CRACKING ANALYSIS OF REINFORCED CONCRETE BEAM ACCORDING TO PROPOSED METHODOLOGY OF PARAMETER SELECTION OF HIGH STRENGTH CONCRETE

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1 3: ( Pior Smrzewki Lblin Univeriy o Tenology ly o Civil Engineering nd Ariere l. Ndbyrzyk Lblin Polnd Correponding or. E-mil: p.mrzewki@pollb.pl Reeived (Orzymno CRACKING ANALYSIS O REINORCED CONCRETE BEAM ACCORDING TO PROPOSED METHODOLOGY O PARAMETER SELECTION O HIGH STRENGTH CONCRETE Te rile preen e nlyi o ilre or reinored onree bem wi low level o reinoremen mde o ig reng onree in e proe o i deormion in omprion o experimenl rel. On e bi o i lime nixil ompreive reng e meodology or deining e prmeer o e oniive model or ig reng onree w developed. Te omprion o e obined rel indie e orrene o e mpion nd oniive model o e ig reng onree nd reinoremen eel nd lo e eeivene o Criield r leng meod nd e Newon- -Rpon ierive olion wi dpive deen. Te nmeril rel o e mered rk pern proved o be qliively greeble regrding e loliion direion nd onenrion wi e experimenl rel. Keyword: inie elemen meod reinored onree bem ig reng onree ANALIZA STANU ZARYSOWANIA BELKI śelbetowej WEDŁUG PROPONOWANEJ METODYKI DOBORU PARAMETRÓW BETONU WYSOKIEJ WYTRZYMAŁOŚCI W rykle przedwiono nlizę n zryowni belki Ŝelbeowej o nikim opni zbrojeni wykonnej z beon wyokiej wyrzymłośi w proeie yznego odkzłni w porównni z wynikmi ekperymenlnymi. Zproponowno meodykę lni prmerów model konyywnego beon wyokiej wyrzymłośi n podwie jego wyrzymłośi n śiknie. Porównni zykny wyników wkzją n poprwność przyjęy złoŝeń odnośnie do modeli beon wyokiej wyrzymłośi i li zbrojeniowej kŝe n kezność meody dłgośi łk Criield i ieryjnego rozwiązni Newon-Rpon z dooownym pdkiem. Wyniki nmeryzne wzorów ry rozmyy wykzły jkośiową zgodność o do loklizji kiernk i konenrji z wynikmi doświdzlnymi. Słow klzowe: meod elemenów końzony Ŝelbe belk beon wyokiej wyrzymłośi INTRODUCTION Hig reng onree re new onrion meril re inreingly ommon in e onrion indry. Te meril i reried by i ig reng low borbbiliy low wer permebiliy nd ig reeze reine wi rel in ig drbiliy. Tey re ed in ig bilding drilling plorm bridge ndergrond nd ydro-enil onrion. I ommon e i indered by lk o normive reglion nd reommendion or deign llion. Admiedly n inreingly iger nmber o implemenion o new meril in e world onrion indry i bed on engineer experiene nd previo experimen o lrger exen rer n on eoreil nlye. Ti rel minly rom e ig omplexiy o ee ompoie nd e properie diingi em rom onvenionl onree. Preie imlion o ee ompoie bevior nder i lod reqire e oniderion o pyil non-lineriie o e onrion omponen meril: ig reng onree nd reinoring eel prilrly meril oening (non-i meril bevior inerion on e on re beween e onree nd eel nd onidering e lol onenrion o lrge deormion inlening onree rking nd ring. In order o olve omplied nmeril k i i neery o onr pil model wi ig deniy meing o onider pil enile ree nd deormion nd o e n pproprie meod o olving imlneo eqion. METHODOLOGY O SELECTING CONCRETE MODEL PARAMETERS Te rile preen e nlye o bevior o model reinored ig reng onree bem in e

2 300 8 P. Smrzewki proe o i deormion. Te inie Elemen Meod llion were perormed ing e ANSYS progrm. Te nmeri model o pil reinored ig reng onree bem e e dimenion o imply ppored renglr BP- bem nlyed nd deribed by Kmińk in er work [] - igre. eel ppor 80 x 50 x 0 eel ple 80 x 50 x 0 BP- irrp φ6 φ0 9 x 00 7 x 00 9 x ig.. Dimenion nd ro-eion o BP- bem wi reinoremen nd loding rrngemen Ry.. Wymiry i przekrój poprzezny belki BP- wrz z kłdem zbrojeni i emem obiąŝeni Conidering e longidinl ymmery o e elemen e bje o modelling w ½ o e bem 700 mm long 50 mm wide nd 300 mm ig. igre preen ro-eion o e BP- bem ino inie elemen wi obervion poin o vriion o deleion nd rin. onree elemen 0 x 5 x 5 eel ppor elemen 0 x 5 x 0 BP- eel ple elemen 0 x 5 x 0 φ0 φ 6 φ0 ε ε d e en nil obining e onvergene o e nmeril olion. In Newon-Rpon eqion ( e ngen ine mrix i deribed m o wo mrie: ( ξ K = ξ K + K T S T i (3 S were: K - en ine mrix T K - ngen ine mrix ξ - dpive deen prmeer. Te meod i bed on eing dpive deen prmeer ξ dring e eqilibrim ierion. Te en ine mrix i genered wile olving liner nd non-liner k onerning pliiy re ine o e onrion wi lrge rin nd ring onree wi oniderion or e po-rking re relxion. In e meod o Criield r-leng preened in igre 3 eqion ( depend on lod or λ : { } λ{ } { } K = T nr i i i (4 In i meod ngeble lod or λ i lled in blne eqion in e proedre o inie elemen nd vle rom e rnge. Te eqion e inremenl orm bep n nd ierion i: { } λ{ } ( λ0 λ { } { } K = + T nr i i i i (5 were λ - inremenl lod or. ig.. EM direizion o BP- bem wi obervion poin o vriion o deleion nd rin Ry.. Podził belki BP- n elemeny końzone z oznzonymi pnkmi oberwji zmin przemiezzeni i odkzłeni lod λ λ II λ I ierion ( λ ierion ( λ ierion 3 ( λ 3 3 Te rel o e nmeril llion mde by e Newon-Rpon meod wi dpive deen nd e deiniion o Criield r leng were ompred o e experimenl d []. Te Newon-Rpon meod i n ierive proe o olving nonliner eqion in e orm o: λ λ λ 3 λ 0 eqilibrim p l K T i nr l onverged poin ( λ nr new onverged poin onrin re were: { } { } { } K = T nr i i i ( T K i { } { } { } i i i + = + ( - - ngen ine mrix i - index orre- nr ponding o e rren eqilibrim ierion { i } veor o reoring lod repreening e elemen inernl lod. Te dpive deen meod preened in e work by Egger e l. [] i bed on e nge o olion p pproximing e limi poin nd revering long 0 3 ig. 3. Criield r-leng meod [3] Ry. 3. Meod dłgośi łk wedłg Criield [3] diplemen Mer [4] proved i old be eoreilly poible o expre e rieri o onree ring in ll poible ombinion o ree wi one prmeer - e nixil ompreive reng o onree. Generlly wi e vle o onree reng nder ompreion i i poible o deine omplee e o meril Compoie Teory nd Prie 3: (03 All rig reerved

3 Crking nlyi o reinored onree bem ording o propoed meodology o prmeer eleion o ig reng onree 9 d neery o mke llion in e non-liner nmeril model. In e ree preened nmeril olion o e BP- bem ll e prmeer o ig-reng onree re lled on e bi o one prmeer - e experimenlly deined nixil ompreive reng. All e prmeer o e reinoring eel were doped on e bi o e vle dped o onrion ey mere nd qoed in e norm or onree onrion deign. In e ir verion o e nmeril olion e model i deribed by ree prmeer ilre re inerpreed ree-xil re o ig reng onree in ompreion nd enion in ordne wi e Willim-Wrnke eory [5 6] (ig. 4 nd by e or own propol o e re evolion in e nion o i deormion (ig. 5 wi w elbored on e bi o nmeril nlye nd experimen on reinored onree bem. igre 4 illre e ree-xil ilre re divided ino e ydroi omponen (deribing volme nge nd e deviori omponen (deribing pe nge. Te longidinl re inlded in e xi o roion = = 3 onie e ydroi ro-eion (ig. 4. Te deviori ro- -eion lie on e norml o e xi o roion. / Te ngle o liner ln eig o e ydroi one re deribed by φ nd φ. Te prmeer o e ilre re z nd r depend on nixil ompreive reng nixil enion reng nd bixil ompreive reng b. Te memil model o e ilre re o onree i reried by rig llion o e model prmeer on e bi o ndrd reng e onvexe (wio bend nd oniniy o e onree ilre re. Te eene o e lw o evolion o e re in e nion o i deormion i o onider e elipli rdening pe nd e meril oening pe in e e o nixil ompreion well e experimenlly onirmed more exenive limi deormion in e onrion member in ombinion wi e deormion o e reorded onrol elemen (ig α /3 E ( α = or ρ<05% α = or ρ>5% φ ε0 ε ε ε = 05 b φ z r r / Θ r b b (b o Hydroi eion ( Θ = 0 ig. 4. Model o ree-xil onree re Ry. 4. Model rójoiowej powierzni beon / 3 / Deviori eion ( = 05 Τ R Te rin devior i deribed by polr oordine r nd θ were r i rdil veor loing e ilre re or ny ngle θ rom e rnge 0 θ 60 o. Angle θ lled Lode ngle i e ngle beween e projeion o e min rin xi ono e devior re wi e direion o iiio devior rin veor on i plne. Te ilre re i deined by e ollowing eqion: τ + = (6 z r ( θ were: τ - verge vle o norml nd ngen ree z - verex o ilre re ompreive reng. - nixil E =E ε k 0Τ ε k ig. 5. Propol o re-rin relionip or onree: in nixil ompreion b in nixil enion Ry. 5. Propozyj zleŝnośi npręŝenie-odkzłenie dl beon: w nie jednooiowego śikni b w nie jednooiowego roziągni In mny model bed on eing mple e deending rgmen o e rve i more inlined o iger reng onree wi prove ig reng onree i more rgile. Te relion i no lwy releed in e onree bevior in onrion elemen mde o reinored onree. Moreover e rel o lborory e owed e er ig reng onree i reried by low pli rin re njiied. Te oberved dvnge re ε Compoie Teory nd Prie 3: (03 All rig reerved

4 0 P. Smrzewki proible or onrion ey. Applying oo low rin vle or e ompreion o ig reng onree in e model in ordne wi e.g. Model Code 90 [7] rel in igniin deree in e ilre rve o e onrion elemen. Te onrion properie o e BP- bem re reried by e ollowing prmeer o oniing model: Hig reng onree - nixil ompreive reng experimenlly deined by Kmińk [] = 8. MP - modl o eliiy lled ording o ACI 363 norm E = E = 3687 MP - nixil enion reng lled ording o ACI 363 norm = 0.54 = 4.87 MP - Poion rio ν = onree deniy ρ = 600 kg/m 3 - ompreive rin in onree pek re ε = 6 - lime ompreive rin in onree ε = - er rner oeiien or open rk β = 0.5 eimed on bi o or nmeril nlye - er rner oeiien or loed rk β = 0.99 Reinored eel - modl o eliiy or φ 6 rod mde o A-II eel φ 0 rod mde o A-III eel E = 00 GP - yield re or φ 0 rod mde o A-III eel y = = 40 MP or φ 6 rod mde o A-II eel y = 355 MP - Poion rio ν = eel deniy ρ = 7800 kg/m 3 Seel ppor nd eel loding ple - modl o eliiy E = 0 GP - Poion rio ν = 03 - eel deniy ρ = 7800 kg/m 3. In e eond verion o e nmeril olion e onree model i deribed by e ive-prmeer ilre re o ig reng onree wi i ompreed nd ened in ordne wi e Willim- Wrnke eory (ig. 6 nd or own propol o e re evolion in e nion o i deormion (ig. 5. Willim nd Wrnke [5] preened more deiled pproximion o e onree ilre re wi ive-prmeer model. In order o deribe e prboli pe o e ilre re meridin e ree-prmeer model w pplemened wi wo ddiionl prmeer. Te rierion o onree rre in e omplex re ondiion i deribed by e eqion: S 0 (7 were: - e nion o e prinipl re e xp yp zp in e direion o e Crein oordine yem x y z S - ilre re dependn on e prinipl ree 3 3 min ( xp yp zp = mx ( xp yp zp were = nd 3 nd ive prmeer: - lime nixil ompreive reng (ing r - lime nixil enile reng (ing ilre b - lime bixil ompreive reng (ing r - lime ompreive reng or e o bixil ompreion perimpoed on ydroi re e (ing r nd - lime ompreive reng or e o nixil ompreion perimpoed on ydroi re e. b ig. 6. Conrion o ive-prmeer ilre re: in prinipl rin pe ( nd in ydroi eion (b Ry. 6. Konrkj pięioprmerowej powierzni grniznej: w obzrze npręŝeń główny ( i w przekroj ydroyznym (b Te onree ilre re i ed rierion o derion ording o e ollowing inerpreion. Te meril i deroyed i ineqliy (7 i lilled. Te e o ilre n be diingied e e o rking i ny prinipl re i ening or in e e o ring i ll e prinipl ree re ompreing. Te ilre re i deined by ive reng prmeer b nd e e o ydroi Compoie Teory nd Prie 3: (03 All rig reerved

5 Crking nlyi o reinored onree bem ording o propoed meodology o prmeer eleion o ig reng onree re. Te deripion o onree ilre i deined in or domin o ree: (ompreion - ompreion - ompreion 0 (enile - ompreion - ompreion (enile - enile - ompreion (enile - enile - enile. In e rnge o rin independen nion 3 4 nd S S S3 S4 deribe e nion e o ree nd ilre re S. Te preened model illre e min ere o onree ilre nd deermine e onil re wi e rviliner ln eig nd e be onied rom ree ellipi egmen. Te vle o e prmeer reriing i model re ey o deermine wi ndrd reng e. I onin ll ree re onn eql o verge re nd ngle θ. Moreover i enre moone o e ilre re in wide rnge o prmeer nd deribe or oer model o ilre re re ed in e problem o onree derion in previoly deermined onn prmeer. Apr rom e bove preened prmeer o oniive model e properie o e onrion meril o e BP- bem re deined by ddiionl prmeer deribing e ilre re o ig reng onree lled wi e eqion preened in e work by Willim nd Wrnke [5]: Addiionl prmeer or ig reng onree - lime bixil ompreive reng =. b = 97.4 MP - lime ompreive reng or e o bixil ompreion perimpoed on ydroi re e =.45 = 7.7 MP - lime ompreive reng or e o nixil ompreion perimpoed on ydroi re e =.75 = 40. MP - mbien ydroi re e deribing verge diribion o norml ree =.73 = = 40.5 MP - mliplier or mon o enile re relxion T =. Te meril model in e ird verion o e nmeril olion re idenil o e bove preened ir propol o e nmeril olion. In e ir wo vrin o e olion Criield r-leng meod w ed. Ye in e ird propoiion e Newon- -Rpon meod wi dpive deen w ed. All e nmeril llion were mde or n eli nd perely pli model o reinored eel (ig. 7 nd e eli nd brile model o onree wi oening in enion (ig. 5b. b +ε ε enion ε y + ig. 7. Sre-rin rve or eel reinoremen ed or nlye o reinored onree bem Ry. 7. ZleŜność npręŝenie-odkzłenie dl li zbrojeniowej zoown w nliz modelowej belki Ŝelbeowej ANALYSIS O AILURE CONDITION E y ompreion igre 8 preen e mp o rk pern or vrio vle o lod obined or ree propol o nmeril olion or model BP- bem. In e ir nd ird nmeril olion e border re o ompreed nd ened ig reng onree i deribed by ree meril prmeer wile in e eond olion e border re i reried by ive prmeer. In bo olion in Criield meod e ir rk re ormed in e eion o onn momen or dering lod r ereore e me momen in e experimenl vle o e dering lod r = 5 kn. Te dierene beween e lled nd experimenl vle o e dering lod i.. 0.7%. Ye in e Newon-Rpon meod wi dpive deen e ir rk i mde in e eion o ypil bending or e lod vle r = 5. kn wi mke.. % dierene in omprion o e nmeril meod o r-leng. In e ir wo nmeril olion e lexrl rk propge in e re o onn momen p o e lod o 0 kn. rer inree e propgion o e rk zone orizonlly owrd e ppor. In e ird olion e proee o rking re qiker p o e momen o eel pliiion. In ll o e preened olion igniin inree in rk n be oberved or e lod o 5 kn er reinoring eel pliiion. Te rk ed by ompreion in e lod ppliion zone nd ring in e bending zone re lerly noieble. In bo olion wi Criield meod idenil imge o mered rk were obined p o e lod o 5 kn. In e inl ge o loding rom e lod o 7 kn p o e derion momen enned devion w oberved in e bending re w reried by new rk in e re perpendilr o e en non-rked re ompnied by relively onn nmber o rk in e pporing re. y E ε y ε ε Compoie Teory nd Prie 3: (03 All rig reerved

6 P. Smrzewki Tree-prmeer ilre re (Criield r leng meod [A-L] = 4.9 kn = 5 kn = 3.3 kn = 7 kn Tree-prmeer nd ive-prmeer ilre re (Criield r leng meod [A-L] = kn = 3.3 kn Tree-prmeer ilre re (Newon-Rpon meod wi dpive deen [N-R d ive-prmeer ilre re (Criield r leng meod [A-L] = 7 kn = 9.7 kn Tree-prmeer ilre re (Newon-Rpon meod wi dpive deen = 5. kn = 5 kn = 3.3 kn = 7 kn = kn = 3.8 kn ig. 8. Mp o mered rk or dieren level o lod vle Ry. 8. Mpy ry rozmyy dl róŝny poziomów obiąŝeni In e nmeril olion e limi vle o lod were reorded or e ollowing vle = 3.3 kn = 9.7 kn nd = 3.8 kn nd were inerpreed e ondiion i obinble or vle o lod or wi e inree in diplemen mll vle o lod inree mke e nmeril olion no on- [] bem verge. Alog in Kmink experimen derion w no oberved e vle o mximm lod lled rom e ondiion o deormion = = 30 kn pproxime e vle o derion lod lled in e nmeril olion. In igre 9 e imge o rel rk in ll BP- bem i preened gin e nmeril imge o mered rk or e le l o e bem wi e me vle o lod = 3 kn. ig. 9. Experimenl [] nd nmeril imge o rking BP- bem or lod = 3 kn Ry. 9. Ekperymenlny [] i nmeryzne obrzy zryowni belki BP- dl iły = 3 kn SUMMARY Te obined nmeril rel ppored by experimenl rel prove e e rgiliy o ig reng onree in lexrl bem i ignii- beween e onree nly limied by e good relion nd e reinoremen. Te ler orrelion o e experimenl nd nmeril rel depend on ppro- nd nonliner meril prie eleion o e liner properie. All e obined nmeril rel re in qliive greemen wi e experimenl rel in erm o eir loion direion nd onenrion. In e e o e Newon-Rpon meod wi dpive deen ligly lrger zone o mered rk w oberved. Te nmeril imge o mered rk obined by Criield r leng meod re be o rele e rel ilre re in erm o qniy. However e imby e Newon-Rpon ge o rk obined meod re reried by beer loion nd oneno rk in e olion rion. Even beer imge obined by Criield meod n be obined by limiing e minimm ep o lod inree wi re- llion. qire longer ime or nmeril REERENCES [] Kmińk M.E. Doświdzlne bdni Ŝelbeowy ele- wyokiej wyrzymłośi. menów pręowy z beon KILiW PAN Łódź 999. [] Egger G.M. Dwon P.R.. Mr K.K. An dpive deen meod or nonliner viopliiy Inernionl Jornl or Nmeril Meod in Engineering [3] Criield M.A. An r-leng meod inlding line er- Jornl or Nmeril e nd elerion Inernionl Meod in Engineering Compoie Teory nd Prie 3: (03 All rig reerved

7 Crking nlyi o reinored onree bem ording o propoed meodology o prmeer eleion o ig reng onree 3 [4] Mer B. W do we need o know bo e repone o plin onree nd i mrix o ombined loding? Proeeding Conerene on e Bevior o Srrl Conree Sbjeed o Combined Loding We Virgini Univeriy [5] Willm K.J. Wrnke E.P. Coniive Model or e Trixil Bevior o Conree. Proeeding Inernionl Aoiion or Bridge nd Srrl Engineering ISMES Bergmo Ily [6] Smrzewki P. Solrki A. Modelownie zowni niepręŝyej belki Ŝelbeowej Bileyn WAT 007 LVI [7] Comié Ero-Inernionl d Beon: Hig Perormne Conree. Reommended o e Model Code 90. Reer Need Bllein d Inormion Compoie Teory nd Prie 3: (03 All rig reerved