The Use Of Mixed Eulerian-Lagrangian Displacement In Analysis Of Prestressed Concrete Bridge Girder

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1 IOSR ounl of Mehnl n Cvl Engneeng IOSR-MCE e-iss: 78-68,-ISS: 3-33X, Volume, Issue 6 Ve. II ov. - De. 7, PP The Use Of Mxe Eulen-Lgngn Dslemen In Anlyss Of Pesesse Conee Bge Ge Bns H. Hn Cvl Engneeng Demen, Bnung Insue of Tehnology, Bnung, Inones As: The e susses he use of mxe Eulen-Lgngn n he nlyss of os-ensone esesse onee ge ge. The hose of he enon s n s onon wh he suounng onee whle he enon n sl onon wh he hose. The sl eween he enon wh he hose s moele y mens of Eulen slemen. The ge onee s moele y usng elemens whle he enon s moele y usng elemens. The qu she of enon s moele y ee-wse lne elemens ngemen. If fne elemen mesh s onsue o follow qu she of enon, hen egul she of elemens my esuls. To vo hs, he noes on elemens e onnee o ge m noes wh g s. Boom ens of s e onnee wh noes. Sl eween enon n g s moele y Eulen slemen. The moele s le o he nlyss of os-ensone esesse onee ge ge. The esuls one e hen ome o nlyss esul usng onvenonl em meho, n he esuls gee well. Keywos:mxe Eulen-Lgngn slemen,esesse onee, os-ensone sysem, nenl n exenl os-enson enon De of Sumsson: De of ene: I. Inouon In os-ensone esesse onee sysem, he enon s feely sls elvely o hose, whle he hose s gly onnee o suounng onee. If onvenonl Lgngn slemen s use, he sl eween enon noe n onee noe my see he mel ons nly oue he sme noe. To ove ome hs olem se, Eulen slemen s use o moel sl eween enon n suounng hose. The one of mxe Eulen-Lgngn slemen s susse n vlle efeene [3]. The eson of efomon s effee o unefome onfguon [] The e els wh he use of mxe Eulen-Lgngn slemen n he nlyss of osensone esesse onee ge ge. The nlyss s e ou y mens of fne elemen meho n he nlyss s s n mx fom. The nlyss lgohm s hen s n omue ge ogm wen n Fon lnguge. The ogm s hen le o he nlyss of os-ensone esesse onee ge, n he esuls e ome o he esuls one y usng onvenonl em meho. II. Mxe Eulen-Lgngn Dslemen Moel Mxe Eulen-Lgngn slemen fel ws fs noue y Hn [3]. Mxe Eulen-Lggn Moel. The followng susson els wh fne elemen fomulon of sysem wh fne slemen. The eomoson of slemen s ese y showng slemen moel n Fgue. A yl mel on n he suul sysem me oues nl onfguon V loon X. Afe long, he mel on evenully oues uen onfguon v loonx A onfguon, whh my e oue y suul sysem ul me whn long oess s hosen s efeenl onfguon n enoe yv The slemen of yl mel on nlly oue loon X, s enoe y q n hs slemen eomose no Eulen slemen w n Lgngn slemenu, q ~ ~ ~ ~ x X ~ x ~ x ~ x X u~ w~ Defomon my e oseve y nseng elongon exeene y yl lne segmen S h me no s, suh h s S x ~ L x ~ followng Eule eson, n DOI: Pge

2 The Use Of Mxe Eulen-Lgngn Dslemen In Anlyss Of Pesesse Conee Bge ~ ~ s S X E X 3 ong o Lgnge eson. The eny E s Geen sn enso gven y z y x x E X X n L s Almns sn enso gven y X X L 5 x x x Fgue Mxe Eulen n Lgngn slemen n whh Ensen summon ule les,.e., summon s o e e ou fo eee nex. The Geen sn enso gves Pol-Khoff sess enso, whle Almns sn enso gves Cuhy sess enso Gnmhe, 975, Hn, 985, Mlven, 969. The slemen gen my e exesse n em of on x x x 6 X x X n whh Ĵ s Lgngn on n x x ; x X 7 s Eulen on gven y Sne he fomulons e nge n ems of efeenl meex hen l evves wh ese o X nee o e nsfome no l evves wh ese ox Fs, s wen h X m 8 x x X X whh, uon nveson gves m 9 X x n whhm s he elemen of nvee mx of he mx fome ym Fuhe, l evves wh ese oxmy e nvee o l evves wh ese o me oonesζ y wng x x n x whh, uon nveson gves DOI: Pge

3 The Use Of Mxe Eulen-Lgngn Dslemen In Anlyss Of Pesesse Conee Bge n x n whh n s he elemen of nvee mx of he mx fome yn. Theefoe, he followng my e one, m m n ; m n X x. Inemenon n Lnezon Tehnque I my e oseve fom he fom of Equons n 5 h he govenng equlum equon s qu n ems of slemen omonens. Theefoe, he olem woul e geomelly nonlne. The govenng equlum equon my e exne n ems of slemen omonens n he exesson my e oxme y enng lne ems. In hs se, suessve eon sheme s le. The followng s nemenon oess of ems. Fs, me he slemen s eomose n Eulen n Lgngn slemen q ~ ~ ~ ~ ~ ~ ~ ~ ~ Sn omonen s gven y E x X x x x X u w 3 Fo me + slemen s gven y q ~ q~ q~ q~ u~ w~ 5 n whh q s nemenl slemen onssng Lgngn nemenl slemen u n Eulen nemenl slemenw. Coesonngly, ol on omonens e nemene 6 n whh nemenl on omonens e gven y 7 m m m m whh my fuhe e wen n ems of Lgngn n Eulen nemenl ons. Sn omonens my lso e nemene y wng E E E E 8 whh esuls n E 9 Eqn. 9 my e wen n mx fom [] ~ ~ E B B ; B u~ ; 3 n whh u ~ s nemenl slemen veo onnng Eulen n Lgngn nemenl slemens. Theefoe, he followng elonsh s eslshe. E B u ~ ; B B B B 3 In he followng, equlum equon s wen n nemenl fom. Fs, me, he equlum onon es Q P K n whh K T E C E v, Q v 3 A me K P whh my e exne n he followng fom, DOI: Pge

4 The Use Of Mxe Eulen-Lgngn Dslemen In Anlyss Of Pesesse Conee Bge K Q Q P P 5 whh, n vew of equlum onon n Equon, oves lneze fom K Q P 6 whee K T B C B v 7 v In he fomulon of glol elemen sffness, he followng elonsh my e use, v x x x mx x x In he followng he, nemenl mx sffness of sevel yes of elemens, n hs se, fou noe some memne n elemens, e eveloe. III. Fne Elemen Fomulon n Comue Pogmmng The followng sussng els wh fne elemen fomulon h my e foun n sevel vlle efeenes []. 3. Fne Elemen Fomulon Due o he lmon on he se, only wo yes of elemens e eveloe,.e., fou noe some memne n elemens, onsee n un n he followng. 3.. Fou noe some memne A fou noe some memne elemen s ee n Fgue. Eh noe onns fou egees of feeom,.e., Eulen n Lgngn slemen omonens n x~ oone. Theefoe, he elemen hs 6 egees of feeom, nge n he fom ~ u u w u w u w u w u~ 9 u w u w u w u w n slemen n nol oones e neole y usng she funons, n whh u, x,, x, u 3,, 3 y η u3 3 x3;y3 v3 η x;y u v ξ -; ; ; -; ξ ; -;- ;- ;- u v x;y u x;y v Fgue Fou noe some memne elemen x Fs, Lgngn n Eulen on omonens e one y lyng Equon 7, n u ; u 3 DOI: Pge

5 The Use Of Mxe Eulen-Lgngn Dslemen In Anlyss Of Pesesse Conee Bge DOI: Pge The B mx hen my e wen n he fom B 33 B mx n he fom B 3 n 3 B mx n he fom 3 B 35 n whh m m m m fo noe. Theefoe, B mx my e onsue y nseng Equons 33, 3 n 35 n Equon. The esul es B 37 n whh fo noe. Fo fl lne memne, sess-sn elonsh s onolle y onsuve equon 36

6 The Use Of Mxe Eulen-Lgngn Dslemen In Anlyss Of Pesesse Conee Bge C E 39 The one mes my e nsee n Equon 7 o onsu elemen sffness mx. The elemen sffness mx s omue y usng Guss numel negon ehnque. 3.. B elemen B elemen s ee n Fgue 3. The elemen hs wo noes n eh noe hs wo egees of feeom nge n he fom ~ u u w u w ~ u u w u w L, EA Fgue 3 B elemen The slemen s foun y neolng nol slemen veo wh she funons, u u n whh,; x L n n L; u x L u, n on omonens my e omue n use o onsu elemen sffness mx. The esul s [ ] EA L 3 3. Comue Pogmmng Fne fomulon ese n evous susson s s n omue ge ogm wen n Fon lnguge. Inemenon ehnque s lso emee n he ogm. Consue ogm s hen vle o ove s uy n oeness, efoe le n se suy s suss n followe. IV Cse Suy Fo se suy, esesse onee I ge of ge s nlyze [5]. The I em hs sn of m, ve no ows n olumns of 3 menson some elemens eng shown n Fg.. Ge menson s x h = 55. m x 3. m. The esessng foe s,8.. The ge s nlyze y usng em heoy o omue sesses n slemens ue o own wegh, suemose, n lve lo. Fgue : Elemen Meshng of Pesesse Conee Ge DOI: Pge

7 The Use Of Mxe Eulen-Lgngn Dslemen In Anlyss Of Pesesse Conee Bge To omue sesses n slemens ue o esessng foe, unfom ulf lo w F = 8F e L = 66. gm s use. The esuls e use s omson o he esuls one y mens of omue ogm. The omson s shown n Tle. Tle : Comson Beween Mnul n Comue Ouu o. Quny Mnul Pogm Eo Peenge Dslemen n Z eon of m sn % Sess n ue fe % 3 Sess n m-hegh fe % Sess n lowe fe % By osevng he le, s onlue h he esuls ou of ogm gee well wh mnul esuls. The elemen meshng s use o nlyze he sme ge wh ol enon wh m sn eeny e = -. m n F =,8.. Fo enon, uue sess.7 f u =6,. MP of hgh enson enon s use.if elemen meshng s en o follow ol she of enon, hen s ene wh he egul she of elemens n hs my evenully use eo ue o she fo of elemen menon. To vo hs olem, le s un elow g s onneng m seon noes wh he le. The gy of he onneng s my e smule y seng lge nume of xl gy of s, o gvng he sme nume of slemens eween ue n lowe noes of he s. The le my sl feely eneh he s. The sl eween le n s moele y Eulen slemen n he slemens of g lowe en s moele y Lgnge slemens. See Fg. 5 s exlnon. The em s hen nlyze usng he omue ogm. The esuls e ome wh he esuls y mnul. The omson s e ou fo sesses n slemens m sn oss seon, s shown n Tle.The le lely shows h he esuls ou of omue ogm gee well wh mnul esuls. Tle : Comson Beween Comue n Mnul Resuls o. Quny Mnul Pogm Eo Peenge Dslemen n Z eon of m sn % Sess n ue fe % 3 Sess n m-hegh fe % Sess n lowe fe % Fgue 5: Tenon Wh Conneng Rg B o Bem Ceneo The ge nlyze evously usng oss seon menson x h = 55 m x 3 m, whh esuls n he use of nenl esessng enon. In he followng, he oss seon menson s hnge o I she wh he hegh m, n ue n lowe flnge hnesses 3 m. ge CTC s mn he los e DOI: Pge

8 The Use Of Mxe Eulen-Lgngn Dslemen In Anlyss Of Pesesse Conee Bge The momens m sn e M g = =5. m M s = =73.6 m M l = = 59. m M = 96, =76. m M =785.8 m q g =. 7 = 7.8 m q s =.7 = 88. m q l =.9 = 36. m P l = 9. = 96,. Bse on llowle omesson sess 35. gm hen y Mgnel sheme, he esessng foe s omue F =,8 n eeny e = 6. m. Sne he eeny flls ouse oss seon, exenl esessng enon s use. By mnul, he slemens m sn e omue s δ g = 5 q g L =.69 m; δ 38 EI s = 5 q sl = 7.8 m 38 EI δ l = 5 q ll = 9.77 m; δ 38 EI = PL3 = 3.55 m 8 EI so h ol slemen ue o los eomes δ = 5.8 m 8 ulf equvlen lo s w F = 8F e = 8.5 gm 9 L esulng n me δ F = 5 w FL =.39 m 5 38 EI so h he ol slemen m sn eomes δ T = 3.3 m 5 Fo he u of sesses, wo sges of long e onsee,.e., nsfe n seve sges. Fo nsfe sge, he sesses e f = F A + F e y + M g y = = +.36MP f = F A + F e y + M g y I = =.96 MP zz n whh he sess oh ue n lowe fes vole llowle sesses hs olem s my e oveome y sessng he enon when he ge s ey se le n suemose lo ley ve. So, M = M g + M s = m 53 The sesses eome f = F A + F e y + M y = = +.7 M f = F A + F e y + M y = = 3.7 MP 5 The ol sesses seve sges e f = F A + F e y + M y = =.97 gm f = F A + F e y + M y = = 78.3 gm 55 The omson s ee n Tle 3, whh emoses h he esuls ou of ogm gee well wh mnul esuls. Tel 3: Comson Beween Comue n Mnul Resuls o. Quny Mnul Pogm Eo Peenge Dslemen n Z eon of m sn % Sess n ue fe % 3 Sess n m-hegh fe % Sess n lowe fe % V Dsussons of he Resuls Sevel osevon ou of he omson of esuls e wn s follows. Fs, fne elemen moel s eslshe fo he nlyss of esesse onee ge ge. Cuve she of enon s DOI: Pge

9 The Use Of Mxe Eulen-Lgngn Dslemen In Anlyss Of Pesesse Conee Bge onnee o he egul engul she of elemen meshng wh g s. The gy of he s e eesene y gvng he sme nume eween lowe n ue slemens. The newly eveloe moel s le o wo ses,.e., nenl le se n exenl le ses. The esuls ou of he ge ogm gee well wh he esuls one y mnul omuon. The sl eween he enon n he lowe en of g s moele y mens of Eulen slemen. VI Conlusons Bse on he esuls one n hs e, sevel onlusons e wn s follows. Fs, newly eveloe fne elemen moel hs een eslshe fo he nlyss of os-ensone esesse onee ge ge.seonly, he moel s nooe n n lgohm nlyss of he ge n he lgohm s se n omue ge ogm wen n Fon lnguge. The moel my e le o oh nenl n exenl enon sysems. The esuls one y he lon of he ogm gee well wh he esuls one y he use of em heoy. Anowlegemens The e s wen ly se on he hess of M. Fl Syfz, e ou Cvl Engneeng Demen, Bnung Insue of Tehnology, n whh he uho ws he eml vso. Tyng n wng of he e wee e ou y M. Donn Cn, o whh he uho exen hs ol n snee eon. Refeenes [] Agys,.M., n Kelsey, S., Enegy Theoems n Suul Anlyss, Buewoh, Lonon 96. [] Coo, R. D., Mlus, D.S., Plesh, M.E., Cone n Alon of Fne Elemen, es eg, ohn Wley & Sons, In., ew Yo 989. [3] Hn, B., Ave Fne Elemen Anlyss onlne Fonl Con wh Mxe Eulen-Lgngn Coones, Ph.D sseon, Unvesy of Illnos Un-Chmgn 985. [] Mlven, L.E, Inouon o he Mehns of Connuous Meum, Pene-Hll, In., Englewoo Clffs, ew esey 977 [5] Syfz, F., Penen Penhn Eule n Lgge Dlm Anlss Gelg Beon Pegng emn, mse hess, Cvl Engneeng Demen, Bnung Insue of Tehnology, Bnung, Inones 7 Bns H. Hn The Use Of Mxe Eulen-Lgngn Dslemen In Anlyss Of Pesesse Conee Bge Ge. IOSR ounl of Mehnl n Cvl Engneeng IOSR-MCE, vol., no. 6, 7, DOI: Pge