Crack width prediction in RC members in bending: a fracture mechanics approach

Transcription

1 Cra widh prediion in RC member in bending: a raure mehani approah S. Saey Aian Proeor in Civil Engineering, B. M. S. College o Engineering, Bangalore, India D. Binoj Po Graduae Suden, B. M. S. College o Engineering, Bangalore, India ABSTRACT: Craing i a very ommon ourrene in reinored onree(rc) ruure. Cra in RC ruure are haraerized by ra widh and ra paing. In he preen udy an expreion i developed uing a oheive ra model having a bilinear rain oening relaionhip o predi ra widh in RC beam. One o he aumpion made in he developmen o he model i ha here i omplee lo o bond beween he bar and he onree. However, hi rude aumpion lead o oo onervaive value or ra widh, ine he riional ore a he bar-onree inerae limi bar lip and onequenly ra widh. Thereore, i i neeary o inrodue a reraining ore o model bond and o ind ra-loing diplaemen. The ra widh value o obained rom he propoed model are ompared wih ode prediion and wih experimenal reul available in lieraure. The reul how ha he propoed approah i ound, onien and realii. INTRODUCTION The ourrene o ra in reinored onree ruure i ineviable beaue o he low enile rengh o onree. Cra orm when he enile re in onree exeed i enile rengh. Craing in reinored onree ruure ha a major inluene on ruural perormane, inluding enile and bending ine, energy aborpion apaiy, duiliy, and orroion reiane o reinoremen. Craing a he ervie load hould no exend o uh a limi ha i poil he appearane o he ruure or lead o exeive deormaion o he member. Thi may be ahieved by peiying an allowable limi on ra widh value. In order o aure a aiaory perormane o he ruure even under ervie load, an imporan limi ae i.e., he limi ae o ervieabiliy (raing) i inrodued ino he limi ae deign proedure. Thi limi ae i aumed o be aiied i ra widh in a onree member are wihin a maximum allowable limi. While he need or a ra limi ae ha been univerally agreed on, he ormulae or prediing he ra widh exenively vary in he variou ode o praie. Inpeion o ra widh prediion proedure propoed by variou inveigaor indiae ha eah ormula onain a dieren e o variable. A lieraure review alo ugge ha here i no general agreemen among variou inveigaor on he relaive igniiane o dieren variable aeing he ra widh, depie he large number o experimenal wor arried ou during he pa ew deade. Taing all he parameer ino aoun in a ingle experimenal program i no normally eaible due o he large number o variable involved, and he inerdependeny o ome o he variable. In hi paper, an aemp i made o predi an expreion or ra widh by inorporaing a bilinear rain oening union and all he variable whih inluene ra widh. The propoed ormula are alo ompared omprehenively wih he e reul available in he lieraure (Hognead, 96; Kaar and Mao, 963; Clar, 956). To ae he relaive perormane o he propoed ra widh equaion, i i ompared wih he inernaional ode o praie. CRACK WIDTH EXPRESSION Gerle e al (99) developed impliied aumpion ha allow analyial oluion or lexural ra in ingly reinored beam in bending while reaining he igniian eaure o he iiiou ra model (FCM) whih wa inrodued by

2 Hilllerborg e al. (976). The FCM ha he poenial o being very ueul in underanding he raure and ailure o onree ruure. I aume ha he raure proe zone a he ra ip i long and ininieimally narrow. The raure proe zone i haraerized by a normal re veru a ra opening diplaemen urve whih i onidered a a maerial propery. The hape o hi re-ra opening diplaemen (oening) urve an be eiher linear/bilinear/ri-linear or a power law. Gerle e al.(99) aumed a linear oening relaion in hi ormulaion and predied he ra widh a a produ o a onan C (a union o brilene o onree or a union o reinoremen) and riial ra opening diplaemen COD r ( a union o he oening urve or he raure energy). Thi expreion or ra widh doe no expliily inlude uh parameer a he diameer/perimeer o reinoremen whih inluene he value o ra widh. In he lieraure, bilinear oening eem o deribe he behavior o onree in enion more appropriaely han he linear oening. An aemp i made here o wor ou an expreion or ra widh baed on bond mehani (bar lip inluded), and ormulaed a he produ o he ra paing ime he mean rain in he reinoremen by inorporaing he bilinear oening union.. Signiiane o he rain oening urve The re-ra opening law or onree in enion i ound o have a deending branh in he popea region. The imple idealizaion or hi behavior i a linear oening relaion, bu i i more realii o onider a bilinear oening relaionhip. A ypial bilinear hape o he oening urve i hown in Figure. Linear oening eem o be an obviou hoie when he daa deribing he aual maerial behavior i limied. However, linear oening proved o overeimae ruural apaiy. Thereore, bilinear urve have been aeped a reaonable approximaion o he oening urve or onree, alhough here eem o be no agreemen abou he preie loaion o he in poin. In he lieraure, everal reearher have given he in poiion (brea poin) on he bai o experimen, and here are quie a ew imple mehod o ideniy any bilinear oening o i pariular experimenal daa (Guinea e al. 994). Briner and Dahl (989) reormulaed he ubruure mehod inrodued by Peeron (98) or he hree poin bending peimen in order o obain omplee load diplaemen relaion. From he eniiviy analyi o heir mehod uing linear, bilinear and ri-linear model, i i eviden ha he hape o he re ra opening diplaemen relaion ha igniian inluene on he reul. However, ri-linear approximaion doe no eem o deviae igniianly rom he bilinear approximaion indiaing he uiieny o he bilinear approximaion. In hi udy, he ra widh i alulaed onidering peii in poiion(brea poin) a uggeed by Briner and Dahl (989), having he value o = and = 0.6. opening di- Figure. Typial bilinear re veru ra plaemen urve. Propoed mehodology uing a lexural raing model whih inorporae he bilinear oening union in enion wih non lineariy o onree in ompreion The main aumpion o he model are a ollow:. Plane eion remain plane beore and aer deormaion wihin he enral elai band.. The beam i onidered rigid ouide he enral elai band. 3. Fiiiou ra urae remain plane aer deormaion. 4. The re veru ra opening diplaemen urve i aumed a bilinear oening in enion. 5. Conree i homogeneou, ioropi and nonlinear elai. 6. The eel ha a perely plai maerial model. 7. The reinoremen an lip wih repe o onree wihin he enral elai band (a). 8. The enroid o he eel i loaed a he boom o he beam and he onree over below he eel level i deliberaely negleed or impliiy in he derivaion o he expreion. A hown in Figure, hi model onider a varying enral elai band whoe widh varie ime he lengh o ra rom he ra urae. The widh o he enral elai band onidered i a, i.e., a a diane o a on eiher ide rom he ra urae, where i a onan and a i he ra lengh. The beam i onidered rigid ouide he

3 enral elai band. The model i apable o prediing he lexural behavior o onree beam Figure 3, how an idealizaion o he deormed hape (grealy magniied) o a ra in a reinored onree beam, ogeher wih normal re diribuion onidering a bilinear re ra opening diplaemen relaionhip. L.3 Normalizaion o Parameer Cra mouh opening diplaemen CMOD C = COD Cra lengh = A = a h Diane rom ra ip o neural axi r S= h Diane rom neural axi o op iber o beam T= h Figure. Shemai diagram o he beam howing enral elai band widh M Figure 3. Shemai o re variaion (Cae I - Sage I) Two ae are onidered and hey are a ollow: i) Cae I, he iiiou ra i no uiienly open o relieve he normal re a i mouh i.e., (CMOD < COD r ). Cae I i urher ubdivided ino wo age, viz. Sage I (ju beore he in), and Sage II (ju aer he in), and ii) Seel P d NA a Span o he beam= L CMOD COD a x Cae II, in whih he iiiou ra i uiien open o relieve he normal re a i mouh (CMOD > COD r ). a [(- )C/ ] M Maerial-ale parameer or onree= h β = E COD. Maerial parameer(or reinoremen), α = ρn r Where, m i he applied momen, n = E /E i he modular raio, ρ i he geomeri reinoremen raio and α i he mehanial reinoremen raio. Deriving on imilar line o ha uggeed by Gerle e al., (99) he expreion or he ra mouh opening diplaemen (CMOD) i obained a variou age o loading a equal o onan C muliplied by riial ra opening diplaemen (COD r ). The CMOD i nohing bu he ra widh a any poin o loading in he reinored onree lexural member a he level o eel. CMOD = C*COD r From he derivaion o he lexural raing model wih bilinear oening in enion and nonlineariy in ompreion, we have he rain a he boom o he beam whih i alo equal o he rain a he level o eel. ( C) ε = ε b = () E ( Rearranging he above equaion, C E ( ) ε (3) = Subiuing (3) in (), CMOD E CODr (4) = ε

4 = ε E = ne = n = E E E A we now, CODrε CODrε CODrε ECODrε.h A β =, α = ρn, ε. πφφ.. ρ = =, E.CODr bh 4. b. h ε = number.o.bar and φ = Diameer o he reinoring bar. Uing he above relaionhip, we ge α CMOD= ρ α h = ρ β β h ε h ε β (5) (6) in eel ( ε ), whih direly inluene he ra widh prediion and hene exhibi a phyially realii expreion. The ra widh in queion i a he level o he reinoremen. The ra widh a he boom o he beam wih he reinoremen over hall be equal o [(h-x) / (d-x)] ime he ra widh a he level o eel, where, h and d are overall deph and eeive deph repeively and x i he neural axi poiion rom he op iber. In he above expreion or ra widh Eq. 7 i derived rom a lexural raing model onidering bilinear rain oening in enion and non-lineariy o onree in enion and one o he aumpion i ha here i a omplee lip beween he ra ae and he eel. However, hi need no be he ae and he prediion made uing Eq. 7 are liely o be onervaive eimae o he ra widh. Furher, i an be underood phyially ha here anno be a omplee lo o bond beween he eel and onree in ordinary bond ondiion. Thereore, in order o aoun or hi anomaly, a reraining bond ore i inrodued again omplee lo o bond beween he eel and onree, o predi ra widh value ha are realii. The ee o hi ra loing ore on he predied value o he ra widh i diued in he ollowing eion. 3 CRACK WIDTH CORRECTION Auming oal lo o bond a he bar-onree inerae wihin he diane + A rom he raed plane i oo onervaive, ine here i he reraining aion due o bond. Thereore, in order o realiially model raing and o avoid any ra-widh overeimaion, i i neeary o inrodue a reraining ore a he level o he reinoremen and o ind he ra-loing diplaemen a ha level. Thi i done uing he expreion a given in he hand boo re ineniy aor (Gdouo 003), ( 4bh α ) ( h CMOD = ε (7) ( επφφβ ) β Where, CMOD = Maximum ra widh a he level o eel & = Kin Poiion(Brea poin). Thereore, Maximum ra widh in he above expreion i a union o nine variable viz: reinoremen raio ( α ), brilene o onree ( β ), enile rengh o onree ( ), re in eel ( ), diameer o bar ( φ ), perimeer o bar ( επφ ), ro eion dimenion o he beam (b, h) and he rain U θ oal / Κ r θ 3θ ( = + in + in 4μ π 3 ν Where, = + ν or plane ree and 3 or plane rain = ( ) 4ν (8) μ = Ε = Rigidiy Modulu, ν = 0. 5 ( + ν ) onree, r = diane o he ra ip. oal Κ = Sre ineniy due o applied load P and enile re o onree.

5 Here, = Coheive ore aing over he iiiou region. Subiuing Eq.0 and Eq., in Eq. 9 Figure 4. Shemai o he reraining ore and he ra loing ree The re ineniy aor K I i alulaed rom Dugdale model (Gdouo 003). Aording o hi model here i a iiiou ra equal o he real ra (L) plu he lengh o raure proe zone (C). The ra i loaded by a reraining ore (P) a he level o eel and an addiional ra loing re whih i equal o he enile rengh o onree. Thereore, he re ineniy aor K (oal) aing a ip o he iiiou ra i expreed a: K (P) +K ( ) (oal) =K (9) K (P) = Sre ineniy aor due o applied load P i ( P) * Ρ Κ = (0) * π * C + L / [ ( )] Where, C = lengh o iiiou ra obained rom he model. L = Real ra obained rom he model. P = Reriing ore P = Β * ( D )* *0. 09 when D/B =. = Β * ( D )* *0. when D/B =. = Β * ( D )* *0. 6 when < D/B < 3. P = Β * ( D )* *0. 3 when 3 < D/B<3.5. = Β * ( D )* *0. 5 when 3.5 < D/B< 4. = Β * ( D )* *0. 4 when D/B = 4. B = Widh o he beam in mm. D = Overall deph o he beam, mm. = Tenile rengh o onree in N/mm. ( K ) = The re ineniy aor due o enile re ( ) aing along he lengh o he raure proe zone. ( K ) / 4* * C = () ( * π ) / [ ( )] ( K (oal) Ρ ) 4* C ( C L ) = * * + π * ( C + L) () Subiuingθ = π, in Eqn. 8, we obain he expreion or he ra loing diplaemen a he level o eel a / Κ r U θ = ( + ) (3) μ π 3 ν where, r = (C+L) = Toal ra, = or + ν plane ree & (3-4ν ) or plane rain μ = Ε + ( ν ) = Rigidiy Modulu, ν ( onree) = 0.5 (oal) Subiuing K rom Eqn. ino Eqn. 3 Thereore, U θ = ( * Ρ) ( 4* * C* ( C+ L) ) = * ( C+ L) *( + ) * ε (4) * μ* π * ( C+ L) * εs The reriing ore P ued in he expreion i alulaed a he produ o he eeive area o onree A around he reinoremen onribuing o hi ee and he maximum enile re o onree. The reriing ore P or variou ombinaion o deph o widh raio (D/b) i alulaed and i i ubiued in Eq (3.0) o obain he re ineniy aor due o an applied load. Thereore he ra i loaded by a reriing ore (P) a he level o eel and an addiional ra loing re whih i equal o he enile rengh o onree and he (oal) re ineniy aor K aing a he ip o he iiiou ra will be he ombinaion o he re ineniy aor due o he applied load and re ineniy aor due o enile re ( ) aing along he lengh o he raure proe zone. Thereore, he maximum ra widh a he level o eel i ompued a he dierene in he ra widh value wih a omplee lo o bond and he ra loing diplaemen due o he reriing ore.

6 i.e., (CMOD) w = CMOD w o - U θ (5) Where, (CMOD) w = Maximum Cra widh wih loing ore CMOD w o = Maximum Cra widh wihou loing ore U θ = Cra loing diplaemen due o loing ore 4 RESULTS AND DISSCUSSION In order o ae he oundne o he propoed expreion (Equaion 7 and 5), hey are ompared wih he e reul available in lieraure (Kaar and Mao, 963; Hognead, 96; Clar, 956) and alo wih he expreion adoped in he inernaional ode o praie. 4. Te reul o Kaar and Mao Kaar and Mao (963) o he Porland Cemen Aoiaion (PCA) modiied he CEB equaion (959) o expre he maximum ra widh a he level o reinoremen on he onree urae. Two ull ale T-beam and a hal and quarer ale model o one o hee beam were eed. The T- beam peimen were loaded by hydrauli ram under he ener diaphragm and were rerained by ie rod near he beam end. Thi loading arrangemen wa ued o imulae a negaive momen region in a oninuou T-beam. A 40-power miroope graduaed in houandh o an inh hydrauli auaor plaed a mid-pan wa ued o meaure ra widh. 4. Te reul o Hognead Hognead (96) eed reinored onree member wih high-rengh deormed bar and onluded ha (i)he mehanim o ra ormaion i uh ha a wide experimenal aer mu inherenly our. (ii) boh maximum and average ra widh are eenially proporional o he re in eel and (iii) he ra widh ha developed in he ae o beam reinored wih ae-o-he-ar deormed bar wa le han one hal o ha or plain bar. He repored ra widh a he enroid o reinoremen or eel ree ranging rom 0000 lb/in (37.9 N/mm ) o lb/in (344.7 N/mm ) or every 0000 lb/in (68.9 N/mm ) inremen. 4.3 Te reul o Clar Clar (956) eed 54 peimen and repored maximum ra widh and paing or eel ree ranging rom 5000 lb/ in (03.4 N/mm ) o lb/ in (30. N/mm ) a every 5000 lb/ in (34.5 N/mm ) inremen. Cra widh on he enile ae were deermined by he ue o Tuerman opial rain gage, rain in he enile reinoremen were meaured wih elerial reiane rain gage. The loaion and exen o ra were oberved and reorded. A number o R/C Slab and beam wih dieren geomerie and bar arrangemen were eed in 4-poin bending. 4.4 Conolidaed e reul The propoed mehod or prediing he maximum ra widh i ompared uing he e reul repored in he lieraure (Hognead 96), Kaar and Mao (963), (Clar 956). For eah beam, he heoreial ra widh obained by mean o he propoed expreion i divided wih he orreponding experimenal ra widh i.e., (W al /W exp ) and he average raio i obained a eel re o 75.8 N/mm (40000 Pi ). The repeive andard deviaion and oeiien o variaion are alo obained a hown in he Table. 4.5 Comparion o he ra widh rom he propoed expreion along wih he Code o Praie wih reerene o he e reul o Hognead In order o ae he relaive perormane o he propoed expreion (Equaion 7 and 5), he average ra widh raio, andard deviaion and oeiien o variaion are obained or he e reul o Hognead (96) and ompared wih he orreponding value obained rom he expreion adoped or ra widh prediion in he inernaional ode o praie. 4.6 Diuion o he e reul From he reul obained (Table & ) he ollowing poin are noed: From Table, i i oberved ha an average ra widh raio o.08 and he oeiien o variaion o.7% i obained a in poiion o (K =0.308 K =0.6) or he e reul o Hognead (96), indiaing ha heoreial value o ra widh ob-

7 ained rom he propoed expreion i loer o he experimenal reul. For he e reul o Kaar and Mao, he propoed expreion produe a ra widh raio o.093 wih a andard deviaion 0.6 and a oeiien o variaion o 0.74% a in poiion o (K =0.308 K =0.6) Thee value indiae ha he ra widh predied by he propoed expreion i onien and reliable, and ha he oeiien o variaion i lower. For he e reul o Clar, he propoed expreion provide a ra widh raio o.54 a in poiion o (K =0.308 K =0.6). The deviaion o heoreial ra widh rom experimenal ra widh wa 0.84 wih a oeiien o variane o 4.64%. From Table, i an be oberved ha he BS 80 equaion undereimae he ra widh by 7.4% or he e reul o Hognead a a eel re o 75.8 N/mm (40000 Pi ) wih an average ra widh raio o 0.76 and a oeiien o variaion o 9.46%. The Model ode equaion 990 alo undereimae he value o ra widh by 38% or he e reul o Hognead a a value o eel re equal o 75.8 N/mm wih an average ra widh raio (W al /W exp ) and oeiien o variaion a 0.60 and 43.55% repeively. The Gergely and Luz equaion whih i baed on a aiial analyi provide an average ra widh raio o 0.89 wih a oeiien o variaion o 3.57% a a eel re o 75.8 N/mm (40000 Pi ) or experimenal value o Hognead. I an be oberved ha even hough he oeiien o variaion i lower, he average ra widh i ill undereimaing by 0.8%. For he e reul o Hognead, he Chinee ode undereimae he value o ra widh by 6.7% wih a oeiien o variaion o 4.0% a a eel re o 75.8 N/mm (40000 Pi). The average ra widh raio i 0.833, whih learly how ha even hough he oeiien o variaion i lower, he ra widh raio ill undereimae. From Table, i i alo oberve ha he propoed expreion provide beer ra widh raio (.08) and oeiien o variaion (.708%). Thee aii indiae ha hi propoed expreion i able o predi onien ra widh value wih a igniianly lower oeiien o variaion a ompared o he ra widh value provided by he ode. Table. Saiial omparion o he propoed expreion a in poiion (brea poin) o bilinear urve (K =0.308 K =0.6) wih he repored e reul. Soure No. o obervaion (Cra widh raio)w al /W exp or Bilinear (K =0.308 K =0.6) Avg Sd. Dev C.O.V Clar Kaar & Mao Hognea d Table. Saiial omparion o variou ode wih he propoed mehod. Soure BS 80 equaion Model ode equaion Gergely and Luz equaion Chinee ode equaion No o obervaion (Cra widh raio)w al /W exp or Bilinear (K =0.308 K =0.6) Avg Sd. Dev C.O.V Bilinear The graphial illuraion o he aiial omparion o he propoed expreion a in poiion (brea poin) o he bi-linear urve ( = & = 0.6) a given in Table i preened in Figure 5 and he graphial illuraion o he aiial omparion o he variou ode wih he propoed mehod a given in Table i preened in Figure 6.

8 Average Cra widh raio, COV HOG K&M CLARK Variou e reul Average Cra Widh Raio Coeiien o variaion Figure 5. Graphial repreenaion o he average ra widh raio and Coeiien o variaion wih reerene o e reul o variou inveigaor. Average Cra widh raio & COV BS EURO ACI CHINESE BILINEAR Variou Mehod Average ra widh raio Coeiien o variaion REFERENCES Briner R. and Dahl H. 989, Fiiiou ra model o onree raure Magazine o Conree Reearh, 4, No. 47, Clar, A. P. 956, Craing in reinored onree lexural member, ACI Journal, proeeding V. 5, No. 8, Dugdale, D. S. 960 Yielding o eel hee onaining li. J. Meh. Phy. Solid, 8, Gergely, P., and Luz, L. A. 968, Maximum ra widh in reinored onree lexural member, aue, mehanim and onrol o raing in onree, SP 0, Amerian Conree Iniue, Deroi, pp Gerle W.H, Parha P.D, Praad N.N.V, Rahulumar P and Ming Xie 99, Cra growh in lexural member A raure mehani approah ACI Journal, Vol. 89, No. 6, pp Gdouo 003, Fraure mehani- An inroduion. Hognead, E. Jan 96, High rengh bar a onree reinoremen Par : Conrol o lexural raing, Journal, PCA Reearh and Developmen Laboraorie, V. 4, No., pp Kaar, P. H., and Mao, A. H. Jan 963, High rengh bar a onree reinoremen Par 4: Conrol o raing, Journal, PCA Reearh and Developmen Laboraorie, V. 5, No., pp Edward G. Nawy O 968 Cra onrol in reinored onree ruure, ACI Journal, pp Figure 6 Comparion o Average ra widh raio and Coeiien o variaion o propoed expreion along wih variou ode o praie uing he e reul o Hognead (96). 5 CONCLUSION In he preen udy an expreion i developed o predi ra widh in R/C beam, aing advanage o he oheive-ra model. Thi expreion i a union o he brilene o onree (a union o he enile rengh o onree, beam deph, elai modulu o onree and he raure energy), reinoremen raio, ra lengh, bar diameer, re in eel and Young modulu o eel. To ae he validiy o he expreion, i wa ompared wih oher e daa on ra widh and ra paing and alo wih variou inernaional ode o praie. The reul how ha he propoed approah obained rom he model uing he bi-linear oening mae i poible o evaluae more auraely he ra widh, ompared o oher ormulaion. Furhermore, he propoed approah ha a raional and mehanially-ound bai, ine i i rooed in onree raure mehani.