Folding and deployment of curved tape springs

International Journal of Mechanical Sciences 42 (2000) 2055}2073

Folding and deployment of curved tape springs

K.A. Se!en1, Z. You2, S. Pellegrino*

Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK

Received 28 November 1997; accepted 26 June 1999

Abstract

This paper presents a study of a special type of tape springs, that are both longitudinally and transverselycurved, as required for the ribs of a novel deployable re#ector. It is shown that, although curved tape springshave much in common with straight tape springs, there is an important di!erence for equal-sense folds withsmall rotation angles. Thus, it is shown that full deployment of the re#ector cannot be guaranteed ifequal-sense folds are used when packaging it. ( 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Deployable structures; Re#ector antenna; Shell structures; Elastic spring

1. Introduction and background

The study reported in this paper is part of a series of preliminary investigations related to a newdeployable membrane re#ector that is being developed by the European Space Agency [1]. Thisnew re#ector, known as the Collapsible Rib-Tensioned Surface (CRTS) re#ector, consists of threemain parts: a central expandable hub; a series of thin-walled foldable ribs connected radially to thehub; and a precision shaped membrane that is supported and tensioned by the ribs. A photographof a small-scale `deploymenta model of the re#ector is shown in Fig. 1. This model consists of sixidentical ribs with the geometry de"ned in Table 1 and length of 530 mm, connected to a centralhub with a diameter of 110 mm. A perforated membrane is used to simulate the re#ectivemembrane without inducing signi"cant air drag and gravity e!ects during deployment. Thismembrane is attached to the ribs using cotton thread loops and cannot be prestressed.

*Corresponding author. Tel.: #44-01223-332-721; fax: #44-01223-332-662.E-mail address: [email protected] (S. Pellegrino).1Current address: Department of Mechanical Engineering, UMIST, P.O. Box 88, Sackville Street, Manchester, M60

1QD, UK.2Current address: Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK.

0020-7403/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 0 - 7 4 0 3 ( 9 9 ) 0 0 0 5 6 - 9

Nomenclature

D #exural rigidityMmax

`, Mmax

~peak moments for opposite-sense and equal-sense bending

MH̀ ,MH~

steady-state moments for opposite-sense and equal-sense bendingR

L,R

Tlongitudinal and transverse radii of curvature of tape spring

RH̀ , RH~

longitudinal radii of curvature of elastic folda angle subtended by cross-section of springh relative rotation of elastic foldsiL,i

Tcurvature changes in longitudinal and transverse directions

iL,0

, iT,0

initial curvatures in longitudinal and transverse directions

Fig. 1. Small-scale `deploymenta model of CRTS re#ector.

Table 1Spring properties

Young's Modulus E 131 000 N/mm2

Poisson's ratio l 0.3Thickness t 0.1 mmLength ¸ 220 mm (bending test and ABAQUS)

530 mm (deployment model)Subtended angle a 2.39 rad (1373)Transverse radius R

T11.5 mm

Longitudinal radius RL

1380 mm

Each rib consists of a thin, slender metallic blade which is both longitudinally and transverselycurved. A key feature of this structural element is that it is continuous, i.e. it contains no mechanicalhinges or other folding devices, and yet it can be folded elastically as will be shown. In this paper, itwill be referred to as a curved tape spring as we have previously referred to straight thin-walledmetallic strips with a curved cross-section as tape springs [2].

2056 K.A. Sewen et al. / International Journal of Mechanical Sciences 42 (2000) 2055}2073

Fig. 2. Schematic M}h relationship for straight tape spring.

The best-known application of standard tape springs is in carpenter tapes, which are usually2}5m long and have a shallow cross-section subtending an angle a+803. Tape springs whosecross-section subtends a bigger angle are increasingly being used in deployable aerospace struc-tures: see Ref. [3] for an extensive bibliography. It has been shown [2] that for the purpose offolding and deployment studies, straight tape springs can be modelled in terms of variable-lengthrigid bodies linked by moving hinges whose moment}rotation relationship matches that of a "nitelength of tape spring. This relationship is shown schematically in Fig. 2. For small rotations, thetape spring bends into a smooth curve and hence the moment varies approximately linearly withthe rotation. For larger rotations its behaviour becomes non-linear, and dependent on the sign ofthe applied moment.

For positive moments, which induce tension along the edges of the tape spring, the cross-sectionof the spring begins to #atten when the moment approaches Mmax

`. Then, suddenly the central part

of the spring snaps through as the deformation localises in a short, longitudinally curved region(fold) while the moment suddenly decreases. Then, as the rotation is further increased, the momentremains approximately constant and equal to MH̀ ; although the arc-length of the central foldincreases, its longitudinal curvature does not vary. When the rotation is decreased, the samehorizontal path is followed, but the reverse snap does not take place until a rotation smaller thanthat at which the initial snap took place. The reverse snap accurs at point D in Fig. 2.

For negative moments, which induce compression along the edges of the tape spring, thelinear behaviour ends much sooner. At Mmax

~there is a bifurcation, usually resulting in

a #exural}torsional deformation mode of the whole tape spring. Then, as the rotation is increased,the moment gradually decreases as the deformation of the spring tends to localise in themiddle. This process comes to an end, and the moment reaches a minimum value, MH

~,

when the deformation has fully localised. The fold that is formed thus is similar to the foldformed under positive moment. From this point onwards, the moment remains constant and equalto MH

~.

K.A. Sewen et al. / International Journal of Mechanical Sciences 42 (2000) 2055}2073 2057

Despite its complexity, this moment}rotation relationship shows that there is only one con"g-uration, the straight one, in which the tape spring can be in equilibrium under zero moment. Hence,if a tape spring is folded by forming any number of elastic folds, it will always attempt to return tothe straight, unstrained con"guration. If the curved tape springs that are used in the CRTS re#ectorbehaved in the same way, it should be expected that the model of Fig. 1 will always deploy back tothe original con"guration, regardless of the way in which it is packaged. To test this conjecture themodel was packaged in accordance with the two main folding schemes identi"ed by You andPellegrino [4], zig-zag folding and wrapping, and its deployment sequence was recorded witha Kodak Ektapro 4500 high-speed digital camera.

In the zig-zag folding technique each rib is folded within a radial plane, by forming localised upand down folds. Fig. 3 shows selected frames from the deployment sequence observed afterintroducing two folds in each rib, one up near the hub and one down about half way along. Notethat the time delay is 24 ms for the "rst 14 frames and 320 ms for the last two. In the "rst threeframes, 0}48 ms, note the compactness of the packaged model. The string that holds the package iscut by hand to trigger deployment, but there is no interference between the model and the handthat cuts this string.

The initial deployment phase proceeds smoothly. Frames 72}96 ms show the upward foldsopening out; frame 120 ms shows the downward folds beginning to open out. Frames 144}168 msshow the ribs twisting as the downward folds continue to open out while `slidinga towards the hub.This sliding motion continues until frame 216 ms. After frame 240 ms all downward folds havereached the hub, and at this point the motion of the model slows down considerably. Theremainder of the deployment sequence involves the slow upward motion of one rib; the other"ve ribs remain bent down. Finally, the model stops in a partially folded con"guration. Thisbehaviour is typical of deployment tests with the concave side of the re#ector facing up. It is incontrast with gravity-assisted deployment, i.e. with the re#ector facing down, which in all testsproduced the correct "nal con"guration.

In the wrapping technique described in Ref. [4] the ribs are supposed to be initially bent andtwisted to form a compact transition region, and then smoothly bent around the hub. Their tipsshould end up alternately above and below the hub plate, in order to form a series of hill and valleyfolds in the membrane. In fact, the ribs in the model are rather short in comparison to the hubdiameter and hence it was not possible to reach the end of the transition region. Selected framesfrom the deployment sequence are shown in Fig. 4. The frames up to 267 ms show the ribsunwrapping, until each rib is fully deployed apart from a localised fold near the hub: the ribs areboth bent and twisted in these hub folds. In frames 267}466 ms the ribs rotate as rigid bodies as thehub folds untwist. Then, the ribs rotate in their own radial planes as a series of cantilever beamsconnected by hinges to the hub. By frame 733 ms the "rst three ribs are fully deployed and by frame933 ms all ribs have reached their deployed con"guration. So, in this case deployment is completedsuccessfully.

In the second folding scheme, the membrane provides a useful coupling to the motion of the ribs.The correct deployed con"guration was reached, without exception and regardless of the orienta-tion of the model with respect to gravity, in all experiments that were carried out.

The above tests suggest that the curved tape springs used in the model of Fig. 1 behave di!erentlyto straight tape springs, at least for some type of folds, and hence the aim of this paper is tostudy the key di!erences between straight and curved tape springs. Due to the complexity of the


Fig. 3. Deployment sequence, from top-left to bottom-right, of CRTS re#ector model packaged according to zig-zagfolding technique.

behaviour of curved tape springs, for example some features of their behaviour are essentiallyindependent of end e!ects but other features are strongly dependent on distance from the ends, thepresent study will focus on springs with a particular set of geometric properties.

The layout of the paper is as follows. Section 2 brie#y describes the way in which curved tapesprings were manufactured, and a series of large-rotation bending tests on these springs. These tests


Fig. 4. Deployment sequence of CRTS re#ector model packaged according to wrapping technique.

show that under opposite-sense bending a curved tape spring naturally forms a localised bend;however, under equal-sense bending it deforms into a #exural-torsional mode. Localised folds ofthe type envisaged in the zig-zag packaging technique of CRTS re#ectors can be formed withmanual intervention for equal-sense bending and, once formed, are stable. Section 3 derives simple,approximate expressions for the longitudinal curvature and the steady-state moments for localisedfolds in curved tape springs. Section 4 presents a series of non-linear "nite-element simulations of


Fig. 5. (a) Curved tape spring with negative Gaussian curvature and subject to M'0; (b) its cross-section.

curved tape springs subject to large-rotation bending. It shows that, whereas for opposite-sensebending the behaviour during loading is essentially the same as during unloading, and imperfectioninsensitive, this is not the case for equal-sense bending. Section 5 presents an analysis of thefold-formation process, using an analytical solution by Mans"eld [5] together with a recentlyproposed localisation approach [2]. This analysis provides further insight into the mechanics of theformation of elastic folds in curved tape springs, but it turns out to be less accurate than thesimpli"ed approach of Section 3. Section 6 discusses the main "ndings of the study, and concludesthe paper.

2. Measurement of M}h relationship

The tape springs that are the object of the present study have negative Gaussian curvature, i.e.the sense of longitudinal curvature is opposite to the sense of transverse curvature. Denoting theuniform longitudinal and transverse radii of curvature respectively by R

Land R

T, the principal

curvatures are 1/RL

and !1/RT, see Fig. 5.

The springs are loaded by a pair of equal and opposite bending couples, M, which cause them tobend in the `softa plane of bending as shown in the "gure. Our sign convention is that a positivebending moment induces tensile stresses along the edges. It is sometimes said that a positivemoment induces opposite-sense bending of a tape spring [6], because it produces a change oftransverse curvature that is of opposite sense to the initial transverse curvature of the tape spring.Conversely, a negative bending moment induces compressive stresses along the edges of the spring,and is also referred to as equal-sense bending. The corresponding rotation, h, of the right-hand endof the tape spring with respect to the left-hand side, de"ned to be positive if anti-clockwise, will bethe key deformation parameter in this study.

Several nominally identical curved tape springs were made by heat treatment of 0.1 mm thicksemi-hard copper}beryllium (Cu}Be) sheet in a mould whose male and female parts were machinedfrom solid mild steel. The two parts have matching longitudinal and transverse curvatures. Averagegeometric and material properties for the specimens are given in Table 1. Note that the longitudinalcurvature of these tape springs is two orders of magnitude smaller than their transverse curvature.


Fig. 6. Curved tape spring with localised folds in the middle: (a) opposite-sense bending; (b) equal-sense bending. It isshown in Section 3 that RH̀ "RH

~"RH. (c) Details of stable fold for h(0.

A preliminary, qualitative comparison between these curved tape springs and a straight tapespring with similar cross-sectional properties was carried out. No signi"cant di!erences werenoticed. In particular, the following observations were made for the curved tape springs.

1. Localised elastic folds can be formed, for bending in both senses. For opposite-sense bendingsuch folds form naturally, once the applied bending moment exceeds a certain value. Forequal-sense bending there is a tendency for the tape spring to form a series of #exural}torsionalfolds, rather than a single, purely #exural fold.

2. The transverse curvature at a well-developed elastic fold is approximately zero, it is non-zero ina very narrow region near the edges.

3. The longitudinal radii of curvature of well-developed folds obtained for opposite-sense andequal-sense bending, Fig. 6, are RH̀ +11.5 mm and RH

~+12 mm, respectively.

4. Once formed, well-developed elastic folds can be easily moved along the length of the tapespring.

5. The bending moment at a well-developed fold is approximately constant.

Subsequently, a detailed series of tests were carried out using the bending apparatus shown inFig. 7. With this apparatus it is possible to apply equal and opposite rotations of up to nearly 903 tothe ends of a tape spring while measuring the corresponding end moments. The rotations areimposed by turning, by hand, two knobs connected to miniature gear boxes with a reduction ratioof 80. One of the gearboxes is mounted on ball bearings, to keep the axial force on the specimenequal to zero throughout the test. The corresponding moments applied at either end of thespecimen are measured by torque cells, and the rotations of the gearboxes are manually adjusted toequalise the two end moments. Further details can be found in Ref. [7].


Fig. 7. Bending apparatus.

Fig. 8. M}h relationship measured from a continuous bending test.

Fig. 8 shows the moment}rotation relationship measured by carrying out a continuous load,unload, reverse load, and reverse unload cycle.

Starting from the unloaded, unstrained con"guration, which corresponds to the origin of theplot, the relative rotation h was gradually increased. After taking up a small amount of initial`slacka the measured response became linear up to a moment of +400Nmm. Then, the responsesoftened and, after reaching a peak at Mmax

`+450Nmm, the spring snapped through to form

a stable, localised fold. As the rotation was further increased, the moment remained constant atMH̀ +50Nmm. Note that the peak moment measured in this test was actually over 10% higherthan any other values measured in subsequent tests. Also, further tests in which larger, positivevalues of h were reached, have shown the steady-state moment to be +40 Nmm.


Fig. 9. M}h relationship for equal-sense bending, measured by releasing an initially folded spring. The range of values ofh is much greater than in Fig. 8.

The unloading path followed the post-snap-through part of the loading path, but thelocalised fold remained in place to smaller values of h than that at which the initial snap hadoccurred. Suddenly, at h+0.22 rad, the spring snapped back to the pre-snap-through con"gura-tion and the moment increased to approximately Mmax

`. Further decreasing h resulted in linear

unloading.The test continued with the reverse moment being applied but, as already expected from

Observation (1) above, the tape spring did not form a single elastic fold in the middle. Instead, itdeformed into a complex three-dimensional #exural}torsional mode. This type of deformation isnot of direct interest to the present study, as a deployable re#ector would be packaged undercarefully controlled conditions, thus minimising unwanted torsional e!ects. However, it should benoted that the peak moment is !130Nmm; this is the maximum bending moment that can becarried by a fully-deployed tape spring, without buckling.

An alternative and, for present purposes, more useful way of measuring the M}h relationship forequal-sense bending is to form by hand a purely #exural, localised fold in the middle of the tapespring and to measure the values of M and h as the rotation is gradually decreased to zero. Thismethod produces, of course, only the stable part of the unloading path. Measurements from threesuch tests are shown in Fig. 9. Despite some scatter in the results, most noticeably at the start of thetests and during the snap-back at the end * due to the di$culty of equalising the end moments* this plot shows clearly that there are some signi"cant departures from the behaviour observed instraight tape springs. First, and most importantly, the steady-state moment MH

~can only be said to

be constant and +!20Nmm for h(!0.8 rad (!463). For h'!0.8 rad (!463), M slowlyincreases and becomes zero for h"!0.3 rad (!173). The reason for this is that, as the elastic foldregion becomes shorter and shorter, it "nally takes the shape of a localised `kinka that is stableunder zero moment, as shown in Fig. 6(c). Second, the variation of M with h during the snap-backshows two separate peaks, which are signs of the existence of separate equilibrium paths, associatedwith di!erent buckling modes.


Fig. 10. Compound M}h relationship, valid only for decreasing rotation amplitudes.

Finally, Fig. 10 shows a moment}rotation relationship that is valid only for decreasing DhD. Thisrelationship was obtained from two separate tests on the same specimen, in which localised folds of$2 rad ($1153) were formed, and the `unloadinga response was measured. This plot can be usedto simulate the deployment dynamics of curved tape springs following the method proposed bySe!en and Pellegrino [2].

3. Approximate values of MH̀ , MH~

and RH

In straight tape springs the longitudinal radius of curvature in a localised elastic fold is equalto the transverse radius of curvature R

T, if end e!ects are negligible [6,8,9]. It will now be

shown that an analogous property holds for curved tape springs, namely localised folds of any signhave approximately the same radius of curvature, RH. An analytical expression for RH will beobtained by an energy method, and simple expressions for the moments MH̀ and MH

~will then be

derived.Based on observations from physical models, it will be assumed that a localised fold in a curved

tape spring consists of a central region with uniform longitudinal curvature and zero transversecurvature, which is connected by transition regions to undeformed parts of the spring. For straightsprings Calladine [9] has pointed out that the transition regions have to stretch as well as bend, butthe central region is in pure bending, as there is no change of Gaussian curvature. Thus, Calladineargued that, because the bending and stretching energies in the transition regions are independentof the longitudinal curvature in the fold, they do not need to be included in the calculation. In thecase of curved tape springs with relatively small longitudinal curvature, as in the present case,see Table 1, one can proceed exactly in the same way. However, it should be realised that, as theoriginal Gaussian curvature of the spring is not exactly zero, the stretching strain energy in the foldis being assumed negligibly small: it is not exactly zero.


From Ref. [10], the bending strain energy in a thin shell element of unit area is given by

D2

(i2L#2li

LiT#i2

T), (1)

where D"Et3/12(1!l2) and iL, i

Tare respectively the longitudinal and transverse changes of

curvature. E, l are the Young's modulus and Poisson's ratio of the shell, respectively, and t itsthickness.

The uniform, but as yet unknown, radius of longitudinal curvature in the fold will be denoted byr. Assuming the transverse curvature to be zero, the curvature changes have the followingexpressions:

(iL, i

T)"A

1r!

1R

L

,1

RTB, (2)

(iL, i

T)"A!

1r!

1R

L

,1

RTB (3)

for folds produced by opposite-sense and equal-sense bending, respectively.To determine the value of r for opposite-sense bending, substitute Eq. (2) into Eq. (1), multiply by

the surface area of the fold region, aRThr , and tidy up to obtain the following expression for the

total strain energy:

;"

aRThD

2 A1r#

rR2

L

!

2R

L

#

2lR

T

!

2lrR

LR

T

#

rR2

TB. (4)

The actual value of the radius of the fold, r"RH̀ , is obtained by minimising ; with respect to r,i.e. by setting

L;Lr

"

aRThD

2 A!1r2

#

1R2

L

!

2lR

LR

T

#

1R2

TB"0. (5)

Then, RH̀ is obtained by solving Eq. (5) for r

RH̀ "

RLR

TJR2

L#R2

T!2lR

LR

T

. (6)

For a fold produced by equal-sense bending one proceeds in the same way, but using Eq. (3)instead of Eq. (2). Then, Eq. (4) is unchanged, apart from the sign of the two constant terms, andhence Eq. (5) is unchanged. Therefore

RH̀ "RH~"RH (7)

within the limits of validity of the present approach.In conclusion, the radius of curvature of any localised folds in a curved tape spring is

RH+R

TJ1!2lb#b2

, (8)


Table 2Key values for tape springs of Table 1

Experiment Section 3 FE Section 5estimate estimate estimate

RH̀ 12 mm 11.5 12.5 mmRH

~11.5 mm 11.5 mm

Mmax`

380Nmm 400Nmm 180NmmMH̀ 40Nmm 37.0N mm 35.6N mm 27.3N mmMmax

~!130Nmm !230Nmm !15.4Nmm

MH~

!20 Nmm !20.2Nmm !20 Nmm

where b"RL/R

T. The bending moments associated with these folds are obtained [10] by

multiplying the bending moment per unit length, equal to the bending sti!ness D times thecurvature change i

L#li

T, by the transverse arc-length of the tape spring, aR

T. Thus

MH"aRTD(i

L#li

T). (9)

Substituting for iL

and iT

their expressions in Eqs. (2) and (3), with r"RH, we obtain

MH̀ "aRTDA

1RH

!

1R

L

#

lR

TB, (10)

MH~"aR

TDA!

1RH

!

1R

L

#

lR

TB. (11)

The values of RH, MH̀ and MH~

obtained by substituting the spring parameters of Table 1 intoEqs. (8), (10), and (11) are given in Table 2. Note that the estimated value of RH practically coincideswith the radii measured in Section 2. Also note that MH̀ and MH

~are within a few percent of the

average measured steady-state moments, shown in Fig. 10.

4. Finite-element computation of M}h relationship

The large-rotation bending behaviour of a curved tape spring with the geometry of Table 1 wassimulated using the "nite-element package ABAQUS [11]. The solution procedure was practicallyidentical to that followed in Ref. [12] for straight tape springs. Separate analyses were carried outfor opposite-sense bending and equal-sense bending, and their results will be shown and discussedseparately.

Fig. 11 shows a plot of M against h for opposite-sense bending. There are three distinct regions ofinterest. The "rst is the initial behaviour, up to the peak moment. As h increases from zero, M alsoincreases and reaches a maximum value of +400 Nmm. The second region extends from this peakvalue to h+0.15 rad. The moment decreases sharply and, near the bottom of the curve, the relativerotation between the ends of the tape spring decreases slightly. In an actual test the rotation would


Fig. 11. ABAQUS prediction of opposite-sense bending response of spring of Table 1.

Fig. 12. ABAQUS predictions of equal-sense bending response of spring of Table 1 with (a) perfect geometry and(b) initial imperfections. The range of rotations in (a) is much greater than in (b).

be monotonically increased, and hence a snap-through would occur. When the end of this phase isreached a localised fold has formed in the spring. Thereafter, M remains at an approximatelyconstant value of MH̀ +35.6Nmm. This simulation fully agrees with the behaviour observed inthe experiments. Furthermore, both the peak and the steady-state moments are within 5% of themeasured values.

Two di!erent simulations of the predicted behaviour for equal-sense bending are shown inFig. 12. Fig. 12(a) shows a simulation of the behaviour of a spring with perfect geometry. Thesudden decrease in the slope of the graph that is predicted by this analysis is triggered by localbuckling of the free edges of the spring, not by the torsional}#exural mode observed in theexperiments. The resulting deformation is symmetric and the associated peak moment is onlyslightly smaller than the peak moment for opposite-sense bending.


Although this analysis is a poor simulation of the loading path, it is actually a rather goodsimulation of the unloading path for a purely #exural fold, as it is readily shown by comparingFig. 12(a) with Fig. 9. In particular, note that Fig. 12(a) shows the following. First, di!erentequilibrium con"gurations are possible for M"0, as observed experimentally. Second, whenh(!0.8 rad a stable fold is formed under an approximately constant bending momentMH+!20Nmm, which agrees with our experiment and also with the simple prediction inSection 3. Third, the snap-back involves a two-peak response.

In the second simulation, the behaviour of the spring under increasing DMD was simulated byseeding a torsional imperfection in the form of a `ripplea along one of the free edges,with wavelength of one "fth the length of the tape spring and amplitude of 0.1 mm. In Fig. 12(b)the initial behaviour consists of a linear phase in which asymmetric torsional buckles formalong the free edges of the curved rib, near the ends. As DhD increases, these buckles growin amplitude and move slowly towards the centre of the tape. As they begin to coalesce, themoment quickly tapers o! at M+!230Nmm and then decreases in magnitude until thetorsional buckles begin to merge into a single fold. However, at h+!0.33 rad the analysisfailed to converge and had to be stopped. A comparison of Fig. 12(b) with the experimentalplot in Fig. 8 shows the predicted peak moment to be 75% higher than the peak measured inthe test.

The simulation shows that the initial behaviour of the imperfect spring is stable as both iL

andthe degree of twisting increase.

5. Analysis of fold-formation process

This section presents an analysis, based on Ref. [5], of the relationship between the bendingmoment, M, and the change of longitudinal curvature, i

L, for a curved tape spring of constant

thickness. Mans"eld's theory does not account for end-e!ects, and hence this analysis is only validfor very long springs, whose deformation can be assumed to be uniform.

The expression relating M to iL

is [5]

M"2RTD sin

a2 GiL

!liT,0

#

K1

(iL#i

L,0)2#F

1C!l2iL#li

T,0!

K1

(iL#i

L,0)2D

#F2Cl2(iL

!iL,0

)!2liT,0

#

K2

iL#i

L,0

#

K3

(iL#i

L,0)2#

i2L,0

i2T,0

(iL#i

L,0)3DH, (12)

where iL,0

"1/RL

and iT,0

"!1/RT

are the initial curvatures of the spring, and

K1"i

L,0i2T,0

#li2L,0

iT,0

!

i2L,0

i2T,0

iL#i

L,0

,

K2"i2

T,0#4li

L,0iT,0

#l2i2L,0

,

K3"!2i

L,0i2T,0

!2li2L,0

iT,0

,


Fig. 13. Moment}curvature plot for spring with geometry of Table 1 and subject to uniform opposite-sense bending.

F1"

2j

cosh j!cos jsinh j!sin j

,

F2"

12j

cosh j!cos jsinh j!sin j

!

sinh j sin j(sinh j#sin j)2

,

with

j"2RT

sina2

4J3(1!l2)SiL#i

L,0t

.(13)

Fig. 13 shows the moment}curvature plot, obtained from Eq. (12), for the opposite-sense bendingof a long tape spring whose cross-section has the geometry de"ned in Table 1. As i

Lincreases,

"rst M increases steeply in a linear fashion; then, as the section begins to #atten, M reaches a peakvalue of +180Nmm and starts to decrease. After levelling out at 17Nmm, it begins to increaseagain.

The formation of localised elastic folds in straight tape springs belongs to a class of problemsknown as propagating instabilities; see the extensive review paper by Kyriakides [13] for anintroduction to this "eld and general references, and Ref. [2] for an application to tape springs. Theformation of a fold begins when the deformation of the tape spring starts to localise, which occurswhen a peak (limit point) is reached in the moment}rotation relationship for the tape spring subjectto uniform bending. For the deformation to start spreading when the rotation is further increased,the moment}rotation relationship has to have a characteristic up}down}up pro"le. Thus, it ispossible to balance the work done by the external moment applied to the tape spring, when thelength of the fold increases by a certain amount, with the associated change of strain energy. Thisbalance sets the value of the required propagation moment.

The moment}rotation relationship for uniform bending of a curved tape spring of unit lengthlooks identical to the moment}curvature relationship shown in Fig. 13 and, since this curve has therequired up}down}up shape, the standard analysis for a propagating instability is carried out.


The peak moment is M`max

+180Nmm here; this value is much lower than all values obtainedpreviously because now we are dealing with an in"nitely long tape spring. The steady-statepropagation moment MH̀ can be obtained by the so-called Maxwell construction [2,13], whereone searches for a horizontal line in the moment}rotation plot such that the area above this line,but under the peak of the curve, is equal to the area below this line and above the curve. The resultof this search, shown by the horizontal line in Fig. 13 is MH̀ "27.3Nmm. As already found forM`

max, this value is lower than all values obtained previously, for the same reason. The correspond-

ing curvature change is iL"0.079 mm~1. The fold radius RH̀ is then determined from the

relationship

1RH̀

!iL,0

"0.079 mm~1 (14)

and hence RH̀ "12.5 mm.Eq. (12) is valid, in principle, for curvature changes of any sign. However, as the numerator in the

second square root of Eq. (13) should always be positive, the signs of iL

and iT

will need to bereversed for equal-sense bending, which corresponds to turning the spring upside down. Moreimportantly, Mans"eld [5] noted that equal-sense bending of straight tape springs tends toproduce snap-through buckling via a torsional mode. As the same type of behaviour occurs incurved tape springs, it would be pointless to generate an unstable moment}curvature relationship.Instead, we have substituted into Eq. (12) the same condition for torsional buckling that was usedby Mans"eld, and thus obtained the following expression for the bending moment at whichtorsional buckling will begin:

DMD"2(1!l)RTD sin

a2 CiT,0

!iL,0

#(1!l)(iL#i

L,0)A1!

F4

F23BD, (15)

where

F3"1!

2j

cosh j!cos jsinh j#sin j

,

F4"1#

sinh j sin j(sinhj#sinj)2

!

52j

cosh j!cos jsinh j#sin j

.

This expression has been plotted against the change in longitudinal curvature in Fig. 14. Notethat the slope dDMD/di

Lis negative for all values of i

L, and hence the torsional mode is unstable at

all times. Therefore, only the maximum value of DMD is of practical interest, which indicates that themaximum moment that can be carried by the curved rib before it snaps-through by twisting isMmax

~"!15.4Nmm. Recall, though, that this analysis is valid only for very long springs.

6. Discussion and conclusions

This preliminary investigation of the folding and deployment behaviour of curved tape springshas shown that they have much in common with straight springs. The di!erences between the two


Fig. 14. Bending moment at which an in"nitely long tape spring buckles in torsion.

types of springs are signi"cant only for equal-sense folds with small rotation angles. In all otherrespects their behaviour is qualitatively the same, and can be simulated using the same kind ofanalytical and computational models that have already been developed for straight springs [2].

Localised folds can be formed without di$culty in curved tape springs and, in analogy withstraight springs, such folds have a characteristic longitudinal radius of curvature and zerotransverse curvature. This holds both for opposite-sense folds and for well-developed equal-sensefolds. For the case of a gently curved tape spring with a transverse radius of 11.5 mm andsubtending an angle of 1373, a minimum rotation of about 503 is required for this result to be valid.Hence, the same packaging techniques that have been developed for deployable structures contain-ing straight tape springs can also be used for structures based on curved tape springs.

Within the limits stated above, opposite-sense and equal-sense folds carry constant moments,which can be accurately predicted from Eqs. (10) and (11).

The maximum opposite-sense bending moment that can be applied to a curved tape spring,without it buckling by snap-through of the cross-section, can be accurately predicted by "nite-element analysis. The maximum equal-sense bending moment has not been predicted accurately bythe present analysis, even when an attempt was made at seeding an initial twist into the mesh.However, this value is not critical in the context of CRTS re#ectors as* after deployment* themembrane would prevent the ribs from buckling under equal-sense moments.

To the question of uniqueness of the deployed con"guration for a folded curved spring, a positiveanswer can be given with certainty only in the case when the spring contains only opposite-sensefolds. Because we now know that equal-sense folds are bi-stable, there is the possibility that a tapespring with equal-sense folds will not deploy fully. Instead, deployment can stop in an alternative,stable con"guration. It is likely that the inertia forces associated with the dynamics of deploymentwill take the spring past any zero-moment con"gurations, but each case will need to be analysed indetail. Also, the sensitivity to small imperfections of any conclusion reached from such an analysiswill need to be tested, as the deployment tests presented in this report have shown that relativelysmall gravity-induced e!ects can be su$cient to prevent the springs from deploying fully.


Acknowledgements

We thank Professor C.R. Calladine and an anonymous reviewer for helpful comments andsuggestions on an earlier version of this paper.

The work presented in this paper was carried out under contract from the European SpaceAgency. Technical advice and support from the Project Manager, Mr. W.J. Rits, has contributed tothe success of this study. Financial support from EPSRC, in the form of an Advanced ResearchFellowship for Dr. You, is gratefully acknowledged.

References

[1] Rits WJ. A multipurpose deployable membrane re#ector. ESA Bulletin 1996; (Issue 88):66}71.[2] Se!en KA, Pellegrino S. Deployment dynamics of tape springs. Proceedings of the Royal Society of London

A 1999;455:1003}48.[3] Se!en KA. Analysis of structures deployed by tape-springs. PhD Dissertation, Cambridge University, 1997.[4] You Z, Pellegrino S. Study of the folding and deployment aspects of a Collapsible Rib Tensioned Surface (CRTS)

Antenna re#ector. Department of Engineering, University of Cambridge, Report CUED/D-STRUCT/TR 144,1994.

[5] Mans"eld EH. Large-de#exion torsion and #exure of initially curved strips. Proceedings of the Royal Society ofLondon A 1973;334:279}98.

[6] Wuest W. Einige Anvendungen der Theorie der Zylinderschale. Zeitschrift fur Angewandte Mathematik undMechanik 1954;34:444}54.

[7] Fischer A. Bending instabilities of thin-walled transversely curved metallic strips. Department of Engineering,University of Cambridge, Report CUED/D-STRUCT/TR 154, 1995.

[8] Rimrott FPJ. Querschnisttsverformung bei Torsion o!nerer Pro"le. Zeitschrift fur Angewandte Mathematik undMechanik 1970;50:775}8.

[9] Calladine CR. The theory of thin shell structures: 1888}1988. Proceedings of the Institute of Mechanical Engineers1988;202:1}9.

[10] Calladine CR. Theory of shell structures. Cambridge: Cambridge University Press, 1983.[11] Hibbit, Karlsson, Sorensen, ABAQUS version 5.8. Pawtucket, RI: Hibbit, Karlsson & Sorensen, Inc. 1998.[12] Se!en KA, Pellegrino S. Deployment of a rigid panel by tape-springs. Department of Engineering, University of

Cambridge, Report CUED/D-STRUCT/TR168, 1997.[13] Kyriakides S. Propagating instabilities in structures. In: Hutchinson JW, Wu TY, editors. Advances in applied

mechanics. Boston: Academic Press, 1994. p. 67}189.


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Folding and deployment of curved tape springs

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