Predicting the pull-out strength of glued-in rods

Transcription

1 Predctng the pull-out strength of glued-n rods Gustafsson, Per J. 1 and Serrano, Erk ABSTRACT The paper presents analytcal and numercal models to predct the pull-out strength of glued-n rods for tmber structures. The analytcal model s a shear lag model based on fracture mechancs and generalsatons of the so-called Volkersen theory. The theory, although based on lnear elastc stress analyss, s capable of accountng for the fracture propertes of the adhesve layer. Ths s accomplshed by lettng the one-dmensonal consttutve law of the adhesve layer be expressed n terms of strength and fracture energy rather than shear stress and stran. The numercal model s a -D FEmodel where a non-lnear fracture mechancs approach s used to model the behavour of the adhesve layer. Ths model ncludes the combned stress states of shear and peel stress actng smultaneously, and also the gradual and progressve damage of the bond layer. Use of ether of the descrbed models requres non-conventonal materal data, such as the fracture energy of the adhesve layer. A test method for obtanng such data, together wth test results from such tests s also presented. INTRODUCTION Glued n rods are used n heavy tmber constructons where jonts wth a large load capacty are needed. An example s portal frames made of glulam and wth strong corner jonts and column to ground connectons acheved by glued-n rods. Knowledge needed for the safe and accurate desgn of glued-n-rod jonts has so far been developed essentally by tests. By test results emprcal and sem-emprcal strength equatons, Acher et al. (1999), have been derved, formng the base for desgn rules used n codes of practse. The central ssue n ths paper s methods for strength analyss of glued n rods wth a ratonal theoretcal bass. Analytcal and numercal models for stress and strength analyss are dscussed and examples of results presented. Theory based ratonal methods of calculaton requre n general not only verfcaton by full scale tests but n most cases also emprcal calbraton of the model. Stll ratonal models are n the long run beleved to be preferable as compared to purely emprcal and statstcal models. The models dealt wth predct the pull-out strength from propertes of the materals and bonds nvolved. The models have ther bass n fracture mechancs and a key materal property s the fracture characterstcs of the decsve fracture area. To ths end a method, Johansson et al. (1995), for testng bond propertes n terms of peak stress, fracture energy and the complete shear stress versus shear slp performance curve has been further developed and results obtaned for three adhesve-adherend combnatons are ndcated. ANALYTICAL SHEAR LAG FRACTURE MODEL The materals nvolved are n the analytcal analyses assumed to be lnear elastc. The elastc propertes of the wood and the rod are assumed to be orthotropc and sotropc, respectvely, wth conventonal stffness parameter values as determned by tests. The stffness of the bond layer s for pull-out strength analyss chosen so that the most essental strength and fracture characterstcs of the bond layer are accurately modelled. Ths means that an effectve lnear stffness value s used, representng the non-lnear performance of the bond layer at hgh stress wth gradual damage and fracture softenng. For shear deformaton of the layer, the effectve elastc stffness, G, s determned from the condton of correct shear fracture energy, G f, defned by G f s ( τ / G ) t = τ ( s) ds = γ G f f τ t d( tγ ) = G (1) 1 Assocate Professor, Dv. of Structural Mechancs, Lund Unversty, Sweden Lcentate n Engneerng, Dv. of Structural Mechancs, Lund Unversty, Sweden

2 here τ s the shear stress at shear deformaton s, and where s s the deformaton at complete fracture of the layer, γ=s/t s the mean shear stran, t s the bond layer thckness and τ f the shear strength. G f and τ f are bond layer materal propertes from whch the effectve shear stffness of layer, G /t, s determned from eq (1), gvng G f f / t = τ /(G ) () Ths effectve stffness s commonly much less, typcally one or two orders of magntude less, than the ntal elastc stffness of the bond layer. Usng the effectve stffness together wth the local bond layer falure crteron τ = τ max f () as ndcator also for global pull-out falure, correct modellng of the local strength as well as the bond layer fracture energy s acheved. Ths s as n non-lnear fracture mechancs or localsed damage modellng, whereas n lnear elastc fracture mechancs modellng only the correct fracture energy of the materal s reproduced and n conventonal maxmum stress base modellng only the correct fracture stress s reproduced, Gustafsson (1987). Lmtng ths presentaton to pull-out strength models where only the shear stress n the bond layer s consdered n the fracture crteron,.e. eq (), the remanng task s stress analyss for the shear stress n the bond layer for gven jont geometry and stffness propertes of the materals. The jont geometry consdered n the analytcal analyss s bascally ax-symmetrc wth the gran drecton of the wood parallel to the rod. In the dervaton of equatons for strength analyss t s however convenent frst to consder an nplane sngle lap jont lap jont wth a boundary condton at the lower edge accordng to Fgure 1. The vertcal dsplacement s prescrbed to zero, correspondng to the symmetry axs for an ax-symmetrc jont beng straght also when the jont s loaded. Results obtaned for the plan jont can then be appled to an ax-symmetrc jont by makng relevant parameter substtutons. Fgure 1. An n-plane sngle lap jont wth a boundary condton at the lower edge. Dfferent equatons wth dfferent degree of complexty can be derved for the stress analyss of the plane jont. Modellng the rod as a bar, the bond layer as a thn layer wth deformaton possble both n shear and n the drecton perpendcular to the layer and the wooden adherend as a Tmoshenko beam takng nto account also stran perpendcular to the axs of the beam, the governng equatons found are 6 th order ordnary dfferental equaton wth the shear stress n the bond layer as the varable together wth a nd order dfferental equaton for dsplacements n the radal drecton. A specal and less complex case s obtaned by assumng the stffness of the bond layer and the wood to be nfnte n the drecton perpendcular to the layer. Ths mples zero curvature of the centre lne of the Tmoshenko beam. Thus the dsplacement n the x-drecton, u, s u( x, y) = u( x,) yθ( x) (4) where the nclnaton of the cross secton, Θ, at zero curvature s equal to mnus the shear stran n the wood, -γ. Havng defned the assumed shape of the dsplacement feld t s possble to derve a governng equaton by means equatons of

3 equlbrum, cross secton stffness propertes and compatblty between the wood, the bond layer an the rod. The equaton obtaned s τ,,,,,, + Sτ + Tτ = (5) wth S = ( G / t )( b /( E A ) + b /( E1A1 ) + e b /( E I ) G A /( E I ) T = ( G / t )( G A ) /( E I )( b /( E A ) + b /( E1 A1 )) The cross secton geometry parameters relevant at analyss of an ax-symmetrc jont are I b = πr, = ( π / ) A1 = π r1, A = π ( r y r ), e = ( / ) ( r y r ) ( r y r ) 4 4 ( r r ) (4π / 9) ( r r ) ( r r ) y y y (6a) (6b) (7a-e) where r 1 s the radus of the rod, r the centre radus of the bond layer and r y and r the outer and nner radus, respectvely, of the wood. Commonly s r 1 r r. The soluton of eq (5) s 4 k x τ = Ce, k1 = k = S / + ( S / ) T, k = k 4 = S / ( S / ) = 1 T (8) The four constants C can be determned from boundary condtons whch n terms τ and dervatves of τ at x= and at x=l can be obtaned from normal force, shear force, secton bendng moment and/or correspondng knematc condtons at x= and L. Snce the calculaton of C nvolve solvng a system of equaton wth four unknowns explct general equatons for the shear stress dstrbuton τ (x) at defned boundary condtons becomes comprehensve and numercal calculaton of C n general more convenent. By lettng the shear stffness of the wood adherend approach nfnty, G A,.e. for zero shear stran n the wood, eq (5) becomes a nd order equaton, recognsed as the well-known governng equaton for bar shear layer lap jonts, studed by Volkersen (198). Fgure shows τ (x) for a jont loaded n "pull-pull",.e. loaded by a tensle load P appled to the rod at x= and an equally large tensle load appled to the wood at x=l. The shear stress s shown for three values of the shear stffness of the wood: G = 7 MPa s a typcal value for wood, G = 5 MPa = E / represents the result f wood was sotropc and G represents the result obtaned by the bar shear lag theory presented by Volkersen. For ax-symmetrc jonts made of an sotropc materal t seems that the lap jont theory of Volkersen gve results that are close to those obtaned by the more advanced theores. Moreover t seems that the orthotropc property of wood wth a low shear stffness gves a sgnfcantly dfferent stress dstrbuton than obtaned for an sotropc materal. The results shown n Fgure were obtaned for P=1 N, A 1 = mm, E 1 = GPa, A =1 mm, E =1 GPa, G vared, t=.5 mm, r= 8. mm, G =1 MPa and L=16 mm. Usng the falure crteron of eq () together wth consderaton of the fracture energy accordng to eq (), the pull-out strength calculatons have been made for varous geometres, materal characterstcs and loadng condtons. The results shown n Fgure for the mean shear stress at falure, P f /(πrl), were obtaned wth A 1 = mm, E 1 = GPa, A =1 mm, E =1 GPa, G =7 MPa, r= 8. mm and τ f =8. MPa. It s evdent that the jont length, bond fracture energy and loadng condtons are predcted to sgnfcantly effect the mean shear stress at falure. For small jont length the shear stress s almost unform at falure, correspondng to a plastc performance of the bond wth the no effect of the fracture energy on the pull-out strength. Only n that case s P f proportonal to τ f. For large jont length there s a strong stress concentraton at the ends of the jont. For ths case wth a strong stress concentraton the results of the present theory are found to concde wth those of lnear fracture mechancs: the pull-out strength P f s proportonal to G f 1/ and

4 not effected by bond shear strength τ f. For glued n rods of common length both G f 1/ and τ f are predcted to sgnfcantly effect P f. The breaks or knee-ponts n the curves for P f versus jont length L shown n Fgure are due to transton of the pont where τ = τ max from one end of the jont to the other. τ, [MPa] P f /(πrl), [MPa].5 G (bar shear lag theory) G =5 MPa ( sotropc ) G =7 MPa ( wood ) 8. τ f = 8. MPa Load: G f = 6. Pull-pull, 8. kj/m Pull-comp., 8. kj/m Pull-pull,. kj/m Pull-comp.,. kj/m Pull-pull,.5 kj/m Pull-comp.,.5 kj/m Poston, x, [mm] Fgure. Shear stress dstrbuton along a jont Glued length, L [mm] Fgure. Mean shear stress at pull out falure. FINITE ELEMENT FRACTURE MODELLING For some cases the peel stress perpendcular to the pull drecton and the actual D geometry of the jont may be of mportance n order to accurately predct the pull-out strength. To account for such features a D-fnte element (FE) model and a nonlnear materal model for the bondlne have been developed. The bass of the materal model s the theory of nonlnear fracture mechancs usng the concept of a cohesve crack. Ths approach makes t possble to account for the stran softenng and gradual and progressve damage of the bondlne. The model s a further development of a twodmensonal model, see Wernersson (1994) and Serrano and Gustafsson (1999). The bondlne model accounts for the effects of combned stress states ncludng two shear stress components and the peel stress n the bondlne. A general mxed mode state of deformaton s descrbed by the two relatve shear slps, s1 and s, and by the relatve normal deformaton n across the bondlne. The stress-deformaton response s assumed to be a pecewse lnear functon for unaxal stress states and t retans a pecewse lnear shape for radal deformaton paths (constant values of ( s1 : s : n )) but vares smoothly wth the degree of mxed mode, expressed by the mxed mode angels ϕ ss and ϕ sn. ϕ ss = arctan ( s1 / s ), ϕ sn = arctan ( s / n ), s s1 s = +. (9) For a gven state of deformaton the breakponts on the pecewse lnear stress-dsplacement curve are obtaned from the relaton s1, s 1, m + s, s, n + n, n, p = 1 (1) where superscrpt stands for the unaxal response and the powers m, n and p are measures of the couplng between the dfferent modes of fracture. The stresses at each breakpont are now calculated accordng to τ = σ = σ. (11) s1,1 s,1 1, τ 1,, τ, = τ,, s1,1 s,1 n,1 n,1

5 Knowng the stresses at the breakponts the stresses at the current state of deformaton can be obtaned by lnear nterpolaton. The current mplementaton of the bondlne model s a so-called smeared crack mplementaton. Ths means that the above stress-dsplacement relatons are transformed nto correspondng stress-stran relaton by dvdng by the wdth of the contnuum fnte element used to model the bond layer. In the present study t has been assumed that m=n=p=. The basc pure loadng stress-deformaton curves have been defned by three lnear parts, where τ 1, = τ 1,1/ and τ 1, = and s1, = 4 s1,1 and s1, = 9 s1, for shear n the s1-drecton. The performance n the s-drecton was assumed to be the same as n the s1-drecton. For the n-drecton σ = σ 1/4 and σ = and n, = n,1 and n, = 179 n,1. Moreover τ 1,1 = τ,1 = τ f and σ 1 = σ f whch together wth G f, s = τ1 d s1 = τ ds and Gf, n = σ dn defnes the bond layer propertes for gven numercal values of the four materal parameters τ f, G f,s, σ f and G f,n. These parameters represent the shear strength and fracture energy n shearng (Mode II) and peel strength and fracture energy n peelng (Mode I) respectvely. The numercal values chosen for the present study were τ f,s =1 MPa, G f,s = J/m σ f =4 MPa and G f,n =4 J/m. The materal model used for the bondlne s general n the sense that t can be used for both the extreme cases of brttle and ductle materals. The model wll gve results that concde wth those of the theores of lnear elastc fracture mechancs and plastcty respectvely. The element subdvson used and the knd of jont geometry studed are shown n Fgure 4. Due to symmetry only one half of the length of the specmen as well as one half of the wdth was analysed. The model conssts of approxmately 14 nodes, 4 degrees of freedom and 1 elements. The bond layer s modelled wth 5 elements n the axal drecton and 1 elements n the crcumferental drecton. The elements representng the wood and the steel are standard, soparametrc 8-node brck elements. The same type of element was used to model the bond layer, but wth reduced ntegraton wth only 1 Gauss-pont n order to avod problems wth ther extreme slenderness rato. In order to check f the fnte element mesh n the plane of the cross secton was fne enough, a mesh wth about 65 nodes and 195 degrees of freedom was used n one calculaton. Comparson wth the load capacty that was calculated wth the 14- node mesh gave a dfference of less than 1%. The loadng was appled by ncrements n dsplacement, not force. Dsplacement controlled loadng can make t possble to trace post peak-load behavour. Fgure 4. The fnte element model of a glued-n rod. The results from the FE-smulatons are dsplayed n Fgure 5 showng the lnear elastc stress dstrbuton n the bondlne and the stress dstrbuton at ultmate load. Clearly the behavour s strongly nonlnear, and about 65% of the bondlne length has become damaged at ultmate load. The results shown are vald for a jont wth L= mm, loaded n pullpull and wth a square cross secton of 1 1 mm, a rod dameter of 16 mm glued nto a hole of 17 mm dameter and wth stffness propertes typcal for softwood and steel. In terms of the parameters defned n Fgure 1: E 1 =1 GPa, A 1 =1 mm, E =14 GPa, G =5/8 MPa (the FE-model accounts for the dfference n radal and tangental elastcty parameters), A =1417 mm.

6 Fgure 5. Stress dstrbuton at lnear elastc state (left) and at ultmate load (rght). = shear stress -- = peel stress. TESTS OF BOND LINE FRACTURE PROPERTIES In order to obtan the parameters needed as nput for the above-descrbed theoretcal bondlne models, tests on small szed specmens were performed. These tests were made to obtan the local shear stress-slp performance of a glued n rod. The tests were performed usng dsplacement control, makng t possble to trace the post peak stress behavour,.e. the descendng branch of the stress dsplacement curve. Small wood prsms of sze mm were cut out from glulam beams. In the centre of the square cross secton of the prsm a hole of 17-mm dameter was drlled. After sealng the holes by means of a synthetc clay and a Teflon flm, steel rods were glued nto the hole. The rods were 9 mm long and had a dameter of 1-mm dameter along 8-mm length, whle at one end there was an 8-mm long part of 16-mm dameter that was threaded (M16). After curng, the synthetc clay and the Teflon flm were removed and the rod could then be fxed drectly nto the hydraulc grps of the testng machne. Wth the rod frmly held by the grps the wood prsms were pressed aganst a self-algnng plate durng the test n order to reduce the nfluence of the rod unntentonally beng glued nto the wood pece at a small angle. A schematc showng a test specmen and the test set-up used s shown n Fgure 6. Fgure 6. Test specmen and set-up used n the expermental study. Three dfferent adhesve types were tested: a fbre-renforced phenol-resorcnol (PRF), a -component polyurethane (PUR) and an epoxy (EPX). These tests were performed at º load to gran angle, cf. Fgure 6, for two dfferent tmber qualtes (C5 and C4) of dfferent denstes. The EPX was also tested for three addtonal load to gran angles (.5º, 45º and 9º) and fnally the EPX was used n a seres wth a glass fbre-renforced polyester (FRP) rod. The results from the tests n terms of stress-dsplacement curves are summarsed n Fgure 7. The fgure shows the results for the three dfferent adhesves glued nto the hgh densty wood (mean densty=458 kg/m dry weght/volume at ºC, 65%RH) at º load to gran angle. The results from the tests wth EPX for the dfferent load to gran angles are also shown. Each curve s a hand-drawn mean curve representng the average response from 6 tests. The deformaton shown s the addtonal deformaton due to the fracturng of the bondlne. Ths deformaton was obtaned by subtractng from the total deformaton measured n the test, the ntal elastc deformaton. The stress s the nomnal shear stress calculated for a

7 cylndrcal fracture surface of 16-mm dameter. There s a sgnfcant dfference n behavour between on the one hand the PRF and on the other hand the PUR and EPX. The PRF had a neglgble adheson to the steel surface, and the falure was n the steel-adhesve nterface. For the PUR the falure was n the adhesve close to the wood. For the EPX the falure was n the wood/adhesve nterface wth a large amount of wood fbres left on the adhesve surface for the º-tests. For the other load to gran angles there was only a small amount of wood fbres on the adhesve surface. Fgure 7. Inflence of adhesve type (left) and nfluence of load to gran angle for EPX (rght). Hand-drawn mean curves. CONCLUDING REMARK The dfferent models dscussed n ths paper are compared n Fgure 8, showng the nfluence of glued length on the nomnal shear strength of a jont. The results denoted deal plastc, lnear elastc fracture mechancs and Volkersen are all specal cases of the present analytcal model, n Fgure 8 denoted Tmoshenko. The numercal and analytcal results are all shown for τ f =1 MPa, G f = J/m, A 1 =1 mm and A =1417 mm. In the fgure s also shown one smple example of an emprcal equaton. Ths and other emprcal results are dscussed n Acher et al. (1999). P f /(πrl), [MPa] 1 Ideal plastc 9 6 Lnear elastc fracture mechancs Tmoshenko Volkersen FE-analyss Emprcal Glued length, L [mm] Fgure 8. Nomnal shear strength as predcted by analytcal models, by the FE-model and by an emprcal relaton.

8 ACKNOWLEDGEMENTS The present paper reports a part of the results from a European research programme Glued-n rods for tmber structures- GIROD. GIROD s co-ordnated by SP-Swedsh Natonal Testng and Research Insttute and the partcpatng nsttutes are: SP, FMPA-Baden Württemberg, Germany, TRADA Technology, UK, Unversty of Karlsruhe, Germany and Lund Unversty, Sweden. The support from the partcpatng partners and the fnancal support from the European Commsson through grant number: SMT4 - CT s gratefully acknowledged. REFERENCES Acher, S., Gustafsson, P.J. and Wolf, M Load dsplacement and bond strength of glued-n rods n tmber nfluenced by adhesve, wood densty, rod slenderness and dameter. In Proc. of 1 st RILEM Symposum on Tmber Engneerng, ed.: L. Bostöm, Stockholm, pp Gustafsson, P.J Analyss of generalsed Volkersen-jonts n terms of non-lnear fracture mechancs. Mechancal behavour of adhesve jonts. ed.: G. Verchery and A.H. Cardon, Edtons Plurals, pp. -8. Johansson, C.J., Serrano, E., Gustafsson, P.J., and Enqust, B Axal Strength of Glued-n Bolts. Calculaton model based on non-lnear fracture mechancs A prelmnary study. In Proc. of CIB-W18. Meetng twenty-eght, Copenhagen Denmark. Serrano, E. and Gustafsson, P.J Influence of bondlne brttleness and defects on the strength of tmber fnger-jonts. Internatonal Journal of Adheson and Adhesves. 19 (1) (1999) pp Volkersen, O De Netkraftvertelung n zugbeanspruchten Netverbndungen mt konstanten Laschenquerschntten. In Luftfahrtvorschung, Band 15, pp Wernersson, H Fracture characterzaton of wood adhesve bonds. PhD thess. Report TVSM-16. Lund Insttute of Technology, Dvson of Structural Mechancs. Lund, Sweden