Songklnkin J. Sci. Tecnol. 0 (, -5, M. - Ap. 008 p://www.sjs.ps.c. Oiginl Aicle Compessive modls of desive bonded bbe block Coeny Decwykl nd Wiiy Tongng * Depmen of Mecnicl Engineeing, Fcly of Engineeing, Pince of Songkl Univesiy, H Yi, Songkl, 90 Tilnd. Received Novembe 007; Acceped 0 Apil 008 Absc Te pesen sdy emined e effec of in desive lye on e modls of n elsic bbe block bonded beween wo ples. Te ples wee ssmed o be igid, bo in eension nd flee, nd sbjeced o veicl compession loding. Te en s ppoc ws sed o obin e nlyic defomions of e bbe nd desive. Te nlyic defomions wee en vlided wi e finie elemen model. Tee ws good geemen beween bo meods. Te modls of e bonded bbe block, defined s effecive modls, ws en sdied. Te effecive modls ws incesed by e fco (+ (/ (6 / + -, wic is composed of e spe fco of e bbe block (/, io of e bonded nd nbonded es, nd e se siffness fco ( /, io of modls nd ickness of bbe nd desive. Te effecive modls does no depend on eie fcos, wen e se siffness of e join is ig o / >0. Keywods: bonded bbe block, ples, mecnicl popeies, nlyicl modeling, finie elemen nlysis. Inodcion An elsic bbe block is widely sed s n engineeing componen fo lod being (en 99. enelly, e bbe block is bonded o seel ples nd is ssmed o ve pefec bonding nd o be igid in eension nd flee o povide veicl siffness. Wen e elsic bbe block is compessed de o lod cying, e bbe epnds lelly. Wen lel epnsion is esiced, e veicl siffness of e bonded bbe block is incesed nd e bbe is ssmed o be incompessible (Ko nd Lim, 00. Te sfce inecion vies depending on e se siffness of e desive lye, wic les e siffness of e elsic bbe block. An ndesnding of e effec of se modls nd e ickness of e desive lye bonding beween e wo sfces will fcilie e design of moe sible desive. Te en s ppoc (en nd Meinecke, 970 is sed o deemine e compessive siffness of bbe block bonded o igid ples (Tsi nd Lee, 998. Tis ppoc ssmes e bbe is in e oionl plne, emins pln nd e veicl lines become pbolic, b e noml sess componens in ll diecions e eql o e pesse, nd c bbe wi Poisson s io ecly eql o 0.5 is ssmed o be incompessible. Tis meod is genelly cceped o deemine e veicl siffness of bonded bbe block. Te siffness deived sing is meod s been veified nd ee is good geemen wen comped wi finie elemen nlysis. In is ppe, we popose o se e en s ppoc o sdy e effec of in desive lye on e defomions, pesse, sess, nd veicl siffness of n elsic bonded bbe block. Te en s ppoc ws sed o deemine e nlyic solions of e displcemens. Te defomions of e bbe nd desive wee lso vlided by finie elemen nlysis. To pesen e effec of e in desive lye in e finie elemen nlysis, e sping elemens wee modeled by e in desive lye nlysis (TALA meod (Decwykl e l., 00. *Coesponding o. Emil ddess: wiiy@me.ps.c.
Decwykl & Tongng / Songklnkin J. Sci. Tecnol. 0 (, -5, 008. Anlyic Meods. Rbbe block bonded wi in desive lye We conside n infinie sip of e bbe block in ecngl Cesin coodine - s sown in Fige. Te bbe lye s wid of nd ickness of. Te boom nd op of e bbe lye e bonded o igid nd siff ples by in desive s wid of nd ickness of. Te meils e ssmed o be isoopic nd line elsic. In is sdy, i is ssmed e desive lye is in comped o e ickness of bbe. Te io of / is bo 0 o 0.. in desive lye ppe ple bbe block lowe ple / / F (0,0 Fige. Tin bbe block bonded wi in desive lyes Te lowe ple is consined in ll diecions. Te ppe ple is consined in e -diecion nd sbjeced o nifom veicl compessive lod F; s en i is fee o nsle only in e - diecion wi e smll displcemen of. In is cse, e ssmpions fom en s ppoc e sed, nd i is lso ssmed ee is no noml sin in e veicl diecion fo e desive lye, becse e desive is vey in nd siff ogo e ickness. Ts, e displcemen fncions of e bbe in e -diecion cn be denoed s w ( wen / < < /. Also, e displcemen fncions of e desive in e -diecion cn be denoed s w (, wen / < < /+ nd -(/+ < < -/. Te displcemen condiions in e veicl diecion e; / / Consideing only > 0, becse of e symmeicl geomey of e displcemen in e -diecion, nd o sisfy ose condiions, e displcemen fncions e: 0 ( = ( + ( = 0 ( ( 4( + 0 ( (4 Becse of e incompessibiliy of e bbe lye, e sm of e noml sins in e nd diecions is eo: w ( + = 0 (5 Sbsie (4 ino (5, en inege nd pply e condiions ( nd (; nd, 0 ( 4 0 ( w ( = ( (6 0 ( 0 ( = ( + (7 Becse w ( is independen of, e ems of 0 (/ nd 0 (/ e consn in (6 nd (7. To deemine ese ems, i is necessy o conside e eqilibim of foces in e -diecion fo e desive nd bbe lyes. Te se of sess dominn in e bbe lye is pesse (p nd se sess (. Now i is ssmed ee is only se sess ( in e in desive lye. Consideing e eqilibim of long sip ny og -/ o /, i is known ; 0 = ( (8 p = = ( w ( + (9 w (/ = w (/ = w (/+ = - ( w (-/ = w (-/ = w (-(/+ = 0 ( Te displcemen fncions in e lel diecion (-diecion e (, fo e bbe nd (, fo e desive. A biy, we cn illse e defomions in e lel diecion s sown in Fige. I is ssmed ee is pefec bonding beween e in desive lye nd e ples, nd e ickness of e desive lyes is in; s, (, is line / < < (/+ nd (/+ < < -/. Te (, is pbolic -/ < < /. Fige. Te defomion of e bbe nd desive lyes in e -diecion
Decwykl & Tongng / Songklnkin J. Sci. Tecnol. 0 (, -5, 008 / / w ( ( + d= d (0 By sbsiing (4, (6, nd (8 ino (0, wee nd e e se modls of e desive nd bbe, especively, we fond ; ( 0 = 4 0 ( ( ( ( Sbsie ( ino (7; = ( ( + 4 0 ( = 4( ( ( ( + 4 0 ( Sbsie ( nd ( ino (6; w ( =[( + 4 4 ( ( ( + 4 4 (+ ( ( ( ] (4 To deemine (, nd (,, we ve o deemine 0 ( nd 0 ( by ineging ( nd ( nd pplying e condiions of 0 (0 = 0 (0 = 0. Tese give; =[4( =[( + 4 ( + 4 ( ( + ( ( 4( + 4 ] ] (5 (6 Consideing Eqions (4 o (6, in e cse of pefec bonding, we cn ssme e se modls of e desive ( is ig comped o e se modls of e bbe (. Ten, ose eqions e simplified o w ( =[( ( ] (7 = 0 (8 = [ ( ( 4( ] (9 Consideing Eqions (4 o (6, in e cse of lbicion, we cn ssme e se modls of e desive ( is eo. Tose give; w ( =[ ( ] (0 ± ( / + = 0 ; ± / ( ( = ( = ( ( Fo e cse of lbicion beween e bbe nd ples, we fond (, is independen of o cn be defined s (. Te desive displcemen (, becomes eo = ±(/+ nd eql o ( = ±/.. Vlidion of nlyic solions Te nlyic solions sown in Eqions (4-(6 wee vlided by e Finie Elemen Meod (FEM. Te commecil finie elemen code ABAQUS (ABAQUS 998 ws sed fo e vlidion. Te FEM model ws composed of wo ps s sown in Fige. Te bbe lye wi e spe fco of / = 5 is ceed in lf wid, becse of e symmey. I is mesed ino -D plne sin (CPE8H; 8-node bi-qdic, ybid wi line pesse. In e finie elemen model, e bbe lye is ssmed line-elsic meil nd e Poisson s io is vey close o 0.5. Te in desive lyes bonding beween e ples nd bbe e ceed wi ickness io of / = 0.. To epesen e esicion of e in desive lye, e TALA meod is sed. Tis meod ses sping elemens o simle e modls nd ickness of e in desive lye in ems of siffness. In is sdy, e se modls io of e bbe nd desive / = 0.5 ws sed. A e ppe ple, e bbe block is compessed wi nifom veicl displcemen of -/ = 0.00.. Deivion of e modls of e bonded bbe block Te compession foce of e bonded bbe block is given by e sm of F nd F (Bnks e l., 00. F is e omogeneos compession foce, wic is obined wen e bbe block is compessed beween flly lbiced sfces. F is e foce eqied o keep poins e oiginl posiion in e plnes of e bonding sfce de o se defomion. To deive F, i is ssmed e defomion of e bbe in e long sip is eo o plne sin (Timosenko nd oodie, 987. Eqion ( nd Hook s lw e pplied nd sed wen e Poisson s io (n of e bbe is 0.5 nd e modls of e bbe is E = fo n isoopic nd incompessible meil. ν ε = ( + = ν σ ( E 4 F = σ A= E (4 To deive F, e pesse is obined fom n eqivlen condiion s sown in Eqion (9 nd pplied wi bondy condiion of pesse e edges of e bbe eo o p(±/=0. I is fond : p( = ( [( ( + 6 ( ] (5 Te p( is posiive wen e bbe is nde compession. Using e sme pinciple, we cn deemine e se sess in e bbe lye;
4 Decwykl & Tongng / Songklnkin J. Sci. Tecnol. 0 (, -5, 008 Ts, = ( ( ( (6 ( + 6 F / = p( d= 6 ( + / (7 Te compessive foce is given by e sm of Eqions (4 nd (7 F = F + F E 6 = 4 [+ ( ( + ] (8 Eqion (8 cn be wien ino e effecive modls (E c of e bonded bbe block: E E c 4 6 = [+ ( ( + ]. Resls nd Discssion. Vlidion of e nlyic solions (9 Te displcemen fncions s sown in Eqions 4 o 6 wee vlided sing finie elemen nlysis. Fige pesens e veicl displcemen og e ickness of e bbe lye =0. Te displcemen is nomlied by, sown in e y-is, nd e veicl posiion is nomlied by, sown in e -is. Te mimm veicl displcemen is e ppe ple ( / = 0.5, wee e bbe block is compessed, nd e displcemen becomes eo e lowe ple, wee e block is consined. A e mid-plne ( / = 0, e veicl displcemen is lf e mimm of e veicl displcemen. Te plo in Fige sows ee is good geemen beween FEM nd e nlyic solions fo veicl displcemen. w ( 0.00-0.0-0.0-0.0-0.40-0.50-0.60-0.70-0.80-0.90 veicl displcemen og e ickness of bbe lye FEM--=0 Anlyicl--=0 -.00 0.5 0.4 0. 0. 0. 0-0. -0. -0. -0.4-0.5 Fige. FEM nd nlyicl vlidion of veicl displcemen w (/ og e ickness of e bbe lye = 0 7 6.5 6 5.5 5 4.5 4.5.5.5 0.5 (,,0 0 lel displcemen e midplne of bbe lye FEM--=0 Anlyicl--=0 0 0 0.5.5.5.5 4 4.5 5.500.400.00.00.00.000 0.900 0.800 0.700 0.600 0.500 0.400 0.00 0.00 0.00 0.000 / lel displcemen long e in desive lye FEM--=/ Anlyicl--=/ 0 0.5 0.5 0.75.5.5.75.5.5.75.5.5.75 4 4.5 4.5 4.75 5 Fige 4. ( FEM nd nlyicl vlidion of e lel displcemen (,0/ e mid-plne of e bbe lye = 0 nd (b FEM nd nlyicl vlidion of e lel displcemen (,// long e in desive lye = / Fige 4 nd b pesen e lel displcemens in e bbe lye e mid-plne nd in e desive lye bonding beween e ppe ple nd bbe, especively. Te lel displcemen is nomlied by sown in e y-is, nd e veicl posiion is nomlied by sown in e -is. Te lel displcemen inceses linely fom / = 0 o e nbonded sfce of e bbe block (/ = 5. Te mimm lel displcemen occs e mid-plne of e bbe block. Te plos eveled ee is geemen fo e lel displcemen beween e FEM nd nlyic solions 0 < / < 4.5. Te discepncies ppenly occ ne e fee o nbonded sfce (4.5 < / < 5, becse is poin is close o e fee sfces of e bbe nd bonding e, wic is e singliy poin.. Effec of in desive lye on e bonded bbe block Te plos of Eqion (9 in Fige 5 sow e effec of e desive on e effecive modls of e bonded
Decwykl & Tongng / Songklnkin J. Sci. Tecnol. 0 (, -5, 008 5 Fige 5. Effec of e desive lye io ( on e effecive modls (E c /E fo diffeen S (spe fco bbe block. S is e spe fco fo e long sip, wic is / (bonded e/fee e, nd e se siffness fco is = / (io of modls nd ickness of bbe nd desive. In is sdy, / is vied fom 0, wic is pefec bonding condiion, o 000. Te spe fco (S is vied fom o 0. 4. Discssion nd Conclsions Te se modls nd ickness of e desive lye e ken ino ccon o nlye e defomions of e bonded bbe block. Te en s ppoc is sed o deemine e nlyic displcemen solions. Eqions (4 o (6 e e defomions of e bbe nd desive, wic wee vlided by e finie elemen nlysis. Tee is n geemen fo e displcemen beween e finie elemen nlysis nd nlyic solions. Tee wee smll discepncies in e lel defomions of e bbe nd desive e cone of e ming sfces beween e bbe nd bonding e, de o singliy e cones. Te nlyic solions indice e se siffness fco /, io of e modls, nd ickness of e bbe nd desive on e sme e ffec e veicl nd lel defomions, pesse, nd se in e bbe nde compessive lod cying. Te siffness fco llows fo e bonding sfce o esis lel nd veicl defomions nd o le pesse nd se sess in e bbe. In e cse of lbiced sfce, i is ssmed ee is no desive, =0 o /. Wen e bbe is com- pessed, e bbe long e inefce e is no consined nd flly moves in e lel diecion. To minin consn volme de o e incompessibiliy of e bbe, e pesse is en consn, nd ee is no se sess in e bbe. In e cse of pefec bonding, i is ssmed ee is vey siff desive, o / =0. Wen e bbe is compessed, e bbe ne e inefce e is flly consined in e lel diecion, wees e bbe in oe es is no, wic en cees se sess nd pesse gdien in e bbe. Te modls of e bonded bbe block defined s effecive modls (E c is incesed by e fco of (+(/ (6 / + -, wic is composed of e spe fco (/ nd e se siffness fco ( /. Wen ee is pefec bonding o / =0 e inefce, e effecive modls is iges, becse e fll consin e inefce eds e bbe epnsion. Te effecive modls of e bbe gdlly deceses nd conves o vle of 4/E, s e inefce is moe complin in e lbiced condiion o /. Te spe fco lso inceses e effecive modls. Te effecive modls depends only on e spe fco o e desive lye wen e se siffness of e join is ig o / >0. Refeences ABAQUS/Sndd Vesion 5.8 Use s Mnl 998. Hibbi, Klsson & Soensen, Inc., Rode Islnd, U.S.A. Bnks, H.T., Pine,.A. nd Yeo, O.H. 00. Anlysis of bonded elsic blocks. Memicl nd Compe Modelling. 6, 875-888. Decwykl, C., Rbin, C.A. nd Hn,.T. 00. Anlysis of e effecs of in seln lyes in icf scl joins. AIAA J. 4, 6-8. en, A.N. 99. Engineeing wi bbe: ow o design bbe componens, Ofod Univesiy Pess, New Yok, U.S.A. en, A.N., Meinecke, E.A. 970. Compession, bending nd se of bonded bbe blocks. Polym. Eng. Sci. 0, 48-5. Ko, C.. nd Lim, H.L. 00. Anlyicl solion fo compession siffness of bonded ecngl lyes. Inenionl Jonl of Solids nd Sces. 8, 445-455. Timosenko, S. nd oodie, J.N. 987. Teoy of elsiciy d Ediion, Mcw-Hill, New Yok, U.S.A. Tsi, H.C. nd Lee, C.C. 998. Compessive siffness of elsic lyes bonded beween igid ples. Inenionl Jonl of Solids nd Sces. 5, 05-069.