FRACTURE MECHANICS BASED BENDING FAILURE ANALYSIS FOR FIBER REINFORCED LIGHT-WEIGHT CONCRETE PANEL CONSIDERING CRACK DISPERSION EFFECT

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1 3 th World Conference on Erthquke Engineering Vncouver, B.C., Cnd August -6, 4 Pper No. 79 FRACTURE MECHANICS BASED BENDING FAILURE ANALYSIS FOR FIBER REINFORCED LIGHT-WEIGHT CONCRETE PANEL CONSIDERING CRACK DISPERSION EFFECT Yoshinori KITSUTAKA SUMMARY In this study, the frcture process of strin softening nd hrdening mteril which hs the crck dispersion nd locliztion chrcteristics ws studied. Numericl nlysis method to clculte the lod displcement reltionship of the member which hs multiple crcking subjected to the bending ws proposed bsed on the frcture mechnics considertion. Bending tests of the fiber reinforced light-weight concrete pnel were performed. Influences of the fiber content nd the ir content on the mechnicl properties nd on the crck dispersing effect were clered. Especilly, the fiber reinforced light-weight concrete contining ir showed high crck dispersion under the bending condition. Tension softening digrm of fiber reinforced light-weight concrete which cn be used s constitutive lw for the numericl nlysis were evluted by the dt of lod - crck mouth opening displcement (CMOD) reltionships obtined by the wedge splitting test. The nlysis results of the bending tests greed well with the test results. INTRODUCTION Active reserch hs been conducted in recent yers regrding subjects relted to the high performnce of concrete mterils. Among such subjects, enhncement of the ductility of concrete by fiber reinforcement is of prmount importnce from the spect of improving the sfety performnce nd durbility of concrete members, s high ductility increses the energy bsorption of concrete during its filure process. The filure of fiber-reinforced concrete is chrcterized by being more ductile thn tht of conventionl concrete, thnks to bridging of fibers, which continues to trnsfer the tensile stress fter crcking. The trnsferred stress my even increse fter crcking, which is known s strin hrdening. In such stte, prtil hrdening in the filure process of member cuses new crcks in other prts of the member, dispersing the frcture process zones. This increses the energy-bsorbing cpcity of the member during filure, while voiding brittleness due to locliztion of crcking, thereby leding to n idel mode of filure. Such behvior of fiber-reinforced mterils during filure in tension cn be experimentlly nd nlyticlly evluted s tension softening properties [], nd such properties hve been reported in regrd to wide vriety of fiber reinforced concretes []. Fundmentl studies on the stress-trnsfer mechnism of fiber-reinforced mterils hve lso been reported [3,4]. Nevertheless, these mostly refer to evlution Professor, Tokyo Metropolitn University, Tokyo, Jpn. Emil: kitsu@comp.metro-u.c.jp

2 on the element level with limited frcture process zone, nd few studies hve modeled the crck dispersion found in the filure of ctul members. Modeling of the process of dispersion/locliztion of frcture process zones is crucil, prticulrly for numericl nlysis of the lod-deformtion reltionship of members. Other pplictions of fiber-reinforced concrete include pre-cst concrete members for buildings, curtinwlls, nd pnel mterils, for which weight reduction is desired from the spect of seismic sfety. This pper reports on n investigtion into the filure of composite mterils exhibiting softening nd hrdening phenomen nd presents simple model of filure dispersion/locliztion. An nlysis method for lod-deformtion reltionship on the member level is proposed s well. The bending filure properties of fiber-reinforced lightweight concrete pnels re lso investigted, with the vlidity of the proposed nlysis method for lod-deformtion reltionship being exmined. ANALYSIS METHOD Models nd nlysis method for dispersion/locliztion of filure Figure shows typicl filure processes when nonliner elements, El nd El, mde of the sme mteril (e.g., fiber-reinforced mteril) hving initil defects of different degrees re serilly rrnged nd undergo tension softening nd hrdening. As the lod increses, crcking grdully occurs in ech element (, ). El hving lrger initil defect, in which crcking proceeds in dvnce of tht in El, begins to grdully soften fter the pek lod (Q b ), with the lod-trnsferring cpcity between the elements reducing (Q b to Q c ). This puts El to n unloding pth (b to c). Due to the plstic deformtion, El does not fully return to point. If El continues to soften, then filure is loclized nd intensely proceeds in El (d ), while crcking in El begins to close (d). Norml softening mterils, such s concrete, shows such locliztion of filure to rupture. However, if El shows hrdening phenomenon from c (c to e ), then El enters re-loding phse (c to b). The filure of El begins to proceed gin fter the lod exceeds Q b (b to e), cusing re-dispersion of the frcture process zones. When the lod increses further, with El pssing the pek lod (Q e ), then El begins to soften (e to f), putting El to n unloding pth (e to f ). The filure of ech element continuously proceeds through these complicted pths, eventully bsorbing gret energy. Also, the chnges in the tension-softening digrm in the unloding process shown s b to c in Fig. (pprent reltionship between the crck mouth opening displcement (CMOD) nd cohesive stress) cn be explined s shown in Fig.. When the lod decreses from Q b to Q c, the cohesive stress, σ, decreses uniformly nd linerly corresponding to the unloding rtio, α = Q c /Q b, to σ (Eq. ()), ssuming tht crcking closes from the tip t constnt rte. The CMOD, δ, cn be expressed s Eq. () by incorporting the plsticity index, p, into. The plsticity index, p, expresses the degree of crck closing when unloded. p = when the CMOD is origin-oriented, wheres p = when the CMOD remins unchnged fter unloding, showing perfect plsticity. During re-loding, the plot returns long the unloding pth to the point where unloding begn, from which it follows norml tension softening digrm (see Fig. ). σ' = ασ () δ' = (α + ( α)p)δ () The CMOD, Eqs. () nd () llow determintion of the CMOD distribution fter unloding nd clcultion of the deformtion of the elements. The lod-deformtion reltionship of the entire member is determined by clculting the deformtion of ech element nd superposing ll, while ssuming tht cross

3 Lod Q sections remin plne nd norml fter deformtion in the stress trnsfer between elements (Nvier s hypothesis). This nlysis flow is shown in Fig. 3. Anlysis method for pure bending filure Frcture of n element of the pure bending member is modeled for the cohesive force model s shown in Fig. 4. The boundry conditions of the crcked element with cohesive forces re provided by the equilibrium of stress intensity fctor nd the equilibrium of COD. K IM () + K Ir () = (3) δ(, x) = δ M (, x) + δ r (, x) (4) Where is the crck length, x is the point on the crck surfce, K IM (), K Ir () re the mode I stress intensity fctors on the crck tip due to the externl moment nd the cohesive stress, respectively, δ(, x) is the COD t x nd δ M (, x), δ r (, x) re the CODs due to the externl moment nd the cohesive stress, respectively. These reltionships cn be clculted by using FEM or BEM, but in the cse of simple bem, the solution cn be obtined using the clcultion results of liner frcture mechnics [5]. K IM (), K Ir () ppered in Eq. (3) re follows. K IM () = 6M / d π F(, d) =M k M () (5) σ (, c) K Ir () = G(,c, d)dc = σ(, c) k π r (,c)dc (6) M is the nominl moment due to the externl lod, d is the height of bem, c is the coordinte indicting the point on crck surfce where cohesive force is cting (see Fig.4), nd F(, d) nd G(, c, d) re weight functions. For the clcultion of COD, Cstiglino's theorem is employed. Displcement of crcked body cn be expressed by stress intensity fctors (K-superposition method) [5]; dy = d + Q K Q =K r σ β = α +(- αp) σ El e e' σ Q e Unloding Unloding ασ b' f Unloding αq αk Q =αk r Q b b El f' σ ασ ' ασ c' ασ Q Softening Hrdening σ = f(δ ) c c FRC σ = f (δ ) d d' pδ βδ δ βδ cr δ cr CDO δ Q locliztion ' Q Q p= (elstic) p = (plstic) El El ασ ασ Fig. Lod-deflection reltionship of two hrdening elements x K(z) K f (z) E f Deflection d f = Cohesive stress αδ cr Fig. Unloding/reloding model of tension softening digrm dz (7) δ cr

4 Where dy is the displcement on x, d is the displcement of uncrcked body, z is the coordinte indicting the crck length for the integrtion, f is the fictitious force cting on the point x, E is the elstic modulus, K(z) is the stress intensity fctor producing the displcement dy, nd K f (z) is the stress intensity fctor due to fictitious force f cting for the direction of dy. Substituting Eqs. (5) nd (6) into Eq. (7) nd clculting the left hnd side of Eq. (4). δ M (, x) = 4M / d F(z, d) G(z, x,d)dz = M d x M (, x) E (8) δ r (, x) = σ(, c) 8 G(z, x,d) G(z,c,d)dz x π z E dc = σ(, c) d r (, x,c)dc (9) Substituting Eqs. (5) nd (6) to Eq. (3), lso Eqs. (8) nd (9) to Eq. (4), nd cnceling the M ppered in Eqs. (3) nd (4), then the simple crck integrl eqution is obtined s Eq. (). H(, x, c) is the weight function clled the H-function, nd it does not depend on the loding condition. δ(, x) = σ (,c) H(, x,c) d ;H(, x,c) = k r (,c) d M (, x)/k M () d r (,x,c) () Tension softening digrm s constitutive lw of mteril is expressed s Eq. (). The cohesive stress σ(, x) is the poly-liner function of crck opening displcement (COD) of δ(, x). σ(,x) = m(δ, x) δ + n(δ,x); δ = δ(, x) () Where m(δ) is the softening inclintion nd n(δ) is the inflection point. Substituting σ(, x) of Eq. () into Eq. () nd expressing in the mtrix form for the totl number of nodes (=n) on the crck surfce, we obtin the simultneous crck eqution on δ(, x). From the solution of the eqution, distribution of COD nd cohesive stress in the cse of crck length re obtined. Substituting σ(, x) into Eq. (6) nd clculting K IM () by Eq. (3) then externl moment M is obtined by Eq. (5). No Lod Q=Q+ Q Element i Yes Deflection d=d+ d Cross sections remin plne Hrdening Unlodig/reloding Plstic Unloding Q =α Q rtio p TSD of fter unloding σ =ασ,δ =(α +(-α )p)δ Deflection clcultion Lod deflection curve Remin plne M -σ b -σ b θ/ c σ (, c) h =K IM +K Ir K I x d δ (, x) θ/ M t = Fig.3 Anlysis flow considering unloding/reloding Fig.4 Cohesive force model of bending element for mode I filure

5 Rottionl ngle θ() of the element is obtined from the ccumultion of the rottionl ngle due to the externl moment nd the rottionl ngle due to the cohesive stress. θ () = θ M () + θ r () () The rottionl ngle θ is expressed s below by using the Cstiglino's theorem. K IM θ = U NoCrck M + E K I d (3) M Substituting Eqs. (5) nd (6) into Eq. (3). θ M () = θ MNoCrck + E K IM (z) K IM (z) dz = Mh M EI + 7πM z F(z, d) dz Ed 4 (4) x dz = 4 σ (,c) F(z, d) G(z,c,d)dz M Ed [ ] dc (5) ( ) M = + E K K Ir (z) IM (z) θ r () = θ MNoCrck M = In θ r (), considering the moment M s fictitious moment s M=. Substituting Eqs. (4) nd (5) into Eq. (), then reltion of M nd θ ()is obtined. From the result of rottionl ngle, curvture k=θ/h of ech element is clculted, nd totl deflection of the bem is clculted by ssuming cross sections remin plne. TEST PROCEDURES Specimens Tble gives the mterils used in the tests. In considertion of the ppliction to pre-cst concrete members, the mterils comprised high-erly-strength portlnd cement, fomed rtificil lightweight ggregte mde from wste glss, nd n ir-entrining nd high-rnge wter-reducing dmixture nd n ir foming gent s the cement, fine ggregte, nd chemicl dmixtures, respectively. Vinylon fibers, which were found effective in improving the ductility nd reducing the weight of concrete, were used s short fibers. These were proportioned with wter-binder rtio (W/B) of %. Norml strength concrete specimens were lso fbricted for reference. Tble Mterils used Symbol Mterils Properties C Cement high-erly-strength portlnd cement, ρ=3.5 SF Silic fume SiO 96.8%, ρ=. VF GL Vinylon fiber (RF4) Wste glss ggregte SP Super plsticizer - AE AE gent - FA Foming gent - S Crshed snd ρ=.6 G Crshed stone ρ=.66 ρ=.3, length=3mm, dimeter=66(μ m), tensile strength=9(mp), Young s modulus=3(gp) #:#:#3=8:: (weight rtio), mximum size=5mm, ρ=.7, wter bsorption=7%

6 Tble gives the proportioning conditions for the specimens. A 7-ml omni mixer ws used for mixing. Three cylindricl specimens (φ mm) for compression nd elstic modulus testing, three prismtic specimens ( mm) for wedge splitting, nd two pnel specimens (3 9 mm) for pnel bending testing were prepred for ech mixture. Specimens were demolded t n ge of one dy nd then wter-cured t C up to n ge of weeks. Tble 3 gives the properties of the mixtures while fresh nd strength properties of the specimens. Tble Mix proportions ) plin concrete W/C Absolute weight( kg / m3 ) Symbol (%) Wter Cement Fine ggregte Corse ggregte NC b) the other concrete Foming Absolute weight ( kg / m3 ) W/B Symbol gent (%) Agent (B %) Wter Cement GL SF VF SP F AE LP LP LF LF Symbol Slump (cm) Tble 3 Properties of specimens Slump flow Specific Air content Compressive (cm cm) grvity (%) strength (MP) Young s modulus (GP) NC LP LP LF LF Wedge splitting testing The tension-softening digrm, which is used s the constitutive lw in the numericl nlysis, ws evluted by wedge splitting testing, whereby mode I filure (tensile deformtion) is obtined with smll specimens. A notch ws cut in the center of ech specimen using dimond cutter with tooth thickness of mm, leving ligment depth of 5 mm, t which tensile filure ws induced by inserting wedge (Fig. 5). A servo-controlled hydrulic tester hving closed loop system (mnufctured by MTS) ws used to chieve ccurte mesurement of the lod-displcement curves. This testing mchine permits control of the displcement rte by djusting the hydrulic servo vlve, while using the clip guge mesurement of the CMOD s the feedbck signl. The displcement rtes by the CMOD were set t. mm/min for norml nd fiber-reinforced concrete specimens, wheres the rte ws set t.5 mm/min for lightweight concrete with no fibers, since this type of concrete tends to led to brittle filure. Sensitive clip guges for displcement control (MTS-63.) were used for the CMOD mesurement. The tensionsoftening digrm ws determined by poly-liner pproximtion nlysis method [6] bsed on the obtined lod-cmod curves.

7 Lod Lod Wedge fixture Notch Clip gge Fixture for lod trnsmission Roller bering Specimen (xx) Fig.5 Wedge splitting test 5 (mm) Pnel bending testing Figure 6 shows the schemtic of pnel bending testing. A -tonf Amsler universl testing mchine ws used for the bending tests by four-point mnully operted loding, with the spn between the supports nd the centrl spn between the loding points being 8 mm nd 4 mm, respectively. The midspn deflections t both bottom edges were mesured. Seven strin guges with guge length of 6 mm were lso glued to the bottom of ech specimen to mesure the stte of crcking nd degree of crck dispersion. Lod Mesuring point LVDT Strin gge 6mm (mm) 9 Fig.6 Bending test of pnel specimen TEST RESULTS AND DISCUSSION Lod-CMOD curves Figure 7 shows the lod-cmod curves obtined from wedge splitting tests by verging the dt for ech mixture. Among the specimens with no fibers, LP contining no foming gent did not chieve stble filure, leding to brittle filure, but LP contining foming gent underwent reltively stble filure, though the strength ws lower thn tht of LP. When compred with norml concrete, NC, the res under the curves of both LP nd LP re smll, indicting their smll energy-bsorbing cpcities. Regrding specimens reinforced with fibers, the pek lods of LF nd LF re similr to those of specimens with no fibers, but subsequent lods re mintined high, indicting substntil improvement in their ductility. It is prticulrly notble tht the lods re-increse fter tenttive post-pek drop, indicting the hrdening phenomenon. The re under the curve nd the pek lod of fomed LF re lower thn those of LF with no foms. This my be becuse the inclusion of foms reduces the strength of the mtrix, while reducing the res of bond between fibers nd pste, thereby reducing the lod-trnsferring cpcity.

8 Lod (kn) LP: Light weight concrete plin NC: Norml concrete LF: Light weight FRC LF: Light weight FRC + F LP: Light weight + F CMOD (mm) Fig.7 Lod ĞCMOD curve (verge) Cohesive stress (MP) LP: Light weight concrete plin LP: Light weight + F LF: Light weight FRC + F NC: Norml concrete LF: Light weight FRC COD (mm) Fig.8 Tension softening digrm Tension softening digrm Figure 8 shows the tension-softening digrms (TSDs) determined by poly-liner pproximtion. Reflecting the lod-cmod curves, the TSDs of mixtures showing high ductility by the lod-cmod curves show moderte softening with lrge res under the lines. The TSDs of specimens with fibers re level or rising towrd the right-hnd side, indicting the stress trnsfer despite the incresed CMOD. The obtined TSDs were linerly pproximted into two to five lines for nlysis. Bending test results of pnel specimens Figure shows the lod-deflection curves obtined from the pnel bending tests. Though the mximum strengths of specimens contining fibers (LF nd LF) re similr to those of specimens with no fibers, their filure is more ductile, without reching filure immeditely fter the mximum lods. The lod on LF with no foms tenttively drops fter the pek lod but grdully recovers. The lod on fomed LF retins rising tendency fter the pek lod, showing severl subsequent peks. Observtion of the pnel crck ptterns fter the bending tests reveled one lrge crck in ech of NC, LP, nd LP. Among the specimens with fibers, LF showed certin fine dispersion of crcks, wheres LF showed significnt dispersion of crcks in contrst to other specimens. Chnges in strin by pnel bending testing Figure 9 shows the strins mesured t the bottom surfces of pnels (one specimen ech). On the NC specimen, the strins re uniformly distributed up to deformtion of.4 mm, which is before the pek lod, but lrge strin develops prtilly when the deformtion is.5 mm ner the pek lod, indicting locliztion of filure. On the LP specimen with foms, the strins re uniformly distributed up to deformtion of pre-pek.4 mm similrly to NC, nd lrge strins develop t two points with ner-pek.5 mm deformtion, indicting filure dispersion. Among the specimens contining fibers, the strins of LF uniformly increse up to deformtion of ner-pek. mm but concentrte fter the pek lod, indicting locliztion of filure. The uniform strins before the pek lod re greter thn those of LP, which cn be ttributed to occurrence of finely distributed crcks due to bridging of fibers. LF is lso chrcterized by the reductions in the strins t other points when the strin is concentrted nd crcking is loclized, indicting substntil unloding t other points. This suggests closing (recovery) of fine crcks. On fomed LF, the strin hs lredy been distributed to severl points by the time when the deformtion reches mm, nd the strin remins distributed fter the pek lod, indicting significnt dispersion of filure.

9 8 6 4 NC: Norml concrete LPD.mm.4mm.5mm 3 NC: Norml concrete Test Anlysis LPD LP: Light weight 4.mm.6mm concrete plin.4mm.8mm LP: Light weight concrete plin Strin (μ ) LPD LP: Light weight 4.mm + F.4mm 3.5mm Lod (kn) Test Anlysis LP: Light weight+ F LF: Light weight FRC LPD.5mm.mm.mm 3.mm 4.mm 5.mm LF: Light weight FRC Test Anlysis Test Anlysis LF: Light weight FRC + F LPD.5mm.mm.mm 3.mm 4.mm 5.mm LF: Light weight FRC + F Test Anlysis Gge No mm Gge plce Deflection (mm) Fig.9 Strins distribution mesured t the bottom surfces of pnels Fig. Lod-deflection curves obtined from the pnel bending tests

10 RELATIONSHIP BETWEEN ANALYSIS AND TEST RESULTS Anlysis conditions Bending nlysis ws conducted by the method proposed in this pper, nd the results were compred with the pnel bending test results. The geometry of specimens nd the loding conditions were the sme s those dopted for the tests. The TSD determined by wedge splitting testing for tension nd pproximted to multiple lines ws used s the mteril constitutive lw. Liner elsticity ws ssumed for the compressive zone. The elstic modulus ws determined from the compression test results of cylindricl specimens. As for the width of elements, h, 5 mm ws dopted, becuse nlysis conducted with h being set t 5 nd mm led to no mrked differences mong mixtures, nd becuse width of 5 mm corresponded better to the test results. The 4-mm centrl spn under pure bending were divided into eight elements. The initil defect ws modeled by rndomly selecting n initil notch with length of to 5 mm with -mm intervls nd ssuming it t the center of ech element. According to the flow shown in Fig. 3, crcking ws llowed to develop from the wekest elements hving 5-mm initil notches, nd the chnges in the rottion ngle of ech element were sequentilly determined ccording to the moment t the time of softening nd hrdening of the wekest elements. The curvture distribution ws then determined from the rottion ngles. Regrding this s the lods, the moment digrm ws determined to obtin the midspn deflection. The plsticity index p ws set t.5, s the index vlues of,.5, nd. led to no mrked differences in the nlysis results. Anlysis results nd discussion Figure shows exmples of clcultion of the moment-rottion ngle of ech element crried out in the nlysis process. In regrd to NC, n element hving lrger initil notch is found to led to smller pek moment nd rech the pek erlier thn other elements. The strin-hrdening phenomenon of LF owing to fibers is dequtely expressed, in which it tenttively softens but begins to hrden on the wy, eventully exceeding the primry pek lod. 8 5 Moment M (Nm) 6 4 =mm =mm =3mm =4mm =5mm Rottion ngle θ ( -3 rd.) Fig. Exmples of clcultion of the moment-rottion ngle of the element 4 3 =mm =mm =3mm =4mm =5mm The reltionship between the nlysis nd test results is lso shown in Fig.. The nlysis nd test results generlly gree well. Though the pek lods for LP nd LP with no fibers re slightly lower thn the test results, the lod-displcement curves show the snpbck phenomenon in which the deformtion turns bck fter the pek lod, ccurtely expressing the unstble filure observed in the tests. The test nd nlysis results of LF nd LF with fibers gree well, prticulrly well-expressing the deformtion cpcity fter the pek lod. The nlysis results express the re-loding of t which ech element reches

11 the pek lod being mrked with sequentil numbers. The upwrd nd downwrd curves found in the test results of LF prticulrly gree well with the nlysis results. CONCLUSIONS The results obtined in this study re summrized s follows: () The progress of filure involving crck dispersion nd locliztion ws modeled with combintions of multiple elements incorporting tension softening. Using this model, n nlysis method for loddisplcement reltionship ws proposed, in which unloding nd re-loding ssocited with the softening nd hrdening of elements re incorported. () Bending tests on vrious concrete pnels reveled tht the inclusion of short fibers disperses the crcks, thereby improving the energy-bsorbing cpcity of the pnels. Crck dispersion ws prticulrly evident in lightweight fiber-reinforced pnels contining lrge mount of foms to reduce the pnel weight. (3) The nlyzed lod-displcement reltionships of vrious concrete pnels under bending nerly greed with the test results. The deformbility of fiber-reinforced mterils ssocited with post-pek softening nd hrdening ws prticulrly well-expressed by the nlysis results. REFERENCES. Li VC, Ling E. Frcture Processes in Concrete nd Fiber Reinforced Cementitious Composites. Journl of Engineering Mechnics, ASCE; (6): , Kitsutk Y, Oh-ok T. Appliction of Ultr-light Weight Fiber Reinforced Concrete for the Incresed Erthquke Resisting Wlls. th World Conference on Erthquke Engineering, Aucklnd, New Zelnd, Pper no. 47,. 3. Evns AG. Perspective on the Development of High-toughness Cermics. Journl of the Americn Cermic Society; 73(): 87-6, Li VC, Leung CKY. Stedy-Stte nd Multiple Crcking of Short Rndom Fiber Composites. Journl of Engineering Mechnics, ASCE; 8(): 46-64, Td H, Pris PC, Irwin GR. The Stress Anlysis of Crck Hndbook. Second Edition: Pris Productions Incorported, Kitsutk Y. Frcture Prmeters by Poly-liner Tension Softening Anlysis. Journl of Engineering Mechnics, ASCE; 3(5): , 997.