Approximate Analytical Solutions for Large Flexural and Shear Deformations of Uniformly Loaded Simply Supported Bimodular Beam

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1 Aoximate Aatica Soutios fo Lage Fexua ad Sea Defomatios of Uifom Loaded Sim Suoted Bimodua Beam Dafe K. Jada 1 Received: 6 Jue 11; Acceted: 11 Seteme 11 Astact I tis ae a aatica aoximate soutio fo age fexua defomatios, sea defomatios ad sea stesses of a imodua uifom oaded sim suoted eam as ee deveoed. Veificatio fo te soutio as ee efomed usig FM aasis wit ANSYS. Te esuts of te ogam wee ve cose te esuts of te aatica soutio eseted i tis ae. Ke Wods: Bimodua eams, age defomatio, sea defomatio. اخلالصة حتليل تلرييب مالحنرافات امكبرية مالحنناء واحنرافات املص نلعتبات مزدوجة معامل املرونه واحملةلمب ورة ة مظتةةلمة وبسيطة الاسناد ظافر خليفة جدعان مد س مساعد يف كسم امهندسة املدنية/ جامعة الانبا يف هذا امبحث مت تطةير حتليل تلرييب مالحنرافات امكبرية امناجتة عن الاحنناء واملص ابالضافة الهجادات املص معتبة مزدوجة معاملل املرونلة ومحمل ورلة ة مظتةةلملة وبسيطة الاسناد. مت تدكيق احلل ابمللا نة مع طريلة امعنارص احملددة ابستخدام برانمج ANSYS 11 حيث اعطى امربانمج نتاجئ كريبه جلدا ملن امنتلاجئ املستحرلب من احلل املطروح يف هذا امبحث. 1. Itoductio Ma eseaces ave studied te oem of age fexua ad sea defomatios, ut u to te kowedge of te auto o oe as studied te aatica soutio of age fexua ad sea defomatios of a uifom oaded imodua eam. Te oems of age fexua ad sea defomatios of uimodua eams ave ee ivestigated ma eseaces. Te age defectio of eams as ee ivestigated Bisso ad Ducke (1945) fo a oit oad o a catieve eam. Timoseko ad Gee (1961) deveoed te soutio fo axia oad o a eam. Rode (195) deveoed te soutio fo uifom oad o a catieve eam. Law (1981) soved te oem fo a oit oad at te ti of te eam ad a uifom oad comied. Most of mateias exiit diffeet tesie ad comessive stais fo te same stess aied i tesio o comessio. Cassica teo of easticit assumes tat mateias ave te same eastic oeties i tesio ad comessio, ut tis is o a simified mode, ad does ot accout fo mateia oieaities. Ma studies ave idicated tat most mateias, icudig cocete, ceamics, gaite, ad some comosites, exiit diffeet tesie ad comessive stais give te same stess aied i tesio o comessio (Ju-i Su et a. (1)). 1 Assistat Lectue, Civi gieeig Det., Coege of g., Uivesit of Aa. 1

2 Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), Tose mateias exiit diffeet eastic modui i tesio ad comessio. Tose mateias ae kow as imodua mateias. Te eastic teo of imodua soids was fist oosed Amatsuma (198), ad Medi (198). B coductig exeimets, Medi (198) eaced te cocusio tat te cuve of stessstai ( ) at te oit of oigi fo mateias wit diffeet modui is oiea. Doog ad Ce (1984) deveoed a metod fo te aasis of diffeet moduus oems ased o a aoximate tigoometic seies. Zag ad Wag (1989) oosed te fiite eemet metod fo diffeet modui stuctue. Siivasa ad Ramacada (1989) aied a imoduus fiite eemet metod to te cacuatio of age defectio of ates. Yag et a. (199) eseted a metod usig te iitia stess-fiite eemet metod fo te aasis of imodua stuctues. Tseg ad Lee (1995) used a fiite sti metod fo te aasis of imodua amiates. Ye (1997) ad Ye et a. (1) deveoed a fiite eemet metod i wic vaiatios of eastic moduus ae diffeet fom tat of Poissiô's atio. Tseg ad Jiag (1998) used te imoduus teo to aase te stess of amiated stuctue. Liu ad Zag () adoted te metod of acceeatig te covegece facto to icease te ate of covegece. I tis eseac, a aoximate aatica aoac as ee adoted to sove te oem of age fexua ad sea defomatios of a uifom oaded sim suoted imodua eam ad te esuts otaied fom tis aoximate aatica soutio ave ee comaed wit umeica esuts otaied fom FM aasis usig ANSYS, ad te esuts foud to e ve cose.. Poem Fomuatio Te sae of te defomed uifom oaded sim suoted eam isies te foowig quatio fo te defomed sae: x v ( a si (1) wee, a :a costat. v ( : te defectio of te eam due to edig stesses o. : te egt of te ojectio of te defomed eam o te x-axis, as sow i Figue 1. q Aig te B.Cs.: v ( x ) & M( x ) v ( x ) & M( x ) Figue 1. A Sim suoted uifom oaded eam. ad tese coditios ae aead satisfied sice ( x ),ad v ( x ) ad fom te smmet of te defomed sae ad te equatio, oe ca cocude tat: x v( x / ) vm a vm v( vm si () wee, v m : te maximum defectio of te eam due to edig stesses. v

3 Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), To simif te cacuatios of te oem, a aoximate fomua fo te adius of cuvatue istead of te exact adius of cuvatue woud e used. Te exact adius of cuvatue is d v 1 dx d v / [1 ( ) ] dx wee, I : fexue igidit of te eam i wic: : moduus of easticit ad, I : momet of ietia of te coss sectio aout te N.A. Usig a aoximate fomua fo te cuvatue aig Macaui seies, as foows quatio () ca e witte i te fom ' 1 ' dv wee 1( v) ad v d v / dx [1 ( ) ] dx Now usig Macaui seies fo aoximatig k ( v ) () 1 d v ' 1( v ) dx (4) 1 ' ' ' ' ' ' '' ' ' 1( v ) 1( v ) 1( v ) v 1 ( v ) v /!... (5) It is eas to get te fist two deivatives of ( v ) ad te sustitute tem ito quatio (5) 1 ' wic ieds te foowig aoximate exessio fo degee of cuvatue 1 '' ' v 1 v (6) O te fist two tems of Macaui seies ave ee take i te aoximatio of te fuctio k ( v ) ecause te auto foud tat te cotiutio of te tid tem was egigie i comaiso 1 ' to te tedious wok added we te tid tem was icuded i te seies. Tis wi e made cea i te comaiso wit te FM aasis of te same oem. Te deivatives of v ( ca e foud usig quatio (1) as foows: ' x '' x v ( vm cos ad v ( ( ) vm si (7) O te ote side, te eam fexua fomua is 1 M ( (8) I Sustitute te two deivatives fom quatio (7) ito quatio (8) to get a exessio fo te edig momet M( i tems of (v m ) x x M( I( ) vm si 1 ( ) vm cos (9) At x=/, M( wi equa to: M( x ) I( ) vm (1) But fo a uifom oaded sim suoted eam te edig momet afte a oizota disacemet occus ca e cacuated as foows

4 Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), x R q Figue. Te eactio R afte te i te defomed eam ad a sectio ( fom te eft ed. Te eactio R wi ot e caged afte te oizota disacemet occus due to te assumtio tat te cuved eam as te same egt as tat of te udefomed eam ad to eig te aied uifom oad sti vetica afte defomatio occus. So tis eactio wi equa to q o q x R ad te sea foce : V( qo (11) o qo qo wee x=, V ( = ad we x=, V( ad sice te sea foce eatio is a iea fuctio aog te eam, so quatio (11) aove wi e suitae to take te effect of te defomed sae o te sea ad ece o te edig momet vaues. So te edig momet at distace ( fom te eft ed equas x qo qx o M( V( t) dt x (1) Hece te maximum edig momet wi equa to qo M( x ) (1) 8 Now equatig te edig momet at te mid-sa i quatios (1 ad 1) ieds, qo qo I( ) vm vm (14) 8 8 I Teefoe te quatio of te defomed sae ecomes qo x v( si (15) 8 I Tis equatio uses te egt () wic eeds te oizota disacemet, v, to e cacuated fist te o v (16) ad, v, ca e cacuated fom te foowig eatio : Te diffeece etwee te egt of te cuve of te defomed sae ad te oizota ojectio of te cuve is equa to te oizota disacemet. Hece v ds 4 ad oe coud imagie tis,as te oe ed moves towad its oigia ositio efoe te defomatios te oadig occu. I cacuus, te fist tem i quatio (17) wic is te egt of te cuve ca e witte i te fom Te quatio (17) takes te fom ds dx 1/ (17) dv 1 ( ) dx (18) dx

5 Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), dv v 1 ( ) dx dx (19) dx Te itegad i te igt ad side of quatio (19) ca e aoximated usig Macaui seies i te same wa used to aoximate quatio (6) ad te esut is 1 dv v ( ) dx dx Agai te aoximatio used tucated Macaui seies wit te fist two tems o. Pat of quatio (7) is estated, ow, fo coveiece ' x v ( vm cos Sustitutig tis deivative ito quatio (), te itegatig, ieds ( v ) v m (1) 4 Now sustitute, v m, fom quatio (14), ad eca tat (= o -v ),ieds q o 5 () 16I Te aove equatio fo a aticua oem ca e soved tia ad eo to fid te vaue of,, ad te te vaues of te vetica ad oizota disacemets ca e evauated usig te quatios (14) ad (1) esective..1 Bimoduait of te Beam I ode to take te imoduait of te eam ito cosideatio, te foowig assumtios must e stated fist 1- Te mateia of te eam is omogeeous aisotoic. - Te egio of te coss sectios sujected to comessio stess as a moduus of easticit caed ad te egio sujected to a tesie stess as a diffeet moduus of easticit caed as sow i Figue. - Staigt aes of te coss sectios of te eam efoe aicatio te oads, emai ae afte tat aicatio. Hece o sea defomatios ae assumed ti ow, ut wi e teated seaate i te ext ase i tis ae). 4- Te stess-stai eatiosi is iiea as sow i Figue. 1/ () Figue. Te stess stai-cuve fo a imodua mateia. Locatig te Neuta Axis Fom eviewig iteatue egadig imodua eams, te issue of ocatig te euta axis ( fom ow o witte N.A.) was ivaia detemied assumig tat te summatio of te axia foces o te coss sectios of te eam equas zeo. Te auto foud it is coveiet to eview tis issue 5

6 Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), (aso te sectio of cacuatig edig stesses) i ee. Ad tis is due to its imotace i udestadig its eatio to te age defomatios as we as te sea defomatios of a imodua eam. N.A. Figue 4. A coss sectio i a imodua eam. Fo te sectio of te eam sow i Figue 4 xd xd d, x Fx Rememeig to set (= we = ad = we = ) ad comiig tis equatio wit te equatio ( ) ad sovig tose two ageaic equatios simutaeous give ad (). Cacuatig Bedig Stesses Takig te sum of momets aout N.A. equas zeo ieds: MN. A. M( xd xd d M( (4) Aso ee it's eeded to set (= we = ad = we = ).Te eaagig tis equatio ad makig use of quatio () ied, 1 M( (5) I wee: I : momet of ietia of te coss sectio of te eam,ad 1 : educed moduus of easticit fo te imodua eams ad equas to 4 ( ) Fo a uimodua sectio ( = =) ad diect sustitutio i quatio (6) gives =. Te oma edig stess at a fie witi te coss sectio ca e foud usig te fexua fomua M M x x. fo tesie stess I I M M (7) x x. fo comessive stess I I (6) 6

7 Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), Now fo te oem of age defomatios discussed i te evious sectio, te exessio of te cuvatue,, is sim eaced wit te ew vaue stated i quatios (6) ad (8), so a tat oe eeds is just to eace te moduus of easticit of te uimodua eam wit te educed moduus of easticit. Hece quatios (15) & () ecome, esective qo x v( si (8) 8 I q o 16 I 5.4 Sea Defomatios ad Sea Stesses Refeig to Figue 5 eow, F 1 ad F ae foces due to edig stesses wee F 1 acts at a distace, x, fom oe ed of te eam ad F acts at a distace, x+dx, fom te same ed. Te smo,, stads fo sea stesses at a distace, x, fom tat ed of te eam. Hoizota equiiium i Figue 5 imies (9) N.A. Figue 5. A coss sectio i te eam sowig sea ad edig stesses. F x F dx 1 F ( d dx ( x d d d dx M(. I M( x d. I 7 d dm( dm( kowig tat M( x d M( dx ad V( is te sea foce ad dx dx sustitutig tese vaues i te itegatio aove, itegatig ad simifig gives Now te quatit dm( dx. d dm( I. I d dm( I d V x d dx. ( ).. d ( ) Q is te fist momet of te aea aove,, to te to suface of te eam at (= ). Hece te fia fomua fo te sea stess at a oit,, aove N.A.(wic wi e deoted ow ( is

8 Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), VQ V ( ( ) () I I It's cea tat maximum sea stess occus at,, o at te N.A., deotig tis maximum sea stess max, usig quatio() to sustitute te vaue of, ecaig tat, max is foud to e max ( ( ) I ( 4 V (1) Afte some simificatios te aove equatio takes te fom V max ( avg () A wee, A= : is te coss sectioa aea. It coud e see tat tis vaue is te same as i te uimodua eam. But oe as to e awae tat te geea distiutio of sea stesses is ot te same as i uimodua eams, ad tis is due to te fact tat te ocatio of te euta axis is diffeet as we as te sea stess vaues ote ta tat at te N.A. is diffeet aso, as it is aaet fom quatio(). Now it's ossie i a simia fasio to deive a fomua fo te sea stess at a oit ude te N.A.. Tis fomua is VQ V ( ( ) () I I V max ( avg (4) A Sice quatios () ad () ae aaoas ad tei vaues at te eds of te coss sectio ae zeos, Figue 6 eow sows sketces eesetig te esuts i tose equatios. ) Figue 6. Sea stess distiutio acoss te sectio of te eam..5 Sea Defomatios of a Bimodua Beam Accodig to Timoseko's eam teo, te tasvese defomatios ae ot o edig defomatios (tose discussed i te ecedig sectios) ut aso sea defomatios ad te atte wi e discussed i tis sectio fo imodua eams. Accodig to Timoseko's eam teo, ae sectios i eams wi o oge emai ae sectios we sea defomatios ae cosideed, ut te wi e cuved as sow i Figue 7 wic sows a defomed eemet i a eam due to sea stesses o. If te vetica sides of te eemets at te euta axis ae assumed to emai vetica afte defomatios occu, te te soe of te defectio cuve of te eam due to sea aoe is aoximate equa to te sea stai at te euta axis. Tis is deicted i Figue 7. Hece te foowig eatio ca e deducted 8

9 Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), V V dv s dx Figue 7. Sea stess defomatios i te sectio of te eam. dvs c c (5) dx G wee, v s : tasvese defectio due to sea stesses o. G : sea moduus of easticit. c : sea stess at te N.A.. I additio to tat, te sea moduus of easticit coud e ased o te moduus of easticit o ased o sice te vaue of G is -deedet accodig to Poisso's eatio: G. (1 ) Te auto suggests te foowig vaue fo : G (6) (1 ) wee: G : te educed sea moduus of easticit (educed ecause its vaue is awas ess ta uimodua eam). Te auto adoted quatio (6) ecause te fexua aasis of te imodua eam ed to te use of te taditioa fexua fomua, quatio (5), ut tis time wit te educed moduus of easticit a fact tat isied te use of Poisso's eatio wit. Late tis wi sow easoae esuts we comaed wit FM aasis. I additio, we te sectio is uimodua sectio te ad te vaue of i quatio (6) eads to te vaue wic is te vaue of sea moduus fo a uimodua eam. Now sustitute te maximum vaue of sea stess, quatio () o (4), ito quatio (6), te dvs c V (7) dx G AG Usig te exessio fo te sea foce V( fom quatio (1), te foowig exessio ca e witte fo te diffeetia sea defectio V o dvs(. 5qo qx dx (8) AG Sustitutig te exessio of V( fom quatio (1), te x q o x v x dv t dt x s( ) s( ) (9) 4 AG Te tota tasvese defectio v t ( is otaied addig te sea defomatio usig quatio (9) to te fexua defomatio usig quatio (15), te fo 9

10 Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), q o x qo x v x x t( ) si (4) 8 I 4 AG At te vaue of ad ad ece te vaue of wi e maximum ad equa to v q q o o t max (41) 8 I 16 AG Kowig tat ad G afte takig ( ) o Poisso's atio equas to zeo, wit some eaagemet i quatio (41), te q o vt max 1 ( ) (4) 8 I Te secod tem iside te aeteses is te cotiutio of te sea defectio to te tota defectio.. Numeica Resuts A sim suoted eam wit a sa of 5. metes was take as a exame. Te eam is uifom oaded 1 kn/m. Te coss sectio of te eam is a ectagua oe wit widt ad tota det of (=15 mm ad =5 mm), esective. To stud te effect of te vaiatio of te modua atio o te stesses ad o te defectio of te eam, diffeet vaues of tis atio ave ee take as foows: 1/.5, 1/., 1/1.5, 1, 1.5/1,./1 ad.5/1. Te esuts otaied usig te fomuas eseted i tis ae ae isted i te foowig taes. Tae 1. Bedig ad sea defectios aog te eam wit diffeet modua atios. ad.5/1 ad./1 ad 1.5/1 ad.5/1 ad./1 ad 1.5/1 1/ / / / / Tae. Tota defectio ad maximum sea stesses aog te eam wit diffeet modua atios. ad.5/1 ad./1 ad 1.5/1 ad.5/1 ad./1 ad 1.5/ / / / / /

11 Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), Tae. Tesie edig stesses aog te eam wit diffeet modua atios. 1/ / / / / To veif te esuts otaied fom te fomuatio eseted i tis ae, a fiite eemet aasis usig te ogam ANSYS as ee efomed fo te same eam wit te same vaues of ad.afte ocatig te N.A. usig quatio () te ue at of te eam is modeed usig moduus of easticit equas wie te owe at of te eam is modeed usig moduus of easticit tat equa. Te esuts ae isted i te taes eow. Tae 4. Comaiso etwee ae ad FM esuts. Pae Resuts FM Resuts 1/ / / / / / Tae 5. Comaiso etwee Bimodua ad Uimodua (FM) esuts. Bimodua Fomuatio Uimodua Fomuatio (wit ) usig FM 1/ / / / / /

12 v (mm) v s (mm) Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), Tae 6. o (%) of ae esuts i comaiso wit FM esuts. o (%) of ae esuts i comaiso wit FM esuts. 1/ / 4. 1/ /1 1.6 / /1 Tae 7. o (%) of Uimodua esuts i comaiso wit Bimodua esuts. o (%) of Uimodua esuts i comaiso wit Bimodua esuts. 1/ / 5. 1/ /1.4 4 / / /=1/.5 o.5/1 /=1/. o./1 /=1/1.5 o 1.5/1 /= /=1/.5 o.5/1 /=1/. o./1 /=1/1.5 o 1.5/1 /= x/ Figue 8. Bedig defectio aog te eam....4 x/ Figue 9. Sea defectio aog te eam. 1

13 σ max (MPa) v t (mm) v t (mm) Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), /=1/.5 o.5/1 1 /=1/. o./1 /=1/1.5 o 1.5/1 Fo a vaues of / /=1 8 1 max (MPa) x/ Figue 1. Tota defectio aog te eam x/ Figue 11. Max. Sea stess aog te eam. 5 Pae Resuts FM Resuts 1 /=1/.5 /=1/. /=1/1.5 /=1 /=1/1.5 /=1/. /=1/ x/ Figue 1. Tesie edig stess aog te eam x/ Figue 1. Pae ad FM esuts fo tota defectio 1

14 v t (mm) σ max (MPa) Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), ae esuts Bimodu Fomuatio FM esuts Uimodua Fomuatio max (MPa) / Figue 14. Pae ad FM esuts fo sea stesses / Figue 15. Bimodua ad Uimodua fomuatio i tota defectio. 11. Bimodua Fomuatio Uimodua Fomuatio 75 Bimodua Fomuatio Uimodua Fomuatio max (MPa) Figue 16. Bimodua ad Uimodua fomuatio i sea stesses / Figue 17. Bimodua ad Uimodua fomuatio i edig stesses. 14

15 (mm) Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), / Figue 18. Te det of te N.A. fom te ottom face.(tota det of te eam=5mm) 4. Cocudig Remaks It's oticeae fom Figues 8, 9 ad 1 tat te defectio of te eam is ot ifueced te modua atio ( / ) ut wit te educed moduus of easticit wit a iea ootioa eatio as it aeas fom quatio (4). I additio, it's cea fom Figue 9 tat te cotiutio of te sea defomatio to te tota defectio is ve sma ad deeds uo te atio of te det of te sectio to te egt of te eam. Tis cotiutio is aout.5% fom te tota defectio if te atio. Hece te sea defectio iceases wit sma vaues of te atio, ut ess ta 1%,te sea defectio is egigae. Aso oe coud see fom Figue 11 tat maximum sea stesses ae ot ifueced te modua atio o wit te educed moduus of easticit ad tis is cea fom quatios ( ad 4) wic is te same case i te uimodua eam. But at ocatios ote ta N.A. te sea stesses ae diffeet fom tose i te uimodua eam, sice te deed o te modua atio as i quatios ( ad ). Figue 1 sows tat te tesie edig stess iceases wit te icease of te modua atio. Hece if it's equied to decease te tesie edig stess (as it is te case i te cocete stuctues), it's o eeded to decease te modua atio istead of deceasig te mateia moduus of easticit. Figues 1 ad 14 sow tat te fomuatio eseted i tis ae is cose to tose gaied fom a FM aasis usig te commecia ackage ANSYS. As sow fom Tae 6 te maximum eo etwee te esuts otaied fom te ae fomuatio ad te FM aasis is (4%) i te tota defectio ad te sea stesses wic ae ve sma ad acceted fo a aoximate aasis. Te eo i te defectio iceases wit te decease i te modua atio ad due to te fact tat te smae te modua atio te smae (comaed to ) ad so te ige extesio i te fies eow te N.A. ad fia iceases defectio eve if te fies aove te N.A. udegoes sma defomatios. O te ote ad, te eo i te defectio deceases wit te icease i te modua atio ad tis is ecause te age te modua atio, te age (comaed to ) ad ece te smae te extesio i te fies eow te N.A. ad fia deceases defectio eve if te fies aove te N.A. udegoes age defomatios. Regadig te tesie edig stess, te eo iceases wit te icease i te modua atio, sice te icease i te modua atio meas 15

16 Dafe K. Jada Iaqi Joua of Civi gieeig 8(1), a icease i i comaiso to, wie te aatica equatio of te imodua tesie stess deas wit wic does ot diffe if te modua atio was (1/.5) o (.5/1), fo exame. Te eo i te edig defectio i te aoximate exessio fo defectio i quatio (1) is due to eatig te defectio i a odes i te eam wit te maximum defectio wit a sie cuve. Aote exessio woud e moe accuate if te same exessio ad two degees of feedom eesetig two defectios, ut te soutio woud e moe difficut te. O te ote ad to sow te imotace of te imodua aasis, te same exame as ee soved usig a uimodua aasis wit a aveage moduus of easticit wee te eo was ve age ad sometimes eaces 6% as sow i Tae 7 ad Figues 15, 16 ad 17. It's oticeae tat te eo i te sea stess vaues wete etwee te imodua ad FM o etwee te imodua ad uimodua aasis was ve sma. Te easo eid tat is tat te maximum sea stess was costat aog te eam ad as te same vaue deived i stegt of mateia fo a imodua eam. Refeeces Amatsuma, S. A., 198, asticit Teo of Diffeet Moduus, W. Ruifeg ad Z. Yuze, eds., Cia Raiwa Pess, Beijig, 11. Bisso, K.., ad Ducke, D. C., 1945, Lage Defectios of Catieve Beams, Quate of Aied Matematics, Vo., Doog, J. L., ad Ce, L. W., 1984, Axismmetic viatio of a iitia stessed imoduus tick cicua ate. Joua of Soud ad Viatio, 94, Ju-i Su, Hai-qiao Zu, Si-og Qi, Da-i Yag ad Xiao-tig He, 1, A eview o te eseac of mecaica oems wit diffeet modui i tesio ad comessio Joua of Mecaica Sciece ad Tecoog, 4(9), Law, Jo H., 1981, Lage Defectio of Catieve Beams, Joua of gieeig Mecaics Divisio, Poceedigs of te Ameica Societ of Civi giees, 17(M1), Liu, X., ad Zag, Y.,, Moduus of easticit i sea ad acceeate covegece of diffeet extesio-comessio eastic moduus fiite eemet metod. Joua of Daia Uivesit of Tecoog, 4, Medi, G., 198, A oiea eastic mode fo isotoic mateias wit diffeet eaviou i tesio ad comessio. Tasactios of ASM: Joua of Aied Mecaics, 14(6), Rode, F. V., Lage Defectio of Catieve Beam wit a Uifom Distiuted Load, Quate of Aied Matematics, Vo. 11, 195, Siivasa, R. S., ad Ramacada, L. S., 1989, Lage defectio aasis of imoduus aua ad cicua ates usig fiite eemets. Comutes ad Stuctues, 1(5), Timoseko, S. P. ad Gee, J. M., 197, Mecaics of Mateias., d ditio, Va Nostad Reiod Coma Ltd., Lodo. Timoseko, S. P. ad Gee, J. M., 1961, Teo of astic Staiit. McGaw-Hi Book Coma, New Yok, ad Tseg, Y.P., ad Jiag, Y.C., 1998, Stess aasis of imodua amiates usig id stess ate eemets. Iteatioa Joua of Soids Stuctues, 5(17), Tseg, Y. P., ad Lee, C. T., 1995, Bedig aasis of imodua amiates usig a ige-ode fiite sti metod. Comosite Stuctues,, Yag, H., Wu, R., Yag, K., ad Zag, Y., 199, Sove imoduus oem usig iitia stess. Joua of Daia Uivesit of Tecoog, (1), Ye, Z. 1997, A ew fiite eemet fomuatio fo aa eastic defomatio. Iteatioa Joua of Numeica Metods i gieeig, 14(4), Ye, Z., Yu, H., ad Yao, W., 1, A fiite eemet fomuatio fo diffeet Youg s moduus we tesio ad comessio oadig. Poc., Cof. o Comutatioa Matematics, Poag Uiv. of Sciece ad Tecoog, Poag, Sout Koea, 5. Zag, Y., ad Wag, Z., 1989, Te fiite eemet metod fo easticit wit diffeet modui i tesio ad comessio. Joua of Comutes ad Stuctua Mecaics Aicatios, 1(6),