Effect of in-plane forces on frequency parameters

Transcription

1 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June I 5-5 Effec of in-lane foces on fequenc aamees A.K.L. ivasava,.r.pande * Deamen of Civil Engineeing,. I. T. Jamshedu- 84, India Absac- Vibaion and buckling chaaceisics of siffened laes subjeced o in-lane unifom and non-unifom edge loading a he lae boundaies ae invesigaed ug he finie elemen mehod. Recangula siffened laes ossesg diffeen bounda condiions, asec aios, vaing mass and siffness oeies and vaing numbe of siffenes have been analzed fo buckling and vibaion sudies. The chaaceisic equaions fo he naual fequencies, buckling loads and hei coesonding mode shaes ae obained fom he equaion of moion. The effecs of he osiion of siffenes and numbe of siffenes, asec aios, bounda condiions, siffenes aamees uon he buckling load aamee and fundamenal fequenc of he siffened laes ae invesigaed. The esuls ae obained consideing he bending dislacemens of he lae and he siffene. Eccenici of he siffenes gives ise o aial and bending dislacemen in he middle lane of he lae. Comaison wih ublished esuls indicaes good ageemen. In he sucue modelling, he lae and he siffenes ae eaed as seaae elemens whee he comaibili beween hese wo es of elemens is mainained. Inde Tems- Finie elemen mehod, iffened lae, uckling and fequenc aamee, iffenes aamees oaions a - Plae dimension in longiudinal diecion b - Plae dimension in he ansvese diecion - Plae hickness E, G - Young s and shea moduli fo he lae maeial b s, d s - web hickness and deh of a -siffene, - on dimensional elemen coodinae A - Coss secional aea of he siffene I - Momen of ineia of he siffene s-secion abou efeence ais {q} - Veco of nodal dislacemen a h node [D P ] - Rigidi mai of lae [D ] - Rigidi mai of siffene [K e ] - Elasic siffness mai of lae [K ] - Elasic siffness mai of siffene M, M - Consisen mass mai of lae, siffene [] - Mai of a shae funcion of a node P c - Ciical buckling load P () - In lane load T - Tosional consan P - Pola momen of ineia of he siffene elemen T I. ITRODUCTIO he dnamic behaviou of siffened laes has been he subjec of inensive sud fo man eas. Fo aeodnamic consideaions, siffene will be ovided inside he hull of aicaf and fo sace sucues he ge can be ovided ouside if ha is moe sucuall efficien. iffened laes ae sucual comonens consig of laes einfoced b a ssem of ibs o enhance hei load caing caaciies. These sucues ae widel used in aicaf, shi, bidge, building, and some ohe engineeing aciviies. In man cicumsances hese sucues ae found o be eosed o in-lane loading. The buckling and vibaion chaaceisics of siffened laes subjeced o unifom and non-unifom in-lane edge loading ae of consideable imoance o aeosace, naval, mechanical and sucual enginees. Aicaf wing skin anels, which ae made of hin shees ae usuall subjeced o non-unifom in-lane sesses caused b concenaed o aial edge loading a he edges, and due o anel siffene suo condiions Diez e al. [] sudied he effec of combined nomal and shea in-lane loads b he Galekin mehod. Tansvese vibaion of ecangula laes subjeced o in-lane foces unde vaious combinaions has been sudied b ingh and De [] b a diffeence based vaiaional aoach. The vibaion and buckling of aiall loaded siml suoed laes wee sudied b Deolasi e al. []. Recenl undesan e al. [4] have sudied he influence of aial edge comession on buckling behaviou of angle l laes fo a few oienaions. A bief lieaue suve eveals ha a vaie of mehods have been oosed o sud he vibaion of siffened laes. The mos common mehod used in eal lieaue was o aoimae he siffened lae as equivalen ohooic laes, ug he smeaed siffene aoach. In moe ecen lieaue, wih he hel of high-seed comues, he lae and siffenes wee eaed seaael. uch numeical mehods as he finie elemen mehod and finie diffeence mehod ae widel used. Aksu [5] has esened a vaiaional incile in conjuncion wih he finie diffeence mehod fo analsis of fee vibaion of uni-diecionall and s-siffened laes consideing in-lane ineia and in lane dislacemens in boh diecions. [K G ] - Geomeic siffness mai

2 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June I 5-5 has and Rao [6] have used he noded confoming elemen and efined beam-bending elemen fo abia oiened siffenes. Olson and Hazell [7] have esened a ciical sud on clamed inegall siffened lae b he finie elemen mehod. The mode shaes and fequencies have been deemined eeimenall ug he eal ime hologahic echnique. The effec of change in ib siffness on vaious modes has been sudied. au Rao e al. [8] have also eoed hei wok on eeimenal deeminaion of fequencies wih eal ime hologahic echnique fo skew siffened canileve laes. Mukhoadha [9] has alied he semi-analic finie diffeence mehod o he sabili analsis of ecangula siffened laes based on he lae beam idealizaion. heikh and Mukhoadha [] alied he sline finie si mehod o he fee vibaion analsis of siffened laes of abia shaes. The analzed he lae of ecangula, skew and annula shaes wih concenic as well as eccenic siffenes. Lage amliude, fee fleual vibaion of siffened laes has been invesigaed b he sline finie si mehod b heikh and Mukhoadha []. The siffene has been eleganl modelled so ha i can be laced anwhee wihin he lae si. Haik and Guo [] have develoed a comound finie elemen model o invesigae he siffened laes in fee vibaion whee he have eaed he beam and lae elemen as he inegal a of a comound secion, and no as indeenden bending comonens. edai [] has sudied he fee vibaion chaaceisics of siffened laes due o laesiffene ooions. He has consideed he lae and he siffene as he discee elemens igidl conneced a hei juncions and he nonlinea sain eneg funcion of he assembled sucue has been ansfomed ino an unconsained oimizaion oblem o which equenial quadaic ogamming has been alied o deemine he magniudes of he lowes naual fequenc and he associaed mode shae Allman [4] has caied ou he analsis of buckling loads of squae and ecangula siffened laes ug iangula elemen. He has esened he esuls boh b including and neglecing he osional siffness of he siffenes. Vibaion of siffened laes wih elasicall esained edges has been analzed b Wu and Liu [5] ug Raleigh-Riz mehod. The fis fou lowe fequencies fo esained laes u o si inemediae siffenes ae calculaed. Mukhoadha [6] has eended he saic and vibaion analsis of laes o analse he sabili of shi laing and allied laed sucues ug he semi-analic mehod. An isoaameic siffened lae bending elemen fo he buckling analsis of siffened lae has been esened b Mukhejee and Mukhoadha [7]. Hee he siffene can be osiioned anwhee wihin he lae elemen and need no necessail be laced on he nodal lines.the geneal sline finie si mehod has been eended b heikh and Mukhoadha [8] o analse siffened lae of abia shae. iffened laes having vaious shaes, bounda condiions and also ossesg vaious disosiions of siffenes have been analzed b he oosed mehod. The sabili of aiall siffened, siml suoed and clamed squae laes is sudied b Ro e al. [9]. A high ecision iangula finie elemen and a comaible siffene elemen ae used in he finie elemen analsis. uckling of siffened laes has been sudied b edai []. An invesigaion on siffened laes has been conduced o deemine he elasic aamees as well as he s-secional dimensions of ecangula siffenes fom eeimenal modal daa and finie elemen edicion, ug model-udaing echnique b Chakabo and Mukhoadha []. A diffeenial quadaue analsis fo he fee vibaion of eccenicall siffened laes is sudied b Zeng and e []. The lae and he siffenes ae seaaed a he ineface wih equilibium and coninui condiion saisfied. Vibaion and dnamic sabili of siffened laes subjeced o in-lane unifom hamonic edge loading is sudied ug finie elemen analsis b ivasava e al. [] consideing and neglecing inlane dislacemens. Fuhe ivasava e al [4] eended hei wok o sud he incial dnamic insabili behaviou of siffened laes subjeced o non-unifom hamonic in-lane edge loading. Vaious mehods used fo vibaion analsis such as Riz echnique, Lev s soluion, finie diffeence mehod, finie elemen mehod, Galekin mehod, diffeenial quadaue mehod and mehod ug bounda chaaceisics ohogonal olnomials (COP) have been eviewed eensivel b Leissa [5]. The effec of he ga beween he siffene i and he suoing edge on he naual fequencies has been invesigaed b ai and Rao [6]. The anel is eesened b iangula lae bending elemens and he siffene b beam elemens. The alied load is seldom unifom and he bounda condiions ma be comleel abia in acice. The oblem becomes comlicaed when he numbes of siffenes ae inceag egadless of he osiion of siffenes no necessa along he nodal lines. Loading is non-unifoml disibued ove he edges and along he siffenes hus affecing he bounda condiions. Analsis of siffened lae is caied ou nomall b eneg mehod b adding enegies due o lae and siffene. The eneg soed in he siffene will deend on is s secion and if a hin walled oen secion, he effec of wig as well as waing have o be included. These sudies fo mos a being concened wih he numeical analsis of he heoeical buckling load and also mosl elaed o unsiffened laes. The esen ae deals wih he oblem of vibaion and buckling of ecangula siffened laes subjeced o in-lane unifom and non-unifom edge loading. Finie elemen fomulaion is alied fo obaining he non-unifom sess disibuion in he lae and also o solve he buckling load and fequenc aamees in vaious modes wih diffeen bounda condiions, asec aios and vaious aamees of siffened laes. The analsis esened deemines he sesses all ove he egion. In he esen analsis, he lae is modeled wih he nine nodded isoeimeic quadaic elemen whee he conibuions of bending and membane acions ae aken ino accoun. Thus he analsis can be caied ou fo boh hin and hick laes. Moeove i can be alied o a sucue having iegula boundaies. The fomulaion of he siffene is done in such a manne so ha i ma lie anwhee wihin a lae elemen. In ode o mainain comaibili beween lae and siffene, he ineolaion funcions used fo he lae ae used fo he siffenes also.

3 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June I 5-5 II. MATHEMATICAL FORMULATIO The govening equaions fo he buckling and vibaion of siffened laes subjeced o in-lane hamonic edge loading ae develoed. The esence of unifom and non-unifom eenal in-lane loads, bounda condiions, siffenes locaions and cuous if an in he lae induce a non-unifom sess field in he sucues. This necessiaes he deeminaion of he sess field as a eequisie o he soluion of he oblems like vibaion, buckling and vibaion behaviou of siffened laes. As he hickness of he sucue is elaivel smalle, he deeminaion of sess field educes o he soluion of a lane sess oblem in he lae skin and siffenes (whee he hickness and beah ae small comaed o lengh). The siffened laes ae modeled and he govening equaions ae solved b finie elemen mehod. In he esen analsis, he lae is modelled wih nine noded isoaameic quadaic elemens whee he conibuions of bending and membane acions ae aken ino accoun. One of he advanages of he elemen is ha i includes he effec of shea defomaion and oa ineia in is fomulaion. Thus he analsis can be caied ou fo boh hin and hick laes. Moeove i can be alied o a sucue having iegula boundaies. Also i can handle an osiion of cuou, diffeen bounda and osiion of in-lane concenaed loads o loading condiions. In ode o mainain comaibili beween lae and siffene, he ineelaion funcions used fo he lae ae used fo he siffenes also. umeical mehods like finie elemen mehod (FEM) ae efeed fo oblems involving comle in lane loading and bounda condiions as analical mehods ae no easil adaable. The fomulaion is based on Mindlin's lae heo, which will allow fo he incooaion of shea defomaion. The lae skin and he siffenescomosie ae modelled as seaae elemens bu he comaibili beween hem is mainained. The nine noded isoaameic quadaic elemens wih five degees of feedom (u, v, w, X and ) e node have been emloed in he esen analsis. The in-lane dislacemens u and v need o be consideed onl when he siffenes ae conneced eccenicall o he lae. If he lae and siffenes ae conneced concenicall, no in lane sesses develo The effec of in-lane defomaions is aken ino accoun in addiion o he defomaions due o bending, which will hel o model he siffene eccenici convenienl. The elemen maices of he siffened lae elemen consis of he conibuion of he lae and ha of he siffene. in he owe of he hickness co-odinae as: The elici evaluaion of inegals involved in he evaluaion of elemen siffness and mass maices of he lae is edious and as such is no aemed. A Gaussian inegaion echnique has been adoed fo his uose fo is high accuac and also i can be imlemened easil. A eac inegaion needs an ode of. Howeve, a educed inegaion oves o be moe effecive and cheae. To inegae he elemen maices a Gaussian inegaion has been adoed, howeve, he ode of inegaion has been menioned. To inegae he elemen maices a Gaussian inegaion has been adoed, howeve, he ode of inegaion has been menioned. The dislacemen a an oin wihin he elemen can be eessed as: u v w 9 = I 5 u v w () ain dislacemen elaion can be wien as: q q ()

4 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June 4 I and P () The genealized sess-sain elaionshi fo a lae elemen is D (4) whee he sess esulan veco is M M M T (5) Ug he isoaameic coodinaes, he elemen siffness mai is eessed as: d d J D K T b (6) The elemen mass mai can be eessed in iso aameic coodinae as: d d J m M T e (7) Geomeic siffness mai eessed in isoaameic coodinaes as: d d J K G P T G P G (8)

5 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June 5 I P (9) Geomeic siffness of siffene The sain mai can be eessed as 9 q q G G G () and G () whee X A () whee A is he aea, F he fis momen of aea abou efeence lane, he second momen of aea abou efeence lane, T he osional consan and P he ola momen of aea of he siffene s-secion. The eession fo he geomeic siffness mai can be fomed b equaing he inenal wok done b he sesses o he eenal wok done b he nodal foces. The geomeic siffness of he siffene elemen can be eessed in iso-aameic co- odinae as: d J K G T G G () The deivaives of and wih esec o ae given b (4) The comonens of genealized sain ve ae obained as follows: u u u (5) (6)

6 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June 6 I (7) We can subsiue he values and finall we ge as: ) ) ( ( ) ( ) ( ) ( ) ( w w v v u u (8) The genealized sain comonens in he siffenes in and coodinaes ae given b T (9) Whee [T] is he ansfomaion mai and is given b T () ) ),( (,,,, ),,(, w w v u v u () 9 () whee 9 ()

7 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June 7 I and (4). ' GA GT E EF EF EA D (5) (6) The equivalen nodal foces ae given b d d J o P F T e (7) The inensi of loading wihin he ach is assumed o be unifom. In such siuaions i becomes necessa o obain he equivalen nodal foces when a concenaed load is acing wihin he elemen. Again, he equivalen nodal foces ae eessed as: J P P T o (8). Govening Equaions The govening equaions fo secified oblems like vibaion, saic and dnamic sabili ae as:. Fee vibaion: q M K b (9)

8 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June 8 I 5-5 Vibaion wihou in-lane load: Vibaion wih in-lane load: O aic sabili o buckling M q K q b M q K b PK G q K P K M q b c K PK q G () () () b G () whee K b, K G, M ae oveall elasic siffness, geomeic siffness, and mass maices esecivel, K, K, q is he dislacemen veco. To evaluae he oveall elasic siffness, geomeic siffness, and mass maices b G M esecivel, i is necessa o use he same shae funcions fo boh lae and siffene elemens The elemen maices fo he lae and siffene ae geneaed seaael and hen added u o fom oveall maices. The equaions ae solved ug he echnique oosed b Co and Jennings [8 ] whee he maices [K], [M] and [KG] ae soed in gle aa accoding o skline soage algoihm. In all he cases, he siffness mai [K] is facoized accoding o Cholesk s decomosiion echnique. Wih his, he soluion fo dislacemen is siml obained b is fowad eliminaion and backwad subsiuion echniques. These dislacemens comonens ae used o find ou he sess field. These sesses ae used o calculae he geomeic siffness maices. The soluions of equaions go hough a numbe of oeaions. Moeove i equies a numbe of ieaions o ge he soluion ce hese equaions come unde he caego of eigenvalue oblem. In such cases, he soluion of eigen veco and eigen value is moe han one whee he diffeen soluions coesond o diffeen modes of vibaion o diffeen modes of buckling. The mode which gives lowes value of he eigen value is quie imoan and i is known as fundamenal mode.. on-dimensionalisaion of Paamees Majoi of he model aamees and esuls ae esened in non-dimensional fom o make hem indeenden of he lae size, hickness, maeial oeies, ec fo he convenience of he analsis. The non-dimensionalisaion of diffeen aamees like vibaion, buckling and eciaion fequenc fo dnamic sabili analsis is aken as given below: Fequencies of vibaion ( ) b D uckling load ( ) () Disibued load b X D Whee D is he lae fleual igidi, D = E ( ), P is he alied load, c () Concenaed load b D P c P is he buckling load, is he densi of he lae maeial and is he lae hickness. In addiion, ceain quaniies ae eessed as he aio of ha quani o some efeence quani. Assuming a geneal case of seveal longiudinal ibs and denoing EI he fleual igidi of a siffene a a disance (D ) fom he edge =, he siffene aamee ems and ae defined as: b aea of he siffene. A = Raio of s-secional aea of he siffene o he lae, whee E I bd = Raio of bending siffness igidi of siffene o he lae, whee he siffene s-secion abou efeence ais. A is he I is he momen of ineia of

9 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June 9 I Poblem Idenificaion The basic configuaion of he oblem consideed hee is a unsiffened and siffened lae subjeced o vaious unifom and nonunifom edge loadings as shown in figues (a-d). The s-secion of he siffened lae is shown in figue 4. fo ecangula ssecion and figue 4. fo kew ecangula lae s secion unde uniaiall loading. b c a (a) Uni-aial loading (b) i-aial loading (c) In-lane shea loading (d) In-lane hea loading Figue (a-d): Plae subjeced o inlane unifom edge loading The oblem consideed hee consiss of a ecangula lae (a b) wih siffene subjeced o vaious es of loading. In Figues (a-d), he laes ae subjeced o unifoml disibued in lane uniaial, biaial, shea edge loading. The lengh (a) of he siffened lae consideed above is vaied keeing is ohe aamees unchanged. The s-secion of he siffene is shown in figue and. Recangula skew laes unde geneal unifoml disibued in-lane edge loading ae consideed also as shown in figue 4. o sud he vibaion and dnamic sabili behaviou of skew siffened laes. A A A b in a b d Figue : iffened lae s secion Figue : kew ecangula lae unde uniaiall loading

10 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June I 5-5 A aameic sud is caied ou hee fo he laes o esen some new esuls in he esen sudies.on he oic ug he esen finie elemen aoach. Diffeen kind of loading cases as wien below ae alied fo buckling, vibaion of siffened laes. III. REULT AD DICUIO. Convegence and validaion sudies wih evious esuls In a finie elemen analsis, i is desied o have he convegence sudies o esimae he ode of mesh size o be necessa fo he numeical soluion. The oblem of isooic ecangula. In-lane Uniaial comession. In-lane biaial Comession. In-lane shea load 4. iaial comession and in-lane shea unsiffened laes wih unifom loading is invesigaed in able fo buckling and vibaion fo ab =,.5,,.5 and validaed wih available esuls of Leissa [5]. As he convegence sud shows ha a mesh size of is sufficien enough o ge a easonable ode of accuac. The analsis in he subsequen oblems is caied ou wih his mesh size.. Table : uckling and Vibaion sudies of ecangula unsiffened lae ab.5.5 ounda Condiion uckling Paamee on dimensional fequenc aamee Refeence Mode o 4.99 Pesen Leissa [ 5] Pesen Leissa [5] C 4.84 Pesen Pesen Pesen C.7 Pesen Pesen Leissa[5] Pesen Leissa [5 ] C 4.6 Pesen A squae lae clamed in all edges having a cenall laced concenic siffene as esened b ai & Rao [6] ug a ackage sif, Mukhajee [7], Mukhoadha [6], and eikh [8] ug FEM, semi analical mehod, and sline finie si mehod esecivel has been analzed esenl in able. eikh [8] has given esuls neglecing and including mass momen of ineia which has been validaed in esen esuls maked as Pesen () fo M.I. eglecing and Pesen () as mass momen of ineia including. The fis si fequencies ae comaed. The ageemen is ecellen. In Mukhoadha [6] in-lane dislacemen is no consideed in he analsis so esuls cause slighl vaing. Table also esen convegence sud showing good convegence of esuls. Plae size = 6mm 6 mm Plae hickness =. mm, mm Poisson s aio =.4 Mass densi =.78e-6 Kg 7 E = 6.87 mm, As = 67. mm, Is = 9 mm 4 Js =. mm 4 Table : Fequenc in (ad s) of clamed siffened lae wih a concenic iffene ouce Pesen Pesen Mode o

11 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June I 5-5 ai and Rao [6] Mukhajee [7] Mukhoadha [6] () eikh [8] () eikh [8] The Effec of he siffene fo he same lae bu wih a siffene of mm b mm size has been solved and esened in able. Dimension of he siffened lae is shown in figue 4. eikh [8] solved his oblem b sline finie si mehod and finie elemen mehod esecivel. The esuls ae comaed and ae found o agee well. ai also solved lacing he siffene a vaious ecceniciies. mm 6 mm A AAA A mm mm 6mm Figue 4: Eccenicall clamed squae siffened lae Table : Convegence of he fequenc wih Mesh Division Mesh Division heikh Mode [8] Mode Pesen Mode Mode uckling of longiudinall siffened lae unde uniaial load.. Validaion udies fo uckling udies of a Cenal concenic siffened lae The esen fomulaion is validaed fo buckling analsis of ecangula siffened lae. The buckling load aamees have been obained fo vaious asec aios, bending siffness igidi and siffene aea aios fo one cenal longiudinal siffene in able 4. The lae hickness aio (ah) and isooic lae and siffene maeial (ν) ae aken as and. esecivel. The dimension a is vaied keeing b as consan o ge diffeen values of asec aio (ab). The lae is subjeced o unifom comession in he -diecion. The mesh division chosen fo he whole lae is. A good ageemen is obseved wih he esul obained b Timoshenko and Gee [7]. In able 4 A is fo Timoshenko and Gee [7]. Table 4: Comaison of buckling load aamees of a ecangula siffened lae wih one longiudinal cenal siffene subjeced o unifom nomal edge loading along - diecion. = 5 = ab =.5 =. =.5 =. Pesen A [7] Pesen A [7] Pesen A [7] Pesen A [7]

12 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June I Afe he validaion of fee vibaion analsis and buckling analsis of siffened laes, deailed sudies has been done fo vaiaion of buckling and vibaion load aamees of siffened laes fo diffeen asec aios, vaious bounda condiions wih vaing load PP c wih diffeen numbe of siffenes. The vaiaion of buckling and fequenc aamees wih PP c fo ecangula eccenic siffened lae wih,, longiudinal equisaced siffenes subjeced o in-lane unifom uni-aial fo vaious asec aios (ab =,.5, ) and bounda condiions (,, C) ae sudied in deails and esened in able 5-8 fo diffeen bounda condiions and asec aios fo he sake of geing vaious inelinking esuls of numbe of siffenes and asec aios. Hee n is numbe of siffenes. I is obseved fom he above sudies ha fequenc aamees a an value of PP c will be moe fo squae siffened lae. These values decease wih he incease of asec aios. imilal i is also concluded ha buckling loads aamees will decease, as he asec aios will incease. I ma be concluded fom above obsevaion ha buckling load aamee and fequenc aamee will incease wih incease of numbe of siffenes. The value will also incease wih he incease of degee of esains. Thus he buckling and fequenc aamee of is moe han and C. In he same manne he buckling and fequenc aamee of C is moe han. Table 5: Vaiaion of fequenc aamee and buckling load aamee ( ) in he case of diffeen bounda condiion and vaious asec aios fo one longiudinal siffene a he cene having ( =. and = 5). ab ounda Condiion Fequenc aamee C C C Table 6: Vaiaion of fequenc aamee and buckling load aamee ( ) in he case of diffeen bounda condiion and asec aios fo wo longiudinal siffene having ( =. and = 5) ab ounda Condiion Fequenc aamee C C

13 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June I 5-5 C Table 7: Vaiaion of fequenc aamee and buckling load aamee ( ) in he case of diffeen bounda condiion and asec aios fo hee longiudinal siffene having ( =. and = 5). ab oundac ondiion Fequenc aamee C C C Table 8: Vaiaion of fequenc aamee and buckling load aamee ( ) fo siffened lae (Equisaced, one siffene, wo siffene, hee siffene) wih diffeen asec aio and bounda condiion having ( =., = 5) o of iffene ab ounda Condiion C C C Unsiffenedsiffened laes unde in-lane biaial load.. Convegence and validaion sud The accuac of he oosed mehod fo unsiffened laes unde in-lane biaial load ae fis esablished b comaing he esuls of vaious oblems wih hose of ealie invesigaos available in he lieaue. The analsis has been done fo buckling load faco fo squae laes having vaious bounda condiions (,, CC) unde bi-aial load consideing comessive in-lane load and validaed in able 9. To sud he asec of convegence, a siml suoed squae lae subjeced o in-lane comessive load in boh diecions has been undeaken in able. In his case, he esul is comaed wih finie diffeence soluion b ingh and De []. The esul is obseved o convege saisfacoil a 8 8 and mesh divisions.

14 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June 4 I 5-5 Fuhe fequenc aamee ( ) fo a siml suoed squae lae subjeced o bi-aial load. ( = = and = ), =., PP c =.5 is analzed and validaed in able 9 wih he esul of ingh and De [] and analical soluion of Diez []. Table 9: uckling load faco fo squae lae unde biaial load ounda Condiions CC on-dimensional buckling load Pesen Ref [] Mode Mode Mode Mode Table : Convegence and validaion sud of fequenc aamee fo a siml suoed squae lae subjeced o biaial load. ( = =, = ), =., PP c =.5 Mode o. 4 Fequenc Paamee Pesen ingh and De [] Analical [] Afe convegence and validaion sud fo buckling and fee vibaion of unsiffened laes in well condiion fo bi-aial edge loading, he analsis is now eended o unsiffenedsiffened laes subjeced o bi-aial load fo buckling and vibaion analsis..4 Vibaion and buckling sudies The vaiaion of fequenc aamee wih in-lane load inensi faco of a lae of vaious asec aio and edge condiion comessed unifoml on all edges ( = = & = ) is sudied in his secion. The fundamenal fequenc aamee fo lae having diffeen bounda condiions (,, C ) ae sudied in able. The end of he esuls fo a siml suoed lae shown in able is almos simila fo laes and C, ece fo is gadien of ise. Fo lae, hee is a see incease in fequencies wih inceag asec aios. The naual fequencies ae found o incease wih deceased magniude of comessive in-lane foces. Table : Vaiaion of fequenc aamee wih in-lane load inensi faco of a lae of vaious asec aio and edge condiion comessed unifoml on all edges ( = = & = ), =.. ab ounda Condiion C uckling Paamee Fequenc aamee a PP c C C

15 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June 5 I 5-5 ow he sud is eended o siffened laes subjeced o biaial in-lane edge loading. The vaiaion of buckling and fequenc aamees wih PP c fo ecangula siffened lae wih,, longiudinal equisaced siffenes (iffene aamees =. and = 5) subjeced o in-lane unifom biaial fo vaious asec aios (ab=,.5, ) and bounda condiions (,, C) ae sudied in deail and shown in able -4 fo he sake of geing vaious inelinking esuls of numbe of siffenes and asec aios. Table : Vaiaion of fequenc aamee wih in-lane load inensi faco of siffened Plae wih one cenal siffene having ( =. and, = 5) comessed unifoml on all edges ( ),. ab ounda Condiion Fequenc aamee a Load inensi faco PP c C C C Table : Vaiaion of fequenc aamee wih in-lane load inensi faco of siffened lae wih wo equisaced siffene having ( =. and, = 5) comessed unifoml on all edges ( ),. ab ounda Condiion Fequenc aamee a Load inensi faco PP c C C Table 4: Vaiaion of fequenc aamee wih in-lane load inensi faco of siffened lae wih hee equisaced siffenes having ( =. and, = 5) comessed unifoml on all edges. ( ) ab ounda Condiion Fequenc aamee A Load inensi faco PP c C

16 on dimensional fequenc aamee Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June 6 I 5-5 C The analsis is also esened below in figue 5 fo siml suoed siffened lae which has been above fo,, siffenes o show as a combined effecs and oveview iml suoed siffened lae Asec aio= = = =.&=5 siffene siffene siffene PP c Figue 5: Vaiaion of on-dimensional fequenc aamee vs PP c fo siffened lae subjeced b unifom comession on all edges..5 Unsiffened and siffened laes uunde shea edge loading.5. Validaion sudies wih unsiffened lae The vaiaion of fequenc aamee wih vaious in-lane load inensi (PP c ) of unsiffened squae lae having bounda condiions (,, CC) subjeced o inlane shea load a all edges have been analzed in vaious modes and he esuls ae obained in able 5 and validaed wih he esul of ingh and De []. Table 5: uckling load fa fo squae lae unde shea load. ounda Condiion CC uckling Load Paamee Pesen Ref [] Mode Mode Mode Mode uckling and Vibaion udies of siffened aes subjeced o hea Load Afe validaing he esuls fo buckling of unsiffened laes unde shea, i is eended fo fuhe sud fo siffened laes in ode o ge deails insigh and wih he view o ge some new numeical esuls unde in-lane shea load. The vaiaion of fequenc aamees wih PP c fo siml suoed ecangula siffened lae wih,, longiudinal equisaced siffenes (iffene aamees =. and = 5) subjeced o unifom shea has been sudied fo (ab=,.5, ) in able Vaiaion of non-dimensional fequenc aamee ( ) vs PPc fo siml suoed siffened lae having ( =. and =5) subjeced b in-lane unifom shea ma be dawn. This is fo he sake of geing vaious inelinking esuls of numbe of siffenes and asec aios. I is obseved fom he above sudies ha fequenc aamees a an value of PP c will be moe fo squae siffened lae. These values

17 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June 7 I 5-5 decease wih he incease of asec aios. imilal i is also concluded ha buckling loads aamees will decease, as he asec aios will incease. Table 6: Vaiaion of fequenc aamee wih in-lane load inensi faco (PP c ) of a siml suoed siffened lae wih one cenal siffene having ( =.,, = 5) subjeced o edge shea onl. ( ),. ab Fequenc aamee a PP c Table 7: Vaiaion of fequenc aamee wih in-lane load inensi faco (PP c ) of a siml suoed siffened lae wih wo equisaced siffenes ( =.,, = 5) subjeced o edge shea onl. ( ),.. ab Fequenc aamee a PP c Table 8: Vaiaion of fequenc aamee wih in-lane load inensi faco (PP c ) of a siml suoed siffened lae wih hee equisaced siffenes ( =.,, = 5) subjeced o edge shea onl. ( ),. ab Fequenc aamee a PP c Table 9: Vaiaion of fequenc aamee wih in-lane load inensi faco (PP c ) and buckling load aamee of a siml suoed siffened lae wih equisaced siffenes ( =. and =5) subjeced o edge shea onl.,. ( ),. o of siffenes ab.5 uckling Paamee Fequenc aamee a PP c

18 Inenaional Jounal of cienific and Reseach Publicaions, Volume, Issue 6, June 8 I IV. COCLUIO The esuls fom he sud of he comessive buckling and vibaion behaviou of a siffened lae subjeced o in-lane unifom and non-unifom edge loading can be summaized as follows. The sabili esisance inceases wih incease of esain a he edges fo all es of loading, siffene aamees and lae asec aios. The sabili esisance inceases wih incease of numbe of siffenes. The vaiaion of buckling load wih he osiion of he concenaed load on he edges is moe onounced fo he siffened laes of he smalle asec aios. The buckling load aamee of unsiffened laes siml suoed along all he edges incease as he loads ae neae o he suo. Fo laes wih small asec aios, he bounda condiion on he loaded edge has he significan effec on he load equied o cause elasic sabili. aual fequencies of siffened laes alwas decease wih he incease of he in-lane comessive load. The fundamenal fequenc becomes zeo a he esecive values of he buckling load. The inclusion of inlane dislacemens educes he fequenc of he siffened laes. The buckling load of he siffened laes educes wih he eccenici of he siffenes. The eccenici of he siffened lae elemen should no be negleced, eseciall fo highe modes of vibaion. REFERECE [] Diez, L., Gianei, C.E. and Laua, P. A. A. A noe on ansvese vibaion of ecangula laes subjeced o in-lane nomal and shea foces. Jounal of ound and Vibaion, 59, 5-58,978. [] ingh, J.P. and De,.. Tansvese vibaion of ecangula laes subjeced o in-lane foces b a diffeence based vaiaional aoach. Inenaional Jounal of Mechanical ciences. (7), , 99. [] Deolasi, P.K., Daa, P.K., and Pabhaka, D.L. uckling and vibaion of ecangula laes subjeced o aial edge loading (Comession o ension). Jounal of ucual Engineeing, (), 5-44, 995. [4] undaesan, P., ingh, G., and Rao, G.V. uckling of modeael hick ecangula comosie laes subjeced o aial edge comession. Inenaional Jounal of Mechanical ciences, 4(), 5-7, 998. [5] Aksu, G. Fee vibaion analsis of siffened laes including he effecs of in-lane ineia. Jounal of Alied Mechanics, Tans of AME, 49, 6-, 98. [6] has,. P. and Rao, G. V. Vibaion of hin ecangula laes wih abiail oiened siffenes. Comue and ucues, 7, 67-65, 977. [7] Olson, M.D and Hazell, C.R. Vibaion sudies of some inegal ib siffened laes. Jounal of ound and Vibaion. 5, 4-6,977. [8] au Rao, M.., Guuswami, P., Venkeeswaa Rao, M. and Pavian,. udies on vibaion of some ib-siffened canileve laes. Jounal of ound and Vibaion, 57 (), 89-4, 978. [9] Mukhoadha, M. Vibaion and sabili analsis of siffened laes b semi analic finie diffeence mehod. Pa I: Consideaion of bending dislacemen onl. Jounal of ound and Vibaion, (), 7-9, 989. [] heikh, A.H. and Mukhoadhaa, M. Fee vibaion analsis of siffened laes wih abia lanfom b he geneal sline finie si mehod. Jounal of ound and Vibaion, 6 (), 47-64, 99. [] heikh, A.H., and Mukhoadha, M. Lage deflecion fee fleual vibaion of siffened laes. AIAA Jounal, 4 (), 996. [] Haik, I.E. and Guo, M. Finie elemen analsis of eccenicall siffened laes in fee vibaion. Comue and ucues, 49 (6), 7-5, 99. [] edai, O.K. Fundamenal fequenc deeminaion of siffened laes ug sequenial quadaic ogamming. Comue and ucues, 99, 88-6, 997. [4] Allman, D. J. Calculaion of he elasic buckling loads of hin fla einfoced laes ug iangula finie elemens. Inenaional Jounal of umeical Mehods in Engineeing, 9, 45-4, 975. [5] Wu, J. R. and Liu, W. H. Vibaion of ecangula laes wih edge esains and inemediae siffenes. Jounal of ound and Vibaion, (), -, 988. [6] Mukhoadha, M. A semi analic soluion fo ecangula lae bending. Comue and ucues, 9, 8-87, 978. [7] Mukhejee, A and Mukhoadha, M. Finie Elemen uckling Analsis of siffened laes. Comue and ucues, 4 (6), 795-8, 99. [8] heikh, A. H. and Mukhoadha, M. Analsis of iffened laes wih abia lanfom b he geneal sline finie si mehod. Comue and ucues, 4(), 5-67, 99. [9] Aun Ro, Y., has,. P. and Rao, G. V., abili of aiall siffened squae laes ug high ecision finie elemen. Comue and ucues, 7 (4), 6-65, 99. [] edai, O.K., A conibuion o he sabili of siffened laes unde unifom comession. Comue and ucues, 66, 55-57, 998. [] Chakabo,. and Mukhoha, M. Esimaion of In-lane elasic aamees and siffene geome of siffened laes, Jounal of ound and Vibaion, (), 99-4,. [] Zeng, H and e, C. W. A diffeenial quadaue analsis of vibaion fo siffened laes. Jounal of ound and Vibaion, 4 (), 47-5,. [] ivasava, A. K. L., Daa, P. K., heikh, A. H., Vibaion and Dnamic insabili of siffened laes subjeced o in-lane hamonic edge loading, Inenaional Jounal of ucual abili and Dnamics,, (), [4] ivasava, A. K. L., Daa, P. K., heikh, A. H., Dnamic sabili of siffened laes subjeced o non-unifom hamonic in-lane edge loading, Jounal of ound and Vibaion,, 6 (5), [5] Leissa, A.W. Recen sudies in lae vibaion: , Pa-I, Classical heo. hock and Vibaion Diges, 9(), -8, 987. [6] ai, P.. and Rao, M.., On vibaion of laes wih vaing siffene lengh. Jounal of ound and Vibaion, 95(), 9-9, 984. [7] Timoshenko,.P. and Gee, J. M. Theo of Elasic abili. McGaw-Hill, ewok, 96. AUTHOR Fis Auho A.K.L. ivasava, Deamen of Civil Engineeing,. I. T. Jamshedu- 84, India, aklsiv.nijs@ahoo.com econd Auho.R.Pande, Deamen of Civil Engineeing,. I. T. Jamshedu- 84, India, avi_suja@ahoo.com