Glued-in rod connections in bending: experiment and stochastic finite-element modelling

Transcription

1 ued-n rod connectons n bendng: experent and stochastc fnte-eeent odeng J. BAROTH, L. BODÉ, Ph. BRESSOLETTE, E. FOURNELY, P. RACHER Cv Engneerng Laboratory, C/U/S/T/ - Base Pasca Unversty, Ceront-Ferrand - France Les Cezeaux, BP AUBIERE CEDEX FRANCE.bode@cust.unv-bpceront.fr - p.racher@cust.unv-bpceront.fr Suary Ths paper presents prenary resuts of a research proect on gued-n rods for gua eeent connectons. The frst part descrbes experents carred out on bea-to-bea connectons. For gven desgn crtera, the effect of the nterna stffness dstrbuton s nvestgated. The experenta resuts exhbt the change n faure echans. Consderng these resuts, a near fnte eeent ode s dscussed to derve the forces transferred by the bars. In a second part, the spectra stochastc fnte eeent ethod s apped n order to anayse the varabty of the structura response of the tested connectons. In these cacuatons, the randoness s taen nto account by a set of ndependent og-nora rando varabes. Ths ode eads to the evauaton of frst statstca oents of the structura response. The coparson of the resuts wth soe Monte- Caro cacuatons show the good convergence of the deveoped approach. Keywords: ued-n rod; bendng connecton; stochastc fnte eeent ode.. Introducton Connectons n tber constructon are osty desgned wth dowe-type fasteners n shear. Ang proveent of ont effcency, gued-n rods have been deveoped for severa years to transfer forces between structura coponents []. ued-n rods present any advantages for aesthetca and econoca reasons. Recenty an European proect, IROD (ued-n rods for tber structures) was carred out to estabsh desgn equatons for the oad-carryng capacty of gued-n rods n tenson or shear. roup effect and ong-ter perforance were aso part of the proect [,3]. Consderng structura safety, t shoud be concuded that desgn equatons are dependent of the nteractng propertes assocated to the tber spece, the gue ne and the stee rod. For gven coponents, the pu-out strength for axay oaded rods parae to the gran s derved fro the Voersen ode. Furtherore, for gued-n rods actng as a sera syste,.e. bea to bea or bea to coun connectons, attenton shoud be pad to addtona forces due to the oca dstrbuton of stffness coponents. A research proect s carred out at our aboratory on oent-resstng onts. The scope of ths wor s to nvestgate dfferent aspects as geoetrca patterns of the rods, edge dstance and varabty of coponent propertes. In order to specfy rues eadng to ducte and reabe connectons, experenta and odeng approaches are deveoped. Ths paper focuses on prenary resuts fro bea connecton tests and anayss usng a stochastc fnte eeent ode. The frst part of the paper descrbes experents carred out on beas n-ne connecton. Usng stee bars of 5 and n daeter, the experenta resuts exhbt the nfuence of the ont coponents on the faure echans. Consderng these resuts, a near fnte eeent ode s dscussed to derve the forces transtted by the bars. Then, a stochastc approach s deveoped usng ths ode: appyng the spectra stochastc fnte eeent ethod proposed by hane and Spanos [4], a spfed approach s presented to anayse the structura response of tested gued connectons. In ths approach, the Young odu of the gued-anated beas and of the gued-n rod coponents are odeed by a set of ndependent ognora rando varabes.

2 . Bendng tests on bea connectons. Specens and experenta set-up Ang to transpose technques used n renforced concrete structures, experents are carred out n order to defne the abty of gued-n rods connecton to sustan the bea capacty. Consderng a gued anated cross-secton of (36x²) and a tber cass of L8h, the ean vaues of the basc bea propertes are: EI=8.4 MN.² and M u =5 N.. Three patterns (see fgure ) were desgned to acheve the above crtera. The stee rods are ndented stee renforcng bars of grade S5 (f y =5 Mpa) wth a daeter of or 5. An epoxy gue was used for a these onts. To reach a stee faure, the gued ength of the rods was desgned to be around 4 tes the daeter for the outerost ayer and was graduay decreased toward the nner ayers. Moreover the pre-bored hoes n the tber ebers have a daeter of 6 and 3. Type T-5 Type T-5 Type T F Fg. Cross-secton of onted beas Fg. Experenta arrangeent. Fgure defnes the set-up of these experents on the testng sab of our aboratory. In the ont area, deforatons are easured by stereovson.. Resuts Four pont bendng test resuts for the three confguratons are suarsed n fgure 3. M 5 (N.) 5 5 Test T3- N '' N T-5 N '' N T-5 (EI) test (MN.²) M u (N.) T - 5 5,8 94,3 T - 5,8 76,7 T3-4,9 38,3 (b) T3- T u () 8 Fg. 3 Test resuts: (a) oent-defecton reatonshp (b) faure odes of the onted beas. In the eastc range, the curves are aost sar up to a oent vaue of 3 N.. The oss of stffness due to the gued ont s ted to 5% wthout sgnfcant effect of the ont pattern. At the oca eve, the depth varaton of the bar ndentatons resuts n two types of pu-out faure of the outer rods (fgure 3b): faure of bond to stee for the rods and shear faure of the tber for the 5 rods. At the connecton scae, the faures are frst caused by tber spttng due to cobned shear and tenson perpendcuar to the gran. The resuts show ceary the benefts n usng a arger nuber of sa rods that reduce the stress concentraton n the tber [5]. In ths experent, the bendng capacty of the ont T3 s % greater than the others and cose to the capacty of the tber bea. In T3 confguraton, wth reduced stresses perpendcuar to the gran, the deforaton capacty s aso proved up to 4% coparatvey to the ont wth bars of 5.

3 3. Stochastc Fnte Eeent odeng 3. Deternstc Fnte Eeent ode The frst step conssts n creatng a fnte eeent ode of the syste. As the ntroducton of randoness w ncrease the sze of the syste, t s portant to obtan a ght but reastc ode. In ths context, we decded to bud a D ode wth ony Naver-Bernou bea eeents. Wth ths nd of eeents, ony the eber center-ne of the bea s eshed. In order to represent the structura eeent and the ont syste, we use three faes of bea eeents (fgure 4): - the eeents () represent the two parts of the gued anated bea, wth Young oduus of the tber parae to the gran (E = E = MPa) and goba bea secton and nerta, - the second group of eeents () represents the stee bars wth rea characterstcs (E =. 5 MPa), secton and nerta of the stee bars, - and the ast eeents (3) n the tber and the stee bars; the Young oduus of these vertca eeents s taen as E 3 = E /, consderng the tber oduus perpendcuar to the gran and the gue ne stffness. In order to represent Naver hypothess, we pose an portant vaue of nerta (we retan the goba bea vaue used for frst group). () (3) gap () Fg. 4 F.E. esh : (a) nta and defored confguraton, (b) zoo on the connecton area. As for the experent, the oadng s four ponts bendng. The whoe esh n type T-5 case (four bars) s depcted on fgure 4(a) (9 nodes, 48 eeents) wth the defored shape for the axu apped oad. As shown on the fgure 4(b), the horzonta stee bars are contnuous and no contact effects are taen nto account between the tber parts. Force 5 (N) Jont type T - 5 Jont type T , 4,3 4,4 4,5 4, 4,7 4,8 4,9 5, 5, x () 5, Fg. 5 Force dstrbuton n the transverse eeents. Wth ths ode, the suatons ead to resuts that are n the sae range than the experenta vaues. A good accuracy between experenta and nuerca dspan defexon s observed. Furtherore, fgure 5 shows, for T and T3 confguratons, the varaton of the forces n the vertca eeents (3) that are ned to the tenson perpendcuar to the gran due to the gued-n rods. A set of sa rods reduces sgnfcanty the tenson perpendcuar to the gran as t was shown n experentatons and entoned n terature [5]. 3. The SSFEM foruaton In a frst stochastc approach, we anayze the echanca probes consderng that the Young odu of each bea coponent are descrbed by ognora rando varabes. For faes of bea eeents, we suppose that the rando varabes, denoted E (=, ), are ndependent. For each ognora rando varabe E, t exsts a aussan rando varabe such as: E = exp( µ + σ ξ ) ()

4 where µ and σ are respectvey the ean and the standard devaton of and ξ s a standard aussan rando varabe. The randoness of these echanca paraeters s taen nto account nto the fnte eeent coputaton by usng the spectra approach due to hane and Spanos [4]. Ths Spectra Stochastc Fnte Eeent Method (SSFEM), ntay proposed for statonary aussan stochastc processes, has been extended ater n the ognora context [6]. The rando varabe E s approxated by an N-order truncated poynoa chaos expanson: ~ E E = N = e { } where Ψ ( ), Ψ ( ξ ) ξ, =,,N- s a set of orthogona one-densona Herte poynoas and e, are σ σ deternstc coeffcents gven by: e, = exp µ + (3)! In order to express the approxatons E ~ on the sae space, P-densona Herte Φ are ntroduced, such that: poynoas { } =, K P ~ E P = y, Φ, = ( ξ, ξ, K ξ ) Fro the FE odeng of the structura probe, the goba stffness atrx [ ] ~ [ K ] = A[ K ] = [ K ] = E [ ] = = = where A denotes the asseby process of the stffness atrx [ ] sae goba coordnates syste. [ ] equa to [ ] K dvded by () (4) K s expressed as: (5) K of the beas, expressed n the s K s the atrx obtaned after the ocasaton of [ ] E ~. Substtutng (4) nto (5) gves: P [ K ] = [ H ] Φ ( ξ, ξ, K, ξ) where [ H ] = y, [ ] = = K, and [ ] The degrees of freedo of the FE-dscretsaton, stored n the vector {q}, are deveoped on the sae bass of -densona Herte poynoas: (6) P P = {} q { q } Φ ( ξ, K, ξ ) q q Φ ( ξ, K, ξ ) for a the coponents q of {} q = Consequenty, the we-nown statc equbru equaton becoes here:, (7) P P [ K ]{} q { F} [ H ]{ q } Φ Φ { F} = = = = Fnay, the resdua of ths equbru syste s posed to be orthogona to each functon (Herte poynoa) Φ r (r=, P-) of the bass used to bud the approxated FE souton. Ths eads to the foowng near syste: P P = = [ H ]{ q } = { F } for r =,, P dr r K (9) where d r =< ΦΦΦ r > and, for a deternstc oadng vector {F}, {F r }={F} f r= and {F r }= f r> and < > denotes the expectaton. (8)

5 3.3 Resuts of the SSFE anayss The d.o.f. q of the esh (the dspaceents and the rotaton of the nodes) are soe rando varabes wth unnown probabty aws. They are approxated by P-order poynoa expansons (see equaton (7)). Then, the second-order characterstcs of these rando varabes (.e. eans and covarance atrx) can be easy approxated: the ean of the th d.o.f. and the covarance of the d.o.f. and are respectvey gven by: < q ean ean >= q, and Cov -5,69E- -5,7E- -5,73E- -5,75E- -5,77E- 48 nuber of suatons P ( ) q q = = 4 chaos expanson order N chaos expanson order N 5 q, q, < Φ In ths secton, we present resuts ony for the type T-5 dsposton at the axu oad eve: there are 6 rando varabes ( = 6 ), one for the tber beas, one for each of the four horzonta rods and one for the vertca beas. For these coputatons, the ean vaues are the ones gven n secton 3.. varance cov=% cov=% cov=3% cov=4% >,E-,E-,E- 8,E- 4,E-,E+ nuber of suatons chaos expanson order N Fg. 6.a Convergence of the ean. Fg. 6.b Convergence of the varance. Fgure 6.a shows the evouton of the ean of the vertca dspaceent of the eft oad appcaton pont (see fg. ) for dfferent orders (N=,,5) of the poynoa chaos expansons of the Young odu E (cf. eq. ()) and a coeffcent of varaton of %. The correspondng vaues, depcted by bac squares on the fgure, are copared to the ean coputed by Monte-Caro suatons (sod grey ne). Except for the case of expansons of order one (that eads to a deternstc cacuaton), the SSFEM resuts are n good accordance wth the suaton ones. But, athough the resuts are cose, the coputaton cost s greaty reduced when the SSFEM s used. In that case, a snge near syste of sze P n (where n s approxatey the nuber of d.o.f. of the esh) has to be soved whereas n s systes of sze n have to be soved n the Monte-Caro case (where n s s the tota nuber of suatons). For exape the axu vaue of n s was here equa to 5 whe the vaue of P ranged fro 7 to 5 ( P = (( N ) ) + ). On the sae coputer, the MC suatons tae 4 hours whereas 8n3 for P=7 to h5 for P=5 are needed when the SSFEM s used. The sae concusons can be drawn fro fgure 6.b for the varance of one d.o.f. (the nu vaue, obtaned for N=, corresponds here agan to the deternstc case). 4 Fgure 7 presents the evoutons of the e ean of the axa force f x n a vertca 4 eeent at the centre of the structure, for dfferent vaues of the order N and of the coeffcent of varaton. Ths ean s 49 expressed by the foowng reatonshp: Fg. 7. Infuence of the coeffcent of varaton. 5 [ < E u > < E u > ] () e S < f x >= () L e where u and u are the axa dspaceents of nodes and of the FE nuber e where the force s coputed. E s the Young oduus and S the secton of the bea that contans the FE e and L e s the ength of ths eeent.

6 As aready notced n [7] for a spe rod probe, we observe here that the saer the coeffcent of varaton, the qucer the convergence. Nevertheess, even for a coeffcent of 4%, whch corresponds to an portant dsperson, the convergence s acheved for N = Concuson Few studes are avaabe on the behavour of oent-resstng onts usng gued-n rods. On the bass of our prenary nvestgatons, a frst concudng rear coud be drawn: gued-n rod connectons coud be effectve to carry bendng oent up to the capacty of the bea but vunerabty for spttng of the tber shoud be prevented and controed. In the experenta approach, a frst contro paraeter s reated to the stffness dstrbuton of the bars over the onted secton n ters of daeter. Facng the faure odes reached wth ndented stee renforcng bars, ths anayss w be extended to threaded rods havng dfferent pu-out propertes. A spe but reastc FE ode based on bea eeents has been proposed. Obtaned nuerca resuts are n good agreeent wth experenta observatons. The SSFEM has been couped wth ths structura ode and the second-order oentu of the response have been copared wth Monte-Caro suatons: sar resuts are obtaned wth a sgnfcanty reduced coputaton cost, even for a arge vaue of the coeffcent of varaton of the rando paraeters of the ode, when the SSFEM s used. The resuts exhbt a ted sensbty to arge changes of the coeffcent of varaton of the atera propertes. They hghght the an portance of the stffness dstrbuton n the cross secton. Ony resuts for one confguraton are presented here but ths approach w aow of studyng the senstvty of ths nd of structure to perfectons or varatons of coponent characterstcs. Ths study coud ead to a better estaton of the reabty of gued-n rods tber structures. 5. References [] Rberhot H., ued bots n gua, Report N, Departent of Structura Engneerng, Technca Unversty of Denar, 986. [] ustafsson J., Serrano E., Acher S., and Johansson C.J., Strength desgn equaton for guedn rods, Internatona RILEM Syposu Jonts n tber structures, Stuttgart,, PRO, pp [3] Bass H.J. and Lasewtz B., Effect of spacng and edge dstance on the axa strength of gued-n rods, CIB W8, raz, 999, paper [4] hane R.., Spanos M. P., Stochastc Fnte Eeents: A Spectra Approach, Sprnger- Verag, New Yor, 99. [5] ehr E., Ducte behavour and group effect of gued-n rods, Internatona RILEM Syposu Jonts n tber structures, Stuttgart,, PRO, pp [6] hane R.., Stochastc Fnte Eeent wth Mutpe Rando Non-aussan Propertes, ASCE Journa of Engneerng Mech., Vo., No., 999, pp [7] Baroth J., Bodé L., Bressoette Ph., Fog M., Nuerca convergence of a Spectra Stochastc Fnte Eeent Method (SSFEM) n ognora context, proceedngs of the 9 th Int. Conf. on Appcatons of Statstcs and Probabty n cv engneerng (ICASP 9), Vo, pp 7-4, Der Kureghan Ed., Mpress, 3