FACTA UNIVERSITATIS Seres: Archtecture and Cvl Engneerng Vol., N o 5, 3, pp. 39-338 DESIGN METHODS OF A TIMBER-CONCRETE T-CROSS-SECTION UDC 64.6:64..+64..3(45)= Radovan Cvetkovć, Dragoslav Stojć Faculty of Cvl Engneerng and Archtecture, Unversty of Nš, E-mal: radovancvetkovc@yahoo.com, dragoslavstojc@yahoo.com Abstract. Ths paper deals wth the composte tmber-concrete structures.. By combnng tmber and concrete n a new type of composte materal and usng the best propertes of both materals (the hgh tensle strength of tmber and the hgh compressve strength of concrete) a new type of composte structure s obtaned, whch can have many applcatons, dependng on the dfferent buldng condtons, due to the certan advantages t has over concrete or steel structures. Here, the desgn procedures accordng to the theory of elastcty based on the exact method and approxmate method are gven n order, and partcularly accordng to the regulatons and recommendatons of the modern concept for desgn of tmber structures and concrete structures, gven n Eurocode 5 and based on the lmt states of bearng capacty and usablty of structures.. INTRODUCTION The analyss of composte tmber-concrete beams requres knowledge of a relatonshp between stress and dormatons for all three components, tmber, concrete and shear connectors. The complexty of problems les n determnaton of ths relatonshp and requres an ntroducton of a large number of parameters, whch complcate the calculatons. For practcal calculatons, certan smplfcatons can be made, and certan assumptons to facltate reachng the soluton n a relatvely easy way. The approxmate calculaton method for sem rgd structures s more approprate for use n engneerng because the calculus procedure s smpler. The calculatons of the desgn of the rgd composte structure are gven, consderng there s no relatve slp n the nterface, va the transformed secton method. In ths method, the concrete secton s transformed n a tmber secton, and the neutral lne remans n the same orgnal poston. The wdth of the cross-secton depends on the rate E c /E t. On the other sde the secton wth two materals n sem rgd structures wll show Receved May 5, 4
33 R. CVETKOVIĆ, D. STOJIĆ yeld neutral lnes. Dependng on theconnecton stffness, we can ntroduce a reducton of the fectve nerta moment "I" and compare t to the momentum of nerta of the rgdly jonted secton.. THE EXACT METHOD The theoretcal analyss, based on equatons of equlbrum, s done accordng to exact method, [5] whch makes possble the accurate determnaton of the strengths, flows and dsplacements. On the other sde, the approxmated method, whch ncorporates smplfcatons n the problem and facltates both the calculatons and desgn procedures, and shows that the smpler equatons can be appled. The slp between mechancally connected tmber and concrete s taken nto account n structural models by means of the slp modulus. The basc assumptons to composte structures, for example, concrete-wood T-beams, are the followng: Dsplacements owng to bendng are small and, therore, the small dsplacements theory s vald; Dsplacements owng to shear dormatons are neglgble n each element; Bernoull-Navers hypothess about plane sectons reman plane and perpendcular to dormed axs of the secton after dormaton s not vald along the whole cross-secton, but t s ndvdually vald for both tmber cross-secton and concrete cross-secton. Tmber and concrete are sotropc elastc materals and Hooks law s vald; Load-slp relatonshp for the connector can be approxmates to elastc-lnear. Connectors are placed at certan dstance and can be regarded as equvalent contnuous connecton. If a beam s subjected to any transversal loadng wth ntensty "q" that may vary along ts axs, f the boundary condtons of the beam do not have the concrete meanng, n regards to the assumed stress and dormaton of coupled cross secton (see fgure.), usng both the prncples of statc equlbrum and dsplacements compatblty, we can obtan the dfferental equaton that dnes the phenomenon dependng on dsplacement "w" [6]: b E,A, I h u=u -u =w (h /+h /) E,A, I h u w x N V b V +dvn +dn M V v M +dm N +dn a u N M M+dM dx V+dV Fg.. System, cross secton, dormaton, stress, element dx.
Desgn Methods of a Tmber-Concrete T-Cross-Secton 33 where: w M M α w = + α (.) ( EI) ( EI) r α = k + + EA E A (.) ( EI) = E A + E A (.3) EAEA ( EI) ( EI) r = + EA + EA (.4) (EI) the bendng stffness of uncoupled cross secton, (EI) the bendng stffness of coupled cross secton, E modulus elastcty of a concrete E modulus elastcty of a tmber, A cross-secton area of a concrete part, A cross-secton area of a tmber part, M, N, V adequate nternal forces of cross secton elements (parts), the others geometrcal sgns are accordng to fgure. If we know the soluton "w" to a specfc set of boundary condtons, the nternal forces for the whole secton and each element of secton are gven by: ( EI ) M = w ( EI) w α α EI q (.5) α V = w EI w q (.6) α M + EI w N = (.7a) r N M + EI w = (.7b) r M M = EIw (.8a) = EIw (.8b) V = M + vr (.9a) V = M + vr (.9b)
33 R. CVETKOVIĆ, D. STOJIĆ + + M EI w V EI w v = = (.) r r When we know the stress from equaton (.) to (.) t s possble to determne normal stress and shear stress. The normal stresses are gven n tmber and concrete, separately observed, by: σ M x N x ( xy, ) = y I + A (.) σ M x N x ( xy, ) = y I + A (.) where y s the dstance between the center of the consdered element and the fber whose stress we want to determne. In order to analyse the shear stress, we can observe the forts n an elementary segment of tmber from a composte tmber-concrete T-beam n fgure : It s possble to wrte Fg.. The forts n an elementary segment of tmber N (, ) * * σ x y da A * = (.3) where: * N resultng normal acton of the stress that acts n the smaller element; * A transversal secton area of the smaller element. Usng equatons (.) and (.3) and by calculatng the equlbrum of the emphaszed elementary segment n fgure, we can fnd:
Desgn Methods of a Tmber-Concrete T-Cross-Secton 333 τ * * * VS v A S ( xy, ) = + r bi b A I (.4) The equaton (.4) makes t possble to determne the shear stress n any one pont on the tmber cross-secton. Analogously, the expresson to the evaluaton of the shear stress n the concrete secton can be obtaned by: τ * * * VS v A S ( xy, ) = + r bi b A I (.5) 3. APPROXIMATE METHOD Ths method provdes approxmate analytc solutons[9]. The basc assumptons n ths method are the same as n exact method presented above. The connecton of tmber and concrete can be made by means of many types of metal connectors: nals, steel dowels, rngs, connected perforate metal plates, etc. The glued jonts are regarded as rgd connectons. The slp between mechancally connected tmber and concrete s taken nto account n structural models by means of the slp modulus K = (3.) u where F s s the shear force n the mechancal fastener and u s the slp n the connecton. The shearng flow "v" that appears on nterface of the materals s yelded by: v = s (3.) where "s" s the spacng between connectors. Usng equatons (3.) and (3.), we fnd: where K s the equvalent slp modulus n the jont. Accordng to the elastc prncples of bendng theory: N N M M v F s F s = ku (3.3) k = K / s (3.4) EAu = (3.5a) EAu = (3.5b) = EIw (3.6a) = EIw (3.6b)
334 R. CVETKOVIĆ, D. STOJIĆ V V = E I w (3.7a) = E I w (3.7b) v = ku = u u + w a = u u + w( h / + h /) (3.8) From the equlbrum of the two elements (tmber and concrete) n both longtudnal and axal drectons, we obtan: N + v = (3.9a) v N + = (3.9b) / M = V vh (3.a) / M = V vh (3.b) V + V = p = V (3.) where "p" s a generc loadng appled to beam. Addng equatons (3.a) to (3.b), takng nto consderaton the equatons (3.) and consequently dfferentatng t n relaton to x, we fnd: M + M + va+ p = (3.) If usng the elastc prncples changes both nternal forces and moments, the followng system of dfferentable equaton s yelded: " E A u + k u u + w a) (3.3) ( = " E A u + k u u + w a) (3.4) ( = ( E I + EI ) w" k( u u + w a) a = p (3.5) In ths way equatons (3.9a), (3.9b) and (3.) are formulated n functon of the dsplacements u, u and v. The practcal applcaton of a system of dfferentable equaton wll be shown at the example of smply supported beam wth a snusodal dstrbuton loadng as shown n fgure 3, so a smple analytcal soluton can be acheved. Ths s due to dormaton forms n the drectons of the axs whch agrees wth both snusodal and cosne functons [9]. Fg. 3. Snusodal dstrbuton loadng.
Desgn Methods of a Tmber-Concrete T-Cross-Secton 335 We can get: p p sn π l x = (3.6) u u cos π l x = u u cos π l x = w w sn π l x = (3.7a,b,c) where: u and u are the horzontal dsplacements at both concrete and tmber centrod, respectvely at the ends of the beam. The maxmum vertcally dsplacement w s at mdpont. These terms, when substtuted n Equatons (3.3), (3.4) and (3.5), produce a system of equatons wth the constants u, u, w, whose soluton allows to determne the stress at both concrete and tmber centers: γema σ = (3.8) where: γema σ = (3.9) γ = (3.) + k a π EA k = (3.) l K γea = a γ E A + E A EA a = a γ E A + E A Stress at the outer fbers of both concrete and tmber: EM σ =, h, h (3.) (3.3) EM σ = (3.4a,b) EM σ =, h EM σ = (3.5a,b), h Although the deductons had been made for mdpont, the expressons for stress calculatons can be extended to others cross-sectons along the length of the coupled beam, beng enough to change M to M(x).
336 R. CVETKOVIĆ, D. STOJIĆ The shearng flow along the length of the coupled beam can be calculated by expresson: γveaa v = (3.6) The elastc lne s gven by: M ( x) v ( x) = (3.7) EI Ths means that, consderng a snusodal dstrbuton loadng, we attaned a dfferental equaton to elastc lne. The equaton (3.7), s well-known from bendng theory, wheren we replace the bendng stffness EI of homogeneous beam by the bendng stffness (EI) f of the case of coupled beams. The soluton of ths equaton s much smpler than expresson (3.). 4. DESIGN METHOD ACCORDING TO EUROCODE 5 The desgn of the composte beams s regulated n the appendx B of the Eurocode 5. The stress calculaton for tmber and concrete and the calculaton of the connectors s to be performed n accordance wth the theory of the elastc compound. Accordng to recommendatons from the appendx B of the Eurocode 5, n consstence wth what has been sad, we can calculate the geometrcal propertes, stresses, and characterstcs of connecton of the cross secton shown by fgure 4, accordng to next steps: b h,5h σ σ m, A I,5h a τ max y A I E b h,5h a h z σ m, σ Fg. 4. Geometrcal propertes and stresses The fectve bendng stffness wll be calculated as follows: = n = ( E I + γ E A a ), (4.) where: number of elements consstng composte (complex) cross secton. In case of T- cross secton, that s. E the average value of modulus of elastcty for concrete and tmber, respectvely A = bh, (4.a)
Desgn Methods of a Tmber-Concrete T-Cross-Secton 337 γ I = bh 3 /, (4.b) γ =, (4.c) ( K ) = + π EAs / a l for = and = 3, (4.d) ( + ) γ ( + ) γ E A h h E A h h 3 3 3 3 = 3 γ EA = For T-cross sectons, h 3 =. The normal stresses are gven by equatons ( EI). (4.e) σ = γ E am/ (4.3a) ( EI) σ, =,5 E hm/ (4.3b) m The shear stress has the maxmum magntude at the pont where normal stresses are equal to zero. Maxmum shear stress at a certan pont along the heght of the tmber element of cross secton should be calculated accordng to the expresson: τ = γ E A a +,5E b h ) V /( b ( EI) ). (4.4),max ( 3 3 3 3 The load of the fastener should be calculated accordng to expresson F = γ EAasV/ EI (4.5) wth = and 3, where s = s (x) dstance between fasteners determned n B.3 and V = V(x). REFERENCES. A. Ceccott, Tmber-concrete composte structures, Structural Tmber Engneerng Proceedngs (STEP ), lecture E3, 995;. H. J. Blaß und M. Schlager, Trag-und Verformungsverhalten von Holz-Beton-Verbundkonstruktonen- Tel, Bauen mt Holz 5/96, pages: 39-399, 996; 3. H. J. Blaß, M. van der Lnden und M. Schlager, Trag- und Verformungsverhalten von Holz-Beton- Verbundkonstruktonen-Tel, Bauen mt Holz 6/96, pages: 47-477, 996; 4. ENV 995--.Eurocode 5: Desgn of tmber structures, Part.: General rules and rules for buldng. European Commttee for Standardsaton. 993; 5. ENV 99--.Eurocode : Desgn of concrete structures, Part.: General rules and rules for buldng, European Commttee for Standardsaton 993; 6. R. Cvetkovć, Behavour of Composte Tmber-Concrete Structures wth Bendng Actons Masters thess, Department of Renforced Concrete and Prestressed Concrete Structures, Ruhr Unversty Bochum, Germany,. 7. B. Stevanovc, Analyss of Composte Tmber-Concrete Structures, Doctoral thess, Faculty of Cvl Engneerng, Belgrade, 3; 8. Demarzo Mauro, Tactano Marcelo:"Semrgd composte wood-concrete T-beams" Proceedngs of World Conference on Tmber Engneerng (WCTE ), P47, Brtsh Columba, Canada, 3. Jul-3. August. 9. Grhammar U. A. and Gopu K. A. 993. Composte beam-columns wth nterlayer slp - exact analyss. Journal of Structural Engneerng, New York, v.9, n.4, p. 65-8, Apr.
338 R. CVETKOVIĆ, D. STOJIĆ METODE PRORAČUNA SPREGNUTIH KONSTRUKCIJA OD DRVETA I BETONA T PRESEKA Radovan Cvetkovć, Dragoslav Stojć U radu je data analza spregnuth konstrukcja tpa drvo-beton. Povezvanjem drveta betona koršćenjem najboljh svojstava jednog drugog materjala dobja se nov spregnut tp konstrukcje za koj se, zavsno od razlčth uslova građenja, može nac mnogo razloga za prmenu s obzrom na određene prednost u odnosu na beton čelk. Ovde su, redom, date proračunske procedure prema teorj elastčnost zasnovane na tačnom prblžnom metodu posebno, prema pravlma preporukama modernog concepta za proračun drvenh betonskh konstrukcja dath u Evrokodu 5 zasnovanh na grančnm stanjma nosvost u upotrebljvost.