GH. Rahimi & AR. Davoodinik

Transcription

1 US ntrnational Journal of nginring Scinc Vol 9 No5-8 Pag 5- HRMAL BHAVOR ANALYSS OF H FUNONALLY GRADD MOSHNKO'S BAM GH Raimi & AR Davoodinik Abstract: intntion of tis stud is t analsis of trmal bavior of functionall gradd bam FGB distribution of matrial proprtis is imitatd ponntial function For trmal loading t stad stat of at conduction wit ponntiall and prbolic variations troug t ticknss of FGB is considrd Wit comparing of trmal bavior of bot isotropic bam and FGB it is appard tat t qualit of tmpratur distribution plas vr important part in trmal rsultant distribution of strsss and strains for FGB So tat for dtcting t particular trmal bavior of FGB t function of at distribution must b sam as function of matrial proprtis distribution n addition n t cas of ponntial distribution of at wit no mcanical loads in spit of t fact tat t bnding is accrud t nutral surfac dos not com into istnc Kwords: imosnko's bam ponntiall distribution functionall gradd matrial trmal bavior ntroduction Functionall gradd matrials FGMs av bn rsarcd and dvlopd in man nginring filds tat nd to b supr at rsistant suc as t outr wall and t ngin parts of futur spac-plans n FGMs matrial proprtis var continuousl from on surfac to t otr spciall from mtal to cramic From tis continuous cang in composition FGMs can witstand ig-tmpratur nvironmnts wil maintain tir structural intgrit Du to ts advantags various rsarcs av bn trid about t modling and application of FGMs for plats and slls tat subjctd to trmal loads Javari and slami drivd t quilibrium and stabilit quations of a functionall gradd rctangular plat undr trmal loads basd on t classical plat tor Buckling analsis of FGM plats undr four tps of trmal loads was carrid out in closd-form solutions [] Najafiad and slami anald t trmal buckling of FGM circular plats undr tr tps of trmal loads nonlinar quilibrium and linar stabilit quations wr drivd using variation formulations [] Sn studid a post buckling analsis for a functionall gradd clindrical panl of finit lngt subjctd to aial comprssion in trmal nvironmnts Matrial proprtis wr assumd to b Papr first rcivd Ma4 7 and in rvisd form Jul 4 9 GH Raimi is wit t Dpartmnt of Mcanical nginring arbiat Modars Univrsit Jalal--Al--Amad p Wa ran ran Raimi_g@irostorg AR Davoodinik is a PD studnt at t sam Dpartmnt Davoodinik@aoocom tmpratur dpndnt and gradd in t ticknss dirction according to a simpl powr law distribution govrning quations wr basd on Rdd's igr ordr sar dformation sll tor wit a von Karman Donnll-tp of kinmatic nonlinarit and including trmal ffcts [] trmal buckling bavior undr uniform or non-uniform tmpratur ris was anald; owvr t tim-dpndnt tmpratur ris was not considrd [4] Kung and Kim studid tr-dimnsional trmo-mcanical buckling analsis for functionall gradd composit structurs tat composd of cramic functionall gradd matrial FGM and mtal lars finit lmnt modl is adoptd b using an 8-nod solid lmnt to anal mor accuratl t variation of matrial proprtis and tmpratur fild in t ticknss dirction For a tim discrtiation rank Nicolson mtod is usd [5] Ravicandran amind t ffcts of t functional form of gradation including t prsnc and structural arrangmnt of monolitic Al O Ni rgions in combination wit t gradd rgion on t trmal rsidual strsss arising from t fabrication of a FGM sstm [6] Howvr for functionall gradd bams FGB rlatd studis ar vr limitd Sankar stablisd a functionall gradd ulr Brnoulli bam modl to trat a static problm of a simpl supportd bam [7] Zong and Yu prsntd act solution for a cantilvr FGB b considring it as an lasticit problm t calculation involvd is fairl cumbrsom [8] abrabort t al dvlopd a nw bam finit lmnt to stud t trmolastic bavior of FGB [9] Li prsntd a unifid approac for analing

2 6 rmal Bavior Analsis of t Functionall Gradd imosnko's Bam FGB wit t rotar inrtia and sar dformation includd n t fr vibration of an FGB wr t dpndnc of t natural frquncis and mod saps on t gradint ind for a simpl supportd bam is givn [] L Yong-dong t al drivd t auc singular intgral quation for t anti-plan fractur analsis of a functionall gradint matrial infinit strip wit finit widt undr t assumption tat t sar modulus is an ponntial function of t spatial coordinat [] Yang and Xiang invstigatd t static bnding fr vibration and dnamic rspons of monomorp bimorp and multimorp actuators mad of functionall gradd piolctric matrials FGPMs undr a combind trmal-lctro-mcanical load b using t imosnko bam tor [] From t litratur surv it is sn tat fw studis av bn mad for an fficint discussion of t ffct of tmpratur distribution on t trmal strsss and strains using imosnko bam tor for FGB objctiv of tis papr is to prsnt t rsponss of FGB undr two tps of trmal loads; t stad stat of at conduction wit ponntiall and prbolic variations troug t ticknss of FGB Firstl t stabilit quations of bam will b driv b assuming trmal loading onl basd on t first ordr sar dformation tor FSD n t act solution of t govrning quations for FGB subjctd to trmal load will b prsnt For vrification of t procdur ulr Brnoulli bam can b analticall rducd from t imosnko's bam tor Aftrwards wit comparing of trmal bavior of bot isotropic bam and FGB subjctd to two functions of at distributions w will b stud t ffct of t tp of tmpratur distribution on t trmal rsultant distribution of strsss and strains For vidnc of t trmal bavior of FGB first t sam as function of matrial proprtis distribution and nt diffr from tat will b considr for function of at distribution Finall rsults of t trmomcanical bavior of t FGB ar prsntd and tn conclusions ar plaind Matrial Gradint of FGM Bams FGM can b producd b continuousl varing t constitunts of multi-pas matrials in a prdtrmind profil most distinct faturs of an FGM ar t non-uniform microstructurs wit continuousl gradd macro proprtis An FGM can b dfind b t variation in t volum fractions Most rsarcrs us t powr-law function P-FGM ponntial function -FGM or sigmoid function S-FGM to dscrib t volum fractions [] n tis stud proprtis distribution is dfind b ponntial function and mtod of problm solution can b tndd for otr tp of distributions onsidr an lastic rctangular cross sction bam As sown in Fig coordinats and dfin t plan of t bam wras t -ais originatd at t middl surfac of t bam is in t ticknss dirction Fig lastic rctangular cross sction FGB onsidr an -FGB wit diffrnt ponntial variations for distributions of trmal pansion trmal conductivit and modulus of lasticit troug t ticknss dirction of bam rspctivl w av [ 4]; A A ln -a k A k A k Ak kk ln -b k A ln -c Wr ar t matrial proprtis in t =- = surfacs rspctivl and t constants A á A K A ù â and ë can b obtaind b boundar conditions Bs kind of matrial proprtis distribution in t ticknss dirction of t -FGB is plottd in Fig Govrning quations n accordanc wit t FSD a point A in t FGB wit a distanc to t middl surfac will b movd to point A' aftr dformation Fig rfor t aial displacmnt at t point A wit distanc from mid-surfac = can b prsntd b [5]; w u u Wr u is t displacmnt at t middl surfac and w is t transvrs dformation w and ar t rotations of vrtical lin AB Fig about Y-ais du to bnding and saring dformations rspctivl and all of tm ar indpndnt of -dirction in FSD [5] Fig Young's modulus distribution troug t ticknss dirction of t -FGB

3 GH Raimi & AR Davoodinik 7 n strain in t -dirction as follows w u Basd on t plan strain condition strss-strain rlation for an FGB tat subjctd to trmal load is [6]; 4-a w u 4-b Wr and ar t aial strss on t surfac wit distanc from mid surfac t Poisson's ratio and tmpratur distribution along -dirction of bam rspctivl As mntiond bfor q 4-b as tr unknown trms wic ar indpndnt of - dirction s trms ma b obtaind b using t quilibrium quations and Bs Fig Aial bnding and saring dformation caractristics of a bam according to FSD 4 Problm Solution 4- Aial Strss strss rsultants pr unit lngt of t middl surfacs ar dfind b intgrating strsss along t ticknss Assuming t trmal loading onl wit distribution in -dirction Wn t bam is in quilibrium t aial rsultant forcs in t -dirction must b ro i l bd F 5 Wr b and l ar t widt and lngt of bam rspctivl Morovr in t absnc of mcanical loads t rsultant momnts about Y-ais appars b trmal ffct onl M n t quilibrium quation for rsultant momnts as follows l M l M M d w M M ; ; 6 Wr tat is t inrtia momnt and M ar dfind as follows [6]; d b M b 7 Morovr M M is prsntd t mcanical momnt about Y-ais Sinc tr ar tr unknown trms in q 4-b w can provid t tird quation wit using Bs rlation Bs of t simpl supportd FGM bam ar; w= M =; = =l 8-a ; bd l 8-b Wr M is t total momnt acting on t bam about Y-ais n wit collocation of q 5 q 6 and q 8-b t paramtrs u w and would b arisn and upon substitution into q 4-b aial strss can b obtaind onsidr convntional dfinitions for simplicit as follows d d d d d d 9-a w u 9-b Wr B substitution q 9-a q 9-b q 7 and q 4-b into q 5 q 6 and q 8-b w can st t quation sstm wit t following form; b n t cofficint is dirctl obtaind and for w av; b b

4 8 rmal Bavior Analsis of t Functionall Gradd imosnko's Bam t is vidnc tat aial strss can b obtain from q 4-b b substituting for and from q and q as follows; 4- ransvrs Sar Strss onsidr an lmnt of a FGM rctangular cross sction bam of t lngt d and t widt b as illustratd in Fig 4 f t tmpratur distribution is assumd to b = t trmal bnding taks plac in t X-Z plan b d b d d b d Wr ô is t transvrs sar strss Rarranging t trms ilds: d 4 Substituting for ó from q ilds: d 5-a Fig4 ransvrs sar strss in t bam quilibrium quation for an lmnt of t cross sction of t FGB is: u w d 5-b Matrial Al O ñkgm 97 ab ampl of Matrial proprtis of -FGM bam [6] GPa í vjkg K kwm K Mlting point K á K Ni For FGB wit simpl supportd nds t sar strss ma b asil sown to b ro to causalit of absnt of scond drivativs of u and tird drivativ of w wil for FGB wit fid-simpl supportd nds lacking of sar strss is not accrud 5 Procdur Vrification ulr-brnoulli bam tor is t spcial cas of t bam toris For vrifing of mntiond mtod w can ignor additional paramtrs of FSD assumptions tat compard wit classical bam tor B and obtain sam rsults n t vnt tat intnd t isotropic classical bam st fort for discussion t paramtrs and ù ë tat prsntd for rotational dformation and non-isotropic proprtis rspctivl must b vanisd n addition t q 5 and q 6 ar satisfid incidntals to obtain t cofficints s considrations and propr dfinition for at distribution will b vntuatd to sam rsults of strss fild for isotropic classical bam as is prsntd in som tts [7] 6Stad stat mpratur Distribution 6- ponntial Distribution of mpratur - Stat Assum t at conduction is on-dimnsional f t ponntial variation for cofficint of at conduction troug t ticknss of -FGB is considrd as q -b t at distribution in ticknss dirction is; A B 6 Wr constants A B can b obtaind from trmal Bs tat ar; rmal Bs ; 7 A B 8 n wit substitution of q 8 in q 6 w av;

5 GH Raimi & AR Davoodinik 9 9 Finall wit q q -a and q -c trmal strss can b obtaind as follows; A B Wr is t strss fr tmpratur 6- Hprbola Distribution of mpratur H- Stat o stud t imprssion of t tmpratur distribution tp in t qualit of trmal rsultants distribution RD of strsss t prbola distribution of tmpratur also is considrd rfor wit solving of on problm for two stats of at distribution t rsponss of bam undr ts two conditions wr comparabl onsidr t prbola distribution of tmpratur as H 4 Wr t tmpratur of boundar surfac at = is = ºK and at =- is ro 7 Numrical ampl onsidr an lastic rctangular cross sction bam Fig wit simpl supportd nds and dimnsions b= = 4m wo tps of bam toris wit two tps of bam constructions ar posd to two stats of tmpratur distributions for driving t RD of strsss and strains dtrmination of trmal strsss distributions is stablisd b B and FSD for isotropic bam kind of Ni will b namd as A-bam and b FSD for -FGB will b namd as B-bam wit matrial proprtis tat is prsntd in abl [8] f t tmpratur Bs ar considrd as follows: = K = K = K Wr and ar indicatd t tmpratur of strss fr stat tmpraturs at surfac =- unmid Ni and surfac = unmid Al O rspctivl Morovr wit considration of matrial proprtis of B-bam abl for ponntial trms in q w av: * a * b * k k c k n t tat is dfind b q 7 convntional dfinitions in q 9-a and -stat from q 9 can b obtaind as follows: =-7 4-a =47- =944-5 =558-6 =979-5 =66-7 = b = c us wit substitution of q 4-a and q 4-b into q and q dtrmination of and ar stablisd so tat: u w Now wit considration of structural Bs as; w l u ; w ; l 5 6 dtrmination of u w and ö ar acivd n t intrim of tis numrical ampl for transvrs dformation w w av 77 w l 7 On t otr and sinc t bam is tin in t vnt tat; if problm assumptions ar varid to A-bam sam rsults will b obtaind for FSD and B in - stat abl Furtrmor in tis numrical ampl t grapical rsults of H-stat assumption will b prsntd for compltion of discussion 8 Rsults 8- Rsults for H-Stat onsidr t H-stat in ticknss dirction of a bam as sown in Fig 5 Wras wit supposing tat bot A-bam and B-bam ar posd to H-stat in ticknss dirction similar rspons for RD of strsss is acivd Fig 6 ndd t H-stat in t ticknss dirction of B-bam is not oprabl bcaus t function of at distribution must b sam as function of matrial proprtis distribution From Fig 6 it is vidnc tat unlik matrial proprtis distribution t kind of at distribution is vr affctiv in t qualit of RD of strsss

6 rmal Bavior Analsis of t Functionall Gradd imosnko's Bam Fig 8 is sown t RD of strss in t bot of tm t is clar tat for bot constructions of bam RD of strsss do not mak an diffrnc in ts trmal Bs tat is dfind wit q Fig 5 Hprbola distribution of tmpratur along t ticknss dirction of a bam Fig7 ponntiall distribution of at in ticknss dirction of a bam Fig6 Non-dimnsional RD of strsss vs ticknss dirction in H-stat dottd; A-bam solid; B-bam 8- Rsults for -Stat Wit considring t -stat in ticknss dirction of a bam Fig 7 t RD of strsss in ticknss dirction for bot A-bam and B-bam ar smmtrical wit rspct to = surfac of t bams rsmbling to H-stat Fig 8 Non dimnsional RD of strsss vs ticknss in -stat dot lin; imosnko's A- bam cross; classical A-bam solid lin; B-bam Ni-Al O FSD 66 B 66 n B is vanisd ab Rsults for imosnko's bam FSD and classical bam B in -stat Wil if wit tmpratur incrasing t nw trmal Bs ar dfind as; =ºK =7ºK =ºK 8 Aftrwards t diffrncs btwn RD of strsss for A-bam and B-bam ar ntirl rvald Fig 9 n spit of tis diffrnc btwn tm t av sam sign positiv for trmal strsss in wol of t ticknss Bcaus wn tmpraturs ar incrasd incrasing in tnsil strsss tat is aris from nd supports ffct is largr from incrasing in comprssion strsss tat is rsult of trmal ffct Furtrmor wit qualit of t subtraction of and in t q and q 8 incrasing in t tnsil strsss arising from nd supports ffct wit no canging in t comprssion strsss arising from trmal ffct is appn 9 Discussions Wit sparating t trmal rsponss of bams to two parts t mor analsis cass ar availabl [6] On part for plaining t trmal bavior of bam witout an nd supports will b nam as part A and anotr part for rprsnting t trmal bavior of bam wit considring t nd support influnc will b nam as part B wn t bam is posd to H- stat or -stat us for ts two parts wit

7 GH Raimi & AR Davoodinik considring t q t following prass can b obtaind; in t opposit sid =5 dcrasing of coupling strss is arisn rfor in t H-stat of trmal loading t outr surfac of bam wit biggr oung modulus must b slctd for subjcting to trnal prssur load n t otr and unlik t RD of strsss t RD of strains is not smmtrical and aftr trmal bnding t natural surfac is appard in -5 wit unsmmtrical final gomtr for bam Fig 9 Non dimnsional RD of strsss vs ticknss in -stat wit ig rat tmpratur dottd; timosnko A-bam cross; classical A- bam solid; B-bam Ni-Al O * * 9-a 9-b 9-c 9-d Fig RD of strsss MPa vs ticknss in H- stat for classical A-bam dottd; strss rsultants of part A ı dottd-das; strss rsultants of part B ı * solid; total strsss ı Wr t ı İ and ı * İ * ar t strss and strain rsultants of part A and part B rspctivl n t total strsss and strains ar as follows; -a -b vn now wit q -a and q -b t distinctions of trmal bavior of A-bam and B- bam can b actl spcifid n t H-stat Fig and Fig ar sown t RD of strsss and Fig and Fig ar prsntd t RD of strains for A-bam and B-bam rspctivl Just as vidnc t distribution qualit of part A and part B of strsss ild t smmtricall distribution of total strss Furtrmor tnsil strss positiv is accrud in t innr sction of ticknss -<< and t comprssion strsss is ild in t outr sction of ticknss <- & > Now considr a bam wit posing to trnal prssur load on t on of boundar surfacs i =-5 Aftr bnding of bam in H-stat t accumulation of trmal strss and mcanical strss in =-5 surfac is ild to incrasing t coupling strsss and Fig RD of strsss MPa vs ticknss in H- stat for B-bam dottd; strss rsultants of part A ı dottd-das; strss rsultants of part B ı * solid; total strsss ı Fig RD of strains vs ticknss in H-stat for classical A-bam dottd; strain rsultants of part A İ dottd-das; strain rsultants of part B İ * solid; total strains İ

8 rmal Bavior Analsis of t Functionall Gradd imosnko's Bam Wn t B-bam is subjctd to -stat wit trmal Bs corrsponding to q RD of strsss and strains would b as Fig 4 and Fig 5 rspctivl According to tm an important point can b acivd tat t nutral surfac is vanisd n t otr word bot t trmal strss and trmal strain ar not contmporanousl qual to ro and t trmal strains ar alwas positiv tnsion mod along ticknss dirction Furtrmor similar to H-stat t distribution qualit of part A and part B of strsss ild t smmtricall distribution of total strss Morovr on t contrar to H-stat tnsil strss positiv is accrud in t outr sction of ticknss <- & > and t comprssion strsss is ild in t innr sction - << Now considr a bam wit subjcting to trnal prssur load on t on of boundar surfacs i =-5 Aftr bnding of bam in -stat t accumulation of trmal strss and mcanical strss in =-5 surfac is ild to dcrasing t coupling strsss and in t rvrs sid =5 t coupling strsss will b incrasd rfor in t -stat for improving of bam rsponss t outr surfac of bam wit smallr oung's modulus must b slctd for subjcting to trnal prssur load Fig 5 RD of strains vs ticknss in -stat for B-bam dottd; strain rsultants of part A İ dottd-das; strain rsultants of part B İ * solid; total strains İ rmo-mcanical Analsis onsidr t B-bam subjctd to bot trnal prssur load on t on of boundar surfacs i =-5 and -stat of trmal loading Fig 6 is comprisd t suprposition of trmal and mcanical strsss along t -dirction of B-bam Fig RD of strains vs ticknss in H-stat for B-bam dottd; strain rsultants of part A İ dottd-das; strain rsultants of part B İ * solid; total strains İ Fig 4 RD of strsss MPa vs ticknss in - stat for B-bam dottd; strss rsultants of part A ı dottd-das; strss rsultants of part B ı * solid; total strsss ı Fig 6 Non dimnsional strsss distribution vs ticknss for tmpratur boundar condition according to q9 at t B-bam as posd to trmo-mcanical load -stat & trnal prssur on =5 solidrigt and; trmal strsss dottdlft and; mcanical strsss solidlft and; total strsss t is sn tat in t outr sction of B-bam ticknss t trmal strsss is alwas positiv tnsil strsss rfor t summation of trmomcanical strsss is causd to improving t stat of strss distribution wit dcrasing t comprssion strsss on t =-5 surfac n t otr and bcaus of positiv mcanical strss on t =5 surfac incrasing t total strss is appard Wr as in lowr tmpraturs tat is causd t trmal strsss smallr tan mcanical strsss tis incrmnt is not considrabl Morovr according to Fig9 in igr tmpraturs t strsss distribution curv av two sctions on wit smallr amount of For dtails s [5]

9 GH Raimi & AR Davoodinik slop tat is appard in largr sction of ticknss sction AB in Fig 9 and anotr wit largr amount of slop in smallr sction sction B in Fig 9 onclusions Som apposit conclusions can b dmonstratd from tis stud as fallows; qualit of tmpratur distribution plas vr important part in trmal rsultant distribution of strsss for -FGB n t otr words for apparing t particular trmal bavior of FGB t function of at distribution must b sam as function of matrial proprtis distribution Wit supposing t prbola tmpratur distribution it is no diffrnc btwn trmal rsponss of isotropic bam and FGB trmal rsultant distribution of strsss in t -FGB wit ponntiall tmpratur distribution is on t contrar wn compard wit prbola tmpratur distribution at mans in t outr sctions of ticknss t tnsil strsss and in t innr sctions of tat t comprssion strsss is accrud Wit tmpratur incrasing in t ponntiall tmpratur distribution stat t trmal strsss would graduall b as tnsil strsss positiv troug t wol of ticknss Until on t wol points of ticknss dirction t positiv strsss is mrl arisn Wn t FGB is coincidd to bot trnal prssur load on t on of boundar surfacs and ponntiall at distribution wit ig tmpratur rat for improving of total strsss distribution t mcanical load must b applid on t sid of ticknss wit smallr slop of trmal strsss distribution curv along of ticknss n lowr tmpratur rat of ponntiall at distribution as t trmal strsss distribution is smmtricall rspct to mid surfac of FGB ticknss if it is also posd to trnal prssur load on t on of boundar surfacs for improving of total strsss distribution t mcanical load must b applid on t sid of ticknss wit wakr matrial proprtis n t cas of ponntial distribution of trmal loading wit no mcanical loads in spit of t fact tat t bnding is accrud t nutral surfac dos not com into istnc Rfrncs [] Javari R slami MR "rmal Buckling of Functionall Gradd Plats" AAA J;4:6 69 [] Najafiad MM slami MR "First-Ordr-or- Basd rmo lastic Stabilit of Functionall Gradd Matrial ircular Plats" AAA J;47: [] Sn HS "Post Buckling Analsis of Aiall Loadd Functionall Gradd lindrical Panls in rmal nvironmnts" nt J Solids Struct;9:599 6 [4] Na KS Kim JH "r-dimnsional rmal Buckling Analsis of Functionall Gradd Matrials" ompos Part B ng; 5: [5] Na KS Kim JH "r-dimnsional trmo mcanical buckling analsis for functionall gradd composit plats" omposit Structurs; 7:4 4 6 [6] Ravicandran KS "rmal rsidual strsss in a functionall gradd matrial sstm" Matr Sci ng A; : [7] Sankar BV "An lasticit Solution for Functionall Gradd Bams" omposits Scinc and cnolog; 6: [8] Zong Z Yu "Analtical Solution of a antilvr Functionall Grad Bam" omposits Scinc and cnolog; 67: [9] abrabort A Gopalakrisnan S Rdd JN "A Nw Bam Finit lmnt for t Analsis of Functionall Gradd Matrials" ntrnational Journal of Mcanical Scinc; 45: [] Li X-F "A Unifid Approac for Analing Static and Dnamic Baviors of Functionall Gradd imosnko and ulr Brnoulli Bams" Journal of Sound and Vibration; 8: 9 8 [] L Yong-dong JA Bin ZHANG Nan DA Yao ANG Li-qiang "Anti-Plan Fractur Analsis of Functionall Gradint Matrial nfinit Strip wit Finit Widt" Applid Matmatics and Mcanics; 76: [] Yang J Xiang HJ "rmo-lctro-mcanical aractristics of Functionall Gradd Piolctric Actuators" Smart Matr Struct; 6: [] Sang-Ho i Yn-Ling ung "Mcanical Bavior of Functionall Gradd Matrial Plats Undr ransvrs Load-Part : Analsis" ntr J of Solids and Structurs; 4: [4] Srkan Dag Suat Kadioglu O Slcuk Yasi "ircumfrntial rack Problm for an FGM lindr Undr rmal Strsss" J of rmal strsss" : [5] Davoodinik AR "Mcanical Bavior Analsis of FGM imosnko's Bam" PD dissrtation arbiat Modarrs Univrsit ran 5 [6] Ugural A "Strsss in Plats and Slls" McGraw- Hill Nw York 98 [7] Nowinski JL "or of rmo lasticit wit Applications" Sijtoff & Noordoff ntrnational Publisrs 978 [8] Noack J Rolfs R ssmr J "Nw Larwis oris and Finit lmnts for fficint rmal Analsis of Hbrid Structurs" omputrs and Structurs;8: 55-58