Next generation calculation method for structural silicone joint dimensioning

Transcription

1 Glass Struct. Eng. (2017) 2: DOI /s S.I. GLASS PERFORMANCE PAPER Nxt gnration calculation mthod for structural silicon joint dimnsioning Pirr Dscamps Valri Hayz Mahmoud Chabih Rcivd: 25 April 2017 / Accptd: 29 May 2017 / Publishd onlin: 23 Jun 2017 Springr Intrnational Publishing AG Switzrland 2017 Abstract Bonding of glass onto aluminum frams, known as structural silicon glazing, has bn applid for mor than 50 yars on facads. Traditionally, th silicon bit is calculatd using a simplifid quation assuming a homognous strss distribution along th salant bit. Du to th complxity of façad dsigns th assumptions bhind simplifid quations ar raching thir limit of validity and rquirmnts to us finit lmnt analysis (FEA) incras sinc it allows to dscrib th local strss distribution within salant volum. Howvr, thr is no standardizd mthodology to run FEA for valuation of SSG. Furthrmor, th complxity of FEA is a limiting factor to its systmatic us as a calculation mthod for all projcts. For ths rasons, a nxt gnration calculation mthod was dvlopd which prdicts dformation of SSG with good accuracy compard to FEA prdictions. Th basis of th mthod was dvlopd 25 yars ago and was includd as annx in ETAG002. Th validation of th mthod was don by comparing xprimntal masurmnts, rsults of FEA modling and outcom of th nw calculation mthod. To furthr improv accuracy, an xtnsion of th rlationship for a nonlinar matrial is proposd, assuming a No-Hookan strss strain bhavior for silicon salant. P. Dscamps V. Hayz (B) M. Chabih Dow Corning Europ SA, Parc Industril Zon C, Ru Juls Bordt, 7180 Snff, Blgium -mail: valri.hayz@dowcorning.com Kywords Finit Elmnt Analysis Calculation mthod Structural silicon ETAG 002 Bonding 1 Introduction Bonding of glass onto aluminum frams, known as Structural Silicon Glazing (SSG), has bn applid for mor than 50 yars on facads with various improvmnts of th tchnology bing mad ovr tim. Silicon salants ar usd in this application bcaus of thir uniqu rsistanc to wathring (UV, tmpratur, moistur, ozon). Thy also provid rsistanc to watr ingrss and thrmal insulation (Klosowski and Wolf 2015). Thir structural rol is to sustain wind loads and to accommodat for diffrntial thrmal xpansion of diffrnt boundd substrats. A considrabl amount of ffort was mad sinc first half of th 20th cntury to undrstand th bhavior of a joint submittd to a dformation. For xampl, Volkrsn (1938), proposd a modl to simulat joint bhavior in lapshar configuration, nglcting th bnding ffct in cas of ccntric load. Starting from Volkrsn s approach, Goland and Rissnr (1944) introducd this bnding ffct. Mor rcnt paprs, building on th us of numrical tools, discuss joint failur critria lik Callwart t al. (2011). Historically, silicon joint dimnsioning is calculatd with a simplifid quation implmntd in various standards for structural glazing (ASTM 2014; EOTA 2012; GB 2005). This quation assums homognous strss distribution along th salant bit whilst

2 170 P. Dscamps t al. Fig. 1 Exampl of local strss distribution in a silicon joint submittd to a combind traction rotation high local strss paks, structur dformation or matrial aging ar includd in a global safty factor. Nw trnds in commrcial buildings includ th us of larg dimnsions glass pans, highr complxity of façad dsigns and strongr nginring prformanc rquirmnts such as high windloads abov 5000 Pa (Hayz 2016; Maniatis and Sibrt 2016). Ths trnds hav rcntly challngd th convntional mthods of joint dimnsioning, sinc using th simplifid quation for ths projcts rsults in conomically unaccptabl larg bit sizs. Furthrmor, incrasing joint bit will not ncssarily incras th safty factor as th simplifid rlationship nglcts important factors such as th joint rotation du to glass pan bnding. Incrasing th salant dsign strss is an option to dcras th bit but this solution is limitd and also rquirs a bttr undrstanding of strss distribution as wll as joint failur mchanisms as was xplaind in Dscamps t al. (2016a, b). This xplains th rcnt incrasd intrst to us Finit Elmnt Analysis (FEA) to hlp dsigning SSG and joint dimnsions. In FEA, th gomtry is dividd in small volum lmnts intrconnctd by points call nods. Applying nrgy consrvation to th whol systm, via strain nrgy calculation at small lmnt lvl, local strss and/or local dformation can b prdictd (Fig. 1). Howvr, thr is no tchnical guidlin or standardizd mthod xplaining how to us FEA in structural joint dimnsioning. Without such guidanc, calculations carrid-out by diffrnt nginring offics may lad to diffrnt absolut valus of th maximum local strss. Th outcom of FEA modl is highly snsitiv to th accuracy of input data such as th paramtrs of th hyprlastic modl slctd for th salant. Th strss volum distribution is also highly msh dpndnt, spcially clos to th intrfacial rgion btwn th salant and th substrat as dmonstratd in Dscamps t al. (2016a) and rcalld latr in this papr. This is mor particularly tru bcaus vn bing asily dformabl, silicon salant is a narly incomprssibl matrial (Wolf and Dscamps 2003). Finally, vn if th maximum local strsss or strains in joint volums ar calculatd in an accurat way, w do not know what is th accptabl valu a joint can sustain whil nsuring long trm durability of façad systms. In fact, thr is no unanimous approach on how to dfin a ruptur critria from local strss and to dtrmin what th bst modl to prdict matrial failur is. Svral critria lik principal strss, Von Miss strss or maximum dformation nrgy ar possibl. Hnc it is difficult to us a local strss distribution for prdicting failur in a macroscopic joint and consquntly us this information for joint dimnsioning (Dscamps t al. 2016a, b). An altrnativ approach is to us FEA rsults to simulat obsrvabl (or nginring) joint dformation bcaus this variabl has a lowr snsitivity to msh configuration. Indd, obsrvabl dformation rsults from th intgral of th strain nrgy ovr th whol joint volum hnc local high strss valus which ar highly msh snsitiv ar avragd. Joint dformation calculatd using FEA for on particular façad can b compard to H-bar tsting rsults for tst pics having th sam gomtry and mor particularly similar joint aspct ratio R (dfind as th ratio btwn joint bit W and joint thicknss ). Whil calculating nginring joint dformation with FEA crats a mor dirct link with salant prformancs masurd on tst pics, carrying out a FEA modl rmains an xpnsiv procdur, rquiring invstmnt in FEA softwar acquisition and nginring rsourcs to run simulations. Hnc this mthodology is difficult to xtnd to small/mdium siz façad makrs who would prfr using a simpl manual calculation mthod. Th goal of this papr is not to provid a dirct contribution to th ffort of joint bhavior undrstanding, but to propos an improvd mathmatical rlationship making a dirct corrspondnc btwn a joint

3 Nxt gnration calculation mthod for structural silicon 171 includd in a façad systm and th bhavior of a tst pic. A history of th mathmatical rlations of joint dimnsioning is prsntd, xplaining thir limit of validity and why it is important to mov to a nw rlationship including additional physics ffcts lik joint rotation which wr nglctd prviously (ASTM 2014; EOTA 2012; GB 2005) and which rprsnt mor accuratly th joint bhavior. Vry rough assumptions hav bn mad for its drivation to kp it simpl. Validation of th proposd rlationship for larg windload is carrid out by confronting prdictions with physical masurmnts and th rsults from FEA modling. Th improvd rlationship was dducd assuming first a linar matrial. This assumption is rlativly accurat as for small joint longation ε = Δ blow 10%, th strss/strain curv dviats vry littl from linarity. To optimiz th corrlation btwn FEA and th quation for largr longations, an xtnsion of th improvd linar modl to accommodat non-linar bhavior is proposd, assuming a No-Hookan modl. All th studis dscribd in this papr wr carrid out using proprtis and xprimntal charactrization of Dow Corning 993 Structural Glazing Salant (Dow Corning 2017), which is a two-componnt nutral alkoxy curing silicon formulation spcifically dvlopd for th structural bonding of glass, mtal and othr building componnts. 2 Idntification of th hyprlastic modl Th FEA Multiphysics softwar packag COMSOL (Comsol 2017) was usd to conduct finit lmnt modling of structural silicon and validat joint dimnsioning rlationships. COMSOL has a solid mchanics packag giving accss to a wid rang of hyprlastic matrial modls lik No-Hookan, Moony Rivlin, Yoh and many othrs. Assuming that silicon matrial is incomprssibl for th small movmnts obsrvd in construction, w obtain a rlationship btwn th macroscopic strss/strain curvs masurd xprimntally and th strtch λ(λ = 1 + ε) for both uniaxial, bi-axial and pur shar tsting. A routin was built in MATLAB (Matlab 2017) to fit diffrnt hyprlastic modls to th xprimntal data masurd on purly uni-axial and biaxial tst pic. Th χ 2 valu (th sum of th squar of th rsidual btwn xprimntal valus and modl Fig. 2 Rprsntation of th tst bi-axial tst pic configuration usd charactriz th Dow Corning 993 bhavior in comprssion prdictions) was calculatd combining th data masurd on th diffrnt typs of tst pics, using hyprlastic modl paramtrs as curv fitting paramtrs. A wight function was usd to prvnt having th fit bing dominatd by high longation valus and guarant that th modl is rprsntativ in a wid longation rang. 2.1 Dtrmination of matrial proprtis To obtain an accurat simulation of joint bhavior undr structural load, accurat strss strain bhavior of silicon matrial is ssntial. Physical proprtis wr masurd via a spcializd laboratory protocol. Th srvics of Axl Products (Axl 2017) wr usd to dvlop accurat uni-axial, qual bi-axial and planar (pur shar) xtnsion for th Dow Corning 993 matrial. Shts of thicknss varying btwn 1 and 2.6 mm wr curd for a priod of 4 wks at room tmpratur ( 20 C) and 75% humidity. Out of thos shts, tst pics for uni-axial tsting [dog-bon ASTM D412 Di D (ASTM 2016)], with an ffctiv gaug lngth of 50 mm wr cut using a di cutting machin. Similarly, bi-axial tsting was carrid-out. For incomprssibl or narly incomprssibl matrials, qual bi-axial xtnsion of a spcimn crats a stat of strain quivalnt to pur comprssion. Although th xprimnt is mor complx than a simpl comprssion xprimnt, a pur stat of strain can b achivd lading to mor accurat matrial modl idntification. Th qual bi-axial strain

4 172 P. Dscamps t al. Tabl 1 Dimnsions of th tst sampls Tsting pic Width (mm) Thicknss (mm) Ara (mm 2 ) Dog-bon Planar tnsil Bi-axial stat may b achivd by radial strtching of a circular disc (Fig. 2) of 75 mm diamtr and an ffctiv ara of 50 mm in diamtr. Finally, planar tsting was prformd on rctangular pics of 150 mm wid and 15 mm tall, th natur of th tst rquiring a width at last 10 tims largr than gaug lngth. Charactristic dimnsions of th diffrnt tst pics ar summarizd in Tabl 1. Thr tst pics wr prpard for ach gomtry and pulld at a rat of 0.01 mm/s. Rsults masurd in uni-axial xtnsion, qual biaxial xtnsion and planar tnsion for Dow Corning 993 ar prsntd in Fig Idntification of th modl paramtrs Th tnsion data, both uni-axial and bi-axial, wr curv-fittd with svral matrial modls in ordr to find a curv fit minimizing th scald rsiduals χ 2 Fig. 4 Fitting of th tnsion masurmnts (uni-axial and biaxial) with a Moony Rivlin modl with 5 paramtrs idntification rsulting from all data providd. Modls of diffrnt ordrs wr tstd. As an illustration a Moony Rivlin (MR) modl with 5 paramtrs is usd (Fig. 4) with th following xprssion for th total strain nrgy dnsity W s in th cas of incomprssibl matrial lik silicon: W s = C 10. (I 1 3) + C 01. (I 2 3) + C 20. (I 1 3) 2 + C 11. (I 1 3). (I 2 3) + C 02. (I 2 3) 2 (1) Fig. 3 Tst rsults in uni-axial, bi-axial and planar tnsion for Dow Corning 993 matrial

5 Nxt gnration calculation mthod for structural silicon 173 Th curv fitting xrcis ld to th following valus of modl cofficints: C 10 = E05, C 01 = E05, C 02 =0.0042E05, C 20 = E05, C 11 =0.007E05 Bst practic consists in slcting th lowst ordr modl nabling to prdict xprimntal bhavior within th rror bar associatd to th masurmnt. Furthrmor incrasing modl complxity is nvr suitabl without first liminating th sourc of data variations, du for xampl to th variability of th sampl s prparation or th rror associatd to th tsting dvic. Sinc thr ar at last 3 ordrs of magnitud diffrnc btwn C 10 and th highr ordr cofficints, w nglct all cofficints xcpt C 10 = 2 1 G.Th MR modls bcoms quivalnt to th simplr No- Hookan modl whrby th strain nrgy dnsity W s bcoms W s = 1 2 G. (I 1 3) (2) With th shar modulus G = 2(1+ν) = 7.2 E5 Pa,ν = Poisson ratio =0.5 for narly incomprssibl matrial. Th top thr 1st ordr hyprlastic modls usd for th simulation of Dow Corning 993 with thir rspctiv constants ar providd in Tabl 2. Thos thr modls provid χ 2 valus vry similar to th ons obtaind with MR modl with 5 paramtrs. For all th studis prsntd in this papr, th No- Hookan modl has bn slctd for its simplicity. W will also not that th modulus is givn by th slop btwn th nginring strss and strain curvs. In th cas of uni-axial tnsion th slop of th curv at zro longation is calld th Young modulus E young = 2.3 MPa, which corrsponds to th G valu listd in Tabl 2, assuming a Poisson ratio of Validation of th modl To validat th modl idntifid in paragraph 2.2, H-bar pics with diffrnt joint dimnsions wr built and corrsponding nginring strss strain curvs masurd. Th tst pics wr prpard using anodizd aluminum substrats and Dow Corning 993 salant was curd in standard conditions (20 C and 75% HR) for a priod of 21 days. Tst pics wr tstd at a load rat of 50 mm/min. Thr joint gomtris wr tstd, with tst pic dimnsions summarizd in Tabl 3. E Tabl 2 Matrial paramtrs coming from curv fitting of Dow Corning 993 data Hyprlastic modl Matrial paramtr Valu Ogdn 1st ordr Matrial constant MU E + 05 Pa Matrial constant A (unitlss) No-Hookan Matrial constant C E + 05 Pa Yoh 1st ordr Matrial constant C E + 05 Pa Tabl 3 List of H-bar gomtris usd to validat th No- Hookan matrial modl Sampl rfrnc Lngth (mm) Width (mm) W Thicknss (mm) Th H-bar configurations ar modlld, slcting th No-Hookan modl out of th summary Tabl 2. This choic is justifid bcaus offring χ 2 valus similar to thos obtaind with MR but with only on curv fitting paramtr, as Poisson ratio was fixd at a valu of 0.49, rflcting th incomprssibl charactr of silicon rubbr matrial. Th diffrnt joint configurations ar modlld using a rctangular msh, th siz of on msh lmnt is kpt qual to 1 1 1mm 3 for th joint configurations of Tabl 3. This msh schm is usd for all rsults rportd in this papr. Figur 5 shows a good corrlation btwn xprimntal data and corrsponding modlld curv. Dviation is mainly du to th manufacturing quality of th H-bars and thir masurmnt. Although th curvs plottd in Fig. 5 wr obtaind on tst pics mad of th sam matrial, w obsrv a clar diffrnc btwn th tst spcimns du to thir gomtry. Th nginring strss strain curvs can b prdictd, by substituting th Young Modulus by a rigidity modulus that dpnds only on gomtry and mor spcifically on th joint aspct ratio. For xampl, w obsrv for th joint with cross-sction 6 36 mm 2, a much mor rapid strss incras for a sam joint longation, than is th cas for th othr joint dimnsions and spcially th mm 2. A structural joint consists of a silicon joint bondd btwn two substrats. If w considr a vry thick

6 174 P. Dscamps t al. Fig. 5 Strss strain curvs obtaind for diffrnt joint gomtris: validation of modl through xprimntal tnsil data (MPa) x12x50mm3 modl 12x12x50mm3 xprimntal Strss rcordd Elonga on (%) 12x36x50mm3 xprimntal 12x36x50mm3 modl 6x36x50mm3 xprimntal 6x36x50mm3 modl H-bar compard to its cross-sction (distanc btwn substrat >>> H-bar x-sction), whn moving away from th plan of adhsion, th bhavior of th tst pic tnds toward th uni-axial tnsion cas, whrby th joint cross-sction dcrass proportional to th Poisson ratio. Th ratio btwn th nginring strss and strain provids th modulus which is for this typ of H-bar configuration minimum, having a valu clos to th matrial Young s modulus. On th othr hand, in a thin layr clos to th intrfac btwn salant and substrat (adhsion plan), th joint cross-sction cannot frly dcras to consrv th volum of narly incomprssibl silicon matrial. Th joint is submittd to uni-axial tnsion but simultanously a forc is bing applid on th orthogonal surfacs which rstrains th joint from dforming so no fr cross-sction rduction can tak plac. Consquntly, a largr nginring forc must b applid to obtain th sam nginring strain. This is obsrvd on H-bars having a small thicknss compard to thir cross-sction. Th modulus calculatd from th nginring strss strain curv is in this typ of gomtry significantly largr than th matrial s Young modulus. Whn th thicknss is minimal, th bhavior of th H-bar rachs th thortical xtrm cas commntd in Fynman t al. (1963). A small cub of linar lastic matrial is considrd, with a pulling forc applid to th top and bottom facs of th cub. Additional forcs ar applid on latral facs of th cub to prvnt any chang in th cub cross-sction. For this xtrm cas, a vry simpl rlationship for th modulus can b obtaind: (1 ν) E = E Young (3) (1 + ν)(1 2ν) For a 100% incomprssibl matrial (ν = 0.5), as no cross-sction rduction is allowd, th volum consrvation rquirs an infinit forc to crat an xtnsion. For a narly incomprssibl matrial such as silicon, whrby ν = 0.49, th modulus bcoms vry larg, approximatly 17 tims th Young modulus. This valu can b sn as an uppr limit of joint modulus, whn having a vry thin tst pic compard to its crosssction. Hnc, H-bars of diffrnt aspct ratio R, will hav a modulus which varis btwn two xtrm valus; E Young for thick H-bars with small cross-sction and 17 E young for vry thin H-bars compard to cross-sction. For all intrmdiat H-bar configurations, modulus valus will b comprisd btwn thos xtrms and can b obtaind through FEA modling, solving th nrgy consrvation in th whol volum. To diffrntiat with th Young modulus of th matrial, which is not influncd by its gomtry, th modulus of an H-bar is calld th rigidity modulus E rigidity. Th rlationship btwn both modulus is calld th rigidity factor f rigidity = E rigidity (4) E Young 2D FEA is usd to modl th nginring strss strain curv for diffrnt H-bar configurations and dtrmin thir corrsponding rigidity modulus (Fig. 6). Th 2D modl is justifid sinc th SSG joint dimnsion along th lngth of th profil is much largr than both joint bit and/or thicknss. Th modl dvlopd in para-

7 Nxt gnration calculation mthod for structural silicon 175 Fig. 6 Rigidity factor as a function of joint aspct ratio for salants obying to a No-Hookan modl graph 2.2 is applid. For larg joint thicknss vrsus bit, as xpctd, rigidity factor tnds to 1, i.. E rigidity = E Young. Th thortical valu calculatd using th quation proposd by Fynman t al. (1963) for aspct ratio R and ν = 0.49 is also rportd. Th rlationship btwn th rigidity factor f rigidity and joint gomtry (aspct ratio R) can b fittd by a scond ordr polynomial: f rigidity = R R (5) 3 Dvlopmnt of th nxt gnration rlationship for joint dimnsioning Bfor introducing a nw quation for joint dimnsioning, w will brifly rcall th diffrnt assumptions and quations usd in th past for joint dimnsioning, starting with th simplst on and xplaining th assumptions bhind succssiv improvmnts. 3.1 Homognous strss distribution along both joint bit and fram lngth Th simplst joint dimnsioning assums that th glass pan is infinitly rigid which mans no dformation of th glass occurs du to dflction and th (soft) salant is not gnrating any local glass dformation at th dg of th glass. Infinit glass pan rigidity implis a fully homognous strss distribution, both along th salant bit (w) and along th profil of th fram. Assuming homognous strss distribution, a simpl balanc of forcs can b don whrby th forc xrtd by th windload (P wind ) on th glass surfac (S glass = a b) should b qual to th raction forc associatd to joint dformation for a salant with dsign strngth σ ds (obtaind as th R u,5 /6 valu of th maximum tnsil strngth at brak) and surfac S joint (w*primtr). P wind S glass = S joint σ ds (6) P wind a b = w 2 (a + b) σ ds (7) Thrfor th joint bit w bcoms w = P wind (a b) (8) 2 (a + b) σ ds 3.2 Htrognous strss distribution along th fram lngth In a scond stp, w do not considr a fully rigid glass pan but assum that it will dform. Glass pan dformation is introducd in th modl as shown in Fig. 7 rprsnting a glass pan: it is assumd that th wind acting on th rd rctangl ara (L*a/2) is fully sustaind by th joint of lngth L and of bit W along th xtrior sid of this rctangl. Th ida bhind is that th strss is largr at this location bcaus du to th flxibility of th glass pan, th joint along th small sid of th glass pan dos not contribut to dcras th strss in th joint at th cntr of th longr sid. Th dflction is small and w assum that glass dformation only influncs th htrognous strss distribution along th profil lngth but not along th salant

8 176 P. Dscamps t al. It dos not includ th salant modulus (as only considring forc balanc), whil glass dformation imposs a movmnt that has to b accommodatd by th joint. Ths aspcts will b addrssd in th nxt paragraphs. 3.3 Htrognous strss distribution along salant bit and thicknss Fig. 7 Trapzoidal dformation of th glass pan. Th htrognous strss distribution along th fram rsults in a htrognous strss in th joint lngth bit whr strss is assumd homognous. Looking along profil dirction, th maximum strss is obsrvd in th middl of th fram (a/2). Applying th quilibrium of forcs, th forc acting on th rd rctangl of ara (L*a/2 m 2 ) is sustaind by th joint of lngth L and of bit W: P wind a 2 L = w L σ ds (9) Rarranging Eq. 9 to calculat th salant bit valu for a dfind dsign strss, w obtain th wll-known joint dimnsioning rlationship usd in most SSG projcts (ASTM 2014; EOTA 2012; GB 2005): w = 0.5 a P wind (10) σ ds Equivalntly, for a dfind valu of joint bit, w can calculat th corrsponding homognous strss in th joint: σ = 0.5 a P wind (11) w This basic Eq. 10 is widly usd by th industry to calculat joint dimnsions whil having svral waknsss: It dos not includ joint rotation associatd to glass dformation It dos not includ th glass pan proprtis, whil w must know glass dformation, and mor particularly, local rotation angl at lvl of th joint. It dos not includ joint thicknss, whil w know th joint gomtry influncs th joint dformation as th rigidity factor dpnds on joint aspct ratio. Undr high glass dflction, th assumption of homognous strss distribution along salant dimnsion is not valid anymor and w must tak into account th dformation imposd on th joint by th rotation of th glass (by an angl α on Fig. 8). Th joint dformation incrass whn moving along th x-axis. Th displacmnt associatd with glass pan rotation ( r ) and th homognous dformation ar indicatd on Fig. 8. Th maximum joint displacmnt max is qual to th sum of and r. max = + r (12) To obtain a joint dimnsioning rlationship that incorporats glass rotation ffct, w mak th following assumptions: Glass is much mor rigid than silicon and silicon dos not influnc joint dformation but follows th dformation imposd by th glass pan dformation. Joint dimnsioning obtaind from calculation prdicts a joint dformation small nough so that w can assum that th salant bhavs as a linar lastic matrial. Evn if not fully accurat, th longation at brak masurd on an H-bar of sam gomtry (H-bar not bing tiltd) is rprsntativ for th maximum dformation of façad joint Each lmntary joint lmnt of lngth dx along x axis bhavs lik a linar matrial having a sam valu of nginring modulus associatd to joint gomtry (E rigidity ). Evn if rough, this assumption allows rtriving th rsult obtaind for an H-bar (nginring strss/strain dpndnc) whn assuming th limit cas whr no rotation taks plac (α 0). Th glass plat dformation and th rotation angl du to th windload ar calculatd assuming a simply supportd boundary assumption. Th assumptions mad in

9 Nxt gnration calculation mthod for structural silicon 177 Fig. 8 Rotation of th joint du to th dflction of th glass paragraph 3.2 ar still valid for forc balanc calculation i.. mass balanc is carrid out on th whol joint surfac rprsntd on Fig. 8. By simpl trigonomtry, joint displacmnt associatd to glass pan rotation is calculatd: r = W tg(α) (13) Prforming th balanc of forcs on half of glass pan using symmtry, w obtain : w 0 E rigidity ( + xtg(α)) dx = P wind a 2 (14) E rigidity w + E rigidity w 2 tg(α) 2 = P wind a 2 E rigidity w E rigidity w 2 tg(α) 2 = P wind a 2 (15) (16) Combining Eqs. 13 and 16 w obtain th maximum joint longation max /: max = P wind a 2 Erigidityw2 tg(α) 2 E rigidity w + W tgα (17) This quation can b furthr dvlopd to calculat th maximum valu of nginring strss σ max sustaind by th joint: σ max = E rigidity max E rigiditywtg(α) 2 = P wind a 2w + E rigidity W tgα (18) σ max = P wind a 2w + E rigidity W tgα (19) 2 σ max = P wind a 2w + f rigidity E young W tgα 2 (20) In Eq. 20, w obsrv that th maximum strss is th sum of two trms having opposit dpndnc of salant bit W and hnc a minimum valu can b idntifid for σ max. Th first trm dcrass whn th joint bit incrass bcaus wind load is sustaind by a largr gluing ara. Th scond trm incrass with bit. It corrsponds to th joint dformation inducd by glass dflction. It is important to work with a salant abl to accommodat th imposd dformation lik a wathrsal joint sinc stiff matrial could lad to vry larg intrnal strss build-up and potntial failur (Dscamps t al. 2016a). Th influnc of th joint gomtry is accountd for by th rigidity factor. To validat this nw quation (Eq. 17) including th ffct of glass bnding, w compar its prdictions to 2D FEA calculations (Fig. 9). Th modlling paramtrs ar summarizd in Tabl 4. Th valu of glass thicknss has bn adjustd to hav a maximum glass dflction at th cntr of glass pan qual to 1%. As discussd a saddl point is obsrvd on Fig. 9 with minimum joint displacmnt. Although many

10 178 P. Dscamps t al. Fig. 9 Comparison of joint dformation along Y dirction calculatd from an FEA simulation and using Eq. 17 that taks into account a htrognous joint dformation along joint imposd by glass pan bnding Tabl 4 Paramtrs for joint maximum longation using FEA simulation Paramtr Valu Glass pan dimnsions 1.9 (distanc btwn joints) (m) Windload (Pa) 4913 Joint thicknss (mm) 9 Glass thicknss (mm) 19 Max dflction at cntr 1% simplifying assumptions wr mad to driv Eq. 17, this rlationship prdicts joint dformation valus wll compard to th rsults coming out FEA simulation. Som discrpancy is obsrvd in th small and larg joint rgions. Th diffrnc obsrvd for small salant bits coms from th fact that w assum a linar rlationship btwn th strss and th dformation whn calculating th forc balanc. Howvr, if small salant bits ar usd whn having a larg windload, th strss imposd on th joint is larg, lading to important dformations and potntially moving out of th linar domain of th strss strain curv. Th diffrnc obsrvd for larg valus of salant bit is du to th assumption of simply supportd glass pans, corrsponding to a glass plat dflction of 1% of th smallr glass sid. Howvr, whn th bit incrass, its contribution in shar along x axis on Fig. 8 contributs to limit glass dflction to a valu blow 1%. For this rason, dformations calculatd for larg bits using th nw modl ar always largr than FEA prdiction. In th nxt stps, w introduc both ffcts in th modl to vrify if furthr improvmnts of th corrspondnc btwn th nw quation and FEA prdictions can b obtaind: Considr that th joint has a non-linar bhavior, following a No-Hookan modl; this will impact rsults in th small bit rgion. Considr that salant joint working in shar contributs to limit glass dflction; this will hav an impact on th rsult corrsponding to larg salant bits. 3.4 Hyprlastic matrial modl: rlationship xtndd for largr joint dformations Equation 20 has bn drivd assuming that th joint dformation nvr xcds th non-linarity thrshold i.. that th strss stays proportional to th strain. Howvr, it is possibl for crtain joint configurations (for a small joint bit, whr th windload is sustaind by a smallr gluing ara, and longation is largr) that a non-linar bhavior occurs. W hav shown in paragraph 2.2 that Dow Corning 993 matrial can b dscribd by a simpl No-

11 Nxt gnration calculation mthod for structural silicon 179 Hookan modl which w will us to introduc nonlinar bhavior whn driving joint dimnsioning rlationship. W hav shown that for a No-Hookan modl, th total strain nrgy dnsity is: Ws = C 10 (I 1 3) (21) whr I 1 is only dpndnt on th strtch λ in ach dirction. Assuming a fully incomprssibl silicon matrial and a pur uniaxial traction, w can calculat th nginring strss σ as a function of λ σ = 2C 10 (λ λ 2) (22) ( σ = G λ λ 2) (23) As for th linar matrial, if w compar th nginring strss/strain curv masurd for a uni-axial tst pic (dogbon tst pic) and H-bar, th curvs diffr, having H-bar curvs apparing stiffr. W will assum again that th bhavior of H-bars can b drivd from uni-axial nginring curv, to which w add a rigidity factor dpndnt on joint gomtry. This is simply don multiplying σ in Eq. 23 by a rigidity factor which is quivalnt to multiplying C 10 by a rigidity factor or rplacing th Young s modulus in th calculation of C 10 by a rigidity modulus (Eq. 4). W can asily vrify that for small longations, w rtriv a linar bhavior. Rplacing λ by (1 + ε) ineq.23, whav: ( ) 1 σ = G 1 + ε (1 + ε) 2 (24) Rplacing th last trm in th quation by its binomial sris, only kping th two first trms as ε is small: 1 = 1 2ε (25) 2 (1 + ε) σ = G (1 + ε 1 + 2ε) = 3εG (26) As for fully uncomprssibl matrial G = E Young 3, w rtriv th nginring strss corrsponding to a uni-axial tst pic: σ = E Young ε (27) If th Young s modulus is rplacd by a rigidity modulus to considr th joint gomtric ffct associatd to H-bar tst pic, w obtain th rlationship allowing to calculat th nginring strss of an H-bar. It is important to mntion that th list of assumptions dtaild in prvious paragraph rmains valid for blow calculations. Doing th forc balanc, ffct of wind load qual to joint raction, w obtain: P wind a 2 = W 0 P wind a 2 = 2C 10{ σ (x) dx (28) W 0 W λ (x) dx 0 λ (x) 2 dx} (29) + xtg(α) With λ =1+εand from Fig. 8, ε = Intgrating abov quation, w obtain th following implicit quation ( a P wind 2 = 2C 10{ 1 + ) W + tg(α) W 2 2 ( ) tg(α) tg(α) W (30) This quation can b solvd graphically with th unknown by plotting th right trm of th quality as a function of displacmnt ; knowing,th maximum joint displacmnt is calculatd from Eqs. 12 and 13 as max = + xtg(α) Th nw rlationship including non-linar bhavior (Eq. 30) is valuatd and compard with FEA rsults (Fig. 10). In comparison with th linar assumption (Eq. 17), Eq. 30 prdicts bttr th strss for a small joint bit, which confirms that non-linar joint bhavior was th rason of th diffrnc btwn FEA and rsults obtaind using th nw rlationship. Bcaus calculations ar carrid-out for on sam larg wind load valu ( 5000 Pa), small bits ar subjctd to largr longations, which could b at th limit of validity of linar assumption. Assuming a non-linar bhavior is howvr a thortical xrcis to dmonstrat th rason of th diffrnc btwn th drivd rlationship and FEA prdiction. Indd, salant bit calculations corrsponding to th prssur usd for this xampl will always rquir largr bits and hnc nvr allow thos rlativ larg dformations. Sinc non-linarity will in ral applications sldom occur, th mor simpl quations (Eqs. 17, 20) dscribing linar bhavior should b prfrrd for practical applications. 3.5 Impact of joint on glass dflction For larg joint bits, th structural joint working in shar limits th glass movmnt along th x dirction and consquntly rducs glass dflction which

12 180 P. Dscamps t al. Fig. 10 Comparison btwn joint longation valus obtaind through FEA simulation, th rlationship including joint local joint dformation associatd to glass bnding for linar (Eq. 17) and non-linar matrial bhavior (Eq. 30) Fig. 11 Glass pan dflction calculatd using simply supportd assumption and assuming th glass pan bing maintaind by joints of diffrnt bit valus. Th rd lin corrsponds th simply supportd glass pan. Whn th joint bit incrass, th joint contributs mor and mor in limiting th glass pan dflction rducs glass rotation angl at its xtrmity. As this angl of tilt α lads to an incras of maximum joint dformation for larg bit valus (saddl point obsrvd for a bit of 27 mm on Fig. 10), it is blivd that not considring th impact of th salant in limiting th glass dflction whn w usd th formula (a sam tilt angl valu is usd for all bits, calculatd assuming simply supportd boundary condition) xplains th diffrnc btwn th prdiction of th formula and FEA for bits largr than 27 mm. To dmonstrat this, w plot th maximum glass pan dflction for diffrnt valus of salant bit (Fig. 11). W also plot th valu of th dflction calculatd assuming a simply supportd glass pan. W obsrv that for small bit valus, th maximum glass dflction is vry clos to th valu calculatd using th simply supportd glass pan, maning that th joint dos not contribut in rducing th glass pan dflction. On th contrary, for larg bits, particularly abov

13 Nxt gnration calculation mthod for structural silicon 181 Fig. 12 Joint maximum longation in function of joint bit: FEA prdiction; quation assuming glass dformation not influncd by joint bit and calculatd assuming a simply supportd glass pan; quation assuming glass dformation is influncd by joint rigidity 25 mm, joint rigidity limits glass dflction in a nonngligibl way. W now us th formula in Eq. 30 but associating diffrnt valus for th tilt angl α dpnding on th joint bit (th angl was calculatd from FEA having joint contributing in limiting glass dflction Fig. 11). This rsults in a good agrmnt with FEA prdictions as shown on Fig. 12. Whil w show why a diffrnc xists btwn th prdictions of th nw quations and FEA simulation, calculating glass pan dformation assuming simply supportd boundary conditions still maks a lot of sns as nglcting th ffct illustratd in this paragraph lads to an rror lowr than 3% for bits of 30 mm and blow 9% for bits of 40 mm. 4 Conclusions As altrnativ to FEA, a nxt gnration quation for joint dimnsioning is proposd giving rsults clos to th prdictions obtaind using FEA simulation. Th approach followd is to calculat for th façad joint th nginring strain (or quivalntly, th nginring strss) and compar it with th strss strain bhavior masurd on an H-bar sampl having th sam aspct ratio. Th diffrnc btwn FEA prdiction and th nw drivd rlationship obsrvd for small bits coms from th hypothsis of matrial linar bhavior. A combination of small bits and high windloads will mov th bhavior out of linarity. Howvr, this is a thortical cas sinc larg windloads will always lad to larg bits. Hnc using linar assumption and th corrsponding quation for ral facad projcts is a bttr option. Th diffrnc btwn FEA prdiction and th nw drivd rlationship obsrvd for vry larg bit coms from th assumption that th joint dos not influnc th glass pan dflction. If th joint cannot influnc a local bnding of th glass nar th glass primtr (bcaus of th diffrnc in matrial rigidity btwn salant and glass), th joint contributs to limit th glass dflction by limiting th translation of th glass xtrmitis along th x axis. Adding this ffct into th quation, an accptabl match btwn FEA and th quation is obsrvd for larg bits valus. Howvr, calculating glass pan dformation assuming simply supportd boundary conditions is accptabl as rrors ar limitd to a fw % for larg bits. Complianc with thical standards Conflict of intrst On bhalf of all authors, th corrsponding author stats that thr is no conflict of intrst Rfrncs ASTM, ASTM C1401, Standard guid for structural salant glazing (2014) ASTM, ASTM D412, Standard tst mthods for vulcanizd rubbr and thrmoplastic lastomrs tnsion (2016)

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