BI-STABLE COMPOSITE SHELLS - California Institute of ...

1 AIAA 2000-1385BI- stable COMPOSITE SHELLSK. Iqbal and S. Pellegrino Department of Engineering, University of Cambridge,Trumpington Street, Cambridge, CB2 1PZ, paper is concerned with a new type of deployablestructures that can be rolled up like a tape measure but,unlike a tape measure, arestablein two different config-urations. The key to the behaviour of these structures isa particular type of COMPOSITE construction based on ananti-symmetric lay-up of glass fibres in a polypropylenematrix. The paper presents a finite element analysis us-ing the ABAQUS package of the process of rolling upa BI-STABLE cylindrical shell . The results of this simula-tion provide considerable new insights into the structuralmechanics of BI-STABLE SHELLS , as well as predicting thestress distribution, curvature variation, etc.

2 In the rolled-up paper investigates a type of shell structures that canbe rolled up like a standard steel tape measure but, unlikea tape measure, arestablein two different configurations,as shown in Fig. 1: Extended and rolled-up configurations of a BI-STABLE COMPOSITE interest in this type of structures arises from theirpotential applications in the field of deployable such application, extendible booms, is illustrated inFigure 2. This photograph shows a boom that is used asa telescopic camera mount for the inspection of nuclearpower stations; its extended length is over 10 m with a di-ameter of about 200 mm. Note that the containment struc-ture around the boom is absolutely minimal, which is ingreat contrast with the deployment cassette of a standardbi-STEM boom, with diameter around 25 mm, shown inFigure 2: Telescopic camera mount based on bi-stabletube (Courtesy of Rolatube Ltd).

3 Figure 3: Bi-STEM boom (Rimrott, 1965).Normally, a cylindrical shell structure that has beenflattened and rolled up, as shown in the examples above, Research Student. Reader in Structural Engineering, Associate Fellow 2000 by S. Pellegrino. Published by the American Institute of Aeronautics and Astronautics, Inc. with Institute for Aeronautics and Astronauticsis in a high-energy, unstable state. Therefore a contain-ment/deployment mechanism is required to prevent thestructure from releasing its stored elastic energy in an un-controlled way. By using BI-STABLE shell structures we canavoid, or much reduce the need for a deployment shell shown in Fig. 1 is strain free in the straight,extended configuration, but is subject to high levels ofstrain in the second, rolled-up configuration.

4 However,it is unable to jump from the second configuration to thefirst without a substantial energy input because the rolled-up configuration corresponds to a local minimum for thepotential energy surface of the layout of this paper is as follows. The next sec-tion gives a brief history of the development of bi-stablecomposite structures and relates their behaviour to theway they are constructed. A simple analytical method,previously developed, is briefly outlined. The follow-ing section describes the simulation techniques that havebeen developed to study the behaviour of BI-STABLE are validated against experimental data from a ten-sion test and a bending test on flat plates. New insightsare gained into the non-linear behaviour observed in thebending test.

5 Next, the simulation of the process of rollingup a cylindrical shell and leaving it rolled up when allexternal loads are removed is described. Results thatare obtained from this simulation include the stress dis-tribution in the coiled-up shell , its longitudinal radius ofcurvature, and the profile of the transverse comparison between the results from the finite elementsimulation, the simple analytical model, and experimentalmeasurements concludes the COMPOSITE SHELLS were discovered by Daton-Lovett (1996). The inventor visited the Deployable Struc-tures Laboratory, in Cambridge, in the summer 1996 andshowed us several of his models. That visit signalledthe beginning of a continuing collaboration with Daton-Lovett.

6 During the last three years we have carried outa number of studies with the aim of developing a betterunderstanding of the structural mechanics of these struc-tures, as well as analytical and computational models topredict their a previous paper (Iqbal et al. 1998) we have pre-sented a simple analytical model that captures the keyfeatures of BI-STABLE cylindrical SHELLS . Because that pa-per provides the background for the present numericalstudy, in this section we summarize its main BI-STABLE behaviour of cylindrical shell structuresoriginatesfromthefactthatthetr ansformationofasurfaceof zero gaussian curvature (recall that the gaussian curva-ture is the product of the two principal curvatures) intoother surfaces of zero gaussian curvature requires onlybending energy, as the transformation of the mid-surfacesis isometric.

7 By arranging stiff fibres in a suitable lay-up,one can create preferential directions of bending for theshell and thus, when the shell is deformed by a sufficientlylarge amount from its original, unstressed configuration,it will flip into an alternative configuration with zerogaussian avoid that the shell ends up in a twisted configu-ration when it flips, Daton-Lovett chose to use an anti-symmetric lay-up, thus almost eliminating coupling be-tween bending and twisting. This makes it possible tocompactly roll-up a cylindrical , this can be seen in theABDmatrix(Hyer, 1998) relating the mid-surface strains and curva-tures to the corresponding stress resultants in the shell NxNyNxyMxMyMxy = ABBD x y xy x y xy (1)Table 1: Properties of PP/Glass unidirectional s Modulus along the fibres GPaYoung s Modulus across the fibres GPaShear GPaMajor Poisson s Poisson s thexandy-axes to coincide with the longitu-dinal and transverse directions of the shell , as shown inFig.

8 4(a), for a 5-ply laminate [+45/-45/0/+45/-45] withthe unidirectional properties listed in Table 1 theABDmatrix is (2)here the units are GPa mm forA,GPamm2forB, andGPa mm3forD. From now on, the ply angle will bedenoted by .Because the coefficientsD13andD23are zero, bend-ing and twisting are now decoupled. Therefore, the longi-tudinal and transverse directions of curvature of the tubewill be principal directions of curvature also in the rolled-up configuration, and hence each turn will overlap theprevious, as it is clearly the case in Fig. 2. Also note thatB =0for the anti-symmetric laminate, but the resultingcoupling between bending and stretching behaviour hasonly a weak effect on the bi-stability of the Institute for Aeronautics and AstronauticsR RNeutralaxis(a) Original configurationxyRR1/yRyMx1/y1/xMx(b) Deformed configurationMyMyFigure 4: Uniform bending of cylindrical simple model for the bi-stability of a shell structurewith uniform transverse curvature,1/R, that is subjectedto uniform curvature changes xand y 1/R, Fig.

9 4,was proposed by Iqbal et al. (1998).The strain energy in the shell has the expressionU=Ub+Us(3)where the bending and stretching energies have the fol-lowing approximate expressionsUb=12 R[D11 2x+2D12 x( y 1R)+D22( y 1R)2](4)Us=A112[ R2 2x 2y+sin( R y)2 2x 3y 4sin2( R y/2) R 2x 4y](5)Figure 5 shows a contour plot of the total strain energy fora shell with =45 as a function of the total curvaturesin the longitudinal and transverse directions, xvs. transverse radius of the shell isR=25mm, and theangle subtended by the cross section is = 160 .As well as having an absolute minimum (U=0)at x=0, y=1/R= 1, which be readily ver-ified on the left-hand side inset plot (Uvs. y), the plotshows the existence of a local minimum at y 0and x 1/36mm 1.

10 This local minimum corresponds tothe rolled-up configuration of the (mm-1)x(mm-1)020406080U (N) (mm-1) (mm-1)Figure 5: Energy plot for shell with =45 andR= simple model is able to capture the main features ofthe folding of cylindrical BI-STABLE SHELLS . However, it is oflimited accuracy in predicting the value of the coiled-upradius of the shell : it typically overestimatedRby about20% for =45 and by up to 50% for larger values of . Also, because the model assumes uniform curvaturechanges, it cannot provide accurate estimates of the peakstrain that occurs during coiling; knowing this value isimportant for design Element ModellingAn extensive computational study of BI-STABLE compositeshells was carried out using the finite-element packageABAQUS (Hibbitt et al.)