FRACTURE OF REINFORCED CONCRETE: SCALE EFFECTS AND SNAP-BACK INSTABILITY

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1 Engineering Fracture Mechanics VI. 35, N. 4/5, pp , 1990 Printed in Great Britain /90 $ Pergamn Press plc. FRACTURE F REINFRCED CNCRETE: SCALE EFFECTS AND SNAP-BACK INSTABILITY C. BSC, A. CARPINTERI and P. G. DEBERNARDI Department f Structural Engineering, Plitecnic di Trin, Crs Duca degli Abruzzi, 24, Trin, Italy Abstract-Remarkable size-scale effects are theretically predicted and experimentally cnfirmed in lw reinfrced high strength cncrete beams. The brittleness fthe system increases by increasing size-scale and/r decreasing steel area. n the ther hand, a physically similar behaviur is revealed in the cases where the nn-dimensinal number: r: b 1 / 2 A Np=-Y_.~ K 1C A is the same, J" being the steel yield strength, K 1C the cncrete fracture tughness, b the beam depth and AsiA the steel percentage. The tensile strength and tughness f cncrete, usually disregarded, are s high in sme cases that the peak bending mment vercmes the bending mment f limit design (hyperstrength). The drp in the lading capacity can hide a virtual sftening branch with psitive slpe (snack-back), which is detected if the lading prcess is cntrlled thrugh the crack width. l. INTRDUCTIN WHEN A small percentage f steel is required t reinfrce high strength cncrete, the crushing failure f the beam edge in cmpressin is usually avided. n the ther hand, ne r mre cracks riginate at the beam edge in tensin and the material is s brittle in this case that the size f the crack tip prcess zne is very small if cmpared with the size f the zne where the stress singularity field is dminant. Fr these reasns, it will be shwn that a Linear Elastic Fracture Mechanics mdel[i-3] is able t capture the mst relevant aspects and trends in the mechanical and failure behaviur f lw reinfrced high strength cncrete beams in ftexure. The present theretical and experimental investigatin aims at evaluating the size-scale effects n reinfrced cncrete members, by means f three pint bending tests. Thirty (30) RiC. beams are tested, with thickness t = 150 mm and depth b = 100, 200, 400 mm, respectively. The span is assumed t be six times the beam depth b. Five different values f the nn-dimensinal number t.-b'!' A Np=-Y--.~ K 1C A are cnsidered (abut equal t 0.00, 0.10, 0.30, 0.75, 1.20), /y being the steel yield strength, K 1c the cncrete fracture tughness and AsiA the steel percentage. Bth size-scale b and steel percentage AsiA are varied. The rati fthe distance fthe bars frm the lwer edge fthe beam, t the beam depth is assumed t be cnstant and equal t 0.1. The lading prcess is carried ut n initially uncracked R.C. beams, by cntrlling the tensile strain n the lwer edge f the beam r, after cracking, the crack muth pening displacement. Even the cmpressi ve strain n the upper edge f the beam and the beam deftectin are recrded. Remarkable size-scale effects are theretically predicted and experimentally cnfirrned. The brittleness f the system increases by increasing size-scale and/r decreasing steel area. n the ther hand, a physically similar behaviur is revealed in the cases where the nn-dimensinal number N; is the same. The phenmena f catastrphical sftening (snap-back) and hyperstrength in lw reinfrced cncrete beams are interpreted accrding t the cncepts f Fracture Mechanics. The tensile strength and tughness f cncrete, usually disregarded, are s high in sme cases that the ultimate bending mment vercmes the bending mment f limi t design The drp in the lading capacity can hide a virtual sftening lad-deftectin branch with psitive slpe, which is detected if the 665

2 666 C. BSC et al. Il t:r - ( ') - -- F F ---- F F Fig. 1. Cracked cncrete beam element. ~ fh l } lading prcess is cntrlled thrugh the crack width. In this way, a phenmenn unstable in nature is made stable in practice[4-6]. 2. THERETICAL MDEL Let the cracked cncrete beam element in Fig. 1 be subjected t the bending mment M and t an eccentric axial frce F due t the statically undetermined reactin f the reinfrcement. It is well-knwn that bending mment M* and axial frce F* induce stress-intensity factrs at the crack tip respectively equal t[7, 8]: (la) (F)_ F*. K j -bl/2.t YF(ç) (l b) where Y M and Y F are given, fr ç ~ alb ~ 0.7, by: YM(ç) = 6(1.99 ç 1/ ç3/ ç5/ ç7/ e/ 2 ) (l c) YF(ç) = (1.99 ç 1/ ç3/ ç5/ ç7/ ç9/2). (l d) n the ther hand, M* and F* prduce lcal rtatins, respectively, equal t[7,8]: cp(m)= À,MM'M* cp(f)= À,MF.F* (2a) (2b) where (3a) Up t the mment f steel yielding r slippage, the lcal rtatin in the cracked crss-sectin is equal t zer: Equatin (4) is the cngruence cnditin giving the unknwn frce F. Recalling that (Fig. 1): (3b) (4) eqs (2) and (4) prvide: M*=M-F (b/2-h) F*= -F F b 1 M (0.5-h/b)+r(ç) (5a) (5b) (6)

3 Fracture f reinfrced cncrete 667 where: f YM(ç) YF(ç) dç r( ç) = "'-0'----:;;-;-1; ---- f Y~(ç)dç (7) If a perfectly plastic behaviur f the reinfrcement is cnsidered (yielding r slippage), frm eq. (6) the mment f plastic flw fr the reinfrcement results: M; = Fç-b [(0.5 - h/b) + r(ç)]. (8) Hwever, it shuld be bserved that, if cncrete presents a lw crushing strength and steel a high yield strength, crushing f cncrete can precede plastic flw f reinfrcement. The mechanical behaviur f the cracked reinfrced cncrete beam sectin is rigid until the bending mment M; is exceeded, i.e., <jj= fr M ~ M«. n the ther hand, fr M > M; the M -<jj diagram becmes linear hardening: After the plastic flw freinfrcement, the superpsitin principi e: K( = K\M) + K\F). Recalling eq. (I) and cnsidering the ladings: M* = M - Fp (b/2 - h) F* = -Fp the glbal stress-intensity factr results: the stress-intensity factr at the crack tip is given by (9) (10) (I la) (llb) (12) The mment f crack prpagatin is then: (13) with: (14) while the rtatin atcrack prpagatin is The crack prpagatin mment is pltted in Fig. 2 as a functin f the crack depth ç and varying the brittleness number N«. Pr lw Np values, i.e. fr lw reinfrced beams r fr small crss-sectins, the fracture mment decreases while the crack extends, and a typical phenmenn f unstable fracture ccurs. Fr N; ~ 0.7, a stable branch fllws the unstable ne, while fr Np ~ 8.5 nly the stable branch remains. The lcus f the minima is represented by a dashed line in Fig. 2. In the upper zne the fracture prcess is stable whereas it is unstable in the lwer ne. Rigid behaviur ( ~ M ~ Mp) is fllwed by linear hardening (M; < M < M F ). The latter stps when crack prpagatin ccurs. If the fracture prcess is unstable, diagram M -<jj presents a discntinuity and drps frm MF t Fç-b with a negative jump. In fact, in this case, a cmplete and instantaneus discnnectin f cncrete ccurs. The new mment Fp b can be estimated accrding t the scheme in Fig. 3. The nn-linear descending law: (15) M = Fp (b - h) cs(<jj/2) (16)

4 668 C. BSC et al. 8 1;; -.c!.l ~-.. u. ::le z 6 i= < (!l <- a: 0- D 4 < a: u.. I- Z w 15 2 ::le (!l z z w cc f b V2 A N = _Y._s P K IC A ~ ~ 4i-\ : ::'). STABILlTY l RELATIVE CRACK LENGTH, ~=a/b Fig. 2. Bending mment f crack prpagatin against relative crack length. is thus apprximated by the perfectly plastic ne: M =Fp b. (17) n the ther hand, if the fracture prcess is stable, diagram M -c/j des nt present any discntinuity. In Fig. 4 the mment-rtatin diagrams are reprted fr ç = 0.1 and five different values f the number N«. Fr N; ~ 0.7, it is Fç-b ~ MF and therefre a discntinuity appears in the M-c/J diagram (Fig. 4a--c). n the ther hand, fr N; < 0.7, the curves in Fig. 21ie n the unstable zne cmpletely. In cnclusin, the abve presented theretical mdel predicts unstable behaviur fr lw cntent f steel and/r fr large beam depth. It is wrth nting that the initial crack f the theretical mdel was nt present in the experimental beams, althugh a crack frmed during the lading prcess In additin, the reinfrcement plastic flw f the theretical mdel, in practice was due als t the steel bar slippage. Nevertheless, the theretical ductile-brittle transitin was cnfirmed by the experiments, as will be shwn in the fllwing sectins Cncrete 3. MATERIAL PRPERTIES The beams f the present experimental investigatin are made f cncrete with crushed aggregate f maximum size Drnax = 12.7 mm. The amunt f cement (type 525) is 480 kg/m ', and Fig. 3. Statical scheme f cmplete discnnectin f cncrete.

5 Fracture f reinfrced cncrete 669 M (a) (b) (c) (d) Fpb MF Mp (e) Fig. 4. Ductile-brittle transitin in the mechanical behaviur f reinfrced cncrete beams, by varying the brittleness number, N p the waterjcement rati is equal t Cnsiderable attentin was spent t avid cracking due t hydratin and shrinkage. The cmpressive strength (after 28 days) was btained with twenty (20) cubic specimens f side 160 mm. The average value resulted t be Rcm = 91.2 Nyrnrrr', with a standard deviatin f 8.8 Njmm", The curing time f the beams was 3 days at 30 C and then a secnd perid fllwed at 20 e. As an average, the tests were carried ut after 20 days frm mulding. The tests f elastic mdulus were perfrmed n three (3) specimens f size 150 x 150 x 450 mm and prvided an average value f the secant mdulus E (between zer and 1j3 f the ultimate l ad) equal t 34,300 Nrmrrr'. The fracture energy G F was determined by three pint bending testing n three (3) specimens f size b = 100 mm, t = 150 mm, 1= 750 mm. The span was equal t L = 720 mm and the beams were pre-ntched n the center-line, the ntch depth being equal t ne half f the beam depth (ajb = 0.5) and the ntch width t 5 mm. The lad-deftectin diagram fr ne fthe beams tested, is reprted in Fig. 5. The average value f the fracture energy results t be G F = Njmm, s that the criticai value f the stress-intensity factr can be evaluated: 3.2. Steel K 1C = J G F E = Njmnr'". The utilized steel bars present a nminai diameter f 4, 5, 8, l mm, respectively. The bars f 4 and 5 mm d nt present yielding and the cnventinal limit, with 0.2% permanent defrmatin, is equal t 637 Njmm? and 569 Njmrrr', respectively. The yield strength fr the bars f 8 and l mm, n the ther hand, is equal t 441 Njmnr' and 456 Nrmm', respectively. 4. DESCRIPTIN F THE R.C. BEAM SPECIMENS Thirty (30) reinfrced cncrete (R.e.) beams were tested, with the crss-sectin f thickness t = 150 mm and depth b = 100, 200, 400 mm. The span between the supprts was assumed t be EFM 35-4/5-E

6 670 C. BSC et al. lad 3.0 (kn) r midspan deflectin ~~~~LL4LLL~~z=g===~==~~--~ Fig. 5. Lad vs defiectin diagram f unreinfrced specimen, fr the experimental determinatin f cncrete fracture energy G F equal t six (6) times the beam depth b and, therefre, t 600, 1200, 2400 mm fr the specimens A, B, C, respectively. The specimens were marked in the fllwing way. -By varying the beam size: (A) beam depth b = 100 mm (t = 150 mm; L = 600 mm); (B) beam depth b = 200 mm (t = 150 mm; L = 1200 mm); (C) beam depth b = 400 mm (t = 150 mm; L = 2400 mm). -By varying the brittleness class: () brittleness number N; = (n reinfrcement); (1) brittleness number Np ~ 0.13 (n the average); (2) brittleness number Np ~ 0.36 (n the average); (3) brittleness number Np ~ 0.72 (n the average); (4) brittleness number Np ~ 1.18 (n the average). The cntent f steel depends n the beam size and n the brittleness number (see eq. 14). It is reprted fr each beam in Table 1. (mm) Table I. Descriptin f the reinfrced cncrete specimens and related lads f first cracking, steel yielding and final cllapse Sizes Nminai Actual Yield limit Actual Cracking Yielding Ultimate Beam Brittleness t x b cntent percentage f steel value lad lad lad size class (mm) f steel f steel (Njmrrr') f Np (kn) (kn) (kn) 150 x I 150 x 100 l <jj X I-l A x 100 2<jJ x I-l x 100 2<jJ x I-l x <jji X I-l x x 200 l <jj X B x <jj X x 200 3<jJ x I-l x <jjl 7.75 X I-l x x 400 2<jJ x C x <jj X x <jj X I-l x <jjl 5.17xl

Fracture r reinfrced cncrete 671 6 ( Fig. 6. Strain gauge n the lwer beam edge.

7 Fracture r reinfrced cncrete ( Fig. 6. Strain gauge n the lwer beam edge. Fig. 7. Deflectin reference bar cnnected with the cncrete beam.

8 672 C. BSC et al. Fig. 8. Brittle enamel applied in the zne where the first crack frmatin is expected.

9 Fracture f reinfrced cncrete 673 The distance f the bars frm the lwer beam edge is, in each case, equal l/i f the beam depth thlb = 0.1). Fr each beam size (A, B, C) and fr each brittleness class (, 1,2,3,4) tw R.e. beams were realized, with a ttal number f 30 specimens. AlI the beams were initialiy unntched and uncracked. The fliwing results, when nt therwise specified, are related t the average value f each case cntemplated by the experimental investigatin. 5. TESTING APPARATUS AND PRCEDURE The experimental investigatin was carried ut at the Department f Structural Engineering fthe Plitecnic di Trin. The three pint bending tests n R.e. beams were realized by a M.T.S. machine. The beams were supprted by a cylindrical rller and a spherical cnnectin, respectively, at the tw extremities. The lad was applied thrugh a hydraulic actuatr and the lading prcess was cntrlled by a strain gauge DD I, placed n the lwer beam edge, parallel t the beam axis and symmetrical with respect t the frce. Its length was equal t the beam depth, i.e. 100, 200 r 400 mm, respectively, fr the beam sizes A, B and e (Fig. 6). The sensitivity f the strain gauge utilized is 1mV/IV/1 mm, fr a feed-vltage f 5 V. The strain rate was impsed at a cnstant and very lw value. n the average, the crack frmatin in the middle f the beam, was achieved after abut 7 min and the steel yielding after abut 45 min. Transducers with a sensitivity f 1 mv/.imm were used t measure the centrai deflectin. The latter was referred t a bar, cnnected with the cncrete beam at the middle f the depth and in crrespndence f the tw supprts. Such a device is shwn in Fig. 7, fr a RiC. beam f depth b = 400 mm (size C). Deflectin and strain gauge defrmatin were pltted authmaticaliy as functins f the applied lad. The lad f first cracking was detected by means f a brittle enamel, applied in the zne where the first crack frmatin is expected (Fig. 8). The lads f first crack frmatin, f steel yielding and f final cllapse, are summarized in Table l. 6. EXPERIMENT AL RESULTS AND DISCUSSIN The lad-deflectin diagrams are pltted in the Fig. 9(a), (b) and (c), fr each beam size and by varying the brittleness class (each curve is related t a single specimen f the tw cnsidered). As is pssible t verify in Table I, the peak r first cracking lad is decidedly lwer than the steel yielding lad nly in the cases 3 and 4, i.e. fr high brittleness numbers Np In the cases and l, the ppsite result is clearly btained. n the ther hand, case 2 demnstrates t represent a transitin cnditin between hyperstrength and plastic cllapse, the tw criticai lads being very clse. Therefre, the same brittleness transitin thereticaliy predicted in Fig. 4, is reprpsed by the experimental diagrams in Fig. 9. The experimental transitin value f the brittleness number N p, results t be between 0.36 and 0.72, whereas the theretical transitin value is apprximately equal t 0.70 (Figs 2 and 4). Specimen e (b = 400 mm, n reinfrcement) presents a very evident snap-back behaviur, the sftening branch assuming even a psitive slpe. It was pssible t fliw such a branch, since the lading prcess was cntrlled by a mntnicaliy increasing functin f time, i.e. the crack muth pening displacement. If the cntrlling parameter had been the central deflectin, a sudden drp in the lading capacity and an unstable and fast crack prpagatin wuld have ccurred[4-6]. The dimensinless bending mment vs rtatin diagrams are pltted in the Fig. 10(a)-(e), fr each brittleness class and by varying the beam size. The lcal rtatin is nn-dimensinalized with respect t the value </>0 recrded at the first cracking, and is related t the central beam element f length equal t the beam depth b. The bending mment, n the ther hand, is nndimensinalized with respect t cncrete fracture tughness K 1C and beam depth b (see eq. 13). The diagrams in Fig. l are significant nly fr cp/cp > I, the strain sftening and curvature lcalizatin ccurring nly after the first cracking. The dimensinless peak mment des nt appear t be the same, when the brittleness class is the same and the beam depth is varying. This is due t the absence f an initial crack r ntch. n the ther hand, the pst-peak branches are very clse t each ther and present the same shape fr each selected brittleness class. The size-scale similarity seems t gvern the pst-peak behaviur, specialiy fr lw brittleness numbers Np (class

10 674 C. BSC et al. lad (kni BEAM SIZE A lad (kn) BEAM SIZE B midspan defleclin (mml (a) (b) midspan _--+ deflectin (mny BEAM SIZE C lad (kn) 60~.- ~.- ~ 4 midspan defleclin (mm) (c) Fig. 9. Lad vs defiectin diagrams f R.e. beams. (a) Beam depth b = 100 mm; (b) beam depth b = 200 mm; and (c) beam depth b = 400 mm , 1, 2, 3), and fr large beam depths b (sizes B, C). In these cases, in fact, it is very likely that the fracture prcess zne is negligible with respect t the zne where the stress-singularity is dminant, s that the Linear Elastic Fracture Mechanics mdel (and the nn-dimensinalizatin in Fig. l) is cnsistent with the experimental phenmena. Frm the present investigatin, the demand transpires f analysing the pst-peak and ductile behaviur f lw reinfrced high strength cncrete beams[9-l2], thrugh the cncepts f Fracture

11 Fracture f reinfrced cncrete dimensi bending M K 1 è b3l2.t n Iess mment BRITTLENESS CLASS (N p = 0.00),.0; J C '.4 : ;3 1/ I /IB ~A.2/// ~ ~ ~.1 W ~ /I '~ dimen r-- --.,~ rtati sinless n <PIPa (a).e dimenslnless bending mment ~ M BRITTLENESS CLASS 1 (N p =0.13).0; C.~ I,I>.a Il~.2 VI1\.1 ~ I::>- Ì'-- I-- r--::: (b) Fig. I. (a) and (b). dimensinle rtat1in <pf< ss

12 676 C. BSC et al..e d' rmensrn ess bendlng mment M BRITTLENESS ~ CLASS 2 < Np= 0.36).5 dimensinless bending mment ~ M.Qj-----,--.4 /Ì\ -il/t: \ V.3 -. V r.1 ~ BRITTLENESS CLASS 3 < Np= 0.72) -r-r-'--t---r--,---~-====--::;i._ 0..~---+~----~--~ ~~~--_r----1_----, 2 3."I-----I1I----7'1I---==t 'I; t---1 l'c \ B A (c) dimensinless bending mment dimensin less rt atin CJl/% V M I/c/ A /" K,i V2 t Q1, ,--~~----_Y~--+-~--t---_4 0.6'T------ff-/--+--/----j,..~//--+v + j 0.5'<! / I # / bf--~--+--y-'----'----' BRITTLENESS 0.4"'!------/'---th"-----t-----l CLASS 4 P 0.3't-+l-l,I/--fV -+ +-_-.,c--<_n =,'_'1_B_)-' -i 0."I--/H t---~----_+----_r----1_ "!-f i dimensin less rtatin CJl/%.,'-tt-----t----t l +-_r~-+--+_ dimensinless +-~_4'---+-_+-r-~t-at~;-n_+p,-Y<-CJlr+-~ ~ W Fig. l. Dimensinless bending mment vs rtatin diagrams. Brittleness number (a) N p = ; (b) N; = 0.13; (c) N; = 0.36; (d) N; = 0.72; (e) N; = Mechanics. As is demnstrated in the present paper, the pssibility f extraplating predictins frm small t large scales, is entrusted t the nn-dimensinal (brittleness) number N; (see eq. 14), where, in additin t the traditinal gemetrical and mechanical parameters, even the cncrete fracture tughness K,c, r the cncrete fracture energy G F, appears. Acknwledgements-The firm RECCHI in Trin, is gratefully acknwledged by the authrs fr the beam mulding. Thanks are due specially t Gem. Pagani, wh attended t the specimens preparatin with cmpetence and carefully avided shrinkage and hydratin cracking.

13 Fracture f reinfrced cncrete 677 REFERENCES [1] A. Carpinteri, A fracture mechanics mdel fr reinfrced cncrete cllapse. IABSE Cllquium n Advanced Mechanics f Reinfrced Cncrete, Delft, pp (1981). [2] A. Carpinteri, Stability f fracturing prcess in RC beams. J. Structural Engng 110, (1984). [3] Albert Carpinteri and Andrea Carpinteri, Hysteretic behaviur f RC beams. J. Structural Engng 110, (1984). [4] A. Carpinteri, Interpretatin n the Griffith instability as a bifurcatin f the glbal equilibrium. NAT Advanced Research Wrkshp n Applicatin f Fracture Mechanics t Cementitius Cmpsites, Evanstn, Illinis, 4-7 September, 1984 (Edited by S. P. Shah) pp Martinus Nijhff, The Netherlands (1985). [5] F. Levi, C. Bsc and P. G. Debernardi, Tw aspects f the behaviur f slight1yreinfrced structures. 25th CEB Plenary Sessin, Trevis, Italy (11-13 May 1987). [6] A. Carpinteri, Catastrphical sftening behaviur and hyperstrength in lw reinfrced cncrete beams. 25th CEB Plenary Sessin, Trevis, Italy (11-13 May 1987). [7] H. kamura, K. Watanabe and T. Takan, Applicatins f the cmpliance cncepts in fracture mechanics. ASTM STP 536, (1973). [8] H. kamura, K. Watanabe and T. Takan, Defrmatin and strength f cracked members under bending mment and axial frce. Engng Fracture Mech. 7, (1975). [9] F. Levi, n minimum reinfrcement in cncrete structures. J. Struclural Engng 111, (1985). [I] E. Fehling, G. Kònig and D. Scheidler, Crack width cntrl and tensin stiffening. A. J. Cncr. Cncr. Struct, Darmstad Cncrete, l, (1986). [Il] G. Kònig, Restraint crack-width cntrl and minimum reinfrcement in thick cncrete members. A. J. Cncr. Cncr. Struct. Darmstad Cndrete, l, 9-22 (1986). [12] J. P. Jaccud and H. Charif, Armature Minimal Pur le Cntròle de la Fissuratin. Rapprt final des essais série "C", Publicatin IBAP n. 114, Ecle Plytechnique Fédérale de Lausanne, Juillet (1986). (Received fr publicatin 22 Nvember 1988)

Dynamic strain softening of concrete in compression under rapid loading K. Fujikake-, J. Mizuno, A. Suzuki*, T. Ohno" & T.

Dynamic strain softening of concrete in compression under rapid loading K. Fujikake*-, J. Mizuno*, A. Suzuki*, T. Ohno & T. Transactins n the Built Envirnment vl 2, 1998 WIT Press, www.witpress.cm, ISSN 174-59 Dynamic strain sftening f cncrete in cmpressin under rapid lading K. Fujikake*-, J. Mizun*, A. Suzuki*, T. hn" & T.