Design methods for steel fiber reinforced concrete industrial floors

Transcription

1 Desig methods for stee fiber reiforced ete idustria foors J.A.O. Barros 1, A. Vetura Gouveia 2, J.M. Sea Cruz 1, J.A.B. Atues 3, A.F.M. Azevedo 4 1 Uiversity of Miho, Portuga; 2 Poytechic Istitute of Viseu, Portuga; 3 Civitest Compay, Portuga; 4 Uiversity of Porto, Portuga Abstract Idustria foors are sti the mai appicatio of stee fiber reiforced ete (SFRC. Crack tro joits are made to cetrate the ack propagatio i these weakess-iduced surfaces, resutig i a foor divided i paes. he desig of a SFRC foor is currety made by meas of the study of oe of these paes, usig the yied ie method (. he utimate oad of the pae depeds o the maximum bedig momet of the sab. Sice i foorig appicatios the tet of stee fibers, i geera, does ot exceed 45 kg/m 3, the maximum bedig momet is oy sighty ieased by the presece of stee fibers. herefore, whe the is used to desig this type of SFRC appicatio the tributio of the fiber reiforcemet caot be accuratey simuated. I the preset work, this deficiecy of the is show ad justified by meas of experimeta ad umerica research. Furthermore, the is uabe to predict the force-defectio reatioship of a ete sab supported o soi. he fiite eemet method (FEM is a powerfu too to aayze this type of structures. However, the accuracy of the aaysis depeds o the quaity of the stitutive mode used to simuate the oiear behavior of the iterveig materias. For this purpose, a appropriate stitutive mode was deveoped ad is briefy desibed i the preset work. his mode is used i the aaysis of SFRC sabs o soi. Usig experimeta resuts ad appyig the rrespodig mputatioa de, a umerica strategy for estabishig desig charts for SFRC sabs o soi is proposed. Keywords: Materia Noiear Aaysis, Stee Fiber Reiforced Coete, Smeared Crack Mode, Fiite Eemet Method, Yied Lie Method. Joaquim Barros, Assistat Prof. Departmet of Civi Egieerig Schoo of Egieerig, Uiversity of Miho Campus de Azurém, Guimarães Portuga Emai: barros@civi.umiho.pt e.:

2 1.0 Itroductio A research project was carried out to characterize the properties ad to deveop a st mpetitive stee fiber reiforced ete (SFRC appicabe to foors of idustria buidigs [1, 2]. I this project the ifuece of the percetage of cemet repaced by fy ash, the ete age, the fiber aspect-ratio ad the tet of fibers o the mechaica properties of the SFRC was aayzed. he mpositios ad the obtaied mai properties are pubished esewhere [1]. he mai objective of the experimeta program was the characterizatio of the post-ackig behavior of this materia. For this purpose, otched beams were subject to three poit bedig tests acrdig to the remmedatios of RILEM C 162-DF [3]. his research project is aso devoted to the deveopmet of a desig approach for idustria foors. I order to avoid the formatio of utroed acks due to shrikage ad temperature variatio, ack tro joits are opeed, dividig the foor ito paes. he desig of a SFRC foor is, i geera, restricted to the aaysis of a represetative pae. he fiber suppiers, i geera, remmed the use of the yied ie method ( to desig this type of appicatios [4]. Acrdig to the the oad carryig capacity of a ete sab depeds o the maximum bedig momet of the sab, M. herefore, the tributio of the fiber reiforcemet ca oy be simuated by the whe M ieases with the tet of fibers, Q f. I order to verify this observatio, the vaues of M were obtaied i ete specimes reiforced with 15, 25, 35 ad 45 kg/m 3 of Dramix RC-65/60 hooked ed stee fibers. I these experimets the mpositios do ot irporate fy ash ad the properties were obtaied at the age of 28 days. he properties of these etes ca be foud esewhere [1, 2]. I the preset work the vaues of M are cacuated with a oss-sectio ayered mode (CSLM that takes ito acut the stitutive aws of the iterveig materias ad the kiematic ad equiibrium ditios [2]. I the CSLM, the SFRC post-ackig behavior is modeed by a stress-ack opeig diagram, that is defied by performig iverse aaysis i order to fit, with acceptabe error, the force defectio reatioships obtaied i the three poit bedig otched beam tests carried out acrdig to the RILEM C 162-DF. he CSLM provides the mometcurvature, M-χ, ad momet-ack opeig, M-w, reatioships of a oss sectio. he performace of the fiber reiforcemet is ot affected by evirometa aggressios whe the ack width does ot exceed 0.3 mm ad orma ditios are sidered [5]. For this reaso the vaue of M used to evauate the oad carryig capacity of the sab acrdig to the is the maximum momet up to a ack opeig of 0.3 mm (M R. he force vaues rrespodig to this oad carryig capacity are referred by F ad are mpared with the resuts obtaied with the fiite eemet method. I this case the force is termed F FEM. I the FEM simuatio a materia oiear aaysis was performed siderig a easto-pastic muti-fixed smeared ack mode [6]. he ifuece of the sab thickess, h, the soi reactio moduus, K s, ad the tet of fibers, Q f, was aayzed by meas of a parametric study where h was sidered equa to 120, 160, 200 or 240 mm, K s equa to 0.01, 0.04 or 0.08 N/mm 3 ad Q f equa to 15, 25, 35 or 45 kg/m 3. I a cases Dramix RC-65/60 hooked eds stee fibers were used. Based o the vaues obtaied with the FEM aaysis, a chart is proposed for the desig of SFRC sabs o soi. 2.0 Easto-pastic muti-fixed smeared ack mode for the FEM simuatios Acrdig to the preset mode, a ete sab is sidered a pae she formuated uder the Reisser-Midi theory [7]. I order to simuate the progressive damage iduced by ackig ad pasticity, the she eemet is disetized i ayers. Each ayer is sidered i a state of pae stress. he iemeta strai vector derived from the iemeta oda dispacemets obtaied uder the framework of a oiear FEM aaysis is demposed i a iemeta ack strai vector, ε, ad a iemeta strai vector of the ete betwee acks, ε. his ast

3 e p vector is demposed i a eastic reversibe part, ε, ad a irreversibe or pastic part, ε, resutig e p ε = ε + ε = ε + ε + ε (1 2.1 Coete stitutive aws he iemeta stress vector ca be mputed from the iemeta eastic strai vector, = D ε (2 where D is the ete taget stitutive matrix, D φ D = (3 φ Ds with D beig the i-pae stiffess matrix ad D the out-of-pae shear stiffess matrix [8]. I s the preset mode ete behavior is assumed iear eastic i terms of out-of-pae shear. herefore, the ete oiear behavior is oy sidered i the D stitutive matrix Liear eastic uacked ete For iear eastic uacked ete, D is desigated by e D beig defied esewhere [8] Liear eastic acked ete I acked ete, with the ete betwee acks i iear eastic state, with e D. his matrix is defied with the foowig expressio [6] ˆ ( 1 e e e e e => = + D D D D D D D D is repaced i (3 where is a trasformatio matrix that depeds o the directio of the acks formed at a sampig poit ad Dˆ is the stitutive matrix of the set of acks. Each ack is govered by the foowig stitutive reatioship = D ε (5 where is the iemeta oca ack stress vector. his vector has the foowig mpoets, = τ (6 t I this equatio, ε is the iemeta ack strai vector, which has the foowig mpoets ε = ε γ (7 t ad D I D = 0 (8 0 DII is the ack stiffess matrix, where D I ad D II are the fracture mode I ad the fracture mode II stiffess moduus of the smeared acks, respectivey. I (8 D I is characterized by the fracture parameters, amey the stress at ack iitiatio, (see Fig. 1, the fracture eergy, G, 1 f, the shape of the softeig aw ad the ack bad width, b. I smeared ack modes the fracture zoe is distributed over b, which must deped o the fiite eemet geometric characteristics i order to assure that the resuts of the FEM aaysis are ot depedet o the fiite eemet mesh [9]. (4

4 herefore, ε = w, where w is the tota ack opeig dispacemet iemet i the ack b bad width. I the preset umerica simuatio b is assumed to be equa to the square root of the area associated with a itegratio poit. Fiber reiforcemet behavior is maiy ifueced by the fracture eergy ad by the shape of the softeig brach. Previous research has show that the triiear ε diagram represeted i Fig. 1 is suitabe for the simuatio of the fracture mode I of the SFRC [2].,1,2,3 ε,2 D 1 D sec 2 D ε,3 g f = G f / b D 3 ε,u ε Figure 1. ri-iear tesie-softeig diagram. _ p _ im 1(κ 0 3 (κ κp _ 2(κ κ im Figure 2. Hardeig/softeig diagram. he fracture mode II moduus, D, is obtaied with the foowig expressio [7] II β DII = G (9 c 1 β where G c is the ete eastic shear moduus ad β is the shear retetio factor, which is defied by p1 ε β = 1 (10 ε u, I this equatio p 1 is a iteger parameter that ca assume distict vaues i order to simuate differet eves of ete shear stiffess degradatio [7] Easto-pastic uacked ete I the easto-pastic uacked ete, the i-pae materia stiffess matrix ep repaced with D. his matrix is obtaied with the foowig expressio [6] f f H H ep D D = H f f h+ H where, 2 1 e 1 f H = D + hc λ 2 κ D of (3 is f is the fow vector ad h c is a scaar fuctio that depeds o the hydrostatic pressure [6]. he aim of h c is the ampificatio of the tributio of λ f to the pastic strai iemet p vector, ε (11 (12

5 = p ε λhc f I (12 λ is the variatio of the pastic mutipier, which was assumed to be equa to the variatio of the hardeig parameter, κ, sice a strai-hardeig hypothesis was assumed. For the amout of fibers used i idustria foors, experimeta research has show [10, 11] that the shape of the yied surface, f, of SFRC uder biaxia stress state is simiar to the yied surface of the rrespodig pai ete. herefore the yied surface proposed by Owe ad Figueiras, 12 ( κ ( ( κ f, = P + q = 0 (14 was adopted i the preset mode, where P is the projectio matrix ad q is the projectio vector [6]. Fig. 2 represets the reatioship betwee the yied stress,, ad the hardeig parameter, κ, used to simuate the hardeig ad softeig phases of pai ete behavior. his reatioship was aso used i SFRC appicatios, sice for the amout of fibers used i foorig appicatios the ete uiaxia mpressio behavior is ot affected by the presece of κ are pubished i [6]. fibers. he expressios of ( i Easto-pastic acked ete For the case of acked ete with ete betwee acks exhibitig a easto-pastic behavior, D of (3 is repaced with ep D. his matrix is defied with the foowig expressio [6] ˆ ( 1 ep ep ep ep ep => = + D D D D D D D where ep D was defied i ( Soi he soi is simuated with sprigs that are orthogoa to the amiate structure. he evauatio of the taget soi reactio moduus ca be performed with pate-oadig tests [12]. he resuts of these tests have reveaed that the soi pressure-settemet reatioship may be simuated with a mutiiear or iear-paraboic diagram [8, 12]. he soi tributio to the stiffess of the whoe structura system is mputed by addig the soi stiffess matrix, ( e so = (16 ( e A s K N K N da ( e to the sab stiffess, where A is the area of a fiite eemet ad N is the vector of the eemet shape fuctios. I (16 K s is the taget soi reactio moduus. he frictio betwee the sab ad the soi is egected. Whe the ete sab oses tact with the soi i a sampig poit, the part of the soi that rrespods to this sampig poit does ot tribute to the stiffess of the sab-soi system. 3.0 versus FEM aaysis 3.1 SFRC fracture parameters ad maximum bedig momet of the sabs oss sectio Appyig a iverse aaysis, as desibed i [13], i order to fit the force-defectio reatioships obtaied i the three poit otched SFRC beam tests carried out acrdig to RILEM C 162-DF remmedatios, the parameters defiig the ε diagram depicted i Fig. 1 were obtaied. hese vaues are idicated i abe 1. (13 (15

6 Q f (kg/m 3 abe 1 Parameters defiig the diagram of Fig. 1 for the st mpetitive SFRC,1 (MPa,2,1,3,1 ε ε,2 u ε ε,3 u w 2 (mm w 3 (mm w 4 (mm G f (N/mm For the evauatio of the maximum bedig momet, M, a CSLM, as desibed i [14], was used. I the CSLM, the ete post-ackig behavior is simuated with a triiear stress-ack opeig -w diagram derived from the ε diagram represeted i Fig. 1. I order to vert ε ito w, the foowig reatioship was assumed: w= ε b. I the preset work b was sidered equa to the square root of the area of the itegratio poits of the fiite eemets that were sidered i materia oiear regime (see Fig. 5, resutig b =100 mm. Acrdig to this strategy, the vaues obtaied for w i are idicated i abe 1. Appyig the CSLM ad the vaues idicated i abe 1, the maximum momet up to a ack opeig of 0.3 mm (M R was obtaied for sabs with distict thickess ad buit with the st mpetitive SFRC proposed i the research project. he obtaied M R vaues are icuded i abe 2. It ca be verified that for Q f < 45 kg/m 3 the maximum momet has occurred for a ack opeig, w R, ess tha 0.3 mm. For Q f = 45 kg/m 3 the maximum momet up to w=0.3 mm occurred at this ack opeig vaue. he vaues icuded i abe 2 idicate that the fiber ifuece i terms of M R is oy sigificat for Q f = 45 kg/m 3. h [mm] abe 2 Maximum momet up to a ack opeig of 0.3 mm (M R. Q f M R w R h Q f M R w R [kg/m 3 ] [kn.m/m] [mm] [mm] [kg/m 3 ] [kn.m/m] [mm] Desig acrdig to the he desig of a SFRC foor is, i geera, restricted to the aaysis of a represetative pae. he fiber suppiers are remmedig the use of the to desig this type of structures. For the most mmo situatios a poit oad i a rer of the pae is the most ufavorabe oad figuratio, see Fig. 3. Acrdig to the, whe a oad F, uiformy distributed i a area rrespodig to a quarter of circe of radius 2a, is appied i the rer of a pae, the utimate oad ca be obtaied from the foowig expressio [4] ( al 2 ( 2M 11γ F = 1+, 3 2 L = ( 4 Eh 12 1 c υ c K, γ = al (17 s 1 al1.8 1 al1.8 where M is the egative maximum bedig momet of the sab, K s is the soi reactio moduus, E c

7 ad ν c are the ete Youg s Moduus ad Poisso efficiet, ad h is the sab thickess. Free edge Ks=0.01 Ks=0.04 Yied Lie F/M Ks=0.08 2a F Free edge Free edge al Figure 3. Yied ie i a ete pae oaded i a rer a/l Figure 4. Reatioship betwee F/M ad a/l. Repacig i (17 M with M R vaues from abe 2, the utimate oad acrdig to the, desigated by F, is obtaied. hese vaues are idicated i abe 3, showig that ieasig the vaue of K s ad keepig stat the vaues of the remaiig variabes, the iease of the F is margia. his ca be justified by the quasi-stat vaue of the F/M ratio for the a/l vaues sidered i the text of the preset work (see (17 ad Fig. 4. he ratio F/M oy ieases sigificaty for vaues of a/l above 0.2, which rrespod to vaues of sab thickess, soi reactio moduus ad oaded area ot used i mmo foorig appicatios. Fig. 4 icudes the exampe of a sab with h=160 mm ad Q f =25 kg/m 3. For the K s vaues sidered, a/l is ess tha 0.1, resutig i a variatio betwee 2.32 ad 2.47 for the reatioship betwee F/M ad a/l. From the aaysis of the vaues of F it is verified that for a give sab thickess, F did ot chage sigificaty up to a fiber tet of 35 kg/m 3. his is justified by the simiar vaues of M R for Q f 35 kg/m 3 (see abe 2. abe 2 shows that for Q f 35 kg/m 3 M R has occurred at a very ow ack opeig vaues. For these w R vaues the fiber reiforcig mechaisms are ot yet activated. herefore, for the amout of fibers used i foorig practice the caot take ito acut, directy, the beefits provided by the additio of fibers to ete. 3.3 Desig acrdig to the FEM he performace of the easto-pastic muti-fixed smeared ack mode has aready bee appraised [7, 8]. his mode was recety impemeted i the reease 4.0 of FEMIX mputatioa de [6], ad severa ehacemets were itroduced i the origia mode i order to iease its umerica robustess. I the curret sectio this mode was appied to aayze the behavior of the 5 5 m 2 pae represeted i Fig. 5, where the adopted fiite eemet mesh is aso show. Sice the eemets outside the square of dashed ie are ot affected by ay ete oiear pheomea, they are assumed to behave ieary. he eemets i the iterior of this square were assumed to have a oiear materia behavior. he pae thickess was demposed i 10 ayers of equa thickess. he SFRC fracture parameters idicated i abe 1 were used to defie the ε triiear diagram adopted to mode the fracture mode I. Average mpressive stregth of 38 MPa ad a Youg's Moduus of 32 GPa were sidered i the aaysis. he mpressive stregth was evauated i cubic specimes of 150 mm edge. Fig. 6 represets the ack patter for the sab with h=160 mm, Q f =25 kg/m 3 ad K s = 0.01 N/mm 3, at a oad eve rrespodig to a maximum ack with of 0.3 mm.

8 abe 3 Load carryig capacity acrdig to the (usig M R vaues ad FEM aaysis (oad rrespodig to a ack opeig of 0.3 mm F (kn F (kn F F h Q f M R (mm (kg/m 3 (kn.m/m K s (N/mm 3 K s (N/mm 3 K s (N/mm x x x x Liear Eemets Noiear Eemets Force Figure 5. Fiite eemet mesh. Mai ack zoe Figure 6. Crack patter i the sab top surface at a oad eve rrespodig to w 0.3 (h=160mm, Q f =25 kg/m 3 ad K s =0.01 N/mm 3. Figure 7 depicts a typica force-rer defectio reatioship up to the momet whe a maximum ack opeig of 0.3 mm (w 0.3 occurred. I this figure oe ca observe the oads for a maximum ack width rrespodig to the w R idicated i abe 2 (desigated by F ad the oads for FEM ( w R w 0.3 ( F. he vaues of F are idicated i abe 3. Sice for Q f 35 kg/m 3 w R are

9 too sma to mobiize the fiber reiforcig mechaisms (see abe 2, the F vaues of the FEM ( w R sabs reiforced with Q f 35 kg/m 3 are simiar. For the F vaues, the effects of the fibers are visibe, i.e., the oad carryig capacity ieases with the tet of fibers, eve for Q f 35 kg/m 3, which was ot the case whe appyig the. 4.0 Strategy to buit charts for the desig of SFRC sabs o soi he force vaues, F, obtaied by FEM aaysis, ad the vaues of the rrespodig variabes of the parametric study (h, K s ad Q f are orgaized i order to defie a chart from which the most eomic soutio for a SFRC sab supported o soi ca be determied. his chart is represeted i Fig. 8 ad its appicabiity is oy restricted to the ceived ad characterized SFRC ad for the oad figuratio sidered i the preset work (rer oad. For istace, whe a oad or 90 kn is actig o the rer of a sab supported o a soi with K s =0.04 N/mm 3, the foowig soutios are obtaied (see Fig. 8: Q f = 15 kg/m 3 ad h = 238 mm; Q f = 25 kg/m 3 ad h = 216 mm; Q f = 35 kg/m 3 ad h = 202 mm; Q f = 45 kg/m 3 ad h = 174 mm. akig ito acut the prices of the ete ad fibers, the most eomic soutio ca be seected. Force (kn h160_k004_qf15 h160_k s0.04_q f15 h160_k004_qf25 h160_k s0.04_q f25 h160_k004_qf35 h160_k s0.04_q f35 h160_k004_qf45 h160_k s0.04_q f45 w R w Dispacemet at rer (mm Figure 7. Reatioship betwee the force ad the defectio at the sab rer up to w R ad w 0.3. h (mm K s =0.01 K s =0.04 K s =0.01 K s = Qf15 f 180 Qf25 f 160 Qf35Q f Qf45 f 140 Dramix RC-65/60-BN 120 w=0.3 mm f cm =38 MPa Force (kn K s =0.01 K s =0.04 K s =0.08 K s =0.04 K s =0.08 K s =0.01 Figure 8. Desig chart. K s =0.04 K s = Cocusios I the preset work the appicabiity of the yied ie method ( for the aaysis of idustria foors buit with stee fiber reiforced ete (SFRC is assessed usig the resuts obtaied i a experimeta research program where a st mpetitive SFRC was ceived ad their properties characterized. For this purpose, the resuts obtaied with the were mpared with those cacuated with a mputer program based o the fiite eemet method (FEM i which a easto-pastic smeared ack mode was impemeted. I this mpariso, the ifuece of the sab thickess (h, the soi reactio moduus (K s ad the amout of fibers (Q f was take ito acut. he utimate force obtaied with the depeds o the maximum bedig momet (M R of the SFRC sab. o evauate M R, a oss sectio ayered mode was used. Sice i idustria foors Q f, i geera, does ot exceed 45 kg/m 3, M R vaues are simiar for SFRC sabs with Q f 35 kg/m 3. Cosiderig that F ad F desigate the oad carryig capacity of a SFRC foor determied with ad FEM, abe 3 shows that F F ieased with Q f ad K s. I

10 geera, F F > 1 for SFRC sabs with Q f >15 kg/m 3 ad supported o a soi with K s >0.01 N/mm 3, showig that is ot capabe of takig ito acut, directy, the beefits provided by fiber reiforcemet i rea foorig ditios. For SFRC sabs with Q f =15 kg/m 3, F F < 1, maiy whe supported o week sois, which meas that the is ot a safe desig approach for these cases. he F F ratio deeased with the thickess of the foor, beig ess tha uit i sabs with h=240 mm, supported o a soi with K s =0.01 N/mm 3, ad reiforced with Q f =25 kg/m 3, which is a curret fiber tet i foorig appicatios. he particuar vaues obtaied i this research are restricted to the desiged SFRC situatios, but it is expected that the observed tedecies ad the mai cusios ca be exteded to ay FRC. 6.0 Refereces 1. Barros, J.A.O.; Atues, J.A.B., "Experimeta characterizatio of the fexura behaviour of stee fibre reiforced ete acrdig to RILEM C 162-DF remmedatios", RILEM C 162 DF Workshop, pp , March Barros, J.A.O.; Cuha, V.M.C.F.; Ribeiro, A.F.; Atues, J.A.B., Post-ackig behaviour of stee fibre reiforced ete", RILEM Materias ad Structures Joura, i press, Vaderwae, L. et a., est ad desig methods for stee fibre reiforced ete - Fia Remmedatio, Materias ad Structures Joura, Vo.35, pp , Noveer Bauma, R.A. ad Weisgerber, F.E., Yied ie aaysis of sabs-o-grade. Jour. Struct. Egrg. ASCE, 1983, 109(7, Magat, P.S.; Mooy, B..; Gurusamy, K. (1989, Marie durabiity of stee fiber reiforced ete of high water-cemet ratio, Fiber Reiforced Cemets ad Coetes-Recet. 6. Sea-Cruz, J.M.; Barros, J.A.O.; Azevedo, A.F.M., "Easto-pastic muti-fixed smeared ack mode for ete", echica report 04-DEC/E-05, Dep. Civi Eg., Uiversity of Miho, 70 pp, Jue Barros, J.A.O., Comportameto do betão reforçado m fibras - aáise experimeta e simuação umérica (Behavior of FRC experimeta aaysis ad umerica simuatio, PhD hesis, Civi Eg. Dept., Facuty of Egieerig, Uiversity of Porto, Portuga, 1995 (i Portuguese. 8. Barros, J.A.O.; Figueiras, J.A., Noiear aaysis of stee fibre reiforced ete sabs o grade, Computers & Structures Joura, Vo.79, No.1, pp , Jauary Bazat, Z.P. ad Oh, B.H., Crack bad theory for fracture of ete. Materias ad Structures, RILEM, 1983, 16(93, Yi, W.S.; Su, E.C.M.; Masur, M.A. ad Hsu,..C., Biaxia ests of pai ad fiber ete. ACI Materias Joura, 1989, 86(3, raia, L.A. ad Masour, S.A., Biaxia stregth ad deformatioa behaviour of pai ad stee fiber ete. ACI Materias Joura, 1991, 88(3, Barros, J.A.O.; Figueiras, J.A., Experimeta behaviour of fiber ete sabs o soi, Joura Mechaics of Cohesive-frictioa Materias, Vo. 3, pp , Cuha, V.M.C.F., Aáise experimeta e umérica do mportameto à tracção de betão reforçado m fibras (Experimeta a umerica aaysis of SFRC", ese de Mestrado, Dep. Egª Civi da UM, Setero de Ribeiro, A.F., Modeos de feda diseta a simuação do mportameto em fexão de betão reforçado m fibras de aço (Disete ack mode for the simuatio of the fexura behavior of SFRC, MSc hesis, Dep. Civi Eg. Uiversity of Miho, Septeer 2004.