CALCULATION OF CRACK WIDTHS WITH THE METHOD

Transcription

1 Tet an Deign Metho for Steel Fibre Reinfore Conrete 9 CLCULTION OF CRCK WIDTHS WITH TH MTHOD Davi Dupont an Luie Vanewalle Katholieke Univeriteit Leuven, Belgium. btrat Durabilit of onrete i in a great extent etermine b the rak with that are forme in the truture an that allow water to infiltrate in the onrete o that orroion of the reinforement bar an our. Thi paper onern the alulation of rak with in beam ontaining longituinal reinforement an teel fibre. Two alulation metho are explaine. The firt i the alulation metho that i propoe b Rilem TC6-TDF. Thi metho i a emi-empirial metho, bae on the approah followe b urooe for normal onrete. The eon metho i a more funamental approah that onier the interation between the onrete an the reinforement bar. Both alulation metho have been ompare with experimental reult of 9 full-ale beam. The invetigate parameter were the reinforement ratio, the fibre ontent, the fibre tpe an the hear pan to epth ratio. The omparion between experimental an theoretial reult how that for both moel a relativel goo preition an be foun of the average rak with.. Introution The etermination of rak in onrete ha alwa been a topi of interet for invetigation b man ientit. In general it i aume that the urabilit of the truture i aure when the rak with are limite to. mm. One of the often-ue approahe to limit the rak with i the ue of teel fibre reinfore onrete (SFRC). Steel fibre onrete i generall known to reue rak with beaue of it pot-raking tenile trength. t the Department of Civil ngineering of the Catholi Univerit of Leuven, Belgium, a tet program ha been exeute on 9 full-ale SFRC beam ontaining longituinal reinforement. ll beam have been tete in four-point bening. The tet were performe in ifferent loa tep until failure of the beam. The loa tep were hoen o that the beam faile after to 5 tep. t eah loa tep the rak with an paing were meaure. The reult of the tet program illutrate the trong benefiial effet of teel fibre on the rak with a well a on the rak paing. The aition of fibre to the onrete an lea to a reution of the rak with of up to 4%. The rak with of the beam of the tet program have alo been alulate b mean of the alulation metho propoe b Rilem TC6-TDF [] a well a with a newl evelope phial raking moel for reinfore SFRC beam. The moel take into aount the bon between the reinforement bar an the SFRC matrix a well a the influene of the teel fibre. omparion

2 RILM TC 6-TDF Workhop, Bohum, German, of the alulate reult an the experimental reult how that there i a relativel goo orrelation between the two.. Calulation metho. Calulation metho propoe b Rilem TC6-TDF []. In the mot reent reommenation of Rilem TC6-TDF the alulation of rak with i one with a emi-empirial metho, whih i bae on the metho ue in urooe for the alulation of rak with in onrete without teel fibre. The mean rak with w m ma be alulate b: w m () m rm with : rm = the average final rak paing (mm); m = the mean teel train allowe uner the relevant ombination of loa for the effet of tenion tiffening, hrinkage, et. The average final rak paing for member ubjete prinipall to flexure or normal fore an be alulate from the equation: b 5 rm 5.5 k k () r Lf f with : 5 Lf f b = the bar ie in mm; k = a oeffiient whih take aount of the bon propertie of the bar; k = a oeffiient whih take aount of the form of the train itribution; r = the effetive reinforement ratio /,eff where i the area of reinforement ontaine within the effetive tenion area,eff ; L f = the fibre length in mm; f = the fibre iameter in mm. The mean teel train m i obtaine from the relation: r m () with : = the tre in the tenile reinforement alulate on the bai of a rake etion a hown in a (MPa); r = the tre in the tenile reinforement alulate on the bai of a rake etion uner loaing onition auing firt raking a preente in Figure a (MPa); = a oeffiient whih take aount of the bon propertie of the bar []; = a oeffiient whih take aount of the uration of the loa or of repeate loaing []; = Young moulu for teel reinforement (MPa).

3 Tet an Deign Metho for Steel Fibre Reinfore Conrete Figure : ume tre itribution for plain onrete (a) an SFRC (b) beam etion. In the alulation the fibre have an impliit influene on m, beaue the fibre reue the teel tree an r. The tre f (b) i taken equal to.45 f R, [], where f R, i etermine b mean of a bening tet on a nothe prim aoring to the new RILM propoal []. The benefiial influene of the fibre on the mean final rak paing i foun in the lat fator of equation ().. lternative alulation metho The Rilem propoal i a rather ea metho that i ver loe to the approah in urooe []. However, the metho i an empirial metho. In earh for a more funamental alulation metho, the author have evelope a new phial moel. In Figure, a fragment of a beam i hown. Setion i a rake etion, while etion i a etion that i jut about to rak. The length L of the fragment i the minimum rak paing for a given moment M. The maximum rak paing i equal to L. It i onlue from thi that the average rak paing i equal to.5 L. The general alulation proeure onit of two tep. In a firt tep the length L in Figure i etermine, auming that the tenile tre in etion i equal to. time the tenile trength f tm an that the tenile tength in etion i equal to.8 time f tm. The fator. an.8 are introue to take the atter on the tenile trength of the onrete into aount. In the eon tep, the length of the beam fragment i fixe at.75 L (half of the average rak paing) an now the tenile tre in etion i variable.

4 RILM TC 6-TDF Workhop, Bohum, German, Figure : Beam part between a rake etion an a etion that ha jut reahe the point of raking. It i aume that the influene of the fibre in etion an be imulate wit a rop-ontant tre-train relation. The rop-ontant relation i oniere to be a peial ae of the two-level moel [4]. The tre f i alulate a.9 f R,. The tre-train relation for the pot-raking behaviour of SFRC implie that there i a full o-operation between the SFRC an the reinforement bar. The tre-train relation wa etermine auming a perfetl linear train itribution over the entire ro etion. In realit however, thi i not the ae. In a rake etion an in the immeiate neighbourhoo, the reinforement bar lip in the onrete matrix. The lip i maximal at the plae of raking an ero at a plae that i far enough awa from the rak. In a etion where a new rak i forme, the lip i alo aume to be ero. Thi i beaue on both ie of the future rak the lip i oppoite. In thi alulation moel the lip i aume to be ero in etion. Setion i an unrake etion. For the alulation of the rak paing it i aume that the poition of the neutral axi an be alulate exatl in etion an etion b uing tati equilibria of axial fore an bening moment in thee etion. For all etion that lie in between etion an, the poition of the neutral axi i unknown. It i aume here that poition of the neutral axi i ontant over a mall interval an an be alulate a follow: x x (4)

5 Tet an Deign Metho for Steel Fibre Reinfore Conrete where: x, an are the poition of the neutral axi in etion x, etion an etion, repetivel; i the lip in etion an x i the lip in a etion x. To tart the alulation a value of mut be aume an afterwar, if thi lip wa not orret, an iteration tep an be performe. In tep one the unknown parameter are: -) the length L; -) the teel train in etion : ; -) the teel train in etion : ; -) the ompreive train in etion : ; -) the ompreive train in etion : ; -) the onrete tenile train in etion : t ; -) the poition of the neutral axi: an ; -) the lip a funtion of x. For the alulation of the length L an the lip, a ifferential equation mut be olve. ll other unknown parameter an be olve b imple tati equilibrium of fore an moment. oring to the hpothei of Bernouilli, the following relation an be written: t (5) t (6) Requiring tati equilibrium of normal fore for etion reult in: b b t t (7) Requiring tati equilibrium of normal fore in etion reult in: t t b b ft b f f (8) t t Requiring tati equilibrium of moment for etion reult in: b b t t M (9) Requiring tati equilibrium of moment for etion reult in: b M b f f t t t b f t t t t From equation (8) an () the train t an the poition of the neutral axi an be foun. Following thi, from equation (7) an (9) the train an an be foun. Thi reult in the omplete knowlege of tree an train in etion an. The onl remaining unknown parameter are the length L an the lip a a funtion of x. If a tati equilibrium of normal ()

6 RILM TC 6-TDF Workhop, Bohum, German, 4 fore an moment i written for an arbitrar etion at plae x, equation () an () are foun: b b t t () M b b t t () xtrating from equation () give: b t t () Subtituting equation () into equation () give: M b b b t t t t (4) From equation (4), t an be written a a funtion of : h b M t (5) Thi leave onl the train to be etermine. The poition of the neutral axi ( an ) in an etion i aume to be the ame a in etion. Sine the teel rebar are lipping relative to the urrouning onrete, the teel train i ompoe of two part. The firt part i the train of the urrouning onrete, while the eon part i the train ue to the lipping of the rebar: x x t (6) quation (5) i ubtitute in equation (6): h b M x x (7) If the erivative of equation (7) i alulate, then equation (8) i foun. It i aume here that the moment M i inepenent of the poition x along the longituinal axi, i.e. the bening moment i ontant in thi one of the beam.

7 Tet an Deign Metho for Steel Fibre Reinfore Conrete 5 x x x b h (8) To olve thi ifferential equation a relation mut be foun between the lip an the train, both funtion of x. The horiontal equilibrium of the teel reinforement between the plae x an the plae x + x reult in (Figure ): x x (9) x x Subtituting expreion (9) in equation (8) reult in: x x x b h Now onl a relation ha to be aume eribing (x) a a funtion of (x). Thi hoen relation i equation (). max e () The etermination of max i explaine in [4, 5, 6]. Reent reearh at the epartment of Civil ngineering of the Catholi Univerit of Leuven ha hown that an an be etermine from pullout tet. a well a are epenent on the onrete over. The aition of teel fiber ha onl an influene on. The reaon i that i to a great extent etermine b the hemial bon between reinforement bar an urrouning onrete. If fiber are ae to the onrete thi ha a negative impat on the egree of ompation. For the onrete over ue in thi tet program it wa foun that an be taken equal to 8, while an be taken equal to.78 for plain onrete an equal to.89 for SFRC with 6 kg/m fiber of tpe RC 65/6 BN. For other fiber oage an interpollation i ue. Subtituting equation () in equation () give: x µ e x max x To olve equation (), a few new parameter are introue: x x max x a quation () an now be written a: () () p ' ()

8 6 RILM TC 6-TDF Workhop, Bohum, German, p p p ' ' ' p ' ' ' µ e ' e If thi lat expreion i integrate, equation (5) i obtaine: ' µ e (5) ' µ e C For the etermination of the integration ontant C the following bounar onition have been taken into aount: For x t x ' a a t In ae the beam i loae with the raing moment, whih we aume for the alulation of the rak paing, the eon bounar onition i equal to. Thi reult in: t C µ (7) a a quation (5) an now be ue to etermine the oure of the lip a funtion of x: t ' µ e a a µ (8) t µ e µ a a Integrating expreion (8) lea to: C ax C (9) t µ e µ a a F (4) (6)

9 Tet an Deign Metho for Steel Fibre Reinfore Conrete 7 The integration ontant C in equation (9) i etermine with the following bounar onition: x () quation (9) annot be olve analtiall. Therefore a numerial proeure ha been worke out. t firt the integration ontant C i etermine. Thi ontant i linke to the tarting value of the integration interval. If the integration i tarte at the point, then C an be foun b ubtituting = in F (the funtion after the integration ign). Furthermore the etermination of the lip a funtion of x i one b mean of an invere proeure. Firt the funtion F i etermine for a lit of value of, tarting with ero. fter that a numerial integration an be performe an point after point for eah value of, a orreponing x an be foun. One the oure of or i known a a funtion of x, the length L an be alulate. Thi i one b oniering the ifferene in fore in the teel reinforement. Thi ifferene mut be equal to the total hear fore between the reinforement bar an the onrete. F xx () L Following the above explaine approah the length L an be etermine for a erie of ifferent moment, eah moment reulting in a ifferent train itribution in etion an an alo a ifferent F an therefore a ifferent L. The rak paing i aume to be equal to.5 L [7].

10 8 RILM TC 6-TDF Workhop, Bohum, German, Figure : Beam part between a rake etion an a etion that lie in between two rak. In the eon tep of the alulation, the rak with are alulate. Therefore the length of the moel i now fixe at half the rak paing or.75l. In etion the bounar onition remain the ame, but in etion, the train itribution i now ifferent. When the beam i loae with bening moment that are larger than the raking moment, the poition of the neutral axi hift upwar. It i therefore aume here that in tep two, where the rak with are alulate for a ertain moment, the poition of the neutral axi i fixe an equal to the poitionof the neutral axi in the rake etion (Fig. ). Furthermore, ine the length of the moel i no longer L, the tre in the bottom fibre of etion i now lower than the tenile trength. Thi mean that the train itribution in etion i unknown. To have a tem that i in a tate of tati equilibrium the train mut be etermine o that at a itane x =.75L the train i equal to. Firt the teel tre i alulate a funtion of x. Thi an be one b the following tep: ) From equation (5) an (6) the following relation an be erive: a t t µ e µ a a a () ) If equation (5) i fille into equation (), an expreion i foun for the train a funtion of x:

11 Tet an Deign Metho for Steel Fibre Reinfore Conrete 9 h b M b h a b h a a e t () Uing equation (), the orret value of an be foun. Thi i one with an iterative proeure. The train i hoen. For thi hoen value of the value of t i etermine b requiring tati equilibrium in etion. With thi new train itribution in etion, equation (9) an be olve numeriall. If thi equation i olve, the value of an be etermine that orrepon to x =.75L. Subtituting thi value in equation (), the new train,new i foun. Thi new train,new mut be equal to the train,ol that wa etermine from equation (8) an (). The proeure i repeate, eah time for a new, until the eman (,new =,ol ) i atifie. One thi ha been one the average rak with i foun a twie the value of the lip at a itane x =.75 L.. xperimental program t the Department of Civil ngineering of the Catholi Univerit of Leuven (Belgium) a tet program wa exeute whih involve four-point bening tet on 9 full-ale beam. ll beam ha a epth of mm an a with of mm. The pan wa alwa equal to mm. The invetigate parameter for the 9 beam were the reinforement ratio, the fibre oage an the fibre tpe. The onrete over on the longituinal reinforement i equal to mm. Detail for all beam an be foun in Table.

12 RILM TC 6-TDF Workhop, Bohum, German, Figure 4: Tet et-up for all beam. piture of the tet et-up i hown in Figure 4. ll beam were tete uner loa ontrol. The ie of the loa tep wa o that the beam faile after to 5 loa tep. fter eah loa tep the efletion wa meaure a well a the rak with on both ie of the beam. The rak with were meaure at ± m above the bottom of the beam, onl in the one of ontant moment (between the loaing point). The meaurement of the rak with wa one with a mall, alibrate miroope. The mallet ale iviion in the miroope orrepon to. mm. Furthermore, ue to the freak hape of the rak it wa ometime iffiult to eie how large the rak with wa. Finall it mut be ai that a rak with an onl be meaure after the rak ha been etete. The mallet rak with that oul be etete ha alrea a rak with of. to. mm. Thi implie that the rak that have a rak with maller than. to. mm are not etete an not taken into aount in the alulation of the average rak with. The reult i that the average experimentall oberve rak with i a little overetimate. The effet of thi beome more important for beam with a high reinforement ratio an a high fibre oage, ine for thee beam the number of ver mall rak that remain unetete i higher. For all thee reaon the author think that, although the meaurement were taken with great are, there i a high egree of unertaint regaring the tet reult.

13 Tet an Deign Metho for Steel Fibre Reinfore Conrete Table : Tet parameter. Beam Fibre oage Fibre tpe Span Reinforement kg/m mm # x (mm) - RC 65/6 BN (*) 6 RC 65/6 BN RC 65/6 BN RC 65/6 BN RL 45/5 BN RL 45/5 BN 6 - RL 45/5 BN 6 RL 45/5 BN 4 RC 65/6 BN RC 8/5 BN RC 8/5 BN RC 65/6 BN RC 65/6 BN RC 65/6 BN 5 (*):R : hooke en fibre - C : fibre are glue in bunle - 65 : apet ratio of fibre (=length/iameter = L/) - 6 : length of fibre (= L in mm) - B : no oating - N : low arbon, i.e. minimum iel trength of MPa. Together with eah beam ube were at to meaure the mean ube ompreive trength f m,ube a well a 8 Rilem -point bening peimen to meaure the pot-raking behaviour [8]. The mean liner ompreive trength f m i taken equal to.8 f m,ube. The Young moulu of teel i taken equal to MPa, while the Young moulu of onrete i alulate with the formula provie in urooe []: 95. (5) f The tenile trength f tm i alulate from the flexural trength b appling the ie fator propoe b Rilem TC6-TDF []: 6 f tm f tm, fl (6) 475 The material propertie for all beam are hown in Table.

14 RILM TC 6-TDF Workhop, Bohum, German, Table : Material propertie for all beam Beam f m f tm f R, f R,4 N/mm N/mm N/mm N/mm The experimental reult a well a the theoretial reult alulate with the new Rilem metho an with the newl evelope phial moel are hown in Figure 5 until Figure. Figure 5: xperimental an theoretial rak with for beam. No teel fibre, Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho

15 Tet an Deign Metho for Steel Fibre Reinfore Conrete Figure 6: xperimental an theoretial rak with for beam. kg/m RC 65/6 BN, Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho Figure 7: xperimental an theoretial rak with for beam. 6 kg/m RC 65/6 BN, Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho

16 4 RILM TC 6-TDF Workhop, Bohum, German, Figure 8: xperimental an theoretial rak with for beam 4. No teel fibre, 6 Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho Figure 9: xperimental an theoretial rak with for beam 5. kg/m RC 65/6 BN, 6 Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho

17 Tet an Deign Metho for Steel Fibre Reinfore Conrete 5 Figure : xperimental an theoretial rak with for beam 6. 6 kg/m RC 65/6 BN, 6 Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho Figure : xperimental an theoretial rak with for beam 7. No teel fibre, 6 Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho

18 6 RILM TC 6-TDF Workhop, Bohum, German, Figure : xperimental an theoretial rak with for beam 8. kg/m RL 45/5 BN, 6 Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho Figure : xperimental an theoretial rak with for beam 9. 6 kg/m RL 45/5 BN, 6 Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho

19 Tet an Deign Metho for Steel Fibre Reinfore Conrete 7 Figure 4: xperimental an theoretial rak with for beam. No teel fibre, Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho Figure 5: xperimental an theoretial rak with for beam. kg/m RL 45/5 BN, Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho

20 8 RILM TC 6-TDF Workhop, Bohum, German, Figure 6: xperimental an theoretial rak with for beam. 6 kg/m RL 45/5 BN, Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho Figure 7: xperimental an theoretial rak with for beam. 4 kg/m RC 65/6 BN, 6 Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho

21 Tet an Deign Metho for Steel Fibre Reinfore Conrete 9 Figure 8: xperimental an theoretial rak with for beam 4. 4 kg/m RC 8/5 BN, 6 Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho Figure 9: xperimental an theoretial rak with for beam 5. 6 kg/m RC 8/5 BN, 6 Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho

22 4 RILM TC 6-TDF Workhop, Bohum, German, Figure : xperimental an theoretial rak with for beam 6. 4 kg/m RC 65/6 BN, Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho Figure : xperimental an theoretial rak with for beam 7. No teel fibre, Crak with (mm) Bening moment (knm) xperimental rak with lternative alulation metho Rilem TC6-TDF

23 Tet an Deign Metho for Steel Fibre Reinfore Conrete 4 Figure : xperimental an theoretial rak with for beam 8. kg/m RC 65/6 BN, Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho Figure : xperimental an theoretial rak with for beam 9. 6 kg/m RC 65/6 BN, Crak with (mm) Bening moment (knm) xperimental rak with Rilem TC6-TDF lternative alulation metho 4. Conluion large tet program ha been exeute on 9 full-ale beam. The invetigate parameter are the reinforement ratio, the fibre oage, the fibre tpe an the onrete trength. It an be

24 4 RILM TC 6-TDF Workhop, Bohum, German, onlue from the tet reult that teel fibre have a trong benefiial effet on rak with. For the ame amount of longituinal reinforement, the aition of 6 kg/m of teel fibre reult learl in a reution of the rak with. The benefit of the teel fibre i more obviou for the beam with 6 than for the beam with. Thi i logial ine the portion of the fibre in the total tenile fore of the beam i higher in thi ae. Two alulation metho have been invetigate. The firt alulation metho i a emi-empirial metho, whih i bae on the alulation metho of urooe. Thi firt metho i alo the new propoal of the Rilem ommittee TC6-TDF. The eon metho i quite ompliate an bae on a phial moel. The phial moel take into aount the bon between the reinforement bar an the urrouning onrete a well a the influene of the pot-raking tenile trength on the teel train in a rake etion. In omparing the alulate rak with with the experimentall etermine rak with it beome lear that, taking into aount ifferent aumption a well a the unertaint on the experimental reult, the newl propoe alternative alulation metho provie goo preition for all beam. lo the emi-empirial moel provie aurate preition of the rak with. Sine the emi-empirial metho i muh eaier to ue an the alulate rak with are goo preition of the experimental rak with, thi metho i b far the mot uitable to be ue a a tanar metho for rak with alulation. The newl propoe alternative alulation metho on the other han i rather ompliate to ue, but in return, it offer a ver goo inight in the mehanim that etermine the formation of rak. With the alternative alulation metho it beome poible not onl to alulate the rak with, but alo the lip at an plae between the reinforement an the urrouning onrete a well a the oure of the teel train along the reinforement. 5. knowlegement Thi tu wa part of the Brite uram projet "Tet an Deign Metho for Steel Fibre Reinfore Conrete", ontrat n BRPR-CT98-8. The partner in the projet are: N.V. Bekaert S.. (Belgium, oorinator), Centre Sientifique et Tehnique e la Contrution (Belgium), Katholieke Univeriteit Leuven (Belgium), Tehnial Univerit of Denmark (Denmark), Balfour Beatt Rail Lt. (Great Britain), Univerit of Wale Cariff (Great Britain), Fertig-Deken-Union GmbH (German), Ruhr-Univerit-Bohum (German), Tehnial Univerit of Braunhweig (German), FCC Contruión S.. (Spain), Univeritat Poltènia e Cataluna (Spain). 6. Lit of mbol oeffiient whih take aount of the bon propertie of - the bar oeffiient whih take aount of the uration of the loa - or of repeate loaing Slip of the reinforement bar relative to the urrouning mm onrete F Differene tenile fore in the reinforement between N etion an etion. Compreion train in the onrete - Compreion train in the onrete in etion -

25 Tet an Deign Metho for Steel Fibre Reinfore Conrete 4 Compreion train in the onrete in etion - t Tenile train in the onrete at the moment of raking - Steel train - Steel train in etion - Steel train in etion - t Tenile train in the onrete - t Tenile train in the onrete in etion - t Tenile train in the onrete in etion - m The mean teel train - b Bar iameter mm f Fibre iameter mm Bon parameter /mm Bon parameter - r ffetive reinforement ratio - The maximum tre in the onrete ompreion one N/mm The tre in the tenile reinforement alulate on the N/mm bai of a rake etion r The tre in the tenile reinforement alulate on the N/mm bai of a rake etion uner loaing onition auing firt raking Bon tre between the reinforement bar an the urrouning onrete N/mm Steel etion mm,eff ffetive tenion area mm b With of the beam mm ffetive epth of the beam mm h Depth of the beam mm Young moulu for onrete N/mm Young moulu for teel N/mm f tm Tenile trength of the onrete N/mm f eq, quivalent flexural tenile trength etermine at a N/mm efletion of.65 mm f eq, quivalent flexural tenile trength etermine at a N/mm efletion of.65 mm f R, Reiual flexural tenile trength etermine at a CMOD of N/mm.5 mm f R,4 Reiual flexural tenile trength etermine at a CMOD of N/mm.5 mm k oeffiient whih take aount of the bon propertie of - the bar k oeffiient whih take aount of the form of the train - itribution L Maximum itane between a rake etion an a etion mm that i jut about to rak L f Fibre length mm M Bening moment knm rm verage final rak paing mm

26 44 RILM TC 6-TDF Workhop, Bohum, German, w m verage rak with mm x Ditane to a etion that i jut about to rak mm Height of the tenile one mm Height of the ompreion one mm 7. Referene [] L. Vanewalle, Tet an eign metho of teel fibre reinfore onrete Reult of the Rilem Committee, Proeeing of the Leipiger Fahtagung Innovationen im Bauween : Faerbeton, 8-9 November. [] NV 99--: 99, urooe : Deign of onrete truture - Part : General rule an rule for builing, 99. [] D. Dupont an L. Vanewalle, Charateriation of teel fibre onrete with a relation, Proeeing of the 4 th international Ph.D. mpoium in Civil ngineering, 9- September, pp 8-4. [4] L. Vanewalle, Hehting tuen wapening met verbetere hehting en beton bij gewone en rogene omtanigheen (in uth), otoral thei, Catholi Univerit of Leuven (Belgium), 988. [5] D. Dupont an L. Vanewalle, Influene of teel fibre in loal bon behaviour, Proeeing of the international mpoium Bon in Conrete, Buapet, November -,. [6] F. De Bonte, Hehtterkte bij Staalveelbeton (in uth), Mater thei, Catholi Univerit of Leuven (Belgium),. [7] K. H. Tan, P. Paramaivam, K. C. Tan, Craking Charateriti of Reinfore teel fibre onrete beam uner hort- an long-term loaing, vane ement bae material, Vol., 995, pp.7-7. [8] L. Vanewalle et al. (): Reommenation of Rilem TC6-TDF : Tet an eign metho for teel fibre reinfore onrete : bening tet, Material an Struture,, Vol., pp.-5.