Laminate circular cylindrical shell

Transcription

1 MAC Web of Coferece 5, 0400 (07) DOI: 0.05/ mateccof/ CSCC 07 amate crcular cylrcal hell va Kormaíková,*, a Kamla Kotraová UK Košce, Cvl geerg Faculty, Vyokoškolká 4, Košce, Slovaka Abtract. he paper eal wth a umercal approach of optmal eg moellg of lamate crcular cylrcal hell. For th-walle hell the clacal hell theory capable of accurately prectg the hell behavour. I the frame of umercal optmato of crcular cylrcal hell there mae the mmato of weght ubject to placemet cotrat. hcke of layer take a eg varable. I the eample there fou the thcke of the hell lamate roof uer cotat preure loag ug Mofe Feable Drecto metho. Itroucto For may year, eger have treate optmato problem volvg compote materal. he eg of a compote lamate tackg equece geerally volve electg crete layer thcke a crete optmato problem. he rectoalty of fbre compote ee the optmato the proce of eg of thee materal. May work have bee oe o optmato of cylrcal hell, for eample, Park et al. [] ue optmato of lamate tackg equece to mame the tregth. Aal a Verjeko [] optme the tackg equece eg for hybr lamate. Soremeku et al. [] ue optmato algorthm for tackg equece bleg of multple compote lamate to mme the weght a the cot of the pael. Weaver [4] ue computatoal tuy for egg the lamate compote cylrcal hell uer aal compreo to mme the ma wth local a global cotrat. Clacal hell theory h-walle lamate hell ca be alo moelle a twomeoal tructural elemet but wth gle or ouble curve referece urface (Fg. ). he moellg a aaly of lamate crcular cylrcal hell fabrcate from fbre compote materal epe o the rau/thcke rato (Fg. ). Fg.. Crcular cylrcal hell uer geeral loag [6] he tra placemet relato for a crcular cylrcal hell ug ove frt appromato are gve by u, v w, R u v, w, w v, R w v. () R he total tra at a arbtrary tace of the mle urface are wrtte by, Fg.. Double curve lamate hell a layout of layer [5], * Correpog author: eva.kormakova@tuke.k he Author, publhe by DP Scece. h a ope acce artcle trbute uer the term of the Creatve Commo Attrbuto cee 4.0 (

2 MAC Web of Coferece 5, 0400 (07) DOI: 0.05/ mateccof/ CSCC 07. () ach vual layer aume to be a tate of geerale plae tre, the Hooke law yel j,, j = (,, ), () j where umber of oe layer a j compoet of elatcty matr efe [7]. he force a momet reultat (Fg. ) are efe by h, h h M,, j = (,, ). (4) h he cottutve equato are wrtte the matr form A M B B, (5) D where A the eteo matr, B the begeteo couplg matr, D the beg matr. he compoet of A, B, D matr are A j j D j where umber of layer., Bj j, j (6) he equlbrum equato for fferetal hell elemet o Fg. are gve by p, M R M p, M M M R p. (7) Subttutg the cottutve equato to the equlbrum equato yel a et of three couple partal fferetal equato for the three placemet u, v, w whch ca be wrtte matr form u p v p w p. (8) he lear fferetal operator j are efe [5]. he goverg equato are olve by the help of Fte lemet Metho (FM). Fte elemet aaly he bac ea of the FM a cretato of the cotuou tructure. he cretato efe by fte elemet meh make up of elemet oe. he tartg pot for elatotatc problem the total potetal eergy. I accorace wth the Rt metho the appromato ue for placemet fel vector by otato u ~ ( ) ( ) v, (9) where () the matr of the hape fucto, that are fucto of the poto vector,, a v the elemet placemet vector. For the tree a tra we obta from q. (9) the q. (0) ( ) ( ) D ( ) v, Du D v Bv. (0) Wth the appromato (q. 9) the total potetal eergy a fucto of all the oal placemet compoet arrage the elemet placemet vector v. he varato of the total potetal eergy lea to v B BvV pv qo () V V Oq Kv f p f 0 v, () q Fg.. Force a momet reultat of the fferetal hell elemet. where p, q are volume a urface loag, repectvely a K the ymmetrc tffe matr gve by

3 MAC Web of Coferece 5, 0400 (07) DOI: 0.05/ mateccof/ CSCC 07 K B BV. () V he vector of the volume force a the urface force are wrtte by f pv, p V f qo. (4) q O q If the compoet of v are epeet of each other, we obta from q. () the ytem of lear equato Kv f, f f p f q. (5) All equato coere above are val for a gle fte elemet a they houl have a atoal e. We have the er elemet eergy U v V B BVv wth the elemet tffe matr v K B BV, V, K v, (6), (7) where the elatcty matr obtae wth utable traformato two tage, frtly from the prcpal materal recto to the elemet local recto a ecoly to the global recto. B the tra matr, the traformato matr wth. (8) Becaue the eergy a calar quatty, the potetal eergy of the whole tructure ca be obtae by ummg the eerge of the gle elemet. By a Boolea matr the correct poto of each gle elemet eterme. he elemet placemet vector v potoe to the ytem placemet vector by the equato v v, (9) the we obta the ytem equato by ummg over all elemet [8-] K v f f. (0) p q he ytem tffe matr alo ymmetrc, but t a gular matr. After coerato of the bouary coto of the whole ytem, K become a potve efte matr a the ytem equato ca be olve. 4 Sg optmato problem he bac problem the mmato of a fucto ubject to equalty cotrat []: Mme objectve fucto Subject to cotrat X Z = F(X) m, () X X =,,...,, () U g j (X) 0 j =,,..., c, () where X a eg varable []. We make ue of the etg repoe at a umber of pot the eg pace to cotruct a polyomal appromato to the repoe at other pot. he optmato proce apple to the appromate problem repreete by the polyomal appromato F a 0 a X b X j j c X X j X (4) where the umber of eg varable, a,b, cj, are coeffcet to be eterme by a leat quare regreo. Whe the objectve fucto a cotrat are appromate a ther graet wth repect to the eg varable are calculate bae o choe appromato, t poble to olve the appromate optmato problem. Ug the Mofe Feable Drecto metho (MFD) the olvg proce terate utl covergece acheve. o f the earch recto, actve a volate cotrat have to be etfe. Covergece to the optmum checke by crtera of mamum terato a crtera chage of objectve fucto. Bee thee crtera, the Kuh-ucker coto eceary for optmalty mut be atfe by ug the agraga multpler metho. he Kuh- ucker coto are uffcet for optmalty whe the umber of actve cotrat equal to the umber of eg varable a f objectve fucto a all of the cotrat are cove. Otherwe, uffcet coto requre the eco ervatve of the objectve fucto a cotrat [4-6]. Covergece or termato check are performe at the e of each optmato loop. he optmato proce cotue utl ether covergece or termato occur. 5 ample a reult here eee to f the thcke of the hell roof fabrcate from lamate [0/45/-45/90] S uer cotat preure loag p the eample (Fg. 4). he materal properte of each layer were ue from homogeato techque: = 4 GPa, = 0. GPa, G = 7. GPa, = 0.7,.58 g.cm -. ach layer of the lamate ha the ame thcke h.

4 MAC Web of Coferece 5, 0400 (07) DOI: 0.05/ mateccof/ CSCC 07 hcke of layer h=? w=v=0 w= u R=50mm y w I v J 00mm 00mm w=0 Fg. 4. Problem ketch a fte elemet moel. w=v=0 Due to ymmetry, a quarter of the hell coere for moellg (Fg. 4). he mathematcal optmato problem we ca wrte: Mme objectve fucto F Subject to cotrat X G h 0.05 h 0.65 m [] -.0 w h0.0 Fg. 5 Varato of the eg varable value thcke urg the optmato proce. Feable oma Ifeable oma where h the thcke of oe layer. he geeral optmato cota:. Ital aaly wth put ata (ab. ).. Mathematcal optmato problem.. ear appromato. 4. Algorthm of MFD metho wth covergece crtera. 5. Covergece or termato check of geeral optmato. Optmato parameter Deg varable - thcke of oe layer Objectve fucto - weght [] Cotrat - placemet w able. Optmato parameter. Ital value Fal value olerace Reult of the optmato proce are lte the able. Fg. 6 Varato of the cotrat value placemet w urg the optmato proce. 6 Cocluo he proceeg eal wth a umercal approach of moellg of crcular cylrcal hell fabrcate from fbre compote materal. he theory of crcular cylrcal hell ecrbe the frame of clacal hell theory. I the paper there are volve the tra placemet relato, cottutve equato a fferetal equlbrum equato. here wa ue homogeato for calculatg the materal charactertc gve the eample [7-9]. I the frame of umercal optmato t wa mae the mmato of weght ubject to placemet cotrat. Deg varable thcke of layer of lamate crcular cylrcal hell. he tal a fal value of eg varable, cotrat a the objectve fucto are how the able. Mamum umber of terato of MFD wa 00. he geeral optmato proce wa toppe after 0 eg et (Fg. 5, Fg. 6), becaue the fferece betwee the curret value a the oe or two prevou eg wa le tha tolerace pecfe the able. he fal value of eg varable h mm (Fg. 5). he total thcke of lamate crcular 4

5 MAC Web of Coferece 5, 0400 (07) DOI: 0.05/ mateccof/ CSCC 07 cylrcal hell 8 h. mm. Fgure 6 how the feable a feable oma the uable eg pace. he optmato very ueful way for eg of lamate tructural elemet clug crcular cylrcal hell. h work wa upporte by the Scetfc Grat Agecy of the Mtry of ucato of Slovak Republc a the Slovak Acaemy of Scece uer Project VGA /0477/5 a /0078/6. Referece. JH. Park, JH. Hwag, CS. ee, W. Hwag, Stackg equece eg of compote lamate for mamum tregth ug geetc algorthm, Compo Struct 5, p. 7, (00). S. Aal, V. Verjeko, Optmum tackg equece eg of ymmetrc hybr lamate uergog free vbrato. Compo Struct 54, p. 8, (00). G. Soremeku, Z. Gural, C. Kaapoglou, D. o, Stackg equece bleg of multple compote lamate ug geetc algorthm, Compo Struct, 56, p. 5 6, (00) 4. PM. Weaver, Deg of lamate compote cylrcal hell uer aal compreo. Compote Part B, p , (000) H. Altebach, J. Altebach, W. Kg, Structural aaly of lamate a awch beam a plate. ubelke owarytwo aukowe, ubl, (00) 8. J. Kralk, Optmal eg of pp cotamet protecto agat fuel cotaer rop, Av. Mat. Re. 688, p. -, (0) 9. M. Kreja, P. Jaa, I. Ylma, M. Marchalko,. Bouchal, he ue of the rect optme probabltc calculato metho eg of bolt reforcemet for uergrou a mg workg, he Scetfc Worl Joural, Artcle umber 6759, (0) 0. M. Zmak, Z. Pelagc, M. Bvoc, Aaly of hgh velocty mpact o compote tructure, Appl. Mech. a Mat. 67, p , (04). J. Melcer, G. ajcakova, Comparo of fte elemet a clacal computg moel of reforcemet pavemet, Av. Mat. Re. 969, p , (04).. Kormakova, I. Mamuc, Optmato of lamate ubjecte to falure crtero, Metal. 50 (), p. 4-44, (0). Z. Güral, R.. Haftka, P. Hajela, Deg a Optmato of amate Compote Matera, J. Wley & So, (999) 4. A. M. Valuev, Moel a metho of multobjectve optmato problem of quarry eg a plag, WSAS ra. o Math.,, p , (04) 5..A. Vorotova, A projectve eparatg plae metho wth atoal clppg for o-mooth optmato, WSAS ra. o Math.,, p. 5-, (04) 6. J. ovíšek, J. Králk, Optmal Cotrol for lato- Orthotropc Plate, Cotr. a Cyber., p. 9-78, (006) 7. M. Sejoha, J. Zema, Mcromechacal moelg of mperfect tetle compote, It. Jour. of g. Sce. 46 (6), p. 5-56, (008) 8. J. Ma, S. Sahraee, P. Wrgger,,. De ore, Stochatc multcale homogeato aaly of heterogeeou materal uer fte eformato wth full ucertaty the mcrotructure, Comp. Mech. 55, Iue 5, p , (05) 9. C. Marucco,. De ore,. Perao, D. Pgao, Computatoal homogeato of fbrou peoelectrc materal, Comp. Mech. 55, Iue 5, p , (05) 5