simulating the behaviour of three dimensional reticular structures

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Cmnpurers ct Ssrncawes Vol. 32, No. 3/4, pp. 661-669, 1989Rimed in Great Britain.

004s7949/89 53.00 l .00g 1989 Maxwell Pergamon Macmillan pit

STRUCTURAL ANALYSIS AND DESIGN OFDEPLOYABLE STRUCTURES

CHARS J. GANTES,~ JEROME. CONNOR,?ROBERT D. LWXHBR~ andYECHIELROSENFELD:

fCivi1 Engineering Department, Massachnsetts Institute of Technology, 77 Massachusetts Avenue,Cambridge, MA 02139, U.S.A.SCivil Engineering Department, Technion, Israel Institute of Technology, Haifa, IsraelAbstmet-Deployable-collapsable structures have many potential applications ranging from emergencyshelters and facilities. through relocatable, semi-permanent structures, to space-station ~rn~nen~. Theirmain advantages are the small volume they occupy during storage and insulation, and their fast andeasy erection procedure. A new concept of self-stabilizing deployable structures featuring stable, stress-freestates in both deployed and collapsed configuration shows even higher promise.

During the deployment phase these structures exhibit a highly nonlinear behavior. A large displace-ments/small strains finite element formulation is used to trace the nonlinear load-displacement curve, andto obtain the maximum internal forces that occur in the members of the structure during deployment.The influence of various parameters that affect the behavior of the structures, such as geometric shape,dimensions of the members, cross-sectional properties and kinematic assumptions is being investigated.

l.iNTRODUCXION

The need for light, compact structures that arecharacterized by their rapid erection procedure andthe possibility of easy disassembling for muse hasexisted since ancient times. Most of these light struc-tures were erected directly in the field and were madeof simple, linear members covered with fab& or rigidpanels. Todays deployable structures have theirmembers connected in the factory, so that they satisfya set of preassigned geometrical constraints. Erectionis then operated by simply articulating the variouscomponents of the structure, resulting in a fast andeasy assembly procedure, as ill~trat~ in Fig. 1.Other advantages are the ease of transportation andstorage, the minimum skill requirements for erection,dismantling and relocation, and the competitive over-all cost.

Some examples of applications include temporaryshelters or bridges for use after ea~hquak~ and otheremergency situations, temporary protective covers inremote construction sites or for curing of concrete incold environments, domes for sport facilities, exhibi-tion structures or shelters for travelling theaters, andsubmarine structures to isolate sea-regions for aqua-culture. Furthermore, deployable structures are ofeven greater interest in the aerospace industry, wheresevere constraints apply to both payload capacity ofspace ships,and to building time in space.Two types of deployable structures have beendesigned and constructed in the past.

(i) Structures that are stress-free in the foldedconfiguration, during deployment, and in the de-ployed configuration. Their disadvantage is that inthe deployed configuration they behave as mecha-nisms, and therefore need to be stabilized by externallocking devices [l-4].

(ii) Structures that are stress-free in the foldedconjuration, but develop stresses during deploy-ment and maintain residual stresses and curved mem-bers in the deployed configuration. As a result, theydo not need external stabilizing, but they are moresusceptible to buckling and their load bearing capac-ity is drastically reduced [5,6].

Fig. 1. Model of a deployable geodesic dome.661


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662 CHAIUS . GANTES er al.

Fig. 2. A typical Scissor-Like Element.A new type of deployable structure that was intro-

duced in [7-91 is investigated in this study. Aninnovative geometric design methodology allows forstructures that exhibit a stable and stress-free state inboth the initial and the final configuration. However,geometric compatibility requirements cause the devel-opment of strains and stresses during the deploymentprocedure. The structural behavior of the structuresduring that phase is highly nonlinear, hence theanalysis presents difficulties for the structuralengineer.

The options of analytical or numerical solution ofthe problem were investigated. Many researchershave worked on second order theory structuralanalysis of frames with analytical methods [lO-131.However, an analytical approach is already too com-plicated for plane structures with simple geometryand few degrees of freedom, and it is practicallyimpossible to use it for these deployable space struc-

tures with their nonregular geometric configuration.On the other hand, substantial work has been donein developing numerical techniques for the geometri-cally nonlinear analysis of structures and in imple-menting them in computer programs that are suitablefor the problem at hand [14-11. After an extensivesearch of the available software, the programADINA, based on the Finite Element Method, wasselected for this study [18].

2. GEOMETRIC CHARACTERISTICS OF DEPLOYABLESTRUCWRES

The basic structural module of deployable struc-tures is the so-called Scissor-Like Element (SLE). Itconsists of two straight bars connected to each otherat an intermediate point with a pivotal connectionand hinged at their end nodes to end nodes of otherSLEs (Fig. 2).

Fig. 3. Planviews and perspective views of polygonal units.


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Structural analysis and design of deployable structures 663

Fig. 4. Planviews of deployable structures consisting of assembled polygonal units.

The SLEs are assembled in such a way that theyform structural units with a planview of normalpolygons. Each side and each diagonal of the polygonis an SLE. The polygons can be either equilateraltriangles, squares, or normal hexagons (Fig. 3). Bycombining several of these Noel-polygon-shardunits, structures of various, flat or curved geometricconfigurations can be created (Figs 4 and 5). Thispaper deals with deployable structures that consist ofone polygonal unit only, and have a flat deployedconfiguration of the form of a double-layer grid [19].Figure 6 illustrates various stages of deployment of astructure consisting of one single square unit.

suggested, and corresponding graphs have been pro-vided. It is assumed that the structures investigated inthe present study satisfy these geometric require-ments.3. RRSFONSE F DEPLOYABLE TRUcfzlRES DURINGDEPLOYMENT3.1. M odeii ng consi derat i ons

Strict geometric constraints have to be satisfied inorder to ensure the deployability of the structure.These constraints have been formulated in [9] and(201,where a geometric design sequence has also been

As mentioned earlier, the structural behavior ofdeployable structures during deployment is of greatinterest due to its highly nonlinear nature. Experi-mental observations lead to the conclusion that thestresses occurring in that phase are very sensitive tosmall changes in geometry or member properties, andcan become quite high. This may result not only inexpensive solutions, but also in malting the f~~bility

Fig. 5. Flat roof and dome in deployed configuration.


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664 CHARIS. GANTIBet al .

Fig. 6. Deployment stages of a square unit.

of deployable structu~s questionable due to practicallimitations during the deployment procedure. There-fore, both a qualitative understanding of the behav-ior, and a quantitative evaluation of stressesoccurring during deployment constitute an integralpart of the design of deployable structures.

Several potential deployment procedures exist, andthe selection of the most convenient one, both froma practical and from a structural point of view, is stillan open question. The method used for this study,illustrated in Fig. 7, is the simplest for structures thatconsist of one polygonal unit only. The lower centernode of the unit is considered fully supported, whilethe upper center node is free to move vertically only,and is subjected to a vertical concentrated load. Allother nodes are free. This deployment procedureoffers the important advantage of symmetry, andtherefore simplifies the analysis considerably. Fig. 7. Method of ~epioyment.


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Structural analysis and design of deployable structures 665

Fig. 8. Planview and perspective view of structure andsimplified model.Due to the symmetry of both the structure and the

loading about the vertical axis connecting the twocenter nodes, all nodes are only free to translateradially and vertically, and to rotate about the tan-gential axis. This reduces the number of degrees offreedom for the problem. Furthermore, only one nthof the structure has to be analyzed, as illustrated inFig. 8. A further simplification of the mode1 could bethe substitution of outer SLEs by springs. However,these springs would have varying stiffness duringdeployment, and therefore could not be represented.

The nature of the strains and stresses that developin the members of the structure during deploymentdefines the type of kinematic assumptions that haveto be made for this problem. These strains andstresses result from compatibility requirements be-tween the members of inner and outer SLEs. Themembers have to deform in such a way that theplanview of the structure remains a normal polygonthroughout the deployment procedure; at the sametime, the vertical distance of upper and lower circum-ferential nodes has to be the same for both inner andouter SLEs. Hence, the resulting strains and stressesare due to second order effects, and a large displace-ments-small strains formulation is appropriate [141.

At the collapsed configuration all nodes of thestructure lie theoretically on a straight line. Further-more, a small deformation has to take place beforethe structure can carry loads. In the light of the abovereasons the deployed configuration was used as initialstate for the analysis, i.e. dismantling was simulatedinstead of deployment.Nonlinear beam elements have been used to mode1inner SLEs, while outer SLEs that are only subjectedto axial stresses were represented by truss elements.After introducing auxiliary coordinate systems, themaster node/slave node technique was used to modelthe pivotal connections. The assumption of equalradial displacements of upper and lower circumferen-

Fig. 9. Member numbering for inner SLEs.

tial nodes was adopted, and its influence on theresponse was analyzed. This analysis provides acomparison of the response of simple polygonal unitsversus assemblages of units.

Since the type of response of deployable structureswas unknown at the beginning of this work, theautomatic step incrementation method was used ini-tially [18,21]. After a better understanding of thebehavior was acquired, the more economical BFGSmethod was employed [14, 18,221. In some cases linesearch was required in order to achieve conver-gence [15-171.3.2. Variation of structural quantit ies duringdeployment

Curves that describe the variation of the requiredexternal load and of the occurring internal memberforces, as the structure deforms from the deployedconfiguration to the collapsed one, are presented inthis section (Fig. 10). These results refer to a deploy-able structure consisting of one simple square unit.The behavior is similar for triangular and hexagonalunits. The response of structures that result as assem-blages of more polygonal units is expected to bequalitatively the same. However, other issues, such asdifferent kinematic assumptions, violation of symmetry conditions, and appropriateness of the suggesteddeployment method have to be addressed.

The load-displacement curve indicates a snap-through type of behavior for the structure. It can alsobe observed that the maximum values of externalload and member forces occur at different time steps.The axial force of member 1 of inner SLEs (Fig. 9)is the dominant load carrying mechanism. The distri-bution of forces among members is very unbalanced,a fact that should be seriously taken into consider-ation during design. The steep slope of the curves atthe collapsed configuration corresponds to the sum ofthe axial stiffness of the members.

4. PARAMETE RS AFFECTING THE STRUCTURALBEHAVIOR DURING DEPLOYMENT

The previous section provided qualitative, physicalinsight into the behavior of deployable structuresduring the deployment procedure. The next point ofinterest for the analyst, and especially for the de-signer, is to obtain information about how severalparameters affect the response. Such data help to


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666 CHAIUS.GANTESt al.

Load Required for Dbmantllng/D8pbyment

Axial Force of Inner SLf8During Depbyment

Bending Moments of Inner SLEsDuring Deployment

I0 SL Dtq&mont :?centor iL* 026- uem8wa *&i .--- uumwa UP-U4

Fig. IO. Variation of structural quantities during deploy-ment.

optimize the design process by taking full advantageof the features of deployable structures, and byminimizing their limitations. Parameters investigatedhere are the geometry of the structure, the cross-section of inner and outer SLEs, and the length tostiffness ratio of the members. The influence of theassumption of equal radial displacements of thecorner nodes is also investigated. The effect of theseparameters on the squired deployment load and theresulting maximum axial force of member 1 of innerSLEs is described by appropriate curves. Similarcurves describinn the influence of the above narame-

ters on the other member generalized forces can easilybe obtained.The geometry of the structure is a very important

factor influencing the structural response. Once thearea to be covered has been specified, and the type ofnormal polygon to be used has been selected, thereare two geometric design parameters that have to bechosen: the height h, and the length ratio x ofmembers (1) and (2). The sensitivity of the responseto changes of these two parameters has been investi-gated for a range of values of both h and x. The resultis that these two parameters are directly related

Influence of the Qeometrk ConfiguratbnOn th8 Required Depbynwnt Load

-6.4 0.8 t2 i.0 2 2.4HolghthInfluence of the Qeom8trk ConflgwatbnOn the Axial Force of Inner SLEo

Height h

influence of the oeOm8trk ~~~Cn the Bending Moment of Inner SLE8

Fig. 11. Influence of geometry on the response duringdeployment.


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Structural analysis and design of deployable structures 667through economy constraints and are therefore notindependent. Treatment of /I and x as independentcan result in very expensive, or even unfeasible,solutions.

The cross-section of inner SLEs is also a veryimportant parameter affecting the response of de-ployable structures during deployment. The sensitiv-ity to even small changes of this cross-section can bevery significant. The obvious conclusion is that thedesigner should be very careful when selecting thecross-section of inner SLEs, and should never overdi-

Influence of X-Section of inner SLEsOn the Required Deployment Load

ld (a)1.6 -1.4 -1.2 -

I.- /-1

,t/, , , , 16 6.6 7 6 6.6 10

Influence of X-Section of Inner SLEsOn the Axial Force of inner SLEe

a- (bl11 -

6-

I-

6-

-6 6.6 7 6 Od 10

Influence of X-Section of Inner SLEeOn the Bending Moment of inner SLEe__

(cl

_..6 6.6 7 H6&&_&& 6.6 l0Fig. 12. Influence of cross-section of inner SLEs on theresponse during deployment.C.A.S.2/ICL

mension. Choosing a large cross-section in order tobe on thesafe side when loading the structure in thedeployed configuration increases the stiffness of thestructure, and therefore also the stresses that developduring deployment. On the contrary, the in!Iuence ofthe cross-section of outer SLEs is almost negligible.The influence of the size to stiffness ratio is exam-ined by changing the radius of the circle that isprescribed around the normal polygon in the de-ployed configuration, while maintaining the samemember cross-sections. As expected, the response

L3

Fig.

Influence of the Size of the StructureOn the Required Deployment Load

1 (al0.6t\

:I \1 l.1 l.2 t6 U I6 l6 t7 U l.6 2Scalina Faotor

Influence of the Size of the StructureOn the Axial Force of Inner SLEe

4.2 -

6.2 -

2.2 e * I 0 0 0 1 l.1 l.2 l6 s4 l6 u 17 u l.6 2SodIng Factor

Influence of the Size of the StructureOn the Bending Moment of Inner SLEo

1 11 u la u t6 u t7 l.6 l.6 2806Ilng Fwtor

13. Influence of size/stiffness ratio on the responseduring deployment.


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668 CHARIs . GANTIBet al.decreases as the radius increases, since the resistanceoffered by the structure to deployment becomessmaller.

As already mentioned, the assumption of equalradial displacements of upper and lower corner nodessimulates the behavior of a structural unit that isconnected to other similar units in ail directions. Theinvestigation of the influence of the above assumptionis of twofold importance. First, it provides an assess-ment of the error made when single structural unitsare analyzed using this assumption. Second, we cansee how representative the results for single units can

Influence of the Motion of Corner NodesCn the D&mantling/DeploymentLoad

-1 --2 -

-0 0.26

Influence of the Motion of Corner NodesCn the Axial Force of Inner SLEe

Influence of the Motion of Comer NodesOn the Sending Moment of Inner SLEa

- Nntrw Noen --CN*NOdUFig. 14. Influence of comer node motion assumption on theresponse during deployment.

be for the behavior of assemblages of more units. Theconclusion is that the effect of this assumption is quiteimportant when analyzing triangular structures, notvery significant for square structures, and absolutelyunimportant for hexagonal structures.

5. SUMMARY AND CONCLUSIONSAnalysis and design issues of a new concept of

deployable structures featuring stable and stress-freestates in both deployed and collapsed configurationshave been investigated. Of particular interest is thehighly nonlinear behavior of these structures duringtheir deployment procedure, which is associated withgeometric compatibility requirements. A large dis-placement-small strain finite element formulationavailable in ADINA was used to obtain the responseof the structures, and to examine the influence ofseveral parameters on that response. Research in thisfield will continue, and other issues such as assess-ment of the sensitivity to initial member and nodeimperfections and to other deployment procedures,as well as comparison of stresses during deploymentand under service loads in the deployed configurationwill be addressed. The feasibility of arch- and dome-shaped structures will also be investigated. The uiti-mate goal is to formulate specific designrecommendations for this type of structure. Thisstudy presents a characteristic example of the poten-tial offered through the use of computers as designtools, by modeling a problem and then performingparametric analyses. Important contributions can bemade by following this approach not only in the fieldof deployable structures, but also in many otherengineering problems.

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Structural analysis and design of deployable structures 66910. A. N. Kounadis, An efficient simplified approach for thenonlinear buckling analysis of frames. J. Am. Inst.

Aeronaut ics Asfro naul ics 23, 1254-1259 (1984).1 . A. N. Kounadis, Efficiency and accuracy of linearized~stb~kling anaiysis of frames based on elastica. Int.J. Solid? Struct. (to be published).

12. C. E. Massonet, The collapse of struts, trusses andframes: a survey of up-to-date problems. In Proceedingsof a Symposium on Collapse : The Buckl in g of Sir ucturesin Theory and Pracrice (Edited by J. M. T. Thompsonand G. W. Hunt). Cambridge University Press, Cam-bridge, U.K. (1982).13. J. W. Butterworth, Analysis of double-layer grids. InAnalysis, Design and Construction of Double-LayerGr ids (Edited by Z. S. Makowski). Applied SciencePublishers, London (1981).14. K. J. Bathe, Fini re Element Procedures in Engineeri ngAnalysis. Prentice-Hall, Englewood Cliffs, NJ (1982).15. C. J. Gantes. Nonlinear analysis of space frames withthe fmite element method: &ctor iieration solutiontechniques. Thesis submitted to the Department of CivilEneineerina of the National Technical University ofA&ens, G&ece, in partial fulfillment of the reqkire-ments for the Diploma of Civil En~n~~ng (1985).16. M. Papadrakakis and C. J. Gantes, Truncated Newtonmethods for nonlinear finite element analysis. In Ciui l -

Camp 87: Thir d ~n~e~at io nal Conference on Civil mdStrucrural Engineering Compuring, special issue of Com-put. St ruct . 30, 705-714 (1988).

17. M . Papadrakakis, and C. J. Gantes, Preconditionedconjugate and secant Newton methods for nonlinearproblems. Ini. J. Numer. M eth. Engng (to be published).18. ADINA-a finite element program for automatic dy-namic incremental, nonlinear analysis. Report ARD84-6, ADINA R&D, Inc., Watertown, MA (1984).19. Z. S. Makowski, Review of the development of varioustypes of double-layer gfids. In Analysis, Design and

Constructi on of Double-Layer Gri ds (Edited by Z. S.Makowski). Applied Science Publishers, London(1981).20. R. D. Logcher, J. J. Connor, Y. Rosenfeld and C. J.Gantes, Preliminary design considerations for flat de-ployable structures. ASCE Structures Congress 89, SanFrancisco (May 1989).21. K. J. Bathe and E. N. Dvorkin, On the automaticsolution of nonlinear finite element equations. Cornput.Swuct. 17, 871-879 (1983).22. K. J. Bathe and A. P. Cimento, Some practical proce-dures for the solution of nonlinear finite element equa-tions. Compur. M eth. appl . mech. Engng 22, 231-277(1980).

simulating the behaviour of three dimensional reticular structures

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