Form finding analysis based on variational method for multistable structure utilizing snap-through behaviour

Transcription

1 IOP Conference Series: Maerials Science and Engineering For finding analysis based on variaional ehod for ulisable srucure uilizing snap-hrough behaviour To cie his aricle: T Kuroawa e al 00 IOP Conf. Ser.: Maer. Sci. Eng Relaed conen - Developen of coroaional forulaed FEM for applicaion o 30 class large deployable reflecor Saoru Ozawa, Yuuichi Fuiwara and Aio Tsuihaa - Dynaic response of srucures wih uncerain paraeers Z H Cai, Y W Yang and Y Liu - A ehodology for analysing laeral coupled behavior of high speed railway vehicles and srucures P Anolín, J M Goicolea, M A Asiz e al. View he aricle online for updaes and enhanceens. This conen was downloaded fro IP address on 0//08 a 6:06

2 WCCM/APCOM 00 IOP Conf. Series: Maerials Science and Engineering 0 (00) 056 doi:0.088/ x/0//056 For Finding Analysis Based on Variaional Mehod for Mulisable Srucure Uilizing Snap-Through Behaviour T Kuroawa, M Shigeasu, and M Kohiyaa Graduae School of Science and Technology, Keio Universiy 3-4- Hiyoshi, Kohou-u, Yoohaa, Japan.aui.noc@gail.co Absrac. This sudy proposed a ulisable srucure using snap-hrough behavior as an adapable srucure, which can change is size and ransfor is configuraion easily. To now he sable configuraion and sress of he srucure, he for finding analysis ehod of he srucure is forulaed. Then, a design suppor syse is proposed based on a calculaion ehod of required forces o ransfor is configuraion. Finally, several configuraions are deonsraed wih a sall scale odel o show he adapabiliy and feasibiliy of he proposed srucure.. Inroducion Environenal probles have becoe serious in recen years. As a soluion of his proble, he echnology of circulaion in consrucion aracs aenion, which consiss of assebly, disanleen, reconsrucion, conversion and reuse. In an archiecural field, a srucural syse using he circulaion echnology has been used for a long ie. For exaple, a yur (a ger) of Mongolians and a ipi of Naive Aericans have been used as a dwelling house for he noadic life, which have he feaure of he porabiliy, lighweigh propery, and flexibiliy. Aferwards aing inernaional exposiions as a urning poin, his ind of srucural syse evolved drasically, and he geodesic doe designed by Bucinser Fuller in Expo 67 in Monreal and he Crysal Palace designed by Joseph Paxon in he Grea Exhibiion of 85 in London are nown as assebly ype consrucion by inroducing he odule syse and prefabricaion (Asaura 993) []. Unil now, various srucural syses have been proposed as a eporary building in a disaser or en even, and hey are used repeaedly in any pars of he world. A srucural syse ha has he above-enioned circulaion echnology has he following hree feaures:. Mebers are ade as disounable coponen pars.. Consrucion process is reversible in disanleen process. 3. Assebly and disanleen are easy. Besides he feaures enioned above, we considered adapabiliy and diversiy in he presen sudy, and paid aenion o he following wo feaures: 4. A shape of he srucural syse is variable. 5. Volue of he inernal space is variable. c 00 Published under licence by Ld

3 WCCM/APCOM 00 IOP Conf. Series: Maerials Science and Engineering 0 (00) 056 doi:0.088/ x/0//056 A srucural syse ha has he fourh feaure in he above is called a ovable srucure. A ypical exaple of a ovable srucure is a scissors srucure, which uses he principle of scissors, and here are any pracical exaples in an archiecural field. However, he scissors srucure does no have a sable for in he process of he for changing. Therefore, i is necessary o add he consrain eber o sabilize he srucure. Oher han he above ovable srucure, a srucural syse ha conains he russ shown in Figure can change is seady for. This srucural syse can shif beween he wo inds of sabile fors A and C in Figure by using snap-hrough, which is he consiuive behavior in bucling. Such a srucural syse has he following wo feaures:. I has wo or ore sabiliy fors.. I is possible o shif he for beween or aong he sabiliy fors by adding a coparaively sall load. This ind of srucural syses hence have he possibiliy ha can ransfor wo or ore desired sable fors freely wihou any consrain ebers o sabilize he srucure. Eploying his echanis, we propose a ulisable srucure syse o enable for change using snap-hrough in his sudy. In addiion, we propose a ehod for for finding analysis, by which he for and sress can be analyzed based on copuaional echanics. F u F A B A B C u C Figure. The shown russ srucure can shif beween he sable fors A and B by using snaphrough. The graph in he righ-hand side shows he load-displaceen relaionship.. Mechanis of he proposed srucural syse Figure shows he iniial plane shape of he eber of a srucural syse ha proposes in his sudy. In Figure 3, i becoes a hree-diensional shape wih wo convex-shaped pars when four pairs of holes wih he sae nuber in Figure are conneced wih a pin respecively. Moreover, he uni is coposed he cobinaion of he pin connecion of four ebers lie a laice. The uni has four convex-shaped pars, and can ae hree inds of fors shown in Figure 4. In addiion, by connecing end holes of uliple unis wih a pin, he size can be freely adused. 3 4 Figure. The iniial shape of he eber. The red circles and nubers show he pair of holes ha will be conneced wih a pin. 3 4

4 WCCM/APCOM 00 IOP Conf. Series: Maerials Science and Engineering 0 (00) 056 doi:0.088/ x/0//056 Figure 3. Pin-connecing process of a eber. As for he convex shape par of a eber, he for is possible o change using snap-hrough behavior. Figure 4. Each uni is coposed of four ebers and can ae hree inds of fors. The pin colour shows upward convex and blue colour downward. 3

5 WCCM/APCOM 00 IOP Conf. Series: Maerials Science and Engineering 0 (00) 056 doi:0.088/ x/0//056 As an exaple of he proposed srucural syse, feasible fors are shown in Figure 5. Four by four unis are pin-conneced in wo direcions. I is possible o expand flaly by connecing wo or ore unis in he for a he upper cenre in Figure 5. The srucure can ransfor ino a desired for by using snap-rough behaviour of he convex shape par of ebers. In his exaple, hree fors are shown in Figure 5: a doe, a cave and a parasie-lie for. The doe for has a wide inernal space, he cave for has coplex inernal space, and he parasie for has an inernal space opening o he ouside. Figure 5. An exaple of he proposed srucural syse. Is shape can be changed ino a doe, a cave and a parasie-lie for. 3. For finding analysis of he proposed srucure In his chaper, we presen a ehod o find a seady for in order o deonsrae he feasibiliy of he proposed srucure based on copuaional echanics. 3.. Modeling This srucure fors an inernal space wih he cobinaion of uliple ebers of a single shape. Because he size of he secion is uch saller han ha of he longiudinal direcion, he ebers are odeled by he Tiosheno bea eleens ha consider a finie deforaion and a finie roaion. Figure 6 shows a odel of a eber wih he Tiosheno bea eleens. 4

6 WCCM/APCOM 00 IOP Conf. Series: Maerials Science and Engineering 0 (00) 056 doi:0.088/ x/0//056 : nodes Tiosheno bea eleens (wo nodes) Figure 6. Modeling of he bea eleens. Because of odeling of he eber, he srain energy of he eber is a oal of srain energy of coposing bea eleens, and i can be wrien as equaion () (Dvorin 988) [] : n bea bea S i : E idv i () Vi i Where n bea, V i, S i and E i are he nuber of bea eleens, he volue of a bea, he second Piola- Kirchhoff sress ensor, and he Green-Lagrange srain ensor, respecively. A eber is conneced by a pin and disorion and bending given o he iniial shape. As a resul, he conneced eber becoes a solid shape ha can change he for by snap-hrough. Here, we siulae he process of pin-connecion, which gives disorion o a eber. Because a connecing pin is insered in wo holes, he posiion and direcion of he holes coincide. Thus, we odel his condiion by aching he coordinaes of he wo nodes and he direcor vecor of a bea eleen as shown in Figure 7. The consrain condiion given by he pin connecion becoes equaions () and (3) when he oal nuber of he pin connecion is n pin. x (,,,n pin ) () x V (,,,n pin ) (3) V Iniial shape : ie 0 V V x x Final shape : ie x: nodes coordinaes V : direcor vecors of bea eleens ha show he direcions of holes Figure 7. Modeling of pin connecion of a eber 5

7 WCCM/APCOM 00 IOP Conf. Series: Maerials Science and Engineering 0 (00) 056 doi:0.088/ x/0//056 Then, we consider he proble o iniize he srain energy of he bea eleen based on he consrain condiion given by he pin connecion. If he gradien ehod is used o solve his nonlinear equaion, i would converge o he soluion of a ain pah ha he eber does no disor or bend, which is differen fro he realiy. Hence, in his analysis, we need o find bifurcaion pah of bucling of he eber. To obain he bifurcaion pah, we propose a new consrain condiion; in he convex shape par, i becoes a geoery relaion as shown in Figure 8. Therefore, he disance beween wo poins is consrained o be a desired disance so ha he eber becoes convex shape. The disance beween wo poins is defined by using he direcor vecors of he bea eleens fro he relaion in Figure 8 as equaion (4): where v l T 3 v V (4) 3, x x (5) By defining he disance beween wo nodes as equaions (4) and (5), he disance is defined as a signed nuber. The disance becoes a posiive value for upward convex, and he disance becoes a negaive value for downward convex. As a resul, consrain condiions in he cases of upward convex and downward convex are expressed by equaions (6) and (7), respecively. Upward convex: l l ˆ 0 (6) Downward convex: l l ˆ 0 (7) Figure 8. Consrain condiions of convex shape. 3.. Forulaion of for finding analysis Through he above enioned odelling, he for finding analysis of he proposed srucure becoes a iniizaion proble of he srain energy of he bea eleens wih equaliy consrain condiions as shown in equaion (8): in s.. x bea x 0, (,,, n pin V V 0, l,,,, n U lˆ, 0, l lˆ,,, n L 0 ) (8) 6

8 WCCM/APCOM 00 IOP Conf. Series: Maerials Science and Engineering 0 (00) 056 doi:0.088/ x/0//056 The poenial energy of he enire syse, which becoes he obecive funcion, is provided by adding he consrain condiion of equaion (8) ino a oal of he srain energy of he bea eleens based on he exended Lagrange uliplier ehod (Konno 978) [3] as follows: where and pin n pin λ T x oal (9) bea pin c x T T c v T x x x x x x λ V V V V V V v c U T cl l lˆ l lˆ L l lˆ l lˆ UL (0) nu n L T UL U () The exended Lagrange uliplier ehod cobines he advanages of he penaly funcion ehod and Lagrange uliplier ehod. This ehod is devised so ha he funcion ay locally becoe convex adding a penaly er in a sense o he Lagrangian funcion. In his analysis, here are any consrain condiions. When using he Lagrange uliplier ehod, which increases he nuber of variables as any as consrain condiions, he Hessian arix of Lagrangian funcion would no be a posiive definie arix and i would be difficul o obain he soluion wih he gradien ehod. On he oher hand, he exended Lagrange uliplier ehod iniizes he Lagrangian funcion wihou increasing he nuber of variables. In he algorih of he exended Lagrange uliplier ehod, he iniu poin ha saisfies consrain condiions is obained by ieraively updaing he Lagrange uliplier hrough he unconsrained iniizaion wihou a Lagrange uliplier reaed as a variable. Based on he principle of virual wor, nonlinear siulaneous equaions are derived fro he saionary condiion of equaion (9). To acquire he for of he srucure and is sress, he ieraive analysis of he Newon-Raphson ehod is eployed using he angen siffness arix obained by differeniaing he inernal vecor (Hisada and Noguchi 995 [4], Goo and Noguchi 008 [5] ). Figure 9 show he flow char of his for finding analysis. Analysis sar Inpu daa define direcor vecor oal nuber of freedo node daa eleen daa pin connecion daa ec. calculae Equivalen nodal force vecor: of liner er Tangen siffness arix: K L Q L Calculae residual force vecor i i i R Q F Derive displaceen / roaion correcion vecor ' i i i u K R Ter of consrain condiion * YES * Convergence es Updae Lagrange uliplier NO Correc displaceen ' i i ' i u u u NO Updae penaly paraeer * calculae Updae direcor vecor Equivalen nodal force vecor: Tangen siffness arix: i K NL i Q NL of nonlinear er Convergence es YES Oupu daa * Pos processing Ter of iniize non-consrain condiion Analysis end Figure 9. Flow char of for finding analysis. 7

9 90 30 WCCM/APCOM 00 IOP Conf. Series: Maerials Science and Engineering 0 (00) 056 doi:0.088/ x/0//056 A physical eaning of he consrain condiion o give convex shape ha he disance beween wo nodes is assigned can be considered as insering he rigid rod beween wo poins. Therefore, he Lagrange uliplier becoes an axial force added o he rigid rod. The load-displaceen relaion can be derived by changing he consrain disance beween wo nodes and he load required o change he srucural for can be obained based on he relaion diagra. 4. Analysis resul An exaple of for finding analysis of a eber is shown below. The analysis odel, condiions and resuls are shown in Figure 0, Table and Figure, respecively. We invesigae a odel wih a feasible size for which consrucion and ransforaion are possible wih huan power. Four inds of fors are analyzed for he all he cases of cobinaion of convex shapes: upward and downward convex shapes. The eber has four paerns of sable fors by snap-hrough [] Figure 0. Analysis odel. 4 Table. Condiions of for finding analysis. Ies Condiions Eleen ype Two-node Tiosheno bea eleen Toal nuber of eleens Refer o Figure 0. Young odulus 0 GPa Poison raion 0.3 Loading condiion No loading Boundary condiion Displaceen and roaion are fixed for a node indicaed by green cross in Figure 0 Pin connecion Four nodes in red circles in Figure 0 Consrain of convex shape Four nodes in blue boxes in Figure 0 8

10 WCCM/APCOM 00 IOP Conf. Series: Maerials Science and Engineering 0 (00) 056 doi:0.088/ x/0//056 Figure. Resuls of for finding analysis. Elevaion and bird s eye view are shown upper and lower figures, respecively. Four inds of fors are analyzed for he all he cases of cobinaion of convex shapes: upward and downward convex shapes. When he lef end of he lef eber is assued o be he origin, he four inds of fors corresponds o he following four cobinaions of wo convex shape pars: upward-downward, upward-upward, downward-upward and downward-downward. I should be noed ha he difference beween upward-downward and downward-upward is he overlapping order of wo bars near a pin connecion. The difference beween upward-upward and downward-downward is also he sae. Figure shows he load-displaceen relaion curve, which is depiced by changing he consrain disance beween wo nodes. The load a he red circles in Figure eans he snap-hrough bucling load ha becoes a required load o change he for of he srucure, and he obained value is 670 N. Figure. Load-displaceen curve. 9

11 WCCM/APCOM 00 IOP Conf. Series: Maerials Science and Engineering 0 (00) 056 doi:0.088/ x/0// Conclusions We proposed a new ulisable srucure syse eploying snap-hrough behaviour and forulaed a ehod o find a saionary for of he proposed srucural syse based on he exended Lagrange uliplier ehod. A design suppor syse is also proposed uilizing a calculaion ehod of required forces o ransfor is configuraion. Several configuraions are deonsraed wih a sall scale odel o show he adapabiliy and feasibiliy of he proposed srucure. In addiion, a feasible size of a eber is derived based on he proposed ehod of he for finding analysis. References [] Asaura N, 993, Design of Teporary Archiecure, Kaia Insiue Publishing, Toyo (in Japanese). [] Dvorin E N, Onae E, Oliver J, 988, On a Non-Linear Forulaion for Curved Tiosheno Bea Eleens Considering Lagrange Displaceen/Roaion Increens, Inernaional Journal of Nuerical Mehods in Engineering, Vol. 6, [3] Konno H, Yaashia H, 978, Nonlinear Prograing Mehod, JUSE Press (in Japanese). [4] Hisada T, Noguchi H, 995, Basics and Applicaions of Nonlinear Finie Eleen Mehod, Maruzen, Toyo (in Japanese). [5] Goo K, Noguchi H, 008, For Finding Analysis of Muli-Reciprocal Eleen (MRE) Space Srucures, Inernaional Conference Copuaional & Experienal Engineering and Sciences. 0