Derivation of Annular Plate Sector Element for General Bending Analysis

Transcription

1 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN 8-78 Divtion of Annul Plt Scto Elmnt fo Gnl Bnding Anlysis Lctu D. Hyd Abdulm Mhdi Civil ngining dptmnt, Engining collg, Al_ Mustnsiiyh Univsity Abstct A nw mthmticl finit lmnt modl suitbl fo th gnl bnding nlysis of nnul plt stuctu is dvlopd in pol coodints systm dpnding on th stin bsd ppoch hs bn divd. Th lmnt is simpl nd contins only th ssntil dgs of fdom. Th lmnt hs 5 dgs of fdom, th t ch nod nd stisfis th ct psnttion of th igid body mods of displcmnt. Th sults obtind by using th poposd lmnt in svl numicl poblms hv shown tht pid convgnc to ct solution cn b obtind with ccptbl dg of ccucy whn only fw lmnt usd. Th lmnt hs th dvntg ov th oth vilbl nnul plt lmnts. Th impovmnt obtind is du to th fct tht ll th displcmnt filds of th psnt lmnt stisfy th ct psnttion of igid body mods of displcmnts thn th shp function o du to igid body mods bcoms zo. Also, th psnt lmnt stisfis th full gomty of th nnul plt du to this point disctiztion o bcoms zo. Finlly th o du to stin mod bcoms vy smll bcus th psnt lmnt stisfis th comptibility qutions of stins nd th th cofficints of stin mod divd ctly fom ptil diffntil qutions of stins. Th numicl solution of svl poblms by using th psnt lmnt povd to b powful in th stuctul bnding nlysis of cicul nnul plts. Its sults btt thn th solution of oth lmnts nd pckgs with spct to nlyticl. Kywods: nnul scto plt, Stin bsd ppoch lmnt 4

2 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN 8-78 الخلاصة: اشتقاق عنصر محدد لتحلیل الانحناء العام للصفاي ح الداي ریة بالا حداثیات القطبیة قسم الھندسة المدنیة م. د. حیدر عبدالا میر مھدي / كلیة الھندسة / الجامعة المستنصریة تم في ھذه الدراسة تطویر عنصر جدید للتحلیل الا نحناي ي للص فاي ح الحلقی ة با س تخدام نظ ام الا ح داثیات القطبی ة یعتمد على طریقة الا نفعال حیث یظم درجات الطلاقة الري یسیة. العنصر یحقق مواصفات الحركة للجسم الصلب بش كل ت ام full gomty of ) كم ا یحق ق كام ل الخ واص للش كل الھندس ي للص فاي ح الحلقی ة (ct igid body mod) comptibility qution ) إضافة إلى ذلك فا ن العنصر یحقق شروط التواف ق لمع ادلات الا نفع ال (nnul plt.(of stin یمتلك ھذا العنصر خمسة عشر درج ة طلاقة حرة fdom) (dg of ثلاث ة ف ي ك ل عق دة ركنی ة وثلاث ة ف ي عق دة الوس ط یعتب ر ھ ذا العنص ر م لاءم للتحلی ل الا نحن اي ي الع ام للص فاي ح الحلقی ة وھ و أفض ل م ن العناص ر الس ابقة لتحلی ل المنشاءات الا نفة الذكر حیث أن الا خطاء التي تظھر في العناصر المحددة السابقة مثل أخط اء التقس یم ) discitiztion (o أخطاء الدوال الشكلیة o) (shp function تختفي في ھذا العنصر نتیج ة للمواص فات الت ي یتمت ع بھ ا. تم استخدام العنصر الحالي في التحلیل العددي لعدد من المساءل المختلفة إذ أنھ ا تب ین وتب رھن أن العنص ر الح الي ممت از وكفو في تحلیل عدة أنواع من التحمیل بش كل أفض ل م ن العناص ر المح ددة لب احثین آخ رین حی ث اظھ رت النت اي ج امكانی ة الاقتراب السریع الى نتاي ج الحل الدقیق solution) (ct وبدرج ة دق ة عالی ة وبا س تخدام ع دد قلی ل م ن العناص ر عن د تمثیل المنشاء. Nottion:, Ɵ Pol coodints D E M, M Ɵ M Ɵ, M Ɵ P q Q Q Ɵ w w w s Bnding igidity. Modulus of lsticity. Th bnding momnts in th dictions of nd Ɵ s, spctivly. Th twisting momnts. Point lod Th unifomly distibutd lod cting on th plt. Th out of pln shing stss sultnt in th -diction of gnl plt thoy Th out of pln shing stss sultnt in th Ɵ-diction of gnl plt thoy Out of pln displcmnt in Z-diction fo ctsin coodint, nd in noml diction fo pol coodints. Th middl sufc noml displcmnts of th nnul plt lmnt du to th igid body pt. Th middl sufc noml displcmnts of th nnul plt lmnt du to th stin pt. 5

3 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN 8-78 { } is th vcto of constnt tms of th displcmnt function { δ } [A] [B] [D] [f] Th tnsfomtion mti. Th stin mti. Th igidity mti. is th mti contining th coodint vibls [K ] Th stiffnss mti of th finit lmnt. P Th vcto contining th nodl lods cting on th finit lmnt. δ ε σ ϕ ϕ Ɵ Th vcto contining th dgs of fdom of th finit lmnt. Th vcto contining th stin (nd cuvtus) of th finit lmnt. Th vcto contining th stsss in th finit lmnt. Rottion in th -diction of gnl plt thoy Rottion in th Ɵ-diction of gnl plt thoy. Intoduction Fo th gnl bnding nlysis of nnul plts bsd on th clssicl thoy of thin plts. In gnl, th following th numicl mthods hv bn usd fo th nlysis of nnul plts [] :. Th finit diffnc mthod (FDM).. Th finit stip mthod (FSM).. Th finit lmnt mthod (FEM). Th pimntl wok bout this subjct w tmly limitd fo simpl css such s point lods nd simpl suppot []. Th finit lmnt mthod of stuctul nlysis is now fimly stblishd s powful tchniqu fo hndling diffnt poblms in solid mchnics []. Th simplst lmnt shps fo nnul plt poblms obviously tingl lmnt with th nods s in wok of Chung t l [4], nd ctngul lmnt with fou nods s in th wok of Ro [5]. Olson nd Lindbg [], dvlopd n nnul sgmnt plt bnding lmnt with fou con nods ch hving th tnl nodl dgs of fdom. Howv, using ths lmnts fo cuvd boundy poblms mns, tht th cuvd boundy is bing ppoimtd by sis of stight lin sgmnts. Hnc, th pps to nd to dvlop nw lmnt by using th pol coodints systm to gt btt psnttion of th cuvd boundis. Th psnt lmnt hv n dvntgous ov th vilbl nnul plt lmnts. Th finl poptis of th psnt lmnt s follows: Th lmnt stisfis th full gomty of nnul plt sgmnt, nd du to this point th discitiztion os tht pp in th cuvd boundy bcoms zo. Th lmnt stisfis th ct igid body mods of nnul plt sgmnt, nd du to this point th shp function o of igid body mod pt bcoms zo. 6

4 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN 8-78 Th stin mod of lmnt is obtind fom intgting of ssumd stin functions stisfying th comptibility qution of nnul plt sgmnt; du to this point th shp function o of stin mod pt bcoms vy smll. Th plicit intgtion is usd to div this lmnt; du to this point th o in numicl intgtion bcoms zo. Accoding to th cntl nod of lmnt th l vlu of stsss in th cnt of lmnt found, not ppoimt stsss du to th mn of fou con nods s in th oth vilbl nnul plt lmnts.. Divtion of Annul Plt Elmnt Using Stin Bsd Appoch. Thoticl Considtion Figu () shows n nnul plt scto. To idliz this nnul plt scto, n lmnt is chosn s shown in Figu (). Fo th gnl bnding nlysis of nnul plt scto und bity loding, th stins (dict stins nd chngs in cuvtu) of th middl sufc divd fom svl thois of nnul plt scto [] : Th ltl dflction (w) is function of () nd (Ө), thn th lplcin opto bcoms [] : w w w w. () Thn th dflction sufc of ltlly lodd plt tnsfoms is bcoms: w w w w q D.....() Du to lod is symmticlly distibutd with spct to th cnt of th plt, th ltl dflction (w) is indpndnt of () nd th bov qution bcoms [] : w w w q D. () Wh q is th pplid lod nd it is givn s function of () nd (), D is th flul igidity, nd quls Et D ( ν ) 7

5 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN 8-78 Th out of pln componnts of th stins cuvtus follows: w χ. (4) w w χ.... (4b) w χ w... (4c) wh: () is th dil coodint msud fom th p of th nnul plt scto, nd ( ) is th ngul coodint msud ound th cicumfnc. Th qution of dil, ngul nd twisting momnts, M, M Ѳ, nd M Ѳ spctivly bcoms: { } M D χ υχ.. (5) { υχ } M D χ... (5b) M ( ν ) D{ χ }... (5c) nd th vticl shing focs Q, nd Q Ѳ spctivly bcoms: Q D [ w]... (6) Q D [ w]... (6b) Th bov th componnts of stins cnnot b considd indpndnt s thy in tms of th ltl displcmnt (w) nd hnc, th stins must stisfy dditionl qutions clld th comptibility qutions. Ths qutions obtind by liminting th ltl displcmnt (w) fom qutions (4). Th finl sults of comptibility qutions s follows: ( χ ) χ 0. (7) ( χ ) ( χ ) χ 0.. (7b) 8

6 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN Displcmnt Functions: Th poposd lmnt is llowd to hv only th ssntil tnl dgs of fdom (w, φ, nd φ ) t ch nod. Pocding s with th usul stin-bsd ppoch, th fist mjo componnt of th displcmnt function is du to (stin-f) igid body mods of displcmnt nd cn b obtind by quting ll th componnts of stins [8], qutions (4), to zo nd intgting th sulting ptil diffntil qutions bcoms: w R cos sin... (8) In this qution w R th igid body componnt of th displcmnt fild w, nd is pssd in tms of th th indpndnt constnts (, ),. Th scond mjo componnt of th displcmnt function is du to stining of th lmnt. If th lmnt is to hv fiftn dgs of fdom (th t ch nod) thn th stins of th lmnt must b ssocitd with twlv dditionl constnts ( 4,, 5 ) of ( ) constnts Assuming stin polynomil functions χ w A 4 A5 A6 A7 A8... (9) χ w w A 9 A0 A A A... (9b) w w 4 A A... (9c) χ 5 Chcking th bov polynomils of stin fo comptibility qutions of nnul plt scto (7, nd 7b). Finlly, th ssumd stin functions of nnul plt scto lmnt which stisfy th quimnt of comptibility qutions th sm of qutions 9, nd 9b cpt qution 9c bcom: χ S 9 4 A A ( A A ) ( A A ) ( A A ) ( A A ) (0) Th bov th qutions 9, 9b, nd 0 intgtd in th sm pocdu tht ws usd to div th igid body mods of nnul plt scto lmnt, thn th finl polynomil function of stin mod bcoms: w S () 9

7 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN Th complt displcmnt fild functions fo th poposd lmnt is bcoms. w w w s sin cos w... () Th ottion φ, nd φ lso givn blow: sin cos φ w... () cos sin φ w... (4). Th Elmnt Stiffnss Mti Figu () shows tht th oigin of th dil coodint of th lmnt () is loctd t th p of th nnul plt scto. Consquntly, th oigin of th ngul coodint () is loctd t th cnt of th lmnt. Sinc, th clcultion of th stiffnss mti is cid out plicitly; this choic of th oigin will simplify th tsk of intgtion, thus: [ ] [ ] ] [ ] ][ [ ] [ A B dd D B A k T T β β... (5) wh [A], [B] nd [D] th tnsfomtion, stin nd igidity mtics of th lmnt, spctivly. Th (5X5) lmnt stiffnss mti [k ], cn now b clcultd using th displcmnt functions () nd th stin displcmnt ltionships (4). On th oth hnd, to kp th stog mmoy smll th stiffnss mti is condnstd fom (55) to () by moving th influnc of th cntl point (nod 5) to th fou con points (nods,,, nd 4) s follows: [ ]{ } { } n n n P K δ... (6) wh: [ ] [ ] [ ][ ] [ ] n K K K K K { } { } n δ δ { } { } [ ][ ] { } ( ) n P K K P P

8 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN Consistnt Lod Vcto Th tnl pplid nodl lods considd in th psnt finit lmnt nlysis clcultd by using consistnt lod vcto. Th consistnt lod vcto is obtind by quting th wok don by th nodl lods on th nodl displcmnts to th wok don by th tnl pplid lod on th ssumd displcmnt function of th lmnt. Th nnul plt lmnt shown in Figu () is considd. If th lmnt is subjctd to svl loding typs such s, distibutd noml pssu (q), concnttd lod (P i ), unifom distibutd momnt (M), nd concnttd momnt (M i ) th lod vcto bcoms [] : β β q { P 5} [ A] [ f ] dd [ A] [ f ] β m β t nod ( i) Pi dd Mi... (7) Fo th lmnt stiffnss mti ft condnstion, th lod vcto is tkn s follows: ( ) n { P } { P } [ K 5][ K ] { P }... (8) φ 4 φ 4 R Z φ w 4 φ 5 φ w5 φ 5 φ β β w w φ φ φ w Y Fig.() An nnul sgmnt lmnt with th coodints systm fo plt bnding poblms.

9 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN 8-78 M4 φ 4 w4 φ 4 M 4 R Z β β w φ M M φ w5 φ M 5 φ M φ 5 φ 5 M5 w φ M M φ w M Y Fig.() Displcmnt nd thi cosponding focs fo nnul plt lmnt.. An nnul plt und point lod Th poblm considd is tht of n nnul plt dfoming unsymmticlly s shown in Figu (). Th plt is clmpd long th inn dg ( b), nd lodd by concnttd lod t th out dg ( ). An ct solution fo this poblm is givn in Timoshnko nd Woinowsky-Kig [6] fo b/ /.5 nd μ0.. Olson nd Lindbg [], Tuki [7] nd Alkhfji nd Mhdi [8] psntd n nnul finit lmnt, to nlyz this poblm. Symmty bout th dimt contining th lod ws usd, so tht, only on hlf of th plt ws to b modld. Th sults fo th ltl displcmnt (w) und th lod fo diffnt msh sizs givn in Tbl (). Whn this poblm is nlyzd with (54) msh using both lmnts, th o is bout 0.45% fo Olson nd Lindbg [], Tuki [7], nd 0.% fo Al-khfji nd Mhdi [8], but th o is bout 0.% fo th psnt lmnt nd with (84) gid fo both lmnts th o is bout 0.7% fo Olson nd Lindbg [], Tuki [7], nd 0.089% fo Alkhfji nd Mhdi [8], but th o is bout 0.06% fo th psnt lmnt. It is cl in Figu () tht th ltl dflctions long th inn dg convg pidly to th ct solution whn th msh siz is find. Olson nd Lindbg [] confind th sults in thi publishd wok to dflctions only (no sults w givn fo th momnts). Figu () shows th distibution of dil bnding momnt in non-dimnsionl fom of (M /P) long th inn dg fo (4) msh. It is

10 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN 8-78 vidnt fom this figu tht th poposd lmnt givs stisfctoy sults fo momnts whn compd with th ct solution givn by Timoshnko nd Woinowsky-Kig [6]. Tbl () shows th convgnc o of fou lmnts, th sult pps tht convgnc of th psnt lmnt is stbl s whn lmnt msh is incsing th o of solution is dcsing continuously, but in th oth lmnts th convgnc of solution is unstbl with incsing msh siz s pping in Tbl (). In gnl, th two typs of convgnc shown in Figu (4). Th son of two typs of os concludd by th igid body mod nd stin mod os, th psnt lmnt stisfis th igid body nd stin mods s this lmnt s sult is stbl, th o is dcsing whn msh siz is incsing, but oth lmnts do not stisfy th bov conditions. Tbl.() Convgnc of th ltl dflction und th pplid point lod Msh siz Olson s nd Lindbg Elmnt Dflction * P/D Tuki s Elmnt Mhdi's Elmnt Psnt Elmnt Ect (Timoshnko nd Woinowsky-Kig 98) Tbl.() Convgnc o of svl mshs of th ltl dflction und th pplid point lod. Msh siz Dflction * P/D Olson s Elmnt Tuki s Elmnt Mhdi's Elmnt Psnt Elmnt 6 0.5% 0.0%.900% 0.% 8.0%.0%.500% 0.95% 0.6% 0.6%.060% 0.5% 44 0.% 0.% 0.40% 0.4% % 0.45% 0.00% 0.% % 0.57% 0.00% % % 0.66% 0.089% 0.06% 4 0.6% 0.54% 0.048% %

11 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN 8-78 C.L Msh (4) Ect Solution of (Ms) F.E.S. Solution of (Ms) b Unit Point Lod t 8.05 mm b mm t 7.6 mm µ 0. E 7.89Gp Rdil Momnt Rdil momnt (M) Angls (dgs) Fig.() An nnul plt und point lod in poblm. Fig.(4) Distibution of dil bnding momnt (Ms) long th inn dg in poblm. () Al-khfji nd Mhdi, nd Psnt lmnt bhvio. (b) Olson s nd Lindbg lmnt nd Tuki s lmnt bhvio. Fig.(5) Gnl typs of convgnc os of finit lmnt solution. 4. An nnul plt und unifomly distibutd lod As n isymmtic poblm, cicul plt with cntl cicul hol is nlyzd. Fou css considd s shown in Figu (6), nd in ch cs, on lmnt in th ngul diction is usd with n includd ngl of 4.5 o. Insting zo vlu th noml 4

12 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN 8-78 ottion φ t ll nods long both th dil dgs, thus stisfying th cicul symmty cition. Tbl () shows th mimum dflctions nd momnts which, obtind by using th poposd lmnt togth with thos obtind by th ct solution givn by Timoshnko nd Woinowsky-Kig (6) nd Swko,s lmnt [9]. Tbl (4) shows th convgnc of mimum dflctions nd momnts. Rfing to this tbl, it is concludd tht th diffnc in th sults is vy smll nd in most css tnds to b stbl, spcilly, fo dflctions. Gnlly, stisfctoy sults cn b obtind with lss thn 0.% diffnc by using 4 dgs of fdom fo dflctions nd 60 dgs of fdom fo momnts, in ll considd css. Th ccucy of this ppoimtion is illusttd in Figus (7) fo vious loding nd boundy conditions. This tbl lso shows th convgnc of th mimum dflction fo ll considd css. Cs q d Cs q Cs q 406.6mm b0.mm 4.5 dg µ0. t7.6mm E7.89 Gp Cs 4 q b Fig.(6) Unifomly distibutd lod nnul plts scto. 5

13 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN 8-78 Tbl.() Mimum dflctions of poblm in Figu (6) Cs No. W m *q 4 /D*0 - Ect Swko's Elmnt Tuki s Elmnt Psnt Elmnt Cs Cs Cs Cs Tbl.(4) Mimum momnts of poblm in Figu (6) Cs No. M m *q /D*0 - Ect Swko's Elmnt Tuki s Elmnt Psnt Elmnt Cs Cs Cs Cs Fig.(7) Msh study fo convgnc of fou css poblm in Figu (6) to ct solution fo (W m *q 4 /D*0 - ) 6

14 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN Unifomly lodd nnul plt scto with ll dgs clmpd. Th finl poblm considd is tht of unifomly lodd nnul plt scto with ll dgs clmpd. Th sktch of this nnul plt nd its poptis givn in th following figu. This poblm ws fist nlysd by Chung nd Chn [0], using th finit stip mthod. Hik [] dvlopd n nlyticl solution to psnt this poblm. Th mimum dflction nd th mimum positiv momnts (dil nd ngul momnts) long th cntl dil lin clcultd by th poposd lmnt fo (9*9) msh, nd fo diffnt vlus of th (b/) tio. Th sults givn in Tbls (5, B, And C), togth with thos of th oth mthods. Convgnc tsts w cid out using th poposd lmnt fo th dflctions nd momnts whn th nnul plt is dividd into mshs nging fom (44) to (66). Tbl (6) shows th convgnc of mimum dflctions nd mimum positiv momnts long th cntl dil lin nd whn th (b/) tio quls (0.75). Clmpd dgs d d8.6 mm 60 υ0. t7.6 mm E68.95Gp. b Fig.(8) Unifomly lodd nnul plt scto with ll dg clmpd nd its poptis. 7

15 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN 8-78 b/ Tbl.(5) Mimum dflctions of poblm in Figu (8) W m *q 4 /D*0-4 Chung nd Chn (9) Hik () Psnt Elmnt Tbl.(5b) Mimum momnts of poblm in Figu (8) b/ M m *q /D*0 - Chung nd Chn (9) Hik () Psnt Elmnt Tbl.(5c) Mimum momnts of poblm in Figu (8) b/ Mq m *q /D*0 - Chung nd Chn (9) Hik () Psnt Elmnt Tbl.(6) Convgnc of th mimum dflctions nd positiv momnts long th cntl dil lin in poblm in Figu (8) fo b/ msh W m *q 4 /D*0-4 M m *q /D*0 - Mq m *q /D*

16 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN Conclusion: Annul plt finit lmnt suitbl fo th gnl bnding nlysis of nnul plt scto hs bn dvlopd. Th lmnt is simpl nd contins only th ssntil dgs of fdom. Th lmnt hs th dvntg ov th oth vilbl nnul plt lmnt. Th impovmnt obtind is du to th fct tht ll th displcmnt filds of th psnt lmnt stisfy th ct psnttion of igid body mods of displcmnts. Also, th displcmnt filds du to stining of th lmnt bsd on indpndnt stins nd stisfy th ct comptibility qutions of stin mods. Th psnt lmnt is usd to nlyz svl typs of poblm. Th numicl sults of th psnt lmnt compd with th nlyticl, nd numicl sults of oth schs. Th sults of th psnt lmnt showd good nd pid convgnc of displcmnts nd stsss with th us fw lmnts. Th os of output sults is lss thn 0.5% of msh siz (6) nd lss thn 0.0% of msh siz (84) fo sttic nlysis of plt poblms. 7. Rfncs. Hik, I. E. "Anlyticl solution to othotopic sctos", J. Eng. Mch., ASCE, Vol. 0, No. 4, Apil 984, pp Coull, A. nd Ds, P. C., "Anlysis of cuvd bidg dsks", Poc. Inst. Civ. Eng., Vol. 7, My, 967, pp Olson, M. D. nd Lindbg, G. M., Annul nd cicul scto finit lmnt of bnding nlysis, Int. J. Mch. Sci., Vol., 970, pp Chung, Y. K., King, I. P., nd Zinkiwiz, O. C, "Slb Bidgs with bity shp nd suppot conditions: A gnl mthod of nlysis bsd on finit lmnts" Poc. Inst. Civ. Engs., Vol. 40, My, 968., pp Ro, S. S., "Th finit lmnt mthod in ngining", nd Ed., Pgmn Pss, Timoshnko, S. P. nd Kig, S. W., Thoy of plts nd shlls, copyight, 98, intntionl dition McGw-Hill book co, Nw Yok. 7. Tuki B., A finit lmnt modl fo bnding nlysis of othogonlly stiffnd nnul plt sctos, MSc. Thsis submittd to Dptmnt of Civil Engining, Collg of Engining, Al-Mustnsiiyh Univsity Al-Khfji, J. M., nd Mhdi, H. A., " Stin Bsd Finit Elmnt Modl fo Conicl Shlls", Intntionl jounl fo ngining nd tchnology, Indi, 0, pp Swko, F., nd Mimn, P. A., "An nnul sgmnt finit lmnt fo plt bnding", Int. J. Num. Mth. In Eng., Vol., 97, pp

17 Jounl of Engining nd Dvlopmnt, Vol. 9, No., Jnuy 05, ISSN Chung, M. S., nd Chn, M. Y. T., "Sttic nd dynmic nlysis of thin nd thick stuctul plts by th finit stip mthod", Comp. nd Stuc., Vol. 4, No., 988, pp El_Eis, H. F., Finit lmnt nlysis of shll stuctus, Ph.D Thsis, 989, submittd to Cdiff Univsity, U. K.. Szild, R., "Thoy nd nlysis of plt", Pintic-Hll, Inc. Englwood Cliffc, N. J., Zinkiwics, O. C., nd Tylo, R. L., "Finit lmnt mthod fo solid nd stuctul mchnics", Sith dition, 005, Elsvi Inc, Ofod, London, U.K. 0