PEF-5750 Estruturas Leves Ruy Marcelo de Oliveira Pauletti ARGYRIS NATURAL MEMBRANE ELEMENT THE NATURAL FORCE DENSITY METHOD

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1 PEF-575 Estutuas Leves Ruy Macelo de Oliveia Pauletti ARGYRIS NATURAL MEMBRANE ELEMENT THE NATURAL FORCE DENSITY METHOD //7 Agyis Natual Membae Elemet Agyis ~974 A membae fiite elemet based o atual defomatios (mesued alog the sides of the elemet), able to cope with lage displacemets ad lage defomatios. Ai to a stai osette plae stess fiite elemet: Mee ~99 A cootatioal desciptio; Small stais. Pauletti ~ a moe cocise otatio; distictio betwee the costitutive ad geometic pats of the elemet taget stiffess; the simplest possible membae fiite elemet : lage displacemets / small stais (a few pecet ) Pauletti (6) fist publicatio o the atual foce desity cocept R.M.O. Pauletti, A extesio of the foce desity pocedue to membae stuctues IASS Symposium / APCS Cofeece New Olympics, New Shell ad Spacial Stuctues, Beijig, 6

2 Refeece, Iitial ad Cuet Cofiguatios Fo Agyis Elemet y, yˆ zˆ ŷ P x P t x P ŷ xˆ u P P ˆx z, zˆ P x, xˆ ˆz Elemet Desciptio x x x x, c, c u u x x u c u v v Side leghts: x x i j i, j,,, (i cyclic pemutatio) v Uit side vectos: v i x e x i e j Uit omal vectos: ˆ v i i

3 Elemet Stess Field ad Vecto of Iteal Nodal Foces Cauchy Plae Stess Teso: p ρ σˆ xˆ xy ˆˆ xˆ xy ˆˆ yˆ yˆ xy ˆˆ ρ Side stess vectos: ρ i σˆ Vecto of iteal odal foces: i p ρ p p ρ ρ e e e t p p ρ ρ e p ρ ρ Vecto of Natual Foces The vecto of iteal foces ca be decomposed ito compoets paallel to the elemet sides: N v P Pp P N v e e Nv Nv e e e p Nv Nv e e N N v v pp c N v P NP v N v P N v P Pp Vecto of Natual Foces N N N N

4 Natual Stesses Compaig both expessio available fo p e : e e Nv Nv ρ ρ e e t N N v v ρ ρ e e N N v v ρ ρ We obtai the Vecto of Natual Foces, as fuctio of Cauchy Stesses, ad we idetity some Natual Stesses (,, ): cos si h yˆ xy ˆˆ si si si si N t cos si N N h yˆ xy ˆˆ si si si si N cos cos cos h xˆ yˆ xy ˆˆ si si si si N h t N N h N h Vecto of Natual Stesses We goup the Natual Stesses ( ) i a Vecto of Natual Stesses:,, σ cos si si si si si xˆ cos si yˆ si si si si cos cos si si si si si xy ˆˆ T σ T σˆ Execise. Veify the above expessio! 4

5 Vecto of Natual Stesses Each atual foce N i ca be udestood as the odal esultat of each atual omal stess field i y y h t N t h i i i l P l N h t h t N P h l x x Ad sice A h i i N i i V i Relatioship betwee the Vectos of Natual Foces ad Stesses I matix fom: N V L - σ Legth matix: L σ Vecto of Natual Stesses 5

6 Vecto of Natual Defomatios The defomatios alog the sides of the elemet ae collected i a Vecto of Natual Defomatios : cos si si cos ˆ x cos si si cos yˆ xy ˆˆ ε Tε ˆ Lieaized Gee Stais Execise. Veify the above expessio! We ema that σ ad ε ae eegetically cojugate. Ideed, by the Piciple of Vitual Wo: T T εˆ σˆ ε σ, εˆ T T T T εˆ σˆ T εˆ σ εˆ T σ, εˆ Thus: T σ T σ ˆ, as deduced befoe. Taget Stiffess Matix fo Agyis Elemet g T C N f t N C u u u u Geometic Stiffess Matix v v N N u u C v v N u u u v v N N u u T g N N N N N N N N N N g I v v I v v I v v I v v N N N N T T T T I v v I vv I vv I v v T T T T T T T T I v v I vv I vv I v v Exact! 6

7 Taget Stiffess Matix fo Agyis Elemet g T C N f t N C u u u u Exteal Stiffess Matix Exteal foce vecto: ext f u V pa weight wid f f f I g I I I I I wid p Λ Λ Λ f ext Λ Λ Λ u 6 Λ Λ Λ z y v v i i z x Λi Sew( li ) i vi v i, i,, y x vi vi Exact! Execise. Veify the above expessio fo ext! Taget Stiffess Matix fo Agyis Elemet g T C N f t N C u u u u Costitutive Stiffess Matix N c C u Defiig the vecto of Natual Displacemets Thee exist some id of elatioship N N a N a c C C C a u N a T a so that Exact! is the Natual Stiffess Matix 7

8 Taget Stiffess Matix fo Agyis Elemet A simplificatio: Liea elastic mateial behavio Thus, a liea elatioship N a exists Whee N a is a x costat atual stiffess matix Ad theefoe c C C T?! A liea elastic simplificatio fo K c yˆ, yˆ yˆ, û ˆv ṷ xˆ, xˆ xˆ uˆ u xˆ x xˆ vˆ v εˆ yˆ yˆ y xy ˆˆ vˆ uˆ xu xu xˆ yˆ xy ε L a T ε T εˆ cos si si cos cos si si cos 8

9 A liea elastic simplificatio fo K c Hooe s Law: ˆ x σˆ ˆ =Dε ˆ σˆ xy ˆˆ yˆ ˆ E D But, ow: σ T σˆ T Dˆε ˆ σˆ T Dˆε ˆ T σˆ T T T T T ˆ T ˆ DT ε σ T T σ That is Whee σ D ε σ D T DT ˆ T A liea elastic simplificatio fo K c Recalig the Natual Foces: N V NV L D ε V L T DT ˆ L a - - T - N V L T DT ˆ a L - Ad we aive to the Natual Stiffess Matix, (cosideig small defomatios): - T - σ L A ode, symmetic matix, that ca be calculated ad stoed at the stat, ad otated at each Newto s iteatio, accodig to the co-otatioal elemet coodiate system: c C C T 9

10 A bechma: a axisymmetic pessuized membae defomed shape, as calculated by SATS compaiso betwee SATS ad theoetical esults THE NATURAL FORCE DENSITY METHOD Vecto of Natual Foces fo Agyis Natual Membae Elemet: N v P Pp P N v pp c N v P NP v N v P N P v Pp The vecto of iteal foces ca be decomposed ito compoets paallel to the elemet sides: e e Nv Nv e e e p Nv Nv e e N N v v

11 e e Nv Nv e e e p Nv Nv e e N N v v N N N N e e e e x x x x N e e e e x x x x N e e e e x x x x e e e e x x x x e e e e x x x x p x x x x e e e e e e I I I x e I I I x e I I I x e d Natual Foce Desity Elemet Stiffess: I I I e d I I I I I I Vecto of elemet iteal foces: p x e e e d

12 Relatioships betwee elemet ad global vectos: b e e et e x A x ; P A p e Natual Foce Desities Global Stiffess: b K d A A e et e e d Equilibium: P F A system of liea equatios: K x d F The Natual Foce Desities,, ca be collected ito a Vecto of Natual Foce Desities: N N N = =L - N

13 Remembeig the elatioship betwee Natual Foces ad Stesses: Natual foce desities ca be calculated accodig to a give geomety ad a iitial stess field: ŷ ˆσ N V ˆx L - σ N V σ = L L - - But Theefoe σ T σˆ T T ˆ = VL - T σ Fo istace: ŷ (,, ) ˆσ (,, ) (,,) ˆx xˆ σˆ yˆ xy ˆˆ Isotopic stess field =

14 x Oce the solutio fo is obtaied, Cauchy stesses at the fial cofiguatio ca be computed accodig to: σˆ V L T - T V L VL σˆ T T σˆ I geeal, eve fo uifom stesses at the efeece cofiguatio, o-uifom stesses esult at the equilibium cofiguatio! This is fully coheet with the oigial foce desity method, fo which omal loads i the equilibium cofiguatio also vay, eve thou iitial omal loads ae uifom! It ca be show that impositio of ˆσ at a efeece cofiguatio coespods to impositio of the d Piola-Kichhoff stesses, associated to the Cauchy stesses ˆσ at the equilibium cofiguatio! R.M.O. Pauletti & P.M. Pimeta, The atual foce desity method fo the shape fidig of taut stuctues Compute Methods i Applied Mechaics ad Egieeig Volume 97, Issues 49-5, 5 Septembe 8, Pages This esult exteds to membaes a coclusio aleady stated by Bletzige & Ramm, fo the oigial foce desity cocept (i.e., fo cables). K.-U. Bletzige & E. Ramm, A Geeal Fiite elemet Appoach to the Fom Fidig of Tesile Stuctues by the Updated Refeece Stategy It. J. Space Stuct. 4 () (999) 45 4

15 Some solutios: fist picipal stess : mi max.88 fist picipal stess : mi. max Some solutios: fist picipal stess : mi.8 max fist picipal stess : mi.796 max

16 Some solutios: fist picipal stess : mi.44 max fist picipal stess : mi.67 max Iteative Natual Foce Desity Method: Although Cauchy stesses at the fial cofiguatio caot be imposed i a sigle foce desity step, d P-K stesses ca be imposed ecusively i - T V L V L V L V L σˆ T T T T σˆ i i i i If a uifom isotopic d P-K stess field is ecusively imposed, the geomety coveges (though a successio of viable shapes) to a miimal suface, with a uifom isotopic Cauchy stess field! σˆi σˆ We ote that a sequece of oliea stuctual aalyses ca also covege to a miimal suface, but though a successio of o-equilibium, uviable shapes! This is a clea advatage of the iteative NFDM, which ca be stopped at ay iteatio, always givig a viable shape! 6

17 Miimal sufaces: Coside the miimal flat squae membae fixed at the coe ad bouded by bode cables: The followig elatioship holds: 4 4 tl T si Uppe limit coditio: T Lowe limit coditio: tl T.4 4 Execise 4. Deduce the elatioship betwee the membae stesses ad the omal foce o bode cables, ad umeically veify the uppe ad lowe limit coditios stated above. Miimal sufaces: C. Isebeg, The sciece of soap films ad soap bubbles, Dove Pub. Ic., New Yo, 99. 7

18 fist picipal stess : mi max Miimal sufaces: fist picipal stess : mi max.644 S S S S4 S4 S S S fist picipal stess : mi max.644 Miimal sufaces: S S S S4 S S4 S S S S4 S S 8

19 Costa s Suface: The Costa suface is a complete miimal embedded suface of fiite topology (i.e., it has o bouday ad does ot itesect itself). It has geus with thee puctues (Schwalbe ad Wago 999). Util this suface was discoveed by Costa (984), the oly othe ow complete miimal embeddable sufaces i R with o self-itesectios wee the plae (geus ), cateoid (geus with two puctues), ad helicoid (geus with two puctues), ad it was cojectued that these wee the oly such sufaces. Rathe amazigly, the Costa suface belogs to the dihedal goup of symmeties. Helama Feguso, 999 / 8 AUSTRALIAN WILDLIFE HEALTH CENTRE Costa s Suface: st ite d ite 5 th ite I I I..4 9

20 Y Z X Costa s Suface: AUG F F F F Symmety & Pattes: F F F F Y F Z X F F F F F F F Z Y X A physical model: No-miimal sufaces Dieto Plaes Z ˆ //Z ˆ ˆx i Nomal Diectos

21 No-miimal sufaces Local base vectos i global coodiates: î v v v ˆ ; v v ˆj ˆ iˆ ˆ ˆx î ˆ ˆ i acsi i ˆ iˆ ˆ Miimal ad o-miimal cooids

22 , m Radial Stess, N/m Compaiso with a aalytical solutio SLADE GELLIN & RUY M.O. PAULETTI - FORM FINDING OF TENSIONED FABRIC CONE STRUCTURES USING THE NATURAL FORCE DENSITY METHOD (i IASS ) Alpha = (NFDM) 6 Alpha = [] Alpha = (NFDM) Alpha = [] Alpha = (Th) Alpha = (Num) Alpha =.5 (Th) Alpha =.5 (Num) Alpha = (Th) Alpha = (Num) Alpha = (Th) Alpha = (Num) Alpha = 4 (Th) Alpha = 4 (Num) Alpha = 5 (Th) Alpha = 5 (Num) z, m Geeatix Pofile Radial Coodiate, m Stess Pofile Miimal saddle suface:

23 No-miimal saddle sufaces: No-miimal saddle sufaces:

24 No-miimal saddle sufaces: 4