Buckling of laminated glass columns

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1 Buckng of amnated gass coumns Johan Baauwendraad Facuty of Cv Engneerng and Geoscences, Deft Unversty of Technoogy, Deft, the Netherands The buckng force of a amnated gass coumn s hghy dependent on the shear stffness of the soft fo whch connects the two gass ayers. The vaue of the buckng force s bounded by the ower mt when no fo s present (two separate gass panes) and the upper mt when the fo s nfntey stff (the gass ayers are deay couped). A formua for the buckng force shoud match these utmate vaues. Three formuas for the buckng force of a amnated gass coumn exst. One of them produces the correct ower and upper mt, one produces the correct ower mt and a conservatve upper mt and one produces a zero ower mt and conservatve upper mt. The way they appear n the terature, the formuas do not provde engneerng nsght and do not contan a dstnct parameter whch contros the transton from the ower to the upper mt. In the present artce, an aternatve formua s derved on the bass of a set of three smutaneous dfferenta equatons, supposton of a sne/cosne dspacement fed, and the formuaton of an egenvaue probem. The new formua s smpe and provdes engneerng nsght. Its dervaton yeds a dmensoness parameter whch contros the transton from the ower to the upper mt. The smartes and/or dfferences wth the exstng formuas are dscussed. Subsequenty, nta deformaton of the gass ayers s taken nto account, transferrng the stabty probem nto a strength probem, so that faure s governed by tense strength. Ths aows straghtforward computaton of the stresses and a unty check to be done. Key words: Lamnated gass, buckng formuas, dervaton 1 Introducton Lamnated gass eements consst of two gass ayers wth a thn fo n between. As such, amnated gass s a speca appcaton of a sandwch structure. In sandwch eements often the faces are thn f compared to the core. In amnated gass the opposte occurs, the gass ayers are reatvey thck and the fo n between s thn. It means that one can ony use the sandwch theory for thck faces. The present paper deas wth one-dmensona coumn buckng of amnated gass panes and restrcts to theoretca work. For nterestng expermenta work t s referred to recent pubcatons ke (Lube, 004) and (Lube et a., 005). When a formua for coumn buckng s requred, researchers refer to (hand)books for sandwch constructon and seect the reevant formua for thck faces. Exampes are found n (Aen, 1969), ( Satter et a., HERON Vo. 5 (007) No. 1/

2 1974) and (Zenkert, 1997). The Satter formua s aso ncuded n (Stamm et a., 1974). It s a check on the vadty of the formua that the formua must produce correct vaues for two boundng cases. A ower mt L occurs when no fo s put n at a. Now the two gass panes behave ndependenty of each other and the buckng force s smpy the sum of the two separate buckng forces, so t s governed by the sum of the bendng stffness of the two ndvdua gass panes. An upper mt U occurs when the fo has extremey arge shear stffness. Now the two gass panes are deay couped and the bendng stffness of the composed cross-secton governs the buckng force. One of the three mentoned exstng formuas correcty meets the requrement to be bounded by these two mt vaues, one does t approxmatey and one fas to do so. But even the formua whch correcty meets the requrement, eaves us troubed wth the fact that no cear dmensoness parameter contros the transton from the one mt to the other. Therefore, an aternatve dervaton s made, startng from the bascs, wth two resuts: () a charmng new formua s obtaned and () the dervaton ncey reveas the dmensoness parameter whch contros the transton. The new formua s consdered to be an mprovement over the od ones from the vewpont of eegance and nsght. Fgure 1 shows the composton of the pane and ts poston wth respect to the x- and z-axs. The gass matera s fuy near eastc wth moduus of eastcty E. The two ayers, whch can be dfferent, are numbered 1 and. These numbers w be used as subscrpts to quanttes reated to the gass ayer cross-sectons. The thcknesses are t 1 and t, the areas A1 and A and the second-order moments I 1 and I respectvey. The fo matera s thn wth regard to the gass ayers and s soft, so just shear stresses are consdered and no norma stresses are taken nto account. The shear moduus s Gs and the fo thckness t s. Though n fact the fo s a vscoeastc matera wth a G s dependng on temperature and oad duraton, t s treated here as a near-eastc matera, whch mpes that the approach s vad for short oad duraton, ke wnd and mpact. The gass pane has a ength and wdth b and s smpy supported at both ends. The buckng oad cr s apped n the neutra pane of the two gass ayers, whch occurs f they are deay connected. Ths pane s on a dstance e 1 to the centre ne of gass ayer 1 and dstance e to the centre ne of gass ayer. These dstances are: A A e1 = e; e 1 = e (1) A A where e= e1 + e ; A= A1 + A () 148

3 In reaty no gass pane w be perfecty straght, so hereafter an nta sne-shaped mperfecton w be assumed wth a md-span vaue w o. When the buckng oad s not apped n the neutra pane, but eccentrcay, the eccentrcty s smpy added to the nta dspacement w o, whch s not totay correct but a suffcenty cose approxmaton n vew of the fact that the vaue of wo anyhow has to be chosen on bass of good judgment. t 1 t s t 1 A 1,I1 A,I e 1 e e x z Fgure 1: Geometrc propertes of a amnated gass eement of two ayers Hereafter, n Secton the three exstng formuas are dscussed and n Secton 3 the procedure for cassca Euer buckng s caed n mnd very brefy as a bass for the dervaton of the new formua n Secton 4. In Secton 5 the new formua s compared to the three exstng ones, Secton 6 dscusses how an nta mperfecton changes the buckng probem nto a strength probem and n Secton 7 an exampe s dscussed. Concusons are drawn n Secton 8. Dscusson of exstng formua In ths secton the buckng formuas of Satter, Zenkert and Aen are reproduced and t w be checked f they correcty match the ower and upper mt. For ths purpose a three formuas are wrtten n the same notaton, and t aso s convenent to ntroduce a few auxary quanttes: DL = EI : summed bendng stffness of the two ndvdua gass ayers, DO = EIO : bendng stffness of the jont two ayers, deay connected, excudng bendng stffness of the ndvdua ayers, DU = EIU : bendng stffness of the jont two ayers, deay connected, ncudng bendng stffness of the ndvdua ayers, D S : shear stffness of the fo. 149

4 Heren: I = I1 + I ; e IO = 1 1 A1 + A ; IU = IL + IO ; Gs be DS = (3) ts Correspondngy we defne buckng forces: EI L = π ; EI O = π O ; EI U = π U ; S = DS (4) where L, O, U are buckng forces when shear deformaton s negected and S appes f ony shear deformatons occur. L s the ower mt of the buckng force, U and O an approxmaton of the upper mt (namey U mnus L ). the upper mt Formuas n terature The buckng formuas as coped from terature, n notaton of ths artce, read: Satter et a., 1974: π ( 1 + α + π αβ) D O cr = 1 + π β (5) where D1 + D α = ; DO Zenkert, 1997: ts D β = O (6) Gs be 4 π DD L O π D + O 4 DS cr = π D 1 + O DS Aen, 1969: O cr = 1 + O Gsb(ts + t) / ts (7) (8) Of these formuas, the ones n (5) and (7) appy generay for non-symmetrc sandwch panes, whe the thrd n (8) appes for symmetrc panes ony. In the form the formuas appear here, t s not easy to recognze whch terms represent the contrbuton of the ndvdua gass ayers and whch the jont acton of fo and gass ayers, even ess whether the ower and upper mt are 150

5 matched correcty. Aso t s hard to compare them. In order to factate that, the formuas can be rewrtten by approprate manpuatons nto: Satter et a., 1974: S U+ L O cr = S + O (9) Zenkert, 1997: S O + L O cr = S + O (10) Aen: S O cr = S + O (11) Now t can be checked whether the ower and upper mts are matched. The three ower mts, occurrng for D S = 0 (so S = 0), are L, L and 0, respectvey. We concude that the formuas of Satter et a. and Zenkert yed the expected ower mt, whereas the formua of Aen predcts the wrong resut. It predcts that the gass ayers can not carry any axa oad f no fo s put n. The three upper mts, occurrng for DS (so S ), are U, O and O, respectvey. Ony the frst one (Satter et a.) s correct. The other two formuas yed an approxmaton for the upper vaue of the buckng force, because the contrbuton of the ndvdua gass ayers to the bendng stffness appears to be negected n the formua for the upper mt. Whereas the dfference for a sandwch constructon wth meta faces may be neggbe, t s not for amnated gass. Then the dfference can be n the order of 0%. For the rest, () the approxmaton s on the safe sde, because the formuas underestmate the buckng force, and () the rea buckng force s n between the ower and upper mt, so the dfference anyhow s smaer then 0%. As for the background of the formuas, Satter et a. pubsh the fu dervaton of ther formua. The Aen formua s obtaned by the we-known approxmaton that cr can be computed from the equaton: = + (1) cr E S where E s the Euer buckng force for pure bendng. Why Aen substtutes O for E and not U s not cear. Zenkert aso starts from formua (1), but after that accounts for the thckness of 151

6 the faces by gvng the formua a sghty dfferent form, 'snce t must be derved from another governng equaton', wthout showng that equaton. 3 rocedure for cassca Euer buckng In Secton 4 use w be made of knowedge whch s borrowed from the procedure for cassca Euer buckng for a snge gass ayer. Therefore, ths matera w be dscussed here brefy. We consder a smpy supported deay straght beam aong the x-axs wth bendng stffness EI, ength, apped force n axa drecton, dstrbuted atera oad f and dspacement w n z- drecton transverse to the beam axs, see Fgure. Ony fexura deformaton of the beam s taken nto account. The orgn of the set of axes s n the eft-hand support. The dfferenta equaton for ths beam s: 4 d w d w EI + = f (13) 4 The frst term n the eft-hand part of the equaton s the we-known frst-order Euer beam contrbuton and the second term represents the second-order effect f the equbrum s consdered n the deformed state. In the dervaton for the amnated gass pane hereafter the frst term w be adapted, but the second term can be borrowed unchanged. z f w w o Fgure : Inta defecton w o and fna defecton w after appcaton of oad n a snge gass eement x For zero oad f, a sne-shaped defecton functon w( x) = w sn( π x / ) (14) satsfes the boundary condtons w = 0 and M = 0 for dspacement and bendng moment respectvey, and transforms dfferenta equaton (13) nto: 15

7 π π EI w= 0 (15) Buckng means that w can be nonzero n absence of oadng. In case of nonzero w the eft-hand member of equaton (15) can vansh ony f the term between brackets s zero, yedng the cassca Euer souton for the crtca oad: π EI cr = (16) In case of a sne-shaped nta defecton wth maxmum vaue wo the dfferenta equaton changes for zero f nto: 4 d (w w o ) d w EI + = 0 (17) 4 No stabty probem occurs anymore and the defecton w can be soved drecty. Agan the souton w(x) s sne-shaped and now the vaue w s soved from π EI π EI w= w o (18) Accountng for equaton (16) we can rewrte (18) nto: (cr )w= cr wo (19) If now the dmensoness parameter n s ntroduced accordng to: n = cr (0) ( aways arger than 1), we sove from equaton (19): n w= w n 1 o (1) 153

8 The term n/(n - 1) s an ampfcaton factor for the defecton. In words, the nta defecton w o has grown to w due to the appcaton of the axa compressve oad, and the stabty probem has changed nto a strength probem. In the next secton ths procedure aso appes for the amnated gass case. A formua dfferent from (16) w be found for cr but equatons (19), (0) and (1) st hod true. 4 rocedure for amnated gass eement The two gass ayers have been drawn n Fgure 3. The fo between the ayers s marked wth dstrbuted strps perpendcuar to the ayers. We defne dspacements u 1 and u n the mdpane of the respectve ayers and dspacement w n the transverse drecton, whch s common to both gass ayers. The dspacements u 1, u and w are ndependent degrees of freedom. Ths mpes that dstrbuted oadng can be apped n each of these three drectons. We ndeed ntroduce a dstrbuted transverse oad f per unt ength of the gass pane n w-drecton and f 1 and f n the drectons of u1 and u. The atter two do not occur n reaty but are apped for N 1 N M 1 M z V 1 V f f1 S q S V f u 1 u dv1 V1 + dn1 N1 + dm1 M1 + dv + dn N + dm M + f w f 1 f u 1 e 1 e Δ u e x dw/ e Fgure 3: Defnton of degrees of freedom (top), quanttes payng a roe n equbrum (bottom eft) and quanttes payng a roe n knematcs (bottom rght). 154

9 the tme beng for reasons of competeness. In course of the dervaton we w put them to zero. The oad f n w-drecton s apped on ayer 1, but w be party transferred to ayer by a dstrbuted compresson force q n the fo atera to the gass ayers. In the ayers norma forces N1 and N, bendng moments M1 and M, and shear forces V 1 and V w occur. The strans ε 1 and ε due to the norma forces and the curvature κ due to the bendng moments (the same curvature n both ayers) are taken nto account, but the deformaton due to the shear forces s negected. A shear force S per unt ength occurs n the md-pane of the fo. The deformaton whch s assocated wth S s a sp Δ between the nner faces of ayer 1 and ayer. The scheme n Fgure 4 summarzes the quanttes whch pay a roe n the amnated gass probem. It shows vectors for dspacements, deformatons, nterna secton forces and externa oadng respectvey. The scheme s of great hep n dervng the needed set of reatons: () the knematca reatonshp between dspacements and deformaton, () the consttutve reatonshp between deformatons and secton forces, and () the equbrum reatonshps between the nterna secton forces and externa oadng. u1 u w ε1 ε κ Δ N1 N M S f1 f f knematc reatons consttutve reatons equbrum reatons Fgure 4: Schematc overvew of quanttes and reatonshps that govern the buckng behavour of a amnated gass pane The eft-hand bottom part of Fgure 3 shows whch forces and moments act on dfferenta gass parts of ength. In ths part of the fgure the fo thckness s exaggerated. The pcture shows the sgn conventon for a quanttes. The rght-hand bottom part shows n whch way the sp Δ s reated to the dspacements u 1, u and dw/. Because the defecton w s common to the two gass ayers, t s convenent to sum up hereafter ther fexura stffness, bendng moment and shear force, ntroducng tota vaues EI, M and V: EI = EI1 + EI ; M = M1 + M; V = V1 + V () 4.1 Set of three reatonshps The knematca reatonshps are: 155

10 du1 du ε 1 = ; ε = ; d w dw κ = ; Δ= u 1 + u + e (3) The consttutve reatonshps are: N1 = EA1ε 1 ; N = EAε ; M = EIκ ; S= ksδ (4) Heren the fo stffness k s s defned by: Gb k s s = (5) ts Where b and t s are defned at page 148 and 149. The equatons of equbrum n x-drecton of the two gass parts can be drecty wrtten: dn1 dn S = f1 ; + S = f (6) The reatonshp n z-drecton between the moment M, shear force S and oad f requres a number of ntermedate steps. Equbrum n rotatona drecton of the two gass parts requres: dm1 es 1 V = ; dm es V 0 + = (7) Force equbrum n z-drecton s satsfed by: dv 1 dv f q+ = 0 ; q + = 0 (8) Summng up the two reatonshps of (7) and smary of (8), accountng for (), yeds: dm es V 0 + = ; dv f + = 0 (9) The ast step s to emnate V from (9), wth the resut: d M ds e f = (30) 156

11 The set reatonshps (3), (4), (6) and (30) are the bass for the dervaton of the buckng formua. 4. Determnaton of buckng force Substtuton of (17) nto (18) makes the secton forces dependent of the dspacements. Subsequenty ths resut s substtuted nto the equbrum equatons (6) and (30), yedng three smutaneous dfferenta equatons: d u 1 dw EA1 k s u1 u e + + = f1 d u dw EA + k s u1 u e + + = f (31) 4 d w du1 du d w d w EI e k 4 s + + e + = f In the thrd dfferenta equaton we have added the second-order term as was done earer n equaton (13). Wrtten n matrx form for zero dstrbuted oadng f1, f and f we obtan: d d EA1 + k s ks eks u1 0 d d ks EA + k s eks u = 0 4 d d d d ek w 0 s eks EI ( e k 4 s) + (3) The tra souton, π x π x u(x) 1 = u1cos ; u (x) = ucos ; π x w(x) = w sn (33) satsfyng the boundary condtons, transforms (3) nto: π k1 + k 1 0 s ks eks u π ks k + ks eks u = 0 π π π eks eks { L + (e ks ) } w 0 (34) 157

12 where L s defned n (4) and the 'stffnesses' k1 and k as foows: π EA k 1 1 = ; π EA k = (35) Buckng occurs f nonzero dspacements u 1, u and w can occur for zero oadng. Ths s the case when the determnant of the stffness matrx n (34) becomes zero. Eaboraton of ths determnant devers (after mutpcaton by π ): (k1k+ k1ks + kk s)l + k1kkse (k1k+ k1ks + kk s)cr = 0 (36) From ths we sove the surprsngy smpe buckng formua: 1 cr = L + e f1 + fs + f (37) where the fexbtes n the denomnator are: 1 f1 = k1 ; 1 fs = ks ; 1 f = k (38) It s easy understood that the frst term n the rght-hand member of equaton (37) s due to the bendng moments and the second term to the norma forces n the ayers. The ower and upper mt vaue of ths formua correspond wth k s = 0 and k s respectvey, so f s and f s = 0 : Lower mt: Upper Lmt: L 1 U = L + e f1 + f (39) It s concuded that formua (37) satsfes the condton that cr = L for zero shear stffness k s of the fo (very arge f s ) and cr = U for very arge shear stffness k s (zero f s ). The atter s easy seen from a comparson wth equatons (3) and (4), remndng the defnton of k1 and k n (35). The structure of reatonshp (37) factates to wrte the buckng force cr as an nterpoaton between the ower and upper mt. If we ntroduce: 158

13 f1 + f ξ = f1 + fs + f (40) the equaton (37) can be transformed nto: cr = ( 1 ξ)l + ξu (41) The transton from L to U occurs neary, see Fgure 5, and ξ s the controng dmensoness parameter. Ths parameter s the rato between on the one hand the sum of the fexbtes of the two gass ayers and on the other hand the sum of the fexbtes of both the two gass ayers and the fo. If the fo stffness k s vares from zero to nfnty, the parameter ξ runs from 0 to 1. cr U L 0 1 ξ Fgure 5: The dmensoness parameter ξ contros a near transton from the ower mt buckng mt to the upper one. 5 Comparson of new formua and exstng formuas In hndsght t appears that the formua of Satter et a. n (5) and the new one n (37) are n fact the same formua. The formua of Zenkert s equa to the new formua n the ower mt, but becomes graduay smaer than the new one for ncreasng fo stffness; the upper mt s L smaer. The formua of Aen produces a vaue of the buckng force that s L smaer than the new one for each vaue of the fo stffness. Ths and that s seen best f we start from the formua of Satter et a. n (9). Substtuton of U = L + O changes t nto: (S + O)L SO S O cr = + = L + S + O S + O S + O (4) 159

14 After dvson of numerator and denomnator of the ast term by S O and subsequent substtuton of (4) we obtan: 1 1 cr = L + = 1 1 L O + S π EIO DS (43) Accountng for the reatonshps (3), (5), (35) and (38) the formua changes nto: e 1 cr = L + = 1 L + e f1 + fs + f + + π EA1 π EA ks (44) Indeed, ths Satter equaton s equa to the freshy derved formua n (37). As a consequence, reatonshp (4) aso s a representaton of the new formua. If compared to formua (11) of Aen, t carfes that he msses the contrbuton L, regardess the vaue of the fo stffness. Summng up, the formua of Satter et a. s correct, but n the opnon of the author the freshy derved formua s preferabe from the vewpont of eegance and nsght. 6 Unty check on stresses The nta mperfecton n rea amnated gass panes makes that faure s not controed by stabty but strength. The compressve stress of gass s very hgh f compared to the tense strength, hence the tense strength governs the appcabe compressve force (Lube, 004). Gven an axa oad and an nta defecton w o md-span, the fna defecton w s cacuated from equatons (0) and (1): n = cr ; n w= w n 1 o (45) where cr s defned n (37). The tota moment md-span: Mtota = w (46) s spt n two contrbutons, one whch conssts of the sum M of the bendng moments and one whch s due to the norma forces N n the ayers : 160

15 Mtota = MM + MN (47) The two parts n the rght-hand member of (44) can be easy determned because ther rato s equa to the rato of the terms n equaton (37) for cr. Here we restrct ourseves to the case of a postve wo vaue and consder the face of the ayers wth postve norma n the z-drecton, here caed bottom face. Tense stresses defntey w occur at the bottom face of ayer, but the stress n the bottom face of ayer 1 shoud aso be examned. The moment MM dvded over ayer 1 and proportona to ther fexura stffness: EI1 EI M1 = MM ; M = MM (48) EI EI The moment MN yeds norma forces: M N1 = Ne ; N = Ne where N N e = (49) e The gass ayers are further oaded by a compressve force whch s spt n 1 and n the rato A 1 to A. The unty check n the bottom face of ayer 1 and ayer s respectvey: Ne 1 M1 N + 1 ; e M + 1 (50) fa t 1 fw t 1 fa t fw t where W1 and W are the secton modu of the ayers and f t s the desgn tense strength. Most key the atter check n (47) s the crtca one. 7 Exampe We consder a smpy supported gass pane 10/1.5/10 mm of ength = 1500 mm and wdth b = 1000 mm. The gass eastcty moduus s E = N/mm and the fo shear moduus G s = 0.5 N/mm. An nta defecton of w o = /400 = 1500/400 = 3.75 mm s assumed, as s recommended n (Lube, 004). The supposed oad s 50 kn and a unty check w be done. Smar to the European code desgn preen :000 E for foat gass, an aowabe tense stress f t = 17 N/mm s chosen. 161

16 e = = 11.5 mm. A 1 = A = = mm². I 1 = I = ³/1 = mm 4. Cacuaton of stffnesses π EA1 π k 1 = = = N/mm π EA π k = = = N/mm Gs b k s = = = N/mm. t s 15. Cacuaton of mts π E( I1 + I ) π L = = = N e U = L f f = = + = N. Cacuaton of parameter ξ and buckng force cr f f ξ = = = = f fs + f cr = ( 1 ξ)l + ξ U = N. cr = = N. Cacuaton of ampfcaton factor cr n n = = = = = n Cacuaton of stresses and unty check n ayer. n w= w o =.. =. mm. n M = w = = Nmm M = MM + MN = = Nmm M N N e = = = N. e

17 W = bt / 6 = / 6 = mm 3. The stresses are cacuated from the equaton: MM N e σ =± + =± + =± N/mm ( = 1, ). W A The resut of the stress cacuaton s shown n Fgure 6. The unty check for gass ayer reads: MM N e = + = 091. < 1 fw t fa t The unty check s satsfed, so the supposed oad can be carred Fgure 6: Stresses n the gass ayers for the dscussed exampe (ength n mm and stress n N/mm²). 8 Concusons Of three consdered formuas for the buckng force of a amnated gass coumn, one produces the correct ower mt (at zero fo stffness) and upper mt (at very stff fo), one produces the correct ower mt but a conservatve upper mt (then mssng the contrbuton of the ndvdua gass ayers), and one aways produces a conservatve vaue (mssng the contrbuton of the ndvdua gass ayers for any fo stffness). As they appear n the terature, the formuas do not provde engneerng nsght and do not show whch parameter contros the transton from ower mt (uncouped gass ayers) to upper mt (deay connected ayers). The freshy derved new formua for the buckng force does t n an eegant way, s smpe and provdes nsght. The dervaton yeded the dmensoness parameter, whch contros the contrbuton of the fo between the two gass ayers. A unty check for tense stress can be done on the bass of a proper choce of an nta defecton. 163

18 References Lube, A., 004, Stabtät von Trageementen aus Gas, doctora thess EFL 3014, Ecoe oytechnque Fédérae de Lausanne, Lausanne. Lube, A., Crsne M., 005, ate buckng of gass panes, Gass roceedngs Days, Fnand. Stamm, K., Wtte, H., 1974, Sandwchkonstruktonen Berechnung, Fertgung, Ausfuehrung, Sprnger Verag, Wen. Satter, K., Sten,., 1974, Ingeneurbauten 3, Theore und raxs, Sprnger-Verag, Wen. Zenkert, D. (edtor), 1997, The Handbook of Sandwch Constructon, Engneerng Materas Advsory Servce Ltd., Cradey Heath, West-Mdands, Unted Kngdom. Aen, H.G., 1969, Anayss and Desgn of Structura Sandwch anes, ergamon, Oxford. UK, Ch