can be carried by the cylinder. Figure 3.23 shows a highly wrinkled Mylar cylinder carrying a pure moment. Stein and Hedgepeth considered a membrane cylinder of uniform radius r and thickness t, made of a material of Young’s modulus E. They showed that for any given P and M, the curvature κ and the extent of the wrinkled region b can be determined from the equations     b b b 2 cos κ (3.59) P = 2Etr sin + π − r r r   2b b 1 3 M = Etr π − + sin κ (3.60) r 2 r

Figure 3.22 Deployable waveguide array antenna with three inflatable aluminum tubes (image used with permission of Costa Cassapakis, 2019).

3.7 Pressurized Membrane Cylinders

85

Figure 3.23 Wrinkling of pressurized Mylar cylinder, from Stein and Hedgepeth (1961) (image courtesy of NASA).

1.0

0.8

0.6 M Pr M

0.4

P

b 0.2

0

2

4

6 2 π r2 Et κ P

8

10

∞

Figure 3.24 Moment-rotation relationship of pressurized cylinder, showing onset of wrinkling (Stein and Hedgepeth, 1961).

where any end pressure load must be included in P . By solving these equations they obtained the moment-curvature relationship in Fig. 3.24 which shows that the cylinder will support twice the moment corresponding to the onset of wrinkling.

86

Tension Structures

3.7.1

Lateral Stiffness Topping (1964) showed that an accurate calculation of the lateral stiffness of a pressurized cylinder should include an additional pressure-related stiffness term. The starting point is that shear deformation can be significant in pressurized cylinders made from shear-compliant membranes, such as plain-weave fabric, in which case shear deformation may need to be included in their analysis, see also Section 6.2. Consider the pressurized cantilevered cylinder shown schematically in Fig. 3.25, which is subject to a tip force W , and assume that: • • • • •

any cross-section of the cylinder remains undeformed; the internal pressure is sufficiently high to prevent local buckling or wrinkling; the displacements of the cylinder are only moderately large; the circumferential strain can be neglected; and there is no twist about the longitudinal axis.

Since we are dealing with a shear-deformable cantilever beam, the natural way of calculating the tip deflection is to add to the standard bending deflection term an additional term resulting from the shear deformation. This approach gives vL ≈

W L3 WL + 3EI kGA

(3.61)

The second term in Eq. 3.61 is obtained by multiplying the uniform shear angle γ =

W kGA

(3.62)

by the length L of the beam. Note that kGA is the shear rigidity of the beam, where G is the shear modulus, A = 2π rt is the cross-sectional area, and k is a numerical factor that depends on the shape of the cross-section. The factor 1/k is known as the shear coefficient of the beam; it is the coefficient by which the average shear stress in the beam, i.e., W/A, has to be multiplied to obtain the shear stress on the neutral axis of bending. For thin-walled circular tubes 1/k = 2, see Gere and Timoshenko (1990). Hence, γ =

W Gπ rt

(3.63)

See Section 6.2 for more details on a beam theory that includes shear deformation.

L

W

r

Figure 3.25 Pressurized cantilevered cylinder.

3.7 Pressurized Membrane Cylinders

87

B W γ

W

B pπr2 B B

(a)

(b)

(c)

Figure 3.26 Two shear-carrying mechanisms in pressurized cylinder shown in (a), through (b) shear stresses and (c) changes of geometry.

However, a pressurized tube can carry shear in two different ways, as depicted in Fig. 3.26: first, through a distribution of shear stresses in the cross section as sketched in Fig. 3.26(b) and, second, through a geometrically non-linear effect known as pressure stiffness, analogous to the geometric loads introduced in Section 4.3.1. The pressure stiffness of the inflated cylinder accounts for the vertical out-of-balance force that accompanies the shear deformation depicted in Fig. 3.26(a). Next, consider the free body shown in Fig. 3.26(c), obtained by cutting through the section B-B. For horizontal equilibrium, the axial force P in the cylinder is P ≈ pπ r 2 Note that this value is independent of the (small) shear angle γ . Resolving vertically and noting that the pressure on the vertical end plate remains perpendicular to it, gives W = pπ r 2 γ

(3.64)

and rearranging γ =

W pπ r 2

(3.65)

Comparing Eq. 3.65 to Eq. 3.62, we conclude that pA is the additional shear stiffness of the pressurized cylinder. Hence, considering both shear stiffnesses of the cylinder, we have W = (pπ r 2 + Gπ rt)γ

(3.66)

In conclusion, the tip deflection of a pressurized cylinder is vL ≈

3.7.2

W L3 WL + 3EI pπ r 2 + Gπ rt

(3.67)

Buckling The result in the previous section also affects the overall buckling load for a pressurized cylindrical column, Fig. 3.27, as the pressure stiffness needs to be included.

88

Tension Structures

P

Figure 3.27 Pin-ended pressurized cylinder under compressive axial load.

The general equation for the elastic buckling of a pin-ended shear compliant column of length L, cross-sectional area A, and second-moment of area I is given in Timoshenko and Gere (1961): Pcr =

PE 1 + PE /(kGA)

(3.68)

where PE is the Euler buckling load of the column, i.e., without including shear deformability effects: PE = π 2 EI /L2

(3.69)

Replacing kGA with the shear stiffness of the pressurized cylinder derived in Section 3.7.1, we obtain (Topping, 1964): Pcr =

PE 1 + PE /(pπ r 2 + Gπ rt)

(3.70)

Based on this result, we can state the two criteria that should be met by a pressurized membrane cylindrical column. The first criterion is that the pressure must be large enough that the column is under longitudinal tension, i.e. P ≤ pπ r 2

(3.71)

The second criterion is that P should not be greater than the buckling load, hence from Eq. 3.70 P ≤

3.8

PE 1 + PE /(pπ r 2 + Gπ rt)

(3.72)

Computational Form-Finding Methods In this section two general methods will be presented for determining the shape of tension structures with a prescribed distribution of prestress. Given a set of fixed boundary nodes and a mesh of nodes connected by specified connecting members (cables or membranes) that are subject to a known prestress, both methods can be used to determine the equilibrium position of the nodes. These methods are normally used in an interactive fashion, e.g. by altering the position of the boundaries or the distribution of prestress until a satisfactory shape is obtained. The two methods are based on a simple finite-element formulation where the only available elements are the constant-stress, two-node cable element and the constantstress, three-node membrane element. The special feature of these elements is that they

3.8 Computational Form-Finding Methods

89

can freely change their size and shape while maintaining a prescribed stress level. In the first method, known as the force-density method, all of the nodal coordinates are determined at the same time by solving a set of linear equilibrium equations. In the second method, known as dynamic relaxation, the full nonlinear equilibrium equations are solved iteratively. The two methods make use of the same expressions relating stress to nodal forces in each type of element. First, these expressions will be derived in the next section, and then the two methods will be described.

3.8.1

Nodal Forces in Cable and Membrane Elements Consider a straight cable element ij , joining node i to node j and carrying an axial force T , as shown in Fig. 3.28(a). The equilibrium equations for node i can be written as in Section 4.3.1 in terms of the length of the element and its projections along the coordinate axes, as follows xi − xj T + · · · = pix L yi − yj T + · · · = piy L zi − zj T + · · · = piz L

(3.73)

where xi , yi , zi , etc. are the Cartesian coordinates of node i, etc. in the global coordinate system x,y,z and L is the length of the cable element, given by L=



(xi − xj )2 + (yi − yj )2 + (zi − zj )2

(3.74)

The dots in Eq. 3.73 correspond to the additional cable elements that are also connected to node i. In form-finding calculations the external forces may be zero or have prescribed values that account for external loads due to pressure, gravity, etc. Hence Eq. 3.73 can be used either to calculate xi , yi , zi , etc. if ζ = T /L is known, or to calculate the out-of-balance force components pix , piy , piz at node i, if xi , yi , zi , etc., T , and L are known. Next, consider the flat membrane element ij k, joining nodes i, j and k. Let Xi , Yi , etc. be the cartesian coordinates of node i, etc., in the local coordinate system X,Y . Note that this system is coplanar with the element. A uniform stress distribution

σX

σY

τXY

T

(3.75)

is applied to the element, as shown in Fig. 3.28(b). From a standard finite element formulation (Cook et al., 2002) the nodal force components, also in the local coordinate system, at the three nodes are

90

Tension Structures

j T

z

i

x

y pkY

(a)

pkX

Tki

Tkj

k

k s X, s Y, tXY

piY Y

sX, sY, t XY

pjY

Tij

pjX

piX j

i X

i

Tji

j Tjk

Tik (c)

(b)

Figure 3.28 Nodal forces in (a) cable element and (b) membrane element; (c) virtual cable forces equivalent to stress components in membrane.

⎡

⎤ ⎡ piX bi ⎢p ⎥ ⎢0 ⎢ iY ⎥ ⎢ ⎢ ⎥ ⎢ ⎢pj X ⎥ h ⎢bj ⎢ ⎥= ⎢ ⎢p ⎥ 2 ⎢ 0 ⎢ jY ⎥ ⎢ ⎢ ⎥ ⎢ ⎣pkX ⎦ ⎣bk

0 cj 0

⎤ ci ⎤ bi ⎥ ⎥⎡ ⎥ σX cj ⎥ ⎥ ⎥⎢ ⎣ σY ⎦ bj ⎥ ⎥ ⎥ τXY ck ⎦

0

ck

bk

pkY

0 ci

(3.76)

where h is the thickness of the element, and bi = Yj − Yk , ci = Xk − Xj ,

bj = Yk − Yi , cj = Xi − Xk ,

bk = Yi − Yj , ck = Xj − Xi .

Equation 3.76 cannot be readily used in form-finding calculations because, although linear in the local nodal coordinates, it becomes non-linear when the nodal coordinates are transformed to a global coordinate system. Hence, it needs to be written in a different form, as follows. Following Przemieniecki (1968) the local load components piX,piY , etc. are transformed into skew components parallel to the edges of the element, Tij , etc. as shown in Fig. 3.28(b). For example, considering node i, 

piX piY



 =

−ck /Lij

cj /Lik

bk /Lij

−bj /Lik



Tij

Tik

(3.77)

3.8 Computational Form-Finding Methods

91

Defining ζij = Tij /Lij and ζik = Tik /Lik , and inverting Eq. 3.77 

ζij

ζik



1 = cj bk − ck bj



bj

cj

bk

ck



piX



piY

(3.78)

Four more equations are obtained by repeating this procedure for nodes j and k. Noting that the area of the element can be calculated from A=−

ci bj − cj bi cj bk − ck bj ck bi − ci bk =− =− 2 2 2

(3.79)

and also noting that, since ζij = ζj i , it is found that only three of the six equations are independent. Finally, substituting Eq. 3.76 for piX,piY , etc. the following transformation is obtained ⎡ ⎤⎡ ⎤ ⎡ ⎤ ζij bi bj ci cj bi cj + bj ci σX h ⎢ ⎥⎢ ⎥ ⎢ζ ⎥ (3.80) ⎣bj bk cj ck bj ck + bk cj ⎦ ⎣ σY ⎦ ⎣ jk⎦ = − 4A ζki bk bi ck ci bk ci + bi ck τXY

3.8.2

Force Density Method The equilibrium equations in any form-finding problem are nonlinear, because the geometry is not known. However, Eqs 3.73 can be transformed into linear equations if, instead of specifying the value of the force in each element, we prescribe the force density (Schek, 1974) ζ = T /L

(3.81)

Consider a cable network. If the force densities of all elements are given, we can determine the coordinates of all unconstrained nodes by solving a system of linear equations of nodal equilibrium. Consider, for example, node 1 in Fig. 3.29. In the absence of any external forces applied to this node, the equilibrium equation in the x-direction is x1 − xB x1 − x2 x1 − x3 x1 − xA T1A + T1B + T12 + T13 = 0 L1A L1B L12 L13

(3.82)

Introducing the force densities ζ1A , ζ1B , ζ12 , and ζ13 – whose values are known –, grouping together terms that contain the unknown coordinates x1 , x2 , etc., and taking all known quantities to the right-hand side Eq. 3.82 becomes (ζ1A + ζ1B + ζ12 + ζ13 )x1 − ζ12 x2 − ζ13 x3 = ζ1A xA + ζ1B xB

(3.83)

If there are n nodes whose coordinates are to be determined, we write three equations of this type for each of these nodes and thus obtain a system of linear equations of the type

92

Tension Structures

2

3 1 A B

Figure 3.29 Node connectivity in a cable net.

Side views:

Top views:

Figure 3.30 Three steps from search for the geometry of a membrane near a high support point (Schek, 1974).

⎤ ⎡ ⎤ x1 b1 ⎢y ⎥ ⎢ b ⎥ ⎢ 1⎥ ⎢ 2 ⎥ A⎢ ⎥ = ⎢ ⎥ ⎣. . . ⎦ ⎣ . . . ⎦ ⎡

zn

(3.84)

b3n

whose solution provides the 3n unknown coordinates. The form of Eq 3.84 does not change if some membrane elements are present, as Eq. 3.80 can be used to calculate three force densities for each triangle and then the triangle is replaced by a set of three virtual cables whose force densities have these values.

3.8 Computational Form-Finding Methods

93

Once all nodal coordinates have been calculated, the lengths of all cable elements are determined using Eq. 3.74; then, the actual cable tensions/membrane stress components are determined from Tij = ζij Lij

(3.85)

for a cable or, for a membrane element, by inverting Eq. 3.80. At this point, if the shape that has been calculated and the corresponding stress distribution are acceptable, one computes the unstressed lengths of the cable elements or the cutting pattern for the membrane. If the shape and/or stress distribution are not acceptable, a new form-finding iteration is performed, after changing the force/stress densities and adding or removing elements, or changing the number and position of the boundary nodes. Figure 3.30 shows an example of such an iteration, reproduced from Schek’s paper. Note that, in the case of membrane elements, the solution of the equilibrium equations is approximate, not exact. This is because the area A of the triangular elements, as well as the coefficients bi , ci , etc. in Eq. 3.80 cannot be calculated exactly before knowing the final shape. Therefore, an iterative refinement of the solution is required.

3.8.3

Dynamic Relaxation This method was first proposed by Day and Bunce (1969) and has been widely adopted (Barnes, 1984). It is also used as a general vector method for the solution of nonlinear systems of equations; it has the advantage of not requiring the set-up and solution of large matrix equations. The idea is to formulate the problem in pseudo-dynamic terms. The procedure will be described for cable nets, as the generalization from cable to membrane equations by means of virtual cable tensions has already been described in Section 3.8.1. Each node of the cable net is assigned a fictitious mass mi and each cable segment is assigned a fictitious stiffness kij . These may be equal to the physical mass associated with the node and the actual stiffness of the cable segment, or may be different in order to speed up convergence. Let Tij0 be the desired prestress of cable ij , whose length in the initial configuration is 0 Lij . Because the equilibrium shape is unknown, the computation begins with the cable net in an arbitrary configuration described by the nodal coordinates xi0 , etc. Equilibrium is not satisfied in this configuration and hence the net is allowed to move towards the equilibrium configuration. Consider the motion from time t to t + t. We assume the coordinates of node xit , etc. at time t to be known, and wish to compute the coordinates xit+t . The current force in cable ij is  Tijt = Tij0 + kij

(xit − xjt )2 + . . . − L0ij L0ij

(3.86)

94

Tension Structures

By generalizing Eq. 3.73 we can define the out-of-balance force components on node i t = rix

 xit − xjt j

Ltij

Tijt − pix

(3.87)

where the summation is extended to all cable elements connected to node i and pix is the external force component on this node, in the x-direction. This out-of-balance force results in an acceleration of node i, whose average x-component is x¨i t+t/2 =

t rix mi

(3.88)

Assuming that the acceleration is uniform through the time increment, the new configuration is defined by xit+t = xit + t x˙it +

t 2 t+t/2 x¨ 2 i

(3.89)

and the velocity at time t + t is t+t/2

x˙it+t = x˙it + t x¨i

(3.90)

Analogous equations can be written also for the y- and z-coordinates. To avoid the build-up of kinetic energy in the system, Day and Bunce (1969) included a fictitious damping term in Eq. 3.89, but it was later found that faster convergence is achieved by setting to zero the kinetic energy of a node (i.e., re-setting all of its velocity components equal to zero) every time that it reaches a peak value (Barnes, 1999). This method avoids the need to search for suitable values of the fictitious damping coefficient. This iterative method simulates the rapidly decaying motion of the cable net towards its equilibrium configuration. Note that the period of the decaying oscillation, and hence the number of computational steps to achieve convergence, depends on the chosen fictitious mass and stiffness; it is not representative of the actual vibration properties of the structure. Barnes (1999) shows that to avoid divergence during a time-stepping solution the following condition has to be satisfied at all nodes  2mi (3.91) t <  j kij In fact, this equation provides an estimate of the fictitious mass that should be introduced at node i for an already chosen t. Note that at the end of the iteration the cable tensions are not exactly equal to the initially required state of prestress, due to the presence of the elastic stiffness term in Eq. 3.86. To avoid excessive difference, it is usually desirable to choose a low value for kij .

3.8 Computational Form-Finding Methods

3.8.4

95

Further Comments Minimal-area surfaces can be readily computed using the dynamic relaxation method. In theory all that should be required is that one simply sets all direct stress components equal to a common value and the shear stress equal to zero everywhere. The problem (Lewis, 1993) is often that the nodes tend to move by large amounts and hence some cables/membrane elements may shrink to a single point. To avoid this, it is common practice to introduce a small elastic stiffness in each element, see Eq. 3.86. The force density method can also be used, but an appropriate iterative strategy should be implemented in order for all stress components to converge to a common value. However, it turns out that minimal-area surfaces are not ideal for membrane roofs or canopies because they tend to produce rather flat regions, which attract rain water and dirt, and narrow, pointed regions near the supports. Designers tend to concentrate on visually pleasing geometric shapes during the initial stages of the form-finding process, and then check if the required stress distribution is acceptable for the membrane.

4

Tension-Stabilized Structures

4.1

Introduction There are several ways of exploiting the inherent stability of slender structural elements in tension. One can design structures with only a few “chunky” compression elements and many slender elements which are pre-tensioned. Alternatively, one can choose the shape of the structure so that the applied loads will induce tension throughout the structure. An extension of this approach is to use thin membranes that are only lightly stressed, to create a functional enclosure that transfers the main loads to a few stiff, uniaxial tension elements. This chapter demonstrates the skilful use of tension elements in the design of high performance, super-efficient structures, such as parachutes and other kinds of decelerators. A further example of tension-stabilized structures is the tension truss antenna presented in Section 9.4. This structural concept features a prestressed triangulated cable net that supports the reflective metallic mesh forming the antenna surface.

4.2

Prestressed Stayed Columns A prestressed stayed column is a tension-stabilized structure consisting of a central column connected to pre-tensioned stays, i.e., wires or cables, by one or more intermediate frames, as shown in Fig. 4.1. The purpose of the stays and frames is to increase the stability of the column. The buckling load of a stayed column is much higher than that of a simple column of equal mass. The idea behind this structural concept is to reduce the wave-length of the buckling mode of an Euler column, discussed in Section B.8, in the way shown in Fig. 4.2. The introduction of an intermediate support, Fig. 4.2(b), eliminates the mode v = sin(π x/L), and thus increases the buckling load of the column from π 2 EI /L2 to 4π 2 EI /L2 . The prestressed wires and frames shown in Fig. 4.1 are a way of providing such intermediate supports. If, for example, we reduce the number of frames to only one, realize a moment-less connection between the frame and the column, and make the tension ties axially very stiff, then we would have realized precisely the support condition of Fig. 4.2(b). Notice, however, that to put the stays under a state of

96

4.2 Prestressed Stayed Columns

97

Column Frame Prestressed stay

Figure 4.1 Atrium roof supported by four prestressed stayed columns. (© Project author: Philippe Samyn and Partners, architects & engineers – Photographer: Marc Detiffe).

P

P

L/2

L/2 x

v

(a)

(b)

Figure 4.2 Buckling modes of an Euler column with and without an intermediate support.

98

Tension-Stabilized Structures

pre-tension would require a corresponding pre-compression in the central strut, which would partially offset the gain made with the stayed column architecture. A simple extension of this approach might suggest that, by increasing the number of frames, it would be possible to raise ad infinitum the buckling load of the stayed column. This is not the case because, in addition to the mass penalty introduced by each additional frame, it is still possible for the stayed column to buckle into an overall mode. To determine the optimal number of frames, including their dimensions and stiffness properties, it is necessary to carry out a complete optimization. Such a study was done by Mauch and Felton (1967) who looked for designs that equalize the local buckling load of the stayed column, i.e., the load required for the short wave-length buckle to form Fig. 4.2(b), to the overall column buckling load, Fig. 4.2(a). Mauch and Felton (1967) assumed that the pre-tension in the ties is of such magnitude that the ties become stress free just on the point of buckling. They also assumed that all of the connections between the frames and the column are pin-jointed, frictionless hinges. Having worked out several complete designs, they compared the mass of an optimally designed simple column, consisting of a thin-walled Al-alloy tube, with the mass of stayed columns with either one or three frames, also made of Al-alloy, see Fig 4.3. For a total length of L = 96 m and a maximum load requirement of 4,450 N, the mass of the simple column was 16.2 kg. The stayed columns with either one or three frames came out much lighter, at 11.5 kg and 7.5 kg respectively. A more general set of results was also obtained by the same authors and the results are plotted in Fig. 4.4, in terms of the variation of the weight ratio between cablestayed and simple columns, with the square root of Wagner’s structural index Pcr /L2 (see Section 24.12 of Shanley, 1957). This structural index is the ratio between the

L/4

L/2

L

Figure 4.3 Side views of stayed column designs considered by Mauch and Felton (1967); each

frame forms an equilateral triangle.

4.2 Prestressed Stayed Columns

Weight ratio

1.0

Euler column

0.8

Column with one frame

0.6 0.4

99

Column with three frames

0.2 0

0

0.05

0.1 Pcr/L

2

0.15

0.2

2 0.5

( N/mm )

Figure 4.4 Comparison of weight ratio between cable-stayed columns and simple Euler column.

critical buckling load, Pcr , and the square of the length, L, of the stayed column; it allows a comparison of structures of different size, in which all dimensions are scaled in proportion to L. Note that this index is dimensional and hence attention needs to be paid to the units used in this plot.

4.2.1

Column with Cross-Arms A study by Smith, McCaffrey and Ellis (1975) provided further insight into the behavior of cable-stayed columns. Instead of using a stiffening frame, as described in the previous section, they used a cross-arm consisting of two perpendicular beams rigidly connected to the column in the middle, Fig. 4.5. Four prestressed cable stays with pre-tension ti (the subscript i denotes initial) completed the structure; note that the cables form an angle α with the column. Denoting by tf (the subscript f denotes final) the cable tensions when the column is on the point of buckling under a critical load Pcr , compatibility of deformation between the stays, the cross-arm, and the column yields the relationship: t f = ti −

(Pcr − 4ti cos α) cos α kc ks

+ 2 kkcac sin2 α

(4.1)

where kc,ks ,kca are the axial stiffness (AE/length) of the column, the stay, and the crossarm respectively. This equation can be used to compute the required initial prestress of the stays once Pcr has been calculated. A quick upper-bound estimate of the buckling load of the cable-stayed column can be made by assuming that both the stays and the cross-arm are infinitely rigid. Thus, each half of the column will buckle as a propped cantilever of length L/2. For this

100

Tension-Stabilized Structures

Column Pretensioned stays

L/2

α

s

α

Cross-arms

L/2

ca

(b)

(a)

Figure 4.5 Cable-stayed column analyzed by Smith et al. (1975).

structure, the effective length is 0.699 of its full length (Timoshenko and Gere, 1961) and hence Pu =

π 2 Ec Ic (0.699L/2)2

(4.2)

where Ec and Ic are respectively the Young’s modulus and second moment of area of the column. Since the Euler buckling load of the column by itself, i.e., without the stays, would be π 2 Ec Ic L2 the maximum possible increase in the buckling load of the stayed column is PE =

Pu 4 = = 8.18 PE 0.6992

(4.3)

(4.4)

A more detailed analysis assumes all components to be elastically deformable, although the cross-arm can be assumed to be axially rigid as it is much stiffer than the stays. The two buckling modes shown in Fig. 4.6(b and c) are considered. For the first buckling mode, Fig. 4.6(b), the elastic stays and the (axially rigid) cross-arm can be replaced by a single translational spring of stiffness 2kh = 4ks sin2 α

(4.5)

Considering only half of the column, Fig. 4.7(a), we obtain a standard buckling problem whose solution – see Timoshenko and Gere (1961) for details – is obtained by solving the equation kh (L/2)3 (βL/2)3 = βL/2 − tan(βL/2) Ec Ic

(4.6)

4.2 Prestressed Stayed Columns

(a)

(b)

101

(c)

Figure 4.6 Buckling modes; (a) mode for upper-bound estimate; (b) mode 1; (c) mode 2.

P

P

P

P

2kr

kh

kr kv

2kh

(a)

P

kv

(b)

Figure 4.7 Simple buckling models.

√ where β = P /Ec Ic . Numerically obtained solutions of this equation yield the values of Pcr for which the column buckles in the mode of Fig. 4.6(b). These values have been plotted in Fig. 4.8 for stayed columns where both the column and the cross-arm are made from a steel tube with internal diameter of 44.5 mm and external diameter of 57.2 mm, and the stays are steel wire ropes with diameter 11.1 mm.

102

Tension-Stabilized Structures

1.0 0.9 Mode 2

0.8 0.7

Mode 1

0.6 Pcr Pu 0.5 L/2

0.4

lca

0.3 0.2 0.1 1

2

3

4

5

6 7 L / 2 lca

8

9

10

Figure 4.8 Buckling loads of column with single cross-arms and cable stays with cross-sectional diameter of 11.1 mm (7/16 ).

For the second buckling mode, Fig. 4.6(c), we can introduce, again, an equivalent beam-and-spring system, but this time the calculation is best done in two steps. First, we replace each stay with an equivalent spring of stiffness kv = 2ks cos2 α

(4.7)

then we obtain a relationship between the rotation θ in the middle of the column and the vertical displacement component, δca , of the tip of the cross-arm δca =

3kca ca θ 2ks cos2 α + 3kca

(4.8)

where kca = Eca Ica /3ca . Then, we replace the two translational springs and the flexible cross-arm with a single rotational spring of stiffness 2kr =

12ks kca 2ca cos2 α 2ks cos2 α + 3kca

(4.9)

Again, we can consider only half of the column, Fig. 4.7(b), and the solution of this standard buckling problem is obtained by solving the equation 4kr (βL/2)2 tan(βL/2) = βL/2 − tan(βL/2) kc (L/2)2

(4.10)

From the numerical solutions of this equation one can compute the corresponding values of Pcr for mode 2, which are also plotted in Fig. 4.8.

4.3 Tensegrity Structures

(a)

(b)

103

(c)

Figure 4.9 Stayed columns with (a) single, (b) double, and (c) triple cross-arms.

This plot shows that the overall buckling (mode 1) corresponds to a lower critical load for more slender columns, but mode 2 becomes dominant as the total cross-arm length, 2ca , is increased beyond L/6 = 0.167L. This transition corresponds to a critical buckling load of over 80% of the upper bound Pu . Thus, with a single, welded cross-arm, a five-fold increase in the Euler buckling load of the column can be expected. Higher increases, in excess of 20 times, can be achieved using multiple cross-arms. See the configurations shown in Fig. 4.9.

4.3

Tensegrity Structures The word tensegrity (= tensile-integrity) was coined by Buckminster Fuller (Fuller, 1962). Fuller did not provide a clear-cut definition but instead gave several descriptions both in the original patent and later writings; among these “a structure having discontinuous compression and continuous tension.” Over the years, a range of definitions have emerged, among these we will list the following three. Pugh (1976) expanded Fuller’s description into: A tensegrity system is established when a set of discontinuous compression components interacts with a set of continuous tensile components to define a stable volume in space. Motro (2006) combined Fuller’s description with those provided by two other inventors of the same structural concept (although different names were given), Kenneth Snelson and David Georges Emmerich, and proposed the following definition: Tensegrity systems are spatial reticulated systems in a state of self-stress. All their elements have a straight middle fibre and are of equivalent size. Tensioned elements have no rigidity in compression and constitute a continuous set. Compressed elements constitute a discontinuous set. Each node receives one and only one compressed element. Skelton and de Oliveira (2009) gave a broader definition in terms of rigid bodies and strings, also allowing for the possibility of compression continuity (in the case of class 2

104

Tension-Stabilized Structures

(a)

(b)

(c) Figure 4.10 Tensegrity structures based on twisted prisms.

or higher, see below): A configuration of rigid bodies is a tensegrity configuration if there exists a string connectivity able to stabilize the configuration. A tensegrity system is composed of any given set of strings connected to a tensegrity configuration of rigid bodies. A tensegrity configuration that has no contacts between its rigid bodies is a class 1 tensegrity system, and a configuration with as many as k rigid bodies in contact is a class k tensegrity system. We will focus on structures made of slender compression elements and tension elements, in which contact between the compression elements is not allowed (class 1 tensegrities). For these structures, all of the above definitions require that, as well as imparting tension to all cables, the state of prestress serves the purpose of stabilizing the structure, thus providing first-order stiffness to its infinitesimal mechanisms. Figure 4.10(a) shows a simple tensegrity structure consisting of four compression elements connected to an outer “net” formed by 12 cable elements. The net consists of two squares lying in parallel planes, but rotated 45◦ with respect to one another, connected by four cables. In this structure, all cables have equal length. We can model the pretensioned cable elements as pin-jointed bars, just as we did in Section 3.2. Hence, we have the following numbers of joints, bars, and kinematic constraints: j = 8, b = 16, and k = 0. Substituting these values into Maxwell’s equation, Eq. 1.2, gives 3 × 8 − 16 − 0 = 8 = m − s

(4.11)

Hence, m − s = 8. Actually, s = 1, as the layout of this structure is identical to Fig. 1.17, apart from the bottom four members which here have been replaced by

4.3 Tensegrity Structures

105

connections to foundations. Because that structure admits a state of self-stress, so does this one. Therefore, Eq. 4.11 gives m = 9. Three of the mechanisms are internal and six are rigid-body translations and rotations. The three internal mechanisms allow only an infinitesimal distortion of the assembly, because the lengths of the four compression elements are a maximum in this particular configuration. Therefore, any finite changes of shape would require that the lengths of these four elements decrease. Two further examples of tensegrity structures based on twisted pentagonal and hexagonal prisms are shown in Fig. 4.10(b and c). They behave in a similar way to the example discussed above: they have s = 1, and 5 and 7 internal mechanisms, respectively. The existence of structures that are “stiff” without satisfying the rule m = 0 in Eq. 1.2 was anticipated by Maxwell (1864). Maxwell stated (the italic words in brackets have been added for clarification) “In those cases where stiffness can be produced with a smaller number of lines (bars) certain conditions must be fulfilled, rendering the case one of a maximum or minimum value (length) for one of its lines. The stiffness of the frame is of an inferior order, as a small disturbing force may produce a displacement infinite in comparison with itself.” To explain Maxwell’s statement and the reason why this particular type of structure exhibits nonzero linear stiffness when prestressed, Calladine (1978) considered a structure of the type shown in Fig. 4.11. In two dimensions, d = 2, it has j = 5, b = 4, and k = 4; hence Maxwell’s equation gives 2×5−4−4=m−s

P

B L

C

L

2L

D L E

A (a) P

B

A L

L

C

E

D L

L

(b) Figure 4.11 Kinematically indeterminate structures whose inextensional mechanisms are (a) finite and (b) infinitesimal.

106

Tension-Stabilized Structures

In the configuration shown in Fig. 4.11(a) this is a structure with s = 0 and m = 2. If an infinitesimally small load P is applied to joint C, as shown in the figure, a finite deflection is produced: there is only one equilibrium configuration, with the structure inverted below the line AE and joint C at the bottom apex of the triangle CDE. Such a displacement is “infinite in comparison with the force,” and hence of the kind described by Maxwell. To find a special configuration that admits only infinitesimal mechanisms, consider shortening the length of member AB while keeping fixed the lengths of all other members. When AB = L, AB has reached its minimum length; the corresponding configuration is shown in Fig. 4.11(b). In this configuration s = 1, as a state of selfstress is clearly possible, where all members carry, e.g., a uniform tension of one unit; hence m = 3. Obviously, only an infinitesimally small motion of the inextensional mechanisms of this structure is possible, as any finite displacement of joints B, C, or D requires the length of at least one member to increase. Now consider the behavior of this structure when a vertical load P is applied at joint C. The following analysis is analogous to that in Section 3.2.1. Denote by δ the vertical deflection at C. Since ABC and CDE will remain collinear in all deflected configurations, the intermediate joints B and D need not be considered in the analysis. The extensions of AC and CE are  δ2 − 2L (4.12) e = 2L 1 + 4L2 Expanding the square root as a Taylor’s series and neglecting higher order terms gives   δ2 δ2 − 2L = e ≈ 2L 1 + 4L 8L2 Then, calculating the corresponding strains in ABC and CDE, their axial forces are found to be T =

AEδ 2 8L2

(4.13)

Resolving vertically the forces at C in the deformed configuration, we find P =

AEδ 3 8L3

(4.14)

which is of the same type as Eq. 3.9, obtained for a straight cable loaded by two forces. Next, consider the behavior of this structure after applying an initial pre-tension T0 , see also Section 3.2.5. Again, a vertical load P is applied at joint C, and δ is the vertical deflection of C. Repeating the above analysis, but now including the initial force T0 in Eq. 4.13, one finds P =

AEδ 3 T0 δ + L 8L3

(4.15)

Plots of Eqs. 4.14 and 4.15 are shown in Fig. 4.12 for L = 1000 mm, T0 = 0, 100,200 N and AE = 50 kN.

4.3 Tensegrity Structures

107

30 T 0 = 200 N

20

P (N)

10

100 N

0

T0 = 0

–10 –20 –30 –100 –80 –60 –40 –20

0

20

40

60

80 100

d (mm) Figure 4.12 Load-displacement relationships for structure of Fig. 4.11(b).

Figure 4.13 Tensegrity structure based on a truncated tetrahedron.

The plot that corresponds to T0 = 0 illustrates what Maxwell meant by “stiffness . . . of an inferior order”: as P is a cubic function of δ, the initial stiffness is dP /dδ = 0. As the deflection increases, so does the stiffness. When the structure is prestressed, its initial stiffness becomes dP /dδ = T0 /L: a tensegrity structure is stabilized by a suitable state of prestress T0 > 0. Note that for this particular example the initial stiffness for T0 = 200 N is greater than the stiffness it would have after a displacement δ = ±100 mm if no prestress had been applied.

108

Tension-Stabilized Structures

Figure 4.14 Kenneth Snelson’s Needle Tower in the garden of the Hirshhorn Museum, Washington, DC. This 28 m high structure, built in 1968, illustrates the discontinuous compression member property of class 1 tensegrity structures (Ian Davies/Shutterstock.com).

Several tensegrity structures with visually striking shapes have been realized. Figure 4.13 shows a structure whose tension envelope is generated by intersecting a tetrahedron with four planes that are perpendicular to the trisectors of its vertices. After arranging 18 cable elements – say, of unit length – along the edges of this polyhedron, six struts of equal length are attached to the 12 joints, and finally the length of these struts is increased until a maximum of 1.468 is reached. An analysis of the nodal coordinates corresponding to this configuration shows that the faces of the truncated tetrahedron that are bounded by hexagons are nonplanar. If the resulting structure is analyzed as a pin-jointed framework it is found that it has s = 1 and m = 13: there are 7 internal, infinitesimal mechanisms, in addition to six rigid-body mechanisms (Pellegrino, 1986). Figure 4.14 shows the tensegrity sculpture Needle Tower of which two were realized by the sculptor Kenneth Snelson.

4.3.1

Geometric Stiffness Prestress plays two key roles in a tensegrity structure. First, it provides a state of pretension to all of its cable, thus enabling the cable elements to carry both tensile and compressive force-changes due to the external loads (of course, any compression has to be smaller than the initial pre-tension). Second, it stabilizes any internal inextensional mechanisms that may be present, by providing additional stiffness to the structure. This new kind of stiffness is associated with small changes of configuration, and hence is called geometric stiffness. Geometric stiffness is the source of the linear term in the loaddeflection relationships of the simple kinematically indeterminate structures considered in Section 3.2.5, see Eq. 3.20, and Section 4.3, see Eq. 4.15. Here, it will be explained in a more general way.

4.3 Tensegrity Structures

109

Consider a general tensegrity structure. Impose a state of self-stress t0 onto it: each joint will be in equilibrium under the action of zero external forces and non-zero internal forces. A general, small inextensional displacement of the structure can be expressed in the form Dα where the matrix D is the mechanism matrix and each of its im columns, dh , represents an independent, internal inextensional mechanism of the structure, obtained in the way described in Section 1.5.3. By choosing different (small) values for the components of the vector α, any (small) inextensional deformation of the structure can be obtained. In the distorted configuration, the prestressed assembly is no longer in equilibrium with zero external loads. Assuming that the prestressing forces remain constant because the deformation is inextensional, then the nodal forces required to restore equilibrium are a linear combination of the geometric loads that are associated with each independent mechanism. The geometric loads are obtained as follows. Let joint i be an unconstrained joint of the structure, connected by cable/bar ij to joint j . The equilibrium equation in the x-direction for joint i, with the structure in its initial prestressed configuration, is:  xi − xj t 0 = pix (4.16) Lij ij j

where xi is the x-coordinate of joint i, Lij is the length of member ij , tij is the prestressing force in cable/bar ij , and pix is the x-component of the initial load on joint i. The summation in Eq. 4.16 is extended to all members connected to joint i. Next, consider the equilibrium equation in the x-direction of joint i, after imposing a small amplitude, αh , of mechanism h. The joint displacements are dh αh . Assuming that all member forces are unchanged from t0 , the new equilibrium equation is obtained simply by replacing the initial joint coordinates in Eq. 4.16 with their updated values; hence xi becomes (xi + uhi αh ), etc. All member lengths remain unchanged because the imposed displacement is inextensional. h is required on the right-hand side to satisfy equilibrium. An additional force δpix Thus, the updated version of Eq. 4.16 is:  (xi + uhi αh ) − (xj + uhj αh ) Lij

j

h tij0 = pix + δpix

Subtracting Eq. 4.16 from Eq. 4.17 we obtain: ⎞ ⎛  uhi − uhj h ⎝ tij ⎠ αh = δpix Lij

(4.17)

(4.18)

j

The summation in brackets gives the x-component of the geometric load at joint i associated with a unit amplitude of mechanism h. Similar expressions are valid also in the y- and z-directions, with uhi,uhj replaced with vih,vjh , and whi,whj , respectively. It is noteworthy that the term in brackets in Eq. 4.18 can be obtained formally from

110

Tension-Stabilized Structures

Eq. 4.16, simply by replacing each nodal coordinate with the corresponding component of inextensional displacement. Defining a matrix of geometric loads, G, whose im columns correspond one by one to the columns of the mechanism matrix, the set of geometric loads that are associated with the inextensional displacement Dα are Gα To find out if the state of self-stress t0 provides positive stiffness to the general mechanism Dα, we examine the sign of the dot-product between Dα and the corresponding geometric load Gα (Calladine and Pellegrino, 1991). For positive stiffness, we require that α T GT Dα > 0

(4.19)

If all mechanisms are to be stiffened, then Eq. 4.19 must be satisfied for any nonvanishing α. This is equivalent to saying that the matrix GT D should be positive definite, which can be tested by checking that its eigenvalues are all positive. The analysis presented in this section checks that a prestressed tensegrity structure is in a state of stable equilibrium in its initial configuration. Only inextensional deformation modes are considered in the analysis since the elastic stiffness of its members will already provide positive stiffness against all extensional modes.

4.3.2

Buckling It is possible for a prestressed tensegrity structure to buckle out of its equilibrium shape, due to the effect of the loads applied to it. Consider, for example, the simple truss structure shown in Fig. 4.15. It consists of √ four bars, 1–3, 1–4, 2–5, and 2–6, of length 2 and with equal axial stiffness, and one bar, 1–2, of length 3 and with much higher axial stiffness. All elements of this structure lie in the x–y plane, but the structure is to be analyzed in three dimensions. For j = 2 and b = 5, Eq. 1.2 yields: m−s =1

(4.20)

Clearly, s = 1 here: the four short bars √ can be in a state of tension, of magnitude t, equilibrated by a compression of − 2t in the long bar. So, from Eq. 4.20, m = 2. Two independent inextensional mechanisms consist of the out-of-plane displacement of joints 1 and 2, say, by a unit amount. Taking first the x, y, and z-deflection components of joint 1, in this order, followed by those of joint 2, the matrix D is:  T 0 0 1 0 0 0 (4.21) D= 0 0 0 0 0 1 First, let us verify that all mechanisms are stiffened by imposing the above state of self-stress. To begin with, we compute the matrix of geometric loads G. In general, the geometric load at joint i associated with the inextensional mechanism h is the sum of

4.3 Tensegrity Structures

F

111

1

1 1

4 1 3 6 1 5

1

y x

2

z Figure 4.15 Two-dimensional prestressed structure in perspective view.

the out-of-balance forces in the bars connected to joint i. These forces arise because of the relative displacements between joint i and the joints connected to it. Taking joint 1, for example, the z-component of the geometric load associated with the first of the two mechanisms in D, i.e., the (3,1) coefficient of the matrix G, is given by: w1 − w2 w1 − w3 w1 − w4 t1-2 + t1-3 + t1-4 L1 - 2 L1-3 L1-4 √ 1−0 2 2 1−0 √ 1−0 (− 2t) + √ t + √ t = t = 3 3 2 2

g1z =

The other coefficients of the matrix G are obtained by similar calculations, giving T √  2 0 0 2 0 0 1 G= t (4.22) 3 0 0 1 0 0 2 Thus, the matrix GT D is:

 2 2 G D= t 3 1 √

T

1 2

(4.23)

with eigenvalues 0.4714 t and 1.4142 t. Here, both eigenvalues are positive if t > 0, and hence GT D is positive definite.

112

Tension-Stabilized Structures

If the structure of Fig. 4.15 is prestressed, the four short elements can be replaced with cable segments which are in a state of pretension, and the resulting tensegrity structure has first-order stiffness against any load condition. Next, consider the response of the prestressed structure when a force F in the (positive) y-direction is applied at joint 1. For F = 0, the axial forces are (prestress only) √



(4.24) t1-3 t1-4 t2-5 t2-6 t1-2 = t t t t − 2t When F is applied, the bar forces are, assuming that the axial stiffness of 1-2 is much larger than the stiffness of the inclined members,   √ F F F F F (4.25) t+ √ t− √ t− √ − 2t + t+ √ 2 2 2 2 2 2 2 2 2 To search for bifurcation points as F varies we test the matrix GT D for positive definiteness. For the axial forces given by Eq. 4.25 we obtain  √ √ 2 2t + 2F 2t − F /2 1 √ √ GT D = (4.26) 9 2t − F /2 2 2t − F We already know that this matrix is positive definite when F = 0. Its determinant becomes zero, and hence the matrix is no longer positive-definite, when √ 2 2 F = (1 ± 2)t (4.27) 3 If the force F applied at joint 1 is upwards (F > 0), the assembly buckles out of plane when F = 2.828 t, i.e., just at the point when the rods 2–5 and 2–6 lose their pretension. The buckling mode, obtained from the eigenvector of GT D that corresponds to the vanishing eigenvalue, is

0 0 0 0 0 1 However, if F is downward, the buckling load is much smaller, F = −0.9428 t, and the corresponding mode is

0 0 0.8944 0 0 −0.4472 It is important to note that in this latter case buckling takes place before bars 1–3 and 1–4 lose their pre-tension.

4.4

Tensegrity Domes Figure 4.16 shows the Tropicana Field Dome in St. Petersburg, Florida (diameter 210 m) during construction in the late 1980s. This was a lightweight membrane roof supported by a prestressed cable-and-strut structure that was invented by David Geiger (Geiger, Stefaniuk, and Chen, 1986). Geiger’s structure was a practical realization of an earlier tensegrity dome concept invented by Buckminster Fuller (1964).

4.4 Tensegrity Domes

113

Figure 4.16 Tropicana Field Dome during construction (image used with permission of Geiger Engineers, 2019).

Tension ring 10 2

Ridge cable

1

3 11 z y

9

4 14 7

8

5 17

6

12

15 x

Diagonal cable

Strut

13 Cable hoop

16

Figure 4.17 Small-scale version of structure of Tropicana Field Dome (Pellegrino, 1992).

This tensegrity dome consisted of 24 radial cable-and-strut trusses prestressed by four cable hoops, two inner tension rings, and a perimeter compression ring (this terminology is defined in Fig. 4.17, for a small scale dome). Each of the radial trusses consists of: a ridge cable that connects the top tension ring to the perimeter, five vertical struts, and five diagonal cables. The struts and diagonal cables join the ridge cable to the cable hoops and also the top tension ring to the bottom one. Note that the cable hoops are connected only to the bottom ends of the struts. The concept of this tensegrity dome is best explained with reference to the small-scale version shown in Fig. 4.17. This structure consists of only twenty-four cable segments and eight struts.

114

Tension-Stabilized Structures

2√2 √2

2√2 2

1 √2

√3 9

–1

√2 1

2√3 √2

√3 5

4√2 -2

2

17

2

1

2√3 √2

2√2 13 (a)

1

1

√2 2

2√2

2√2 (b)

Figure 4.18 Analysis of free bodies to determine state of self-stress: (a) radial truss (b) top view of tension rings and cable hoop (Pellegrino, 1992).

It will now be shown that this structure has a single state of self-stress and thirteen independent inextensional mechanisms, and that a state of prestress can stiffen all of the mechanisms of the dome. The stability of the prestressed structure under load will also be investigated. First, we consider a pin-jointed truss model of the structure and analyze its static and kinematic properties. Substituting the number of unconstrained joints, j = 16, and the number of pin-jointed bars, b = 36, into Maxwell’s equation, Eq. 1.2, 3j − b = m − s

(4.28)

m − s = 12

(4.29)

we obtain

It can be shown that s = 1. Consider the radial truss shown in Fig. 4.18(a), taken in two forces of magnitude √ isolation from the rest of the dome. It is in equilibrium if√ 2 are applied at joints 1 and 5, and a force of magnitude 2 2 is applied at joint √ 13. 3 in All forces act radially inwards. This loading induces tensile forces of magnitude √ the ridge element 1–9 and the diagonal element 5–9, and of magnitude 2 3 in 9–17 and 13–17. It also induces compressive forces in the vertical elements 1–5 and 9–13, respectively of magnitude 1 and 2. A similar set of three radial forces can be applied to the other three cable trusses in the dome. Now, turning to the two tension rings and the hoop, see Fig. 4.18(b), we note that they are in equilibrium if radial √ forces of equal magnitude are applied at each corner. Outward forces of magnitude 2 would induce tensile forces of magnitude 1 in the hoops, etc. However, these external forces can be transmitted by the joints between the beams, the rings, and the hoop, and therefore the set of axial forces obtained above is in equilibrium without any external loads.

4.4 Tensegrity Domes

115

(b)

(a)

(c) 1

3

9 7

5

7 13

(d1)

9

11

5

15

1

3

11

13

15 (d2)

Figure 4.19 Four types of mechanisms of simple dome structure (Pellegrino, 1992).

Having found one state of self-stress for the structure, it is natural to ask whether any more independent states of self-stress can be generated by a similar process. For equilibrium of the rings and the hoop, all states of self-stress must be four-fold rotationally symmetric. In addition, the forces applied to each radial truss can only be radial and in the ratio 1:1:2, as shown in Fig. 4.18(a), or in-plane equilibrium would be violated. With these constraints, there is only one independent solution, given zero external forces. In conclusion, s = 1. For s = 1, Eq. 4.29 gives the number of independent mechanisms, m = 13, which can be divided into the following four groups. •

• •

Out-of-plane displacement of the mid-ridge joints. There are four independent mechanisms of this type, which involve displacements of joints 9, 10, 11, and 12. For instance, in one of these four mechanisms – shown in Fig. 4.19(a) – joint 9 moves by equal amounts in the directions x and −y. Rotation of the two tension rings and the hoop about the z-axis, see Fig. 4.19(b). These three mechanisms are independent. In-plane distortion of the square “four-bar” links formed by two corresponding elements of the top and bottom inner rings, and the two struts which connect them. The four joints at the corners of the square move in a diagonal direction, alternately in and out as shown in Fig. 4.19(c). Although the dome contains four

116

Tension-Stabilized Structures

•

different square “four-bar” links, only (any) three mechanisms of this type are independent. The fourth mechanism can be obtained as a linear combination of the other three mechanisms in this set, and of the two ring rotations in the previous set. Global mechanisms, which include a rigid-body rotation of the prism formed by the inner rings and the four struts joining them, about any horizontal axis through the center of the prism. There are two independent mechanisms of this type of which one is shown in Fig. 4.19(d1). Another mechanism involves an in-plane distortion of the bottom inner ring into a rhombus and a similar distortion of the hoop which also twists out-of-plane, while the top inner ring does not move; this mechanism is shown in Fig. 4.19(d2).

Pellegrino (1992) computed the thirteen vectors of geometric loads for the prestress in Fig. 4.18, and analyzed the resulting 13 × 13 matrix GT D: its eigenvalues are all positive, and hence the matrix is positive-definite. The largest and smallest eigenvalues are 46.54 and 2.46, respectively. Thus, we conclude that the state of self-stress stiffens all of the inextensional mechanisms of the tensegrity dome. Pellegrino (1992) carried out a study of the response of this structure to various load conditions, including a set of experiments on a physical model with diagonal span of 1.725 m that was prestressed with ≈ 33 N in the inner ring and corresponding values in the remaining members. For the particular case of four equal downward forces applied at joints 5–8, GT D loses its positive definiteness when the loads have a magnitude of 27.8 N. At this point, joints 1–4 undergo fairly large horizontal displacements just before the upper tension ring goes slack.

4.5

Parachutes and Other Decelerators Decelerator structures are used for many applications. One of the most common decelerator structures is the parachute, used as a high-drag device for a variety of applications, including planetary entry. Common requirements for a decelerator structure are that it needs to be tightly packaged for transportation and, once deployed, it should form a stable structure with high drag. The drag on a structure is calculated from the equation D = CD qA

(4.30)

where CD is the drag coefficient, which is related to the shape of the structure, A is the projected area of the structure onto a plane perpendicular to the direction of motion, and q is the dynamic pressure, given by 1 2 (4.31) ρv 2 Here, ρ is the density of the atmosphere and v is the velocity of the decelerator. In a typical application requiring the use of a decelerator, the value of q is chosen on the basis of the values of ρ and v, determined by the particular application, for which the maximum dynamic pressure is obtained. q=

4.5 Parachutes and Other Decelerators

117

According to Eq. 4.30, the design of a structure that has high drag D could be accomplished either by choosing a shape with high CD , or by increasing its area A; that however would increase the mass of the structure. In practice, though, the range of values for CD is relatively small (typically 0.5–1) and hence the only relatively “free” parameter is A. In practice, the designer of the decelerator has to achieve a compromise between the values of CD and A (Anderson, Bohon, and Mikulas, 1969).

4.5.1

Shape of Parachutes Figure 4.20 shows a typical design of a parachute structure that meets the requirements described above: obviously A is large and, for this shape, CD ≈ 0.7. This design was arrived at by the end of World War I, at which time G.I. Taylor wrote the classical paper “On the shapes of parachutes” (Taylor, 1919). The design of a parachute consists of many load-carrying meridional cords overlaying a fabric envelope. When the parachute is fully deployed, the fabric forms a series of pleats, or lobes, separated by the cords. The reason why parachutes have this shape – rather than the smooth shape of the balloons in Section 3.5 – is that this kind of lobed construction decreases the level of stress in the fabric, provides a continuous path for

Figure 4.20 Test of NASA parachute structure, conducted at the U.S. Army’s Yuma Proving Ground. The main parachute was dropped from a U.S. Air Force C-17 aircraft with a 40,600-pound load at an altitude of 17,500 feet (NASA image).

118

Tension-Stabilized Structures

B C s A B C

C'

s

C' B'

A

A'

r

A'

θ

B' p T 2

T2 (a)

(b)

(c)

Figure 4.21 Simplified geometry of parachute: (a) flat fabric; (b) perspective views of single panel, flat (top) and deployed (bottom); (c) much enlarged view of section A C  B  through an inflated lobe.

the loads – from the weight suspended from the parachute up into the canopy – and also reduces the complexity of manufacturing the whole system. Taylor’s analysis of the shape of parachutes began by showing that the circumferential stress in the fabric of a parachute can be assumed to be zero. His argument was as follows: consider a flat, circular piece of fabric and attach a large number, n, of cords to its circumference. Then, weave into the fabric a series of radial, inextensible wires that go from the center of the fabric to the point of attachment of each cord. If the ends of the cords are attached to a common point and an initial pressure p is applied under the fabric, the parachute deploys and the fabric bulges out between successive wires. The cross-sections of the pleats formed by the fabric can be approximated by circular arcs. Consider a circle of radius s drawn on the flat fabric, Fig. 4.21(a): the length of arc ACB is 2π s/n. When the parachute deploys, this circle will deform approximately into a series of circular arcs (this approximation neglects the meridional curvature of the parachute), Fig. 4.21(b). In this deformed configuration, the arc ACB moves to A C  B  and its chord length is A B  = 2π r/n, where r is the radius of a cylindrical surface through A and B  . Since n is large, A C  B  can be approximated by a circular arc forming an angle θ with A B  , see Fig. 4.21(c), and it can be shown that A B  r sin θ = ≈    θ s ACB

(4.32)

from which it follows that θ is independent of n. This simple geometric description of the parachute surface can be used to determine the stress resultants in the fabric. The surface has n-fold symmetry about the axis of the parachute but we will still use the stress resultants Nφ∗ in the meridional direction and Nθ∗ in the circumferential direction, although the stress distribution is not exactly axisymmetric. Here a ∗ denotes the actual stress resultants in the parachute fabric, to distinguish them from a limiting axisymmetric surface that will be considered in Section 4.5.2.

4.5 Parachutes and Other Decelerators

119

The meridional stress in the parachute fabric is negligible, Nφ∗ = 0. Equilibrium of a strip parallel to A C  B  and of unit width, in the direction perpendicular to the chord A B  , gives pA B  = 2Nθ∗ sin θ

(4.33)

Hence, since θ is independent of n, as shown above, when n increases both A B  and will decrease. Therefore, in the limit for an infinite number of lobes, the parachute takes up a shape in which there is no circumferential stress. Taylor proposed a parachute design based on the limiting condition that the circumferential stress is zero. This approach produces the flattest possible design and, since there is no compressive stress anywhere in the fabric, parachutes designed in this way will not wrinkle in the hoop direction. In practice, the stress resultant Nθ∗ in the fabric can be made as small as required by choosing n sufficiently large. The corresponding value of A B  , and hence of n, is obtained from Eq. 4.33. The role of the fabric is simply to transfer the pressure load onto the cords. The shape of the meridional cords can be approximately obtained from an axisymmetric surface that equilibrates the pressure p without any circumferential stress; this is also the shape of the lowest possible axisymmetric pneumatic dome considered in Section 3.6. The details of an analytical solution are presented in the next section. Nθ∗

4.5.2

Analysis Details We consider the problem of finding the shape of an axisymmetric membrane loaded by a uniform internal pressure p and such that the circumferential stress resultant is zero everywhere. This surface, which was already encountered in Section 3.6, is known as the isotensoid. It has already been shown that the geometry of the isotensoid is intrinsically defined by Eq. 3.58, which is reproduced here: R1 =

R2 2

(4.34)

where R1 is the (principal) meridional radius of curvature and R2 the second principal radius of curvature, see Fig. A.21. To integrate this equation analytically, it is best to transform it into the (r, φ) coordinate system defined in Fig. A.21. Substitute Eq. A.33 into Eq. 4.34 to obtain R1 =

r 2 sin ψ

(4.35)

Then, from inspection of Fig. A.21, we write ds = R1 dψ

(4.36)

dr = ds cos ψ

(4.37)

120

Tension-Stabilized Structures

1.0

0.5

r/r0

0

ψ 0.2 z/r0 0.4 0.6

Figure 4.22 Axisymmetric parachute profile (Taylor, 1919).

and these two equations are combined to obtain R1 =

1 dr cos ψ dψ

(4.38)

Substituting Eq. 4.38 into Eq. 4.35 dψ dr = r 2 tan ψ

(4.39)

Equation 4.39 can be integrated to obtain log cr =

1 log sin ψ 2

(4.40)

where c is a constant of integration. From Eq. 4.40 we obtain  sin ψ

(4.41)

c2 r 2 = sin ψ

(4.42)

cr = which gives

The constant c can be found by imposing the boundary condition r = r0 for ψ = 0, which sets r0 as the radius at the equator. This gives r2 = sin ψ r02

(4.43)

4.5 Parachutes and Other Decelerators

121

Taylor (1919) obtained the following expressions for the radius, r, and vertical distance from the apex, z, r = r0 cn(u) "

r0 ! z = √ 2 E − E(u) − (K − u) 2 where

√ s 2 u=K− r0

(4.44) (4.45)

(4.46)

Here cn(u) is a Jacobi elliptic function and E(u) is the incomplete elliptic integral of √ the second kind. For modulus 1/ 2, K = 1.8541 and E = 1.3506. s is the arc-length along the meridian, measured from the axis. These equations can be evaluated using the Matlab functions ellipj, ellipke, and EllipticE (Mathworks, 2018). A meridian of the isotensoid has been plotted in Fig. 4.22.

4.5.3

Other Types of Decelerators Parachutes of similar shape to those derived in the previous section can also be used at supersonic speed provided that some air flow is allowed through the canopy. In the example shown in Fig. 4.23(a), the canopy consists of separate ring sails with a hole at the apex. Figure 4.23(b) shows an alternative, more stable solution known as a ballute, which is inflated by ram air inlets that face in the direction of the flow.

(a)

( b)

Figure 4.23 Shapes of supersonic decelerators (Anderson et al., 1969).

122

Tension-Stabilized Structures

Aft surface Burble fence

Ram air inlet

Front surface

Entry vehicle Figure 4.24 Inflatable decelerator attached to an entry vehicle (Mikulas and Bohon, 1969).

0.6 T=1.0

0.4

0.75 0.64

0.2

0.53 0.48

z/r0 0

0.46 0.36 0.28 0.42

–0.2 –0.4 –0.6 –0.8 –1.0

0

0.2

0.4

0.6

0.8

1.0

r/r0 Figure 4.25 Profiles of attached inflatable decelerators (Mikulas and Bohon, 1969).

An even more stable configuration, obtained by attaching the decelerator directly to the entry vehicle, is shown in Fig. 4.24. This kind of attached inflatable decelerators was developed specifically for use at supersonic speeds (Mikulas and Bohon, 1969). The main reason for bringing the aft (rear) surface towards the front is in order to reduce the volume of gas required for inflation, and hence decrease the inflation time, thus

4.6 Tension Field Beams

123

increasing stability. The main purpose of the “burble fence” at the edge between front and aft surfaces is to increase the aerodynamic stability at subsonic speeds. A set of preliminary profiles obtained for zero hoop stress in the fabric, as in the isotensoid, but with the internal pressure equal to twice the dynamic pressure on the front surface is shown in Fig. 4.25. Different values of the non-dimensional cord load parameter for the rear surface, T =

nT pπ r0

(4.47)

have been considered in the plot. Note that n is the number of cords, T is the tension in the rear meridional cords and p is the pressure difference. For T = 1, the isotensoid profile is obtained. In the plot, the rear surfaces that do not intersect the axis of the decelerator are shown terminated at their intersection with the front surface curves.

4.6

Tension Field Beams This section introduces a structural concept in which lightweight tension elements carry the shear load in lightweight beams. An example is the thin-walled cantilever beam shown in Fig. 4.26. It consists of top and bottom flanges at a distance h, connected by equally spaced vertical stiffeners and by a thin web (membrane) of thickness t. The beam is loaded at the tip by a shear force Q. For small values of Q, the web behaves as a plate subject to in-plane loading and hence it is subject to a uniform state of pure shear τ ≈ Q/ ht. The corresponding principal stresses are ±σ , at 45◦ to the horizontal, as shown in Fig. 4.27(a). As Q increases, at some point the web buckles and so it is unable to carry any more compressive stresses. However, the tensile principal stress can be further increased, subject to an appropriate redistribution of the stress state. Thus, for moderately large Q the web carries some load in pure shear and the remainder in pure tension, as shown in Fig. 4.27(b). If Q is further increased, the web wrinkles completely, as shown in Fig. 4.28, and its ability to carry shear stress becomes negligible. However, the beam

d

d

d

A

t

h

Section A-A Q

Figure 4.26 Thin-walled cantilever beam.

A

124

Tension-Stabilized Structures

σ = –τ σ=τ PS (a) 45o σ = –(1 –k) τ σ = (1 –k)τ

τ PS

T

PS

(b)

2kτ σw= _____ sin 2θ

θ 45o

T

(c)

2τ σ w = _____ sin 2θ

θ Figure 4.27 States of stress in web as the shear force increases; (a) pure shear, (b) incomplete diagonal tension field, (c) diagonal tension field (k = 1).

70°

47°

60°

55°

50°

50°

55°

47°

60°

70°

Figure 4.28 Distribution of wrinkles in rectangular panel with edge stiffeners (Mansfield, 1970).

4.6 Tension Field Beams

125

x Ft

σw

σxy σxx

Q θ

Fb

Figure 4.29 Stress components on tip section of cantilever beam with wrinkled web.

can still carry substantial loads very efficiently, purely by uniaxial tension σw in the web and uniaxial compression in the vertical stiffeners, see Fig. 4.27(c). This load-carrying mechanism is called tension field and the resulting beam concept is known as a “Wagner beam” from the name of its discoverer (Wagner, 1929). This approach has been extended to general panel shapes and to stiffened cylindrical shells (Kuhn, 1956). A detailed account of the development of tension field theory can be found in Chapter 9 of Mansfield (1989). Designs based on tension fields are used in the design of aircraft structural members that are not exposed to the air stream, such as wing spars (Peery and Azar, 1982). They are also used in the construction of super-light Japanese “shoji” screens, consisting of a wooden lattice attached to a web made of paper and used as room dividers, etc. We will explain the key aspects of this structural concept with reference to the beam shown in Fig. 4.26, based on Megson (1999). It will be assumed that the magnitude of the shear load Q is such that the web is completely wrinkled and is in a state of pure tension σw at an angle θ to the horizontal, as shown in Fig. 4.27(c). Consider the free body obtained by cutting vertically through the beam of Fig. 4.26 at an arbitrary distance x from the tip, as shown in Fig. 4.29. The stress components on this cross section can be obtained using the transformation in Eq. B.7. In the present case, the stress components are σx  = σw , σy  = 0 and σx  y  = 0. Substituting into Eq. B.7 and doing the matrix multiplications gives   σx σxy σw sin θ cos θ σw cos2 θ = (4.48) σxy σy σw sin2 θ σw sin θ cos θ where it should be noticed that the shear stress has been denoted by σxy for consistency with the general expression in Section B.2, but here σxy = τ . Assuming that the flanges do not carry any shear and resolving vertically Q = σxy ht = σw ht sin θ cos θ

(4.49)

Solving for σw gives σw =

Q 2Q 1 = ht sin θ cos θ ht sin 2θ

(4.50)

126

Tension-Stabilized Structures

The axial forces in the top and bottom flanges, Ft and Fb , respectively, are Qx Q − cot θ h 2 Qx Q − cot θ Fb = − h 2 Ft =

(4.51) (4.52)

In the above equations, note that the first terms represent equal and opposite forces that equilibrate the bending moment Qx, whereas the second terms represent equal compressive forces that equilibrate the horizontal component of the resultant tension in the web, σw th cos2 θ = Q cot θ . Also note that the flanges are subject to a transverse, uniform load given by σy = σw sin2 θ . Hence, they act as beams supported by the stiffeners; the corresponding stresses have not been included in the present analysis. Each vertical stiffener carries a compressive force V that equilibrates the vertical component of the web tension, σw td sin2 θ. Substituting Eq. 4.50 for σw V =

Qd tan θ h

(4.53)

Substituting the mean shear stress, τ = Q/ ht, we obtain the following expressions for the stresses in the web, flanges (note that the stresses due to the first terms of Eqs 4.51–4.52 are not included), and stiffeners 2τ sin 2θ τ ht cot θ σf = − 2Af σw =

σs = −

(4.54)

τ dt tan θ As

where Af ,As are the cross-sectional areas of the flanges (assumed to be equal) and the cross-sectional areas of the stiffeners. Recall that the stresses σs are in addition to the bending stresses in the beam, and hence the flanges of a tension-field beam have to be bigger than those of a standard I-beam. The angle θ of the diagonal tension can be determined by noting that the wrinkles will arrange themselves in a way that maximizes the stiffness of the beam. This requires that the strain energy in the beam should be minimized. Consider one bay of the beam, of length d. The strain energies that correspond to the above stresses are: web: flanges: stiffener:

σw2 htd 2E σf2 Uf = 2 Af d 2E σ2 Us = s As h 2E Uw =

(4.55)

4.6 Tension Field Beams

127

Minimizing the total strain energy $ d # Uw + Uf + Us = 0 dθ

(4.56)

gives tan4 θ =

1 + ht/2Af 1 + td/As

(4.57)

from which the angle θ can be calculated for any given beam. Note that for beams with very thin webs in comparison to the flanges, and hence ht and td small in comparison with Af and As , respectively, the right-hand side of Eq. 4.57 tends to 1 and hence θ = 45◦ . In the above analysis it has been assumed that the wrinkles are all parallel but it can be seen in Fig. 4.28 that in reality a fan of wrinkles forms near the corners of an edge-stiffened panel. A more detailed analysis that captures this effect can be found in Mansfield (1989).

5

Shell Structures

5.1

Introduction Shell structures are continuous, thin-walled, surface-like structures. Their continuity allows them to transmit forces in all directions, unlike lattice or skeletal structures such as space frames. Their curvature allows them to equilibrate external loads by a suitable distribution of in-plane forces, instead of bending and transverse shear. Shells are ubiquitous in nature, Fig. 5.1, and are also widely used in man-made constructions, Fig. 5.2, as they combine structural and environmental functions in a thin layer of material. For example, the shell of an egg is very stiff and strong and it also forms a habitat in which a chick can grow. Similarly, the concrete domes and shells shown in Fig. 5.2 both support the roof and provide themselves the roofing element. If a space frame had been used instead, an additional covering layer would have been needed. But the efficiency of thin shell structures goes much beyond the combination of structural and environmental functions. Two-dimensional curvature results in a double arch effect that, in contrast to one-dimensional curved structures such as cable or arches, allows the shell to carry several smooth load cases by membrane action alone (Ramm, 1992). This represents a huge advantage, provided that both the shape and the boundary conditions have been carefully designed for the loading. Engineered shell structures have been used for many centuries as roofs covering churches and assembly halls; these classical shells had a slenderness (radius of curvature/thickness, R/t) of up to 50 and regular, e.g., spherical or cylindrical, shapes. The Romans built many cylindrical shells (“barrel vaults”) and also some large domes, such as the hemispherical Pantheon dome, which has a diameter of 43 m and a thickness of 1.2 m. Large diameter masonry vaults feature in most major Gothic cathedrals and Renaissance churches (Heyman, 1995). Figure 5.2(a) shows the Byzantine dome of Hagia Sophia, in Istanbul. The main dome forms a hemisphere with radius of 16.3 m. It is built of brick masonry and has thickness varying between 0.69 m and 0.61 m. Melaragno (1991) tells the story of the remarkable way in which this dome has survived for almost 1500 years, despite violent earthquakes which on occasion have caused it to partially collapse. The extreme of efficiency combined with remarkable elegance has been achieved by the Swiss engineer Heinz Isler, who in 1959 changed the outlook of concrete shell design by proposing shells of “natural shapes” based on physical experiments

128

5.1 Introduction

(a)

(b)

(c)

(d)

129

Figure 5.1 Shells in nature (a) sea shells (image reproduced from Nicepik), (b) walnut

(Alexapicso/Shutterstock.com), (c) eggs (photo by Daniele Levis Pelus on Unsplash), (d) nautilus (image reproduced from Nicepik).

(Isler, 1959). Only shapes amenable to analytical mathematical descriptions had been used previously, most famously the hyperbolic paraboloidal shells of Felix Candela (Abel and Oliva, 2010), but the structural behavior of these shape-constrained shells was not ideal. Throughout his career Isler perfected the art of free-form shells whose optimal surface profiles were obtained from scaled experimental models (Isler, 1993). An example of Isler’s structures is shown in Fig. 5.2(b) and more examples are described in Chilton (2000). Computational methods that mimic the experimental shell form-finding methods of Isler were developed by Ramm (1992) with his collaborators (Ramm, Bletzinger, and Reitinger 1993; Ramm and Wall, 2004). A construction technique for mass produced axisymmetric reinforced-concrete shells is the Bini-shell method invented by Dante Bini. Key stages of this process are shown in Fig. 5.3: lightweight concrete being poured over a flat rubber membrane, the membrane being inflated, and the completed dome. Over 2,000 Bini-shells have been constructed worldwide. More general shapes for concrete shells have been constructed by injecting concrete between stitched membrane skins (Pronk, Dominicus and ten Hoope, 2010) and masonry domes have been constructed by Guastavino using tiles and without any kind of scaffolding (Ochsendorf and Freeman, 2010). Shell structures are also used for a wide variety of applications. For example, they are used in the food industry as containers for fluids, in the car industry as body panels for cars, in aerospace for airplane bodies, etc., see Fig. 5.4.

130

Shell Structures

(a)

(b) Figure 5.2 (a) The brick masonry domes of Hagia Sophia (Luciano Mortula – LGM/ Shutterstock.com) and (b) elegance and efficiency combined in the concrete shells of the Deitingen Service Station (Switzerland) by Heinz Isler (image reproduced with permission of John Chilton, 2019).

5.2 Why Shell Structures?

131

Figure 5.3 Construction of a Bini-shell (images reproduced with permission of Dante Bini).

Figure 5.4 Examples of engineered shell structures: (a) Coca Cola drinks can (Antony

McAulay/Shutterstock.com) and (b) Ferrari 250 Testa Rossa (Photo © Sjo / iStock Unreleased / Getty Images).

5.2

Why Shell Structures? The three simple examples in Fig. 5.5 illustrate different ways in which different types of shells carry loads. These shells are based on analytical surfaces that have been cut by two parallel planes: the sphere in Fig. 5.5(a) has positive gaussian curvature, the cylinder in Fig. 5.5(b) has zero gaussian curvature, and the hypar (= hyperbolic paraboloid) shell in Fig. 5.5(c) has negative gaussian curvature. The three shells are unsupported and without any stiffeners, and they are loaded by two equal and opposite “pinching” forces. All three shells ovalize at the ends, as a large inextensional deformation can take place on account of the free ends. The distribution of the mid-plane stresses, in the lower part of the figure, shows the Mises stress slowly decreasing along two characteristic directions (straight lines) in the case of the hypar, and only one direction in the case of

132

Shell Structures

(a) Κ > 0

(b) Κ = 0

(c) Κ < 0

Figure 5.5 Effects of curvature on load carrying behavior of different types of shells: (a) spherical

shell; (b) cylindrical shell; (c) hypar shell. The deformed shells are shown at top and the mid-plane Mises stress distribution is plotted on the undeformed configuration at the bottom (Ramm and Wall, 2004).

the cylinder. The stress distribution is again different in the spherical shell, where the stress is localized near the loading points. To quantify the efficiency of shell structures, we can compare the small-displacement, linear response of the five structures shown in Fig. 5.6. The first structure, Fig. 5.6(a), is a 1 m × 1 m flat plate of thickness 0.5 mm, made of steel, whose edges are simply supported, i.e., only constrained against translation in all directions. The remaining four structures are not flat, but all fit within an envelope of height δ = 10 mm, corresponding to a height-to-span ratio of 1%. Two other cases, with δ/S = 5% and 10%, are also considered. All structures are subject to the same support conditions as the flat plate. The structure in Fig. 5.6(b) is a folded plate structure, i.e., a faceted shell consisting of two flat plates fully connected along the ridge. The structure in Fig. 5.6(c) has a cylindrical shape. The structure in Fig. 5.6(d) is a spherical cap whose edges lie in vertical planes, and the last structure is a hypar shell whose edges have a characteristic “twisted square” shape. For all three structures we have computed using finite-element software (i) the deflection caused by a vertical force of 1 N applied at the center point, and (ii) the fundamental natural frequency of vibration. Both of these parameters provide a measure of stiffness but the point stiffness is a more local measure, and depends on the particular point chosen, whereas the natural frequency is a more global measure. The results of our analysis are shown in Tables 5.1 and 5.2, in terms of ratios between the performance of each shell structure and a flat plate.

5.2 Why Shell Structures?

133

S=1m S=1m (a)

(b)

d

(c)

d

d

(d)

(e) d/2

d/2 Figure 5.6 Simple shell structures with equal span S and fitting within the same height envelope δ.

Table 5.1 shows that for δ/S = 1% the stiffness varies by more than two orders of magnitude. The flat plate structure has the lowest stiffness (1/4.8 mm/N) and also the lowest fundamental natural frequency (0.078 Hz). The shell structures are all ten or more times stiffer. They also have significantly higher natural frequencies than the flat plate, with the lowest (2.5 times higher) being the folded plate shell.

134

Shell Structures

Table 5.1 Stiffness ratios. δ/S a b c d e

Flat plate Folded plate Cylindrical Spherical Hypar

1%

5%

10%

1.0 177.8 18.5 25.3 10.0

1.0 2400.0 65.8 141.2 39.3

1.0 12000.0 111.6 320.0 73.8

Table 5.2 Fundamental frequency ratio.

a b c d e

δ/S

1%

5%

10%

Flat plate Folded plate Cylindrical Spherical Hypar

1.0 2.5 7.9 14.2 6.5

1.0 2.4 17.8 66.8 19.0

1.0 2.4 25.0 128.2 29.5

In terms of static stiffness, the “best” structure is the folded plate shell, followed by the spherical shell, but this result is a bit misleading because the stiffness of the folded plate structure would be much lower if the force were to be moved away from the ridge line. In fact, if one ranks the structures in terms of their natural frequencies, the best one – i.e., the structure with the highest frequency – is the spherical shell followed by the hypar shell. The fundamental natural frequency of the folded plate shell is not so high because it admits a natural mode of vibration similar to the second mode of the flat plate, which does not take advantage of the central rise. Conversely, the static stiffness of the hypar shell is not as good as its dynamic performance because its curvature is smaller than the cylinder or the sphere. With a rise of only 1%, these shell structures would visually appear almost identical to the flat plate and yet, as it has been shown, their behavior is remarkably different. Since introducing small shape changes in the geometry of the initial flat plate has such large effects, there are obvious questions about the importance of geometric accuracy and the sensitivity of these results to fabrication errors. Also, the hypothesized boundary conditions of full translational fixity along the edges may be difficult to realize in practice: even a small compliance in the edge supports would change these results. We have increased the rise δ to 50 mm and to 100 mm (with the height-to-span ratio increasing to 5% and 10% respectively), repeated the analysis, and compared the static stiffness and fundamental natural frequencies of the various structures, again, to the stiffness and natural frequency of the flat plate. The results are presented as additional columns in Tables 5.1 and 5.2. They show that the rank order is essentially unchanged, although some of the ratios become even bigger. In Table 5.1 note that there is no change in the rank order as δ increases. In Table 5.2 note that, for the reason explained above, the natural frequency of the folded plate shell remains practically constant. Also note that

5.3 Membrane vs. Bending Action

135

the frequency of the spherical shell increases at a faster rate than those of the cylindrical and hypar shells. Finally, note that the hypar becomes the second-best, instead of the cylinder, once the rise is greater than 1%. In conclusion, it has been shown that shell structures are potentially much more efficient than flat structures, and that spherical shells are the most efficient. The two other shapes that have been considered, cylindrical and hypar shells, appear to be roughly equivalent from our comparison, but in fact cylindrical shells are more sensitive to geometric imperfections when loaded in compression, hence they are not always the best choice. Furthermore, it should be noted that the definition of rise used for our comparison, based on overall envelope size rather than rise at the center, has penalized the hypar shells in comparison with the cylindrical shells.

5.3

Membrane vs. Bending Action The main reason why the shell structures analyzed in the previous section have turned out to be so much stiffer than the flat plate is that they rely on a structurally efficient load-carrying mechanism. This mechanism is known as the membrane action of a shell, as it is associated with the in-plane stress resultants Nx ,Ny ,Nxy defined in Fig. 5.7(a). Membrane action, which has already been encountered when dealing with continuum models for single-layer space frames in Section 2.3 and balloons in Section 3.5, acts in addition to the bending action, Fig. 5.7(b), which we have encountered when dealing with double-layer space frames in Section 2.5. Note that the definition of the stress resultants for shells is equivalent to that for a flat plate, see Section B.5.1 and B.5.2, because the curvature effects are negligible in the case of thin shells. A thin shell has R/t > 50, where t and R are the thickness and the smaller of the two principal radii of curvature, respectively. Such structures have only a very limited ability to carry loads by bending action and hence if they are properly supported, see later, their stiffness is mainly due to the membrane action. For such structures it is usual to neglect the stress resultants associated with bending action, as a first approximation. This is the so-called membrane theory for shell structures (Flugge, 1966, Den Hartog 1952), in which all external loads applied to the shell are assumed to be equilibrated by the in-plane stress resultants only. For any given shell structure subject to prescribed loads, the membrane equations of equilibrium will yield a mid-plane stress distribution that is in equilibrium with the load. However, under certain load conditions this solution is a poor approximation to the actual stress distribution, as (i) the associated deformation of the shell does not satisfy geometric compatibility, particularly at the boundaries and in regions where there is a discontinuity in shape or curvature of the mid-surface of the shell, or (ii) there is a discontinuity in thickness or in the loading. Both effects are related to limitations of the membrane theory and can be eliminated by a using a theory that accounts for both membrane and bending effects. Alternatively, the shape of the structure and the applied boundary conditions can be changed in such a way that the membrane solution becomes

136

Shell Structures

py

pz

y

px

z

x

Nxy

Nxy

Nx

(a)

Ny

Qx y

(b)

Qy

z

x

Myx

Mxy

Mx

My Figure 5.7 Stress resultants associated with (a) membrane action and (b) bending action

in a shell.

more dominant, which requires the solution of a shape optimization problem, leading to a free-form shell. Increasing the slenderness of the shell is a way of moving in this direction. Consider a rigidly-clamped hemisphere of radius R and thickness t loaded only by its own selfweight. Ramm and Wall (2004) have computed the fractions of membrane, bending and shear strain energy in the full solution, as a function of the slenderness R/t, see Fig. 5.8. For R/t ≈ 50 the bending boundary layer around the clamped edge takes up almost 90% of the total strain energy, but the situation is reversed for R/t ≈ 1000. A more detailed understanding of these effects can be obtained by studying a weightless, open top, cylindrical tank, as shown in Fig. 5.9. It is filled up to the rim with a liquid of density ρ. We will compare the distribution of stress resultants predicted by the membrane theory to the distribution that accounts for the constraint imposed by the rigid base. Let H,R, and t be respectively the height, radius, and thickness of the tank. The liquid applies to the tank a pressure that increases linearly with depth. Determining the membrane stress distribution in terms of the cylindrical coordinates (x,θ) introduced in the figure is straightforward. Because the tank is weightless Nx = 0 and because the loading is axisymmetric Nxθ = 0. The only nonzero stress resultant is Nθ , which can be determined by modifying the axisymmetric equation of equilibrium in the normal direction, Eq. B.44, by noting that here R1 = ∞ and R2 = R. Also, the axes of the coordinate system are x,θ instead of φ,θ , and pn = ρg(H − x).

5.3 Membrane vs. Bending Action

137

1.0 bending energy

W / Wtot

0.8

0.6

0.4 membrane energy 0.2 shear energy 0.0 0

200

400

600

800

1000

R/ t Figure 5.8 Variation of strain energy fractions in clamped spherical shell (Ramm and Wall, 2004).

H x q

Figure 5.9 Cylindrical tank and definition of cylindrical coordinate system.

Hence, Eq. B.44 becomes Nθ Nx + = ρg(H − x) ∞ R

(5.1)

Nθ = ρgR(H − x)

(5.2)

which has the solution

Associated with this membrane stress distribution there will be a corresponding deformation of the tank. Since the hoop strain θ is zero at the top and increases linearly with depth, there will be a radial displacement proportional to H − x, which is maximum at the bottom of the tank. There will also be an axial displacement, resulting from Poisson’s ratio effects, but this is not important for the present discussion.

138

Shell Structures

r

F

M

Figure 5.10 Superposition of membrane and bending solutions.

10

10

Membrane x (m) 5

Membrane + bending

5

0

0 0

1000

-5

Nθ (kN/m)

0

10 15

5

Mx (kNm/m)

Figure 5.11 Distribution of stress resultants in cylindrical tank.

Obviously, the radial displacement at the bottom of the tank violates the base constraint. To restore compatibility two edge-loading solutions are superposed onto the above solution, Fig. 5.10. The magnitudes of the axi-symmetric base reaction force F and moment M are determined by imposing the conditions that both the radial displacement and the rotation of the bottom edge of the tank should be zero. Thus, it is found (Flugge, 1966, p. 274) that the correct stress distribution in the tank is in fact 

  1 −αx cos αx + sin αx Nθ = ρgR H − x − H e −H e α    ρg 1 −αx −αx Mx = cos αx + H e sin αx − H e α 2α 2 −αx



(5.3) (5.4)

 √ where α = 4 3(1 − ν 2 )/ tR. Figure 5.11 shows plots of Eqs. 5.2, 5.3, and 5.4 for a steel tank with R = 10 m, H = 10 m, and t = 0.05 m, filled with a liquid of density ρ = 1,000 kg/m3 . A comparison of the uncorrected and corrected membrane stress distributions shows that Nθ becomes zero at the bottom. Due to the small thickness of

5.4 Shell Morphology

139

the shell, the “secondary bending” dies down within 2 m from the base and yet there is a sizable bending moment at the bottom of the tank.

5.4

Shell Morphology Shell structures are unique in that the shape of the envelope provided by the structure, which is related to its function, is closely related to the structure itself. Thus, for example, the overall shape of an aircraft or that of a car are determined mainly by the need to provide efficient envelopes for transportation, as well as by aerodynamic considerations. Similarly the shape of a large dome covering a sports stadium will be chosen both on the basis of architectural and environmental considerations. A study of shell structures that satisfy specific constraints on the shape of the envelope, its aerodynamic characteristics, or its visual impact are specialized topics beyond the scope of this book. Therefore, in this section we discuss a series of applications of shell structures where the nonstructural constraints are such that the aim of the structural morphology study can be stated simply as finding the most efficient shape for a shell structure capable of carrying out the prescribed function. As discussed in Sections 5.2 and 5.3, efficient shell behavior is directly linked to designing a structure that carries loads by membrane action, and hence by minimizing the bending action. We will now show how this can be done by means of several examples. For analytical simplicity, the majority of our examples are axi-symmetric, by which we mean that both the shape of the shell and the applied loads are axi-symmetric. The equilibrium equations for this problem are available in Section B.6.

5.4.1

Spherical Dome Loaded by Self-Weight We begin by considering a shell of given shape, with an easily described geometry and assuming the wall thickness t to be uniform. The structure is subject to the vertical load distribution shown in Fig. 5.12.

R

ρgt

φ

Figure 5.12 Spherical dome loaded by self-weight.

140

Shell Structures

We can apply the equations in Section B.6 to determine the membrane stress distribution in this dome. The principal radii of a sphere are R1 = R2 = R and the normal and tangential components of the load in Fig. 5.12 are pn = −ρgt cos φ

(5.5)

pφ = ρgt sin φ

(5.6)

Nφ can be obtained from Eq. B.45. The self-weight of a spherical cap is given by the surface area, 2π R 2 (1 − cos φ), multiplied by ρgt. Hence, in Eq. B.45 we substitute: F = −ρgt2π R 2 (1 − cos φ)

(5.7)

and simplify to obtain Nφ = −

ρgRt 1 + cos φ

(5.8)

Then, substituting Eqs. 5.5 and 5.8 into Eq. B.44, substituting R1 + R2 = R, and solving for Nθ gives   1 Nθ = ρgRt − cos φ (5.9) 1 + cos φ It is interesting to discuss this force distribution in some detail. At the apex, φ = 0, Eqs. 5.8 and 5.9 give Nφ = Nθ = −ρgRt/2: the meridional and circumferential directions are equivalent at this point and hence the two stress resultants are interchangeable. The meridional force is negative, i.e., compressive, throughout, but Nθ gradually decreases in magnitude until it changes sign at φ = cos−1

2 √ = 51.82◦ 1+ 5

The complete distribution has been plotted in Fig. 5.13. Thus, a dome forming a hemisphere embracing a total angle of over about 100◦ will be in tension in the hoop direction, which might suggest that masonry domes such as Hagia Sophia would not be in equilibrium. This paradox is resolved by noting that the above membrane solution was derived for shells of vanishingly small thickness, because it was assumed that Nφ and Nθ lie on a spherical surface. However, in domes of finite thickness the equilibrium surface is not uniquely defined and hence it is possible to find alternative stress distributions which involve only compression everywhere and lie fully within a spherical shell of given thickness (Heyman, 1995).

5.4.2

Boiler End In this section we allow the shape of the shell to vary, in a problem that is typical of pressure vessel design, such as the liquid propellant tank of a rocket engine, as shown in Fig. 5.14. We consider a cylindrical pressure vessel of radius R and uniform thickness t with internal pressure p: we want to determine the shape of the end caps for which the

5.4 Shell Morphology

141

1

0.5

Nθ

+

Nφ , Nθ 0 ρgRt

–0.5 Nφ –1 0

+

51.2 10

20 30

40 50 60 φ (deg)

70

80 90

Figure 5.13 Distribution of stress resultants in a spherical dome (Heyman, 1995).

Satellite

Third and fourth stages

Second stage

First stage

Figure 5.14 Rocket engine system for ISAS M-5 rocket, showing two boiler end fuel tanks in third and fourth stages of rocket (JAXA image).

membrane stress distribution is the actual solution. In other words, we are looking for a solution that eliminates secondary bending stresses. The pressure loading corresponds to pn = p and pφ = 0. Also, in the cylindrical part R1 = ∞ and R2 = R. Hence, from Eq. B.44 we obtain: Nθ = pR

(5.10)

This is the expression for Nθ in the cylindrical part of the shell. If the end caps are chosen to be hemispherical, R1 = R2 = R and hence, noting that in a sphere Nθ = Nφ for symmetry, from Eq. B.44 we obtain: Nθ = Nφ = pR/2

(5.11)

Comparing Eq. 5.10 with Eq. 5.11 it can be seen that there is a sudden discontinuity in the hoop stress resultant Nθ at the junction between the cylindrical and hemispherical parts. Hence, if the cylindrical and spherical shells have equal thickness, there would be a discontinuity in hoop strain and radial displacement at the seam between the two shells. This situation is analogous to the cylindrical tank discussed in Section 5.3 and, again, it induces an additional distribution of bending moments, as shown in Fig. 5.15. The discontinuity in the hoop forces at the junction between the cylindrical shell and the

142

Shell Structures

− +

Mx

Mφ

+ Nφ

Figure 5.15 A sudden transition from cylindrical to hemispherical shells of equal thickness leads to a step variation in Nθ that induces the bending moments shown (Flugge, 1973).

z

r

Figure 5.16 Set of Cassini curves, with the curve given by Eq. 5.12 shown by a solid line

(Flugge, 1960).

end cap results from the sudden change of curvature in the meridional direction from zero to 1/R. The discontinuity in the hoop stress resultants can be avoided by choosing a shape for the meridian of the end cap that is second-order continuous with the cylinder, i.e., by choosing a meridian profile with both first- and second-order derivatives equal to zero at the transition point. Among the many profiles that satisfy this condition is the family of Cassini curves (Shikin, 1995), and Flugge (1960) chose the following equation for discontinuity-free boiler end shells: (r 2 + α 2 z2 )2 + 2R 2 (r 2 − α 2 z2 ) = 3R 4

(5.12)

This is the curve shown as a solid line in Fig. 5.16, where it should be noted that the z-coordinate is along the axis of the pressure vessel and z = 0 corresponds to the interface with the cylindrical shell. The scaling parameter α in Eq. 5.12 is used to shorten

5.4 Shell Morphology

143

z

R r

Nφ

Nθ

Figure 5.17 Stress resultants in boiler end pressure vessel with meridian profile based on Cassini curve.

the curve in the z-direction, to avoid profiles that are too elongated. According to Flugge, acceptable shapes are obtained for α ≈ 2. Flugge (1960) obtained the following analytical expressions for the principal radii corresponding to the axisymmetric profile given by Eq. 5.12, from which the profile itself can be determined: [r 2 (R 2 + α 2 z2 ) + α 4 z2 (R 2 − r 2 )]3/2 3α 2 R 3 (R 2 − r 2 + α 2 z2 )  r 2 (R 2 + α 2 z2 ) + α 4 z2 (R 2 − r 2 ) R2 = 2R R 2 + r 2 + α 2 z2

R1 = 2

(5.13) (5.14)

The stress resultants in this shell are then given by:  Nφ = pR  Nθ = Nφ

r 2 (R 2 + α 2 z2 ) + α 4 z2 (R 2 − r 2 ) R 2 + r 2 + α 2 z2

3α 2 R 4 (R 2 − r 2 + α 2 z2 ) 2− 2 (R + r 2 + α 2 z2 )[r 2 (R 2 + α 2 z2 ) + α 4 z2 (R 2 − r 2 )]

(5.15)  (5.16)

An example has been plotted in Fig. 5.17. In the figure, note that Nθ is continuous throughout, and in particular at the transition between the end cap and the cylindrical shell. Of course, Nφ is also continuous. In Fig. 5.17 also note that there is a narrow region of compressive Nθ , which however can be avoided by choosing α < 1.9 for the shell profile. Returning to Fig. 5.14, it should be noted that in this particular rocket the first and second motor chambers consist of a cylinder with two hemispheres on the ends, all made of steel. The third and fourth motor chambers, which kick the satellite into its final orbit, were designed to follow approximately Eq. 5.12, and were built by filament winding of a composite material.

5.4.3

Drop-Shaped Tank We consider the problem of finding the shape of a fully stressed axisymmetric tank for storage of liquids. Tanks designed according to this approach are called echinodomes, because of their similarity to the Sea Urchin (in Latin, Common Echinus); they were

144

Shell Structures

r z

h

φ

r

H

pn

R

z Figure 5.18 Drop-shaped tank (Flugge, 1960).

developed for oil storage by the Chicago Bridge and Iron Company in the 1910s (Royles et al., 1980). It is assumed that the tank is completely filled with liquid of density ρ and the pressure at the top is equivalent to an additional height h of liquid, Fig. 5.18. The self-weight of the tank will be neglected. The idea is to shape the tank in such a way that the hoop and meridional stress resultants are equal and uniform throughout, i.e., Nφ = Nθ = A

(5.17)

where A is a positive constant. The shape that we are looking for is identical to the shape of a drop of liquid resting on a plane surface, and in equilibrium under the surface tension produced by capillary action, see also the discussion of soap films in Section 3.3. To determine the shape of the meridian it is convenient to choose the origin O of our coordinate system at a level where the pressure is zero, i.e., at a height h above the top of the tank, Fig. 5.18. Hence, the load components on the shell are pφ = 0 and pn = ρgz

(5.18)

Substituting Eq. 5.17 into Eq. B.44 1 ρgz 1 + = R1 R2 A

(5.19)

and then substituting Eqs. A.38 and A.39 gives sin φ ρg d sin φ + = z dr r A

(5.20)

The variable φ can be expressed in terms of r and z using tan φ =

dz dr

(5.21)

5.4 Shell Morphology

145

Figure 5.19 Shapes of drop-shaped tanks with different pressure heads (Flugge, 1960).

and this yields a nonlinear differential equation similar to Equation 3.54, which can be integrated numerically using the shooting technique described in Section 3.5.2. Figure 5.19 shows a set of solutions obtained by Flugge (1960). The figure shows also the pressure head h that corresponds to each shape. As the ratio between the head and the height of the tank increases, the tank becomes closer to a sphere.

5.4.4

Tension Shells Tension shells were developed as a concept for supersonic, planetary decelerators. They are an alternative scheme to the attached inflatable decelerator discussed in Section 4.5.3 but use a design approach normally used for membrane structures. Anderson et al. (1965) noted that the structural design of most aircraft and entry vehicles is controlled by loading conditions that cause buckling and therefore the stresses in the material are often much lower than the material strength. In order to take better advantage of the strength of the material, they developed a structural concept that resists the primary loads by tensile stresses in the main part of the vehicle surface. The structural concept that emerged from this approach, Fig. 5.20, consists of an axisymmetric tension shell of negative gaussian curvature attached to a compression ring at the rear; a representative payload is shown in the picture. Note the catenary-like meridional profile of the shell, resulting from the requirement that the external pressure be carried mainly by tension in the meridional direction. To determine the shape of a tension shell one assumes a relationship between hoop and meridional stress resultants, in analogy to the approach followed for hot-air balloons in Section 3.5.2 and for parachutes in Section 4.5. The main difference is that here the presence of a compression ring allows us to make use of a different part of the equilibrium surface.

146

Shell Structures

Ring

Payload

Tension shell

Air flow

Figure 5.20 Tension shell concept (Anderson et al., 1965).

The normal stress resultants are assumed to have a constant ratio throughout the shell Nθ = αNφ

(5.22)

where α is a positive constant and Nθφ = 0 for symmetry. Substituting these expressions into the meridional equation of equilibrium for an axisymmetric shell, Eq. B.43, gives for pφ = 0: d (R2 Nφ sin φ) − αR1 Nφ cos φ = 0 dφ

(5.23)

Substituting Eqs. A.38 and A.39 for the curvatures, this becomes dr d (Nφ r) − α Nφ = 0 dφ dφ

(5.24)

dNφ dr r + (1 − α)Nφ =0 dφ dφ

(5.25)

which can be rearranged into

and, multiplying by dφ/rdr we obtain the differential equation dNφ 1−α + Nφ = 0 dr r

(5.26)

Nφ r 1−α = constant

(5.27)

The integral of Eq. 5.26 is

5.4 Shell Morphology

147

where the integration constant is found by substituting the radius rb at the rear of the shell and the corresponding meridional stress resultant Nφ = N0 ; N0 is the load applied by the tension shell to the compression ring. This gives: Nφ r 1−α = N0 rb1−α

(5.28)

Equation 5.27 defines the variation of Nφ and, through Eq. 5.22, also Nθ . The corresponding shape of the tension shell is obtained by substituting Eqs. 5.28, A.38, and A.39 into the equation of equilibrium in the normal direction, Eq. B.44, to obtain α p d sin φ + sin φ − dr r N0



r rb

1−α =0

(5.29)

Anderson et al. (1965) considered two (axisymmetric) pressure distributions. The first distribution results from Newtonian impact theory, i.e., it is assumed that each air particle that hits the decelerator loses its component of momentum normal to the surface of the decelerator and then moves along the surface with its tangential component of momentum unchanged (inelastic collision) hence the normal pressure p depends on the dynamic pressure q, multiplied by a coefficient k: p = −kq cos2 φ The second pressure distribution is uniform: p = p0 . For the first pressure distribution Eq. 5.29 becomes a nonlinear differential equation, whose solution has been plotted in Fig. 5.21 for α = 0 and different values of the parameter A2 =

kqrb 2N0

(5.30)

For uniform pressure Eq. 5.29 can be solved to find   2−α p0 rb rb α r sin φ = − 2N0 r rb

(5.31)

and this equation implicitly defines the shape of the meridian. Figure 5.22 compares the shape of a catenary curve (derived in Section 3.4) to those obtained from the above formulations assuming α = 0. It turns out that the assumption α = 0 yields the highest drag coefficient and hence these curves provide the best aerodynamic design for a decelerator. The plots in Fig. 5.22 assume A2 = kqrb /2N0 = 1.4 for the Newtonian flow solution and p0 rb /2N0 = 1 for the uniform pressure case. Note that the comparison has been limited to the range 0.25 < r/rb ≤ 1.0 because for aerodynamic reasons a blunt nose is used near the tip of the decelerator, instead of the shape derived from the tension shell solution.

148

Shell Structures

r/rb 0

0.2

0.4

0.6

0.8

1.0

0.2 0.5

0.4 0.8 0.6

1.0

z/rb

1.2

0.8

1.4

1.0 1.6 1.2

1.4 Figure 5.21 Shape of tension shells with Nθ = 0 assuming Newtonian flow, for different values of the parameter A2 given by Eq. 5.30 (Anderson et al., 1965).

r/rb 0

0.2

0.4

0.6

0.8

1.0

0.2

z/rb

Catenary 0.4

Newtonian flow Uniform pressure

0.6

0.8 Figure 5.22 Comparison of shapes of tension shells; the Newtonian flow solution is for the case A2 = 1.4 and the uniform pressure solution assumes p0 rb /2N0 = 1.

5.4 Shell Morphology

149

Figure 5.23 Application of hanging model principle over the last 300 years (image courtesy of Ramm and Wall, 2004).

(b) (a) Figure 5.24 Comparison of (a) hanging model by Isler with (b) optimized shape obtained by Ramm and Wall (2004).

5.4.5

Free-Form Shells The pioneer of experimentally based form-finding methods for thin shell structures, Heinz Isler, used a variety of experimental techniques to design over 1,500 thin-walled concrete domes (Isler, 1993; Chilton, 2000). One of the techniques used by Isler was the hanging model technique that had already been used by several notable predecessors for the design of both masonry domes and membrane structures, Fig. 5.23. Figure 5.24(a) shows a wet membrane hanging from four points. It has been frozen to obtain a shape that, when turned upside down, provides the shape of a shell that is in compression everywhere when it is loaded by self-weight. Figure 5.24(b) shows the corresponding solution obtained by Ramm and Wall (2004), by minimizing the total strain energy and imposing the constraint that no tension is allowed anywhere.

Shell Structures

However, optimized shells tend to be sensitive to imperfections and shells that are optimized for one particular load condition are particularly sensitive to variations in loading. It is important to consider more than one load conditions in the optimization analysis, which cannot be done when the profile of the shell is determined experimentally. Numerical methods of shape optimization provide much greater freedom that can be used to generate more robust designs (Ramm and Wall, 2004). Here there is a strong link with form-finding of membrane structures, discussed in Section 3.8, since in both cases we are after a surface-like structure that carries loads without relying on bending. Ramm and Wall (2004) recommend to use a model that includes potential geometric imperfections to achieve optimized designs that are less imperfection sensitive. For example Ramm and Wall (2004) considered the shell shown in Fig. 5.25(a), supported at four points and subject to dead load only. Their optimization study maximized the critical buckling load, obtained from a geometrically nonlinear analysis. The overall mass was set to remain constant, while the shape and thickness distribution of the shell were allowed to vary. Figure 5.25(b) shows the load-displacement behavior of the perfect shell, with a buckling load factor λc = 84.3, as well as the behavior of several shells with different geometrical imperfections. The lowest buckling load factor of these

100

A

80

load

54.3

40

B

A

84.3

60

20 0 0

(a)

0.5

1.0

1.5

displacement

B

(b)

Perfect

Imperfect

100

0.112

0.110

B–B 0.412

0.372

83.1

80

A–A load

150

74.0

60 40 20 0

(c)

0

0.5

1.0

1.5

displacement

(d) Figure 5.25 (a) Shell supported at four points and subject to dead load only; load-displacement behavior of optimized shells that were (b) initially perfect and (d) initially imperfect shells; (c) cross-sections of two optimized structures (Ramm and Wall, 2004).

5.5 Polyhedral Shells

151

imperfect shells is λc,min = 54.3, indicating a substantial sensitivity to imperfections of this particular design. The results of the alternative optimization approach, which included in the objective function the performance of both perfect and imperfect structures, with prescribed geometrical deviations based on the first eigenmode, are shown in Fig. 5.25(b). For this design the buckling load factor for the perfect shell was to found to be λc = 83.1, but when the analysis was repeated with several different imperfections the lowest load factor was λc,min = 74.0. Thus it can be seen that the sensitivity of this optimized thin shells has been significantly reduced. Figure 5.25(c) shows the shape and thickness variation along two cross-sections of the two optimized structures; the perfect shell is shown on the left and the imperfect shell on the right. This approach has been applied by Ning and Pellegrino (2015, 2017, 2018) to the design of super-stable cylindrical and near-spherical shells. These studies have shown that, by introducing optimized deviations in the shape of cylindrical and spherical shells, it is possible to both increase the buckling limit of these structures, as well as reduce their sensitivity to imperfections.

5.5

Polyhedral Shells Imagine a pattern of straight creases in a thin sheet, which is then kinked along these creases to form a polyhedral shell, i.e., a nonsmooth, faceted surface consisting of interconnected planar elements. If the edges of the shell are allowed to deform, then its geometry can change as, for example, in the retractable roof shown in Fig. 5.26. Another example is the folded plate cantilever roof in Fig. 5.27. A wide range of shapes can be realized with polygonal facets, whose shape, size and arrangement can be varied to achieve different effects. The key difference in behavior between a thin plate and its folded-plate version is that even a small kink 2α along the fold results in a much deeper structure, whose bending stiffness is considerably greater. To compare the bending deflections of the two

(a)

(b)

Figure 5.26 Retractable roof based on polyhedral shell concept (Karni and Pellegrino, 2007).

152

Shell Structures

Figure 5.27 Folded plate roof for grandstand of Groenendael hippodrome, near Brussels, designed by Andre Paduart (photo by Yoshito Isono, reproduced with permission of Structurae, 2019).

B/2

B

B/2

L L (a)

(b)

Figure 5.28 Simple comparison of bending stiffnesses of thin plate vs. folded-plate cantilever.

cantilevered structures shown in Fig. 5.28 we determine the second moments of area of their cross-sections, which are I1 =

Bt 3 12

and

I2 =

B 3 t sin2 α 48

where t is the plate thickness. If B = 50 mm, t = 0.1 mm and the kink angle is α = 10◦ , then the stiffness ratio is given by: I2 /I1 ≈ 475 This increase in bending stiffness was exploited by Miura (1969, 1970) in the concept of pseudo-cylindrical concave polyhedral (PCCP) shells. An example of PCCP shell is the well-known Yoshimura pattern, which was first proposed as an illustration of the postbuckling deformation mode of thin cylindrical shells under axial compression (see Fig. 5.29). An important characteristic of the Yoshimura pattern, as well as of PCCP shells in general, is that it can be obtained by a process of inextensional deformation of a cylindrical surface. This is demonstrated by the fact that the card model shown in Fig. 5.29(b) is made from a single sheet, without making any cuts or additions. The PCCP surface is a concave polyhedral surface consisting of identical repeating modules. Both diamond and hexagonal modules were considered by Miura (1970) but here we will restrict our attention only to diamond-shaped repeating modules made from

5.5 Polyhedral Shells

(a)

153

(b)

Figure 5.29 Buckling pattern of a circular cylindrical shell subjected to axial loading (a) experiment (b) Yoshimura pattern.

R’

π /N Repeating module

D γ C

λ α

B

F E

α

L

A

Figure 5.30 Geometry of PCCP shell.

equilateral triangles as shown in Fig. 5.30. Hence the geometry of the PCCP surface is defined by the following parameters: L α N γ R

base length of triangle, in circumferential direction; base angles of triangle (α = 60◦ in present case); number of circumferential triangles; inclination of triangles with respect to axis of cylinder; and radius of equivalent cylinder, where R = N L/2π .

Note that the radius of the equivalent cylinder is smaller than the radius R  of the cylindrical surface passing through the vertices of the PCCP surface.

154

Shell Structures

Therefore, to define a PCCP shell we only need to define N and R, plus of course the wall thickness, t. If N is small, such as 6 to 12, the inclination γ is substantial and hence the concavity of the shell is significant. As N is increased to 20 or higher, the PCCP shell becomes closer to a smooth circular cylinder. PCCP shells have two main advantages over smooth cylindrical shells. First, they consist of flat elements which, in large scale applications, can be assembled from prefabricated flat panels. Second, their shape is not inherently unstable, unlike a “perfect” cylindrical shell and hence their behavior is similar to cylinders with stiffeners. However, unlike stiffened shells, this more stable behavior is achieved by avoiding the complexity of separate stiffening elements. The polyhedral geometry of PCCP shells makes it difficult to obtain simple analytical descriptions of their properties and hence we have to rely mostly on numerical solutions or experiments to understand their behavior. The first thing to note is that the stress distribution in a PCCP shell typically involves a combination of membrane and bending actions, where both membrane and bending stresses concentrate along the polyhedral edges. For example, Figs. 5.31–5.32 show stress distributions, computed by Miura and Tanizawa, in a PCCP shell consisting of isosceles triangles and loaded by axial compression, Fig. 5.31, and by hydrostatic pressure, Fig. 5.32. In both sets of plots the direction and magnitude of the principal stresses are denoted as follows: ←→ denotes tension and → ← denotes compression. It is interesting to note that for both loading conditions the maximum principal membrane stress is compressive along the inclined edge BD and tensile along the circumferential edge BE. Also note that the center region of each facet is stressed only lightly: both types of load are, in effect, being carried by a space frame whose members coincide with the creased edges of the shell. It is a general feature of structures made σm/σc D

σb/σc 0

D

0.5

B

E (a)

0

0.25

B

E (b)

Figure 5.31 Numerically obtained distribution of (a) principal membrane stresses σm and

(b) principal bending stresses σb in two half-facets of PCCP shell with R/t = 478, N = 6, α = 45◦ loaded by axial compression. σc denotes the longitudinal stress in the equivalent cylinder, under the same loading.

5.5 Polyhedral Shells

σ m/σ c D

0 1.0

E

σ b/σ c

D

3.5

0 1.0

B

155

5.5

B

E

(a)

(b)

Figure 5.32 Numerically obtained distribution of (a) principal membrane stresses σm and

(b) principal bending stresses σb in two half-facets of PCCP shell with R/t = 104, N = 6, α = 45◦ loaded by hydrostatic pressure. σc denotes the hoop stress in the equivalent cylinder, under the same loading.

1.0 N=20

0.8 EPCCP 0.6 ECC 0.4

N=10 N=6

0.2 0

40

80

120

160

200

R/t Figure 5.33 Axial stiffness ratio of PCCP shells along line AFD, see Fig. 5.30 (Knapp, 1977a).

from interconnected plates that the higher stiffness of the junction between the plates causes significant stress increases there; conversely, the stresses decrease away from the junctions. This effect is called shear lag. The plots of principal bending stresses on the surface of the PCCP shells show a similar variation, which indicates that – like the membrane stresses – the bending stresses are also concentrated near the crease lines of the shell. It is instructive to compare the axial and bending stiffnesses of a PCCP shell to its equivalent circular cylindrical shell. Clearly the axial stiffness will be much smaller because the longitudinal sections of the PCCP shell are not straight and so axial loads induce significant bending action. Knapp 1977 has carried out a detailed finite-element study of this problem, which has shown that the axial stiffness is not uniform in the

156

Shell Structures

circumferential direction. Figure 5.33 is a plot of the ratio between the lowest stiffness and the axial stiffness of the equivalent cylindrical shell, as a function of R/t, for different numbers of circumferential triangles. The figure shows that the axial stiffness can be reduced by as much as a factor of 10, for very slender PCCP shells with small N.

(a)

(b) Figure 5.34 Proposals for (a) undersea habitats (Knapp, 1977b) and (b) undersea structure

(Knapp, 1998).

Figure 5.35 PCCP-type beverage can produced by Toyo-Seikan Co.

5.5 Polyhedral Shells

157

Applications of PCCP shells for undersea habitats have been proposed by Knapp (1977b, 1998) and some examples are shown in Fig. 5.34, although the first actual application was in depressurized drinks cans (Miura, 2002). Drinks containing milk need to be stored at lower-than-atmospheric pressure and hence the storage can is loaded by an external superpressure. Using a PCCP shell, as shown in Fig. 5.35, has led to a reduction of 30% in the required thickness and so a considerable saving in material.

6

Sandwich Structures

6.1

Introduction A sandwich structure is a particular kind of composite plate or shell that consists of two thin and stiff skins attached to a lightweight and thick core, as in the human skull (Fig. 6.1), which consists of two hard layers connected by softer bone. In beams, the bending efficiency is increased by replacing the solid cross-section with an I-section, whose two equal area flanges are connected by a thin, deep web. Both the bending stiffness and the strength of the beam are increased by increasing the distance between the flanges, up to the point where the buckling load of the web is reached. The wing of an aircraft is a hollow body with a comparatively thin skin which carries heavy structural loads and yet has to provide a stable, aerodynamically clean, smooth, streamlined surface. As the speed of the aircraft increases, the aerodynamic loading on the wing also increases and there is the danger of local buckling or wrinkling of the skin. A lightweight skin is required to reduce the overall weight, and yet its second moment of area must be high to avoid buckling. In stressed skin structures, shown in Fig. 6.2(a), the upper and lower skins of a wing are separated by a series of thin spars carrying the shear load. To avoid buckling of the skin, a large number of reinforcing spars is required, and hence an obvious drawback of this type construction is that it requires many small parts and many connections, which are expensive. Sandwich construction provided a practical and very effective replacement to stressed-skin construction. It first appeared in 1924 (von Karman and Stock, 1925) and was first used in an aircraft at the end of the 1930’s. It was a huge success and was the key “new technology” in the DeHavilland Mosquito aircraft, Fig. 6.3 (Hoff and Mautner, 1944). Suddenly, the number of parts required to make key components of an aircraft could be drastically reduced, as one can simply glue a continuous core to two thin, stiff skins, see Fig. 6.2(b), to form a wing structure. Two aileron structures, the original stressed-skin Al-alloy design and its replacement using sandwich construction, are shown in Fig. 6.2(c and d). During the early years of sandwich construction natural lightweight materials were chosen for the core, such as balsa wood and paper, Fig. 6.4. Nowadays, Al-alloy honeycomb is a frequent choice; paper honeycomb is also used.

158

6.1 Introduction

159

Figure 6.1 Section of human skull (Hodgson, 1973).

Face skin Skin

Spar cap

Honeycomb

(a)

Adhesive

(b) Face skin

(c)

(d)

Figure 6.2 (a) Three spar wing (Niu, 2006); (b) honeycomb sandwich; (c-d) comparison of

Aluminum alloy and sandwich construction (Hoff and Mautner, 1944).

A potential problem of sandwich construction is that, while making the core lighter and lighter, a point is reached where the structural behavior of a panel is no longer dominated by bending deformation, but by shear. Then any increase in bending stiffness and strength would be offset by shear flexibility. Therefore, in the next section we will present a theory to identify the transition from bending-dominated to shear-dominated behavior as a function of various structural parameters. Then, in the rest of the chapter we will deal with the morphology of the core, and how it affects its shear modulus.

160

Sandwich Structures

Figure 6.3 Mosquito aircraft (Alex JW Robinson/Shutterstock.com).

Figure 6.4 Young’s modulus E plotted against density ρ (Ashby, 2010).

6.2 Theory of Sandwich Plates

161

S+dS M

M

M+dM

M S

(a)

(b)

(c)

Figure 6.5 Deformation of sandwich beam element shown in (a), due to (b) bending, and

(c) combination of bending and shearing.

6.2

Theory of Sandwich Plates To analyze the deflection of a sandwich beam we need to consider both bending and shear deformations. Figure 6.5 shows the deformation of a short element of a sandwich beam. In Fig. 6.5(b), this element is subject to a pure bending moment, M, and hence the top face becomes longer while the bottom face becomes shorter; both face plates also bend but, because of their thinness, the bending moment needed to bend them is rather small. The core does not carry much force or moment because it is much more compliant than the two face plates; its main function is to maintain the separation of the face plates and preventing the thin face plates from buckling or wrinkling. Figure 6.5(c) shows how the same element deforms when it is subject to both a bending moment M and a shear force S. The deformation pattern described previously is altered because the overall plane sections of the sandwich beam no longer remain plane, due to the shear deformation of the core, however the shear deformation of the face plates is negligibly small because they have a much higher shear modulus than the core. Based on these observations, the following hypotheses are made when setting up the standard theory of sandwich beams: • • • •

the face and core materials behave linear-elastically; the deflections and rotations are small; the modulus of the face plates, Ef , is much larger than the homogenized modulus of the core, Ec∗ ; and the shear modulus of the face plates, Gf , is much larger than the homogenized shear modulus of the core, G∗c .

Hence, the key effects that are included in the theory are the bending and stretching of the face plates and also the shearing of the core, which is assumed to be uniform through the thickness. Note that G∗c is the overall shear modulus of the core, measured on test samples much larger than the elements that make up the core structure. The relationship between G∗c and the morphology and mechanical properties of the constituent materials is discussed in Section 6.3.

162

Sandwich Structures

1

W

t y

(a)

b

x

t L u γ1 (b) u

x

v γ2

(c) v

Figure 6.6 (a) Geometry of sandwich cantilever beam and assumed deflection component fields

due to (b) x- and (c) y-components of core deflection.

The particular geometry that is analyzed, shown in Fig. 6.6, is a cantilever beam of length L, face plate thickness t, core thickness b, and unit width. As before, the relevant elastic properties are the modulus of the face plates, Ef , and the shear modulus of the core, G∗c . The cantilever is loaded by a single transverse force W at the tip. The approach presented here is based on a solution by Hoff and Mautner (1948). In general, problems involving the shear deformation of a beam need to be approached using a two-dimensional formulation (Timoshenko and Goodier, 1970); hence one would introduce two scalar functions u(x,y) and v(x,y), which represent the x-and y-components of the displacement field, respectively. However, with the hypotheses listed above, u and v are general functions of only x and linear functions of y, as shown in Fig. 6.6(b–c). The system of differential equations relating u and v can be found from an energy approach, as follows. •

First, calculate the strain energy as the sum of the following three contributions. i.

Membrane energy of face plates. The normal strain, assumed to be constant through the thickness, is =

∂u = u ∂x

6.2 Theory of Sandwich Plates

163

and the corresponding axial force is T = (EA)f  = Ef tu Hence the membrane strain energy in the two face plates is:   L   L 1 T dx = Ef t (u )2 dx UM = 2 2 0 0 ii.

(6.1)

Bending energy of face plates. The curvature is given by: κ=

∂ 2v = v  ∂x 2

and the corresponding bending moment is: M = (EI )f κ = (EI )f v  where (EI )f =

1 Ef t 3 12

(6.2)

Hence, the bending strain energy in the two face plates is   L   L 1 Mκdx = (EI )f (v  )2 dx UB = 2 2 0 0 iii.

(6.3)

Shear energy of core. The shear strain, uniform through the thickness, is γ =

2u ∂u ∂v + = − v ∂y ∂x b+t

and the corresponding shear force is S = (GA)c γ = G∗c b



2u − v b+t



Hence, the shear strain energy is 2   L 1 L 1 ∗ 2u  US = Sγ dx = Gc b dx −v 2 0 2 b+t 0 •

Second, calculate the potential energy of the load V = −W vL

•

(6.4)

(6.5)

where vL = v(L,y). Third, assemble the total potential energy of the structure, which has the expression E = UM + UB + US + V where the individual terms are given by Eqs. 6.1, 6.3–6.5.

(6.6)

164

Sandwich Structures

The problem of minimizing E with respect to the (unknown) functions u and v can be approached using the Calculus of Variations, thus transforming the energy minimization into a system of differential equations. This procedure, analogous to that described in Section B.9, leads to the following system of differential equations:   2G∗c b 2 −2Ef tu + u − v = 0 b+t b+t (6.7)   Ef t 3  2 ∗   v + Gc b u −v =0 6 b+t subject to the natural boundary condition   Ef t 3  2 ∗  v + Gc b u−v +W =0 6 b+t

(6.8)

The general solution of Eq. 6.7 has the form: u= 

b+t 2



A1 A2 sinh px + 3 cosh px + A3 x 2 + A4 x + A5 p3 p 1 − p2 k 2 1 − p2 k 2 cosh px + A2 sinh px+ 4 p p4   3 x2 x + A3 − 2k 2 x + A4 + A5 x + A6 3 2

v = A1

(6.9)

where we have defined p2 =

G∗c b (EI )s

and

k2 =

(EI )1 G∗c b

(6.10)

and (EI )1 = (EI )s =

1 t(b + t)2 Ef 2 1 1 (EI )1

+

1 2(EI )f

≈ 2(EI )f

Note that (EI )s is the “in series” flexural stiffness of the sandwich beam, i.e., the reciprocal of the sum of the bending compliance of the face plates, 1/(2(EI )f ), and of an I-beam with a web of depth b + t, 1/(EI )1 . The six constants of integration A1, . . . ,A6 are determined from Eq. 6.8 and from the additional boundary conditions u = v = v  = 0 at x = 0 u = v  = 0 at x = L Substituting these boundary conditions, we obtain general expressions for the u,v. Then, setting x = L we find the tip deflection, that can be expressed in the form: vL = ψ

W L3 3(EI )p

(6.11)

6.2 Theory of Sandwich Plates

165

where (EI )p is the ordinary flexural stiffness, i.e., the “in parallel” flexural stiffness of an I-beam whose web has negligible thickness: (EI )p = (EI )1 + 2(EI )f = and 3 ψ =1+ (pL)2

1 1 t(b + t)2 Ef + t 3 Ef 2 6



tanh pL 1− pL



(6.12)

(EI )1 2(EI )f

(6.13)

Note that Eq. 6.11 has the same structure as the standard formula for the tip deflection of a cantilever loaded by a tip force, apart from the correction factor ψ. An alternative, much simpler approach (Allen, 1969) is to neglect the interaction between shear and bending, and to simply add the bending deflection v1 to the shear deflection v2 . The bending deflection is obtained from the standard formula for the tip deflection of a cantilever, with (EI )p given by Eq. 6.12: v1 =

W L3 3(EI )p

(6.14)

The shear deflection is obtained by multiplying the shear strain by the span, hence v2 = γ L =

τL WL = ∗ ∗ Gc Gc b

(6.15)

Adding Eqs. 6.14 and 6.15 vL ≈

W L3 WL + ∗ 3(EI )p Gc b

(6.16)

This approach has been extended to other statically determinate beams, and also to simple statically indeterminate beams with different support conditions (Gibson and Ashby, 1997) to obtain the following formula for the maximum deflection vmax ≈

W L3 WL + B1 (EI )p B2 (G∗c b)

(6.17)

where the values of B1 and B2 are given in Table 6.1 for beams with different end conditions. Table 6.1 Constants for maximum deflection of beams of length L (Gibson and Ashby, 1988). Support

Mode of loading

Cantilever Cantilever Simply supported Simply supported Ends built in Ends built in

tip force W uniformly distributed load q = W/L central force W uniformly distributed load q = W/L central force W uniformly distributed load q = W/L

B1

B2

3 8 48 384/5 192 384

1 2 4 8 4 8

166

Sandwich Structures

Table 6.2 Mechanical properties of honeycomb (HEXCEL, 2018). Type

Material

Relative density α (%)

CR III-1/4-5052-.002” CR III-1/4-5052-.0007” A1-48-3

Al-alloy Al-alloy Aramid paper

2.5 1 7

G∗cL (MPa)

G∗cW (MPa)

G∗c (MPa)

455 145 40

206 76 25

330 110 32.5

L 0.5 mm 25 mm 0.5 mm 1 Figure 6.7 Sandwich beam cantilever studied in Section 6.2.1.

6.2.1

Example The sandwich cantilever beam shown in Fig. 6.7 has a general length L and a fixed cross-section, with face plates of thickness t = 0.5 mm, core of thickness b = 25 mm, and unit width. We will consider four different types of core: two different Al-alloy honeycombs, a paper honeycomb, and a polyurethane foam. Hexcel honeycombs are denoted by a standard four-part code: Material - Cell size - Alloy type or weight of paper - Foil thickness where the cell size and material thicknesses are in inches or mm, and sets of actual values for several honeycombs are given in Table 6.2. Since honeycombs are anisotropic, as explained in Section 6.5, two values of the shear modulus are given, G∗cL in the longitudinal direction, L, and G∗cW in the transverse direction, W . The average shear modulus of the core, given in the last column of the table, has been calculated from G∗c =

G∗cL + G∗cW 2

(6.18)

The only difference between the two Al-alloy honeycombs is in the foil thickness, which results in a density reduction from 69 kg/m3 for 0.05 mm thick foil, to 26 kg/m3 for 0.018 mm thick foil. The paper honeycomb has a density of 48 kg/m3 ; it is heavier than the lighter Al-alloy honeycomb despite its lower shear modulus. The fourth type of core is a polyurethane foam with a relative density (density of foam divided by density of solid material) α = 10%. It is shown in Section 6.4, Eq. 6.32, that

6.2 Theory of Sandwich Plates

167

Deflection vL (mm)

102 101 100 Gc*= 6

10-1 10-2

Gc*= 32.5 Gc*= 110 Gc*= 330

10-3 101

102 Length L (mm)

103

Figure 6.8 Variation of deflection with beam length for four different sandwich beams; the dotted line shows the deflection due to bending effects only. Values of G∗c in MPa.

the shear modulus of a foam made from material with modulus Ec can be calculated from the equation G∗c =

3 Ec α 2 8

and, taking Ec = 1600 MPa for solid polyurethane (Gibson and Ashby, 1997) G∗c =

3 1600 × 0.12 ≈ 6 N/mm2 8

Figure 6.8 shows plots of the deflections vL , calculated from Eq. 6.11, of sandwich beams with the four core types and varying lengths. For L < 200 mm the shear deflection is significant. For example, for L = 100 mm the deflections are 0.0414, 0.0654, 0.1507, and 0.6750 mm for G∗c respectively equal to 330, 110, 32.5, and 6 MPa. Hence, the tip deflection increases by 58%, 364%, and 1,630% as one goes from the first type of core respectively to the second, third, and fourth. To explore further the effects of the core shear stiffness, Fig. 6.9 shows plots of the tip deflection of beams with a fixed length of L = 100 mm for varying G∗c . The deflection becomes constant for G∗c > 300 MPa, thus characterizing the range where bending dominates. Lastly, Fig. 6.10 compares predictions from the “exact” (Eq. 6.11) and simplified (Eq. 6.16) shear deformation theories for the beams with the highest and lowest values of G∗c . This figure indicates that the complexity of the “exact” theory is not justified in the present case study, as the results from two theories show some small difference only for the cantilever with the softest core and a span L ≈ 20 mm or less, i.e., when there is a significant interaction between the bending of a single flange and the shear distortion of the core.

168

Sandwich Structures

Deflection vL (mm)

102

6 101

32.5

100 100

110

330

101 102 Shear modulus Gc* (MPa)

103

Figure 6.9 Variation of deflection with shear modulus, for sandwich cantilever beams with

L = 100 mm.

102

Deflection vL (mm)

101 Gc*= 6 100

Eq. 6.15 Gc*= 330

10-1 Eq. 6.11 10-2 10-3 101

Eq. 6.10 ≅ Eq. 6.16 102 Length L (mm)

103

Figure 6.10 Comparison of exact and simplified theories for core shear deformation.

6.3

Shape Efficiency of Core In the previous section it has been shown that the low effective shear modulus of some low-density cores can cause significant reductions in the overall stiffness of a sandwich beam. To make a rational comparison between different types of cores, we need the value of G∗c for each core and also the corresponding density. An analogous situation was encountered for space frames, in Sections 2.3–2.7, and in analogy with space frames, we will introduce an efficiency parameter to better make this comparison. The key difference is that here we will focus on shear, instead of extensional and bending stiffness.

6.3 Shape Efficiency of Core

169

To understand the key effects that need to be analyzed, it is useful to think initially of making a lightweight core out of a given material in the following idealized way. Consider a block of homogeneous, isotropic material of shear modulus Gc and density ρc . To reduce its weight, imagine making many small holes in this block and let ρc∗ be the reduced density of the block. The relative density of the block is defined as α=

ρc∗ ρc

(6.19)

If the holes are only a small proportion of the total volume, i.e., α ≈ 1, the shear modulus of the block is reduced in proportion to α G∗c ≈ αGc

(6.20)

however, as more and more material is removed by increasing the number and/or size of the holes, then G∗c decreases at a faster rate than α, because the block starts behaving as a space frame instead of a continuum. Hence, for α  1 the value of G∗c depends on the morphology and detailed properties of the core; for example, it will depend on the way in which the loads are carried by the core, whether in an axial, bending, or shear mode. We are usually interested only in rather small values of α, typically in the range 0–10%, and it is then useful to quantify the difference between G∗c and αGc by means of the additional parameter, μ=

G∗c αGc

(6.21)

The parameter μ is a measure of the shape efficiency of the core. Comparing Eq. 6.21 with Eq. 2.17 we find that the two definitions are analogous, only the extensional modulus has been replaced by the shear modulus. The effective shear modulus can then be obtained from G∗c = μαGc

(6.22)

Comparison of Eq. 6.20 with Eq. 6.22 highlights the fact that μ is a knock-down factor on the purely geometric reduction of the shear stiffness. Ideally, of course, it would be desirable for this factor to be as close as possible to one but in general this is not possible. The most efficient types of core, such as honeycomb, have μ ≈ 0.5, and the reason why larger values of μ are not possible can be understood from the following analysis of four different, idealized types of core. First, consider an open cell core made of rod-like elements forming the edges of small, open-walled cells, as found in many foams. It will be shown in Section 6.4 that the shear modulus of such cellular materials has the general expression G∗c = α 2 Gc

(6.23)

Substituting Eq. 6.23 into Eq. 6.21 μ=

α 2 Gc =α αGc

(6.24)

which states that the shape efficiency of foams increases linearly with its relative density.

170

Sandwich Structures

(a)

(b)

(c)

(d)

Figure 6.11 (a) Shear loading of core, (b) two-way and (c) three-way grid of webs, and (d) space frame type core.

As a second type of core, consider the two-way grid of thin shear webs shown in Fig. 6.11(b). Neglecting the out-of-plane bending stiffness of these webs, the uniform shear load shown in Fig. 6.11(a) is carried by uniform shearing in one set of webs only. Because these webs represent only one half of the total, the effective shear modulus is obtained by multiplying Gc by 0.5α 1 G∗c ≈ αGc (6.25) 2

6.3 Shape Efficiency of Core

171

Hence, comparing Eq. 6.25 with Eq. 6.22 we find that for the second type of core μ≈

1 2

(6.26)

As an alternative to the two-way grid of webs, consider a three-way tessellation of orthogonal webs, as shown in Fig. 6.11(c). In this case, only one third of the core material resists the load shown in Fig. 6.11(a) and hence μ≈

1 3

(6.27)

As a fourth case, consider a space frame core based on the square-on-square space frame in Fig. 2.8(a). This core type is shown in Fig. 6.11(d) and the rods are all of equal length and lie in two sets of parallel planes. It can be shown that the maximum shear stiffness of this core type space frame is obtained when the rods are inclined at 45◦ to the faces of the sandwich plate, and in this case: μ≈

1 3

(6.28)

The shape efficiency predictions in Eqs 6.24–6.28 have been plotted in Fig. 6.12, together with a series of experimental values for Al-alloy honeycombs (HEXCEL, 2018) and polyurethane foams (Gibson and Ashby, 1982). For honeycombs the mean value of μ in the L and W directions, see Fig. 6.19, has been plotted. It can be noted from the figure that the data for polyurethane foams is well within the theoretical range for sandwich core made from rod-like elements. Also, the data for honeycombs agree quite

Two-way webs

Shape efficiency μ

0.5 0.4

Three-way webs & space frame

0.3

Open cell polyurethane Closed cell polyurethane Al-alloy honeycomb

0.2 Foams 0.1 0

0

2 4 6 8 Relative density α (%)

10

Figure 6.12 Shape efficiency of sandwich plates with different types of core. Experimental data for polyurethane foam from Gibson and Ashby (1982) and for honeycomb from HEXCEL (2018).

Sandwich Structures

Table 6.3 Morphologies of sandwich core. Space frame

Shear web

Porous continuum

1

Shape efficiency μ

172

0.8

two-way webs

0.6 three-way 0.4 webs foams

0.2 0

0

0.2

0.4 0.6 0.8 Relative density α

1

Figure 6.13 Shape efficiency of sandwich plates with cores of different types.

well with the theoretical estimate for two-way webs: this type of core is much more efficient than the other types. The above analyses are valid for relatively small values of the relative density α. If α is increased, it might be expected that load-carrying modes that were neglected in the analysis will tend to become gradually more significant, thus leading to μ increasing at a faster rate. This is the case for all types of core other than foams (Gibson and Ashby, 1982). The expected variation of μ over the full range of α has been plotted in Fig. 6.13. In conclusion, the different morphologies of sandwich core that have been discussed in this section are summarized in Table 6.3. Within each type there is, of course, considerable freedom and, also, different types can be combined together, as will be seen in Sections 6.5 and 6.6.

6.4 Foam Core

173

Figure 6.14 Metal sandwich panel with polyurethane foam core (image used with permission of The Dow Chemical Company).

6.4

Foam Core Structural foams are a commonly chosen core material. For example, Fig. 6.14 shows a sandwich panel for a roofing system, with skins made of metal, and a polyurethane foam core. In fact, foams can be made out of almost anything: metals, plastic, ceramics, and even glass. Foaming techniques vary from the introduction of gas bubbles into hot polymers to the mixing of organic beads into liquid metals, which are burnt off during cooling; a review is provided in Gibson and Ashby (1997). Figure 6.15 shows the cell structure and size of different foams: the first three are made from polymers, the next two from ceramics, and the last one from glass. The difference between open and closed cells is clearly seen in Fig. 6.15(a and b). In an opencell foam the solid material is mainly concentrated into struts which form the edges of the cells. The struts join at vertices, usually four struts at each vertex. In a closed-cell foam, the cells have thin walls that become thicker from the center to the edges. Let us look more closely at the last of these microstructures, Fig. 6.15(h). It closely resembles the space filling tessellation of truncated octahedra, in Fig. A.6(e). The truncated octahedron was first proposed by Lord Kelvin (Thomson, 1887), to describe the space-filling pattern of soap bubbles. Kelvin observed that the bubbles take the shape of truncated octahedra, and pack with four edges meeting at each vertex, and three faces meeting at each edge. Each bubble is in equilibrium under internal pressure and surface tension, a situation that closely resembles the forming process of many foams. This explains why the structure of most foams is well described, in an average sense, as a tessellation of truncated octahedra. Although in practice the shape and size of the cells varies somewhat, it is found that the average number of faces is about 14. Up-to-date reviews of the state of the art in the shape of foams can be found in Weaire and Hutzler (1999) and Chapter 2 of Gibson and Ashby (1997). In most foams the cells are either open or with very thin walls, because during foaming the surface tension draws the material towards the edges of the cells. Hence, to develop a simple model for the properties of a foam we can idealize the foam as a space frame consisting of beams of length L and uniform, square cross-section t × t, arranged along the edges of a tessellation of tetrakaidecahedrons. This model is shown in Fig. 6.16(a) and will be used to estimate the efficiency of a foam. First, we estimate α and G∗c . The volume enclosed within a tetrakaidecahedron is given in Table A.4 V = 11.31L3

174

Sandwich Structures

Figure 6.15 Three-dimensional cellular materials (a) open-cell polyurethane, (b) closed-cell polyehtylene, (c) nickel, (d) copper, (e) zirconia, (f) mullite, (g) glass, (h) polyether foam with both open and closed cells (Gibson and Ashby, 1997).

6.4 Foam Core

175

while the volume of material in the beams is given by the volume of one beam, multiplied by the number of edges of the tetrakaidecahedron, from Table A.4, divided by three since each beam is shared by three cells, that is V∗ =

36Lt 2 3

Hence, the relative density α can be computed as follows: α=

 2 t V∗ 12Lt 2 ρ∗ = 1.06 = = ρ V L 11.31L3

In conclusion, α is proportional to the square of t/L: α∝

 2 t 

(6.29)

In order to estimate the homogenized shear modulus of the core, G∗c , we need to identify the main load-carrying mode of an assembly of cells of the type shown in Fig. 6.16(a). If the assembly is idealized as a pin-jointed truss, Maxwell’s rule shows that this structure would have many mechanisms. For example, consider a single cell: the number of joints is j = 24 and the number of bars is b = 36. Hence, substituting

L

4xF t a b

c d

e f

4xF

(b) (a) L

βF βFL 3 12EI

βF (c) Figure 6.16 Geometry and deflection of truncated octahedron core.

176

Sandwich Structures

these values into Equation 1.2, we obtain – for d = 3 and k = 0 (since there are no constraints) 3 × 24 − 36 − 0 = m − s Hence this pin-jointed structure has 36 independent mechanisms at least, i.e., if we assume that s = 0, of which 30 are internal mechanisms. With so many mechanisms, it is unlikely that a general load condition would not excite any one of them, and hence we conclude that the cell is unable to carry loads in a purely extensional mode. Therefore, we abandon the pin-jointed truss model and consider a model with stiff connections between the beams. Now, the main effect to be considered is the bending deformation of the cells. Consider the unit cell shown in Fig. 6.16(b), loaded by four equal and opposite forces at the top and the bottom joints. Due to symmetry, only one quarter of the structure needs to be analyzed and, if joint rotations are assumed to be zero because adjacent cells are equally loaded, the extension δ of the cell can be obtained by superposing the bending deflections of the four beams a–f on the load path between each pair of equal and opposite forces. Each of these beams deforms according to the simple shear-bending mode shown in Fig. 6.16(b), and hence the total cell extension δ has an expression of the type: −δ ≈

4  i=1

βi

4 F L3 F L3  βi = 12Ec I 12Ec I

(6.30)

i=1

where Ec is the Young’s modulus of the core material and I is the second moment of area of a beam. The coefficients βi depend on the orientation of the beams and the number of beams carrying the force F . Equation 6.31 can be extended to different directions of the load and to a larger number of cells. The general expression for loads and deflections in any direction of an assembly of cells of any size is of the type: δ∝

F L3 Ec I

(6.31)

Therefore, an expression for the effective shear modulus is obtained by dividing the shear stress, i.e. the ratio between the shear force over the area, by the shear strain, i.e. the ratio between the deflection δ and the beam length. We substitute Eq. 6.31, tidy up, make the substitution I ∝ t 4 , and finally substitute Eq. 6.29 to obtain  4 F /L2 Ec I F /L2 t ∗ ∝ 4 ∝ Ec ∝ Ec α 2 ∝ Gc ∝ 3 1 F L δ/L L L Ec I L

The final result is G∗c =

3 2 α Ec 8

(6.32)

where the constant of proportionality has been determined experimentally, see Fig. 6.17.

6.5 Honeycomb Core

177

1

Relative shear modulus G*/Ec

◊ Moore et al. (1974) PPC Gibson and Ashby (1982) PU (R) o Gibson and Ashby (1982) PU (F)

10-1

10-2

10-3 G* = 3 α2 Ec 8

10-4

10-5 10-3

10-2 10-1 Relative density α

1

Figure 6.17 Relative shear modulus vs. relative density for open cell foams (Gibson and Ashby,

1997).

It is useful to express the Young’s modulus Ec in terms of the shear modulus. From Eq. B.35: 8 Gc 3

(6.33)

G∗c = α 2 Gc

(6.34)

Ec = and, substituting into Eq. 6.32,

which is the expression that was used in Section 6.3 to compute the efficiency of the foam core.

6.5

Honeycomb Core A honeycomb is an array of prismatic hexagonal cells forming a regular tessellation, see Appendix A.2 and Fig. A.5(a). A wide range of materials and core configurations are used for honeycombs. Plastic honeycombs are made from a variety of thermo-forming processes – an example involving bonded tubes is shown in Fig. 6.18 – but are rarely used for structural applications. Structural honeycombs are made from Al-alloy, Nomex, Kevlar, carbon-fiber composites. They have high stiffness and strength when loaded out of plane, both with and without the stabilizing skins. In a sandwich panel this feature is

178

Sandwich Structures

Figure 6.18 Polycarbonate honeycomb (image reproduced with permission of Plascore, Inc.).

Un-expanded block W

T

T L

Figure 6.19 Expansion process of honeycomb manufacture (HEXCEL, 2018).

important in resisting concentrated transverse loads. Also important in sandwich panels is the out-of-plane shear stiffness and strength of honeycomb, which becomes available only after the honeycomb has been stabilized by the skins. A commonly used expansion method for manufacturing a honeycomb is shown in Fig. 6.19. Sheets of the cell material, on which adhesive lines have been printed, are stacked together and the adhesive is cured to form a block, which is then sliced to the required thickness T and finally expanded. Note that the wall thickness is doubled along the seams and also the cells are not perfectly hexagonal, Fig. 6.20; these effects are sources of significant anisotropy. Denoting by V the overall volume of a hexagonal cell, by V ∗ the actual volume of material within the cell (taking into account the cell side walls that are shared with neighboring cells), and by Vp the volume of material in a single side wall α=

2Vp + 4(Vp /2) 4Vp V∗ ρ∗ = = = ρ V V V

(6.35)

6.5 Honeycomb Core

Overexpanded cell

Hexagonal cell T

179

T W direction

L direction

L direction

Figure 6.20 Honeycomb is not always perfectly hexagonal (HEXCEL, 2018).

b a

c

γ

φ W

c

a

γ

b

L Figure 6.21 Unit cell of honeycomb under out-of-plane shear.

To estimate the homogenized shear stiffness of the cell, consider an out-of-plane shear strain γ , in the plane defined by the arbitrary direction φ shown in Fig. 6.21, and a direction orthogonal to the plane of W and L. The shear strains in the side walls of the cell are as follows:

π  γa = γ cos +φ 3 γb = γ cos φ

π  γc = γ cos −φ 3

(6.36)

The strain energy in the side walls is equal to the strain energy in an equivalent continuum of shear modulus G∗c 1 ∗ 2 1  2 (γi Vi ) Gc γ V = Gc 2 2

(6.37)

i

where Vi is the volume of side wall i. Since there are two parallel side walls in the cell and each wall is shared between two cells, the summation in Eq. 6.37 can be extended to three walls only, provided that their full volume is considered. We substitute Eq. 6.36 into Eq. 6.37, let Vi = 2Vp for plates b, and simplify to find:  & V %

π

π p + φ + 2 cos2 φ + cos2 −φ G∗c = Gc cos2 3 3 V

(6.38)

180

Sandwich Structures

Then, noting that, cos2

  3

π + φ + 2 cos2 φ + cos2 − φ = + cos2 φ 3 3 2

π

Eq. 6.38 becomes G∗c

 =



Vp Gc V

(6.39)

 3 1 2 + cos φ αGc 8 4

(6.40)

3 + cos2 φ 2

and finally substituting Eq. 6.35 for Vp /V gives G∗c

 =

This expression for the homogenized shear stiffness of honeycomb core can be used to find its shape efficiency. Comparing Eq. 6.40 with Eq. 6.22 leads to: μ=

3 1 + cos2 φ 8 4

(6.41)

This expression has been plotted in Fig. 6.22, which shows that there is a considerable anisotropy in honeycomb. The shape efficiency varies between the two extreme values μL = 0.625, for φ = 0, and μW = 0.375 , for φ = 90◦ , where the subscripts denote the directions L and W , defined in Fig. 6.21. It is interesting to note that the average efficiency is μ = 0.5, corresponding to φ = 45◦ . A comparison of the two extreme values, μL and μW , with experimental results for Al-alloy honeycomb provided in a Hexcel datasheet (HEXCEL, 2018) is presented in Fig. 6.23.

Figure 6.22 Polar plot of shape efficiency of honeycomb vs. direction.

6.6 Corrugated Core and Zeta-Core

181

Shape efficiency μ

1 0.8 μL 0.6 0.4

μW

0.2 0

0

2

8 4 6 Relative density α

10

Figure 6.23 Shape efficiency of Al-alloy honeycomb core vs. density.

6.6

Corrugated Core and Zeta-Core This section deals with sandwich cores that can be made from a single sheet of thin material. We begin from a widely used, low-cost type of sandwich panel, cardboard. Then, we use cardboard as inspiration for an alternative type of core that, unlike cardboard, behaves isotropically, and which is a strong competitor to standard honeycomb.

6.6.1

Single-Corrugation Core Single-corrugation sheets are developable surfaces obtained by creasing a flat sheet along a series of parallel fold lines. This is a simple and low-cost process that is used, for example, to make steel floor deck plates, where the creases are localized, and cardboard, where the creases are distributed to form a smoothly curved cross-section. Examples of single-corrugation sheets are shown in Fig. 6.24. Consider a sandwich panel with a core consisting of a single-corrugation sheet with a zig-zagging cross-section forming a series of isosceles triangles with sides inclined at an angle θ to the face plates, Fig. 6.25(a). This core is an extreme version, with sharp creases, of a cardboard-type core. This type of core is clearly anisotropic, as its geometry is so different in the longitudinal direction, L, i.e. the direction parallel to the creases, from the transverse direction, T . We will quantify the anisotropy of this core type by evaluating the shape efficiency in the directions L and T . Denoting by V the overall volume of a repeating cell of the core, defined as the rectangular prism containing a single inclined web element and having unit length in the L-direction, as shown shaded in Fig. 6.25(b), we have V =h

h ×1 tan θ

(6.42)

182

Sandwich Structures

Figure 6.24 Examples of single-corrugation core in single-wall and double-wall cardboard (Olivier Le Moal/Shutterstock.com).

θ

T L

h (a)

1 γL (b) Figure 6.25 (a) Sandwich panel with single-corrugation core, the webs are inclined at an angle θ; (b) definition of unit cell (shaded) and imposed strain γL for calculation of G∗L .

whereas the volume of web material in the unit cell is V∗ = t

h ×1 sin θ

(6.43)

Therefore, the relative density of the core is α=

ρ∗ V∗ t = = ρ V h cos θ

(6.44)

6.6 Corrugated Core and Zeta-Core

183

To estimate the homogenized shear stiffness of the core in the L-direction we impose an out-of-plane shear strain γL , as shown in Fig. 6.25(b). The shear strain in each web is γ = γL sin θ In analogy with Section 6.5, the strain energy balance is 1 ∗ 2 1 GL γL V = Gc γ 2 V ∗ 2 2 and, substituting the above expression for γ and simplifying V∗ V Then, the shape efficiency in the L-direction is calculated as G∗L = Gc sin2 θ

μL =

G∗L = sin2 θ Gc α

(6.45)

(6.46)

where it should be noted that μL is independent of α. In practice, the largest feasible inclination of the web is θ = 60◦ and hence the maximum shape efficiency in the longitudinal direction is μL = 0.75. However, some narrow strips of core material parallel to the face plates are required to bond the core to the skin; these strips have been neglected in the above analysis. Hence, in practice about 20% of the core material is not available to carry shear and a corresponding reduction of μL can be expected. In conclusion, the estimated shape efficiency in the L-direction is: μL ≈ 0.6

(6.47)

A more detailed analysis of a continuously curved web (Libove and Hubka, 1951), of the type used in cardboard, gives the result μL = 0.608, which is in good agreement with our simple analysis. Regarding the shape efficiency in the T -direction, the calculation is more complex because the webs are smoothly curved, instead of forming a triangulated cross section. They are loaded mostly out-of-plane and hence bending is the dominant load-carrying mode. A detailed analysis was carried out by Libove and Hubka (1951), and their analysis shows that for θ = 60◦ μT ≈ 18α 2

(6.48)

The two values of μ have been plotted in Fig. 6.26, which confirms the strong anisotropy of this type of core.

6.6.2

Zeta-Core Despite the attractiveness of single-sheet core fabrication, the single-corrugation core is not satisfactory for the many applications in which isotropic stiffness and strength are required. The key reason for this deficiency is the lack of stabilizing elements in the T -direction, which results in an inefficient load-carrying mode in this direction. A simple, although rather hypothetical, way of solving this problem is to consider a core made from two single-corrugation sheets in orthogonal directions, as shown in Fig. 6.27(a).

Sandwich Structures

S

184

R Figure 6.26 Shape efficiency of single-corrugation core with θ = 60◦ .

θ T

L

T (a)

L (b)

Figure 6.27 (a) Hypothetical double-corrugation core and (b) concept of Zeta-Core. The face plates are shown dotted for clarity.

Since it is assumed that the two sheets are fully connected along each intersection line, according to the previous argument, this core will have an effective shear modulus of at least the value given by Eq. 6.45 in all directions and hence, since this hypothetical core has twice the amount of material as the original single-sheet core, its shape efficiency can be expected to be about the half the value in Eq. 6.46, and hence μ∼

1 2 sin θ 2

In fact, this double surface is able to carry shear loads through in-plane action in both of its constituent singly-corrugated surfaces, which results in an additional shear stiffness contribution. It will be shown later in this section that μ=

1 2 3 sin θ + sin2 2θ 2 8

(6.49)

6.6 Corrugated Core and Zeta-Core

185

From this expression it can be shown that μ≥

1 2

for 35.3◦ < θ < 90◦

and the maximum value is μ = 2/3, for webs inclined at an angle 1 θ = cos−1 √ = 54.7◦ 3 Hence, it has been shown that this hypothetical type of core is potentially more efficient than honeycomb. A practical solution is to remove half of the facets of the double-surface core in a way that produces a different kind of single-sheet core. The solution proposed by Miura (1972) is isotropic and is called Zeta-Core. It is shown in Fig. 6.27(b). Note that half of the Zeta-Core is parallel to the L-direction, identifying elements of the core that originally belonged to the first single-corrugation sheet. The other half is parallel to the T -direction, and identifies the elements of the core that originally belonged to the second single-corrugation sheet. Also note that it is possible to combine two Zeta-Core sheets shifted by the pitch of the corrugation, and re-obtain the original double-corrugation surface of Fig. 6.27(a). Figure 6.28 shows a modified Zeta-Core where flats have been added along the ridges and valleys for ease of fabrication, to allow better bonding to the face plates of the sandwich. In this case only a percentage, β, of the material is used to carry the shear action in the core. It should be noted that, whereas the original Zeta-Core (with β = 1) is developable, this modified Zeta-Core (with β < 1) is not developable and hence stretching deformation has to be imposed during manufacturing of this core, in addition to bending of the sheet. A review of manufacturing techniques for metal corrugated sheets has been compiled by Schenk (2011). In the following analysis we will consider the case β = 1.0, corresponding to the case in which there are no flats. We will evaluate the shape efficiency of the Zeta-Core and also show that the isotropy of the double-corrugation core in Fig. 6.27 is maintained in this new configuration.

T L Figure 6.28 Perspective view of Zeta-Core with flats to allow bonding to the core (Miura, 1975).

186

Sandwich Structures

γ γ

c

1

c 2

1

d

d

2

b h

θ

90o

a

T

z L

90o

T

z

b

a L (b)

(a)

Figure 6.29 (a) Perspective view of Zeta-Core unit, showing directions 1, 2 for plate b; (b) top view.

The repeating cell, shown in Fig. 6.29, consists of four identical 45◦ parallelograms a, b, c, and d, all inclined at an angle θ to the face plates. It is enclosed in a prism with base given by the horizontal projection shown in Fig. 6.29(b) and height h defined in Fig. 6.29(a). Denoting by V the overall volume of this cell and by Vp the volume of material in a single inclined plate, the relative density is α=

4Vp V∗ ρ∗ = = ρ V V

(6.50)

To determine the homogenized shear stiffness of this unit cell, consider a shear strain γ in the plane defined by the directions T and z. This deformation corresponds, in analogy with Section 6.6.1, to a shear strain γ sin θ in the plates denoted as a and d in Fig. 6.29(a). Hence, the strain energy in each of these plates is 1 (6.51) Gc γ 2 sin2 θ Vp 2 Plates b and c are subject to an in-plane normal strain in the 1-direction, defined in Fig. 6.29(b), 1 γ sin 2θ (6.52) 2 Note that the normal strain in the 2-direction in plates b and c is zero due to the edge constraint applied by plate a and by the corresponding plate in an adjacent unit. Hence, the strain energy in plates b and c is obtained by computing the stress, σ1 , corresponding to the strains 1 from Eq. 6.51 and 2 = 0. Equation B.37 gives 1 = γ sin θ cos θ =

σ1 =

Ec 1 1 − ν2

(6.53)

For ν = 1/3, and using the approximation in Eq. B.35, Eq. 6.53 gives: σ1 ≈ 3Gc 1

(6.54)

6.6 Corrugated Core and Zeta-Core

187

Hence, the strain energy in each of plates b and c can be calculated as follows, using 6.54 and then Eq. 6.52: 1 3 3 σ1 1 Vp = Gc 12 Vp = Gc γ 2 (sin2 2θ )Vp 2 2 8

(6.55)

The strain energy balance that defines the homogenized shear modulus in the T -direction G∗T is as follows 1 ∗ 2 1 3 G γ V = Gc γ 2 (sin2 θ)2Vp + Gc γ 2 (sin2 2θ )2Vp 2 T 2 8 and simplifying and tidying up G∗T



3 = sin θ + sin2 2θ 4 2



2Vp Gc V

(6.56)

(6.57)

Then the shape efficiency in the T -direction can be calculated, by substituting Eq. 6.50 for α, μT =

G∗c 3 1 = sin2 θ + sin2 2θ Gc α 2 8

(6.58)

The same analysis could be repeated for a shear strain acting in the L-direction, to find exactly the same result. Thus it is concluded that 1 2 3 sin θ + sin2 2θ 2 8

μL = μT =

(6.59)

and in fact, it can also be shown that the same result holds for any direction, leading to the conclusion that the Zeta-Core is isotropic.

1

Shape efficiency μ

0.8 0.666

β = 1.0

0.6 0.533 β = 0.8

0.4 0.2

54.7o 0

0

20

40

60

Inclination angle θ (deg) Figure 6.30 Shape efficiency of Zeta-Core.

80

188

Sandwich Structures

Figure 6.31 Aluminum-alloy Zeta-Core sandwich plate.

Since the Zeta-Core contains half the plate elements of the double-corrugation core of Fig. 6.27, it has exactly half the relative density and also half the shear stiffness of the double-corrugation core. Hence it follows that the shape efficiency μ of the Zeta-Core and the double-corrugation surface are identical, which proves the validity of Eq. 6.49. Figure 6.30 is a plot of μ vs θ for zero-width ridge and valley folds, i.e. β = 1.0, and also for folds of finite width, β = 0.8 to allow for 20% of the core material to be bonded to the skin of the sandwich. The expression for μ in the latter case can be derived by extending the approach above. The plots show that, as already noted earlier, the maximum value of μ occurs for θ ≈ 55◦ . Preliminary tests on a series of sandwich panels, Fig. 6.31, have produced μ ≈ 0.4 (Miura, 1975).

7

Packaging of Membranes

7.1

Introduction Membranes are a type of thin, surface-like structure that can be folded tightly due to its small (theoretically zero) bending stiffness. A simple, unstructured way of folding a membrane is to form random creases by crumpling the membrane as one pushes it into a container. However, this folding method does not allow us to precisely answer questions such as: how densely can the membrane be packed, or what is the maximum force required to pull it out of the container, or even how much residual deformation will remain in the membrane after unpacking it? A more predictable approach is to adopt geometrically defined packaging schemes. Although some of the fine details are beyond the scope of this chapter, significant progress can be made by investigating fold patterns that minimize the volume of voids left in the package and also allow smooth and regular unfolding. This last requirement is particularly important in applications where unpacking is to be carried out automatically. An important parameter is the packing efficiency, defined as the ratio between actual volume of material and the volume of the package, including any voids. Generally, it is hard to achieve a packing efficiency better than 50%. Nature is a source of inspiration for studies of packaging of thin membranes. For example, flowers grow almost completely, and quite slowly, in the bud, and then blossom quickly, Fig. 7.1. This particular example resembles a wrapping scheme for flat membranes that will be presented in Section 7.5, although the flower grows into its “packaged” shape, without the use of a folding machine. Examples of packaging occurring in natural membranes have been described by Kresling (1993). It should also be noted that engineered structures are subject to specific performance requirements that often result in complex shapes requiring a different folding scheme for each part. An example is the inflatable antenna structure shown in Fig. 7.2 and described in more detail in Section 9.6. In this experiment the three legs of the reflectors were folded separately from the canopy. This chapter presents the fundamental geometric concepts of membrane folding, leading to two different types of solution to the problem of packaging a membrane using periodic distributions of fold lines. The solutions presented are for flat membranes, which are the most commonly used type of man-made membrane structures. However, it should also be noted that the overall shape of a membrane structure consisting of flat pieces is not restricted to being flat. For example, the curved surface in Fig. 7.3(a) 189

190

Packaging of Membranes

Figure 7.1 The moonflower plant puts out numerous new flowers every evening at dusk. The

flowers take an hour or two to open, and they drop off in the morning. A viscous sap-like liquid between the spiraled petals controls the rate of the deployment process. On cool evenings the opening may take two or three hours while on a hot evening the deployment may occur in half of an hour. The flowers are up to 15 cm in diameter (Images reproduced with permission of Martin Mikulas, 2019).

Figure 7.2 Unfolding in space of an inflatable membrane reflector (Freeland et al., 1997).

7.2 Geometric Conditions

191

Figure 7.3 Removal of gaussian curvature in a polygonal curved surface (shown in a.i and a.ii) by

prefolding the surface along straight lines.

consists of flat pieces with a curved edge profile; because the gaussian curvature of these surfaces is localized along the seam lines, it can be “hidden” by introducing a set of pre-folds as shown in Fig. 7.3(b) and can then be packaged using a scheme for flat membranes.

7.2

Geometric Conditions Because of their small bending stiffness, membranes are usually modeled as inextensional plates of zero thickness in folding studies. This model does not allow the membrane to wrinkle when it is folded, although wrinkling could itself be a feature of a packaging scheme; it actually occurs in flowers and can be seen in Fig. 7.1. The problem with allowing wrinkling is that it is difficult to overlay precisely wrinkles in different parts of a membrane to avoid leaving large gaps. A condition on the deformation of an inextensional plate can be obtained by considering its gaussian curvature, an intrinsic property of surfaces introduced in Section A.4.4. Calladine (1983) has shown that the in-plane deformation of a flat plate can be linked to its change of gaussian curvature through the equation: dK = −

∂ 2 y ∂ 2 γxy ∂ 2 x + − ∂x∂y ∂x 2 ∂y 2

(7.1)

Here, x ,y ,γxy are the extensional and shear strains referred to a cartesian (x,y) coordinate system in the plane of the sheet.

192

Packaging of Membranes

Figure 7.4 Two developable surfaces S1 and S2 produced by folding an inextensible sheet along

the curve C (Duncan and Duncan, 1982).

For the deformation to be inextensional x = y = γxy = 0 and hence dK = 0 and therefore an initially flat membrane, whose gaussian curvature is zero everywhere, can only be folded to deform in such a way that its gaussian curvature remains zero everywhere. The next thing to note is that packaging schemes for flat membranes make use of straight, localized folds. This is because a curved fold would divide the membrane into two singly curved, concave, and convex parts, as shown in Fig. 7.4, that cannot be folded back to back to achieve gap-free packaging, whereas a distributed fold of width π r would form a cylindrical surface of radius r and hence would lead to a separation of 2r between the two parts of the membrane. It follows from these considerations that all folding patterns for flat membranes consist of localized, straight fold lines (sharp creases) that satisfy some general conditions at every point of intersection of two or more creases. These conditions will be derived next. It is shown in Section A.4.4 that a given amount of gaussian curvature is geometrically represented by an area over a sphere of unit radius, and hence zero gaussian curvature is associated with a zero area. For the case considered here, this means that the area on the unit sphere associated with any change of configuration of the membrane during packaging should be zero. This observation has several important practical implications. Consider the flat membrane shown in Fig. 7.5(a); it is divided into two parts by a single, straight fold. The image on the unit sphere of the patch drawn on the membrane is the single point labeled 1, 2, since the rule of mapping is that the radius from the center to the point on the sphere is parallel to the normals n1 , n2 . Hence the area on the unit sphere associated with this patch is A = K = 0. Next, rotate down part 2 of the membrane, through 90◦ . The spherical image of part 1 is unchanged, whereas the image of part 2 moves along an arc of a great circle, as shown

7.2 Geometric Conditions

193

1,2 n1

n2

K=0

2

1

(a)

n1

1 n2

ΔK = 0

1

2

2 (b) Figure 7.5 Flat membrane with a single fold and its spherical image, (a) before and (b) after

folding.

in Fig. 7.5(b). The change of gaussian curvature of the membrane during this operation is zero, because the area of a circular arc is zero. Next, consider the membrane shown in Fig. 7.6(a), which is divided into three parts by three straight creases that meet at a common point. The spherical image of any chosen patch on the membrane is, again, a single point, but this time if part 2 of the membrane is folded down through 90◦ without moving part 1 – this motion fully defines the direction of the normal vector n3 – the spherical image of the chosen patch has non-zero area (i.e., the area of a spherical triangle). This implies that the gaussian curvature has to increase, i.e., if parts 1 and 2 do not deform, part 3 of the membrane has to shear. As an extension of the previous case, consider the membrane shown in Fig. 7.7(a), divided into four parts by four straight folds meeting at a common point. Its spherical image is a spherical quadrangle and so, in general, the gaussian curvature has to change. However, the zero area-change condition can be satisfied by forming three hill, or convex folds and one valley, or concave fold, as shown in Fig. 7.7(b). As in the previous cases, rotate part 2 down by rotating the first crease through 90◦ . It is now possible to determine the magnitude by which the crease between parts 3 and 4 should be rotated so that all the parts fit together, as shown in Fig. 7.7(b). Next, consider the spherical image of the patch marked on the membrane: it is a skew quadrangle whose edges are arcs of great circles. In calculating its area one has to be careful about signs, because to go from point 1 to point 2 and then to 3 on the sphere the curve is traced in an anticlockwise sense, but to go from point 3 to point 4 and then to 1 it is traced in the opposite sense. The area enclosed by the curve 1,2,3,4 on the surface is given by: K = A1 + A2

194

Packaging of Membranes

1,2,3 n1 n2

n3 K =0

3

1 2

(a) n1 1

n3

3

1

ΔK > 0

3 2

n2

2

(b) Figure 7.6 Flat membrane with three folds and its spherical image, (a) before and (b) after folding.

n1

n4 n2

1,2,3,4 n3

K=0

4

1 3

2

(a) n1 n3

1

2

3 4 n4

n2

ΔK = 0

1 A1

A2 2

4

3

(b) Figure 7.7 Flat membrane with four folds and its spherical image, (a) before and (b) after folding.

but A1 = −A2 (Miura, 1989) and hence K = 0. Of course, it is possible to reverse the directions of all folds, thus forming three valley folds and one hill fold; the same argument holds in this case.

7.2 Geometric Conditions

195

–

2

3 a3

a2

+

a1

O

a4-a1 = a3-a2

+

a1

a4

1

4 +

1

Convex (hill) Concave (valley)

(a)

(b)

Figure 7.8 Flat folding.

In conclusion, inextensional folding of a membrane requires that whenever different creases meet at a point there should be at least four folds, of which three have one sign, and one fold has the opposite sign. Having explored the conditions that are imposed by inextensionality on the design of the folding pattern, an additional condition is imposed by the requirement that the membrane should be packed without leaving any voids; this condition is known as flat foldability (see Tachi, 2009 for several references on this topic). For example, in the case of four folds meeting at a point O, Fig. 7.8, to avoid any voids in the packaged configuration the difference between two adjacent angles has to be equal to the difference between the remaining two angles α4 − α1 = α3 − α2 Rearranging, we obtain α1 + α3 = α2 + α4

(7.2)

In origami, the ancient art of paper folding, the condition expressed by Eq. 7.2 is  known as Husimi’s theorem (Husimi and Husimi, 1979). Since αi = 360◦ Eq. 7.2 becomes α1 + α3 = α2 + α4 = 180◦

(7.3)

this is a key condition for the design of folding patterns that provide flat, i.e. void-free, packaging.

7.2.1

Thickness Effects So far, it has been assumed that the thickness of the membrane is zero. This is generally acceptable in preliminary studies of the folding pattern, but then the analysis needs to be refined by considering two different effects arising from the actual thickness of the membrane.

196

Packaging of Membranes

Figure 7.9 Effects of membrane thickness on folding (Miura, 1978).

In packaging schemes based on fold patterns that are translationally periodic in two perpendicular directions, e.g., the Miura-ori scheme in Section 7.4, thickness effects are relatively small because the folds are not tightly nested and hence thickness effects are not compounded as the number of repeating units is increased. However, in the case of packaging schemes that are not periodic in two directions, e.g., the wrapping scheme in Section 7.5, thickness effects are more significant because they are compounded and hence the folding pattern obtained by assuming a membrane of zero thickness is incorrect. Another important issue associated with thickness is illustrated in Fig. 7.9. Due to the thickness of the membrane, the fold regions are subject to highly localized bending strains, but not uniformly so, because the curvature is highest along a fold that is folded inside other folds. Also, very high shear strains occur at the point of intersection of four (or more) folds, due to the existence of biaxial strain states associated with the reverse bending near such points. In the case of tightly packaged membranes these high strain levels are the source of plastic deformation, resulting in a permanent residual creasing of the membrane after deployment. For example, Papa and Pellegrino (2008) measured a residual creasing of 27.6◦ in a 25 μm thick aluminized Kapton sheet.

7.3

Unfolding of Tree Leaves A tree leaf is an example of a natural membrane that grows tightly packaged in the bud, and quickly deploys in the spring season. Kobayashi, Kresling and Vincent (1998) did a detailed study of the unfolding of hornbeam leaves, shown in Fig. 7.10. They observed that the creases in hornbeam leaves are along the veins, which are approximately parallel and at an angle α ≈ 40◦ to the mid-rib, which divides the leaf into two symmetric parts. Thus, they investigated the possibility of a link between the angle of the veins and the

7.3 Unfolding of Tree Leaves

(a)

(b)

197

(c)

(d)

Figure 7.10 Unfolding of hornbeam leaves (Kobayashi et al., 1998).

unfolding mechanism of the leaves; their conjecture was that the value of this angle might be optimized such that the leaf area exposed to sunlight would be as large as possible during the early stages of unfolding. The unfolding of a rectangular leaf model consisting of two interconnected corrugation patterns based on parallel hill and valley folds is shown in Fig. 7.11(a). The angle β, defined in Fig. 7.11(b), between the inner edge of the leaf and the central plane of symmetry is the single parameter that controls the unfolding process. More details on folding schemes based on corrugation patterns are provided in Section 7.4. Initially (β ≈ 0) the leaf expands mainly in the transverse direction but then, as β increases, expansion in the longitudinal direction begins. Therefore the shaded area, representing the area of leaf exposed to the sun, increases slowly at first and later at a faster rate. Kobayashi et al. computed the relationship between area exposed and the angle β, for different values of the vein angle α. Note that for α < 30◦ the packaging becomes very inefficient, and hence such values were not considered. The results of their computations are shown in Fig. 7.12: this plot shows that leaf models with α = 30◦ to 45◦ expose over 50% of the leaf area to the sunlight once β > 35◦ .

198

Packaging of Membranes

x’’ z x x’

α = 30°, 45°, 60°, 75°, 85° 30° z

20 mm

30°

α

x

β

z L = 108 mm

β

Hill fold Valley fold

(a)

x β

β

(b)

β

y

Figure 7.11 Rectangular leaf model: (a) crease geometry and (b) unfolding behavior (Kobayashi et al., 1998).

1.0

Leaf area ratio A*

0.8 30°

0.6

45°

60°

0.4

75° 85°

0.2 0.0 0

20

40 60 Opening angle b (deg)

80

Figure 7.12 Leaf area ratios for different values of α.

7.4

Biaxial Folding: Miura-Ori There are several ways of folding a membrane into a small, flat package. In the well known scheme called booklet folding, a rectangular sheet is folded about its center line, the folded sheet is then folded about its own center line, and the same operation is repeated, say, five times. By folding a sheet in this way, we set up the crease pattern

7.4 Biaxial Folding: Miura-Ori

199

(a)

(b)

Figure 7.13 Biaxial folding schemes, (a) booklet folding and (b) map folding.

shown in Fig. 7.13(a). Note that: (i) the three-to-one condition on the sign of the folds meeting at the 21 points of intersection of the folds, Section 7.2, is satisfied, and (ii) the arrangement of hill and valley folds appears fairly random. The reason for calling this scheme booklet folding is because if we trim three edges of the folded sheet, the fourth edge forms the spine of a 32-page booklet. This is a traditional bookbinding technique: the 32 pages of the booklet are printed on both sides of the original sheet, in the correct orientations, and then the sheet is creased, folded and trimmed as described. The unfolding of a sheet packaged according to the booklet folding technique is a complex process, which would be difficult to carry out automatically. The unfolding process involves a multistep sequence where the motion of different sets of folds has to start and stop, which would be difficult to implement through a global actuation scheme. An alternative to this packaging is the map folding scheme, in which one first z-folds the sheet, by forming a series of parallel, alternating hill and valley folds. This produces a strip with a width of one-eighth the original width, its full height, and eight times the thickness of the sheet. The strip is then z-folded in the perpendicular direction, setting up the fold pattern shown in Fig. 7.13(b). Note that the map folding scheme also satisfies the three-to-one sign rule for all sets of folds meeting at any point; the difference from

200

Packaging of Membranes

Figure 7.14 Developable double corrugation surface known as Miura-ori.

the booklet folding scheme is that the distribution of the folds repeats periodically, both horizontally and vertically. Unfolding a sheet packaged according to the map folding scheme is much simpler, as the sheet can be deployed completely by pulling apart the two corners of the sheet that are right at the top and at the bottom of the package. As these two corners are pulled apart, the sheet deploys first in one direction and then, rather suddenly, in the other direction. The deployment process mirrors closely the two-step folding process. A fundamentally different solution to the problem of biaxial packaging of membranes is based on the polyhedral surface shown in Fig. 7.14, first described as a developable double corrugation surface and widely known as Miura-ori. Depending on the reader’s viewpoint, this surface is the simplest origami pattern, a symmetry-rich tessellation, an infinite concave polyhedron, a wrinkle pattern, a deployable structure, or a piece of art. Its invention, described in Miura (2009), began with the observation that the Yoshimura buckling pattern for a cylindrical shell, see Fig. 5.29(b), is an inextensional deformation of the original surface. Later on, Miura (1970) put forward the hypothesis that this periodic concave polyhedral surface is the extreme outcome of the buckling of a vanishingly thin, elastic, flat plate subject to biaxial compression. This hypothesis was later confirmed computationally by Tanizawa and Miura (1978) and experimentally by Mahadevan and Rica (2005). Figure 7.15 shows a series of snapshots from a numerical simulation of

7.4 Biaxial Folding: Miura-Ori

(a)

(b)

(c)

(d)

201

Figure 7.15 Snapshots from numerical simulation of the periodic buckling and postbuckling of a thin plate (Tanizawa and Miura, 1978).

the buckling and postbuckling of such a thin plate, from Tanizawa and Miura (1978). Figure 7.15(a) is the initial buckling mode, assumed to be periodic. The numbers of the contour lines denote the largest downward (contour 1) and upward (contour 9) deflection. The next three figures show the evolution of the out-of-plane deflection of the plate as both edges are compressed by equal amounts. The total strain energy density in the plate increases from (a) to (b) and then decreases from (b) to (d). Figure 7.15(d) shows a low strain energy, fully developed displacement patterns that matches the repeating unit of Miura-ori. Miura-ori satisfies Husimi’s theorem, Eq. 7.2, at each of its vertices, and hence leads to a flat package. The continuous, inextensional folding of Miura-ori is shown in

202

Packaging of Membranes

Figure 7.16 Miura-ori inextensional transformation of a flat sheet (Miura, 1980). Image reprinted with permission from IAF.

Fig. 7.16. Unlike the map-folding scheme, here the expansion is biaxial and hence, if the sheet is unfolded by pulling two opposite corners apart, it gradually expands both in the direction of pulling and in the direction orthogonal to it. As it will be seen at the end of this section, changing the fold angles only by small amounts has the effect of coupling the motions in the two directions. In deployable structures, introduced in Chapter 8, one distinguishes between two different types of deployment behavior: synchronous deployment for a structure that deploys as a whole, vs. sequential deployment for a structure that deploys one piece at a time. A fundamental difference between map-folding and Miura-ori is that maps deploy in a kind of sequential fashion, whereas Miura-ori deploys synchronously. Miura-ori is based on a repeating unit cell consisting of four identical parallelograms with side lengths a and b, and with internal angles α and 180◦ − α. The four parallelograms are shown in Fig. 7.17(a) for two different configurations of Miura-ori. The geometry of the partially folded unit cell is shown in Fig. 7.17(b). The overall dimensions of the repeating unit are defined by its height, H , measured in the z-direction, and the x and y distances, respectively 2S and 2L, between the corners of the unit, lying in the x–y plane. A further dimension, v, is defined in Fig. 7.17(b). It identifies the offset of the center nodes of the unit cell with respect to the edge nodes. In Fig. 7.17(b) the dihedral angle β between any of the four parallelograms and the x-y plane is defined as the deployment angle of the unit cell. Hence, β = 0◦ when the unit cell is fully deployed, i.e., it is flat, and β = 90◦ when the unit is fully folded, i.e.

7.4 Biaxial Folding: Miura-Ori

y

b z

A

b C

a

203

α

a b b

F

M

N

v0

2L0

α

2S0

D

180-α α

G

a

x

(a)

z

y

G

b

C a a H

M

E

D

γ

b F P 2S

A

v

Q

β γ

2L

N x (b) y G

C B

γ

A α

L

b E

D γ

(c) Figure 7.17 Repeating unit of Miura-ori (a) definition of geometry of basic parallelogram showing definition of angle α, (b) geometry of partially folded unit cell showing definition of deployment angle β, and angle γ , and (c) detail of figure (b). Image adapted from Schenk (2011).

the four parallelograms overlap. An alternative deployment angle, γ , is also defined in Fig. 7.17(b). γ is the angle between the crease lines that are straight in the flat configuration, e.g., DM and MN, and a line parallel to the y-axis. Figure 7.17(c) shows a detailed view of only two parallelograms in the unit cell.

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Packaging of Membranes

A simple relationship between the angles β and γ is found as follows. First, consider the right-angle triangle AGL in Fig. 7.17(c); the height H of the unit cell has the expression: H = GL = a sin γ

(7.4)

Next, consider the right-angle triangle MPQ in Fig. 7.17(b) to obtain the alternative expression: H = MQ = MP sin β = a sin α sin β

(7.5)

Then equate Eqs. 7.4 and 7.5 to find sin γ = sin α sin β

(7.6)

which is the required relationship between γ and β. Since α is defined for any chosen Miura-ori, for any value of the deployment angle β, the alternative deployment angle γ can be determined from Eq. 7.6. An expression for the length of the unit cell is found as follows. Consider the rightangle triangle MQP in Fig. 7.17(b). It has hypotenuse of length a, base length L and height H , and hence a 2 = H 2 + L2

(7.7)

Substituting Eq. 7.5 for H and rearranging, we obtain the half-length of the unit cell:  L = a 1 − sin2 α sin2 β (7.8) It is useful for later on to find an expression for tan γ . Consider again the triangle MQP and note that H = L tan γ

(7.9)

Substituting Eqs. 7.5 and 7.8 and rearranging we obtain: tan γ = 

sin α sin β

(7.10)

1 − sin2 α sin2 β

An expression for the width of the unit cell is found as follows. First, define point C in Fig. 7.17(a) as the perpendicular projection of vertex D onto the line AG, and also define point E, the projection of point C onto the line AF in Fig. 7.17(b). Consider the right-angle triangle ACD in Fig. 7.17(a) and note that CD = AD sin α = b sin α

(7.11)

AC = AD cos α = b cos α

(7.12)

and

7.4 Biaxial Folding: Miura-Ori

205

Also consider the right-angle triangle ACE in Fig. 7.17(c), for which CE = AC tan γ

(7.13)

Substitution of Eq. 7.12 into Eq. 7.13 gives CE = b cos α tan γ

(7.14)

Finally, consider the right-angle triangle CDE in Fig. 7.17(c), for which 2

2

CD = CE + DE

2

(7.15)

Substituting DE = S

(7.16)

into Eq. 7.15 and also substituting Eqs. 7.13 and 7.14, and rearranging we obtain the following expression for S:  S = b sin2 α − cos2 α tan2 γ (7.17) Substituting Eq. 7.10 and rearranging leads to the final expression: S = b

tan α cos β 1 + tan2 α cos2 β

(7.18)

In conclusion, the three dimensions of the Miura-ori unit cell can be obtained from Eqs. 7.5, 7.18, and 7.8 (see also Schenk and Guest, 2013): H = a sin α sin β 2S = 2b 

tan α cos β 1 + tan2 α cos2 β

 2L = 2a 1 − sin2 α sin2 β

(7.19) (7.20) (7.21)

The dimensions of the flat unit cell are 2S0 by 2L0 , which can be obtained by substituting β = 0 into Eqs. 7.20–7.21. The ratios between the dimensions of the unit cell in a general configuration and the flat configuration have the expressions: cos β S  = S0 cos α 1 + tan2 α cos2 β  L = 1 − sin2 α sin2 β L0

(7.22) (7.23)

Plots of these expressions for deployment angles varying between 90◦ and 0◦ can be used to study the deployment coupling of Miura-ori. Several values of α have been considered in Fig. 7.18. The general trend is for the packaged unit cell to initially expand mostly in the x-direction and then mostly in the y-direction. An extreme behavior (decoupling) is achieved for α → 90◦ , as Miura-ori approaches the map folding in

Packaging of Membranes

90

β=70

88

0.8

90

1

0.6

86

β=50

β=40

84

β=80

82

S/S0

β=60

0.4

80

206

0.2

β=90

0

0

0.2

0.4

0.6

0.8

1

L/L0

Figure 7.18 Biaxial expansion of Miura-ori unit cell with different angles α = 80◦,82◦, . . . ,90◦

(shown in the figure without any label). Different degrees of deployment are identified by the lines β = 90◦, . . . ,40◦ .

Fig. 7.13(b). For α = 90◦ the plot of S/S0 vs. L/L0 no longer consists of a smooth curve, but instead it breaks into two perpendicular straight lines. For practical applications, α = 84◦ is a good compromise between deployment coupling, which tends to get better as α gets smaller, and packaging efficiency, which gets worse as the packaged unit cell tends to “grow” in the y-direction for smaller α’s. For α = 84◦ , up to 80% of full expansion in the x-direction is reached with only 20% expansion in the y-direction. In the second deployment phase, expansion is mainly in the y-direction. The crease pattern for this particular value of α and a = 82 mm, b = 84.5 mm is shown in Fig. 7.19(a) for an A2-sized sheet of paper. The repeating unit cell of Fig. 7.17(b) is shown in the figure. Note that the horizontal folds are straight but consist alternately of hill and valley folds, whereas the vertical folds form a zig-zag line at angles of ±6◦ to the vertical. Also note that in Fig. 7.19(b) the packaged configuration for this model measures 112.5 mm by 130 mm. A positive effect of allowing the packaged membrane to grow in one direction, which happens by choosing α < 90◦ , is that the creases become offset by a small amount in the packaged configuration. With reference to the membrane thickness effects discussed in Section 7.2.1, it should be noticed that Miura-ori naturally leads to nested folds. Hence, there is no innermost fold that needs to be folded very tightly in order to reduce the overall thickness of the package. Miura-ori has found many applications in large membranes (Miura, 1980) and space structures, including a novel retractable solar array concept that was launched on the

7.4 Biaxial Folding: Miura-Ori

207

y x

420

84

96

82

82

84o 88

82

82 594 (a)

12o 130 84 84o

6

106.5 (b)

Figure 7.19 Miura-ori. The dimensions given, in millimeters, are for a model made from an A2 √ sheet. A model can be made also from an A4 sheet, but all dimensions should be divided by 2.

208

Packaging of Membranes

Figure 7.20 Solar array based on Miura-ori (Natori et al., 1995).

Figure 7.21 Miura-ori map of Venice. A value α ≈ 84◦ was used in this map (Miura, 1978).

Space Flyer Unit in 1995 and retrieved by the Space Shuttle in 1996 (Miura and Natori, 1985c; Natori et al., 1995). The array consisted of 20 Kapton panels with dimensions of 0.6 m × 0.6 m, and connected by piano hinges. A photograph of the flight unit, before launch, is shown in Fig. 7.20. It has also found applications in the design of metamaterials, as discussed in more detail at the end of this chapter. Another application of Miura-ori has been in the design of city maps (Miura, Sakamaki, and Suzuki, 1980) see Fig. 7.21.

7.5 Biaxial Folding: Wrapping

7.5

209

Biaxial Folding: Wrapping A different way of packaging a flat, thin membrane is to wrap it around the center, as shown by means of two different examples in Fig. 7.22. Both of these folding patterns consist of rotationally symmetric sets of hill and valley folds, and the folding action consists in wrapping the membrane by winding it in an anti-clockwise sense, as shown in the figure. This packaging technique was invented in the early 1960s. Huso (1960) invented a sheet reel for folding compactly a car cover. Huso’s device consisted of a fixed part, connected to the car roof, attached to a rotatable hub connected to the car cover: when the hub was rotated, the cover was gradually wound onto it. The technique was refined by Lanford (1961) who patented the folding apparatus shown in Fig. 7.23, where a regular spacing of hill and valley folds is achieved by means of guiding wires tensioned by weights. This produces a fully-wrapped sheet with a regular saw-tooth edge. A further contribution was made by Scheel (1974), whose patent envisaged a set of straight flexible ribs and major folds approximately tangent to the hub; a further set of

(a)

Figure 7.22 Two ways of wrapping a thin, flat membrane: (a) wrapping scheme and (b) origami Flasher (Shafer, 2001).

210

Packaging of Membranes

Figure 7.23 Lanford’s folding machine (Lanford, 1961).

Figure 7.24 Crease pattern by Scheel (1974) (hill folds are solid, valley folds are shown dashed).

intermediate folds along which the sheet is folded in alternate directions, intermediate folds bisect the angles between adjacent major folds. Additional minor folds, parallel to the major folds and shown in Fig. 7.24, were introduced to reduce the height of the packaged membrane. This pattern leads to a series of pleats and sub-pleats, which are wrapped around a circular hub, as in the previous schemes. This folding pattern was adopted by Temple and Oswald (Cambridge Consultants, 1989) for the packaging of a solar sail. Their plan was to launch the sail wrapped around the body of a spacecraft, about 4 m in diameter. Once in orbit, the sail would deploy into a 276 m diameter disk and would collect enough solar pressure to sail to Mars. For such a large application it became necessary to work out the crease pattern in more detail. Guest and Pellegrino (1992) showed that for the theoretical case of a membrane of zero thickness that is wrapped around a regular prismatic hub with 2n sides the n hill folds and n valley folds are straight. An example with n = 3 is shown in Fig. 7.25(a).

7.5 Biaxial Folding: Wrapping

211

Hill fold Valley fold

d hub d

a

d

β

c

b

c

c

b

a

b a G

O δ

α A γ

β

(a)

hub

O B

D

F

β C

E

A

hub C γ B

β

β D F

(c) E (b) G Figure 7.25 (a) Wrapping crease pattern for a theoretical membrane of zero thickness; (b) detail showing two major folds and hub corners, in flat configuration; (c) wrapped configuration of membrane panel shown in (b) (Guest and Pellegrino, 1992).

212

Packaging of Membranes

This basic pattern can be modified to account for the actual thickness of the membrane, as will be described later. To determine the details of the wrapping folding pattern, we start at the hub vertex, Fig. 7.25(b). Given the hub interior angle α, we calculate the angles β, γ , δ defined in the figure. Since the sum of the interior angles in a polygon with 2n sides is (2n − 2)π , the interior angle at a vertex of a regular polygon is   1 α = 1− π (7.24) n Also, obviously α + β + γ + δ = 2π

(7.25)

To obtain two more equations, consider the fully wrapped configuration of the membrane, a part of which is shown in Fig. 7.25(c). Because the membrane has zero thickness, it wraps around the hub without increasing its thickness. Hence, the wrapped membrane coincides, in plan view, with the edge of the hub. Hence, its geometry is very simple; BC, DE, F G end up vertical, which implies ' = δ = π/2 ABC

(7.26)

Because, by symmetry, the angles at vertex A are equal to the corresponding angles at B, Fig. 7.25(b), and BC is vertical after wrapping, we have the following fourth condition, see Fig. 7.25(c): γ − β = π/2 Given Eq. 7.24, the solution of the system consisting of Eqs. 7.25–7.27 is   1 π 1 π , γ = + π, δ = β= 2n 2 2n 2

(7.27)

(7.28)

which defines completely the wrapping pattern in the region next to the hub. Note that AC bisects the angle between side AB and the line of side OA. To define the rest of the crease pattern, note that the folds BC, DE, F G end up parallel in Fig. 7.25(c) and hence have to be parallel, since they are coplanar, also in the flat crease pattern, in Fig. 7.25(b). They are also equidistant because in Fig. 7.25(c) they pass through adjacent vertices of the hub. With reference to Fig. 7.25(a), this shows that type b folds are parallel and equidistant. By symmetry, the same is true for type d folds. Finally, note that B, D, F and, similarly, A, C, E, G, are collinear because type c folds pass through the intersections of folds b and d, in Fig. 7.25(a). At this point we are ready to draw the complete crease pattern on a flat sheet. First we draw a 2n-sided regular polygon representing the hub: its sides are alternate hill and valley folds. Then, we draw 2n major fold lines, each forming an angle β = π/2n with a side of the polygon. Finally, we draw the 2n sets of equally spaced, parallel folds b and d, orthogonal to the sides of the hub. The fold patterns derived above can be modified to account for a small membrane thickness t. Obviously, the wrapped membrane will no longer coincide with the edge of the hub, a consideration which greatly simplified the analysis in the case t = 0. Guest

7.5 Biaxial Folding: Wrapping

213

2n = 6, t = 2 mm

50 mm

Figure 7.26 Wrapping crease pattern for a thick membrane (Guest and Pellegrino, 1992).

and Pellegrino (1992) assumed that the wrapping of a membrane with t = 0 around an n-sided polygon is similar to the wrapping of a membrane with t = 0 and whose vertices, after folding, have to lie on helical curves whose radius increases at a constant rate, based on t. Thus, they set up a more general version of Eqs. 7.24–7.27, which can be solved numerically for any value of t. The folding pattern for 2n = 6, assuming a rather thick membrane (t = 2 mm) is shown in Fig. 7.26. To make a model, the reader should enlarge this pattern, so that the 50 mm line has approximately this length. Because the thickness assumed in the calculation of the folding pattern is much greater than the thickness of the paper, the sheet will not package very tightly. Note that the main effect of non-zero thickness is that the major fold lines are no longer straight.

214

Packaging of Membranes

(a)

(b)

Figure 7.27 Crease pattern for origami Flasher with 2n = 6, (a) assuming zero thickness and (b) a thickness equal to 10% the edge length of the central polygon (Zirbel et al., 2013).

A circular hub can be seen as a limiting case for 2n → ∞. However, as the number of major hill and valley folds increases, what happens in practice is that, near the hub, localized stretching and wrinkling of the membrane create a smooth transition from the hypothesized, prismatic hub shape and the actual circular hub. In conclusion, we return to the second wrapping pattern shown in Fig. 7.22. The Flasher was invented by Shafer and Palmer (Lang, 1997) and a calculation of thickness effects, similar to that outlined above for the wrapping fold pattern, was carried out by Zirbel et al. (2013). A comparison of the crease patterns for the theoretical, zerothickness case and the finite-thickness case is shown in Fig. 7.27. Note the similarity of this crease pattern to that envisaged by Scheel (1974) and presented in Fig. 7.24. The main difference between the two wrapping patterns is that in the original one the cylindrical length of the packaged membrane increases linearly with the diameter of the flat membrane, whereas in the Flasher the cylindrical length remains approximately constant. This is a significant advantage for applications in deployable spacecraft structures, although the number of layers wrapping around the central polygon, which increases linearly with the diameter in the original pattern, increases quadratically in the Flasher (Zirbel et al., 2013).

7.6

Foldable Cylinders One of the basic packaging problems for membrane structures is to fold a cylinder axially all the way to its extreme, flat configuration while maintaining its crosssection, like a bellows. This problem is encountered in many applications, such as sunshields for precision deployable booms or deployable telescopes. An example is the International X-ray Observatory (IXO) for which Northrop Grumman has proposed a structural concept based on a cable structure prestressed by deployable booms, as shown in Fig. 7.28(a). To keep the structure thermally stable and thus avoid thermally

7.6 Foldable Cylinders

215

Graphite Fixed Truss

Highly Preloaded Graphite Cable/Link Hexapod Truss Tensioned After Boom Latch-out

Telescoping Graphite Booms Deploy and Latch under Low Load

(a)

3

Depth (m)

2 1 0 −1 −2 −3

−2

−1

0 1 Width (m)

2

(b) Figure 7.28 (a) IXO structural architecture concept and (b) envelope for packaged sunshield (Wilson et al., 2013).

induced deformation, the structure has to be enclosed within a deployable sunshade that fits inside the faring of the launcher and clears the telescope structure. These constraints define an annular cylindrical envelope with the cross-section shown in Fig. 7.28(b). Wilson et al. (2013) did a study of deployable sunshades for this application, in which they considered two packaging schemes, as described in this section. Other applications require cylinders that fold by decreasing the size of their crosssection. An example is the esophageal stent shown in Fig. 7.29; this deployable metal stent is delivered to a patient’s esophagus with an endoscope and is then expanded in-situ. This type of deformation will not be considered in the following discussion.

216

Packaging of Membranes

Figure 7.29 Deployable stent (image reproduced with permission from Zhong You, 2019).

d2 Pattern repeats

d1 href

δ

nd2 Figure 7.30 Crease pattern for Kresling folding scheme; valley folds are dotted, hill folds are solid. The pattern shown generates a cylindrical polyhedron with n = 6 and δ = 60◦ (Wilson et al., 2013).

7.6.1

Kresling Folding Scheme This folding scheme was first proposed by Kresling (2008), as a special case of Guest and Pellegrino (1994); the analysis presented in this section is from Wilson et al. (2013). A cylindrical polyhedral surface is constructed from a flat sheet with a repeating crease pattern based on Fig. 7.30. The crease pattern has six folds meeting at each vertex, four hill folds (solid lines) and two valley folds (dotted), defined by the three variables δ, d1 /d2 , and n. Here, n is the number of parallelograms wrapped around the cross-section of the polyhedron. A more general folding scheme based on fold lines that lie on set of

7.6 Foldable Cylinders

217

Figure 7.31 Cylindrical polyhedral surface consisting of two pairs of modules, at various stages of deployment (Wilson et al., 2013).

helices was proposed by Guest and Pellegrino (1994), in which folding is achieved by an overall torsional rotation of the structure. The repeating module of the deployed polyhedral surface, shown in Fig. 7.31(a), is obtained by joining the side edges of the crease pattern in Fig. 7.30. Note that the module consists of a pair of ring structures, skewed alternately in clockwise and anticlockwise directions. The skewness of these rings results from the arrangement of the parallelograms forming the crease pattern, which can be imagined as produced by shearing the central horizontal line in an initially rectangular lattice. This polyhedral surface can be folded, as shown in Fig. 7.31(b and c). Each ring structure behaves essentially independently of the others, and the folding sequence shown in the figure assumes – as an example – that all rings twist by the same amount. As each module is folded, it becomes strained. The exact amount of strain and its variation with the degree of folding are a function of the three design variables, and can be estimated with a simple wireframe analytical model, where the hill folds are represented by pin-jointed rods of fixed length and the valley folds by springs. The strain experienced by the springs is a qualitative measure of the overall level of strain in the polyhedral surface. In addition, if the spring strain is zero then so too is the strain energy in the actual structure. Another assumption of the model is that every parallelogram unit deploys at the same rate. With this assumption the wire frame model allows for only a single degree of freedom, given by the height h of a repeating half-module. All vertices of the wire frame model lie on a cylinder of radius R. A geometrical relationship between the strain experienced by the valley fold and the height of the ring structure can be found as follows. From Fig. 7.32, the x–y projections of the valley fold and hill fold are given by:  (7.29) g1 = d12 − h2 g3 = 2R sin

α+β 2

(7.30)

218

Packaging of Membranes

z d2

D

y

x

C

d2

D d1

d1

df

d3

B

B

(a)

g1

(b)

C

d1 B y

h

g1 (d)

x β

R

α

C d3

C h

g1

g3

A

B

g3 (e)

d2 (c)

h

d1

d2

A A

C

A

Figure 7.32 (a) Single parallelogram unit developed on flat surface; (b) isometric view of same unit; (c) top view of same unit; (d) hill fold BC (=AD) and (e) valley fold AC and their projections (Wilson et al., 2013).

Thus, the valley fold length d3 is d3 = =

 h2 + g32 



h2 + 4R 2 sin2

α+β 2

 (7.31)

and the cylinder radius is R=

d2 2 sin β/2

(7.32)

Finally, α and β can be found from Fig. 7.32, α = 2 arcsin β=

2π n

g1 2R

(7.33) (7.34)

7.6 Foldable Cylinders

Valley Fold Strain

0.2

δ=70

0.2

δ=70

δ=60

δ=60 0.1

δ=50

0.1

δ=50

δ=40 δ=30

δ=40 δ=30

0

219

0

−0.1

−0.2 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Normalized Deployed Height

Normalized Deployed Height (a)

(b)

Figure 7.33 Strain in the valley folds during deployment assuming d1 /d2 = 1, for (a) n = 12 and (b) n = 20. The height has been normalized by href , defined in Fig. 7.30 (Wilson et al., 2013).

Hence, the valley fold length is only a function of the height h and the original parallelogram parameters d1 , d2 , and δ. The valley fold strain is given by: =

d3 − df df

(7.35)

For each value of n, changing δ alters the strain in the valley folds predicted by the numerical model described above, allowing the behavior of the cylindrical polyhedron to be tailored to fit the design requirements. Figure 7.33 shows that as n increases so too does the valley fold strain in the packaged configuration. It also reveals that decreasing δ also decreases the initial strain in the packaged configuration. In choosing the geometric parameters of the polyhedral surface, the internal constraint imposed by the telescope structure has to be considered together with the outer constraint imposed by the launcher fairing. As n decreases the size of the triangles increases and it becomes more difficult to satisfy these constraints. In Fig. 7.34 it can be seen that for d1 /d2 = 1 the lowest number of sides is n = 12. This simple analytical model is a qualitative tool to compare different designs, but it should be realized that the actual strain levels that are predicted are not accurate because in reality the whole structure can deform, not only the valley folds. Furthermore, a thin film can wrinkle much more easily than it can stretch. Therefore, small buckles in compression areas of a foldable cylinder are more likely to occur than the predicted yielding of the material. A 1:10 scale prototype of the IXO sunshield was designed by Wilson et al. (2013). The valley fold strain was set to zero in the fully deployed configuration, and arbitrarily set to 1% in the fully packaged configuration. An optimization study led to n = 10, d1 /d2 = 0.952 and δ = 38.54◦ (see the crease pattern in Fig 7.35(a)), for which the total number of modules would be 19. A proof of concept model, consisting of 3 modules made of 50 μm thick Kapton is shown in Figs 7.35(b and c).

Packaging of Membranes

2

2

1

1

Depth (m)

Depth (m)

220

0

0

−1

−1

−2

−2 −2

−1

0 1 Width (m) (a)

2

−2

−1

0 1 Width (m)

2

(b)

Figure 7.34 Top views of Kresling cylinder in packaged configuration for (a) n = 12 and (b) n = 20, for d1 /d2 = 1 (Wilson et al., 2013).

Figure 7.35 Proof of concept model of 1:10 scaled of IXO sunshield. (a) Crease pattern; (b) packaged configuration and (c) deployed configuration (Wilson et al., 2013).

7.6 Foldable Cylinders

221

a b

180-α

α

Q

P

m1

Figure 7.36 Basic Miura-ori pattern of fold lines for module m1.

Q’

Q

P

m1

P’

m2 (a)

P

Q

Q’

P’

m3 (b)

P

Q

Q’

P’

(c)

Figure 7.37 Process of synthesizing a foldable building block through symmetry operations: (a) generation of m2 as mirror image of m1; (b) generation of m3 by rotation of m1 + m2; and (c) joining of m1 + m2 and m3.

222

Packaging of Membranes

7.6.2

Miura–Tachi Folding Scheme Tachi (2010) discovered a general principle for constructing rigid-foldable polyhedral cylinders based on the Miura-ori membrane packaging scheme described in Section 7.4. Based on this principle, a cylindrical polyhedral surface that can be folded without any strain is synthesized by means of symmetry operations. This work on foldable cylinders led to foldable three-dimensional cellular structures (Miura and Tachi, 2010). Consider the basic Miura-ori module m1 shown in Fig. 7.36; it is composed of nine identical facets (parallelograms). Note the central line P Q drawn on the left middle facet. Next, fold this Miura-ori module to reach a deployment angle β of 45◦ (recall that the deployment angle of Miura-ori has been defined in Fig. 7.17): the new configuration is shown shaded in yellow in Fig. 7.37(a). Then, consider the mirror image m2 of this module through a vertical symmetry plane: this mirror image is also shown in Fig. 7.37(a). Note that at the intersection between m1 and m2 both the fold lines and the facets match fully. Eliminating any overlapping areas, modules m1 and m2 can be integrated into a single module m1 + m2 that can be folded without any strain, because it is assembled from modules which are themselves foldable. Next, a second symmetry operation is carried out, a 180◦ rotation of the module m1 + m2 about the line P QQ P  which produces the module m3 shown in Fig. 7.37(b). m3 is also a structure that can be folded without any strain. Because the left and right sets of facets of m1 + m2 are rotationally symmetric about P QQ P  the facet planes and the fold lines of modules m3 and m1 + m2 coincide. Hence, the left and right edge facets of modules m3 and m1 + m2 can be joined without leaving any gaps, as shown in Fig. 7.36(c). For ease of construction it may be convenient to retain some overlapping parts. Through these symmetry operations we have formed the basic building block for a foldable structure that encloses an inner space and has a single degree of freedom. Removing the overlapping parts and repeating the reflection operation vertically results in the rigid-foldable cylindrical polyhedron shown in Fig. 7.38. This figure shows the two extreme configurations of this foldable cylinder, as well as the intermediate ones.

Figure 7.38 Miura–Tachi foldable cylinder with α = 45◦ , shown in different configurations with β = 89◦, 67.5◦, 45◦, 22.5◦, 1◦ (Tachi and Miura, 2012).

7.6 Foldable Cylinders

223

Figure 7.39 Folding of three-dimensional cellular structure, based on Miura–Tachi cylinder (Tachi and Miura, 2012).

The foldable cylinder can be replicated in a regular, two-dimensional tessellation that forms the cellular structure shown in Fig. 7.39. This result can be obtained by translating the foldable cylinder in two directions and, importantly, is valid for any folded state of the cylinder. Hence this three-dimensional cellular structure provides a synchronized one-DOF rigid-folding motion for a tessellation of foldable cylinders, which can be used as the basis for the design of variable shape metamaterials.

8

Concepts for Deployable Structures

8.1

Introduction The structures described in previous chapters of this book operate in a single configuration and can be broadly described as passive. Starting from this chapter, we consider active structures. Deployable structures, considered in this chapter and the next, are a type of transformable structure able to autonomously change geometric configuration between a compact, packaged state, in which it can be stored or transported, and an extended, deployed state, in which the structure is operational. The transformation from the packaged configuration to the deployed configuration is called deployment and the reverse transformation is called retraction. The oldest example of deployable structure is the umbrella, Fig. 8.1, which over many years has evolved into a spring-loaded structure that opens automatically at the push of a button, Fig. 8.2. Many deployable structures, e.g., beach chairs, clothes drying racks, tape measures, etc. are used in our daily lives, but the focus of this chapter is on how to conceive and design engineered deployable structures. These structures achieve extreme performance, such as maximum stiffness in the deployed configuration with minimal mass, or high geometric precision at low cost. This chapter presents the main underlying challenges, and general approaches to address them, whereas the following chapter presents several example applications. The requirements that have to be met by a deployable structure in its operational configuration, e.g., providing shelter from rain or the sun in the case of the umbrella, or forming an accurate reflective surface in the case of a deployable reflector antenna for telecommunication, are different from the requirements in the packaged configuration, which usually should have the smallest possible volume, and in some cases is also subject to specific constraints. For example, the launch vibration acceleration of a packaged solar array should be smaller than a safe limit, and the packaged sunshade for a telescope may need to fit around the primary mirror of the telescope. A general requirement is that the deployment process should be robust, predictable and damage-free for the structure. Another requirement of engineered deployable structures is that it should be possible to predict the folding/deployment process in sufficient detail that the margin against any kind of failure should be known and considered to be acceptable. The field of deployable structures is broad and cannot be approached with a single, general concept or unified theory. Instead, careful study of established structural

224

8.1 Introduction

225

Figure 8.1 Depiction of Assyrian ruler Assurbanipal, showing the first known sun umbrella in

13th Century BC (Gordon, 2011).

Figure 8.2 Fully open umbrella (New Africa/Shutterstock.com).

concepts leads to the identification of different feasible regions in a discontinuous design space. Then, to find viable solutions to a specific design problem, one can either operate within a feasible design region, or work to expand the boundaries of the feasible region by exploiting advances in materials, actuators, numerical simulation techniques, etc.

8.1.1

Packaging, Deployment, and Stabilization For passive structures, the approach presented in previous chapters of this book was to choose a basic structural form and develop structural concepts that make use of that chosen structural form to efficiently carry out a specific primary function. Deployable structures are different. In addition to carrying out their primary function, they have to fulfill three underlying functions specifically connected to the process of efficient shape transformation, namely packaging, deployment, and stabilization. The first underlying function of a deployable structure is packaging. It consists in substantially decreasing at least one physical dimension of the structure and often also the volume enclosed by the structure. Defining the packaging ratio as the volume of material

226

Concepts for Deployable Structures

within the structure divided by the volume of the convex envelope of the structure, the ultimate objective is achieving a packaging ratio equal to one. This corresponds to storing the structure into a package whose volume equals the volume of the material, i.e., there are absolutely no voids. Whereas achieving this limit of one is practically impossible, although rolled up membrane structures can get close to this limit, the packaging ratio is a useful way of comparing the volume efficiency of different packaging schemes. The second underlying function is deployment. It includes both the actuation of deployment, for example through electric motors or by releasing the elastic strain energy stored in discrete springs or continuously within the structure, and the process of transforming the shape from the packaged configuration to the deployed configuration. Deployment is required to be highly deterministic in some applications, such as car roofs and large space telescopes, but can also be an almost random transition between two extreme shapes, as in inflatable space antennas. Lastly, stabilization is the process that “locks” a deployable structure in its fully deployed configuration and provides the stiffness required to enable it to carry out its primary function. Stabilization can be achieved in many different ways, such as by powering off electric motors, to hold a powered joint at a specific angle, or by activating

Figure 8.3 Example of kinematically controlled deployment in the retractable roof of the Ferrari California T (images used with permission of William Weber, 2019).

8.2 Packaging through Elastic Deformation

227

Figure 8.4 Example of dynamic boom deployment in the Autonomous Assembly of

Reconfigurable Space Telescope (AAReST).

a chemical reaction that increases the elastic modulus of the material that makes up some key components of a structure. An example of a kinematically controlled deployment is shown in Figure 8.3. The car roof consists of two pieces that are stowed on top of one another; each piece is supported by a four-bar linkage actuated by an electric motor. An example of a dynamic deployment is the camera boom for the AAReST spacecraft (Underwood et al., 2015), shown in Fig. 8.4. The 1.4 m long boom contains four pairs of tape spring hinges built by cutting slots in a thin-walled carbon and glass fiber tube. During launch, the boom is tightly folded near the center of the spacecraft. In orbit, the boom is deployed in two stages by cutting two restraining cables which cause the elastic strain energy stored in the hinges to be released, letting the boom deploy from the folded configuration in Fig. 8.4(a) to the configurations in (b) and then (c). Lastly, the umbrella, Fig. 8.2, provides an example of stabilization through prestressing of a membrane, driven by the snap back of an over-centered mechanism. The following sections present three approaches for packaging structures, through (i) smooth elastic deformation of the structure, (ii) localized deformation at mechanical joints, or (iii) localized deformation within the structure itself. Different approaches to the remaining two functions, namely deployment and stabilization, are then presented.

8.2

Packaging through Elastic Deformation The general idea is to elastically deform (bend) a continuous structure into a small volume while ensuring that the structure remains everywhere within its elastic limit, to avoid damage or permanent deformation. The most basic example is that of a slender rod packaged into a coil, which is then generalized to this cylindrical shells known as tape springs.

228

Concepts for Deployable Structures

8.2.1

Coiling a Rod A slender rod with circular cross-section of radius r can be coiled uniformly into a circle of radius R. The tightness of the coil is limited by the yield/failure strain of the material of the rod, y , and the cross-sectional radius. From standard beam theory, the maximum bending strain in an initially straight rod due to a curvature change κ =

1 R

(8.1)

is given by: max = rκ = r/R

(8.2)

Setting max = y in Eq. 8.2 and solving for R provides the smallest radius of coiling for which the rod remains elastic: Rmin = r/y

(8.3)

An example of a “deployment cassette” to house the coiled rod is shown in Fig. 8.5. For example, consider an S2 glass fiber rod, i.e., a rod made by pultruding glass fibers and an epoxy resin, with a diameter of 2 mm. For glass fiber max is well in excess of 2% and hence it is assumed that Rmin = r/max ∼ 50 mm. For a high-strength steel rod of the same diameter max = 0.5 % and hence Rmin ∼ 200 mm. Because the transition region between the fully coiled part of the rod and the straight part is invariant, this mode of coiling is a steady-state process that requires a constant force Fp . Here, the subscript p denotes the planar rod configuration. Fp can be obtained by equating the external work required to fully coil an additional piece of rod, of length δL, to the corresponding change of strain energy. Denoting the flexural stiffness by EI , the energy balance equation is: 1 EI (κ)2 δL 2 Substituting Eq. 8.1 for κ and dividing by δL, gives: Fp δL =

(8.4)

EI (8.5) 2R 2 An alternative coiling scheme relies on a fundamental property of isotropic rods, e.g., rods with circular cross-section, first investigated by Kirchhoff and reported in Love Fp =

Fp rod

R>Rmin

roller

Figure 8.5 Schematic diagram of planar coiled rod and its housing.

8.2 Packaging through Elastic Deformation

229

Fh=0

Fh R L0

R α L

Fh=0

Fh

Figure 8.6 Helically coiled rod.

(1944). Any elastic rod can be held in a helical shape by applying two equal and opposite forces Fh at either end through rigid elements of length R, as shown in Fig. 8.6. Here the subscript h denotes the helical coiling mode. The rod can be elastically deformed into a uniform helix and the pitch of the helix can be continuously decreased from α = 90◦ to α = 0◦ , respectively corresponding to the fully extended and the fully packaged rod. The relationship between Fh and the pitch α of the helically deformed rod is (Love, 1944): Fh = EI

cos2 α R 2 sin α

(8.6)

Note that the radius of the helix is set at R. The relationship between the length L0 of the rod and the current height L of the helix is, neglecting the torsional deformation of the rod, L = L0 sin α

(8.7)

The variation of Fp and Fh with the non-dimensional height L/L0 , for the two modes of coiling described above, is shown in Fig. 8.7, which was obtained by plotting Eqs 8.5 and 8.6 vs. sin α. The figure shows that the planar coiling mode corresponds to a constant and relatively small force that opposes coiling, whereas the helical coiling mode starts with a tiny force, when the rod is approximately straight, and rapidly increases as the helix gets flatter. However, it should be noted that the torsional compliance of the rod,

230

Concepts for Deployable Structures

10 9 8 7 6 FR2 5 EI 4 3 2 1 0

helical coiling

planar coiling 0

0.2

0.4

0.6

0.8

1.0

L/L0 Figure 8.7 Comparison of forces required to coil a rod.

t

L

R α Figure 8.8 Tape spring geometry.

which has been neglected in the analysis, would in fact limit the value of Fh for α → 0◦ to a finite value.

8.2.2

Coiling a Tape Spring The circular coiling of a rod, described in Section 8.2.1, becomes much more efficient if the compact cross-section of the rod is replaced with a thin, circular arc. This is the basic idea of the tape spring, i.e., a cylindrical shell of uniform thickness t and transverse radius of curvature R, subtending a uniform angle α, as shown in Fig. 8.8. The first use of tape springs was as steel tape measures, invented in the late 1920s (Calladine, 1988). Tape measures can be wound in a small case, and are strained elastically in this configuration, but they become strain-free when they are extended. Typical geometric parameters of a steel tape measure are R ≈ 20 mm, α ≈ 50 deg, i.e., 0.87 rad, and L ≈ 3 m. The values of α are given in degrees, for clarity, but note that this value is converted to radians for the calculation of the arc-length of the tape spring cross-section.

8.2 Packaging through Elastic Deformation

r

R

z

r

231

z x

x

y

y (a)

R

(b)

Figure 8.9 (a) Equal-sense and (b) opposite-sense coiling of a tape measure.

Tape springs are a prototypical deployable structure. Their simple geometry makes them easy to study and yet they display the key behaviors that are seen in many structures that are deployed by releasing stored elastic strain energy. The present discussion will focus on the geometry-change effects that allow tape springs to coil with relatively small bending strains; other important features of tape springs are discussed in Section 8.6. A key feature of tape springs is that their transverse curvature can change, to allow easy coiling. Figure 8.9 shows the two different ways in which a tape spring can be coiled onto a cylindrical spool of radius r. These two modes of coiling are called equalsense coiling and opposite-sense coiling, respectively, if the centers of curvature are on the same side, Fig. 8.9(a), or on opposite sides of the mid-surface of the shell, Fig. 8.9(b). In this figure, note that the same coordinate system is used for both coiling modes, and hence the z-axis points in different directions with respect to the curvature of the cross-section. The curvature changes due to the two coiling modes can be obtained by subtracting the initial curvatures  κx0 = 0  κy0 = ±

(8.8) 1 R

(8.9)

where the + sign refers to opposite-sense coiling and the − sign to equal-sense, from the curvatures in the coiled configuration, which are:  κx = −

1 r

 κy = 0

(8.10) (8.11)

Hence, the curvature changes are: 1 r 1  κy = ∓ R  κx = −

where the − sign refers to opposite-sense and the + sign to equal-sense coiling.

(8.12) (8.13)

232

Concepts for Deployable Structures

The corresponding bending strain components can be obtained from the curvature changes: z r z y =  κy z = ∓ R κx z = − x = 

(8.14) (8.15)

where the − sign refers to opposite-sense and the + sign to equal-sense coiling. The stresses can then be obtained from Hooke’s law, see Eq. B.37 written in terms of σx ,σy :    σx 1 ν x E = (8.16) 1 − ν 2 ν 1 y σy For equal-sense coiling the maximum and minimum longitudinal and transverse stress components, corresponding to z = ±t/2, are:   E R t − +ν (8.17) σx = ± r 2(1 − ν 2 ) R   E R t σy = ± 1 − ν (8.18) r 2(1 − ν 2 ) R For opposite-sense coiling the expressions for the maximum bending stresses at z = ±t/2 are:   E t R +ν (8.19) σx = ∓ 2(1 − ν 2 ) R r   E R t σy = ∓ 1 + ν (8.20) r 2(1 − ν 2 ) R From these equations, one can determine the smallest possible ratio between crosssectional radius and thickness as a function of r/R (Rimrott, 1965). The value of t can be determined by requiring that this beam is on the point of yielding (e.g., according to the Tresca yield condition) when its cross-section is flattened and the beam is longitudinally coiled. For opposite-sense coiling, this condition requires E (r/R) + ν R = t 2σy (1 − ν 2 )(r/R)

(8.21)

where ν is the Poisson’s ratio. For equal-sense coiling the limits are: R 1 + r/R E = t 2σy (1 + ν)(r/R)

(8.22)

E (r/R) − ν R = t 2σy (1 − ν 2 )(r/R)

(8.23)

for 1 < r/R < 1/ν

for 1/ν < r/R. For high yield strength steel (E/σy = 200) these limits have been plotted in Fig. 8.10.

8.2 Packaging through Elastic Deformation

233

Tube radius / thickness, R/t

160 150 Permissible region 140 130 Opposite-sense coiling boundary

120 110

Equal-sense

100 Nonpermissible region

90 1

2

3 4 5 6 7 8 Spool / tube radius, r/R

9

10

Figure 8.10 R/t limits for elastic coiling of tape springs with E/σy = 200 and ν = 0.3.

Tape measures and most deployable structures built around the concept of the tape spring, are generally designed in such a way that in the packaged configuration r/R ≈ 1. This is because an isotropic cylindrical shell has a natural tendency to form elastic folds with a natural radius of folding R, and hence a tape spring forms smooth coils with a radius of ≈ R. More details about the natural radius of folding of a tape spring are provided in Section 8.6. To make a direct comparison of coilable structures based on a compact cross-section vs. tape springs, consider the case of a tape spring with a fully circular cross-section, α = 2π , i.e. the slit tube shown in Figure 8.11. The radius of coiling, r, is given and, following Tan and Pellegrino (2006), we compare the stiffest possible elastically coilable boom with a rectangular cross-section to a tape-spring boom. Both the total cross-sectional area and the material of the two booms, and hence also their mass, are equal. In the rectangular cross-section boom, with breadth b and height h, the maximum stress in the coiled configuration is: σ =

Eh 2r

(8.24)

where E is the Young’s modulus. If the maximum allowable value for σ is the yield stress σy , then the maximum value of h is: h = 2rσy /E

(8.25)

and the bending stiffness of this maximally stiff rectangular-section beam is – without considering the modulus, which is the same for the two booms –

σ 3 bh 2 y IR = = br 3 12 3 E 3

(8.26)

234

Concepts for Deployable Structures

b h r (a)

d t

r

2R (b) Figure 8.11 Coilable booms with (a) rectangular and (b) tubular cross-sections (Tan and Pellegrino, 2006).

Next, consider a tubular boom with cross-section of radius R and thickness t, split along the top. To make its cross-sectional area equal to that of the previous boom, set bh = 2π Rt

(8.27)

Solving for R and substituting Eq. 8.25, yields the following expression for the radius of the tape-spring boom R = brσy /π Et

(8.28)

The largest possible thickness of the tape-spring boom is determined by the yield condition. Hence, substituting Eq. 8.28 into Eq. 8.21, one obtains a quadratic equation in t, which has the solution: ⎞ ⎛

ν 2 2(1 − ν 2 ) r σ ν ⎠ yb + − (8.29) t =⎝ 2π π b 2π E When this boom is uncoiled, its bending stiffness – without considering the Young’s modulus – is: IT ≈ π R 3 t

(8.30)

where R and t can be obtained from Equations 8.28 and 8.29. To compare the static stiffness of these two booms, consider the ratio IT /IR . It has the expression: ⎞2 ⎛  2 (

ν 2 2(1 − ν 2 ) c IT 3 ν E ⎠ ⎝ = + (8.31) − IR 2π π b 2π 2π 2 σy For equal mass booms the ratio IT /IR is inversely proportional to the radius of coiling, r, divided by the breadth of the original rectangular section, b, and is directly proportional

8.3 Articulated Space Frames

235

Stiffness ratio, IT / IR

105

104

103

102 10-1

100 r/b

101

Figure 8.12 Variation of flexural stiffness of maximally stiff tubular beam with respect to stiffness of rectangular cross-section beam, for E/σy = 100 (Tan and Pellegrino, 2006).

to the squared ratio of the modulus divided by the yield stress. The ratio E/σy has values in the range 50-200 for most structural materials. Equation 8.31 has been plotted for E/σy = 100 in Figure 8.12. The plot shows that the stiffness ratio is in excess of 200, even for loosely coiled booms, and much larger for tightly coiled booms. This example has shown that a boom with a rectangular cross-section is orders of magnitudes less stiff than a tape spring boom. Tape springs with non-circular cross-sections could provide even stiffer designs, but the example presented above has demonstrated the potential advantages of thin-walled foldable structures able to elastically flatten to allow folding, and which can increase their stiffness by elastically returning to their original geometry, in the deployed configuration. A simple example is the steel tape measure. Specifically, the Stanley FatMax 33– 735 is housed in a 8 cm diameter cassette, has a transverse radius of 21.5 mm at the center and is flat near the edges. It has a flattened width of 25 mm and wall thickness of 130 μm.

8.3

Articulated Space Frames Alternative approaches to packaging are applicable to structures consisting of stiff rods/plates connected by mechanical articulations (hinges). The number and position of the hinges have to be such that the structure has one or more internal degrees of freedom, which allow it to move in a strain-free manner. Also, the motion needs to lead to a compact packaged configuration.

236

Concepts for Deployable Structures

8.3.1

Structural Mechanisms Deployable structures that belong to the mechanisms category are assemblies of stiff elements connected by hinges, arranged in such a way that a continuous transformation from the packaged configuration to the deployed configuration is possible (You and Chen, 2014). Mechanism theory was first developed in mechanical engineering, where the parts of a mechanism are often massive, but recently it has become connected to lightweight structures through the need for large deployable structures. Links to classical mechanisms can be identified for several of the concepts that will be presented. It was already noted that the retractable roof of the Ferrari California T is based on a classical four-bar planar linkage. Taking a cubic space frame as an example, two simple folding principles that are found in many deployable space frames are shown in Fig. 8.13. In Fig. 8.13(a) each of the four vertical members has been fitted with three parallel cylindrical hinges which allow it to fold within a diagonal plane. Thus, the upper square is lowered as the center hinges in the vertical members move inwards. It would also be possible for the central hinges to fold outwards, to allow the horizontal plates to fold closer but then the envelope of the packaged structure would get larger. Note that the same folding principle works for hinges in any horizontal direction: the only condition is that all three hinges in each vertical member are parallel. In the alternative concept shown in Fig. 8.13(b), each of the four vertical members has been fitted with two spherical joints, and thus it is possible to lower the upper square provided that it is also rotated at the same time. When the upper square reaches the level of the bottom square, it has rotated through 90◦ .

(a)

(b)

Figure 8.13 Folding principles for cubic space frames.

8.3 Articulated Space Frames

237

Figure 8.14 Pactruss module (Hogstrom and Pellegrino, 2016).

Figure 8.15 Pactruss module deployment (Hogstrom and Pellegrino, 2016).

A more complex articulated space frame, used as a module in many deployable structures concepts, is the Pactruss invented by Rhodes and Hedgepeth (1986), and recently selected by Lee et al. (2016) for the robotic assembly of a space telescope consisting of modular deployable structures. Fig. 8.14 shows a hexagonal Pactruss module. It is a tessellation of triangular prisms where each horizontal edge of the prism is connected by revolute joints at both ends to the vertical edge members. Of these, eight horizontal edge members (four in the top plane and four in the bottom) have a revolute joint in the middle. There are also eight diagonal members, on vertical faces of the prisms, with revolute joints both at the ends and in the middle, for a total of 39 members. Fig. 8.15 shows the deployment of this structural module. Detailed analyses of these structures, along the same lines as in Section 1.5, but allowing for different types of joints and modeling each rod as a body with 6 degrees of

238

Concepts for Deployable Structures

l l (a)

(b)

Figure 8.16 (a) Two-dimensional pantograph structure (lazy-tong); (b) three-dimensional pantograph structure forming a triangular prism.

Figure 8.17 Deployment of 5-module pantograph mast (You and Pellegrino, 1996).

freedom (Denavit and Hartenberg, 1955; Waldron and Kinzel, 2004; Uicker, Pennock and Shigley, 2003; Gan and Pellegrino, 2006) would show that all of these examples are both kinematically indeterminate, with multiple degrees of kinematic freedom, and also statically indeterminate. There are a number of practical implications to this observation. First, to achieve a controlled deployment, it is necessary to introduce additional kinematic constraints, such as gears in the hinges or synchronization cables. Second, these mechanisms may jam due to geometric lack of fit. A simpler structural mechanism is the pantograph, whose simplest, two-dimensional form is the lazy-tong, an assembly of straight rods of equal length connected by pivots in the middle and at the ends. The pivot in the middle of each pair of rods is called scissor joint, Fig. 8.16(a). The geometric configuration of this structure is defined by the deployment angle, θ . Fig. 8.16(b) shows a foldable, three-dimensional structure consisting of six rods, arranged in three pairs, connected by spherical joints at the ends. We are assuming this type of joint for simplicity, but in fact two revolute joints at 120◦ would be sufficient to provide the required kinematic freedom. This basic module can be folded and deployed freely: its configuration is defined by the single parameter θ . If several modules are stacked on top of one another, the parameter θ has the same value for all modules to fit together. Hence, the entire pantograph has only one mechanism. This means that the whole structure deploys synchronously, i.e., at the same rate, see Fig. 8.17.

8.3 Articulated Space Frames

239

Pantographs are widely used, for the simplicity of their joints and their intuitive kinematics. However, pantographs are statically indeterminate structures, although they are a bit more difficult to analyze than the pin-jointed structures discussed in Section 1.2, and their geometry has to be carefully chosen to find a geometry that works for a particular application. Two very useful examples of pantographs are structures forming a closed ring, in which identical pantograph modules have to fulfill the condition that each module subtends a constant angle, for variable deployment angle θ . Two different approaches are shown in the next sections.

8.3.2

Ring Pantographs Figure 8.18 shows a ring-like pantograph formed by the three sets of pairs of rods shown in Fig. 8.19. The elements of type (a) lie on the inside of the ring, the elements of type (c) lie on the outside, and the elements of type (b) provide a connection between the other two. In the elements of type (a) and (c) the pivot is located half way between the end connectors, whereas in the elements of type (b) the pivot is at distance  from one end and L from the other end. A top view of the repeating module of this structure is shown in Fig. 8.20; note that AB, AD, and CD correspond respectively to elements of type (a), (b), and (c). For the ring pantograph to fold, we require that a variation in the angle θ between the rods does not change the angle α, because the angle subtended by each unit must remain constant for the whole ring to fit together. An expression for α can be obtained as follows. The projected rod lengths, shown in Fig. 8.20(b), have the following expressions: AB = 2AD sin β

(8.32)

CD = 2AD sin(α + β)

(8.33)

Expressing AB, AD, CD in terms of , L and θ : AB = 2 sin(θ/2)

(8.34)

AD = ( + L) sin(θ/2)

(8.35)

CD = 2L sin(θ/2)

(8.36)

Substituting Eqs. 8.34, 8.35 into Eq. 8.32, and Eqs. 8.35, 8.36 into Eq. 8.33, and dividing by sin(θ/2) gives the following two equations: ( + L) sin β = 

(8.37)

( + L) sin(α + β) = L

(8.38)

From Eq. 8.37 we obtain: β = arcsin

 +L

(8.39)

240

Concepts for Deployable Structures

Figure 8.18 Ring-like pantograph (You and Pellegrino, 1997a).

Dividing the left- and right-hand side of Eq. 8.37 by the corresponding sides of Eq. 8.38 gives, after simple manipulations:  1 L = sin α − 1 + cos α  sin2 β

(8.40)

8.3 Articulated Space Frames

241

L

l

l

L

l L

Type (a)

Type (c)

Type (b)

Figure 8.19 Components of ring pantograph (You and Pellegrino, 1997a).

D

C

D

C β

A

2(α+β)

β

B A

2α

B

2α (a)

(b)

Figure 8.20 Top views of ring pantograph (You and Pellegrino, 1997a).

Substituting Eq. 8.39 for β and rearranging, yields a quadratic equation in L/, cos2 α

 2

L L − 2 cos α + sin2 α + cos2 α = 0  

(8.41)

whose solutions are: √ L cos α + sin2 α ± sin α 1 + 2 cos α =  cos2 α

(8.42)

It can be shown that these two solutions are one the reciprocal of the other, and thus produce identical structures. This equation is the key relationship for the design of this particular type of foldable ring. The pantograph structure shown in Fig. 8.18 has 12 sides, hence α = 15◦ , and from Eq. 8.42, L/ = 1.5821.

242

Concepts for Deployable Structures

In the above analysis it has been assumed that the joints of the ring structure have zero size. In practice, they have a finite size and the joint eccentricities have to satisfy geometric conditions similar to Eq. 8.42 (You and Pellegrino, 1997a).

8.3.3

Pantographs with Angulated Elements In the solution presented in the previous section, each rod of the pantograph rotates about an axis perpendicular to the axis of n-fold symmetry of the structure. Therefore, in the theoretical packaged configuration (θ = 0) all of the rods become collinear with the axis of symmetry. There is an alternative design for pantographs where each rod rotates about the axis of n-fold symmetry of the structure. First, it should be noted that a circular pantograph made from straight rods lying in a plane, cannot be folded. This can be shown by considering a pair of identical straight rods, hinged by a scissor hinge at E, as shown in Fig. 8.21(a). The end connectors A, B, C, and D define two straight lines OP and OS. This element is symmetric with respect to the line OQ. A relationship between the angle subtended by this element, 2α, and the deployment angle, θ , can be obtained by noting that CG − BF = F G tan α

(8.43)

where θ 2 θ θ BF = BE sin = AE sin 2 2 θ F G = AC cos 2 Substituting Eqs. 8.44–8.46 into Eq. 8.43 gives: CG = CE sin

(CE − AE) sin

θ θ = AC cos tan α 2 2

(8.44) (8.45) (8.46)

(8.47)

from which one obtains: tan α =

CE − AE

θ 2

(8.48) AC It is obvious from Eq. 8.48 that α varies when θ varies. Hence, because the positions of OP and OS are fixed by the rest of the ring, it can be concluded that it is impossible to mobilize the pantographic element ABCD, i.e., to vary the angle θ , if A, D and B, C are allowed to move only along the lines OP and OS (Zanardo, 1986). This difficulty can be resolved (Hoberman, 1990, 1991) by using nonstraight, angulated rods, and hence by moving the pivot E to the new, special position shown in Fig. 8.21(b). It can be shown that for this element: tan α =

CF − AF AC

tan

tan

EF θ +2 2 AC

(8.49)

8.3 Articulated Space Frames

S

S

C Q

Q

243

C α

G F P D

θ/2

E

B

E

P

θ/2

D

F

α

A α

B θ/2 θ/2 α

A α

O

(a)

O

(b)

Figure 8.21 Elements of circular pantographs (You and Pellegrino, 1997b).

For AF = CF , i.e. F is half-way between A and C, the first term on the righthand side vanishes. Hence α becomes constant for all θ ’s, and it is now possible to mobilize the pantographic element with A, D and B, C moving on the lines OP and OS, respectively. Therefore, a deployable ring pantograph has to satisfy the following two conditions: AF = CF and α = arctan

EF AF

from which it follows: ' = ACE ' =α CAE Therefore, the two angulated rods that form the basic element of a circular pantograph have equal semilengths (AE = CE = BE = DE), are kinked by equal amounts, and the kink angle – measured from straight – is equal to the angle subtended by the pairs of angulated rods. This simple and general result allows the design of flat ring pantographs made from angulated elements. A simple example is the ring pantograph shown in Fig. 8.22. It consists of n = 8 identical pairs of angulated rods, each subtending an angle 2α = 45◦ . In the two extreme configurations, in Fig. 8.22(a,c), the geometry of the angulated elements can be intuitively worked out. The basic angulated rod element can be replaced with a multi-angulated rod, with equal kinks at every joint, as shown in Fig. 8.23(a), to obtain two-dimensional ring pantographs with multiple connections. Figure 8.23(b) shows an example with 24-fold symmetry, where each kink has an angle 2α = 15◦ . Variants of this geometry can be found in You and Pellegrino (1997b).

244

Concepts for Deployable Structures

Figure 8.22 Eight-fold symmetric pantograph consisting of 16 angulated rods (You and Pellegrino, 1997b).

8.3.4

Spherical Pantographs A ring pantograph consisting of angulated elements can be designed to subtend an angle smaller than 2π , in which case the ring can only be closed by allowing it to take a threedimensional shape. This basic idea leads to a structure that lies on a cone with angle β, and hence with an angular defect of:   β 2π 1 − sin 2

(8.50)

as shown in Fig. 8.24. An angulated pantograph ring that subtends a constant angle of 2π sin(β/2) can then be designed, by an extension of the method in Section 8.3.3. In fact, the pantograph can be designed so that it approximates to a sphere of radius R, as shown in Fig. 8.25. Consider the pair of angulated rods AEC and BED: joints A, B, C, D lie on a spherical surface of radius R. The connectors at A, B, C, D, E are all cylindrical hinges perpendicular to the plane ABCD and, therefore, the pair of angulated rods is kinematically equivalent to a flat angulated element. When the angle

8.3 Articulated Space Frames

245

P4 A4 P3 A3 S3

P2

C3 A2

P1

S2

C1

α

S1 A1

P0

θ/2 S0 α

A0

α α

αα O

(a)

(b) Figure 8.23 (a) Multiangulated rod and (b) circular pantograph consisting of 48 multiangulated rods (You and Pellegrino, 1997b).

246

Concepts for Deployable Structures

O

O

2p sin(b/2)

b b/2 flattening

r

r

r sin(b/2)

2p r sin(b/2) Figure 8.24 Flattening of conical surface with cone angle β. O

O

O

2π sin(β/2)

β/2

R sin(β/2)

B

A P

B

C A

E Q

D

P (a)

R sin(β/2)

β

C

B

R S

(b)

E A

2πR

D (c)

Figure 8.25 Spherical pantograph dome concept.

between the angulated rods is varied, the connectors A, B move on OP whereas the connectors C, D move on OQ. Note that the connectors of the angulated elements adjacent to this one are perpendicular to a different plane. Hence, the four pins in a general joint of this pantograph ring, which is shared by four pairs of angulated elements, are not parallel.

8.3 Articulated Space Frames

247

Figure 8.26 Iris Dome displayed at the World’s Fair in Germany in 2000 (images used with permission of Hoberman Associates, 2019).

Figure 8.25(b) shows a section of the circular cone defined by A, B, C, D, etc. If the angle subtended by AB is β, then AO =

R sin(β/2)

and ' = β/2 SOP Since the developed surface of the cone shown in Fig. 8.25(c) subtends an angle of 2π sin(β/2), the angle subtended by each angulated element is β 2π sin n 2 This is also the value of the required kink angle for the angulated rods. By designing a series of matching conical pantographs, Hoberman (1991) designed the Iris Dome, shown in Fig. 8.26. The structure shown in Fig. 8.26 consists of six layers of angulated elements and each layer forms a complete pantograph ring lying on a different conical surface.

8.3.5

Hinged Panel Structures A structural concept for a deployable/retractable structure consisting of a series of rectangular panels connected by cylindrical hinges on parallel edges, is shown in Fig. 8.27. In the packaged configuration these panels stack alongside the support structure. In the deployed configuration they form a single, flat structure cantilevering out of the support, and perpendicular to it. Such an arrangement of panels is called a concertina because it resembles the structure of the bellows in an accordion. This structure has multiple degrees of freedom (one free rotation for each panel), but its motion can be controlled in a unique way by synchronizing the motion of alternate panels through cable loops, and by controlling the overall deployment with a continuous cable that runs over free-turning pulleys connected to the hinges of the structure, and connected to a motor at one end. This structural concept is based on the design of the European Retrievable Carrier (EURECA) (de Kam, 1986).

248

Concepts for Deployable Structures

Motion

Figure 8.27 Deployment concept for structure consisting of rectangular panels.

Δφ2 D

φ2 B Motor

Δφ

Δφ2

Δφ1 A φ1 Δφ1

Motion Δφ3 C φ3

(a)

Δφ3

B

D

Δφ1 Motor

A Δφ3 = Δφ1

(b)

C

Figure 8.28 Deployment scheme; (a) deployment/retraction cable; (b) closed cable loop.

8.4 Packaging through Elastic Creasing

249

Figure 8.28 explains how the rotary motion of a single motor, on the left-hand side, is converted into the linear motion of the panels by means of a continuous cable. If the motor turns through an angle φ, thus shortening the cable, the panels AB, BC, and CD will have to turn through angles φ1 , φ2 , and φ3 , respectively, such that φ = φ1 + φ2 + φ3

(8.51)

Note that the only parts of the continuous cable that can vary their lengths are those subtending the angles φ1 , φ2 , and φ3 . Because the lengths of the continuous cable applies applies only one constraint on the three degrees of freedom of the structure, φ1 , φ2 , φ3 , two uncontrolled degrees of freedom are left. Hence, additional constraints are added in the form of two closed cable loops (CCLs). Each CCL is mounted alongside a panel and looped over pulleys that are free to rotate with respect to that panel, but are fixed to the adjacent panels. For example, in the CCL shown in Fig. 8.28(b) the pulley at hinge B is fixed to panel AB, whereas the pulley at hinge C is fixed to panel CD. Provided that this CCL does not slip over the two pulleys, panels AB and CD are thus constrained to remain parallel. Similarly, a second CCL is applied over panel AB.

8.4

Packaging through Elastic Creasing The basic idea is to apply to thin shell structures the type of packaging schemes that were introduced for membranes in Chapter 7, and to use in the fold regions a material that does not suffer any plastic deformation or damage when a localized crease is formed. A possible approach is to embed soft regions within a stiff structure, but potential issues related to structural continuity and interface damage would need to be addressed. An alternative approach is to build dual-matrix thin-walled, fiber-composite deployable structures where the matrix in the crease region is changed to a low-modulus elastomer. The shear modulus in the crease region should be at least 10 times lower than elsewhere. Geometrically, the key feature of this approach is that the fibers on the tension side of the crease form an outer envelope for the material of the crease and the fibers under compression form microbuckles. Note that the thickness of the crease region increases, and there are large shear strains in the matrix. Figure 8.29 shows a sketch that explains this behavior in a thin sheet of stiff unidirectional fibers that has been folded back-to-back over a length of approximately π times the sheet thickness. Forming localized creases without any plastic deformation or damage requires a three-dimensional material deformation instead of the state of plane strain of a smooth elastic bend, of the type discussed in Section 8.2. Actually, the buckled fibers lie within the compression surface of the bent sheet, i.e., buckling is in-plane, not out-of-plane, as shown in the micrographs in Fig. 8.30. For example, the packaging scheme for a dual-matrix conical shell made of plain R is shown in Fig. 8.31(a) and the actual shell, deployed, weave fabric of Astroquartz  flattened and z-folded is shown in Fig. 8.31(b).

250

Concepts for Deployable Structures

Stress Profile lo = π (r + t 2 )

ε=

Tensile Stresses s

lo − li π (r + t 2 ) − π (r − t 2 ) t = = l r πr

l =πr

t

Compressive Stresses

li = π (r − t 2 )

Neutral Axis

Figure 8.29 Fiber microbuckling and stress profile in a heavily bent fiber composite sheet (Murphey et al., 2001).

Figure 8.30 Micrographs of carbon-fiber reinforced silicone sheets, folded 180◦ : (a) side view

(Lopez Jimenez and Pellegrino, 2013); (b) view of inner (compression) surface (Lopez Jimenez and Pellegrino, 2012).

The mechanics of fiber-reinforced elastomers with applications to deployable structures was studied by Francis et al. (2006, 2007). These authors considered the behavior of thin sheets of uniaxial carbon fibers embedded in a soft matrix. They assumed that the fibers buckle in a mode described by: y = a sin

πx λ

(8.52)

where a is the amplitude and λ the half-wavelength, i.e. the distance between two consecutive peaks or troughs in the wave. They determined the following expression for the initial buckle wavelength λ0 from energy considerations, assuming plane sections to

8.4 Packaging through Elastic Creasing

251

(a)

(b)

Figure 8.31 (a) Packaging scheme for a conical shell. (b) Shell made from Astroquartz fabric reinforced with epoxy resin (panels) and silicone rubber (hinge lines).

remain plane. They also assumed the fibers to have a circular cross-section with radius  R and to be arranged according to a square lattice with spacing b = R π/Vf , and obtained ⎛ ⎞1 4 9π 3 Vf t 2 EI ⎝ ⎠

 λ0 = (8.53) 8R 2 log 3tb G where Vf is the fiber volume fraction, t the thickness of the sheet, E and I the modulus and second moment of area of the fibers and G the shear modulus of the matrix. Lopez Jimenez and Pellegrino (2013) studied the failure of brittle fibers in a thin sheet, using weak-link theory. The failure probability P of a fiber subject to uniform strain  is given by a Weibull distribution that is characterized by carrying out multiple tensile tests on individual fibers  m    V (8.54) P (,V ) = 1 − exp − V0 0 where V is the volume of a specific fiber, V0 is the volume of the fiber used for the characterization, 0 is a reference strain for which a fiber of volume V0 has a probability of failure of P = 1 − exp (−1) ≈ 0.632, and m is the Weibull modulus, whose value controls the spread of strength variation. The failure probability in tension of AS4 fibers (AS4 is a PAN-based, high strength, high strain fiber produced by Hexcel) was investigated, and for these fibers the Weibull modulus m = 10.397 with 0 = 1.898 was obtained.

252

Concepts for Deployable Structures

0.8

3.2

0.6

kt

3.1 emax (%)

kt 0.4

3 ef

0.2 0.5

1

1.5 l0 (mm)

2

2.9 2.5

Figure 8.32 Values of κt and f for creases with a single buckle and a failure probability of 1%, for fibers with Weibull modulus m = 10.397 (Lopez Jimenez and Pellegrino, 2013).

For this specific value of m, Lopez Jimenez and Pellegrino generated the crease design chart shown in Fig. 8.32, where it has been assumed that a failure probability of 1% in the most compressed layer of fibers is acceptable. For any value of λ0 , which can be computed from Eq. 8.53 for any given set of material/geometric properties of the sheet, this plot provides the value of κt and also the corresponding value of the maximum strain in the fibers, max , reached at the point of maximum curvature of the buckled fibers. In computing the probability of failure it was assumed that there is only one buckle, in order to eliminate the dependence on the thickness t.

8.5

Concepts for Deployment and Deployment Actuation Deployment actuation concepts can be broadly divided into the following two categories. •

•

Constraint-driven, which includes varying the angle at a joint by turning an electric motor or other kind of rotary actuator, or decreasing the distance between two or more points on the structure by shortening the length of an active cable, or increasing the volume enclosed by an envelope (in the case of an inflatable structure). This type of actuation may be reversible, e.g., by reversing the direction of a motor, but unilateral actions such as cable shortening or volume inflation are not reversible. Energy-driven, in which case deployment has the effect of decreasing the total potential energy of the structure. Examples include the release of strain energy stored throughout the structure or within its joints, or the decrease of the potential energy of the loads acting on the structure (as in the radial expansion of a spinning structure). This category of actuation is typically nonreversible.

There are several kinds of deployment. One of the most intuitive kinds is seen in pantographs structures, where there is a single deployment angle in all pantograph units

8.5 Concepts for Deployment and Deployment Actuation

(a)

253

(b)

Figure 8.33 (a) Two-dimensional pantograph consisting of rods with noncentral joints. (b) Four snap-shots from deployment sequence (Kwan and Pellegrino, 1994).

and a single deployment actuator can be used to control the variation of this angle. If the elastic deformation of the pantograph rods is neglected, the geometric deformation is entirely defined by the deployment angle, through a purely geometric analysis that mimics the actual deployment of the structure. Kwan and Pellegrino (1994) present a derivation of the equilibrium and kinematic matrices, see Section 1.5, for pantograph elements and use these matrices to carry out a series of analyses of a pantograph structure as it deploys. Figure 8.33 shows four snapshots from the deployment of a pantograph that forms a quadrant of a circle in its fully deployed configuration, shown in Fig. 8.33(a). This requires noncentral joints; the pivot is located at distances of 570.4 mm and 370.8 mm from the end joints. All structural mechanisms with a single degree of freedom behave essentially this way, and it is straightforward to determine the relationship between the torque required to deploy the structure and all resisting actions (external loads acting on the structure, friction torques at the joints, etc.) The synchronous deployment of pantograph structures is a special case of controlled deployment, which generally describes a quasi-static process that can be stopped at any step. Inertia forces do not play any role in these processes.

8.5.1

Calculation of Deployment Torque in a Pantograph As a specific example, an analysis of the deployment motor torque for a pantograph structure is presented. More details on this analysis can be found in You and Pellegrino (1996). Figure 8.34 shows an n-module triangular mast consisting of rods of length L. In the present case n = 5. There is a single active cable, going over pulleys attached to the joints of the pantograph, that drives the deployment of the structure. The route of the active cable is shown in Figure 8.35. Three constant-force retraction springs are mounted at the base of the pantograph.

254

Concepts for Deployable Structures

17

16 5

18 14

4 6

13 (b)

15 11

1 2 3

10 12

8

7

L/2

9

L/2 θ

5

(c) L/2

4 6

x

(a)

1 2

L/2

z y

3 Figure 8.34 Triangular pantograph mast: (a) pantograph structure; (b) bottom module; (c) detail of rod pair (You and Pellegrino, 1996).

17

16 18

14

13 15

11

10 12

8

7 9

5

4

1 2 3 Figure 8.35 Route of active cable in pantograph mast (You and Pellegrino, 1996).

It is assumed that the pantograph deploys slowly upwards, against gravity. The tension in the active cable, ta , can be obtained using the Principle of Virtual Work, Section B.7.1. Because the accelerations are small, inertia forces can be neglected; hence, the active cable tension is in equilibrium with the retraction spring forces, the friction torques at the joints, and the weight of the mast (represented by concentrated forces at the joints).

8.5 Concepts for Deployment and Deployment Actuation

255

Consider the mast in a generic configuration, defined by the angle θ . To calculate the value of ta required for static equilibrium in this configuration, a small configuration change of the mast is considered, defined by the change dθ in the deployment angle, together with the associated virtual work. Consider a pair of active cable forces acting on a pair of joints of the pantograph that are directly linked by a variable-length segment of active cable. The external work done by these forces is equal to ta times the change of distance between the two joints, which is L sin(θ/2)(dθ/2). Because there are 6n such pairs of forces acting on the pantograph, and each pair does the same amount of work, the total virtual work done is 6ta nL sin

θ dθ 2 2

(8.55)

The work done by a constant force spring connected to one of the base joints is equal √ to the magnitude of the force, fs , times the displacement of the base joint (L/ 3) sin(θ/2)(dθ/2). Hence, the total virtual work done by the three springs is: √ 3fs L sin(θ/2)(dθ/2) (8.56) Next, the virtual work done by the weight W of the mast is calculated. The z components of the joint displacements are zero for the base joints 1, 2, and 3; they are L cos(θ/2)(dθ/2) for joints 4, 5, and 6, etc. Calculating the corresponding work contributions and adding them up gives: −W (n/2)L cos(θ/2)(dθ/2)

(8.57)

Finally, the work done by the friction torques acting at the rod-joint hinges can be obtained simply by multiplying the total friction torque Tf by the hinge rotation dθ/2, where Tf is the sum of the friction torques at all joints. This gives −Tf (dθ/2)

(8.58)

Friction in the pulleys supporting the active cable is neglected, assuming that low friction bearings would be used. The total virtual work for the pantograph is obtained by adding up the four terms just obtained. The internal work is zero because the rods of the pantograph do not deform at all during the configuration change. The equation of virtual work is therefore, 6ta nL sin

n θ dθ dθ θ dθ √ θ dθ − 3fs L sin − W L cos − Tf =0 2 2 2 2 2 2 2 2

(8.59)

Solving for ta : √ ta =

Tf 3fs W θ + cot + 6n 12 2 6nL sin(θ/2)

where the friction torque can be calculated from: ) )  ) ) fi ) |fi | ∼ μr ) Tf = μr

(8.60)

(8.61)

256

Concepts for Deployable Structures

600 500

μ=1

ta (N)

400 300 μ=0.1

200 100 μ=0.01

0 0

5 θ (deg)

10

Figure 8.36 Variation of active cable tension during initial deployment of pantograph structure, for different values of the friction coefficient (You and Pellegrino, 1996).

where μ,r are the friction coefficient and radius of the joint pins, and fi is the axial force in rod i. Using the Principle of Virtual Work, You and Pellegrino (1996) obtained the expression ) ) nW ) ) (8.62) fi ) = 2nta + ) sin(θ/2) The active cable variation for small values of θ is shown in Figure 8.36 for a pantograph with n = 5,L = 400 mm, r = 2.5 mm, W = 20 N, fs = 18 N, and a total deployed height of 1.401 m. In the figure, note that ta is very large for values of θ on the order of a few degrees. In reality, due to the physical size of the rods of the pantographs, it is impractical to build pantographs that fold down to such small values of θ .

8.5.2

Unconstrained Deployment of Coiled Tape Springs An example of unconstrained deployment (from Seffen and Pellegrino (1999)) is provided by a tape spring of length L, coiled on a free-turning circular spool whose radius is equal to the transverse radius of curvature R of the tape, to match the conditions discussed in Section 8.2.2. The deploying tape spring consists of two parts, as shown in Fig. 8.37: the first part is a variable-length, straight body which is always tangent to the spool; the second part is a uniformly coiled tape spring, also of variable length. Thus, a general configuration of the system formed by the spool and the tape spring can be described by means of the angles θ and ζ defined in Fig. 8.37. ζ is the angle between point C, at the attachment of the tape spring to the spool, and the horizontal x-axis, which is horizontal. θ is the angle subtended by the part of the tape spring that is coiled around the drum, and hence the length of coiled tape at time t is Rθ .

8.5 Concepts for Deployment and Deployment Actuation

257

λL

A

B

P e*

y

θ e ζ O x

ξL C g

R

Figure 8.37 Schematic diagram of a tape spring coiled around a free-turning spool (Seffen and Pellegrino, 1999).

The equations of motion for a coiled tape spring can be derived from Lagrange’s equations (Synge and Griffith, 1970)   d ∂L ∂L = Qi (8.63) − dt ∂ q˙i ∂qi In this equation, qi are the generalized coordinates of the system, i.e., in the present case θ and ζ . Qi are the generalized forces corresponding to any non-conservative forces acting on the system; and L is the Lagrangian of the system, i.e., the difference between total kinetic and potential energies. The total kinetic energy of the system can be obtained by adding the kinetic energy of the straight, extended part of the tape, AB, to the kinetic energy of the coiled part, BC, and to the kinetic energy of the spool. Define a unit vector e going radially from the origin O to the point B separating the extended part of the tape from the coiled part. Also define a unit vector e∗ perpendicular to e. The position vector of a general point P on AB, whose curvilinear coordinate from C is ξ L, is rP = Re + (ξ L − Rθ )e∗

(8.64)

˙ + R ζ˙ e∗ r˙ P = −(ξ L − Rθ )(ζ˙ + θ)e

(8.65)

Differentiating Eq. 8.64

The kinetic energy of AB is given by: TAB =

ρL 2



1 Rθ L

r˙ 2P dξ

(8.66)

where ρ is the mass per unit length of the tape spring. Substituting Eq. 8.65 into Eq. 8.66 and integrating yields   (L − Rθ )3 ρ 2 2 ζ˙ R (L − Rθ ) + (ζ˙ + θ˙ )2 (8.67) TAB = 2 3

258

Concepts for Deployable Structures

A general point within the coiled part of the tape spring, BC, has velocity R ζ˙ tangential to the spool. The kinetic energy of BC is then given by: 1 3 2 ρR θ ζ˙ 2 The kinetic energy of the spool is given by: TBC =

(8.68)

1 2 (8.69) I ζ˙ 2 where I is the polar moment of inertia of the spool. Combining Eqs. 8.67, 8.68, and 8.69 yields the total kinetic energy of the system,   3 1 ρ ˙2 2 2 (L − Rθ ) 3 ˙2 ˙ ˙ (8.70) ζ R (L − Rθ ) + (ζ + θ ) + R θ ζ + I ζ˙ 2 T = 2 3 2 Ts =

The potential energy of the system is given by the strain energy stored in the coiled region, BC, plus the gravitational potential energy of the system. The strain energy stored in the transition regions near A and C will be neglected, as these regions remain unchanged in a steady state deployment. The strain energy in the coiled region is calculated from the strain energy per unit area of tape spring (strain energy density). Since the geometry change of the tape spring is from a developable surface to another developable surface, there is no mid-plane strain and hence only the bending strain energy is non zero. The strain energy density has the expression: ⎤ ⎡  κx

⎢ 1 ⎥ (8.71) μ= κy ⎦ Mx My Mxy ⎣  2  κxy where the moments are obtained from Eq. B.39, and the curvature changes from Eqs. 8.12, 8.13 (for r = R). Substituting these expressions into Eq. 8.71 gives: ⎤⎡ 1⎤ ⎡ 1 ν 0 −R

⎢ 1 1 ⎥ ⎢ 1 ⎥ D(1 ∓ ν) 1 μ= (8.72) 0 ⎦ ⎣∓ R ⎦ = − R ∓ R 0 D ⎣ν 1 2 R2 1−ν 0 0 0 2 and hence the strain energy in the coiled region is obtained by multiplying Eq. 8.72 by the area of the folded region, θ R × αR. Recall from Section 8.2.2 that the − sign corresponds to a tape spring coiled in equal sense and the + sign to a tape spring coiled in opposite sense. The gravitational potential energy is calculated taking the centre of the spool as a reference and assuming that gravity acts vertically down, as shown in Fig. 8.37. Hence, the gravitational potential energy is equal to the sum of the potential energy of the extended tape spring, AB, plus the potential energy of the coiled part, BC. The first term is   L − Rθ cos(θ + ζ ) (8.73) VAB = ρg(L − Rθ ) R sin(θ + ζ ) + 2 and the second term is VBC = 2ρgR 2 sin

  θ θ sin ζ + 2 2

(8.74)

8.5 Concepts for Deployment and Deployment Actuation

259

The total potential energy of the system is obtained by adding Eqs. 8.72, 8.73, and 8.74:   θ θ + V = μR 2 αθ + 2ρgR 2 sin sin ζ + 2 2   L − Rθ + ρg(L − Rθ ) R sin(θ + ζ ) + cos(θ + ζ ) (8.75) 2 Hence, the Lagrangian of the system is given by   (L − Rθ )3 ρ 2 2 1 L=T −V = ζ˙ R (L − Rθ ) + (ζ˙ + θ˙ )2 + R 3 θ ζ˙ 2 + I ζ˙ 2 + 2 3 2   θ θ 2 − D(1 ∓ ν)αθ − 2ρgR sin sin ζ + + 2 2   L − Rθ − ρg(L − Rθ ) R sin(θ + ζ ) + cos(θ + ζ ) (8.76) 2 Finally, differentiating L with respect to θ , and also differentiating with respect to θ˙ and then t, and then substituting into Eq. 8.63 gives the first equation of motion: ρ ρr (L − rθ )3 (ζ¨ + θ¨ ) + (L − rθ )2 (ζ˙ 2 − θ˙ 2 )+ 3 2 ρ (8.77) + D(1 ∓ ν)α − g(L − rθ )2 sin(θ + ζ ) = Q1 2 A second, independent equation of motion is obtained by differentiating L with respect to ζ , and also to ζ˙ and t:   1 2 ¨ 3 ¨ 2˙ ˙ ¨ ˙ ρ R Lζ + (L − Rθ ) (ζ + θ) − R(L − Rθ ) θ(ζ + θ ) + I ζ¨ + 3   θ θ + ρgR(L − Rθ ) cos(θ + ζ )+ + 2ρgR 2 sin cos ζ + 2 2 ρg (8.78) (L − Rθ )2 sin(θ + ζ ) = Q2 − 2 The generalized loads on the right-hand side, Q1,Q2 , represent nonconservative loads. For deployment in air, Seffen and Pellegrino (1999) derived the expression: Q 1 = Q2 = −

C ˙ ζ˙ + θ|(L ˙ (ζ˙ + θ)| − Rθ )4 4

(8.79)

where the air drag coefficient C  is given by: C =

1 α CD ρair 2R sin 2 2

(8.80)

and CD is the drag coefficient for the cross section, ρair is the density of air and 2R sin α2 is the width of the cross-section. Equations 8.77 and 8.78 form a system of nonlinear differential equations that can be integrated numerically. As a special case, consider a tape spring deploying from a fixed spool in a gravityfree, vacuum environment. Hence, substitute ζ = 0, ζ˙ = 0 and ζ¨ = 0 into Eq. 8.77,

260

Concepts for Deployable Structures

and also set g = 0 and Q1 = 0 (because there is no air drag). The differential equation becomes ρ ρR (L − Rθ )3 θ¨ − (L − Rθ )2 θ˙ 2 + D(1 ∓ ν)α = 0 3 2

(8.81)

This equation can be integrated (see Seffen and Pellegrino (1999) for details) to obtain θ˙ 2 =

6D(1 ∓ ν)α θ0 − θ ρ (L − Rθ )3

(8.82)

where θ0 is the angle subtended by the coiled tape spring at time t = 0. Then, denoting by λL the deployed part of the tape spring, see Fig. 8.37, gives θ = (1 − λ) θ˙ = −

L R

(8.83)

˙ λL R

(8.84)

and, assuming that at t = 0 the tape is fully coiled on the spool, i.e., θ0 =  6D(1 ∓ ν)Rα 1 λ˙ = λ ρL4

L R , one obtains

(8.85)

This equation can be integrated to give  6D(1 ∓ ν)Rα √ λ= 4 2t ρL4

(8.86)

Figure 8.38 shows the results of two sets of deployment experiments on a steel tape spring with the properties given in Table 8.1. 1

1

zero air drag

0.8

0.8

CD=1.94

0.6

λ

λ 0.4

0.4

0.2

0.2

0

0

0.5

1 t [s]

(a)

1.5

CD=1.24

0.6

0 0

0.5

1

1.5

t [s]

(b)

Figure 8.38 Deployment plots of tape spring coiled on a fixed spool: (a) opposite-sense bending; (b) equal-sense bending. Solid lines correspond to theoretical predictions, × show experimental measurements (Seffen and Pellegrino, 1999).

8.5 Concepts for Deployment and Deployment Actuation

261

Table 8.1 Properties of steel tape spring. L R t α ∗ M+ ∗ M− CD opposite-sense CD equal-sense

500 mm 15.1 mm 0.086 mm 2.36 rad (135◦ ) 37.1 Nmm 20.2 Nmm 1.94 1.24

In the first experiment, the tape spring was coiled in the opposite sense. A plot of the length of the deployed section against time is shown in Fig. 8.38(a). This figure shows the deployment response predicted by Eq. 8.86 (i.e., without gravity and air drag) which is not very accurate, as well as the best numerical predictions based on Eqs. 8.77–8.80 with ζ = 0, ζ˙ = 0, and ζ¨ = 0. The value of CD = 1.94 provided a good fit with the experimental measurements. In fact, among these two effects air drag is by far the most significant. The effects of gravity are quite small. In the second experiment, the tape spring was coiled in equal sense bending and the air drag coefficient that best fits the experimental results is CD = 1.24. Comparing Fig. 8.38(a) and Fig. 8.38(b), it is interesting to note that the two plots are almost identical, as the higher value of D(1 + ν) for opposite-sense bending, which tends to speed up deployment, is almost exactly compensated by a higher value of CD , which tends to reduce the speed of deployment.

8.5.3

Actuation by Means of a Buckled Column The deployment of as structure can be actuated by pushing apart two points of the structure by means of a buckled, pin-jointed column. This type of actuation is based on an interesting property of the post-buckling behavior of pin-jointed columns, whose compression force varies slowly in the postbuckling regime. Approximately free-free boundary conditions for the dynamic testing of structures can also be achieved through soft springs based on buckled columns, as proposed by Fichter and Pinson (1989). The basic mechanics of these systems are derived from the postbuckling equations for slender, elastic columns, presented in Section B.8. For the elastic column of Fig. B.10, Bazant and Cedolin (1991) derived the following relations for the maximum lateral deflection, vmax , and end shortening of the buckled column, L, assuming small end rotations, θ0 :  L PE (8.87) vmax = θ0 π P     PE 1 (8.88) L = 2L 1 − 1 − θ02 16 P

262

Concepts for Deployable Structures

P / PE e=0 1.0 e=L/100 0.5

0.2

0.4

vmax / L

Figure 8.39 Load-deflection response of axially loaded initially curved cantilevers with different initial tip rotations and zero eccentricity.

Rearranging Eq. 8.87 and substituting into Eq. B.58 they obtained:   π2 2 P = PE 1 + v 8L2 max

(8.89)

which is plotted in Fig. 8.39. The central, transverse deflection of a column with an initial eccentricity e = L/100 is also plotted, and more details on the analysis of this case can be found in Bazant and Cedolin (1991). These authors also derived the relationship: L =

π2 2 v 4L max

from which the relationship between axial force and axial deflection is:   L P = PE 1 + 2L

(8.90)

(8.91)

This result, plotted in Fig. 8.40, is most important from the viewpoint of using this structure as a deployment actuator. In the range P /Pe > 1 the force vs. axial deflection relationship is approximately linear, with slope equal to one half in the nondimensional plot in Fig. 8.40. Substituting the expression for PE , Eq. B.54, into Eq. 8.91 gives the following expression for the axial stiffness of a buckled column: π 2 EI dP = dL 2 L3

(8.92)

For comparison, recall that the transverse stiffness of a cantilever of length L, loaded transversally at the tip is: dP EI =3 3 dv L

(8.93)

8.6 Concepts for Stabilization

263

P / PE 0.5 1.0

1

ΔL / L Figure 8.40 Load-shortening response of axially loaded curved column.

8.6

Concepts for Stabilization Stabilization is the third of the underlying functions required in a deployable structure. The most basic stabilization technique, mainly applicable to articulated space frames (described in Section 8.3) consists in the activation of one or more mechanical latches to lock the deployment degrees of freedom of the structure. Alternatively, if deployment is driven by electric motors, the structure can be stabilized simply by powering off the motors. If the space frame is a single degree of freedom mechanism (m = 1), then changing the kinematic properties of a single hinge is sufficient to making the structure rigid. However, to achieve sufficient stiffness in a large structure requires the activation of multiple latches, distributed throughout the structure. The latches may be mechanical devices, such as spring-loaded pins or clips that lock moving parts of the structure, but may also be structural components that perform an equivalent function, as described in the next two subsections. An alternative approach is to induce a state of prestress in the structure by applying stabilizing forces to it. For example, a very thin structure can be stabilized by centrifugal forces, as discussed in the last subsection.

8.6.1

Stabilization through Snap Back: Tape Springs The prototypical lightweight deployable structure that exploits snap back is the tape spring, already introduced in Section 8.2.2. By an obvious extension of equal sense and opposite-sense coiling, defined in that section and shown in Fig. 8.9, a tape spring can be bent either in equal sense, or in opposite sense, as shown in Fig. 8.41. For equal sense bending (longitudinal and transverse curvatures are of equal sign) the edges of the tape spring are subject to compressive stresses and the applied bending moment is defined to be negative. Conversely, for opposite-sense bending (longitudinal and transverse curvatures are of opposite sign) the edges of the tape spring are subject to tensile stresses and the applied bending moment is defined to be positive.

264

Concepts for Deployable Structures

Small rotations

Large rotations M

M

M Opposite sense (M > 0, θ > 0)

M

M Equal sense (M < 0, θ < 0)

M M

M

Figure 8.41 Two ways of bending/folding a tape spring.

Figure 8.42 Apparatus for measuring the moment-rotation relationship of a tape spring.

The details of the moment-rotation relationship for a tape spring have been studied with the large-rotation test apparatus shown in Fig. 8.42. 200 mm long tape springs were potted with epoxy resin into acrylic blocks. The left-hand-side block was connected to a gear box fixed to the base of the apparatus. The right-hand-side block was connected to an identical gear box, but mounted on a linear bearing so that it could move towards the first gear box when the large-deflection bending of the tape causes the distance between the end blocks to decrease. The adaptors that connect the blocks to the gear boxes consist of a top U-shaped element supported by a short, thin-walled circular shaft with strain gauges mounted on its surface. These strain gauges measure the torque in the shaft, which is equal to the bending moment applied to the tape spring.

8.6 Concepts for Stabilization

500

265

Mmax +

400

M [Nmm]

300 200 100 0

M+*

M*-

–100 Mmax -

–200 –3

–2

–1

0

1

2

3

[rad] Figure 8.43 Moment-rotation relationship for a beryllium-copper tape spring with R = 13.3 mm, α = 106◦ , and t = 0.1 mm. The solid line shows numerical simulation results; experimental

measurements are denoted by “×” (loading) and “+” (unloading). (Seffen and Pellegrino, 1999).

The elastic compliance of the apparatus is small in comparison with that of a shallow tape spring. The input shafts of the gear boxes are turned manually, through small knobs and the angle turned through by each block is displayed on a dial. The tape spring rotation, θ , is defined as the rotation of the right end of the tape spring with respect to the left end, and is equal in magnitude to the sum of the two gear box rotations. Figure 8.43 shows the measured behavior of a 200 mm long tape spring and the test measurements are overlaid with a finite-element simulation (Seffen and Pellegrino, 1999). When an initially straight tape spring is subject to gradually increasing equal and opposite end rotations, initially it takes a uniform longitudinal curvature. Its momentrotation relationship is linear for sufficiently small rotations. In the case of oppositesense bending, when the end rotations are increased the tape spring suddenly snaps and forms an elastic fold that is straight in the transverse direction and has uniform longitudinal curvature, Fig. 8.43. Then, if the rotations are further increased, the arc-length of the fold increases while its curvature remains constant. If, however, the tape spring is subject to equal-sense bending, it deforms by gradually twisting over two separate regions whose lengths grow until the two folds merge into a single, localized fold, Fig. 8.43. Once this single fold has formed, further increasing the end rotation results – again – only in an increase of the arc-length of the fold region. Important characteristics of the described behavior are that: (i) the elastic folds for both senses of bending have a longitudinal radius of curvature that is independent of the end rotations imposed on the ends of the tape spring; (ii) the peak stresses and strains in a folded tape spring are largely independent of the end rotations imposed on the ends; (iii) upon unfolding, a tape spring snaps back into the straight configuration, this is most

266

Concepts for Deployable Structures

noticeable for a tape spring that has been bent in the opposite sense. This last characteristic makes tape springs ideal self-latching components in deployable structures. For each sign of M, in Fig. 8.43 there are two characteristic values of the moment, the maximum moment M max and the steady-state moment M ∗ at which the fold grows in length under a constant moment. Both of these values can be obtained from the expression of M(θ ) for uniform longitudinal bending. An analysis of uniformly deformed, laminated shells (Wuest, 1954; Yee et al., 2004) leads to the following expression for the end moment:  s/2 M( κx ) = (Mx + Nx w)dy −s/2

= sD11



β 1 + −β r R



β 1 + R r





βR F1 + 1 + r

2

r F2 R2

(8.94)

where s = 2R sin(α/2), β = D12 /D22 , and 2 cosh λ − cos λ λ sinh λ + sin λ F1 sin λ sin λ − F2 = 4 (sinh λ + sin λ)2 ns λ= r F1 =

(8.95) (8.96) (8.97)

and Dij are the bending stiffnesses of the laminate, see Appendix B.5.3. Figure 8.44 shows a plot of M based on Eq. 8.94, for a specific tape spring. Note that the function M( κx ) has a characteristic up-down-up relationship with a local max . This maximum maximum, which corresponds to the first desired parameter M+ moment occurs when a localized fold begins to form, and this value coincides with

Figure 8.44 Moment vs. curvature plot showing Maxwell construction for 0.42 mm thick, 2-ply plain weave T300 carbon fiber reinforced plastic, subtending an angle α = 80◦ (Seffen and Pellegrino, 1999).

8.6 Concepts for Stabilization

267

the maximum moment for purely cylindrical deformation of the shell. Therefore, the max can be calculated by finding the maximum of M with respect maximum moment M+ to  κx (= 1/r). The steady-state propagation moment has a value such that the two shaded areas in Fig. 8.44, A1 and A2 , are equal. This equal-area construction, known as ∗ . Maxwell’s construction explains Maxwell’s construction, determines the value of M+ why the moment remains constant when the tape-spring is folded and the rotation of the ∗ , which elastic hinge is varied. However, there is a more direct way of calculating M+ will be described later on. ∗ , for opposite-sense folding, can be obtained The negative steady-state moment, M− max by the same method, but M− cannot be obtained from this method because the torsional instability is not captured. ∗ and M ∗ can be obtained by observing that Simple analytical predictions of M+ − the fold region is approximately cylindrical, i.e. the transverse radius of curvature in the fold region is zero, and from the following analysis, based on this observation (Calladine, 1988). Consider a tape spring based on a symmetric laminate (of which an isotropic tape spring is a special case). These laminates have B = 0, and hence Eq. B.42 can be decoupled into membrane and bending effects. During folding the tape spring undergoes changes of principal curvature (note that there is no twist along the longitudinal and transverse directions, i.e.  κxy = 0):   1 1 κy ) = ∓ , (8.98) ( κx , r R where the positive and negative signs in the first term correspond to equal and opposite sense bending, respectively. The total bending strain energy over the area of the fold region, equal to Rrθ α, is 

 κx Rθ α (8.99) U= M x My 2  κy Substituting Eqs. B.39 and 8.98 into Eq. 8.99, and minimizing the total bending energy with respect to r yields the following expression for the radius of the fold region (Schulgasser, 1992):  D11 R (8.100) r= D22 The steady-state moments for equal and opposite sense bending of a tape spring are thus obtained by substituting Eqs. 8.100 and 8.98 into Eq. B.39, and multiplying by the arc length of the cross-section, Rθ :  ∗ = ( D11 D22 + D12 )α (8.101) M+  ∗ M− = (− D11 D22 + D12 )α (8.102)

Concepts for Deployable Structures

800 α = 100 deg

700 Moment (Nmm)

268

600 500 80 deg

400 300

60 deg

200 40 deg

100

Flat strip

0 0

0.02

0.04 0.06 Curvature (1/mm)

0.0

.1

Figure 8.45 Moment vs. curvature relationships for tape springs with different subtended angles but equal cross-sectional arc length (Yee et al., 2004)

Mmax +

A E

∗ M+

B M −∗

θ

max

M−

Figure 8.46 Simplified moment vs. rotation relationship for tape spring.

If, in addition to being a symmetric laminate, the tape spring has equal bending stiffnesses D11 = D22 , which of course is the case for isotropic tape springs, then, Eq. 8.100 yields r = R.

(8.103)

Figure 8.45 shows the effect of varying the subtended angle for a particular laminate (two-ply of plain weave carbon-fiber fabric [0◦,90◦ ]2 made of T300-3K/913, while maintaining the arc length of the cross-section constant and equal to Rα = 14 mm; the thickness of this laminate is 0.42 mm). The characteristic tape spring behavior for opposite-sense bending, involving a sudden snap accompanied by the formation of an elastic fold, requires the moment-curvature relationship to have the up-down-up shape of Fig. 8.44. Note that when the subtended angle is small, typically less than ∼ 40◦ , the “down” part of the curve disappears for this particular laminate, and hence the tape spring behaves like a flat strip. Figure 8.46 shows an idealized version of the moment-rotation diagram originally shown in Fig. 8.43. In addition to capturing the fold localization behavior already

8.6 Concepts for Stabilization

269

L

end block tape spring h

Figure 8.47 Self-deploying and self-latching hinge consisting of two tape springs (Pellegrino, 2015. Image reproduced with permission of CISM, 2019).

described above, corresponding to the downward jump AB, this diagram highlights a key feature of tape springs. When θ is decreased, the upward jump DE corresponds to the tape spring snapping back to a near-straight configuration, i.e., effectively self-latching. This simple diagram emphasizes the asymmetric behavior of tape springs, which are “strong” for positive moments and “weak” for negative moments. To achieve a symmetric behavior, elastic hinges that are both self-deploying and self-locking are formed from multiple tape springs, mounted “in parallel” and with the convex sides facing in opposite directions. A simple version is shown in Fig. 8.47 and more details about self-locking hinges of this kind are available in Watt (2003).

8.6.2

Stabilization through Prestressing An alternative to the termination of deployment by the activation of mechanical latches or snap-back devices is the activation of unilateral constraints and the preloading of these constraints, to produce a structure that has a linear load-deflection behavior within the load range for which the reactions at the constraints do not exceed the preload. This explanation may seem rather abstract and, as a practical example, we consider the push chain. This is a mechanical device consisting of interlocking chain links that can freely rotate in one direction, allowing the links to form a compact coil, but in the opposite direction the links cannot rotate beyond the straight configuration. Rotation of the links beyond the straight configuration is prevented by stops embedded in the links, as shown in Fig. 8.48(a). This example illustrates the use of compression-only elements to terminate deployment of a coilable boom, as shown in Fig. 8.48(b). A complementary approach is to use tension-only elements, such as cables or membranes. This approach is illustrated by the cable-stiffened pantograph mast (Kwan and Pellegrino, 1993; You and Pellegrino, 1996). Here, the starting point is a pantograph structure, of the kind already discussed in Section 8.3.1. In addition to the pantograph structure, a cable-stiffened pantograph contains a set of short passive cables that become taut when the pantograph is fully deployed. The passive cables are pretensioned by an active cable, which was also already discussed in Section 8.5.1 as an actuator for the deployment of pantographs,

270

Concepts for Deployable Structures

(a)

(b) Figure 8.48 (a) Interlocking chain links of a push chain; (b) push-chain boom made of carbon

fiber links.

but in fact the route of the active cable can be chosen in a special way, such that it achieves the prestressing of the passive cables. Functionally, the passive cables replace the latching system required in Section 8.6.1. In order to understand how the pantograph, the active cable and the passive cables interact to set up a state of self-stress, consider initially the one-module two-dimensional pantograph shown in Fig. 8.49, with two identical rods of length L, connected by a scissor joint in the middle. During deployment θ varies from θi (the initial deployment angle, which is close to zero) in the folded configuration to θf in the fully deployed configuration. Joints A and B and joints C and D get closer during deployment and may thus be connected by active cables while joints A and C and joints B and D get further apart, and so can be connected by passive cables. As the pantograph has only one mechanism, one passive cable is sufficient to terminate deployment. However, the structure would then have to rely on the bending stiffness of the rods to resist a general applied load, which in general is not an efficient use of the structural members. To increase the stiffness of the fully deployed structure it is often better to have many passive cables, and to ensure that all cables become prestressed when the structure is fully deployed. For example, consider using two passive cables and one active cable. The passive cables, of length L sin(θf /2), link joints A and C, and joints B and D. The active cable runs continuously from joint A to B, to C, and to D.

8.6 Concepts for Stabilization

B

A

A

tp θf

C

D

fr

tp C

D

(a)

C

ta

ta

fr+ta

B

A

271

t fr+ta a fr

ta

tp

ta D

ta

(b)

B

tp

(c)

Figure 8.49 Single-module, two-dimensional cable-stiffened pantograph: (a) folded; (b) deployed; (c) state of self-stress.

A

B

C

D

E

F

(b)

(a)

(c)

Figure 8.50 Two-module, two-dimensional cable-stiffened pantograph: (a) pantograph and passive cables; (b and c) different routes for active cable.

To have a pretension tp in the passive cables, vertical equilibrium of node A, see Fig. 8.49(b), requires a compressive force fr in rod AD: fr =

tp sin(θf /2)

(8.104)

Horizontal equilibrium of A then gives the tension ta in the active cable: ta = tp cot(θf /2)

(8.105)

Next, consider a pantograph with more than one module; the above solution can easily be modified for these more complex structures. Consider, for example, the two-module structure shown in Fig. 8.50(a). Four passive cables connect joints A to C, C to E, B to D, and D to F. Figure 8.50(b and c) shows two possible routes of the active cables. Figure 8.50(b) simply reproduces the one module solution for both the top module

272

Concepts for Deployable Structures

ABCD and the bottom module CDEF. In this case there are two active cables running from C to D providing a total tension of 2ta ; the factor 2 is because a tension ta is needed for equilibrium of both the top and bottom modules. Figure 8.50(c) shows an arrangement that uses only one active cable. In this case, the tension 2ta is provided by a cable loop which runs twice between nodes C and D. A natural extension of this approach can be followed to determine the active cable routes for three-dimensional pantographs (You and Pellegrino, 1996). For the specific case shown in Fig. 8.35, it can be shown that this particular active cable route leads to uniform pretensioning of all vertical passive cables.

8.6.3

Minimum Prestress Level A key question in the case of structures stabilized by imposing a state of prestress is: How large should the prestress be? Of course, the prestress should be sufficiently large to avoid that any preloaded interface loses its pre-compression, and any cable element loses its pretension, during operation of the structure. This will avoid a sudden decrease in the stiffness of the structure, due to one or more joints becoming loose or cables going slack. A more subtle situation arises, for example, in the case of thin membranes, which are often packaged by forming creases in the material. The fundamental issue is that, once a material crease has been formed in a thin film, no amount of subsequent prestressing can fully “iron out” the crease (Greschik and Mikulas, 1996), and hence neither the film surface will ever regain its pristine smoothness, nor its stiffness will ever return to its original value. For the case of tightly packaged membranes that are only lightly loaded after deployment, it can be assumed that there is no plastic deformation of the film after deployment of the membrane. Papa and Pellegrino (2008) carried out experiments on a 25 μm thick aluminized Kapton, 500 × 500 mm2 square membrane, tightly creased according to a five by five Miura-ori pattern, see Section 7.4. The residual half crease angle, Fig. 8.51, after deployment but with the membrane still unloaded, was measured as θ0 =

180 − η0 = 13.8◦ 2

(8.106)

The deployed membrane was loaded with equal diagonal forces at all four corners and both the corner load-displacement relationship, Fig. 8.52, and the shape of the membrane, Fig. 8.53, were measured. These measurements showed that the membrane stiffens considerably, although it is still much more compliant than a pristine membrane, after a displacement of about 13 mm, when the corner loads are approximately 0.02 N. A similar trend has been observed also in randomly creased membranes (Murphey and Mikulas, 1999). These tests have shown that the out-of-flatness of the membrane can be decreased by preloading it with corner forces, and the remaining out-of-flatness depends on the level of preload.

8.6 Concepts for Stabilization

273

10

Scan Amplitude (mm)

8 6 4 2 0 –2 –4

θ0

–6 –8

–10 175

180 185 190 195 Distance Perpendicular to Crease (mm)

Figure 8.51 Measured profile of a crease in 25 μm thick Kapton (Papa and Pellegrino, 2008).

0.2

T (N)

0.16 0.12 0.08 0.04 0

0

2

4 6 8 10 Displacement (mm)

12

14

Figure 8.52 Load-displacement relationship for corner of creased membrane; insert shows magnified response over a very small displacement range (Papa and Pellegrino, 2008).

T

T

T

T

Figure 8.53 Deformed Kapton membrane, with out-of-plane deformations amplified by a factor of 10 (Papa and Pellegrino, 2008).

274

Concepts for Deployable Structures

8.6.4

Creased Beam Model A simple model to predict the in-plane stiffness of a membrane with periodically distributed creases, e.g., arranged according to the Miura-ori pattern, can be based on the following three assumptions. First, based on the observation that the maximum stresses in the membrane are well below the yield stress of the material, linear-elastic material behavior is assumed. Second, stretching of the membrane mid-plane is neglected, and hence only bending deformation mode considered. Third, it is assumed that parallel edges of the membrane remain parallel throughout, and hence that the behavior of the creased membrane can be captured using only the displacement components dx and dy , defined in Fig. 8.54. The model consists of two uniform creased beams, parallel to the x and y axes. Consider the creased beam ABC shown in Fig. 8.55. Its deflection-force relationship, δ = δ(P ), can be determined as a function of the length, L, rise, a0 , and flexural stiffness EI of the beam. The beam is assumed to be inextensional. Given δ, one can determine the average longitudinal strain along AC and then the force P . It is sufficient to analyze only the half-beam AB, subject to appropriate boundary conditions at B. Denoting by w(ξ ) the deflection of the beam, which includes the initial T

T

45o

o

45

dy/2 dy

T

kX dx kY

dy/2 45o

45o

T

dx/2

dx/2

(b)

T

(a) Figure 8.54 (a) Miura-ori membrane deformation due to corner loads T (thicker lines) and

its initial configuration for T = 0 (thinner lines); (b) simple two-parameter model (Papa and Pellegrino, 2008).

B

w

A

P, δ

ao

ξ

C L

Figure 8.55 Creased beam model (Papa and Pellegrino, 2008).

8.6 Concepts for Stabilization

275

deflection w0 = 2a0 ξ/L and the additional, elastic deflection w1 due to P , the bending moment is given by   2a0 ξ M = Pw = P + w1 (8.107) L Substituting the moment-curvature relationship for small rotations of a beam, M = EI

d 2 w1 dξ 2

(8.108)

gives d 2 w1 P = 2 EI dξ



2a0 ξ + w1 L

 (8.109)

This equation can be simplified by introducing the nondimensional ratio between P and the Euler buckling load of a straight column of length L, from Eq. B.54,  L P α= (8.110) π EI Substituting Eq. 8.110 into Eq. 8.109 and rearranging gives   2a0 L 2 d 2 w1 − w1 = ξ 2 πα L dξ

(8.111)

The solution of this equation, having included the boundary conditions w1 = 0 at ξ = 0 and dw1 /dξ = 0 at ξ = L/2, is   2a0 sinh(παξ/L) 2a0 − ξ (8.112) w1 = π α cosh(π α/2) L Hence, the deflected shape of the beam is given by   2a0 sinh(παξ/L) w = w0 + w1 = π α cosh(π α/2)

(8.113)

The end displacement of the beam can then be found by integrating along ξ the strain due to w and then subtracting the initial displacement due to creasing, −2a02 /L. This gives  L/2   2a 2 du δ=2 dξ + 0 (8.114) dξ L 0 where du/dξ can be found by noting that, since the Green strain, G , along ξ has to vanish in an inextensional beam,   du 1 dw 2 =0 (8.115) + G = dξ 2 dξ Equation 8.115 can be rewritten as du 1 =− dξ 2



dw dξ

2 (8.116)

Concepts for Deployable Structures

which can then be substituted into Eq. 8.114. Differentiating w in Eq. 8.113, and then solving the integral gives:   sinh(π α) + π α a02 δ = 2− (8.117) π α cosh2 (π α/2) L Then, dividing by L gives the average strain along AC:   δ sinh(π α) + π α a02  = = 2− L π α cosh2 (π α/2) L2

(8.118)

To predict the load-displacement behavior of a corner of the previously tested √ 500 mm × 500 mm Kapton membrane, it can be assumed that P = T / 2, i.e., each corner force is shared equally between the two sets of creased beams, with T varying between 0 and 0.2 N. The bending stiffness of the creased beams can be computed assuming a rectangular cross-section beam with the breadth and thickness equal to the full width and thickness of the membrane, respectively. Hence, 500(25 × 10−3 )3 wt 3 (8.119) = 3530 × = 2.298 N mm2 12 12 The geometry of the creased beam depends on the panel side lengths l(= 100 mm) of the Miura-ori pattern and on the fold semi-angle θ = θ0 = 13.8◦ . The details of the calculation can be found in Papa and Pellegrino (2008). Figure 8.56 shows the variation of this prediction of the total corner displacement d with T . The corresponding displacements obtained from a finite-element solution are shown for comparison. Compared to the finite element model, the theoretical model overestimates the total corner displacement by less than 6%. For the case of membranes that are more highly loaded after deployment, this analysis could be extended to consider elasto-plastic effects. Intuitively, one would expect that an EI = E

0.2 Theoretical FE

0.16 T (N)

276

0.12 0.08 0.04 0

0

5

10

15

d (mm) Figure 8.56 Corner displacement predicted by theoretical model and finite element analysis (Papa and Pellegrino, 2008).

8.6 Concepts for Stabilization

277

σt Elastic Deformation

ΔL/2

Plastic Deformation

0.4

t = tape thickness E = Young’s modulus

Crease tape 0.3

σt = normal stress σy = yield stress

ΔL t√E/σt

σt

0.2

ΔL/2

ΔL= tape shortening

0.1

0 0.1

0.2

0.5

1.0

2.0

5.0

10

20

50

100

σtE σy2 Figure 8.57 Change of length of elasto-plastic tape with initial crease θ0 = 90◦ , due to an applied

tension (MacNeal and Robbins, 1967).

elasto-plastic membrane can be pulled sufficiently hard that the creases can be “ironed out.” This intuition was confirmed by an analysis of the shortening, L, from straight of a tape made of elastic-perfectly-plastic material and with an initial crease angle of 90◦ , under a remote tensile stress σt (MacNeal and Robbins, 1967). This analysis shows that L decreases asymptotically with σt , as shown in Fig. 8.57.

8.6.5

Stabilization of Membranes through Spinning To conclude this chapter, we consider a thin membrane disk that is deployed and stabilized by centrifugal action, thus exploiting the purely tensile distribution of stress in a flat disk, whose radial and circumferential components are given by: 3+ν 2 2 ρω (b − r 2 ) 8   3 + ν 2 2 1 + 3ν 2 σθ = ρω b − r 8 3+ν σr =

(8.120) (8.121)

and have been plotted in Fig. 8.58. However, there are many deployable structures applications, such as solar sails, in which a transverse load is applied to the disk, as shown in Fig. 8.59, and in this case the circumferential stress around the edge, which already has the lowest value, is further decreased, and can approach zero if the magnitude of the load is sufficiently large.

278

Concepts for Deployable Structures

1 0.8 σθ 8σ (3+ν)ρω2b2

0.6 σr 0.4 0.2 0

0

0.2

0.4 0.6 r/b

0.8

1

Figure 8.58 Variation of stress components in spinning disk.

ω

b

q θ a

h

Figure 8.59 Geometry of transversely loaded spinning membrane (Delapierre et al., 2018).

Therefore, depending on the magnitude of the transverse load q, there is a minimum angular velocity that is required to avoid edge compression (wrinkling). The solution of this problem can be described in terms of three nondimensional parameters, related to the geometry of the membrane, the angular velocity, and the transverse loading. They are defined as: α= = G=

a b 

(8.122) ρh 2 b ω D

b4 q Dh

(8.123) (8.124)

where D = Eh3 /12(1 − ν 2 ) is the flexural stiffness of the membrane. The load G at which the circumferential stress on the outer edge becomes negative, which corresponds to edge wrinkling in the case of a membrane without any bending

8.6 Concepts for Stabilization

279

ν = 0.1 ν = 0.2 ν = 0.3

0.3

0.2 kcrit 0.1

0

0.2

0.4

0.6

0.8

1

α Figure 8.60 Variation of kcrit with α and ν.

10 6

Membrane Plate Finite elements

10 4 2 G 10

10 0 10–2 –2

10

–1

10

0

10 Ω

10

1

10

2

Figure 8.61 Comparison of buckling loads of spinning membranes, based on membrane theory, plate theory, and finite element simulations (Delapierre et al., 2018).

stiffness, is proportional to the cube of  and depends only on the nondimensional outer radius α, and on the Poisson’s ratio of the membrane (Simmonds, 1962):  kcrit (α,ν) (3 + ν)3 3 # $  (8.125) G= 192 1 − ν 2 where kcrit has been plotted in Fig. 8.60. For thicker membranes, bending effects need to be accounted for, and wrinkling becomes independent of G for  ∼< 10 (Delapierre et al., 2018). The transition in behavior between membranes and plates is shown in Fig. 8.61, which also shows that the pure membrane theory and the plate theory converge at values of  >∼ 10.

9

Applications of Deployable Structures

9.1

Introduction A series of examples from applications of deployable spacecraft structures of three specific kinds, masts and booms, reflector antennas, and large mirrors, illustrate the practical implementation of the basic concepts presented in Chapter 8. Packaging through elastic deformation, a concept presented for both rods and shells in Section 8.2, is embodied in coilable masts and booms (Section 9.2.1–9.2.2) and elastically foldable reflector antennas (Section 9.5). The concept of space frames whose internal mechanisms allow a desired packaging motion, presented in Section 8.3.1, is used for deployable masts in Section 9.2.3, and the concept of using a motor-controlled active cable to drive the deployment of a space frame, presented in Section 8.5.1, is adopted in the AstroMesh antenna in Section 9.4. The concept of stabilization of the internal mechanisms of a space frame, by prestressing a series of cables, presented in Section 8.6.2, is adopted in the tension truss antenna, in Section 9.4. Lastly, the concept of stabilization through spinning, presented in Section 8.6.5, was used in the Znamya flight demonstration and in the IKAROS solar sail, as discussed in Section 9.7.

9.2

Deployable Booms and Masts Deployable masts are slender deployable space frames that have a variety of applications, such as supporting RF antennas or deploying large solar arrays. Deployable booms are tubular deployable structures for similar applications, and they are also used as deployable RF antennas. This section present four different types of deployable booms and masts, and is concluded by a comparison of structural and mass efficiency.

9.2.1

Coilable Masts An efficient deployable mast was invented by Webb and Mauch of the Astro Research Corporation (Webb and Mauch, 1969). The inventors observed that a lattice column consisting of three longitudinal rods (longerons) braced at regular intervals has low dead weight and can support high loads. The chosen bracing consists of equally spaced triangular battens forming identical, transverse bays, and of pairs of prestressed cables

280

9.2 Deployable Booms and Masts

281

Longerons Bracing cables

h

Batten

Figure 9.1 Schematic diagram of coilable mast.

(a)

(b)

Figure 9.2 Coilable masts that are deployed by: (a) releasing a cable and (b) rotating the inner

part of a canister (images reproduced with permission of Northrop Grumman).

bracing the three faces of each bay. The resulting structure is shown schematically in Fig. 9.1. Note that, because the longerons have small cross-sectional areas, the wind resistance of the mast is minimized. Figure 9.2 shows two practical realizations of the coilable mast. Note that in Fig. 9.2(b) the battens in the extended part of the mast are in a buckled shape, as

282

Applications of Deployable Structures

they have been built longer than the distance between the longerons. This intended “imperfection” has the effect of introducing a state of precompression in the battens and a corresponding state of pre-tension in the bracing cables. Also note that during coiling the battens have to buckle to an even greater extent. The use of buckled battens in this structure provides an example of using a buckled Euler column to apply an approximately constant compression force between two points, as discussed in Section 8.5.3. How is the height, h, of the bays decided? First, for the bracing members to work efficiently, the faces of each bay should be approximately square, i.e. h/ ≈ 1. Second, the longerons should ideally be coiled without leaving any gaps and hence one would like to arrange the joints between the longerons and the battens on a helix, as can be seen in Fig. 9.2(b), rather than stacked on top of one another. This undesirable stacking would occur, for example, for h = 2π R/3, which in fact corresponds to rather tall bays. Third, the local buckling load of the longerons should be a bit smaller than the overall buckling load of the structure, loaded under axial compression. In practice, in mast design this is the most critical requirement and, typically, it is assumed that h/ ≈ 2/3 (Crawford, 1971). As an example, consider a coilable mast, with 4 mm diameter S-glass longerons at the corners of an equilateral triangle, that is loaded in axial compression. The circumscribed circle to the triangle has a diameter of 2R = 250 mm; from geometry  = 216 mm. If there are 50 bays of height h = (2/3) = 143 mm, the total length of the coilable mast is L = 7150 mm. The “global” buckling load of such a mast – for the whole structure buckling as a cantilever with an effective length of 2L – is P1 =

π 2 EIC π 2 × 52000 × 2.9 × 105 = = 728 N (2L)2 (2 × 50 × 143)2

(9.1)

Here, IC is the second moment of area of the three longerons about an axis through the centroid of the equilateral triangle whose corners coincide with the longerons, and E = 52 GPa is the composite modulus of S-glass rods. The “local” buckling load of the mast for the three longerons in a bay buckling into an S-shape is obtained, assuming that all three longerons buckle simultaneously P2 = 3

π 2 EIL π 2 × 52000 × 12.6 =3 = 949 N 2 h 1432

(9.2)

where IL is the second moment of area of a single longeron. Therefore, the maximum axial load that can be carried by this particular mast is given by the lower of P1 and P2 , i.e., 728 N. This value could be increased, for example, by reducing the height of the bays, but the number of bays would have to be increased correspondingly. Having established how to design the main load-carrying members of a coilable mast, it is important to understand the behavior of the coilable mast during deployment and retraction. A set of detailed experimental observations was made by Natori et al. (1986) on a shorter mast than that described above, but with similar properties.

9.2 Deployable Booms and Masts

283

1.0

P/Pcr 0.8 2 0.4 1

2

3

4

3 0.2

0.0 0 5

6

7

4 56 7 8

1 d/2πr

8

(a)

2

1 (b)

Figure 9.3 (a) Photographs of folding sequence of a coilable mast and (b) corresponding

force-displacement plot (Natori et al., 1986).

A series of photographs taken during the course of a compression test is shown in Fig. 9.3, together with a plot of the measured load-displacement response. The photographs show that coiling of the mast is initially triggered by local buckling near the tip. Then, its deformation localizes in the last bay, which snaps through and collapses in torsion. As the mast is further compressed, further bays next to the one that has buckled snap through and gradually a uniform helical region forms in the tip part of the mast. After further compression, this helical region forms a fully compacted coil that is joined to the lower, relatively undeformed part of the mast, by a transition region of complex shape. From then on, further compression induces a smooth “growth” of the coiled region, as the transition region is pushed further down the mast. Note, Fig. 9.3(b), that the axial load on the mast reaches an initial peak but, having rapidly decreased as soon as the snap-through sequence begins, it remains approximately constant. A detailed study, including numerical simulations, of the behavior of a similar mast was carried out by Eiden et al. (1987). Coilable masts have been extensively used on spacecraft. A provider of these structures lists applications (Northrop Grumman, 2018) ranging from high-precision optical systems to very long (>100 m) ultralight (boom linear density 0 2 4π R

(9.18)

Then, the minimum required pressure in the torus can be obtained by setting Nθ = 0 in the above equation, and then taking an adequate margin. The size and inflation pressure of the inflatable struts for the inflatable antenna are determined by considering the natural frequency of the whole reflector (Mikulas and Cassapakis, 1995).

9.7

Deployable Mirrors and Solar Sails Due to the ease with which membrane structures can be compactly packaged, they have been chosen for many applications where high shape accuracy is not a key requirement, most notably space mirrors and solar sails. A series of orbiting deployable mirrors to reflect sunlight and extend the hours of daylight in northern parts of Russia were developed under the Znamya project, during which two flight tests were carried out in 1993 and 1999. The first one was the Znamya 2 experiment, a 20 m diameter space solar mirror made of 5 μm-thick Mylar film consisting of 8 gores which were separately folded around hub mechanisms. This membrane mirror was deployed by spinning the hub at about 20 rpm. Figure 9.29 is a photograph, taken from the MIR space station, showing the shape that was taken up by the membrane after deployment: note that the edges of the gores are not straight because,

304

Applications of Deployable Structures

Figure 9.29 Mirror membrane deployed by centrifugal forces in the orbital experiment Znamya 2 (Melnikov and Koshelev, 1998).

due to the discontinuity of the membrane in the hoop direction, wrinkling occurs near the edges of each gore (Melnikov and Koshelev, 1998). The mirror deployed successfully, and, when illuminated, produced a 5 km wide bright spot, which traversed Europe from southern France to western Russia at a speed of 8 km/s. The bright spot had a luminosity equivalent to approximately that of a full moon. The mirror was deorbited after several hours and burned up while reentering the atmosphere, over Canada. The second test structure was Znamya 2.5, deployed on February 5, 1999. It had a diameter of 25 m, and was expected to produce a bright spot 7 km in diameter, with luminosity between five and ten full moons. However, soon after deployment, the mirror became caught on an antenna on the MIR space station, and was ripped. A larger structure, Znamya 3, with a diameter of around 60 m was later planned but did not materialize. Another application of thin membranes is solar sails, large ultra-thin mirrors that obtain a propulsive force from the momentum of radiation pressure exerted by sunlight. The pressure on a perfectly reflective sail that faces the sun at a right angle, which decreases quadratically with distance from the sun, has a magnitude of 9.08 × 10−6 N/m2 at 1 AU (AU = Astronomical Unit, i.e., the mean distance between the center of the sun and the earth) (Friedman et al., 1978; Wright, 1992). This pressure induces an acceleration of x¨ = 9.08η/ρ

(9.19)

for a solar sail of areal density ρ (in units of g/m2 ), where the acceleration units are mm/s2 and η is an efficiency factor, smaller than 1, that accounts for reflectivity and planarity induced losses. This acceleration is a significant source of propulsion for

9.7 Deployable Mirrors and Solar Sails

5 rpm

305

2 rpm

(a)

25 rpm

15 rpm 5 rpm

(b)

1~2 rpm (c)

Figure 9.30 Staged deployment of IKAROS solar sail (Sawada et al., 2011).

sufficiently light spacecraft, and can enable a wide variety of missions (McInnes, 2013). High-energy laser beams or microwaves can be used as an alternative light source to exert much greater propulsive force than it would be possible using sunlight (Forward, 1984, 1985). Lubin (2016) has proposed using such a scheme for interstellar flight. Solar sailing was first demonstrated in 2010 by the successful deployment of the Interplanetary Kite-craft Accelerated by Radiation Of the Sun (IKAROS) (Sawada et al., 2011). A square, 7.5 mm thick Mylar membrane with a side length of 14 m was wrapped around a cylindrical spacecraft bus with a diameter of 1.6 m and height of 0.8 m. The total spacecraft mass was 310 kg, including 15 kg for the solar sail. The overall areal density of IKAROS was ρ = 775 g/m2 , providing a maximum, potential acceleration – from Eq. 9.19 – of x¨ = 0.010 mm/s2 . About one third of this value was actually measured during the IKAROS mission (JAXA, 2010). The sail was deployed by the centrifugal action induced by spinning the spacecraft to a maximum angular velocity of 25 rpm. The full sequence is shown in Fig. 9.30. It involved a staged process that included the spin up of the spacecraft to 5 rpm and the release of four tip masses of 0.5 kg at the corners of the membrane, as shown in Fig. 9.30(a). This initial mass-separation phase was followed by the first deployment stage involving the uncoiling of the wrapped sail in Fig. 9.30(b). The second deployment phase was initiated by the release of four rotation guides, and the deployment of the sail in Fig. 9.30(c).

10

Adaptive Structures

10.1

Adaptivity Adaptivity generally indicates a relation between design and function, in order to achieve a better fit to a purpose. Simple examples of adaptivity in everyday life include photochromic lenses for eyeglasses, that become darker upon exposure to ultra-violet light, and smart glass windows that change from translucent to transparent through the application of an electric field. In structures, adaptivity indicates an ability to change specific properties in response to changes in the environment. For example, an adaptive structure subject to varying loads can change its shape to reduce the maximum stress level, while a structure subject to different dynamic excitations can change its stiffness or damping to reduce the vibration amplitude. The best examples of adaptivity are found in nature, where the most slender structures can also be the strongest. In parts of the world that are subject to hurricanes, palm trees are able to survive enormous wind loads by recoverable deformation involving large tip rotations that greatly reduce the maximum stress in the trunk of the palm, Fig. 10.1. The ultimate adaptive structure is the human body, capable of a myriad different shapes as illustrated in Fig. 10.2. The difference in performance between a static structure and a shape-adaptive structure can be examined by considering a palm tree in a hurricane force wind, Fig. 10.3. Since the Young’s modulus of a palm tree lies in the range 0.01–30 GPa (Gibson, 2012), its behavior will change drastically depending on which extreme value is chosen. On one extreme, we consider a very stiff palm tree, as shown in Fig. 10.3(b), and on the other extreme a highly compliant tree, as shown in Fig. 10.3(c), or a shape adaptive structure of the kind that will be discussed later in this chapter. The wind loading acting on both structures can be analyzed simply as a horizontal pressure calculated from: p=

1 2 ρu cd 2

(10.1)

Here, ρ is the density of air (ρ = 1.2 kg/m3 ), cd is the drag coefficient assumed to be constant and equal to 1.5, for simplicity, and u is the wind velocity which will be assumed to be 28 m/s(= 100 km/h) to model a hurricane. The wind boundary layer will be neglected, thus assuming u to be uniform with height. Therefore, p = 1,411 Pa uniform with height. 306

10.1 Adaptivity

307

Figure 10.1 Palm trees in a hurricane (Photobank gallery/Shutterstock.com).

Figure 10.2 Shape adaptivity of the human body (mtkang/Shutterstock.com).

Figure 10.3(b) shows a stiff structure of height h1 and width w1 supporting a perforated rectangular prism of height h2 , and width and depth w2 . It is a simple model for a highly stiff palm tree with a cluster of leaves at the crown. For h1 = 20 m, w1 = 0.2 m, and h2 = w2 = 2 m, the wind pressure resultants are: F1 = w1 h1 p = 5.644 kN

(10.2)

F2 = 0.5w2 h2 p = 2.822 kN

(10.3)

308

Adaptive Structures

w2

F2

h2

F2 p

h1

F1 2h1

p

(a)

π

F1

w1

(b)

(c)

Figure 10.3 Comparison of slender structures modeling a tall palm tree under wind loading, shown in (a) for scale: (b) stiff and (c) shape adaptive structural models.

where 0.5 is an assumed porosity coefficient. The moment at the root is: Mstiff = F1 h1 /2 + F2 (h1 + h2 /2) = 116 kN m

(10.4)

In contrast, Fig. 10.3(c) shows a bent structure, representing a highly deformed palm tree that has bent 90◦ under the wind loading. The perforated prism modeling the crown of the tree is also assumed to have rotated 90◦ . The wind pressure resultants are: F1 = w1

h1 p = 3.593 kN π/2

F2 = 0.5w22 p = 2.822 kN

(10.5) (10.6)

where 0.5 is again the porosity coefficient. The moment at the root is then: h1 2h1 + F2 = 59 kN m (10.7) π π This example has shown that, by heavily deforming under the applied loading, the more compliant structure has lowered the moment at the root by a factor of two, which highlights the scale of potential gains that can be made through adaptivity. The three key components of an adaptive structural system are: the sensors, the decision logic, and the actuators, connected through a closed-loop control architecture as depicted in Fig. 10.4. The first implementations were in tall buildings under the action of wind and earthquakes, in the pioneering study by Yao (1972) that drew the system architecture in Fig. 10.4. Later on, the same approach was applied by Geiger to large-span air-supported fabric roof systems, see Fig. 10.5, whose stiffness could be increased by raising the internal over-pressure when strong wind conditions were expected. Once the strong winds had subsided, the over-pressure would be reduced to its normal value. Msoft = F1

10.1 Adaptivity

309

Earthquake or wind excitation nth storey accel./vel./disp.

Tall building 1st storey accel./vel./disp.

Controller Actuator 1

Actuator ....

Actuator n

Figure 10.4 Proposal for a feedback control system based on Yao (1972).

Figure 10.5 Tokyo Dome, designed by David Geiger and built in 1988 (image used with permission of Geiger Engineers).

Recently, the Stuttgart SmartShell project (Neuhaeuser et al., 2013) has built a doubly curved timber dome with a span of 10 m×10 m, with three of the four supports provided by hydraulic actuators. Structural control is used to achieve the most uniform stress distribution for any load case, and 15 strain gauges distributed through the structure are used as sensors. In aircraft structures, morphing structures in the form of ailerons, trailing edges, etc. have been used since the early days of aviation. A significant development was

310

Adaptive Structures

Figure 10.6 Stuttgart SmartShell (Neuhaeuser et al., 2013).

Figure 10.7 Variable sweep wing configuration of the F-111 aircraft (US Air Force image).

the variable sweep wing configuration of the F-111 aircraft, developed by General Dynamics to optimize performance under different flight conditions, Fig. 10.7. In aerospace structures, shape adaptivity has many important applications, such as space platforms that can be reconfigured every few years for new experiments, or space telescopes that are built in stages to spread the cost of expensive infrastructure, while allowing the telescope to “evolve” over time (Polidan et al., 2016)

10.2 Actuators

311

This chapter is concerned mainly with shape adaptivity, and in particular with the morphology of shape-adaptive structures. We also briefly touch upon some of the key hardware required in these systems.

10.2

Actuators The field of actuators is rather specialized, and hence here we briefly review a restricted class of actuators most suitable for structures that are required to execute large shape changes. A muscle is a one-way actuator capable of shortening its length while applying a pulling force; it can pull but not push. Figure 10.8(a) shows the antagonistic arrangement of two muscles in the human arm: the arm bends up when the biceps is activated, and it is extended when the triceps is activated. This arrangement of two muscles illustrates the general point that two-way actuation can be obtained by coupling two one-way actuators. A more subtle type of coupling is found in the body of the Hydra, Fig. 10.8(b), whose tubular body and tentacles, surrounding the mouth, are contracted or extended by two perpendicular sets of muscle fibers, circular and longitudinal. The mechanical coupling between these two sets of muscles is provided by a fluid-filled internal cavity, through the constraint that the volume of the fluid must remain constant. Hence, contraction of the circular muscles must be accompanied by an extension of the longitudinal muscles. Man-made actuators are less advanced than the Hydra, but efforts are being made to develop more versatile systems (Bar-Cohen, 2004). Hydraulic actuators tend to be large and heavy and, apart from a few exceptions (e.g., applications in flight simulators), are unsuitable for adaptive structures.

(a) Figure 10.8 (a) Antagonistic muscles in the upper arm (NoPainNoGain/Shutterstock.com); (b) use of longitudinal and circular fibers in Hydra (Aldona Griskeviciene/Shutterstock.com).

312

Adaptive Structures

(a)

(b)

Figure 10.9 (a) Telescopic actuator, (b) cable actuator.

A frequent choice is a motor-driven screw mechanism. In this case, the inner part of a “telescopic” member is a threaded rod (screw) which is driven in and out of an outer element by an electric motor, through a set of gears and a nut. Figure 10.9(a) shows a typical actuator based on this principle, with a small electric DC motor mounted alongside the telescopic member. The motor turns a nut, which causes the left-hand part of the actuator to move left or right. Figure 10.9(b) shows an actuator used by You and Pellegrino (1997a) to vary the length of a cable, based on a similar principle: a 0.4 m long Al-alloy screw with 20 mm diameter and a pitch of 4 mm, mounted coaxially with a DC motor and gear set, drives a brass nut supporting a small pulley. A small fork prevents the nut from turning. As the screw turns, the nut moves up and down along the screw and the wire rope is wound/unwound on the screw itself. A length of up to ≈1.6 m of cable can be wound on this screw. While the actuator in Fig. 10.9(a) is capable of both pulling and pushing (two-way action), the actuator in Fig. 10.9(b) can pull only (one-way action), and hence can be considered as a mechanical equivalent of a very long-stroke muscle. The actuators discussed above are mostly suitable for low speed applications, and in this case the key performance characteristics are the maximum actuation stress, which is divided by the density of the actuator to obtain a specific metric, and the maximum actuation strain. A general comparison of actuators, based on these two metrics and including several types of actuators, was made by Huber et al. (1997); it is summarized in Fig. 10.10. In addition to the linear actuator described above, another type of actuator that is frequently used in adaptive structures is the rotary actuator. Usually, this consists of a heavily geared down motor (Musser, 1967) which is used to control the angle between two structural elements connected by a hinge. An application of this type of actuator is in manipulator arms. For example, Fig. 10.11(a) shows the articulation of the 17 m long International Space Station remote manipulator system (Canadarm2) that was built

10.2 Actuators

0

g

Shape memory alloy

–1

1J

Hydraulic

–1

kg

10

10

–1

Jk

g

–1

Jk

Thermal expansion (100K)

0.

10

0k

10

10

10

Thermal expansion (10K)

–2

Magnetostrictor Pneumatic

–3

High strain piezoelectric Low strain piezoelectric

Piezoelectric polymer Muscle

10

–4

Solenoid 10

10

–5

Moving coil transducer

–6 –6

10

10

–5

10

–4

10

–3

10

–2

10

–1

0

10

Figure 10.10 Specific actuation stress, σ/ρ, vs. actuation strain for various actuators (Huber et al., 1997).

Wrist Joint

Elbow Joint

Shoulder Joint

(b)

(a)

Figure 10.11 International Space Station remote manipulator system: (a) kinematic layout;

(b) astronaut anchored to the foot restraint attached to manipulator system during mission STS-116 (NASA, 2007).

313

314

Adaptive Structures

by MacDonald, Dettwiler and Associates, and deployed in 2001. It makes use of seven rotary actuators, three in the shoulder joint, one in the elbow, and three in the wrist, has a diameter of 35 cm, a mass of 1,800 kg, and can handle payloads up to 116,000 kg. Most existing actuators are one dimensional, i.e., either capable of extending and contracting only in one direction, or to turn about only one axis. Hence, when thinking about concepts for adaptive structures, the practical options are restricted to two main types: adaptive truss structures, based on linear actuators, and manipulator arms, based on rotary actuators. Since the morphology of manipulator arms is straightforward, in the next two sections we focus on adaptive truss structures.

10.3

Structural Adaptivity to Increase Material Utilization Conventional structural design practice for traditional, passive structures ensures that the strength and deformation of the structure meet the requirements associated with the worst load cases. For deterministic loads, such as self-weight, there is little that can be done by transitioning to an adaptive structural design. However, there are many applications where structural design is dominated by extreme events, such as strong wind storms or earthquakes for a civil engineering structure, or launch loading for a spacecraft. In such cases, the structure is significantly overdesigned for most of its working life. Senatore et al. (2011) showed, for the specific case of a roof truss, that replacement of selected passive members with active elements (actuators) that are only activated beyond a certain load threshold, leads to the design of a structure that carries low levels of load passively. Above the threshold, actuation allows the structure to resist high but less frequent loading scenarios. Active load-bearing capacity is provided by controlled length changes of the actuators, which modify the distribution of internal forces and reduce the maximum stresses while keeping the maximum displacements within desired limits. Senatore et al. (2018a) presented a cost analysis of five case studies, and demonstrated that adaptive structures based on this approach can save on average 70% of the wholelife energy of a structure. The whole-life energy is defined as the sum of the energy of building the structure (embodied energy), the energy of operation and maintenance of the monitoring and control system, and the energy to operate the actuators (operational energy). In this approach, the design of an adaptive structure consists in determining the cross-sectional areas of the passive elements as well as the optimal number and position of the actuators to minimize the total energy. This optimal design lies between two extreme cases: an entirely passive design that has high embodied energy and zero operating energy; a highly adaptive design with low embodied energy but very high operational energy. Senatore et al. (2013) proposed a method to investigate systematically the performance of structures lying between these two extremes, and corresponding to different values of the load threshold at which the actuators start working. The situation is shown

Energy (MJ)

10.3 Structural Adaptivity to Increase Material Utilization

315

Operational Embodied Total Optimum

100%

75%

50% MUF

Active design

25% Passive design

Figure 10.12 Schematic diagram showing variation of embodied and operational energy in an

adaptive structure, in terms of the material utilization factor (MUF) (Senatore et al., 2018b).

schematically in Fig. 10.12, which shows the embodied energy increasing and the operational energy rapidly decreasing as the structure becomes more passive. The parameter on the abscissa is the material utilization factor (MUF), which is a scaling coefficient (≤1) for the material strength parameters. The optimum design corresponds the lowest point in the curve (minimum energy).

10.3.1

Analysis Method A design method for pin-jointed truss structures was proposed by Senatore et al. (2013). It consists of two nested optimization steps, as shown in Fig. 10.13. The first stage of the design consists in minimizing the total volume of material in the structure, V , subject to satisfying the equations of equilibrium at all joints of the structure, for all load cases to be considered (superscript k), as well as inequality constraints that require all bar forces, tIkJ , to be such that the material of the bars does not yield and no buckling occurs. Mathematically,  min V = AI J LI J subject to: (10.8) bars

At = pk k

(10.9)

if tIkJ > 0 :

tIkJ ≤ σt MU F AI J

if tIkJ < 0 :

−

tIkJ ≤ σc MU F and − tIkJ < (Pe )I J AI J

(10.10) (10.11)

Here, Aij , Lij are the cross-sectional area and the length of the bar between joint I and J, and Eq. 10.9 is equivalent to Eq. 1.13. σt ,σc are the stress yield limits for bars in tension and compression, respectively (including any required safety margins), and (Pe )I J = π 2 EI /L2I J is the Euler buckling load of bar IJ. Setting the material utilization factor MU F = 1 yields absolute minimum mass designs for the structure.

316

Adaptive Structures

Inputs Layout, material & element type

Design load & probability distribution

Control nodes & serviceability limits

Whole-life energy optimization

Load path & material optimization

Actuator layout optimization

0% < MUF(i) < 100%

Operational energy computation

Load path redirection & shape control

Load activation threshold detection

Minimum whole-life energy design Optimal material distribution Embodied energy optimization

Optimal load-paths Operational energy computation

Optimal actuator layout

MUF = Material Utilization

Figure 10.13 Flowchart of design methodology for structures that minimize whole-life energy

(Senatore et al., 2018b).

This optimization problem is nonlinear because the self-weight of the structure included in the nodal loads pk , and the Euler buckling loads for the bars vary with the bar cross-sections. It was solved using sequential quadratic programming (Senatore et al., 2013). The solution of the optimization problem determines the cross-sectional areas of the bars, corresponding to a particular load case (maximum load). For all other cases, a set of bar forces that satisfies all constraints requires that suitable actuator extensions are imposed. The next stage in the design process involves finding the best location for the actuators, after choosing a set of c nodes whose displacements must be controlled. The total number of required actuators is equal to the number of states of self-stress (recall Section 1.5) plus the number of controlled displacements, s + c. The position of the actuators may be subject to practical limitations, such as being located in the interior of the structure, or generally it can be chosen such that the sum of the overall actuator extensions is minimized:    |ei | (10.12) min bars

where ei is the extension of the actuator in bar i. A detailed formulation of this problem can be found in Kwan and Pellegrino (1992), Kawaguchi et al. (1996), and You (1997). The solution requires linear programming and least squares minimization.

10.3 Structural Adaptivity to Increase Material Utilization

10.3.2

317

Case Study Senatore et al. (2018) analyzed a number of case studies that highlight potential applications of the above adaptive structure concept. One of the case studies considered by these authors is a tall building with a square plan of side length of 20 m and height of 100 m. Its 5 m deep core consists of four planar vertical trusses, as shown in Fig. 10.14. Due to symmetry, the structural design needs to consider only one of the planar trusses. The serviceability requirement for comfortable use of the building limits the horizontal drift to the value: height/500 = 0.2 m. To analyze this problem, the horizontal displacements of all unconstrained nodes of the truss were chosen as controlled displacements, hence c = 40. The controlled nodes

L1 L2

L3

z y

z y x

(a)

(b)

Figure 10.14 (a) Tall building with four symmetric vertical trusses forming the central core. (b)

Three load cases and definition of controlled nodes (denoted by circles) (Senatore et al., 2018a).

Adaptive Structures

are indicated by circles in Fig. 10.14(a). Because the degree of static indeterminacy is s = 20, the minimum number of actuators to control exactly all selected displacements is 60, as discussed in Section 10.3.1. Three load cases were considered. LC1, vertically downward self-weight plus a uniformly distributed dead load of 3 kN/m2 . LC2 and LC3, horizontal live distributed loads (wind loading) in opposite directions with intensity varying quadratically with height and maximum intensity of 0.6 kN/m2 , corresponding to a wind maximum velocity of 30 m/s with equal distribution functions for the two cases. The load activation threshold from passive to active behavior was set at 18 m/s, providing a total actuation time of 5.2 years (over a life span of 50 years), and 2.6 years for each of LC2 and LC3. The embodied, operational, and total energy as a function of the MUF are plotted in Fig. 10.15(a). The optimal adaptive configuration is found for a MUF of 45% whereas the passive structure corresponds to a MUF of 17%. This means that the optimized adaptive structure and the passive structure are designed so that the maximum stresses under the worst load combination are 45% and 17% of the yield stress, respectively.

Energy (MJ)

10

10 6

8 6 Operational Embodied Total Adaptive Passive

4 2 0

100%

45%

17%

MUF

(a) 12

106 Embodied Operational

10

Energy (MJ)

318

8 6 4 2 0

Passive

Adaptive

(b) Figure 10.15 (a) Variation of embodied and operational energy in tall building, in terms of

material utilization factor. (b) Comparison of total energy for passive vs. adaptive structure (Senatore et al., 2018a).

10.3 Structural Adaptivity to Increase Material Utilization

319

Figure 10.16 Comparison of bar forces in (a) passive vs. (b) adaptive solution. (c) Layout of

actuators. Deflected shape (d) with active control, and (e) without. (f) Bar forces under load case LC2 (Senatore et al., 2018a).

Compared with the passive structure, the adaptive structure achieves savings of 63% and 43% in mass and total energy, respectively, see Fig. 10.15(b). Figure 10.16 compares the optimized passive structure with the adaptive structure. The actuators are integrated in the tubular elements that make up the structure. The biggest and smallest diameters of the tubular elements for the adaptive design are 1,280 mm and 100 mm for the lowest vertical and horizontal elements, respectively. For the passive design, these values are more than double. Figure 10.16(d) shows the controlled shape after displacement compensation and the noncontrolled shape under LC2. Without shape change, the tip deflection is 633 mm, which is well beyond the serviceability limit of 200 mm. Under live load, the maximum length change is a 4 mm shortening in the bottommost horizontal actuator. The two vertical actuators

320

Adaptive Structures

at the bottom of the structure apply the greatest tensile (2,200 kN) and compressive (14,000 kN) forces, respectively. This case study, representative of a large class of real systems, confirms that, for stiffness-governed problems, total energy savings are significant even when the live load has a magnitude substantially lower than that of the dead load. In terms of monetary cost, the authors of the study concluded that the passive solution is 16% cheaper than the adaptive one. The control system and maintenance cost shares amount to 42% and 27% of the total cost, respectively. Although the adaptive structure is more expensive, it not only uses much lower whole-life energy than the passive structure but is also a competitive solution in terms of energy-saving potential.

10.4

Concepts for Shape Adaptive Trusses Adaptive spacecraft structures that can be launched in a rocket and, once in orbit, can operate as cranes that assemble a space station, or serve as reconfigurable space platforms, have been developed between 1980 and the mid-1990s. The requirements for these structures are extreme. They need to be packaged efficiently, as a deployable structure, and need the dexterity of a robot. Hence, the guiding principles (constraints) governing the design of a shape adaptive truss are rather unique. First, the truss must be kinematically determinate, see Section 1.2, in all configurations, so that high precision shape control is possible without inducing any strain in its members. Second, the configuration with the smallest number of actuators is required, in order to minimize the cost and complexity of the hardware. Third, the actuators should not lie on the main load path, to minimize the load carried by these delicate and sensitive components, and to limit the effects of friction, backlash, etc. on shape accuracy. This third principle also addresses the need that the structure should be able to maintain its shape when the actuators are powered off.

10.4.1

Two-Dimensional Trusses As an initial example, we consider a simple two-dimensional adaptive truss. Figure 10.17(a) shows a triangulated truss with N = 8 bays. The task is to design an adaptive truss which is both foldable and able to compensate for any thermal distortion of the upper cord members to maintain the bottom joints perfectly aligned. In addition to the three guiding principles already listed, the size of the packaged truss will also be considered when comparing different solutions. Figure 10.17(b) shows the deformation resulting from a uniform temperature rise +T of the upper members if there are no actuators. Since the layout of the structure is already known, only the layout of the actuators has to be determined. There are three different positions for the actuators, see Fig. 10.17(c). If the actuators are built into the diagonal members their number will be minimal (N ), but only the subsidiary task of folding can be fulfilled, while the main task of joint alignment cannot be fulfilled. Also, diagonal actuators would lie in the main load path of the truss if a transverse tip force is applied.

10.4 Concepts for Shape Adaptive Trusses

321

(a) +ΔT

+ΔT

+ΔT

+ΔT +ΔT +ΔT +ΔT +ΔT

Diagonal actuators (N)

(b)

2L

Longeron actuators (2N)

√2L

Batten actuators (N+1)

√2L

(c)

(d)

Figure 10.17 Adaptive truss with N bays; the number of actuators in each group is shown

in brackets.

If the actuators are built into in the longerons, then both tasks of folding and deployment can be fulfilled, but actuators are required both in the upper and bottom cords of the truss. Hence, the total number of actuators is high (2N ) and, furthermore, the actuators are again on the main load path. Finally, if the actuators are built into the battens their total number is small and both tasks can be fulfilled. Actually, only alternate actuators are required for the main task, but folding the truss requires shortening of all battens. In conclusion, only the configuration with batten actuators can accomplish both tasks while also satisfying √ all of the constraints. It is interesting to note that the extension ratio of the actuators is 2 for all three solutions.

322

Adaptive Structures

Note that in some cases it may be impossible to fulfill all requirements while also satisfying all the constraints. In such cases, a trade-off between performance and constraint satisfaction has to be carried out. For example, if it is required that the adaptive truss maintains both the top and bottom joints aligned when the upper cord members are subject to thermal extensions, then the only possible solution is to mount the actuators in the cord members, and therefore the actuators lie on the load path.

10.4.2

Three-Dimensional Adaptive Trusses Real structures operate in three-dimensional space and, while it would be possible to give structural depth to the simple two-dimensional adaptive truss discussed in the previous section, so that it can carry out-of-plane loads, adaptivity in three dimensions can only be achieved if we consider fully three-dimensional truss layouts. Figure 10.18 illustrates some typical applications of adaptive trusses in space, to capture satellites, reconfigure space station modules, etc. These space cranes (Card, 1983), should be able to change the position and orientation of the end effector, within a specified workspace, and also to act as support structures/pointing mechanisms for large reflectors, cameras, etc.

Figure 10.18 (a) Adaptive truss on a space platform (Chen and Wada, 1993); (b) first proposal for

a space crane based on adaptive truss (Card, 1983).

10.4 Concepts for Shape Adaptive Trusses

323

Mathematically, we need to consider an adaptive truss with six degrees-of-freedom at the tip, three translation and three rotation components, and the largest possible workspace. Also, it is essential that these structures should be foldable, so that they can be assembled on the earth and launched into orbit. One of the best known three-dimensional structures with six degrees of motion is the Stewart platform (Stewart, 1965), originally developed for the training of helicopter pilots and now widely used for flight simulators and other applications. Figure 10.19(a) shows a Stewart platform with six hydraulic jacks. Three actuators, connected directly to the moving platform, are connected to the ground by two-axis joints. The motion about the axis perpendicular to the jack is controlled by a lower jack, also connected to the ground by a two-axis joint, while the motion about the other axis is left uncontrolled; this motion is determined by the platform itself. Within the amplitude limits set by the extension and contraction of each jack, the position and orientation of the platform are uniquely determined by the lengths of the six jacks. In some particular configurations of the platform certain motions are not possible (Stewart, 1965), and when one of the three main actuators becomes aligned with the centroid of the platform support points, the assembly becomes kinematically indeterminate. The principle of the Stewart platform can be readily adapted to the design of truss structures with six-degrees of freedom, simply by connecting six linear actuators directly to the platform itself. The usual arrangement is shown in Fig. 10.19(b). Six actuators are connected to a triangular base and to a triangular platform, in an arrangement that resembles the octahedral truss in Section 1.2.1. The general properties of the Stewart truss can be analyzed using Maxwell’s equation, Eq. 1.2. Treating the three joints connected to the bottom triangle as fixed and assuming that the six actuators are locked, then j = 6, b = 9 of which six bars contain the actuators and the remaining three are the edges of the platform, and k = 9. Hence 3j − b − k = 0

(10.13)

5 1

3

4

6 z y

(a)

2 x

(b)

Figure 10.19 (a) Stewart platform (Stewart, 1965) and (b) Stewart truss with elements of

equal length.

324

Adaptive Structures

and hence, from Eq. 1.2, m − s = 0. It can be shown, Section 1.5, that s = 0, apart from a few special configurations. Excluding these configurations, the Stewart truss is statically determinate and hence the length of any of its legs can be changed freely, without inducing any stresses in any leg. It is also kinematically determinate (m = 0) and hence the motion of the platform is uniquely determined by the length changes of the legs. For example, consider the Stewart truss of Fig. 10.19(b) whose upper and bottom triangles have a side length of 1, and whose legs are assumed to have an initial length of 1, thus forming an octahedral truss. Starting from this configuration, we can use the computational method of Section 1.5 to determine the relationship between the smallamplitude translations of the platform and the corresponding sets of actuator extensions. They are as follows (α is an arbitrary constant, α  1): translation in the x-direction translation in the y-direction translation in the z-direction

α [ 1 –1 0 1 –1 0 ] α [ –1–1 2 –1–1 2 ] α[ 1 1 1 1 1 1]

Similarly, pure rotations of the platform about the centroid of the upper triangle are obtained from the following sets of actuator extensions: rotation about axis parallel to the x-axis rotation about axis parallel to the y-axis rotation about z-axis

α [ –2 –2 1 1 1 1 ] α [ 0 0 –1 –1 1 1 ] α [ 1 –1 1 –1 1 –1 ]

For a different initial configuration of the assembly we can use, again, the inverse of the compatibility matrix, Section 1.5, in that particular configuration to update these relationships. The special configurations of the platform where the compatibility matrix becomes singular should be avoided. The working space of the Stewart truss is relatively small because no telescopic actuator can do better than doubling its length, but within this space the Stewart truss provides six degree-of-freedom control. This is perfectly adequate for flight simulators, which require large accelerations but only small velocity and displacement, but a space crane needs a much larger working space. A natural extension of the Stewart platform concept is shown in Fig. 10.20(a): showing four identical trusses stacked on top of one another. The work space of this adaptive structure is much larger than that of a single Stewart truss. The main problem, though, is that it requires a very large number of actuators. A further problem is that the actuators lie in the main load path of the structure which, actually, is quite difficult to avoid in the case of three-dimensional trusses. In analogy with the two-dimensional adaptive truss discussed earlier, the number of actuators can be reduced by noting that usually there is no need to provide the complete set of degrees of freedom within each module of the structure. Thus, a better solution is to consider the truss shown in Fig. 10.20(b), where the actuators are located in the battens, which reduces the number of actuators to less than half. It is interesting to note that both adaptive trusses shown in Fig. 10.20 are derived from the rod-like truss based on the octahedral truss, discussed in Section 1.3. Of the

10.5 Variable Geometry Truss

(a)

325

(b)

Figure 10.20 Adaptive trusses based on (a) Stewart platform and (b) VGT concept.

two architectures shown in Fig. 1.11, adaptive trusses based on the tetrahedral truss would require many more actuators as well as joints with different shapes (only in a stack of octahedra all joints are identical). The main advantage of adaptive trusses based on the tetrahedral truss would be that their inverse kinematics, i.e. the calculation of a set of actuator extensions that produces a specified position and orientation of the end effector, are much easier to formulate.

10.5

Variable Geometry Truss The Variable Geometry Truss (VGT), Fig. 10.21, is the first and most widely studied adaptive structure. The concept was first proposed in 1984 (Miura, 1984a, 1984b; Rhodes and Mikulas 1985) and later developed mainly in Japan and Canada (Miura and Furuya, 1985a; Miura et al., 1985b; Miura and Matsunaga, 1989; Hughes et al., 1990; Chen and Wada, 1993). Several applications of the VGT as a space crane have been proposed, where the most significant advantage of a truss manipulator over a conventional mechanical arm is the superior stiffness-to-weight ratio. Topologically, the VGT consists of a series of octahedral truss modules whose triangular interfaces (battens) have been fitted1 with actuators and encoders. In the upright configuration, Fig. 10.21, all members have equal √ length L. To fold the VGT all actuators are extended to their maximum length, 3L, at which point in each octahedral truss the battens become coplanar with the six diagonal members. Different folding modes can be obtained, for example by extending all actuators simultaneously, at equal rates, or by extending the actuators sequentially one batten at a time (Miura et al., 1985; Chen and Wada, 1993). To obtain nonstraight configurations of the VGT the actuators in the same batten are extended by different amounts. Thus, each batten translates and rotates relative to the

326

Adaptive Structures

z

x y

(a)

(b)

Figure 10.21 Variable Geometry Truss (a) layout of actuators (Miura et al., 1985b) and

(b) demonstration of shape adaptation (Miura and Matsunaga, 1989).

previous batten and a wide range of forms and shapes can be produced. The key problem is to find a relationship between the shape of the VGT or, more precisely, the position of the centroids of its triangular battens, and the lengths of its actuators. This is generally known as an inverse kinematics problem (Sciavicco and Siciliano, 1996). Finding an explicit solution to the inverse kinematics of a VGT is quite hard and also the solution is often not unique. For this reason all existing analytical solutions are based on some kind of simplifying assumption. For example, Naccarato and Hughes (1991) considered VGT’s with actuators only in alternate battens. Thus, a VGT consisting of 2N octahedral trusses is in fact made up of N identical modules, like that shown in Fig. 10.22(a). The top and bottom battens of each module are equilateral triangles of fixed size while the central batten can change its shape. With this simplification an elegant analytical solution of the inverse kinematics problem was obtained by Naccarato and Hughes (1991). A general shape of the VGT is defined by assigning the positions of the centroids of the fixed shape battens. Hence, N + 1 points are required. For example, if the VGT has to follow the shape of a curve of given equation, usually N + 1 evenly spaced points would be placed on the curve. Consider the one-module VGT shown in Fig. 10.22. Points G1 and G3 correspond to two consecutive points on the curve, hence their coordinates are assumed to be known. The unit normal to the bottom batten, n1 , would have been determined by analyzing the previous module. All other quantities and in particular the actuator lengths are calculated as follows. The position of each point, e.g. G1 , is identified by a position vector, g1 , whose origin is at an arbitrary point O, not shown in the figure. Let d be the vector that defines the position of G3 with respect to G1 d = g3 − g1

(10.14)

10.5 Variable Geometry Truss

327

n3 n3 B3

G3

C3 G3

A3

d B2

C A2 C2

n1

n1

B1 G1

G1 A1

C1 (b)

(a)

Figure 10.22 One-module VGT.

In any configuration of the module, the plane containing the three actuators shown in Fig. 10.22(b), is a plane of symmetry for the module, hence d is perpendicular to this plane and the unit normals to the bottom and top batten, respectively n1 and n3 , will form equal angles φ/2 with d. This rotation magnitude can be calculated by taking the dot-product of n1 and d/||d||: cos

n1 · d φ = 2 ||d||

(10.15)

The vector n3 is obtained by reflecting n1 across the plane of the actuators. Using the standard expression for the reflection matrix in a direction defined by a unit vector (Strang, 1980), if the unit vector is d/||d|| we obtain:   ddT n3 = 2 − I n1 (10.16) ||d||2 where I is the identity matrix. Now, the position of the top batten can be obtained by translating the bottom batten by d and then rotating it through an angle φ about a vector a which is perpendicular to both n1 and n3 . This unit vector is given by: a=

n1 × n3 ||n1 × n3 ||

(10.17)

and the rotation can be represented by the vector φa. However, what is needed here is the rotation matrix R, which rotates n1 into n3 n3 = Rn1

(10.18)

and, similarly, any other element of the bottom batten into the corresponding element of the top batten. This matrix is obtained from Rodrigues formula (Crisfield, 1997), which gives the following expression: R = I + a × I sin φ + (1 − cos φ)aaT

(10.19)

328

Adaptive Structures

B2 A2

d/2

α A1

C2

n1 B1 m G1 C1

Figure 10.23 Bottom half of VGT module.

Hence, the position vector of a joint of the top batten can be obtained by adding the translation d and the rotation-induced displacement (given by the rotation R times the distance from the centroid of the batten) to the position vector of the corresponding joint of the bottom batten. For example, joint A3 is obtained from A1 , from a3 = a1 + d + R(a1 − g1 )

(10.20)

The actuator lengths are calculated from the actuator end points A2 , B2 , C2 . For example,  LA2 B2 = (a2 − b2 )T (a)2 − b2 ) (10.21) Figure 10.23 shows, for example, how to obtain the position of A2 . Starting from the midpoint of side A1 B1 , one considers the components along m and n1 of the height of the equilateral triangle A1 B1 A2 . Hence, √ 3 (a1 + b1 ) (10.22) + L(cos αm + sin αn1 ) a2 = 2 2 where L is the (fixed) length of the diagonal bars. In Eq. 10.22 the angle α is found by noting that the plane of the actuators is perpendicular to d. Hence, the dot-product of between the vector v defined in the figure and n must vanish, which gives:   1 T a2 − g1 − d d = 0 (10.23) 2 Substituting Eq. 10.22 for a2 into Eq. 10.23 gives an equation in α, which can be solved analytically. The positions of B2 and C2 are obtained in a similar way, and then the actuator lengths are calculated from Eq. 10.21, etc. By moving along a VGT with any number of modules, its inverse kinematics can be solved. Figure 10.24 shows two different ways of setting up this type of VGT into a circular arc configuration. The configuration in Fig. 10.24(a) is obtained by extending two actuators by equal amounts in all extendible battens, whereas the sequence Fig. 10.24(b) is obtained by extending only one actuator in each batten. The apparent contradiction between being able to solve “uniquely” the inverse kinematics problems and yet having found two different sets of actuator extensions that

10.6 Homologous Structures

329

(a)

(b) Figure 10.24 Two different ways of forming a circular profile, by extending (a) two actuators or (b) one actuator in each batten (Miura et al., 1985).

appear to give VGT’s with exactly the same shape, is due to the fact that in this example no constraint was set on the length of the centroidal line of the truss, which leaves one uncontrolled degree of freedom in each module. More general shapes of the VGT can be determined with the same approach, see Fig. 10.25.

10.6

Homologous Structures A special kind of adaptive structure concept has been successfully used in the design of ground-based reflector antennas, Fig. 10.26. These structures are space frames whose upper surface joints lie on a paraboloidal surface. Here, keeping the shape of the reflector sufficiently close to a paraboloid, in order to maintain the required accuracy in the

330

Adaptive Structures

Figure 10.25 VGT simulating a complex space curve (image used with permission of J. Onoda).

Figure 10.26 Deep space antenna in Usuda, Japan.

flatness of the reflected wavefront (Levy, 1996), imposes a more severe design constraint than that imposed by the requirement that the material of the structure should not yield. This point is illustrated by Fig. 10.27, showing the limits on the wavelength of a reflector that is tilted under gravity as a function of its diameter. The two lines that define

10.6 Homologous Structures

1000

331

Stress Limit

300 100

3 1

lL ona

ti

vita

Gra

Th e (Δ rm T= al 5 o Lim C) i t

10

t

imi

D [m] 30

1

3

10

30

100 300 1000

λ [mm] Figure 10.27 Limits of conventionally designed tiltable antennas, compared to eight actual design

points (based on data from Hoerner, 1967).

the shape limits imposed by excessive gravitational deformation and excessive thermal deformation were computed by Hoerner (1967), who showed that all previously built or designed reflector antennas, shown as black dots in the figure, were within these limits. A way to go beyond these limits is to consider the distribution of the deformation, rather than its absolute magnitude. For example, in the case of a parabolic reflector dish much larger deformations can be allowed if the deformed shape is of the same type, i.e., paraboloidal, as the original surface. Thus, Hoerner searched for a structure whose surface deforms into a surface of the same type. This type of deformation he called homologous deformation, and structures that deform in this way are now known as homologous structures. Hoerner showed that by adopting the concept of homologous structure one can go beyond the size limits of conventional steerable antennas. The concept of homologous structures is not limited to antennas; it is linked to optimal structural design (Yoshikawa and Nakagiri, 1994) and has also been applied to the design of surfaces that are required to maintain a prescribed degree of flatness under moving loads (Hangai and Harada, 1993).

10.6.1

Gravitational Limit of Large Antennas Figure 10.26 shows the 64 m diameter antenna at the Deep Space Center, in Usuda, Japan. This fully steerable antenna was built for telecommunication with spacecraft and is also used as a radio telescope. This antenna has two axes of rotation, the horizontal and elevation axes. Rotating a reflector with a mass of about 1000 tonne around the horizontal axis – from about 5◦ to the horizontal to 90◦ – causes a change in the gravityinduced loading on the reflector. The accuracy requirement for a reflector, see also Section 9.4, is usually stated in terms of the root-mean-square (RMS) deviation of the reflector surface having to be smaller than a given value for the full range of operational configurations of the reflector.

332

Adaptive Structures

x

L

Figure 10.28 Length change of uniform elastic bar under gravity.

The size, mass and required precision of a reflector, to go beyond the gravitational limit, pose a typical design problem for a large space frame. In the conventional way of designing a structure, excessive deformation is suppressed by increasing the stiffness of the structural members, but this requires additional weight and further increases the deformation. This process is a converging one in space frames that are below the gravitational limit, an example can be found in Medwadowski (1989), but leads to a nonconverging increase of stiffness-weight-deformation for space frames that lie beyond this limit. To estimate the gravitational limit imposed by the elastic deformation of a structure of Young’s modulus E and density ρ, consider the length change of the uniform bar, as shown in Fig. 10.28. The bar has an unstressed length L; when it is turned through 90◦ the stress at a distance x from the top is σ = ρgx, where g is the acceleration due to gravity (g ≈ 10 N/kg). The corresponding elastic strain is  = ρgx/E and the overall length change is  L ρgx 1 ρgL2 L = dx = (10.24) E 2 E 0 Note that L increases with the square of the bar length L, but is independent of the cross-section. For a steel bar (ρ = 7,800 kg/m3 and E = 210 GPa) this gives: L = 1.86 × 10−7 L2 m and it can be calculated that the self-weight shortening of a 100 m long, steel bar of uniform cross-section is 1.86 mm. Note that the deflection is smaller in a tapered bar, with a bigger cross-section at the bottom. Defining a geometric shape factor, γ , Eq. 10.24 can be written more generally in the form: L = γ

ρgL2 E

(10.25)

To find an expression for the geometric shape factor for a large space frame antenna, Hoerner (1967) considered a structure whose overall shape is an octahedron with

10.6 Homologous Structures

333

F B

B

O

A

C

O

A

-1.4

10.7

D

D d

-4.1

C 10.7

-4.1

(b)

(a) Figure 10.29 (a) Simple model of steerable reflector; (b) gravity deflections, in mm, of points

A, B, C, D, O for a steel structure with d = 100 m.

diagonals of length d, as shown in Fig. 10.29(a). Of course, this is a highly idealized model of an antenna, because it has only five joints to which a reflective surface could be attached. Furthermore, these joints lie in a plane, instead of a paraboloid, unless the position of the point where the diagonals cross is moved. However, it is a simple structure to analyze, and has the key features that are required for a tiltable structure, namely, it is rigid; it can be rotated about a horizontal axis by moving one of its joints; and it provides a support for the antenna feed, or a subreflector (in the case of a Cassegrain architecture), at the focal point. Hoerner computed approximate values for the deflections of points A, B, C, D, O, with respect to the deflection of point F, for a diameter d = 100 m of the truss. These values are shown in Fig. 10.29(b). The out-of-plane rms error of the five points that represent the reflective surface is 3.4 mm, which Hoerner amplified by a factor of 1.5 because out-plane bending of the members had been neglected in the deflection calculation. Considering that the length change of a bar is proportional to the square of its length, Eq. 10.25, the following relationship is obtained between the rms error, in millimeters, and the diameter of the frame, in meters: δrms = 5.1 × 10−4 d 2

(10.26)

To relate this result to antenna design, it can be assumed that the diameter of the antenna is related to that of the octahedral frame by D = 1.25d. Denoting by λg the shortest operational wavelength, in millimeters, of the antenna, and requiring δrms = λg /16 (wavelength divided by 16 is a typical accuracy requirement, see Levy (1996)) we arrive at the following expression for the gravitational limit: λg = 53 × 10−4 D 2

(10.27)

334

Adaptive Structures

For deep space telecommunication the diameter of the Usuda reflector antenna was set at 64 m. For communication in S-band and X-band the shortest wave length is 35 mm. Since Eq. 10.27 gives a value of 21.7 mm for the shortest wave length, which is rather too close to the requirement, an alternative design approach was pursued for this structure.

10.6.2

Going beyond the Gravitational Limit Hoerner (1967) examined three ways of going beyond the gravitational limit for tiltable antennas: (i) opposing the deformations by the use of motors; (ii) canceling the deformations with levers and counterweights; and (iii) allowing deformations which do not affect the performance of the antenna. In relation to the third approach, he proposed to use a homologous structure. In the case of a paraboloidal dish, the deformation is homologous if the points defining the reflective surface always lie exactly on a paraboloid, whose axis and focal length are allowed to vary, Fig. 10.30(a). These changes can be compensated for by moving the position of the antenna feed with a motor, according to the elevation angle.

F' F

Original surface

Homologous surface

x z (a)

F'

x

h2

F

z

h1

z

F

f0 g

h3 h4

dφ F'

Original surface Surface after deformation

(c)

(b) Figure 10.30 (a) Homologous deformation; (b) symmetric deformation; (c) antisymmetric

deformation.

g

10.6 Homologous Structures

335

The equation of an undeformed paraboloid of focal length f0 is: z = −(x 2 + y 2 )/4f0

(10.28)

If gravity acts in the z-direction, i.e., the antenna points to the zenith (elevation angle of 90◦ ), only two homologous changes are permitted: a downward translation by an

64 m

20 m

Figure 10.31 Structural layout of 64 m Usuda antenna.

336

Adaptive Structures

1.0

-1.0

0.0 1.0

1.0 0.0

0.0

1.0

0.0

-2.0 -1.0

-2.0 -1.0 1.0

0.0 0.0

(a)

1.0

0.0

0.0

0.0

0.0

0.0

(b)

0.0

1.0 0.0

-1.0 -2.0 -3.0

(c)

Figure 10.32 Surface error of Usuda antenna, in millimeters, due to gravitational loading. The elevation angles are (a) 7◦ ; (b) 35◦ ; (c) 90◦ .

amount dz and an increase in focal length df0 , as shown in Fig. 10.30(b). If gravity acts in the x-direction, i.e. the antenna points to the horizon (elevation angle of 0◦ ), the two homologous changes that are permitted are a downward translation by dx and a rotation about the y-axis through dφ. The corresponding four homology parameters are: h1 = dz h2 = 2df0 h3 = dx h4 = 2f0 dφ Note that the factor 2 that appears in the expressions for h2 and h4 accounts for the fact that when the focus of the reflector moves, the total length of the radiation path changes by twice the focal displacement, because both the length of the incoming and reflected radiation need to be considered. This change of path length is responsible for a frequency shift in the radiation, which sets the shortest wavelength at which the reflector can operate (Levy, 1996). Note that, although only two configurations of the reflector have been considered when defining the four homology parameters, because small deflections can be superimposed, homology holds for any elevation angle. The deformations modes considered in defining the homology parameters are the modes for which one wishes to cancel the effect of gravity-induced deflections; other deformation modes cannot be cancelled. To minimize these higher-order effects, it is usually best to aim for a structure of uniform stiffness. The structural design of the Usuda antenna was based on a homologous design, including also the effects of wind loading. The massive space frame (1050 tonne) is shown in Fig. 10.26. Figure 10.32 shows the distribution of surface deviation from the best-fit paraboloid of revolution, for different elevation angles. The maximum rms error due to gravity effects was estimated at 1.55 mm, which is close to the gravitational limit of 21.7/16 = 1.3 mm from the simple octahedron model.

Appendix A Geometric Foundations

A.1

Introduction Two topics that are particularly relevant to the study of structural concepts are discussed in this appendix: the subdivision of space and the curvature of surfaces. The first topic is based on the important concept of tessellation. Tessellations of polyhedra are the geometric forms from which we can assemble three-dimensional structures, such as space frames, structural foams, and honeycomb sandwich structures, using repeating elements. Tessellations of polygons are also introduced, as a way of subdividing surfaces. The second field is the geometry of surfaces, which are used to describe the shape of shell structures and membrane structures. The idea of curvature is introduced, and the way in which it varies at a particular point of a surface, along different directions on the surface. The reader interested in more details should consult the classic textbooks by Coxeter (1980) and Hilbert and Cohn–Vossen (1983).

A.2

Polyhedra and Tessellations

A.2.1

Regular and Semi-Regular Polyhedra Coxeter (1980) defines a polyhedron as a finite region of space enclosed by a finite number of planes. The part of each plane that is cut off by other planes is a face, and any common side of two faces is an edge. A topological property of convex polyhedra, i.e., a property that is independent from their geometric details, is that they satisfy Euler’s equation: F −E+V =2

(A.1)

where F is the number of faces, E the number of edges, and V the number of vertices. Among the infinity of polyhedra that can be constructed (Huybers, 2015), two particular sets are particularly important: the five regular polyhedra, or platonic polyhedra, shown in Fig. A.1 and the thirteen semi-regular polyhedra, or archimedean polyhedra, shown in Fig. A.3. The five regular polyhedra satisfy the conditions that they are bounded by equal, regular polygons, and the same number of edges meet at each vertex. Thus, the 337

338

Geometric Foundations

Tetrahedron

δ

θ

α δ

θ

Cube

α

δ

θ Octahedron α

Dodecahedron

θ

δ

α

δ Icosahedron

θ α

Figure A.1 The regular polyhedra.

tetrahedron is a triangular pyramid whose four faces are equilateral triangles. Of the remaining polyhedra, the cube and the dodecahedron have square and pentagonal faces, respectively, whereas the octahedron, the octahedron and the icosahedron have triangular faces. Figure A.1 shows, together with each regular polyhedron, the cutting pattern from which it can be made. Table A.1 defines a number of topological parameters of regular polyhedra. The last column defines the rigidity, or lack of it, of a truss model of each regular polyhedron, where the edges consist of pin-jointed bars. The table states that only the tetrahedron and the icosahedron – whose faces are triangles – are rigid. Details about this behavior are provided in Section 1.2.1. For each polyhedron it is useful to calculate the values of the following angles, whose values are listed in Table A.2.

A.2 Polyhedra and Tessellations

339

Table A.1 Regular polyhedra: topological parameters and properties. Polyhedron Tetrahedron Cube Octahedron Dodecahedron Icosahedron

Faces F

Vertices V

Edges E

Edges per vertex

Edges per face

Rigid

4 6 8 12 20

4 8 6 20 12

6 12 12 30 30

3 3 4 3 5

3 4 3 5 3

Yes No Yes No Yes

Table A.2 Angles (in degrees) of regular polyhedra. Polyhedron Tetrahedron Cube Octahedron Dodecahedron Icosahedron

•

•

δ

α

β

θ

180 90 120 36 60

60 90 60 108 60

60 60 90 60 108

70.53 90 109.47 116.57 138.18

The angular defect at each vertex, δ, is equal to 360◦ minus the sum of the angles of the polygon surrounding that vertex. It corresponds to the angle that needs to be cut away at each vertex, when making a model of the polyhedron from a flat sheet, see Fig. A.1. The internal angle of the polygonal faces of the polyhedron, α, which is related to the number of edges per face by the relationship α=

• •

180(p − 2) p

(A.2)

where p is the number of edges of the polygon. The angle of the vertex figure, β, which is the polygon obtained by joining the mid-side points of all edges connected to a particular vertex. The dihedral angle, θ, which is the (internal) angle between two adjacent faces of the polyhedron.

Regular polyhedra are rotationally symmetric about a number of axes. Their symmetry is called n-fold rotational symmetry, meaning that they are geometrically invariant after rotation through the angles 360◦ /n,2 × 360◦ /n, . . . ,(n − 1) × 360◦ /n. In the case of the cube the axes of symmetry are: any line through the centers of two opposite faces, any space diagonal (line joining opposite vertices), and any line through the centers of opposite edges. These symmetry axes are shown in Fig. A.2, and the number of rotations about each axis that produce identical configurations of the cube is listed in Table A.3.

340

Geometric Foundations

Table A.3 Regular polyhedra: symmetry properties. Polyhedron Tetrahedron Cube Octahedron Dodecahedron Icosahedron

2-fold

3-fold

4-fold

5-fold

Planes of reflection

3 6 6 15 15

4 4 4 10 10

6 3 3 – –

– – – 6 6

6 9 9 15 15

(a)

(b)

(c)

Figure A.2 Symmetry of the cube.

The last column in the table is related to a further type of symmetry of regular polyhedra, called reflection symmetry. This means that half of the polyhedron reflects into the other half in a mirror that lies on the plane of reflection. The thirteen semiregular polyhedra are obtained if the condition that all polygons must be equal is relaxed, while maintaining the condition that the vertices must still be identical. The semiregular polyedra are shown in Fig. A.3. The majority contain only two different polygons. For example, the faces of the truncated octahedron are squares and hexagons, see also Fig. A.4. Dual polyhedra are also defined (Huybers, 2015); they can be obtained by placing identical pyramids on the faces of the regular polyhedra.

A.2.2

Tessellations in 2D and 3D Polyhedra can be used to subdivide two-dimensional and three-dimensional space. A two-dimensional tessellation, also known as a tiling, is defined as a collection of plane figures that fills the plane with no overlaps and no gaps. There are only three different plane filling tessellations of equal and regular polygons; they are called regular tessellations and are shown in Fig. A.5(a). There are also eight semiregular tessellations, Fig. A.5(b), each consisting of two or more regular polygons. In both regular and semiregular tessellations all vertices are identical.

A.2 Polyhedra and Tessellations

Truncated Tetrahedron

Truncated Cube

341

Truncated Octahedron

Truncated Icosahedron

Truncated Dodecahedron

Icosidodecahedron Cuboctahedron

Small Rhombicuboctahedron

Great Small Rhombicuboctahedron Rhombicosidodecahedron

Snub Cube

Figure A.3 The semiregular polyhedra.

Figure A.4 Truncated octahedron.

Snub Dodecahedron

Great Rhombicosidodecahedron

342

Geometric Foundations

(a)

(b) Figure A.5 (a) Regular and (b) semi-regular tessellations.

Turning to three dimensions, there are only five space filling arrangements consisting of identical polyhedra. They are shown in Fig. A.6, and can be classified into two different families. The first family, Fig. A.6(a–c), extends to three dimensions the regular tessellations of triangles, squares, and hexagons, and consists of space filling prisms. The second family contains two fully three-dimensional solutions, based on the rhombic dodecahedron, a polyhedron with twelve diamond-shaped faces Fig. A.6(d), and the truncated octahedron, or tetrakaidecahedron, a body with six square and eight hexagonal faces, Fig. A.6(e). In Fig. A.6(d) it can be seen that the dodecahedron, drawn with thick lines, fully encloses the central cube, plus it encloses one-sixth of the volume of the adjacent six

A.2 Polyhedra and Tessellations

343

Table A.4 Geometric properties of tetrakaidecahedron of side L. Number of faces F Number of edges E Number of vertices V Surface area Volume

14 36 24 26.80L2 11.31L3

(b)

(a)

(d)

(c)

(e)

Figure A.6 Space filling regular tessellations based on (a) triangular, (b) square, and (c) hexagonal

prisms, and (d) rhombic dodecahedra and (e) truncated octahedra (tetrakaidecahedra).

cubes (only four of which are shown in the figure). Adjacent dodecahedra (not shown) in a cartesian x,y,z arrangement would enclose the remaining five pyramids and so it can be seen that the tessellation is space filling. The space-filling nature of the truncated octahedron is of particular interest in the study of foams, see Section 6.4, but is a bit more difficult to visualize. It is best seen by thinking about sets of identically oriented polyhedra, where each set is translated by half the side length of the cubic lattice shown in Fig. A.6(e). The number of faces in the truncated octahedron is F = 14, eight faces of the original octahedron plus six additional faces obtained when the corners are cut off. The number of edges E and vertices V are given in Table A.4, together with other useful geometric properties. Substituting the values from Table A.4 we verify that Euler’s equation is satisfied: F − E + V = 14 − 36 + 24 = 2

(A.3)

344

Geometric Foundations

(a)

(b)

Figure A.7 Examples of semiregular space filling tessellations.

Figure A.8 Unit cell of Weaire–Phelan space-filling tessellation (image used with permission of

Guy Inchbold, 2019).

The number of semiregular space filling tessellations is very large. Figure A.7 shows two semiregular tessellations that fill the space by means of identical plate-like layers. These layers consist of square pyramids and tetrahedra in Fig. A.7(a), and octahedra and tetrahedra in Fig. A.7(b). Another example of a space-filling tessellation is the Weaire–Phelan structure. Figure A.8 shows its unit cell, consisting of two irregular pentagonal dodecahedra (12-sided) and six tetrakaidecahedra (14-sided) forming a unit that is repeated by translation in a cubic lattice. The shape of the faces is best understood from the cutting patterns for the two polyhedra, shown in Fig. A.9. Note that the dodecahedra do not touch each other, but are entirely surrounded by tetrakaidecahedra.

A.2.3

Tessellations of the Sphere Consider the problem of forming a tessellation over a curved surface. This is an important practical problem relevant to many different fields, such as the design of dome-like space structures or large segmented mirrors, also the shape of viruses in biology, etc. Consider, for example, the problem of designing a geodesic dome of spherical shape. In fact, in many applications only a part of a sphere would be required, but it is simpler to think first about a complete sphere, which can be truncated as required.

A.2 Polyhedra and Tessellations

(a)

(b) Figure A.9 Cutting patterns for Wearie–Phelan tessellation (a) dodecahedra and (b) tetrakaidecahedra (images used with permission of Guy Inchbold, 2019).

345

346

Geometric Foundations

Figure A.10 Order b = 3 subdivision of icosahedron; black dots mark the vertices of the icosahedron.

A simple and frequently-used approach to the design of such domes is to start from a regular polyhedron with triangular faces, e.g., an icosahedron, and to subdivide each face into a specified number of equilateral triangles; the frequency b of this subdivision – 22 = 4,32 = 9 – depends on the degree of smoothness that is required in the dome. Then, all intermediate vertices of the triangular grid are projected onto the sphere. Thus, we obtain a series of smaller triangles, whose vertices lie on the surface of the sphere. These projected triangles are no longer identical or equilateral. For example an icosahedron with order two sub-division (b = 2) can be constructed from two sets of bars of length 1 and 0.8843; an icosahedron with order four subdivision requires six sets of bars with length varying between 1 and 0.7793. Figure A.10 shows an icosahedron where each edge has been subdivided into three parts. There are several other ways of designing geodesic domes; a readable introduction can be found in van Loon (1994) and more specialized articles can be found in Tarnai (1990). The review article by Clinton (2002) provides a comprehensive review of tessellations on a spherical surface. If one is interested in designing a geodesic dome where all member lengths are identical or nearly identical, subdivisions based on the icosahedron do not work. Numerical solutions of this problem, which is related to the classical problem, known as Tammes problem, of finding the maximum radius of n circles lying on a sphere, have been obtained by Tarnai and Gaspar (1983) and Tarnai (1993). Two examples of these solutions are shown in Figs. A.11–A.12. In both cases the circles are packed in a skew triangular lattice, shown on the right: to go from one vertex of the polyhedron to a neighboring vertex one takes b steps in one direction, along one set of edges of the triangular lattice, then changes direction and takes c steps. Note that, according to this more general definition the subdivisions shown in Figs. A.10–A.12 have b = 3 and c = 0,b = 2 and c = 3, and b = 2 and c = 1, respectively.

A.3 Surfaces

347

c=3

b=2 (a)

(b)

Figure A.11 Packing of 72 equal circles on a sphere with octahedral symmetry; the skewness of the triangular lattice is defined by the parameters defined in (b) (Tarnai, 1992).

c=1

b=2 (a)

(b)

Figure A.12 Packing of 72 equal circles on a sphere with icosahedral symmetry; the skewness of the triangular lattice is defined by the parameters defined in (b) (Tarnai, 1992).

A.3

Surfaces A surface is a two-dimensional locus of points in three-dimensional space, which in this section will be assumed to be smooth and defined by a single analytical expression. Two different descriptions will be used, either in terms of two curvilinear coordinates lying on the surface, u and v, or in terms of a constraint equation, typically in the form f (x,y,z) = 0, that prescribes a condition to be satisfied by a general point with cartesian coordinates x,y,z.

348

Geometric Foundations

z

z

z

y

x

x (a.1)

y

x

y (a.2)

(a.3)

z

y

x

(c)

(b)

Figure A.13 Examples of developable surfaces: (a) circular, elliptic, and parabolic cylinders, (b) cone, and (c) tangent surface.

The developable surfaces, shown in Fig. A.13, have this name because they can be transformed or “developed” into a plane without stretching. Among these, the generalized cylinder is obtained by translating the general two-dimensional curve with equation f(u) in the direction specified by the general vector q. Thus, the equation of the generalized cylinder is p = f(u) + qv

(A.4)

where f and q are given in Table A.5 for the three specific examples shown in the figure. The generalized cone is defined by linearly expanding the general two-dimensional curve, with equation f(u), starting from a fixed point defined by the vector q. Its general equation is p = q + v(f(u) − q) where f and q are given in Table A.5 for the case of a circular cone.

(A.5)

A.3 Surfaces

349

Table A.5 Example equations of developable surfaces.

Cylinder

Cone

Vectors f, q

Equation

i

circular

f = (cos u, sin u,0) q = (0,0,1)

p = (cos u, sin u,v)

ii

elliptic

f = (a cos u,b sin u,0) q = (0,0,1)

p = (a cos u,b sin u,v)

iii

parabolic

f = (au2,u,0) q = (0,0,1)

p = (au2,u,v)

f = (cos u, sin u,0) q = (0,0,1)

p = (v cos u,v sin u,1 − v)

circular

The third and last kind of developable surface is the tangent surface, defined by the set of lines tangent to a smooth three-dimensional curve, with equation g(u). The equation of the tangent surface is p = g(u) + vg (u)

(A.6)

where g (u) is the derivative dg/du. An example is the tangent surface obtained by considering the tangents to a helix, shown in Fig. A.13(c). The equation of the helix is g(u) = (cos u, sin u,u) and so g = (− sin u, cos u,1); substituting these expressions into Eq. A.6 we obtain p = (cos u − v sin u, sin u + v cos u,u + v)

(A.7)

These and many other surfaces can be generated by computer graphics, Gray et al. (2006) gives a readable introduction to this field. The proper quadrics, which have this name because they have quadratic equations but are not cylindrical or conical surfaces (also known as improper quadrics), are shown in Fig. A.14. The simplest possible equations, in the specially chosen set of cartesian coordinate system also shown in the figure, are given in Table A.6. The quadrics of revolution, obtained by setting a = b in these equations, are axisymmetric around the z-axis and are used for a variety of practical applications. For example, the axisymmetric elliptic paraboloid has the property of focusing to a single focal point any set of incoming parallel electromagnetic waves. It is the shape used for antenna reflectors, see Sections 9.3 and 10.6, solar concentrators and telescope mirrors. The axisymmetric hyperbolic paraboloid provides the characteristic saddle shape used for lightweight roofs, see Figs. 1.19 and 2.8(d). The ellipsoid with a = b = c is, of course, a sphere of radius a. More general surfaces of revolution can be generated by a 360◦ rotation of any twodimensional curve about an axis that is coplanar with the curve. Two examples that have interesting structural applications are shown in Fig. A.15. The first is the torus, generated by rotating a circle of radius b around an axis at distance a from the center of the circle. Its equation in cylindrical coordinates is: (r − a)2 + z2 = b2

(A.8)

350

Geometric Foundations

z z

y

x (a)

(b)

y

x

z

y x (c) z

z

y

x

y

x (d)

(e)

Figure A.14 The proper quadric surfaces.

The second is the catenoid, generated by rotating the catenary curve around the z-axis. Its equation is: z (A.9) r = a cosh a which is derived in Section 3.4.

A.4 Curvature of Surfaces

351

Table A.6 Equations of proper quadrics. Gaussian curvature

Equation a

Ellipsoid

b

Elliptic paraboloid

c

Hyperbolic paraboloid

d

Hyperboloid of one sheet

e

Hyperboloid of two sheets

2 x2 + y 2 2 a b 2 x2 + y 2 2 a b 2 x2 − y 2 2 a b 2 x2 + y 2 2 a b 2 2 y x + 2 2 a b

2 + z2 = 1 c

>0

− 2z = 0

>0

− 2z = 0