ISSN: [Silva* et al., 7(6): June, 2018] Impact Factor: 5.164

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1 IJRT ITRATIOA JOURA OF GIRIG CIC & RARCH TCHOOGY UMRICA AAYI I FIIT MT OF AMIATD PAT I T APPID I DIFFRT TYP OF COMPOIT MATRIA MODD AAU MODIG AD COMPARD WITH AAYTICA Welgo Val da la *, ro do obrho, Clero Adré la Freas, reda A. Táora Rbero, Fracsco F.. de Moras Júor, Rao. Y. R. Cosa la 4 * Docorae rcres, Depare of Cl geerg, Uers of rasíla U. Asa ore, Caps Darc Rbero, rasla, DF, ral Professor PhD Cl geerg, UFCA Federal Uers of Carr, A. eea Rado Rocha / ad Uers C of Jaero do ore, Ceará, C, ral Cl egeerg sde, UFCA Federal Uers of Carr, A. eea Rado Rocha / ad Uers C of Jaero do ore, Ceará, C, ral 4 Docor rcres, Depare of Cl geerg, Uers of rasíla U. Asa ore, Caps Darc Rbero, rasla, DF, ral DOI:.58/eodo.995 ATRACT The srcres fored b laaed coposes hae grea applcao cl egeerg. Mal coerg of dsres h rapeodal le h acosc slao. Aoher applcao of he copose sse s he gree laers of he roof. o hs paper dscssed a forlao for sabl aalss of laaed plaes hrogh he Fe lee Mehod. The forlao preseed allos he se of raglar ad qadrlaeral elees h dffere erpolao orders. I he ercal eaples, he qadrac fe elee h qadrac erpolao CD8R as sed o sd s behaor. The redced egrao echqe as sed o alleae he shear block proble. The behaor of hese elees as ealaed b coparg he resls obaed b FM h aalcal solos based o he classcal laaed plae heor. The resls sho ha he elees proded ecelle precso resls for cross-laer laaes ad sgle ad baal agles ad loads. KYWORD: aaed Plaes, Fe lees, o Geoerc ear ad AAU I. ITRODUCTIO I he desg of srcres, lear odels are sall sed for sress aalss. Hoeer, lear aalss s ald ol he he srcre sffers js sall deforaos ad dsplacees ad s aeral has lear elasc behaor, obeg Hooke's a. Whe he dsplacees ad deforaos crease or he behaor of he aeral preses pheoea sch as plasc ad crackg, he olear effecs ga porace. I he geeral case, boh geoerc olear de o large dsplacees ad phscal o-lear de o aeral s echacal behaor eed o be cosdered. Hoeer, sleder srcres a lose sabl ho sffce appled loads o case aeral degradao. Ths, he sd of he sabl of hese srcral elees ca be assed ha he aeral reas he lear elasc rege ad he loss of sabl s cased ol b geoerc o-lear. Plae sabl aalss oles he solo of paral dffereal eqaos (Redd, 996). e he case of hoogeeos ad soropc plaes, hese eqaos ol hae eac (closed) solos for sple geoeres, loads ad bodar codos. I order o elae hs lao ad allo he solo of praccal cases, he Fe lee Mehod (FM) as sed. [5]

The focs of hs paper ll be he sabl aalss of laaed plaes sg FM.

ahe, 996; Cook e al., ) ad he o-lear effecs ll be ake o acco hrogh he se of o Kárá's heor (Crsfeld, 99).

The forlao preseed here as pleeed he Malab sofare, hch s a free sofare for srcral aalss hrogh Fe lee Mehod

The ercal resls ll be copared h he aalcal solo of soe classc probles fod he lerare (becharks) order o aldae

Copose aerals Copose aerals are fored b o or ore aerals cobed o a acroscopc scale, ag o oba a aeral h

2 The focs of hs paper ll be he sabl aalss of laaed plaes sg FM. I ll be sed fe elees based o he frs order shear deforao heor, also ko as Resser-Mdl's plae heor (Redd, 996; ahe, 996; Cook e al., ) ad he o-lear effecs ll be ake o acco hrogh he se of o Kárá's heor (Crsfeld, 99). Parclarl, ll be sded he behaor of a qadrac qadrlaeral elee of egh kos CD8R AAU. The forlao preseed here as pleeed he Malab sofare, hch s a free sofare for srcral aalss hrogh Fe lee Mehod ha ses he phlosoph of Objec Oreed Prograg. The ercal resls ll be copared h he aalcal solo of soe classc probles fod he lerare (becharks) order o aldae he proposed forlao ad pleeao. II. MATRIA AD MTHOD. Copose aerals Copose aerals are fored b o or ore aerals cobed o a acroscopc scale, ag o oba a aeral h desred properes o obaed he ddal copoes (Redd, 996). Copose aerals are geerall classfed hree dffere pes: parclae coposes, fbros coposes or laaed coposes (Joes, 999), as sho Fgre. A) Copose aeral h fber: Cocree h seel fber ) Copose fored b parcles: Rocks fored b crsals (C)Tpes of coposes sed dsral roofg [5]

(D) aaed Copose Fgre - Dffere pes of copose aerals The srcral ebers fabrcaed sg laaed coposes are fored b a seres of laers joed o ork soldar. A coo eaple of a copose obaed hs a s plood.

I hs case, each blade s fored b a se of fbers of hgh sregh ad sffess ebedded he base aeral or ar of loer sregh ad sffess, b al of loer cos.

Mal coerg of dsres h rapeodal le h acosc slao. Aoher applcao of he copose sse s he gree laers of he roof.

eeral cees o se hs sse are beg pleeed rope ad he Ued aes, led b oprof phlahropc orgaaos.

3 (D) aaed Copose Fgre - Dffere pes of copose aerals The srcral ebers fabrcaed sg laaed coposes are fored b a seres of laers joed o ork soldar. A coo eaple of a copose obaed hs a s plood. Hoeer, for he fabrcao of hgh perforace srcral elees, he os del sed aerals are fber reforced coposes (Joes, 999; Redd, 996). I hs case, each blade s fored b a se of fbers of hgh sregh ad sffess ebedded he base aeral or ar of loer sregh ad sffess, b al of loer cos. I hs a, each laer behaes acroscopcall as a orhoropc aeral, hose a drecos are parallel ad perpedclar o he fbers.. aaed Copose The srcres fored b laaed coposes hae grea applcao cl egeerg. Mal coerg of dsres h rapeodal le h acosc slao. Aoher applcao of he copose sse s he gree laers of he roof. Gree roofs offer a are of beefs for he resde ad he c, cldg redced eerg coss, eeded roof lfe, raaer reeo ad proed ar qal. eeral cees o se hs sse are beg pleeed rope ad he Ued aes, led b oprof phlahropc orgaaos. Oe s he Aacosa Waershed oce, a o-prof orgaao ha faclaes he Gree Roof Rebae progra of he Dsrc erg Depare, hch coers a cera cos per sqare foo of s gree roof. I hs a, he laer represes he fdaeal block of laaed coposes (Redd, 996). The laers hae hgh sregh ad odls of elasc he dreco of he fbers ad lo ales he orhogoal dreco o he fbers. The laaed coposes are cosrced accordg o a laer srcre, hch s defed b he ber of laers, oreao of he fbers (θ) ad hckess of each laer, as sho Fgre. Fgre - eeral laers of he copose aeral o he referece as [5]

De o he orhoropc characersc of each laer, he coposes ca be fabrcaed sch a a as o oba a a adaage each specfc sao b placg he dreco of he fbers alog he os reqesed drecos (pah of he loads).

. Cose codos aaed coposes are fored b seeral laers ad each oe has fbers oreed dffere drecos.

(,, ), As sho Fgre. Fgre - aers of a laae. Redd (4) I laers h drecoal fbers, he as s oreed parallel o he fbers, akg a agle θ h he prcpal as.

4 De o he orhoropc characersc of each laer, he coposes ca be fabrcaed sch a a as o oba a a adaage each specfc sao b placg he dreco of he fbers alog he os reqesed drecos (pah of he loads). I hs a, ca be obaed fro hghl oped projecs b chagg he ber, hckess, ad seqece of he laers, as ell as he oreao of he fbers of each laer.. Cose codos aaed coposes are fored b seeral laers ad each oe has fbers oreed dffere drecos. The, he aalss of laaed srcres reqres he se of a global coordae sse (,, ) hch he dffereal eqaos of eqlbr ad keac codos are re, hle he cose codos of he aeral are re he local coordae sse or of he aeral (,, ), As sho Fgre. Fgre - aers of a laae. Redd (4) I laers h drecoal fbers, he as s oreed parallel o he fbers, akg a agle θ h he prcpal as. The as s perpedclar o he fbers he plae of he plae ad he as s perpedclar o he plae of he plae, as sho Fgre. ce he aeral of each laer s cosdered hoogeeos ad orhoropc, s ecessar o esablsh a relao beee he coordae sse of he plae aeral (,, ) ad he global coordae sse (,, ). I s erfed epereall (Joes, 999) ha sal serce codos he echacal behaor of he laers ca be cosdered as elasc lear, sg he Hooke's a geeraled o represe he sress-sra relaoshp. For orhoropc aerals he sress-sra relaoshp each laer s ge b. σ C Where C s he elasc cose ar. C ( ) (a) jkl j kl k jl l jk Where λ ad μ are he aé cosas. The cosa μ s eqale o he rasersal odls of elasc, ha s, μ G = ( + ν). O he oher had, he cosa λ relaes o he logdal odls of elasc ad he Posso coeffce ν hrogh he follog epresso λ = ν ( + ν) (-ν). Therefore, he sra erss sra rao s ge b: j C jklkl. I pracce, hs ar s obaed b erg he sress-sra relaoshp: () [54]

5 44 55 σ. () I hs eqao s he ar of flebl (or coplace) of he aeral, hose coeffces are ge b: ; ; ; ; 44 G ; ; 55 G ; ; G Where, ad are he odl of elasc he prcpal drecos,,,,,, are he Posso coeffces ad G, G e G are he shear odl of elasc. I s pora o oe ha orhoropc elasc aerals hae ol 9 depede cosas, sce he Posso coeffces ( ) s sasf he relao: j j j ce he ar s serc., j,,, (4) The relao beee he coordaes of he aeral (,, ) ad he global referece sse (,, ) s defed fro he oreao of he fbers (θ) h respec o he -as of he global sse, as sho Fgre. The chage of coordae sse s perfored hrogh he rasforao: cos se se cos A. (5) The rasforao of he, ad dsplacees s carred o a aalogos a: cos se se cos A. (6) De o he sall hckess, s coo o asse ha each laer of he plae s sbjeced o a Plae ress ae (P). Ths, all sress copoes parallel o he -as are ero. I hs case, he deforao copoes of eres are ge b. j () [55]

6 [56] e. (7) Usg qaos (6) ad (7) ca be sho ha he ebrae deforaos ca be rasfored beee he sses hrogh he ar relao: T se se se se se se se cos cos cos cos cos cos cos. (8) The relao beee esos ad ebrae deforaos ca be obaed fro qao () b elag he ll ers: (9) here j coeffces are ge b qao (). Fall, b erg qao (9), he sress-sra relaoshp ca be re as σ, () here he j coeffces are calclaed fro he properes of he aeral, as follo:,,, G. () I relael hck plaes s ecessar o ake o acco he effec of raserse shear, hch eas ha he τ ad τ sresses shold be cosdered. I hs case, he shear deforaos he o sses are ge b e γ γ. () Usg qaos (6) ad (), s deered he rasforao of he shear deforaos beee he o sses

7 cos se se cos γ T γ. () s O he oher had, he sress-sra relaoshp de o raserse shear s ge b: τ s γ, (4) Where 44 G e G. (5) 55 Usg he Prcple of Vral Work ca be sho (Cook e al., ) ha he rasforao of local sse sresses o he global sse s doe hrogh: σ. (6) Ts σ e τ Ts τ Fall, sbsg qaos (), (8), (4) ad (), respecel he frs ad secod qao (6), e ge: σ T s T τ T T γ Where s s σ τ s γ (7) e s (8) The j coeffces are deered b perforg he follog ar operaos: cos ( ( ( ( 4 4 se ( 55 cos se ( cos se cos ) se cos se ) se cos ( se cos ) ) se cos ( ) se cos ( 4 ) se cos ( se cos ) 4 se ) se cos cos 4 ) se cos ) se cos (9) [57]

8 The ar s called he rasfored sffess ar, sce s he global coordae sse. s () 4. Aalss of laaed plaes The deforaos he plae, a a dsace fro he eda plae of he plae are deforaos de o bedg. Tha ca be represeed b he paral deraes, accordg o lard () ad hs o be deered he eqaos ha goer he laaed plae ad hck plae, accordg o Fgre 4: eral as [58]

Fgre 4 - Defored bsppored plae h dsrbed loadg (lard, ) Accordg o he classcal plae heor (Krchhoff's heor) a oral le o he eda plae of he plae ll rea oral o he eda plae srface afer deforao.

9 Fgre 4 - Defored bsppored plae h dsrbed loadg (lard, ) Accordg o he classcal plae heor (Krchhoff's heor) a oral le o he eda plae of he plae ll rea oral o he eda plae srface afer deforao. Ths ples ha he raserse shear effec s o cosdered, sce he aglar deforaos (γ ad γ ) are ero. Ths heor s approprae ol for hoogeeos plaes of sall hckess. I he case of hck plaes ad laaed plaes, raserse shear cao be egleced, reqrg he se of aoher plae heor. I Mdl's heor, he shear effec s cosdered roghl. The basc hpohess of hs heor s ha a oral le o he eda plae of he plae ll o ecessarl rea oral o he eda plae of he plae afer he deforao, as sho Fgre 5. Fgre 5 Deforaos Mdl s heor [59]

10 [54] Accordg o hs fgre s erfed ha he dsplacees a a po of he plae (, ad ) are ge b:,,,,,,,,, () here, ad are he raserse dsplacees of he eda plae of he plae ad e are he roaos of he oral le abo he ad aes respecel. Whe he dsplacees are oderael large, here s a eraco beee he effecs of ebrae ad bedg de o raserse dsplacees. Vo Kárá's heor corporaes hs effec sg a splfed for of he Gree-agrage deforaos obaed b eglecg he olear ers assocaed h he dsplacee copoes he drecos of he plae s plae, ad (Crsfeld, 99). I hs case, he deforao copoes a a po o he plae are:. () Replacg he dsplacee ge b q. () q. (), e hae ha he deforaos a plae parallel o he eda plae of he plae are ge b: () The frs o poros correspod o he ebrae deforaos ad he hrd poro de o he bedg copoes. These deforaos ca be re arcall as: κ (4) here s he deforao of he eda plae s ebrae ad s assocaed h he crare of he plae. Mebrae deforaos are coposed of o poros, he frs de o lear behaor ad he secod de o he olear effec of he raserse dsplacees of he plae, as sho q. (5). Usg he ar oao, e hae: (5) O he oher had, he deforaos de o he rasersal shear are ge b:

11 γ (6) For a geerc laer k h drecoal fbers, as sho Fgre, he esos are ge b q. (7) ad he copoes of k ad k s are ge b q. (9). The reslg forces ad oes (geeraled esos) are obaed b egrag he sresses alog he hckess of he plae: h M h / / h / d, M M d, d (7) h / M h / h / I s pora o oe ha, accordg o qs. () ad (4), he deforaos are coos alog he hckess of he laer, regardless he aeral arao, hch does o occr h he sress copoes, as sho q. (7). I geeral, hs s de o he chage he properes of each laer. Usg qs. (7) ad (4) (7), ca be re ers of eral sresses ad geeraled deforaos: σ C A M D κ G γ here he sb-ar copoes are ge b A j l k j l l G j j l l k j k j k l l j l l Dj k j, j 4e 5 l l l, j, e6 (8) (9) ad k j s he correco facor of shear sresses (Redd, 996). I s fod ha f he laaed plae s sercal, he ar s ero. 5. o-lear geoerc aalss eeral probles of praccal eres ca be easl soled cosderg lear odels, hch prese sasfacor approae resls he dsplacees ad deforaos are sall. Hoeer, he he dsplacees ad deforaos beg o crease, he effecs of geoerc o-lear s be cosdered he srcre aalss. Ths s also he case for srcres ha dergo bcklg b bfrcao ee he he dsplacees are sall, sch as perfec cols ad plaes ad loaded her o plae. I o-lear geoercal probles, he aalss ca be perfored sarg fro he al cofgrao of he srcre ad deerg he dsplacees, sresses ad sresses as he load s creased. Ths sraeg s ko as oad Corol (Crsfeld, 99), here erae load crees are appled order o deere he eqlbr pah of he srcre. De o he olear of he respose s ecessar o se a erae procedre o deere he dsplacees correspodg o he appled load. Oe of he os sed ehods for hs prpose s he eo-raphso s ehod (Crsfeld, 99, ahe, 996, Cook e al., ), sce hs oe preses qadrac coergece. [54]

12 The eqlbr eqaos of he srcral sse ca be re as (Crsfeld, 99): r() g() f here s he odal dsplacee ecor, r s he ecor of balaced forces (resdal), g s he ecor of eral forces, f s he ecor of eeral forces, ad λ s he load facor. oe ha hs eqao he oal load s ge b he prodc beee he load facor ad he referece loads. Icreasg he load facor, s possble o corol he arao of he oal eeral force. Usg he Prcple of Vral Works (PVW), e hae: W W e The eeral ral ork leads o he ecor of eqale odal forces a process ha s esseall he sae as he oe sed lear probles. O he oher had, eral ork leads o he deerao of he ecor of eral forces W σdv g g σdv () V here he ar s defed hrogh he relao V () () () Usg he eo-raphso ehod, he ecor of eral forces g() s leared h respec o he dsplacees d. Therefore: dg V dσ dv V d σ dv K d here K s he age sffess ar. Usg qs. () ad (), s erfed ha he sress crease s ge b: dσ (5) Cd C d ad fro qao (5): d d (6) bsg qs. (5) ad (6) o q. (4), e hae: d g CdV d dσdv ( K K ) d V V e K K e K here K e s ko as he elasc sffess ar ad K s ko as he geoerc sffess ar or he al sresses (Crsfeld, 99; Zekec ad Talor, ). Fro qs. (5) ad (4) ca be see ha hs ar ca be re as C K (8) V K e dv K Ths, elasc sffess ar s obaed b sg he al sffess ar ( K ) ad he sffess ar de o he preos dsplacees ( K ). (4) (7) [54]

13 6. Isoparaerc elees I s pora o oe ha he elasc ar sho q. (8) s depede of he fe elee sed. Hoeer, he sae does o occr h he ad K arces. Ths, he coplee defo of he arces o be sed he aalss of he sabl of laaed plaes reqres he specfcao of he fe elee sed. I hs ork, soparaerc fe elees ere sed (Cook e al., ). I s pora o oe ha, de o he se of Mdl's heor, ol erpolao fcos eed o be coos beee he elees. Therefore, co fcos C ca be sed (Cook e al, ), hch greal splfes he forlao of FM. I he soparaerc forlao, he dsplacees he fe elee are erpolaed depedel fro he odal dsplacees. Accordg o qaos () ad (4), he elees hae 5 degrees of freedo per ode, hch are he raserse dsplacees, ad ad he roaos ad. ke hs, (9) here s he ber of odes ad are he shape (erpolao) fcos of he elee. Fgre 6 Isoparaerc fe elee I order o allo for cred elees, as sho Fgre 5, he soparaerc forlao erpolaes he geoer of he elee b eas of he sae erpolao fcos of he dsplacee. ke hs: e (4) Fro q. () s erfed ha he deforaos are he s of a lear poro ad a qadrac poro. Usg a ar oao, he oal deforaos ca be re as () (4) here s a cosa ar ha relaes he dsplacees o he fesal deforaos ad relaes he olear poro of he deforaos o he odal dsplacees (). The ar depeds learl o he odal dsplacees. Usg qaos (4), () ad (9), ca be sho ha he ar s ge b: b s (4) [54]

14 [544] here s he poro of ebrae, b s he poro of bedg ad s s he poro of shear. I he soparaerc forlao hese arces are coposed of a se of sbarces correspodg o each ode of he elee. Ths, cosderg ol he sbarces referrg o ode, e hae: (4) κ κ b (44) γ γ s (45) As for he olear poro of he deforao, sg qaos (6), () ad (9), s erfed ha he ar s ge b:, (46) ce, accordg o o Kárá's heor, ol he ebrae deforaos deped qadracall o he dsplacees. Fro q. () hese deforaos ca be re as: Aβ, (47) Where he ecor, hch coas he deraes of he raserse dsplacees, ca be re as:

15 [545] G β G β (48) The, accordg o qaos (47) ad (48), e ca re: (49) The ar relaes he deforao crees sde he elee h he cree of odal dsplacees. Fro qs. (), (4), (4) ad (46) e easl ge () (5) Wh hs ar, all he ers eeded o deere he sffess ar hae alread bee obaed. I order o oba he age sffess ar, s ecessar o deere he geoerc sffess ar. Accordg o qs. (7) ad (46): da d dv d d T A V σ σ K (5) Fro q. (46) e hae dag d (5) The, he sbso of qao (5) qao (5) leads o da d d A T T σ A G K (5) I s ko ha: β σ A d d d d d d d d T (54) here β d s fod b derg qao (48) relao o he dsplacees. G β d d (55) bsg qao (55) o qao (54), e hae:

16 T da σ G d G d (56) Fall, replacg qao (56) qao (5), e oba he epresso of he geoerc sffess ar: K A T G G da I s erfed ha hs ar s serc sce he ar s sercal b cosrco. (57) 7. Calclao of crcal loads If he eral forces ecor ad he age sffess ar are deered, he olear pah of laaed plaes ca be obaed b sg approprae ehods sch as he oad Corol Mehod or he Arc egh Mehod (Crsfeld, 99). Hoeer, hs ork he objece s ol o oba he leared crcal load of he srcre hrogh he solo of a geeraled egeale proble (ahe, 996; Cook e al., ). I hs case, a lear aalss of he srcre sbed o he f referece load s carred o all o deere he esos ad he K correspodg geoerc sffess ar. Assg ha he pre-crcal dsplacees are eglgble, he sresses dsrbo reas cosa f he load s lpled b a load facor λ, h he correspodg sresses beg obaed b lplg he referece sresses b hs facor. The sae occrs h he geoerc rgd: K K (58) I s also oced ha, eglecg he pre-crcal dsplacees, he elasc sffess ar redces o he al sffess ar K. I s ecessar ha he age sffess ar be sglar for bcklg o occr: K K K. (59) cr Alerael, he crcal pos are characered b he codo ha a leas oe egeale of he age sffess ar s be ll. Therefore, he crcal load ca be calclaed b solg he geeraled egeale proble: ( K cr K ), (6) I hs eqao, he λ cr egeale represes he crcal load facor ad he egeecor represes he assocaed bcklg ode. III. UMRICA XAMP The forlao descrbed he preos es as pleeed he Malab sofare (Marha ad Paree, ). I hs seco are ge eaples of calclao of crcal loads of laaed plaes, hese plaes ca represe cerac le appled o fla srface. The resls of he prese forlao ll be copared h he aalcal resls obaed fro aer's solo for h plaes (Redd, 996) ad he AAU fe elee coercal sofare. aaed plaes sbjeced o ol loads her o plae shall be cosdered. Accordg o Fgre 6, he forces cosdered are: [546]

17 ˆ ˆ o, ˆ ko, k ˆ (6) oh aal ad baal copresso plaes ll be cosdered, as sho Fgre 6 (a). For plaes sbjeced o aal loadg, qao (6) s ge b: ˆ, ˆ, k (6) o ad for baal loadg, Fgre 6 (b), qao (6) s ge b: ˆ, ˆ k, k (6) a) Uaal load b) aal load Fgre 7 oad cases I all eaples, a esh coposed of qadrac qadrlaeral elees (8) as sed, as sho Fgre 7. I order o redce shear lockage, he redced egrao echqe h Gass pos as sed (Cook e al., ). Plaes fored of o pes of aerals ere aaled ad he sae hckess as sed for all eaples. [547]

Recallg ha, for he aal loadg facor k = ad for baal loadg k =, o aerals h he follog relaos ere cosdered: G G.5 G.

18 hear sress (MPa). ercal aaed Plae Fgre 8 Fe elee esh The plae of Fgre 7 s sqare (a / b = ), sde eqal o., h hckess eqal o., fored b laers of he sae hckess ad aeral. Recallg ha, for he aal loadg facor k = ad for baal loadg k =, o aerals h he follog relaos ere cosdered: G G.5 G..5 (64) 5 G G.5 G..5 (65) 4 For serc plaes here s o ebrae coplg, sce ar s ero de o ser. The obaed resls are desoless as cr b D Malab for dffere laar plaes as sho Fgre 9., (Redd, 996). The aalcal solo as prograed hear ress plae Mesh he dreco Mesh he dreco [548]

hear sress laaed board hear sress -/ Mesh he dreco Mesh he dreco Fgre

he elee pleeed Malab coerges perfecl for he aalcal solo, boh for

I s erfed ha, hs case, a ecelle agreee h he heor s obaed, as epeced ge

I s also obsered ha he resls are er close o hose obaed b AAU, aldag he

19 hear sress laaed board hear sress -/ Mesh he dreco Mesh he dreco Fgre 9- Aalcal solo sg Malab Fgre 9- Aalcal solo sg Malab The resl obaed b he elee pleeed Malab coerges perfecl for he aalcal solo, boh for Maeral ad Maeral. I s erfed ha, hs case, a ecelle agreee h he heor s obaed, as epeced ge s a h plae. I s also obsered ha he resls are er close o hose obaed b AAU, aldag he forlao ad pleeao of he elee.. Cross-pl a-sercal laaed plae For a cross-pl a-serc laao plae, as sho Fgre 9, here s ebrae-bedg coplg becase he a-ser cases soe elees of ar o be o-ero. [549]

Fgre 9 - Modelg of he laaed board h dffere odls of elasc Table shos he resls of he aalcal solo, AAU s solos for he o pes of aeral ad loads for a cross-pl (/9) laaed plae.

20 Fgre 9 - Modelg of he laaed board h dffere odls of elasc Table shos he resls of he aalcal solo, AAU s solos for he o pes of aeral ad loads for a cross-pl (/9) laaed plae. The obaed resls are desoless as cr b D (Redd, 996). Table Aalcal solo, Malab e AAU s solo for he plae (/9). / Aalcal Galerk Abaqs rror Galerk rror Abaqs, I s erfed ha he resl obaed b he elee pleeed Abaqs for hs case also coerges perfecl for he aalcal solo, for boh aerals. I s also oed ha he resls sho good agreee h hose obaed b AAU, aldag he forlao ad pleeao of he elee. IV. COCUIO Ths paper dscssed a forlao for sabl aalss of laaed plaes hrogh he Fe lee Mehod. The forlao preseed allos he se of raglar ad qadrlaeral elees h dffere erpolao orders. I he ercal eaples, he qadrac fe elee h qadrac erpolao CD8R as sed o sd s behaor. The redced egrao echqe as sed o alleae he proble of shear lockg. The behaor of hese elees as ealaed b coparg he resls obaed b FM h aalcal solos based o classcal heor of laaed plaes. The resls sho ha he elees proded ecelle accrac resls for boh cross-pl ad agle-pl laaed plaes ad ad baal loadg. The eaple h sercal laaed plaes preseed ecelle resls, shog good agreee h he classcal plae heor, as epeced. I s eresg o oe ha b aag he hckess of he board, aeral ad ber of laers, ol chagg he laer srcre he crcal load of he sercal laae s greaer ha he crcal load of he cross-pl a-serc laae. Aoher eresg aspec s he lo redco of he crcal load he he / rao s creased for boh laaed srcres. Ths aspec s pora defg he desg of he laer, hch ll be deered fro he characerscs ad reqrees of he projec. The eaples h agle-pl a-sercal laaed plaes shoed good resls for boh aeral ad loadg. oe ha he creasg he ber of laers, fro o o egh, hle aag he hckess of he plae, here s a cosderable crease he ale of he crcal load. Ths s a er pora obserao for he desg of laaed plaes sbjeced o copresse loads. [55]

21 I s obsered ha he crcal load also creases for he agle-pl sercal ad a-sercal laaed srcre, he he / rao creases, coras o he cross-pl a-serc laaed srcre ha decreases. Aoher pora obserao s abo he pe of loadg, here s erfed ha he crcal load for aal loadg s alas ce he crcal load he loadg s baal. The resls sho ha he elee prodes sasfacor resls for praccal se. Therefore, hese elees ca sere as a bass for oher sdes sch as, aalss ad opao of plaes sg robs ad effce ehods, sch as geec algorhs. V. ACKOWDGMT The ahors are graefl o he Cl geerg rcres Grop - GPC of he Federal Uers of Carr - UFCA ad for he facal sppor of CAP, CP ad Fcap. VI. RFRC [] Arekar, V. A. e hasar,.. Aalcal d of MRO Coecor Doble aer Grd rcre, (), p [] Maksk, Z.. srcras espacales de acero. o ed. doral Gsao Gl, arceloa. 97. [] ahe, K. J., (996), Fe lee Procedres. Prece-Hall. [4] elo, I. M., (6), Aálse fcee de Copósos aados Plaos lado-se a Forlação de leeos Fos corrgda a-pror se os feos do Traaeo. Dsseração (Mesrado e gehara Mecâca), Pofíca Uersdade Caólca do Paraá, Paraá. [5] Cook, R. D., Malks, D..; Plesha, M.. W, R. J., (), Coceps ad Applcaos of Fe lee Aalss. 4h do, Joh Wle & os. [6] Crsfeld, M. A., (99), o-lear Fe lee Aalss of olds ad rcres, ol., Joh Wle ad os. [7] Holada, A.., (), Aálse do qlíbro e sabldade de placas co Resrções de Coao. Tese (Doorado e Cêcas e gehara Cl), Pofíca Uersdade Caólca do Ro de Jaero, Ro de Jaero. [8] Joes, R. M., (999), Mechacs of Copose Maerals. ed. Phladelpha: Talor & Fracs. [9] Marha,. F.; Paree Jr.,. (), A Objec-Oreed Fraeork for Fe lee Prograg. Proceedgs of Ffh World Cogress o Copaoal Mechacs (WCCM V), Vea:, pp.. [] Redd, J.., (996), Mechacs of aaed Copose Plaes Theor ad Aalss. e York: CRC Press. [] Vasle, V. V.; Moroo,. V., (). Mechacs ad Aalss of Copose Maerals, lseer cece d. [] Zekec, O. C.; Talor, R.., (), The Fe lee Mehod. 5ª ed. erorh-heea. CIT A ARTIC Da la, W., obrho,., la Freas, C., Táora Rbero,., De Moras Júor, F., & Cosa la, R. (8). UMRICA AAYI I FIIT MT OF AMIATD PAT I T APPID I DIFFRT TYP OF COMPOIT MATRIA MODD AAU MODIG AD COMPARD WITH AAYTICA. ITRATIOA JOURA OF GIRIG CIC & RARCH TCHOOGY, 7(6), [55]

Analytical modelling of extruded plates

Analytical modelling of extruded plates paper ID: 56 /p. Aalcal modellg of erded plaes C. Pézera, J.-L. Gader Laboraore Vbraos Acosqe, INSA de Lo,5 bs a.j. Cappelle 696 VILLEURBANNE Cede Erded plaes are ofe sed o bld lgh srcres h hgh sffess.