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Iii flexural analysis of reinforced concrete
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Iii flexural analysis of reinforced concrete Iii flexural analysis of reinforced concrete Document Transcript

  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa III. viPaKFñwmebtugGarem:rgkarBt;begáag Flexural Analysis of Reinforced Concrete Beam 1> karsnμt; Assumption ebtugGarem:CasMPar³minEmnsac;mYy BIeRBaHvaekIteLIgedaysMPar³BIrRbePTKW ebtug nigEdk. dUc enHeRKOgbgÁúMebtugGarem:EdlkMNt;edayersIusþg;cugeRkay RtUvkMNt;tamkarsnμt;xageRkam³ - bMErbMrYlrageFobrbs;ebtug nigEdkRtUvEtmantMéldUcKña mann½yfaPaBs¥itrvagebtug nigEdkman tMélRKb;RKan;. - bMErbMrYlrageFobrbs;ebtugRtUvEt smamaRteTAnwgcMgayBIGkS½NWt - m:UDuleGLasÞicrbs;Edk RtUvEtyk E = 2 ×10 MPa . kugRtaMgrbs;EdkkñúgtMbn;eGLasÞicRtUvEt s 5 mantMélesμIplKuNrvag bMErbMrYlrageFobCamYynwgm:UDuleGLasÞic. - muxkat;Rtg;enAEtRtg;eRkayeBlrgkarBt; - ersIusþg;Tajrbs;ebtugRtUv)anecal BIeRBaH ersIusþg;TajebtugmantMéltUcCagersIusþg;sgát;dl; eTA 10 dg ehIysñameRbHrbs;ebtugRtUvsnμt;faKμanT§iBl nigmü:ageTotmuneBleRbH muxkat;ebtugTaMgmUl manRbsiT§PaBkñúgkarTb;nwgm:Um:g;xageRkA. - sac;lUteFobGtibrmarbs;ebtugkMritRtwm 0.003 - rUbragénkarBRgaykugRtaMgsgát;rbs;ebtug snμt;manragctuekaNEkg 2> RbePTénkar)ak;edaykarBt; nigEdnkMNt;sac;lUteFob k> kar)ak;edaykarBt; eRKOgbgÁúMrgkarBt; Gac)ak;edaybIkrNIGaRs½yeTAnwgPaKryEdkEdl)andak;enAkñúgmuxkat;ebtug³ - EdkGaceTAdl;cMnucyarmunebtugeFVIkardl;ersIusþg;Gtibrma. kñúgkrNIenH kar)ak;bNþalmkBI sac;lUteFobrbs;EdkmantMélFMCagb¤esμI 0.005 . muxkat;manbrimaNEdktic ehIyRtUv)aneKeGayeQμaHfa muxkat;rgkarTaj tension-controlled section . - EdkGaceTAdl;cMnucyarGMLúgeBlEdlebtugeFVIkardl;ersIusþg;GtibrmaEdr. muxkat;RtUv)aneK eGayeQμaHfa muxkat; balanced section . - ebtugGacEbkmuneBlEdlEdkeFVIkardl;cMnucyar bNþalmkBIPaKryEdkeRcInenAkñúgmuxkat;. kñúg krNIenHebtug)aneFVIkardl;ersIusþg;Gtibrma ehIymansac;lUteFobGtibrma 0.003 Edr b:uEnþkugRtaMgrbs; Flexural Analysis of Reinforced Concrete Beam 18
  • T.Chhay NPIC EdkmantMélticCagersIusþg;KNna Edl f < f . sac;lUteFobrbs;EdkmantMéltUcCagb¤esμI 0.002 . s y muxkat; enHRtUv)aneKeGayeQμaHfa muxkat;rgkarsgát; compression-controlled section. eK)ansnμt;faebtugEbkedaysarkMlaMgsgát; enAeBlEdlsac;lUteFobrbs;ebtugmantMél 0.003 Etkñúgkar BiesaFn_tMélenHERbRbYlBI 0.0025 → 0.004 . kñúgkarKNnaFñwm eKeRCIserIsykmuxkat;rgkarTaj edayeGayEdkeFVIkardl;ersIusþg;KNna muneb tugEbk. sñameRbHrbs;ebtugrIkFMeLIg² EdlCasBaØaRbkasGasnñmuneBlebtugEbk ehIyrcnasm<n§½)ak; Ebk. kñúgkarKNnaFñwm edayeRCIserIsykmuxkat;rgkarsgát; nigbalanced section ebtugEbkPøam² rcnasm<n§½ )ak;EbkmYyrMBicedayKμankarRbkasGasnñ. kareRCIserIsmuxkat;EbbenHRtUv)aneCosvag. x> EdnkMNt;sac;lUteFobsMrab;muxkarrgkarTaj nigrgkarsgát; karKNnapþl;eGaycMeBaHkarKNnaebtugGarem:sMrab;muxkat;rgkarTaj b¤sgát;. muxkat;TaMgBIr RtUv)ankMNt;edaysac;lUteFobsuT§ net tension strain (NTS). elIsBIenHlkçxNÐBIreTot)anekItKW lkçxNÐbMErbMrYlrageFobtulüPaB balanced strain condition niglkçxNÐkñúgtMbn; transition region condition. lkçxNÐTaMgbYnenHRtUv)ankMNt;dUcxageRkam³ - muxkat;rgkarsgát; Camuxkat;Edlsac;lUteFobsuT§ (NTS) sMrab;EdkrgkarTajEpñkxageRkAbMput mantMél tUcCagbMErbMrYlrageFobrgkarsgát; enAeBlEdlbMErbMrYlrageFobrbs;ebtugrgkarsgát;mantMél esμI 0.003 . krNIenHekIteLIgCaTUeTAcMeBaH ssrEdlrgbnÞúktamGkS½ nigm:Um:g;. - muxkat;rgkarTaj Camuxkat;Edlsac;lUteFob (NTS) sMrab;EdkrgkarTajEpñkxageRkAbMputman tMélFMCag b¤esμI 0.005 kñúgkrNIEdlebtugmanbMErbMrYlrageFobdl;EdnkMNt; 0.003 . - muxkat;Edlsac;lUteFoblUteFob (NTS) sMrab;EdkrgkarTajEpñkxageRkAbMputmantMélsßitenA cenøaHmuxkat;rgkarsgát; nigmuxkat;rgkarTaj KWenAcenøaH 0.002 → 0.005 Camuxkat; transition region - lkçxNÐbMErbMrYlrageFobtulüPaB ekItmanenAkñúgmuxkat; enAeBlEdlbMErbMrYlrageFobEdktMbn; TajmantMélesμI ε = E kñúgxN³Edlebtugrgkarsgát;manbMErbMrYlrageFobmantMélesμI 0.003 . f s y s Section condition Concrete strain Steel strain Note ( f y = 400MPa) Compression-controlled 0.003 ε t ≤ f y Es ε t ≤ 0.002 Tension-controlled 0.003 ε t ≥ 0.005 ε t ≥ 0.005 Transition region 0.003 f y E s ≤ ε t ≤ 0.005 0.002 ≤ ε t ≤ 0.005 Balanced strain 0.003 ε s = f y Es ε s = 0.002 Transition region 0.003 0.004 ≤ ε t < 0.005 0.004 ≤ ε t < 0.005 viPaKFñwmebtugGarem:rgkarBt;begáag 19
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa > > Flexural Analysis of Reinforced Concrete Beam 20 View slide
  • T.Chhay NPIC 3> emKuNbnÞúk bnÞúkEdlmanGMeBIelIeRKOgbgÁúMRtUv)anKuNCamYynwgemKuNbnÞúk edIm,IkarBarkar)ak;Pøam² nigpþl; nUvkarKNnamYyEdlmanlkçN³esdækic©. emKuNbnÞúkGaRs½ynwgRbePTbnÞúk nigkarbnSMbnÞúk. emKuNbnÞúk sMrab;bnÞúkGefr KW 1.6 ÉemKuNbnÞúksMrab;bnÞúkefr KW 1.2 . dUcenHkarbnSMbnÞúksMrab;bnÞúkGefr nigbnÞúkefrKW U = 1 .2 D + 1 .6 L Edl U - bnÞúkKNnacugeRkay L - bnÞúkGefr D - bnÞúkefr 4> emKuNkat;bnßyersIusþg; emKuNkat;bnßyersIusþg; φ mantMéltUcCag 1 . emKuNkat;bnßyersIusþg;GaRs½ynwgRbePTén eRKOgbgÁúM³ - sMrab;muxkat;rgkarTaj φ = 0.90 - sMrab;muxkat;rgkarsgát; k> CamYyEdkkgvNÐ φ = 0.70 x> CamYyEdkkgdac;² φ = 0.65 - sMrab;ebtugsuT§ φ = 0.55 - sMrab;kMlaMgkat; nigkMlaMgrmYl φ = 0.75 - sMrab;RTnab;enAelIebtug φ = 0.65 - sMrab;KMrU strut and tie φ = 0.75 5> karEbgEckkugRtaMgsgát;smmUl kugRtaMgEbgEckkñúgebtugrgkarsgát;enAxN³eBl)ak;RtUv)ansnμt;famanragctuekaNEkg ctuekaNBñay ExSekag)a:ra:bUl b¤ragNamYyepSgeTotGaRs½yedaykaryl;RBmKñaenAeBleFVIBiesaFn_. enAeBlEdlFñwmerobnwg)ak; srésEdk)aneFVIkardl;cMnucyarmun RbsinebImuxkat;enaHmanbrimaN Edktic under-reinforced section ehIykñúgkrNIenHsrésEdkeFVIkardl;kgRtaMgKNna. RbsinebImuxkat; u manEdkeRcIn ebtugEbkmun ehIybMErbMrYlrageFobRtUv)ansnμt;faesμI 0.003 . kMlaMgsgát; C ekItmanenAkñúgtMbn;sgát; ehIykMlaMgTaj T ekItmanenAtMbn;TajEdlsßitenAelInIv:U Edk. eKsÁal;TItaMgénkMlaMg T BIeRBaHvamanGMeBIRtYtsIuKñanwgGkS½TIRbCMuTMgn;rbs;Edk. ÉTItaMgrbs;kMlaMg viPaKFñwmebtugGarem:rgkarBt;begáag 21 View slide
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa C eKGacsÁal;)an luHNaEteKsÁal;maDéntMbn;sgát; ehIyeBlenaHeKGackMNt;)annUvTItaMgTIRbCMuTMgn; )an. RbsinebIeKsÁal;TItaMgrbs;kMlaMgTaMgBIr enaHeKGackMNt;nUvRbEvgédXñas; EdlCacMgayBIkMlaMgTaj mkkMlaMgsgát;. RbsinebIebtugEbk enAbMErbMrYlrag ε ' = 0.003 ehIyRbsinebIEdkeFVIkardl;cMnucyar f = f enaH c s y muxkat;Camuxkat; balanced section. kMlaMgsgát; C RtUv)ansMEdgedaymaDénbøúkkugRtaMg EdlmanragminÉksNæan ekItmanelIépÞctuekaNqñÚt bc . maDénkugRtaMgsgát;snμt;esμI C = bc(α f ' ) 1 c Edl α f ' CakugRtaMgmFüménbøúkkugRtaMgminÉksNæan 1 c TItaMgrbs;kMlaMgsgát; KWmancMgay z BIsrésEpñkxagelIbMputEdlGaccat;TukCaEpñkéncMgay c ¬cM gayBIsrésEpñkxagelImkGkS½NWt¦. z = α 2c α1 = 0.72 sMrab;ebtugEdlmanersIusþg; f 'c ≤ 28MPa . α RtUv)ankat;bnßyeday 0.04 ral; 7 MPa sMrab;ebtugEdlmanersIusþg; f ' c > 28MPa . 1 α1 = 0.425 sMrab;ebtugEdlmanersIusþg; f ' ≤ 27.6MPa . c α RtUv)ankat;bnßyeday 0.025 ral; 7 MPa sMrab;ebtugEdlmanersIusþg; f ' c > 28MPa . 1 edIm,IsMrYldl;karKNnakMlaMgkñúgénmuxkat; ACI code )anyknUvkugRtaMgEbgEckkñúgmuxkat;rag ctuekaNEkg EdlmantMél 0.85 f ' BRgayesμIelItMbn;sgát;smmUl EdlxNнedaybnÞat;RsbnwgGkS½NWt c EdlmanRbEvg a = β c . 1 β = 0.85 sMrab;ebtugEdlmanersIusþg; f ' ≤ 28MPa . 1 c f ' −28 β = 0.85 − 0.05( 1 c ) sMrab;ebtugEdlmanersIusþg; 28MPa < f ' ≤ 56MPa . c 7 β = 0.65 sMrab;ebtugEdlmanersIusþg; f ' > 56MPa . 1 c Flexural Analysis of Reinforced Concrete Beam 22
  • T.Chhay NPIC sMrab;muxkat;ragctuekaNEkg RkLaépÞtMbn;sgát;mantMélesμI ba ehIytMélkugRtaMgBRgayesIμKW 0.85 f ' Edlpþl;nUvmaDkugRtaMgsrubesμInwg 0.85 f ' ab ehIyRtUvKñanwgkMlaMgsgát; C . sMrab;muxkat;epSg c c BIragctuekaNEkg kMlaMgsrubesμInwgplKuNRkLaépÞtMbn;sgát;CamYynwg 0.85 f ' . c 6> srésEdkrgkMlaMgTajénmuxkat;ctuekaNEkgrgkarBt; PaKryEdkenAkñúgmuxkat;ebtugkñúglkçxNÐ balanced RtUv)aneKeGayeQμaHfa balanced steel ratio ρ EdlCapleFobrvagmuxkat;Edk A nigmuxkat;RbsiT§PaB bd b s As ρb = bd Edl - TTwgmuxkat;eRKOgbgÁúMtMbn;sgát; b d - cMgayBIsésrEpñkxageRkAbMputmkTIRbCMuTMgn;EdkrgkMlaMgTaj ¬kMBs;RbsiT§PaB¦ smIkarlMnwgBIr EdlCaeKalkarN_kñúgkarviPaK nigKNnaeRKOgbgÁúMehIymantMélRKb;muxkat; nigRKb; RbePTbnÞúkKW³ - kMlaMgsgát;RtUvmantMélesμIkMlaMgTaj C = T - ersIusþg;m:Um:g;Bt;xagkñúg M esμIeTAnwgplKuNrvagkMlaMgsgát; b¤kMlaMgTajCamYynwgédXñas; n M = C (d − z ) = T (d − z ) nig M = φM Edl φ emKuNkat;bnßyersIusþg; n u u viPaKFñwmebtugGarem:rgkarBt;begáag 23
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa kareRbIR)as;nUvsmIkarTaMgenHRtUv)anBnül;sMrab;muxkat;ragctuekaNEkgCamYyEdktMbn;Taj. mux kat;GacCa muxkat; balanced section muxkat;Edktic muxkat;EdkeRcIn GaRs½yedaykareRbIR)as;nUvPaKry Edk. k> muxkat; balanced section CMh‘anTI1³ BIdüaRkamsac;lUteFob eyIg)an cb 0.003 = d − cb fy Es c 0.003 ⇒ b = d fy 0.003 + Es edayCMnYs E s = 200000MPa 600 ⇒ cb = ( )d 600 + f y CMh‘anTI2³ BIsmIkarlMnwg eyIg)an C = T ⇒ 0.85 f 'c ab = As f y As f y ⇒a= 0.85 f 'c b Edl a - CaRbEvgbøúkrgkarsgát; mantMélesμInwg β c 1 b edaysarvaCamuxkat; balanced section dUcenHPaKryEdkRtUv)aneRbIKW As ρb = bd ⇒ As = ρ bbd CMnYs A eTAkñúgsmIkarxagelI s ⇒ 0.85 f 'c ab = ρ bbdf y Flexural Analysis of Reinforced Concrete Beam 24
  • T.Chhay NPIC 0.85 f 'c a 0.85 f 'c ⇒ ρb = = ( β1cb ) f yd f yd CMnYstMél c b =( 600 600 + f y )d eTAkñúgsmIkarxagelI eyIg)an f 'c 600 ρ b = 0.85β1 ( ) f y 600 + f y CMh‘anTI3³ BIsmIkarlMnwgénm:Um:g;xagkñúg eyIg)an M n = C (d − z ) = T (d − z ) sMrab;muxkat;ragctuekaNEkg cMgay z = a 2 a a ⇒ M n = C (d − ) = T (d − ) 2 2 sMrab;muxkat; balanced section b¤muxkat;EdlmanbrimaNEdktic T = As f y dUcenH M = A f (d − a ) n 2 s y m:Um:g;kñúgxagelIEdl)anKNna RtUvkat;bnßyedayemKuN φ As f y ⇒ φM n = φAs f y (d − ) 1.7 f 'c b smIkarenH sresredayCab;GBaØti ρ ρbdf y ρf y ⇒ φM n = φf y ρbd (d − ) = φf y ρbd 2 (1 − ) 1.7 f 'c b 1.7 f 'c eyIgGacsresrsmIkarxagelIenHCa φM n = Ru bd 2 Edl R = φf ρ (1 − 1.ρff ' ) u 7 y y c pleFobrvagRbEvgbøúkkugRtaMgsgát;smmUl a nig kMBs;RbsiT§PaBénmuxkat; d a ρf y = d 0.85 f 'c x> PaKryEdkGtibrma PaKryEdkGtibrma ρ EdlGaceRbIenAkñúgmuxkat;ebtugEdlmanEtEdkrgkMlaMgTaj QrelIeKal max karN_sac;lUteFobsuT§enAkñúgEdkrgkMlaMgTaj PaKryEdk balanced nigersIusþg;rbs;Edk. TMnak;TMngrvagPaKryEdkenAkúñgmuxkat; ρ nigsac;lUteFobsuT§ ε t viPaKFñwmebtugGarem:rgkarBt;begáag 25
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa fy fy 0.003 + 0.003 + ρ = Es ρ b 0.003 + ε t b¤ ε t =( ρ Es ) − 0.003 ρb sMrab; f y = 414MPa nigsnμt; f y / Es = 0.00207 - sMrab;muxkat;rgkMlaMgTaj ⇒ ε ≥ 0.005 snμt; ε = 0.005 ¬b¤ dc ≤ 0.375 ¦ t t t d - cMgayBIsésEpñkxageRkAbMput eTAGkS½EdkTajCYrTI1 t ρ 0.00507 = ρb 0.008 kñúgkrNIEdl ρ = ρ max ⇒ ρ max = 0.63375ρ b PaKryEdkenHeFVIeGayFñwmmanlkçN³yWtRKb;RKan;munnwg)ak; Casegçb³ sMrab; f = 276MPa ⇒ ρ = 0.5474ρ y max b f y = 345MPa ⇒ ρ max = 0.5905ρ b f y = 517 MPa ⇒ ρ max = 0.6983ρ b sMrab;muxkat;rgkMlaMgTaj φ = 0.9 - sMrab;muxkat;enAkñúgtMbn; transition region snμt; ε t = 0.004 ¬minRtUvtUcCag 0.004 ¦ b¤ 0.6 > d > 0.375 c ρ 0.00507 = ρb 0.007 kñúgkrNIEdl ρ = ρ max t ⇒ ρ max t = 0.724 ρ b Flexural Analysis of Reinforced Concrete Beam 26
  • T.Chhay NPIC sMrab;muxkat;enAkñúgtMbn; transition region φ < 0.9 250 ⇒ φ = 0.65 + (ε t − 0.002)( ) 3 ]TahrN_1³ sMrab;muxkat;dUcbgðajkñúgrUb k> kMNt;muxkat;Edk balanced section x> muxkat;EdkGtibrmaEdlGnuBaØatieday ACI Code sMrab;muxkat;rgkMlaMgTaj nig sMrab;muxkat;enAkñúg tMbn; transition region K> TItaMgGkS½NWt nigRbEvgbøúkkugRtaMgsgát;sMrab;muxkat;rgkMlaMgTaj smμtikmμ³ f ' = 28MPa nig f = 400MPa c y dMeNaHRsay³ k> kMNt;muxkat;Edk balanced section f 'c 600 ρ b = 0.85β1 ( ) f y 600 + f y eday f ' = 28MPa c f y = 400MPa nig β 1 = 0.85 28 600 ⇒ ρ b = 0.852 ( ) = 0.030345 400 600 + 400 muxkat;EdkEdldak;kñúgmuxkat;ebtugedIm,I)anlkçxNÐ balanced KW Asb = ρ bbd = 0.030345 × 40 × 65 = 78.897cm 2 x> muxkat;EdkGtibrmasMrab;muxkat;rgkMlaMgTaj fy 0.003 + Es ρ max = ( ) ρb 0.003 + ε t sMrab; ε t = 0.005 0.005 ⇒ ρ max = ρ b = 0.625ρb = 0.625 × 0.030345 = 0.019 0.008 viPaKFñwmebtugGarem:rgkarBt;begáag 27
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa ⇒ As max = ρ b maxbd = 0.019 × 40 × 65 = 49.4cm 2 sMrab; φ = 0.9 muxkat;EdkGtibrmasMrab;muxkat;kñúgtMbn; transition region fy 0.003 + Es ρ max = ( ) ρb 0.003 + ε t sMrab; ε t = 0.004 0.005 ⇒ ρ max = ρ b = 0.714 ρb = 0.714 × 0.030345 = 0.0217 0.007 ⇒ As max = ρ b maxbd = 0.0217 × 40 × 65 = 56.42cm 2sMrab; φ = 0.817 K> TItaMgGkS½NWt nigRbEvgbøúkkugRtaMgsgát;sMrab;muxkat;rgkMlaMgTaj As max f y 49.4 × 400 amax = = = 20.76cm 0.85 f 'c b 0.85 × 28 × 40 cMgayBIsésrEpñkxagelImkGkS½NWtKW a 20.76 c= = = 24.42cm β1 0.85 ]TahrN_2³ kMNt;ersIusþg;m:Um:g;KNna nigTItaMgGkS½NWténmuxkat;ctuekaNEkgdUcbgðajkñúgrUbxageRkam. RbsinebIeKeRbIEdk 3DB30 ersIusþg;ebtug f ' = 20MPa nig f = 400MPa c y dMeNaHRsay³ muxkat;Edk 3DB30 ⇒ A = 21.195cm s 2 PaKryEdkeRbIR)as;kñúgebtug ρ = bd = 30.× 55 = 0.0128 A 21 195 s PaKryEdk balanced kñúgebtug ρ = 0.85β ff ' ( 600 + f ) = 0.021675 600 b 1 c y y × 400 RbEvgbøúkkugRtaMgsgát; a = 0.85 ff ' b = 021.19520 × 30 = 16.62cm A s .85 × y c TItaMgGkS½NWt c = β = 16..85 = 19.55cm a 0 62 1 fy 0.003 + sac;lUtEdksuT§ εt = ( ρ Es ) − 0.003 = 0.0055 > 0.005 ρb ⇒ muxkat;rgkMlaMgTaj ⇒ φ = 0.9 ersIusþg;m:Um:g;xagkñúgKNna a 16.62 φM n = φAs f y (d − ) = 0.9 × 21.195 × 400 × (55 − ) × 10 −3 = 356.25kN .m 2 2 ]TahrN_3³ kMNt;ersIusþg;m:Um:g;KNna nigTItaMgGkS½NWténmuxkat;ctuekaNEkgdUcbgðajkñúgrUbxagelI. Flexural Analysis of Reinforced Concrete Beam 28
  • T.Chhay NPIC EteKeRbIEdk 3DB32 vij ersIusþg;ebtug f ' = 20MPa nig f = 400MPa c y dMeNaHRsay³ muxkat;Edk 3DB32 ⇒ A = 24.1152cm s 2 PaKryEdkeRbIR)as;kñúgebtug ρ = bd = 24.1152 = 0.0146 A 30 × 55 s PaKryEdk balanced kñúgebtug ρ = 0.85β ff ' ( 600 + f ) = 0.021675 600 b 1 c y y RbEvgbøúkkugRtaMgsgát; a = 0.85 ff ' b = 2485 × 20××400 = 18.91cm A 0. .1152 s y 30 c TItaMgGkS½NWt c = β = 18..85 = 22.25cm a 0 91 1 fy 0.003 + sac;lUteFobEdksuT§ εt = ( ρ Es ) − 0.003 = 0.0044 < 0.005 ρb ⇒ muxkat;enAkñúgtMbn; transition region ⇒ φ = 0.65 + (ε − 0.002)( 250 ) = 0.85 3 t ersIusþg;m:Um:g;KNna φM = φA f (d − a ) = 0.85 × 24.1152 × 400 × (55 − 16262 ) ×10 n 2 s y . −3 = 373.43kN .m sMrab;muxkat;rgkMlaMgTaj ε = 0.005 t 0.005 ⇒ ρ max = ρ b = 0.625ρb = 0.625 × 0.021675 = 0.01355 0.008 As max = ρ max bd = 0.01355 × 30 × 55 = 22.3575cm 2 < 24.1153cm 2 RbEvgbøúkkugRtaMgsgát; a = 0.85 ff ' b = 2285 × 20××400 = 17.535cm A 0. .3575 30 s y c a 17.535 ⇒ φM n = φAs f y (d − ) = 0.9 × 22.3575 × 400 × (55 − ) × 10 −3 = 372.11kN .m 2 2 eyIgeXIjfa tMélénersIusþg;mantMélesÞIresμIKña EdleKGacTTYlyk)an. K> PaKryEdkGb,brma RbsinebIm:Um:g;Gnuvtþn_mkelIFñwmmantMéltUc ehIyTMhMénmuxkat;FMCagGVIEdlRtUvkarsMrab;Tb;Tl;nwg m:Um:g; enaHkarKNnanwgbgðajeGayeXIjmuxkat;EdktUc b¤k¾Kμan. RbsinebImindak;sésrEdk Fñwmrgm:Um:g; nwgkar)ak;Pøam². ACI Code kMNt;nUvmuxkat;EdkGb,brma A s min b d nig ≥ f' 1.4 A = s min c w b d w 4f y f y sMrab;krNIFñwmragGkSr T EdlsøabrgkMlaMgTaj enaHmuxkat;EdkRtUvyktMéltUcCageKevagsmIkar xagelI nigxageRkam viPaKFñwmebtugGarem:rgkarBt;begáag 29
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa f 'c As min = bw d 2 fy Edl bw = b sMrab;muxkat;ragctuekaNEkg bw CaTTwgsøab 7> muxkat;lμm muxkat;EdlmanlkçN³lμm RbsinebIersIusþg;m:Um:g;kñúgénmuxkat;FMCag b¤esμIm:Um:g;xageRkA φM ≥ M . viFIsaRsþGacsegçbdUcxageRkam³ n u - KNnam:Um:g;xageRkAEdlGnuvtþn_mkelIeRKOgbgÁúM M u M u = 1.2M D + 1.6M L - KNna φM sMrab;muxkat;EdlsésrEdkrgkMlaMgTaj n + RtYtBinitüfa ρ < ρ < ρ min max + kMNt; a = nigRtYtBinitü ε sMrab; φ A f s y t 0.85 f ' b c kMNt; φM = φA f (d − a ) + n 2 s y - RbsinebI φM ≥ M enaHmuxkat;manlkçN³lμm n u ]TahrN_4³ eKmanFñwmTMrbgáb;mYyEdlmanRbEvg 2.5m . FñwmenHmanmuxkat;ragctuekaNEkgdUcbgðaj kñúgrUb. FñwmRTbnÞúkefr EdlrYmmanbnÞúkpÞal;xøÜn rbs;vasrub 22kN / m nigbnÞúkGefr 13kN / m . edayeRbI f ' = 28MPa nig f = 400MPa c y cUrepÞógpÞat;fa FñwmenHmansuvtßiPaBRKb;RKan;kñúg Flexural Analysis of Reinforced Concrete Beam 30
  • T.Chhay NPIC karRTbnÞúkxagelI dMeNaHRsay³ bnÞúkKNna Wu = 1.2 D + 1.6 L = 1.2 × 22 + 1.6 × 13 = 47.2kN / m m:Um:g;KNna L2 2.52 M u = Wu = 47.2 = 147.5kN .m 2 2 muxkat;Edk As = 11.3982cm 2 PaKryEdkenAkñúgmuxkat;ebtug As 11.3982 ρ= = = 0.012256 bd 20 × 46.5 PaKryEdk balance f 'c 600 ρ b = 0.85β1 ( ) = 0.030345 f y 600 + f y RbEvgbøúkkugRtaMgsgát; As f y 11.3982 × 400 a= = = 9.578cm 0.85 f 'c b 0.85 × 28 × 20 TItaMgGkS½NWt a 9.578 c= = = 11.268cm β1 0.85 sac;lUteFobEdksuT§ fy 0.003 + εt = ( ρ Es ) − 0.003 = 0.00938 > 0.005 ⇒ muxkat;rgkMlaMgTaj φ = 0.9 ρb ersIusþg;m:Um:g;xagkñúgKNna a 9.578 φM n = φAs f y (d − ) = 0.9 × 11.3982 × 400 × (46.5 − ) × 10− 3 = 171.155kN .m 2 2 muxkat;manlkçN³RKb;RKan; φM n > M u ]TahrN_5³ eKmanFñwmmuxkat;mYymanRbEvg 6m . FñwmenHmanmuxkat;dUcbgðajkñúgrUb. edayeRbI f ' = 20MPa nig f = 400MPa c y kMNt;bnÞúkGefrrayesμIGnuBaØati. FñwmenHmin viPaKFñwmebtugGarem:rgkarBt;begáag 31
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa manbnÞúkefrGVIeRkABITMgn;xøÜnvaeT. dMeNaHRsay³ TMgn;pÞal;rbs;Fñwm WD = 30 × 52.5 × 10 −4 × 24 = 3.78kN / m muxkat;Edk As = 14.71875cm 2 RbEvgbøúkkugRtaMgsgát; As f y 14.71875 × 400 a= = = 17.32cm 0.85 f 'c b 0.85 × 20 × 20 PaKryEdkeRbIR)as;enAkñúgmuxkat;ebtug As 14.71875 ρ= = = 0.009345 bd 30 × 52.5 PaKryEdk balanced kñúgebtug f 'c 600 ρ b = 0.85β1 ( ) = 0.021675 f y 600 + f y sac;lUteFobEdksuT§ fy 0.003 + εt = ( ρ Es ) − 0.003 = 0.0086 > 0.005 ⇒ muxkat;rgkMlaMgTaj φ = 0.9 ρb ersIusþg;m:Um:g;xagkñúgKNna a 17.32 φM n = φAs f y (d − ) = 0.9 ×14.71875 × 400 × (52.5 − ) × 10 −3 = 232.3kN .m 2 2 edayeGay M = φM u n mü:ageTot M = 1.2Mu D + 1.6M L 3.78 × 6 2 W 232.3 = 1.2( ) + 1.6( L × 6 2 ) = 20.412 + 7.2WL 8 8 232.3 − 20.412 WL = = 29.43kN / m 7.2 ]TahrN_6³ RtYtBinitümuxkat;dUcbgðajkñúgrUbxageRkam edIm,ITb;Tl;nwg m:Um:g;KNna 41kN.m . edayeRbI f ' = 20MPa nig f = 235MPa . c y dMeNaHRsay³ muxkat;Edk 7.5cm As = 3.3912cm 2 Flexural Analysis of Reinforced Concrete Beam 32
  • T.Chhay NPIC PaKryEdkeRbIR)as;enAkñúgmuxkat;ebtug As 3.3912 ρ= = = 0.00377 bd 20 × 45 PaKryEdkGb,brmaeRbIR)as;enAkñúgmuxkat;ebtug f 'c 1.4 ρ min = max( , ) = max(0.004756,0.00596) = 0.00596 4 fy fy ⇒ ρ < ρ min ⇒ As min = 0.00596 × 20 × 52.5 = 6.258 dUcenHeKRtUveRbIEdk 3DB18 ⇒ A = 7.63cm s 2 > 6.258cm 2 RbEvgbøúkkugRtaMgsgát; As f y 7.63 × 235 a= = = 5.274cm 0.85 f 'c b 0.85 × 20 × 20 ersIusþg;m:Um:g;xagkñúgKNna a 5.274 φM n = φAs f y (d − ) = 0.9 × 7.63 × 235 × (45 − ) ×10 − 3 = 68.4kN .m 2 2 ⇒ φM n > M u dUcenH Edk 3DB18 RKb;RKan;edIm,ITb;Tl;nwgm:Um:g;KNnaxageRkA . 8> bNþúMénEdk enAeBlEdlkarKNnamuxkat;EdkRtUvkarsMrab;ebtugmanbrimaNeRcIn ]TahrN_ enAeBlEdl ρ max RtUv)aneRbI eBlenaHeKBi)akkñúgkarBRgayEdkeTAkñúgmuxkat;ebtug. ACI Code )anGnuBaØatieGayEdk beNþayGacdak;CabNþúMEdl manTMrg;dUcbgðaykñúgrUb.bNþúMénEdkcab;BIbYn GaceFVIeTA)anedayman EdkkgBT§½ CMuvij. kareFVIbNþúMEdkkgenHk¾GacRbRBwtþeTA)ansMrab;ssr. bNþúMénEdk RtUv)ancat;TukCaEdkmYyedImsMrab; kMNt;KMlatEdk nigkMras;karBarebtug. Ggát;p©it énEdkeTal RtUv)anbMEbkBIRkLaépÞsmmUlrbs;bNþúMEdk. segçb³ karkMNt;EdkrgkMlaMgTajsMrab;muxkat;ctuekaNEkg 1> kMNt;PaKryEdkeRbIR)as;enAkñúgebtug ρ = bd A s 2> kMNt;PaKryEdk balanced ρ = 0.85β ff ' ( 600 + f ) nigPaKryEdkGtibrma b 600 1 c y y fy 0.003 + ρ max = ( 0.008 Es ) ρb sMrab;muxkat;rgkMlaMgTaj. dUcKña kMNt;PaKryEdkGb,brma viPaKFñwmebtugGarem:rgkarBt;begáag 33
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa f 'c 1.4 ρ min = max( , ) 4 fy fy 3> RbsinebI ρ < ρ < ρ kMNt; a = 0.85 ff ' b / c / ε nig φ = 0.9 . RbsinebI ρ < ρ min max A s y t min c PaKryEdkEdleRbIR)as;kñúgebtugminRKb;RKan; eTaHCay:agNaPaKryEdkEdleRbIR)as;kñúgebtug RtUvEt ρ ≥ ρ . RbsinebI ρ ≥ ρ enaH φ < 0.9 . min max 4> kMNt;ersIusþg;m:Um:g;xagkñúgKNna φM = φA f (d − a ) n 2 s y 9> muxkat;ctuekaNEkgCamYyEdkrgkMlaMgsgát; enAkñúgmuxkat;ebtug muxkat;EdkEdlTb;nwgm:Um:g;Bt; RtUv)ankMNt;ecjBIbnÞúkxageRkAEdlmanGMeBI elIeRKOgbgÁúM edayeFVIy:agNaeGayersIusþg;m:Um:g;xagkñúgFMCag b¤esμInwgm:Um:g;xageRkA. b:uEnþenAeBlEdlmux kat;ebtug ¬TTwg nigkMBs;RbsiT§PaB¦ mantMéltUcenaH ρ RtUv)aneRbI. RbsinebIm:Um:g;xageRkAFMCag max ersIusþg;m:Um:g;xagkñúg enaHbrimaNEdksgát; nigEdkTajRtUv)anbEnßm. Edksgát;RtUv)aneRbI enAeBlEdlmuxkat;ebtugRtUv)ankMNt;edaymUlehtusßabtükmμ. pl RbeyaCn_rbs;Edksgát;KW kat;bnßyPaBdabry³eBlyUr nigedIm,IgayRsYldak;Edkkg. muxkat;EdkDubmanBIrkrNIEdleKRtUvBicarNa GaRs½yeTAnwgkareFVIrbs;Edkdl;cMnucyar b¤Gt;. k> enAeBlEdksgát;eFVIkardl;cMnucyar Flexural Analysis of Reinforced Concrete Beam 34
  • T.Chhay NPIC m:Um:g;xagkñúgGacRtUv)anEckecjCaBIr dUcbgðajkñúgrUb M Cam:Um:g;EdlekItBIkMlaMgsgát;rbs;ebtug u1 nigkMlaMgTajsmmUlrbs;Edk A sMrab;muxkat;eKal. M Cam:Um:g;bEnßmEdlekItBIkMlaMgsgát;enAkñúg s1 u2 Edksgát; A' nigkMlaMgTajenAkñúgEdkrgkMlaMgTajbEnßm A . s s2 m:Um:g; M Cam:Um:g;Edl)anBImuxkat;sMrab;EdkrgkarTajeKal u1 T1 = Cc ⇒ As1 f y = 0.85 f 'c ab As1 f y ⇒a= 0.85 f 'c b a M u1 = φAs1 f y (d − ) 2 fy 0.003 + karkMNt; M RtUveGay ρ < bd nigtUcCag b¤esμI ρ = ( 0.008E ) ρ sMrab;eGaymuxkat; A u1 1 s1 max s b rgkarTajeKal. BicarNaelIm:Um:g; M edaysnμt;fa muxkat;Edkrgkarsgát; A' eFVIkardl;cMnucyar u2 s M u 2 = φAs 2 f y (d − d ' ) M u 2 = φA' s f y (d − d ' ) d' - CacMgayBIsésEpñkxageRkAbMputeTAGkS½Edkrgkarsgát; kñúgkrNIenH A = A' begáItnUvkMlaMgesμIKñaTisedApÞúyKña s2 s m:Um:g;srub esμInwgplbUkénm:Um:g; M nig M u1 u2 a φM n = M u1 + M u 2 = φ[ As1 f y (d − ) + A' s f y (d − d ' )] 2 muxkat;EdksrubEdleRbIsMrab;karTajCaplbUkénbrimaNEdk A nig A s1 s2 dUcenH A = A + A = A + A' s s1 s2 s1 s ⇒ As1 = As − A' s ( A − A's ) f y ⇒a= s 0.85 f 'c b dUcenHeK)an φM a = φ[( As − A' s ) f y (d − ) + A' s f y (d − d ' )] n 2 fy 0 . 003 + nigeyIgman ρ 1 = ( ρ − ρ ' ) ≤ ρ max = ρ b ( (1) 0 . 008 Es ) sMrab; f = 414MPa enaH ( ρ − ρ ' ) ≤ 0.63375ρ / φ = 0.9 nig ε = 0.005 kar)ak;rbs;FñwmbNþal y b t mkBIEdksrubrgkarTajeFVIkardl;cMnucyar ehIykarEbkPøam²rbs;ebtugRtUv)aneCosvag. viPaKFñwmebtugGarem:rgkarBt;begáag 35
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa RbsinebI ρ 1 = ( ρ − ρ ' ) > ρ max enHmuxkat;sßitenAtMbn; transition region Edl fy 0.003 + ( ρ − ρ ' ) ≤ ρ max,t = ρ b ( 0.007 Es ) kñúgkrNIenH φ < 0.9 sMrab; M nig φ = 0.9 sMrab; M enaH u1 u2 eK)an a φM n = φ[( As − A' s ) f y (d − )] + 0.9 A' s f y (d − d ' ) 2 cMNaMfa ( A − A' ) ≤ ρ bd s s max,t enAkñúgtMbn;sgát; kMlaMgEdkrgkarsgát;KW C = A' ( f s s y − 0.85 f 'c ) edayKitfaépÞebtugEdlCMnYsedayépÞEdk A' enaH s T = As f y = Cc + C s = 0.85 f 'c ab + A' s ( f y − 0.85 f 'c ) ⇒ As f y − A' s f y + 0.85 f 'c A' s = 0.85 f 'c ab eday 0.85 f ' ab = A c s1 fy ⇒ As f y − A' s f y + 0.85 f 'c A' s = As1 f y EckGgÁTaMgBIrnwg bdf y ⇒ ρ − ρ ' (1 − 0.85 f 'c fy ) = ρ1 Edl ρ 1 = As1 bd ≤ ρ max fy 0.003 + dUcenH ρ − ρ ' (1 − 0.85 ff ' ) ≤ ρ c max = ρb ( 0.008 Es ) (2) y PaKryEdkrgkarTajsrubGtibrma ρ EdleRbIenAkñúgmuxkat;ctuekaNEkg enAeBlEdlEdkrgkar sgát;eFVIkardl;cMnucyar Maxρ = ( ρ max + ρ ' ) mann½yfa muxkat;EdkrgkarTajsrubeRbIenAkñúgmuxkat;ctuekaN enAeBlEdkrgkarsgát;eFVIkardl; cMnucyar MaxA = bd ( ρ + ρ ' ) s max edIm,IeGaydwgfa Edkrgkarsgát;eFVIkardl;cMncyar eyIgRtUvBinitüsac;lUteFob edayeGay u fy ε 's ≥ ε y = Es Flexural Analysis of Reinforced Concrete Beam 36
  • T.Chhay NPIC tamrUbxagelI eyIg)an c 0.003 600 = = d' fy 600 − f y 0.003 − Es 600 ⇒c=( )d ' 600 − f y eyIgman A f = 0.85 f ' ab s1 y c b:uEnþ A = A − A' nig ρ = ρ − ρ ' s1 s s 1 dUcenHeyIg)an ( A − A' ) f = 0.85 f ' ab s s y c ⇒ ( ρ − ρ ' )bdf y = 0.85 f 'c ab f 'c a ⇒ ( ρ − ρ ' ) = 0.85( )( ) fy d eday a = β c = β ( 600 − f 1 600 1 )d ' y dUcenH ( ρ − ρ ' ) = 0.85β ( ff ' )( d ' )( 600 − f d 1 600 c )=K y y RbsinebI ( ρ − ρ ' ) ≥ K enaHEdkrgkarsgát;eFVIkardl;cMnucyar. eyIgeXIjfa enAeBlEdlbrimaNEdkrgkarTajeKal A ekIneLIg enaH T nig C k¾mantMélkan; s1 1 1 EtFMEdr ehIyGkS½NWtnwgFøak;cuH eBlenaHsac;lUteFobrbs;Edkrgkarsgát;k¾ekIneLIg rhUtdl;cMnucyar. viPaKFñwmebtugGarem:rgkarBt;begáag 37
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa ]TahrN_7³ FñwmctuekaNEkg EdlmanTTwg 30cm nigkMBs;RbsiT§PaB d = 60cm . EdkrgkarTajman 6 DB 28 tMerobCaBIrCYr ÉEdkrgkarsgát;man 2DB 22 . kMNt;ersIusþg;m:Um:g;xagkñúgRbsinebIeKeRbI f ' = 28MPa nig f = 400MPa . c y dMeNaHRsay³ muxkat;EdkrgkarTaj A = 36.93cm PaKryEdkrgkarTaj ρ = 30 ×93 = 0.02052 s 2 36. 60 Flexural Analysis of Reinforced Concrete Beam 38
  • T.Chhay NPIC muxkat;Edkrgkarsgát; A' = 7.6cm PaKryEdkrgkarsgát; ρ ' = 307×660 = 0.0042 s 2 . muxkat;EdkrgkarTajeKal A = 29.33cm PaKryEdkrgkarTajeKal ρ = 30 ×.33 = 0.01629 s1 2 29 60 1 f 'c d ' 600 28 6 600 K = 0.85β1 ( )( )( ) = 0.852 = 0.01517 f y d 600 − f y 400 60 600 − 400 eday ( ρ − ρ ' ) ≥ K enaHEdkrgkarsgát;eFVIkardl;cMnucyar sMrab; f ' = 28MPa nig f = 400MPa ⇒ ρ = 0.030345 ⇒ ρ c y b max = 0.019 eday ( ρ − ρ ' ) < ρ ⇒ φ = 0.9max ersIusþg;m:Um:g;xagkñúg a φM n = φ[( As − A' s ) f y (d − ) + A' s f y (d − d ' )] 2 ( As − A's ) f y ⇒a= ⇒ a = 16.43cm 0.85 f 'c b 16.43 ⇒ φM n = 0.9[29.33 × 400 × (60 − ) + 7.6 × 400 × (60 − 6)] × 10−3 = 694.5kN .m 2 viFImü:ageTot epÞógpÞat;faetIEdkrgkarsgát;eFVIkardl;cMnucyarb¤enA a 16.43 c= = = 19.33cm 0.85 0.85 sac;lUteFobEdkrgkarsgát; ε ' = c −c d ' × 0.003 = 1919.33 6 × 0.003 = 0.00207 .33 − s sac;lUteFobrbs;Edk ε = 0.002 y eday ε ' > ε ⇒ Edkrgkarsgát;eFVIkardl;cMnucyar s y dt − c (60 + 6) − 19.33 εt = ( )0.003 = × 0.003 = 0.007 > 0.005 c 19.33 b¤ d c 19.33 = 60 = 0.322 < 0.375 muxkat;EdkrgkarTajsrub MaxAs = bd ( ρ max + ρ ' ) = 30 × 60 × (0.019 + 0.0042) = 41.76cm 2 > As RtwmRtUv x> enAeBlEdksgát;eFVIkarmindl;cMnucyar dUckarbkRsayxagelI RbsibebI ( ρ − ρ ' ) < 0.85β ( ff ' )( d ' )( 600 − f d 600 1 c )=K y y enaHEdksgát;eFVIkarmindl;cMnucyareT. enHbgðajfa RbsinebI ( ρ − ρ ' ) < K EdkrgkarTajeFVIkar dl;cMnucyarmun ebtugmansac;lUteFobGtibrma 0.003 ehIyEdkrgkarsgát;k¾eFVIkarmindl;cMnucyarEdr. pleFob d ' c kan;EtFM mann½yfakalNaeKdak;Edkrgkarsgát;enACitGkS½NWt enaHsac;lUteFobrbs;Edk rgkarsgát;kan;EttUc. viPaKFñwmebtugGarem:rgkarBt;begáag 39
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa RbsinebIEdksgát;eFVIkarmindl;cMnucyar dMeNaHRsayTUeTAGaceFVIeTA)anedayQrelIeKalkarN_ sþaTic. c − d' c − d' ε ' s = 0.003( ) f ' s = E s ε ' s = 600( ) c c edayeGay C c = 0.85 f 'c β1cb c − d' C s = A' s ( f ' s −0.85 f 'c ) = A' s [600( )0.85 f 'c ] c edaysar T = A f s y = Cc + C s enaH c − d' As f y = 0.85 f 'c β1cb + A' s [600( )0.85 f 'c ] c ⇒ (0.85 f 'c β1b)c 2 + [(600 A' s ) − (0.85 f 'c A' s ) − As f y ]c − 600 A' s d ' = 0 smIkarenHmanTMrg; A c + A c + A = 0 1 2 2 3 eRkayeBlKNna c KNna f ' = 600( c −c d ' ) KNna a = β c KNna C A' [600( c −c d ' )0.85 f ' ] nigKNna s 1 s s c Cc = 0.85 f 'c β1cb a φM n = φ[Cc (d − ) + C s (d − d ' )] 2 enAeBlEdksgát;eFVIkarmindl;cMnucyar/ f 's < f y nigEdkTajsrubRtUvkarsMrab;muxkat;ctuekaN EkgKW³ f 's ρ ' f 's MaxAs = ρ max bd + A' s = bd ( ρ max + ) fy fy edayEckGgÁTaMgBIrnwg bd eyIg)anPaKryEdk MaxAs f' Maxρ = ≤ ρ max + ρ ' s bd fy b¤ ( ρ − ρ ' ff ' ) ≤ ρ s max y − kñúgkrNIenH a = A 0f.85 fA'' bf ' s y s s c a φM n = φ[( As f y − A' s f ' s )(d − ) + A' s f ' s (d − d ' )] 2 segçb³ viFIsaRsþviPaKmuxkat;CamYyEdkrgkarsgát; 1> kMNt; ρ / ρ ' / ( ρ − ρ ' ) dUcKñakMNt; ρ / ρ max min 2> kMNt; K = 0.85β ( ff ' )( d ' )( 600 − f ) d 1 600 c y y Flexural Analysis of Reinforced Concrete Beam 40
  • T.Chhay NPIC 3> RbsinebI ( ρ − ρ ' ) ≥ K enaHEdkrgkarsgát;eFVIkardl;cMnucyar f ' = f . RbsinebI s y ( ρ − ρ ' ) < K enaHEdkrgkarsgát;eFVIkarmindl;cMnucyar f ' < f . s y 4> RbsinebIEdkrgkarsgát;eFVIkardl;cMnucyar k> BinitüemIl ρ ≥ ( ρ − ρ ' ) ≥ ρ b¤ ε ≥ 0.005 / eRbI φ = 0.9 max min t − x> kMNt; a = ( A .85Af'' )bf 0 s s y c K> kMNt; φM = φ[( A − A' ) f (d − a ) + A' f (d − d ' )] n s 2 s y s y X> muxkat;EdkrgkarTajGtibrma A EdlGaceRbIenAkñúgmuxkat;KW s MaxAs = bd ( ρ max + ρ ' ) ≥ As 5> RbsinebIEdkrgkarsgát;eFVIkarmindl;cMnucyar k> KNnacMgayGkS½NWt c edayeRbIsmIkar T = C + C s c x> kMNt; f ' = 600( c −c d ' ) s K> RtYtBinitü ( ρ − ρ ' ff ' ) ≤ ρ b¤ MaxA EdlGaceRbIenAkñúgmuxkat; RtUvEtFMCagb¤esμI A s max s s y Edl)aneRbI f 's MaxAs = bd ( ρ max + ρ ' ) ≥ As fy − X> kMNt; a = A 0f.85 fA'' bf ' b¤ a = β c s y s s 1 c g> kMNt; φM = φ[( A f − A' f ' )(d − a ) + A' f ' (d − d ' )] n s 2 y s s s s ]TahrN_8³ kMNt;ersIusþg;m:Um:g;kñúgénmuxkat;dUcbgðajkñúgrUb edayeRbI f ' = 35MPa / f = 400MPa . eK c y eRbIEdkrgkarsgát; 3DB25 Edl A' = 14.72cm nigEdlrgkarTaj 6DB32 Edl A = 42.39MPa . s 2 s viPaKFñwmebtugGarem:rgkarBt;begáag 41
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa dMeNaHRsay³ kMNt; ρ = bd = 35 ×.39 = 0.02125 / ρ ' = bd = 35 ×72 = 0.00738 / ( ρ − ρ ' ) = 0.01387 A 42 s 57 A' 14. 57 s eday f ' = 35MPa ⇒ β = 0.85 − 0.05( f ' 7−28 ) ⇒ β = 0.85 − 0.05( 35 − 28 ) = 0.8 c 1 c 7 1 kMNt; K = 0.85β ( ff ' )( d ')( 600 − f ) = 0.85 × 0.8( 400 )( 6.5 )( 600600400 ) = 0.020355 1 d 600c 35 57 − y y eday ( ρ − ρ ' ) < K enaHEdkrgkarsgát;eFVIkarmindl;cMnucyar f 'c 600 ρ b = 0.85β1 ( ) = 0.0357 f y 600 + f y 0.005 ρ max = 0.0357 = 0.02231 0.008 muxkat;rgkarTaj ⇒ φ = 0.9 ( ρ − ρ ' ) < ρ max kMNt;cMgayGkS½NWt c C = 0.85 f ' ab eday a = β c = 0.8c ⇒ C = 0.85 × 35 × 0.8c × 350 = 8330c c c 1 c C s = A' s ( f ' s −0.85 f 'c ) c − d' c − 65 c − 65 eday f ' s = 600( c ) ⇒ C s = 1472[600( c ) − 0.85 × 35] = 883200( c ) − 43792 T = As f y = 4239 × 400 = 1695600 N c − 65 ⇒ 1695600 = 8330c + 883200( ) − 43792 c ⇒ 8330c 2 − 856192c − 57408000 = 0 ⇒ c = 149mm = 14.9cm ⇒ a = 0.8 ×14.9 = 11.92cm c − d' 14.9 − 6.5 kMNt; f ' s = 600( c ) ⇒ f ' s = 600 14.9 = 339MPa kMNt; C = 0.85 f ' ab ⇒ C = 0.85 × 35 ×119.2 × 350 = 1241170 N = 1241.17kN c c c kMNt; C = A' ( f ' −0.85 f ' ) ⇒ C = 1472(339 − 0.85 × 35) = 455216 N = 455.216kN s s s c s edIm,IkMNt;ersIusþg;m:Um:g;kñúg eKRtUvKitm:Um:g;eFobGkS½EdkTaj A s a 0.1192 φM n = φ[Cc (d − ) + C s (d − d ' )] = 0.9[1241.17(0.57 − ) + 455.216(0.57 − 0.065)] 2 2 φM n = 863.38kN .m RtYtBinitü ( ρ − ρ ' ff ' ) ≤ ρ s max ⇒ (0.02125 − 0.00738 339 400 ) = 0.015 < 0.02231 y kMNt;muxkat;EdkTajGtibrma MaxA s = bd ( ρ max + ρ ' f 's fy ) Flexural Analysis of Reinforced Concrete Beam 42
  • T.Chhay NPIC MaxAs = 35 × 57(0.02231 + 0.00738 339 400 ) = 56.99cm 2 > 42.39cm 2 RtwmRtUv c = 14.9 d t 57 + 9 − 6.5 = 0.25 < 0.375RtwmRtUv d −c εt = t c 0.003 = 0.009 > 0.005 muxkat;rgkarTaj 10> viPaKmuxkat;GkSret T nigmuxkat;GIu I CaFmμtakMralxNÐ nigFñwmRtUv)aneKcak;CamYyKña edIm,IbegáItCaeRKOgbgÁúMEtmYy monolithic structure. kMralxNÐmankMras;esþIgCagFñwm. eRkamGMeBIénkugRtaMgBt; EpñkénkMralxNÐEdlCaEpñkrbs;Fñwm rgnUvkugRtaMgsgát; GaRs½yeTAelITItaMgGkS½NWt. EpñkénkMralxNÐEdleFVIkarCamYyFñwmRtUv)aneKeGay eQμaHfa søab flange EdlbgðajkñúgrUbedayépÞ bt . EpñkénFñwmEdlenAsl; EdlbgðajedayépÞ (h − t )b w RtUv)aneKeGayeQμaHfa RTnug stem b¤ web. sMrab;muxkat;GkSr I mansøabBIr KWsøabrbkarsgát; EdlcUlrYmeFVIkar nigsøabrgkarTaj EdlKμanRb siT§PaB BIeRBaHvaenABIeRkamGkS½NWt ehIyEdlminRtUv)aneKykvamkKit. dUcenH karviPaK nigkarKNna Fñwmmuxkat; I manlkçN³dUcKñanwgFñwmmuxkat; T . k> TTwgRbsiT§PaB sMrab;muxkat;GkSr T EdlsøabmanRbEvgEvg kugRtaMgsgát;manragCa):ar:abUl EdltMélGtibrmasßit enAelIFñwm ehIytMélGb,brmasßitenAcMgay x BImuxrbs;Fñwm. ehIykugRtaMgk¾ERbRbYlBIsésEpñkxagelI søab viPaKFñwmebtugGarem:rgkarBt;begáag 43
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa mksésEpñkxageRkamsøab BIGtibrma mkGb,brma. tMélbMErbMrYlenHGaRs½yeTAnwgTItaMgGkS½NWt. kugRtaMgsmmUl CakugRtaMgBRgayesμImanGMeBIelITTwgsøabsmmUl b . TTwgRbsiT§PaB b RtUv)an e e eKkMNt;edayGnuKmn¾eTAnwg³ - RbelaHElVg s 1 - TTwgRTnug b w - TMnak;TMngrvagkMras;kMralxNÐ nigkMBs;srubrbs;Fñwm - lkçxNÐTMrrbs;Fñwm ¬samBaØ b¤Cab;¦ - lkçxNÐbnÞúk ¬BRgayesμI b¤cMcMnuc¦ - pleFobrvagRbEvgFñwmcenøaHm:Um:g;sUnü nigTTwgRTnug nigcMgayrvagRTnug ACI Code )ankMNt;nUvTTwgRbsiT§PaBedaykMNt;yktMélGb,brmaénsmIkarxageRkam³ -b = e L 4 Edl L CaRbEvgFñwm - b = 16t + b Edl t kMras;kMralxNÐ nig b TTwgRTnug e w w Flexural Analysis of Reinforced Concrete Beam 44
  • T.Chhay NPIC - b = b Edl b cMgayBIcenøaHGkS½kMralxNÐ e muxkat;ragGkSr T b¤muxkat;ragGkSr I GacRtUvviPaKCaragctuekaNEkg b¤ragGkSr T GaRs½yelITI taMgGkS½NWt. x> muxkat;GkSret T RtUv)anKitCaragctuekaNEkg kñúgkrNIenH kMBs;énbøúkkugRtaMgsmmUl a sßitenAkñúgsøab a ≤ t begáIt)anCaépÞkugRtaMgsgát;esμI nwg b a . muxkat;ebtugBIeRkamGkS½NWtRtUv)aneKsnμt;faKμanRbsiT§iPaB ehIymuxkat;RtUv)aneKKitfaman e EdkrgkarTaj Edl)anBnül;BIxagelI edayRKan;EtCMnYs b eday b . e dUcenH a = 0.85 ff' b A s y c e nig φM = φA f (d − a ) n s y 2 RbsinebI kMBs; a ekIneLIgeday a = t enaH φM n t = φAs f y (d − ) 2 kñúgkrNIenH t = 0.85 ff' b b¤ A = 0.85 ff ' b t A s y s c e c e y sMrab;karviPaKenH A ≤ A nig ε s s max t ≥ 0.005 viPaKFñwmebtugGarem:rgkarBt;begáag 45
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa K> viPaKmuxkat;ragGkSret T kñúgkrNIenH GkS½NWtsßitenAelIRTnug. EpñkxøHrbs;ebtugenAkñúgRTnugmanRbsiT§PaBkñúgkarTb;Tl; nwgm:Um:g;xageRkA. kMlaMgsgát; C = 0.85 f ' [b t + b (a − t )] c e w TItaMgrbs; C sßitenAelITIRbCMuTMgn;rbs;épÞragGkSr T enAcMgay z BIsésEpñkxageRkAbMput. Flexural Analysis of Reinforced Concrete Beam 46
  • T.Chhay NPIC karviPaKmuxkat;ragGkSr T manlkçN³RsedogKñanwgkarviPaKmuxkat;ebtugEdlEdkrgkarsgát; edaycat;TuképÞebtug (b − b )t smmUleTAnwgEdksgát; A' . karviPaKenHEckecjCaBIrEpñkdUcbgðajkñúg e w s rUbxageRkam³ - muxkat;eKalragctuekaNEkg b d nigmuxkat;Edk A . kMlaMgsgát; C = 0.85 f ' ab nigkMlaMg w s1 1 c w T = A f ehIyRbEvgédXñas; (d − ) . a 1 s1 y 2 - muxkat;Edlmansøabebtugsgxag 2 × [(b − b )t ] / 2 begáIt)anCakMlaMgsgát;edayKuNCamYy e w 0.85 f ' nigRbEvgédXñas;esμInwg (d − ) . RbsinebI A Camuxkat;EdkTajEdlbegáItkMlaMgesμInwg t c sf 2 kMlaMgsgát;EdlbegáItedayebtugsøabsgxag dUcenH A = 0.85 f ' ft (b − b ) sf c e w y muxkat;Edksrub A EdleRbIkñúgmuxkat;GkSr T KW³ A = A + A s s s1 sf b¤ A = A − A s1 s sf muxkat;GkSr T sßitkñúgsßanPaBlMnwg dUcenH C = T / C = T nig C = C + C 1 1 2 2 1 2 = T1 + T2 + T BicarNaelIsmIkar C = T sMrab;muxkat;eKalctuekaNEkg eK)an 1 1 A f = 0.85 f ' ab b¤ ( A − A ) f = 0.85 f ' ab s1 y c w s sf y c w A − ) dUcenH a = (0.85 fA' b f s sf y c w cMNaMfa b RtUv)aneRbIedIm,IkMNt; a . w ersIusþg;énm:Um:g;kñúgénmuxkat;CaplbUkénm:Um:g;BIr M nig M u1 u2 φM n = M u1 + M u 2 viPaKFñwmebtugGarem:rgkarBt;begáag 47
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa a a M u1 = φAs1 f y (d − ) = φ ( As − Asf ) f y (d − ) 2 2 ( As − Asf ) f y Edl As1 = As − Asf a= nig 0.85 f 'c bw t M u 2 = φAsf f y (d − ) 2 a t φM n = φ[( As − Asf ) f y (d − ) + Asf f y (d − )] 2 2 BicarNaelImuxkat;RTnug b d / sac;lUteFobsuT§ ε GackMNt;BI a / c nig d dUcxageRkam³ w t t RbsinebI c = βa nig d = h − 6.5cm bnÞab;mk ε = 0.003 (c −cd ) sMrab;muxkat;rgkarTajenAkñúg t t t 1 RTnug/ ε ≥ 0.005 . t karKNnaersIusþg;m:Um:g;kñúgsMrab;muxkat;GkSr T b¤muxkat;GkSr I GacKNnaedayeRbIsmIkarxagelI EteKcaM)ac;RtUvRtYtBinitülkçxNÐxageRkam³ - PaKryEdkTajsrubeFobRkLaépÞRbsiT§iPaBRTnugRtUvFMCag b¤esμI ρ min As ρw = ≥ ρ min bw d f 'c 1.4 ρ min = ≥ 4 fy fy - RtYtBinitü sac;lUteFobsuT§FMCag b¤esμI ε ≥ 0.005 sMrab;muxkat;rgkarTaj t - muxkat;EdkGtibrma MaxA enAkñúgmuxkat;GkSr T RtUvEtFMCag b¤esμI muxkat;EdkEdl)aneRbI A s s sMrab;muxkat;rgkarTaj CamYy φ = 0.9 MaxAs = Asf ( flange) + ρ max (bw d )( web) 1 MaxAs = [0.85 f 'c t (b − bw )] + ρ max (bw d ) fy PaKryEdkeFobnwgRTnug ρ w = As bw d ≤ ( ρ max + Asf bw d ) ⇒ ρ w − ρ f ≤ ρ max smIkarTUeTAsMrab;KNna MaxA enAkñúgmuxkat;GkSr T enAeBl a > t GackMNt;tam s C = 0.85 f 'c [(be − bw )t + abw ] sMrab; ε = 0.003 nig ε = 0.005 / d = 0.003.003.005 = 0.375 sMrab;RTnug c c 0 t +0 dUcenH a = β c = 0.375β d 1 1 muxkat;EdkGtibrmaesμInwg C f y Flexural Analysis of Reinforced Concrete Beam 48
  • T.Chhay NPIC dUcenH MaxA s = 0.85 f 'c fy [(be − bw )t + 0.375β1bw d ] segçb³ viFIsaRsþviPaKmuxkat;GkSret T b¤GkSrGil L páab; 1> kMNt;TTwgRbsiT§PaB b nigkMNt; ρ / ρ e max min 2> kMNt; a = 0.85 ff' b A s y c e 3> RbsinebI a < t enaHmuxkat;eFVIkarCaragctuekaNEkg - kMNt; φM = φA f (d − a )n 2 s y cMNaMfa³ c = βa nig ε = 0.003 (c −cd ) ≥ 0.005 sMrab;muxkat;rgkarTaj φ = 0.9 t t 1 - RtYtBinitü ρ w = As bw d ≥ ρ min - MaxA s = 1 fy [0.85 f 'c t (b − bw )] + ρ max (bw d ) ≥ As 4> RbsinebI a > t enaHmuxkat;eFVIkarCaragGkSret k> kMNt; A = 0.85 f ' ft (b − b ) sf c w y ( As − A' s ) f y x> kMNt; a = 0.85 f ' b c K> RtYtBinitü ρ − ρ ≤ ρ eFobnwgRkLaépÞRTnug w f max Edl ρ = bAd nig ρ = bA d w s f sf w w b¤RtYtBinitü MaxA s = 0.85 f 'c fy [(be − bw )t + 0.375β1bw d ] ≥ As / sMrab; φ = 0.9 A − ) X> kMNt; a = (0.85 fA' b f s sf y c w g> kMNt; φM = φ[( A − A ) f (d − a ) + A f (d − 2 )] n 2 t s sf y sf y ]TahrN_9³ FñwmebtugGarem:EdlmanRbEvg 4.5m ehIymanKMlatBImYyeTAmYyRbEvg 2m . FñwmenHRTkM ralxNÐEdlmankMras; 10cm . kMNt;nUversIusþg;m:Um:g;kñúgrbs;FñwmkNþal. eKeRbI f ' = 20MPa nig c f = 400MPa . y viPaKFñwmebtugGarem:rgkarBt;begáag 49
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa dMeNaHRsay³ kMNt;TTwgRbsiT§iPaB L 450 be = min{16t + bw ; ; b} = min{16 × 10 + 25; ;200} = 112.5cm 4 4 kMNt;kMBs;bøúkkugRtaMg a= A f s y 0.85 f ' b / A = 14.72cm s 2 c e 14.72 × 400 a= = 3.08cm < t 0.85 × 20 × 112.5 dUcenHeyIgRtUvKNnaCaragctuekaNEkgEdlmanTTwg b = 112.5cm e PaKryEdkGb,brma ρ = 4 ff ' ≥ 1f.4 ⇒ ρ = 0.0035 min c min y y PaKryGtibrma ρ max = 0.625 × 0.85β 1 f 'c fy ( 600 600 + f y ) = 0.01355 PaKryEdkeFobnwgépÞRkLaRTnug ρ w = As = 14.72 bw d 25 × 40 = 0.01472 > 0.0035 TItaMgGkS½NWt c = βa = 3..08 = 3.62cm 0 85 1 sac;lUteFobsuT§rbs;Edk ε = 0.003( d c− c ) = 0.003( 403−.62.62 ) = 0.03 > 0.005 ⇒ φ = 0.9 t 3 t KNna φM = φA f (d − a ) = 0.9 ×1472 × 400(400 − 30.8 ) = 203807232 N .mm = 203.81kN .m n s 2 y 2 epÞógpÞat;muxkat;Gtibrma MaxA = f [0.85 f ' t (b − b )] + ρ (b d ) ≥ A 1 s c w max w s y Flexural Analysis of Reinforced Concrete Beam 50
  • T.Chhay NPIC MaxA = 37.22cm 2 > As RtwmRtUv ]TahrN_10³ KNnaersIusþg;m:Um:g;kñúgénmuxkat;GkSr T dUcbgðajkñúgrUb edayeRbI f 'c = 25MPa nig f = 400 MPa . y dMeNaHRsay³ eKeGay b = b = 90cm / b = 25cm / d = 43cm nig A e e s = 36.93cm 2 × KNna a = 0.85 ff' b = 036.9325400 = 7.72cm > t A s y .85 × × 90 c e eday a > t sikSaCaragGkSr T KNna A = 0.85 f ' ft (b − b ) = 24.17cm sf c w 2 y ⇒ As1 = As − Asf = 12.76cm 2 epÞógpÞat; ε t As1 f y 12.76 × 400 a ( web) = = = 9.6cm 0.85 f 'c bw 0.85 × 25 × 25 a( web) c= = 11.29cm β1 d t = 52 − 6.5 = 45.8cm dt − c ε t = 0.003( ) = 0.00917 > 0.005 ⇒ φ = 0.9 c RtYtBinitü A s min = ρ min bw d = 0.0035 × 25 × 43 = 3.76cm 2 < 36.93cm 2 RtwmRtUv KNna φM a t = φ[( As − Asf ) f y (d − ) + Asf f y (d − )] n 2 2 96 70 φM n = 0.9[(3693 − 2417)400(430 − ) + 2417 × 400(430 − ) 2 2 φM n = 519172920 N .mm = 519.173kN .m 11> TMhMénmuxkat;FñwmGkSr T Éeka eBlxøH FñwmGkSr T Éeka RtUv)aneRbIedm,IbEnßmépÞrgkarsgát;. muxkat;enHRtUv)aneKeRbIsMrab;Fñwm EdleKcak;TukCamun. viPaKFñwmebtugGarem:rgkarBt;begáag 51
  • Department of Civil Engineering viTüasßanCatibec©keTskm<úCa ACI Code )anENnaMnUvTMhMmuxkat;sMrab;GkSr T ÉekadUcxageRkam³ - kMras;søab t RtUvFMCag b¤esμIBak;kNþalTTwgRTnug b w - TTwgsrubrbs;søab b RtUvEttUcCag b¤esμIbYndgTTwgRTnug b w 11> muxkat;GkSr L páab; Fñwmmuxkat;GkSr L páab;CaFñwmEdlRTkMralxNÐEpñkxageKbMput. TTwgRbsiT§PaBrbs;muxkat;enHRtUv )ankMNt;nUvtMélGb,brmaénsmIkarxageRkam³ - (b − b ) ≤ 12 e w L - (b − b ) ≤ 6t e w - (b − b ) ≤ 2 e w l Edl L - RbEvgFñwm l - KMlatFñwm Flexural Analysis of Reinforced Concrete Beam 52
  • T.Chhay NPIC viPaKFñwmebtugGarem:rgkarBt;begáag 53