Self-Equilibrium state of V-Expander Tensegrity Beam

Transcription

1 Slf-Equilibrium stat of V-Expandr nsgrity Bam Pilad Foti 1, guinaldo Fraddosio 1, Salvator Marzano 1, Gatano Pavon 1, Mario Danil Piccioni 1 1 Dpartmnt of Civil Enginring and rchitctur, Politcnico of Bari, Italy BSRC In this papr, w study an innovativ class of tnsgrity bams, obtaind by a suitabl assmbly of lmntary V-Expandr tnsgrity clls along a longitudinal axis in th thrdimnsional spac. nsgrity structurs, mad by struts in comprssion and cabls in tnsion, ar an innovativ structurs by itslf: thy ar similar only in apparanc to convntional pin-joint structurs (trusss), and thir mchanics is strongly rlatd to initial fasibl slf-strss stats inducd in absnc of xtrnal loads. In particular, from a kinmatical point of viw ths slfstrss stats avoid th activation of possibl infinitsimal mchanisms. By a numrical study, w analyz th fasibl slf-strss stats for lightwight tnsgrity bams mad by a suitabl assmbly of V-Expandr lmntary clls. Morovr, w analyz th influnc on th fasibl slf-strss stats of th addition of struts or cabls starting from th simplst V-Expandr configuration. Kywords: tnsgrity bam; slf-quibriulium; forc dnsity mthod; numrical mthods INRODUCION nsgrity structurs ar an innovativ class of lightwight structurs, which hav gaind th intrst of rsarchrs in many diffrnt filds, including but not limitd to nginring. In particular, th intrst for tnsgrity structurs in structural nginring, as wll as in architctur, is du to thir asthtic valu, thir larg stiffnss-to-mass ratio, th possibl dployability, togthr to thir rliability and controllability. h tnsgrity concpt has found applications within architctur and civil nginring, such as towrs, larg dom structurs, stadium roofs, tmporarily structurs and tnts [1]. nsgrity is a pin-connctd fr-standing framwork composd of struts in comprssion and cabls ncssarily in tnsion. Usually, th structural analysis of a tnsgrity prliminarily rquirs a form-finding procss, sinc th shap of a tnsgrity structur is strictly rlatd to th slf-strss in its lmnts. In this papr, w show a numrical study of a class of tnsgrity bams, obtaind by a suitabl assmbly of lmntary V-Expandr tnsgrity clls along a longitudinal axis in thrdimnsional spac [2]. h ovrall prformanc of this kind of bam is strongly dpndnt on th way th diffrnt lmntary cll ar connctd. By applying a numrical mthod, w study th slf-quilibrium stats for th V-Expandr tnsgrity bam, and w analyz th structural bhavior as th pattrn of th lmnts changs. 126

2 FORM-FINDING OF ENSEGRIY SRUCURES Basic assumptions nsgrity structurs can b dfind as a discontinuous st of lmnts in comprssion within a continuous ntwork of tnsil lmnts; this dfinition is dscribd by wll-known xprssion: island of comprssion in an ocan of tnsion [3]. h following assumptions ar considrd in this study: (i) lmnts ar connctd by pin joints; (ii) mmbrs of th structur ar rctilinar; (iii) th connction btwn th struts is possibl only at thir xtrmitis; (iv) topology, i.. th connctivity btwn th nods and lmnts, and th gomtrical configuration ar known; (v) slf-wight of lmnts is nglctd and no xtrnal load is applid; (vi) buckling of th strut is not considrd; (vii) th structur is fr-standing. From th assumptions, only axial forcs ar carrid by th lmnts, i.. thr ar only two typs of lmnts: struts in comprssion and cabls in tnsion. h gomtrical configuration of th structur is dscribd in trms of nodal coordinats. Sinc th structur is fr-standing no supports ar ndd. tnsgrity structurs is a systm in a stabl slf-quilibratd stat. h slf-quilibrium stat rfrs to th initial mchanical stat of th structur bfor any load, vn gravitational, ar applid. In this initial stat, thr is a slf-strss stat in th lmnts. Furthrmor, if tnsgrity structur posssss any infinitsimal mchanisms, ths ar stabilizd by th slf-strss stat in th lmnts. h stability of th structur is dfind as th ability of th systm to rturn in quilibrium configuration aftr a small prturbation [4]. Gomtry and topology In thr-dimnsional spac, a tnsgrity structur has lmnts: c cabls and s struts, (c + s = ). h lmnts of th structur ar jointd at n nods. In th cabl-nt structurs, apparntly similar to th tnsgrity structurs, thr xist som fixd nods du to fact that only tnsion is carrid into th cabls. nsgrity structurs ar fr-standing, for assumption, and thrfor thr xist only fr nods in thr-dimnsional spac. In ordr to dfin th n gomtrical configuration of a tnsgrity structurs w dfin x, y and z ( ) as th nodal coordinat vctors of th fr nods in thr dirctions x, y, z of an orthogonal rfrnc systm x, y, z O. h topology of th tnsgrity structurs can b dfind by a connctivity matrix C ( n ). If mmbr k conncts nods i and nod j, with i < j, thn in th kth row of C w st 1 and -1 at th ith and jth position, rspctivly. hrfor, th connctivity matrix can b dfind as follows: 1, 1 if p i C if p j k p (1) 0 othrwis. Lt u, v and w ( dirctions rspctivly: ) th vctors of coordinat diffrncs of lmnts k in x, y, z 127

3 u = Cx v = Cy w = Cz (2) and lt l th vctor ( ) which collcts th lngths of th lmnts. Lt U, V, W and L ( ) b th diagonal form of u, v, w and l, rspctivly. h diagonal matrix L can b xprssd: L U V W (3) In this way, th gomtrical configuration and th topology of th tnsgrity structur ar compltly dfind. Slf-Equilibrium stat Whn gomtrical configuration and topology of tnsgrity structur ar dfind, th quilibrium quations in ach dirctions can b st as dvlopd by Schck [5]. h nonlinar quilibrium quations for unknown locations of th nods ar transformd to a st of linar quations by introducing th so-calld forc dnsity q k as th intrnal axial forc to lngth ratio for ach k lmnts. Not that q k 0 for cabls and q k 0 for struts. his condition is associatd to th unilatral mchanical bhaviour of lmnts, i.. cabl ar in tnsion, and struts ar in comprssion. In absnc of th xtrnal loads, th slfquilibrium quations for a gnral pin-jointd structur can b writtn as: C QCx 0 C QCy 0 C QCz 0 (4) whr Q ( ) is th diagonal matrix collcting th forc dnsitis ratios of all nn lmnts. By introducing th forc dnsity matrix D, ( ) as D C QC (5) th quilibrium quations in (4) can b writtn as Dx 0 Dy 0 Dz 0 (6) 128

4 Noting that diag( b) f diag( f) b, with b and f ar a gnric vctors, and by introducing th so calld quilibrium matrix ( 3n ), (4) can b rwrittn as C diag( Cx) C diag( Cy) q 0 C diag( Cz) (7) whr th quilibrium matrix is dfind as C diag( Cx) C diag( Cy) C diag( Cz) (8) From th quations in (7), th unknown basis of th vctor spac of th unknown forc dnsitis in th lmnts li in th null spac of. h fasibl slf-strss stat q, ( ) is dfind as a stat of slf-strss that satisfis (7) and should b in accordanc with th unilatral bhaviour of th lmnts. In [6] it is shown that q, ( ), which can b xprssd as a linar combination of th basis of th vctor spac of th forc dnsitis, can b writtn according to th gomtrical symmtry of th structur. In particular, lmnts in symmtric position hav th sam forc dnsity and thn thy can b collctd in a group. Lt r and r, th rank and th dimnsion of th null spac of th quilibrium matrix, hnc, thr xist r indpndnt stats of slf-strss. r r (9) s abov mntiond, q can b writtn as q 1q 1 2q 2... q r r (10) whr i, i=1,2,..., r, ar ral cofficints. Furthrmor, lt h, th numbr of groups of th symmtry, vctor q can also b writtn as q q q q (11) h h whr i ( ), i=1,2,, h, is a vctor composd of a unit in th ith position if th lmnt blongs to th group and zro othrwis. From (10) and (11), a nw matrix G ( r h ), and nw vctor β ( r h ), can b writtn, which collcts th vctors q i and i, and th ral i and q i, rspctivly as 129

5 G 1, 2,..., r, 1, 2,..., h q q q (12) β 1, 2,... r, q1, q 2,..., q h (13) In this way, in ordr to find th fasibl slf-strss stat, can b writtn th follows quation Gβ 0 (14) Singular Valu Dcomposition (SVD) should b carrid out to find all solutions of (14). hs solutions li in th null spac of G. If th dimnsion of th null spac of G is unit, a tnsgrity structur possss a singl initial mod of slf-strss which is compatibl with th unilatral bhaviour of th lmnts, and is in according to th symmtry of th structur in th slf-quilibrium. If th dimnsion of th null spac of G is qual to zro (14) has only trivial solutions; if th dimnsion of th null spac of G is mor than unit (14) has mor than on non-trivial solutions. It is clar that th first r trms of β ar th ral cofficints of th linar combination in (10) and th last h trms ar th forc dnsitis in th groups. h initial forc vctor, in th slf-quilibrium stat, f i ( ) can b xprss as f Lq (15) h Euclidan norm of th vctor of unbalancd forc f u ( th accuracy of th slf-quilibrium conditions 3n ) can b usd to valuat fu q (16) Infinitsimal mchanisms 3n Lt ε ( ε ), and d ( d ) b th vctor of th axial strains of th lmnts and th vctor of th nodal displacmnts, rspctivly. By th principl of virtual works d ε (17) 3n Infinitsimal mchanism d m ( d m ), is th vctor of th nodal displacmnts which ar rlatd to null axial strains, i.. d m 0 (18) 130

6 By (18), infinitsimal mchanisms li in th null spac of th transpos of th quilibrium matrix. h dimnsion of th null spac of, which is th multiplicity of infinitsimal mchanisms is r 3n r (19) Rank dficincy conditions Lt D rank of th forc dnsity matrix, th dimnsion of th null spac of D is r n r (20) D D h dimnsion of th null spac of D, in ordr to crat a spac of th solution of (6) with at last four dimnsions, should b qual or mor than four. Furthrmor, dimnsion of th null spac of should b qual or mor than unit. hs conditions nsur th possibility to build a non-dgnrat slf-quilibratd tnsgrity structur in thr-dimnsional spac. V-Expandr tnsgrity bam In this papr, w study how th fasibl slf-strss stat q changs whn th numbr of th lmnts incrass. h V-Expandr nsgrity bam is obtaind by assmbling thr V22- Expandr tnsgrity cll as shown in Fig. 1. Hr, th V-Expandr nsgrity bam is studid as th gomtrical paramtrs d (m) and h (m) chang. hrfor, th fasibl slf-strss-stat is calculatd and plottd for d = 1 m and h = 0.5 m. h first analysd V-Expandr bam is composd of sixtn struts and twnty-svn cabls; twnty-two nods connct th lmnts (nods and lmnts ar lablld in viw of gomtrical symmtry of th structur). hn, mor complx V-Expandr bams ar analysd, with th following outlin: Cas 1, forty-thr lmnts, Cas 2, fifty-thr lmnts, Cas 3, fifty-svn lmnts, Cas 4, fifty-nin lmnts, Cas 5, sixty-thr lmnts, Cas 6, sixty-svn lmnts, Cas 7, svnty-on lmnts. In all th abov cass, th tnsgrity bam is capabl of bing nclosd in a paralllipd, th dimnsions of which ar 6d, 2d and h, in x, y and z dirction rspctivly. For xampl, in Figur 1 is shown th top viw of th V-Expandr bam of Cas 1. In th V-Expandr nsgrity bam, as shown in Fig. 2, rfrrd to Cas 7, th initial struts ar lablld from 48 to

7 Figur 1 op viw of V-Expandr bam in Cas 1. Figur 2 Prspctiv viw of V-Expandr bam in Cas 7. Whn th numbr of th lmnts incrass, th rank of th quilibrium matrix incrass and, thn, th numbr of th indpndnt slf-strss stats also incrass. Furthrmor, th numbr of th infinitsimal mchanisms of th V-Expandr tnsgrity bam dcrass. In particular, Cas 1 posssss ightn infinitsimal mchanisms whn th six rigid-body motions in thr-dimnsional spac ar opportunly constraind; Cas 7 dos not possss any infinitsimal mchanisms whn th rigid-body motions ar xcludd. h first cas analysd, Cas 1, is composd of: wnty-svn cabls (1, 2, 5, 8, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41) and sixtn struts, which ar lablld as mntiond abov. h othr cass analysd ar obtaind, starting form Cas 1, by addition of th lmnts (addition rspcts gomtrical symmtry of th structur) as follow: Cas 2, tn lmnts (3, 4, 6, 7, 24, 25, 26, 27, 28, 29); Cas 3, four lmnts (14, 15, 16, 17); Cas 4, two lmnts (42, 43); Cas 5, four lmnts (44, 45, 46, 47); Cas 6, four lmnts (64, 65, 66, 67); Cas 7, four lmnts (68, 69, 70, 71). [a] [d] 132

8 [b] [] [c] [f] [g] Figur 3 Fasibl slf-strss stats of V-Expandr bam [a] Cas 1, [b] Cas 2, [c] Cas 3, [d] Cas 4, [] Cas 5, [f] Cas 6, [g] Cas 7. In Figur 3 ar listd th fasibl slf-strss stats of th svn cass analysd; notic that as th complxity incrass, som forc dnsitis assum valus nar to zro. In Figur 4 is shown th numbr of th indpndnt slf-strss stats and th numbr of th infinitsimal mchanisms for vry V-Expandr bam studid. 133

9 Figur 4 Numbr of indpndnt slf-strss stats s and numbr of infinitsimal mchanisms m of V-Expandr bams. In abl 1 ar listd th proprtis of th svn V-Expandr bams studid. abl 1 Proprtis of th svn V-Expandr bams. Cas Proprtis of V- Expandr bam C ( n ) ( 3n ) 43 x x x x x x x x x x x x x x 71 r In abl 2 ar listd th ral cofficints of th linar combination in (10) for Cas 7. abl 2 Ral cofficints of th linar combination for Cas 7. Cas 7 Ral cofficints λ i (i=1,2,,11 ) CONCLUSIONS In this papr, w analys th influnc on th fasibl slf-strss stats of th addition of lmnts (struts and/or cabls) starting from an initial configuration of a tnsgrity V-Expandr bam. In Cas 1 thr ar possibl twnty-four indpndnt infinitsimal mchanisms which li in th krnl of th transpos of th quilibrium matrix of th structur. Furthrmor, in this cas, thr xists only on indpndnt slf-strss stat. 134

10 From Cas 1 to Cas 7 w obsrv a rdistribution of th forc dnsitis in th lmnts; in particular, w find that th forc dnsitis in th vrtical cabls, i.. lmnts 5, 12, 13, 22, 23, dcras from 0.3 to Simultanously, w obsrv that th forc dnsitis in th struts incras until thy rach th valu , starting from a valu, in Cas 1, qual to dditional lmnts stiffn th V-Expandr tnsgrity bam: indd, disrgarding rigidbody motions, th numbr of infinitsimal mchanisms dcrass, and it bcoms zro in Cas 7 (th most complx xamind bam). s natural xtnsion of th prsnt work, th mchanical bhaviour of th V-Expandr tnsgrity bam undr th action of xtrnal loads will b analysd in forthcoming paprs. REFERENCES [1] Gilwsky, W. and Klosowska, J. and Obara, P. (2015) pplications of tnsgrity structurs in civil nginring. Procdia Enginring, 111, [2] Raducanu, V. and Motro, R. (2011) SBLE SELF-BLNCING SYSEM FOR BUILDING COMPONEN. Patnt, WO [3] Fullr, R. B. (1975) Synrgtics. Explorations in th Gomtry of hinking. Macmillan Publishing, Nw York, NY, US. [4] Motro, R. (2003) nsgrity: Structural Systms for th Futur. Kogan pag Scinc, London, UK. [5] Schck, H. J. (1975) h forc dnsity mthod for form finding and computation of gnral ntworks. Comput. Mthods ppl. Mch. Eng., [6] ran, H.C. and L, J. (2010) Initial slf-strss dsign of tnsgrity grid structurs. Computrs & Structurs, 88, [7] ran, H.C. and L, J. (2010) Slf-strss dsign of tnsgrity grid structurs with xostrsss. Intrnational Journal of Solids and Structurs, 47, [8] Foti, P. and Fraddosio,. and Marzano, S. and Pavon, G. and Piccioni, M. D. (2016) On slf-strss dsign in V-Expandr tnsgrity structurs. in prss. [9] Zhang, J. Y. and Ohsaki, M. (2007) Stability conditions for tnsgrity structurs. Intrnational Journal of Solids and Structurs, 44, [10] Gómz-Jáurgui, V. (2010) nsgrity structurs and thir application to architctur. Univrsidad d Cantabria. Srvicio d Publicacions. 135