Iv flexural design on reinforced concrete beams

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Iv flexural design on reinforced concrete beams

  1. 1. T.Chhay NPIC IV. karKNnaFñwmebtugGarem:rgkarkac;begáag 1> kMBs;RbsiT§iPaBsMrab;Fñwm nigkMralxNÐ kMBs;rbs;Fñwmx<s; b¤TabGaRs½yeTAnwgRbEvgrbs;Fñwm nigbnÞúkxageRkAEdlvaRtUvRT. edIm,IepÞógpÞat; CamYYyPaBdab xageRkamenHCarUbmnþsMrab;kMNt;kMBs;FñwmGaRs½y nwgRbEvgFñwm. kMralxNÐ L/20 L/24 L/28 L/10 mYyTis Fñwm L/16 L/18.5 L/21 L/8 TTwgFñwmRtUv)ankMNt;ecjBIkMBs;rbs;Fñwm edaykMNt;enAcenøaH 1 d → 1 d . 3 2 2> muxkat;ctuekaNEkgCamYyEdkrgkarTaj xageRkamCarUbmnþsMrab;karKNnamuxkat;rgkarTaj f 'c 600 ρ b = 0.85β1 ( ) f y 600 + f y fy 0.003 + Es ρ max = ρ b ( ) 0.008 sMrab; f y = 400MPa ⇒ ρ max = 0.625ρ b ehIy β 1 sMrab;ebtugEdlmanersIusþg; f ' ≤ 28MPa . = 0.85 c f ' −28 β = 0.85 − 0.05( 1 ) sMrab;ebtugEdlmanersIusþg; 28MPa < f ' ≤ 56MPa . c c 7 β = 0.65 sMrab;ebtugEdlmanersIusþg; f ' > 56MPa . 1 c tMélénsac;lUteFobrbs;Edkkan;EtFM bgðajBIPaBsVitrbs;ebtugGarem:kan;EtFM )ann½yfaenAeBl muxkat;ebtugkan;EtFMCamYyPaKryEdkticPaBsVitkan;EtFM pÞúymkvijenAeBlmuxkat;ebtugkan;EttUcCamYy PaKryEdkkan;EtFMenaHPaBsVitrbs;ebtugkan;EttUc. xageRkamCataragbgðajBIPaKryEdksMNUmBrGa Rs½yersIusþg;ebtug nigersIusþg;Edk. Flexural Design on Reinforced Concrete Beams 53
  2. 2. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa taragTI1³ PaKryEdksMNUmBr ρ s f 'c ( MPa) f y (MPa) %ρ s 20 235 1.4 400 1.2 28 400 1.4 500 1.2 35 400 1.6 500 1.4 sac;lUteFobEdksuT§ ε edIm,IeGaymuxkat;rgkarTajRtUv)ankMNt;FMCag 0.005 enaH φ = 0.9 . t ) sac;lUteFobEdksuT§ ε enAcenøaH 0.004 → 0.005 . 250 φ = 0.65 + (ε − 0.002)( t t 3 smIkarKNnaersIusþg;m:Um:g;kñúgmanTMrg;dUcxageRkam³ φM n = M u = Ru bd 2 Edl R = φρf (1 − 1.ρff ' ) = φR u 7 y y n c Edl φ = 0.9 sMrab;muxkat;rgkarTaj nig φ < 0.9 sMrab;muxkat;enAkñúgtMbn; transition region dUcenH φM = M = φA f (d − 1.A f f' b ) n u 7 s y s y c dUcKña φM = M = φρf bd (− 1.ρff ' ) n u 7 y 2 y c kñúgkrNIEdleyIgsÁal;m:Um:g;KNnaxageRkA nigsÁal;ersIusþg;sMPar³ eyIgenAsl;GBaØtibIeTotEdl minTan;sÁal; kñúgenaHman TTwgFñwm b kMBs;RbsiT§iPaB d nigPaKryEdk ρ . dMeNaHRsayGaceFVIeTA)anluH RtaEteyIgRtUveFVIkarsnμt;cMeBaHGBaØtiBIr. CaTUeTA ρ RtUv)ansnμt; ¬edayeRbI ρ ¦ nigdUcKña b k¾RtUv)an max snμt;. xageRkamCaviFIsaRsþkñúgkaredaHRsayedaysÁal; M / f ' nig f ³ u c y - RbsinebI ρ RtUv)ansnμt; ¬edayeGayenAcenøaH ρ nig ρ ¦ enaHeyIgGackMNt; 1 2 max 1 2 b ρf y Ru = φρf y (1 − ) 1.7 f 'c Mu ⇒ bd 2 = Ru CaTUeTApleFob d ERbRbYlBI 2 → 3 ¬eKniymyk 2 ¦ b dUcenH eKGackMNt; b nig d dUcKña eyIg)an A = ρbd s - RbsinebI b nig d RtUv)aneGay enaHPaKryEdkEdlRtUvkarkMNt;tamrUbmnþxageRkam³ karKNnaFñwmebtugGarem:rgkarkac;begáag 54
  3. 3. T.Chhay NPIC 0.85 f 'c 4M u ρ= (1 − 1 − ) fy 1.7φf 'c bd 2 ⇒ As = ρbd - RbsinebI b RtUv)ansnμt;bEnßmBIelI ρ enaHeKRtUv KNna R = φρf (1 − 1.ρff ' ) u y 7 y c KNna d = Mu Ru b RbsinebI d = 2 enaH d = b 3 2M u Ru b ⇒ As = ρbd 3> KMlatEdk nigRsTab;karBarEdk k> KMlatEdk sésrEdkRtUv)antMerobedaymanKMlaty:agticbMputesμInwgGgát;p©itEdk b:uEnþminRtUvtUcCag 25mm edIm,IeGayeKGacbgðab;ebtug)any:aggayRsYlkñúgeBlcak;ebtug. KMlatEdksMrab;kartMerobEdkbBaÄrEdl maneRcInCagmYyRsTab; minRtUvmantMéltUcCag 25mm Edr. RbsinebIKMlatEdkmanTMhMtUcenaHl,ayeb tugminGacBT§½CMuvijEdk)anl¥eT. x> RsTab;karBarEdk RsTab;karBarEdk Casac;ebtugEdlenAcenøaHépÞxageRkA nigépÞrbs;Edk. eKcaM)ac;RtUvkarRsTab;kar BarEdkeRBaHvamanplRbeyaCn_bYny:ag³ Flexural Design on Reinforced Concrete Beams 55
  4. 4. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa - edIm,Ietags¥itsésrEdkeTAnwgebtugEdleFVIeGaysMPar³TaMgBIreFVIkarCamYyKña. T§iBlrbs;PaB s¥itGaRs½yeTAnwgkMras;RsTab;karBar. - edIm,IkarBarsésrEdkeTAnwgERcHsIuEdk. - edIm,IkarBarkar)at;bg;ersIusþg;EdkEdlbNþalmkBIkMedA. kMras;RsTab;karBar 20mm GacTb;Tl; nwgePøIgeqH)an 1em:ag. - sMrab;yanþdæan eragcRk cMNtrfynþ RsTab;karBarbEnßmRtUv)aneKdak;BIelIkMralxNÐEfmeTot edIm,IkarBarkarswkercrwlEdlbNþalmkBIcracrN_. kMras;RsTab;karBarEdkGaRs½yeTAnwgmCÄdæanEdleRKOgbgÁúMenaHsßitenA. xageRkamCatarag bgðajBIkMras;RsTab;karBarGb,brma³ mCÄdæan kMras;karBarEdk (mm) ebtugcak;pÞal;nwgdI 75 ebtugcak;pÞal;nwgdI b¤halxül; + Ggát;p©itEdkFMCag 16mm 50 + Ggát;p©itEdktUcCag 16mm 35 ebtugmincak;pÞal;nwgdI b¤minhalxül; - kMralxNÐ CBa¢aMg + Ggát;p©itEdkFMCag 36mm 35 + Ggát;p©itEdktUcCag 36mm 20 - Fñwm ssr 35 - kMralekag + Ggát;p©itEdkFMCag 20mm 20 + Ggát;p©itEdktUcCag 20mm 15 K> TTwgGb,brmarbs;muxkat;ebtug smIkarTUeTAedIm,IkMNt;TTwgGb,brmarbs;muxkat;ebtugGacsresrdUcxageRkam³ b = nD + (n − 1) s + 2(φEdkkg ) + 2(RsTab;karBarEdk ) min Edl n - cMnYnEdkbeNþay D - Ggát;p©itEdkEdlFMCageK s - KMlatEdk karKNnaFñwmebtugGarem:rgkarkac;begáag 56
  5. 5. T.Chhay NPIC X> kMBs;Gb,brmarbs;muxkat;ebtug kMBs;rbs;muxkat;ebtugRtUv)ankMNt;edayGaRs½ynigRsTab;Edk. - EdkmYyRsTab;³ h = d + D + 50mm 1 2 1 - EdkBIrRsTab;³ h = d + D + 60mm 2 2 ]TahrN_1³ kMNt;muxkat;Edk nigmuxkat;ebtugedIm,ITb;Tl;nwgm:Um:g;KNna 490kN .m edayeRbIPaKryEdk Gtibrma ρ sMrab;muxkat;rgkarTaj. smμtikmμ³ f ' = 20MPa nig f = 400MPa . max c y dMeNaHRsay³ eday f ' = 20MPa / f = 400MPa / β = 0.85 nig φ = 0.9 sMrab;muxkat;rgkarTaj c y 1 f 'c 600 ρ b = 0.85β1 ( ) f y 600 + f y 20 600 ρ b = 0.85 2 ( ) = 0.021675 400 600 + 400 fy 0.003 + Es ρ max = ρ b ( ) = 0.625 ρ b = 0.01355 0.008 ρ max f y 0.01355 × 400 ⇒ Ru (max) = φρ max f y (1 − ) = 0.9 × 0.01355 × 400(1 − ) = 4.1MPa 1.7 f 'c 1.7 × 20 Mu 490 × 10 6 ⇒ bd = 2 = = 119.5 ×10 6 mm 3 Ru (max) 4.1 dUcenHsMrab;karsnμt; b / kMNt; d nig A = ρbd s b = 200mm / d = 773mm / A = 20.95cm s 2 b = 250mm / d = 691.5mm / A = 23.42cm s 2 b = 300mm / d = 630mm / A = 25.61cm ¬ 6DB 25 ¦ s 2 b = 400mm / d = 546.5mm / A = 29.62cm s 2 * kareRCIserIskMBs;RbsiT§iPaBGaRs½ynwgktþaxageRkam³ - kMBs;bnÞb;³ TTwgrbs;muxkat;tUcpþl;nUvkMBs;FñwmFM Edlkat;bnßylMhrkMBs;. elIsBIenH FñwmeRCA ceg¥ótkat;bnßyersIusþg;m:Um:g;edaykarxUcRTg;RTayxag lateral deformation. - brimaN nigkarBRgaysésrEdk³ Fñwmceg¥ótRtUvkarsésrEdkeRcInCagmYyRsTab; dUcenHvabegáIt kMBs;Fñwm. - kMras;CBa¢aMg³ RbsinebIbøúksIum:g;t_RtUv)aneRbI TTwg b RtUv)aneRCIserIsesμInwgkMras;CBa¢aMg. sMrab; GKarCBa¢aMgxagRkas;CagCBa¢aMgkñúg. Flexural Design on Reinforced Concrete Beams 57
  6. 6. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa * kareRCIserIsmuxkat;EdkGaRs½ynwgktþaxageRkam³ - kartMerobEdkRKb;RKan;enAkñúgmuxkat; CaTUeTA mYyRsTab; b¤BIrRsTab; nigbMeBjtamlkçxNÐ ACI Code sMrab;KMlatEdkGb,brma. - épÞmuxkat;EdkeRCIserIsRtUvmantMélEk,rbMputépÞsésrEdktMrUvkar dUcenHeyIgeRCIserIsyk b = 300mm / d = 630mm / A = 25.61cm ¬ 6 @ DB25 ¦ RtUvtMerobBIrRsTab; s 2 kMNt;kMBs;Fñwm h = d + D + 60mm = 630 + 25 + 60 = 715mm dUcenHyk h = 750mm ⇒ d = 665mm ]TahrN_2³ edaHRsay]TahrN_1 edayeRbI ρ RbEhl 1% nig b = 35cm dMeNaHRsay³ eday f ' = 20MPa / f = 400MPa / ρ = 0.01355 sMrab;muxkat;rgkarTaj c y max ρf y 0.01 × 400 Ru = φρf y (1 − ) ⇒ Ru = 0.9 × 0.01 × 400(1 − ) = 3.1765MPa 1.7 f 'c 1.7 × 20 M u 490 × 106 ⇒ bd 2 = = = 154257830mm3 Ru 3.1765 sMrab; b = 35cm ⇒ d = 665mm As = 0.01 × 35 × 66.5 = 23.275cm 2 edayeRCIserIs 4DB28 mYyRsTab; A = 24.62cm s 2 epÞógpÞat;TTwg b = nD + (n − 1)s + 95mm edayyk s = D min ⇒ b = 7 D + 95mm = 7 × 28 + 95 = 291mm < 35cm epÞógpÞat; min kMNt;kMBs;Fñwm h = d + D + 50mm = 665 + 28 + 50 = 729mm yk h = 75cm min 2 2 edayEdkEdldak;mantMélFMCagEdkKNna dUcenHeyIgGacbnßykMBs;FñwmBI 75cm → 72cm sMrab;karERbRbYlkMBs;Fñwm A = 23.275( 72 ) = 24.24cm < 24.62cm s 75 2 2 d = 720 − 64 = 656mm RtYtBinitüersIusþg;m:Um:g; ρ= 24.62 35 × 72 = 0.0098 < ρ max ⇒ muxkat;rgkarTaj As f y 24.62 × 400 a= = = 16.55cm 0.85 f 'c b 0.85 × 20 × 35 a 165.5 φM n = φAs f y (d − ) = 0.9 × 2462 × 400 × (656 − ) = 508.083 × 106 N .mm = 508.083kN .m > 490MPa 2 2 RtYtBinitüsac;lUteFobEdksuT§ ε t karKNnaFñwmebtugGarem:rgkarkac;begáag 58
  7. 7. T.Chhay NPIC 0.005 εt = ( ρ ) − 0.003 ρb 20 600 ρ b = 0.85 2 ( ) = 0.021675 400 600 + 400 ρ 0.0098 = = 0.452 ρb 0.021675 εt = 0.005 0.452 − 0.003 = 0.008 > 0.005 epÞógpÞat; mü:ageTot c= a = 16.55 0.85 0.85 = 19.47 c 19.47 dt = 65.6 = 0.3 < 0.375 epÞógpÞat; ]TahrN_3³ kMNt;muxkat;EdksMrab;muxkat;eGayxageRkam b = 25cm nig h = 50cm EdlTb;nwgm:Um:g; KNna M = 200kN .m . smμtikmμ³ f ' = 28MPa nig f = 400MPa . u c y dMeNaHRsay³ edaysnμt;Edk DB25 mYyRsTab; d = 500 − 70 = 430mm = 43cm eday f 'c = 28MPa / / f y = 400 MPa β1 = 0.85 f 'c 600 ρ b = 0.85β1 ( ) f y 600 + f y 28 600 ρb = 0.852 ( ) = 0.030345 400 600 + 400 f 0.003 + y Es ρ max = ρb ( ) = 0.625ρb = 0.019 0.008 ρ= 0.85 f 'c fy (1 − 1 − 4M u 1.7φf 'c bd 2 ) edaysnμt; φ = 0.9 0.85 × 28 4 × 200 × 106 ρ= 400 (1 − 1 − 1.7 × 0.9 × 28 × 250 × 4302 ) = 0.01356 < ρ max muxkat;rgkarTajBitR)akd As = ρbd = 0.01356 × 25 × 43 = 14.58cm2 DB 25 ⇒ As 25 = 4.9cm 2 A ⇒n= s =3 As 25 dUcenH A = 3DB25 = 14.7cm s 2 ]TahrN_4³ kMNt;muxkat;EdkcaM)ac;sMrab; b = 35cm RbsinebIvaRbQmnwgm:Um:g;KNna M u = 425kN .m . smμtikmμ³ f ' = 28MPa nig f = 400MPa . c y Flexural Design on Reinforced Concrete Beams 59
  8. 8. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ sMrab; f ' = 28MPa / f = 400MPa / β c y 1 = 0.85 ρ = 0.030345 / ρ = 0.019 b max sMrab;muxkat;rgkarTaj φ = 0.9 edayeRbI ρ = 0.019 max ρmax f y 0.019 × 400 ⇒ Ru (max) = φρmax f y (1 − ) = 0.9 × 0.019 × 400(1 − ) = 5.75MPa 1.7 f 'c 1.7 × 28 Mu 425 × 106 bd 2 = = = 73.913 × 106 mm3 Ru (max) 5.75 sMrab; b = 35cm ⇒ d = 46cm ⇒ As = 0.019 × 35 × 46 = 30.59cm 2 edayeRbIEdk φ 32 4edIm 4DB32 = 32.154cm 2 EdkRtUv)antMerobmYyCYr ⇒ b = nD + (n − 1)s + 95mm min edayyk D = s ⇒ b = 7 D + 95mm = 7 × 32 + 95 = 319mm < 350mm RtwmRtUv min kMBs;Fñwm h = d + D + 50mm = 460 + 32 + 50 = 526mm yk 53cm 2 2 BiPakSa³ edaysarsésrEdkeRbIR)as; 32.154cm FMCagsésrEdktMrUvkar 30.59cm 2 2 32.154 ⇒ρ= = 0.02 > ρ max 35 × 46 sac;lUteFobEdksuT§ εt = ( 0.005 ) − 0.003 = 0.0046 > 0.004 ⇒ ρ muxkat;sßitkñúgtMbn; transition region ρb 250 ⇒ φ = 0.65 + (ε t − 0.002)( ) = 0.867 < 0.9 3 As f y 32.154 × 400 a= = = 15.44cm 0.85 f 'c b 0.85 × 28 × 35 a 154.4 φM n = φAs f y (d − ) = 0.867 × 3215.4 × 400(460 − ) = 426.86 × 106 N .mm = 426.86kN .m ≈ 425kN .m 2 2 eyIgeXIjfa sMrab;bMErbMrYlmuxkat;Edk 32.154 − 30.59 = 1.546cm RbEhl 5% enaHersIusþg;m:Um:g;man 2 tMélRbEhlKña edaysar φ fycuH. dUcenHkarKNnamuxkat;edayeGaymuxkat;rgkarTaj φ = 0.9 man lkçN³esdækic©. karKNnaFñwmebtugGarem:rgkarkac;begáag 60
  9. 9. T.Chhay NPIC 4> muxkat;ctuekaNEkgCamYyEdkrgkarsgát; RbsinebIm:Um:g;KNnamantMélFMCagersIusþg;m:Um:g;kñúg enaHmuxkat;RtUv)armuxkat;EdkbEnßmsMrab;tMbn; rgkarTaj nigrgkarsgát;. sésrEdkrgkarsgát;pþl;nUvkMlaMgsgát;bEnßmeTAelIkMlaMgsgát;rbs;ebtug. k> edaysnμt;EdkrgkarTajmanmYyRsTab; viFIsaRsþKNnasMrab;muxkat;ctuekaNEkgCamYyEdkrgkarsgát; enAeBleKsÁal; M / f ' / b / d u c nig d ' mandUcxageRkam³ - KNnaPaKryEdk balanced ρ nigPaKryEdkGtibrma ρ edayeRbIsmIkarxageRkam³ b max fy 0.003 + ρ b = 0.85β1 f 'c ( 600 f y 600 + f y ) nig ρ max = ρb ( 0.008 Es ) kMNt;muxkat;EdkGtibrmasMrab;rgkarTaj A s1 = ρ maxbd - KNna R edayeRbI ρ (φ = 0.9) u (max) max ρ max f y Ru (max) = φρmax f y (1 − ) 1.7 f 'c - KNnaersIusþg;m:Um:g;kñúgEdlekItedaysarmuxkat;EdkrgkarTaj M edayeRbI R u1 u (max) M u1 = Ru (max)bd 2 + RbsinebI M u1 enaHRtUvkarEdkrgkarsgát; < Mu + RbsinebI M > M enaHminRtUvkarEdkrgkarsgát;eT. KNnaPaKryEdk ρ tamrUbmnþ u1 u ) nigKNna A = ρbd . 0.85 f 'c 4M ρ= (1 − 1 − u 1.7φf ' bd 2 s fy c - KNna M = M − M CaersIusþg;m:Um:g;EdlekItBIEdkrgkarsgát; u2 u u1 - KNna A BI M = φA f (d − d ' ) nigbnÞab;mkKNna A = A + A s2 u2 s2 y s s1 s2 - KNnakugRtaMgenAkñúgEdkrgkarsgát;dUcxageRkam c − d' + KNna f ' = 600( s )≤ f y c + b¤KNna ε ' BIdüaRkambMErbMrYlrageFob nig f ' = ε ' E . RbsinebI ε ' = ε enaHEdkrgkar s s s s s y sgát;yar ehIy f ' = f . s y + KNna A' BI M = φA f ' (d − d ' ) .RbsinebI f ' = f enaH A' = A . EtebI f ' < f s u2 s2 s s y s s2 s y enaH A' > A ehIy A' = A ( ff ' ) . s s2 s s2 y s - eRCIserIsmuxkat;EdksMrab; A nig A' edIm,IeGaysmlμmnwgTTwg b . CaTUeTA Edk A eRcIntM s s s erobCaBIrRsTab; É A' eRcIntMerobCamYyRsTab;. s Flexural Design on Reinforced Concrete Beams 61
  10. 10. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa - KNna h = d + 65mm sMrab;EdkrgkarTajmYyRsTab; nig h = d + 90mm sMrab;Edkrgkar TajBIrRsTab;. RtYtBinitüfa [ ρ − ρ ' ( ff ' )] < ρ edayeRbI d fμI b¤RtYtBinitü s max y As (max) = bd [ ρ max + ρ ' ( f 's fy )] ≥ As eday ρ = bd nig ρ '= bd' . karRtYtBinitüenHmincaM)ac;eT A s A s RbsinebI ρ RtUv)aneRbIenAkñúgmuxkat;eKal. max - RbsinebIcaM)ac; eKRtUvKNnaersIusþg;m:Um:g;énmuxkat;cugeRkay φM ehIyeRbobeFobCamYy M n u edayeGay φM ≥ M . n u - RtYtBinitüsac;lUtEdksuT§ ε = ( d c− c )0.003 ≥ 0.005 t t x> edaysnμt;EdkrgkarTajmanBIrRsTab; kúñgkrNIEdkrgkarTajmanBIrRsTab; eKGacsnμt;fa d = h − 90mm nig d = h − 65mm = d + 25mm . eKmanviFIBIry:agkñúgkarkMNt;muxkat;sésrEdk³ t - vIFITI1³ edaysnμt; sac;lUteFobEdksuT§enAnIv:UTIRbCMuTMgn;EdkTajesμI 0.005 b¤ ε = 0.005 s enAnI v:U d . kñúgkrNIenH sac;lUteFobEdksuT§ sMrab;sésrEdkRsTab;eRkambMputmantMélFMCag 0.005 . d −c ε =( t )0.003 > 0.005 . dUcenHeyIgedaHRsaydUckrNIxagelI EdkrgkarTajmanmYyRs t c Tab;. - vIFITI2³ edaysnμt; ε = 0.005 enAnIv:UEdkRsTab;eRkameK d . kñúgkrNIenH sac;lUtEdkeFob t t enAnIv:UTIRbCMuTMgn;EdkTajmantMéltUcCag 0.005 ³ ε = ( d c− c )0.003 < 0.005 EdlenAEtGac s t TTYlyk)an. dMeNaHRsaytamviFITI2enH segçbdUcxageRkam³ + KNna d = h − 65mm / c = ( )d nig a = β c 3 t t 1 8 + KNnakMlaMgsgát;enAkñúgebtug C = 0.85 f ' ab = T = A f 1 c 1 s1 y kMNt; A . KNna M = φA f (d − a ) . ρ = bd / φ = 0.9 s1 2 u1 A s1 y 1 s1 + KNna M = M − M edaysnμt; d ' = 65mm u2 u u1 + KNna A BI M = φA f (d − d ' ) eday f ' = f . muxkat;EdkrgkarTajsrub s2 u2 s2 y s y As = As1 + As 2 + KNnakugRtaMgenAkñúgEdkrgkarsgát;dUcxageRkam c − d' × KNna f ' = 600(s )≤ f y c karKNnaFñwmebtugGarem:rgkarkac;begáag 62
  11. 11. T.Chhay NPIC b¤KNna ε ' BIdüaRkambMErbMrYlrageFob nig f ' = ε ' E . RbsinebI ε ' = ε enaHEdkrg × s s s s s y karsgát;yar ehIy f ' = f . s y × KNna A' BI M = φA f ' (d − d ' ) . RbsinebI f ' = f enaH A' = A . EtebI s u2 s2 s s y s s2 f ' < f enaH A' > A ehIy A' = A ( ) . f y s y s s2 s s2 f' s ]TahrN_5³ eKmanFñwmmYymanmuxkat; b = 25cm nigkMBs; h = 55cm EdlRtUvRTnUvm:Um:g;Bt;KNna M = 300kN .m . KNnasésrEdktMrUvkar. smμtikmμ f ' = 20MPa nig f = 345MPa . u c y dMeNaHRsay³ - kMNt;ersIusþg;m:Um:g;EdlekItBIEdkrgkarTajCaeKalsMrab;lkçxNÐmuxkat;rgkarTaj sMrab; f ' = 20MPa nig f = 345MPa / β = 0.85 c y 1 f 'c 600 20 600 ρb = 0.85β1 ( ) = 0.852 ( ) = 0.0266 f y 600 + f y 345 600 + 345 fy 345 0.003 + 0.003 + Es ρ max = ρb ( ) = 0.0266( 200000 ) = 0.0157 0.008 0.008 ρ max f y 0.0157 × 345 Ru (max) = φρmax f y (1 − ) = 0.9 × 0.0157 × 345(1 − ) = 4.1MPa 1.7 f 'c 1.7 × 20 edaysnμt;EdkrgkarTajmanBIrRsTab; d = h − 90mm = 550 − 90 = 460mm ⇒ φM n = Ru (max)bd 2 = 4.1 × 250 × 4602 = 216.89 × 106 N .mm = 216.89kN .m < M u = 300 MPa dUcenHmuxkat;enHRtuvkarEdkrgkarsgát;edIm,ITb;Tl;nwgm:Um:g;EdlenAsl;. - KNna A / M nig M s1 u1 u2 As1 = ρ maxbd = 0.0157 × 25 × 46 = 18.06cm 2 M u1 = φM n = 216.89kN .m ⇒ M u 2 = M u − M u1 = 300 − 216.89 = 83.11cm 2 - KNna A nig A' Edl)anBI M edaysnμt; d '= 6cm s2 s u2 M u 2 = φAs 2 f y (d − d ' ) Mu2 83.11× 106 As 2 = = = 669mm2 = 6.69cm2 φf y (d − d ' ) 0.9 × 345 × (460 − 60) EdkrgkarTajsrub A = A + A = 18.06 + 6.69 = 24.75cm s s1 s2 2 Edkrgkarsgát; A' = A = 6.69cm eFVIkardl;yar s s2 2 - RtYtBinitüEdksgát;eFVIkardl;yar Flexural Design on Reinforced Concrete Beams 63
  12. 12. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa fy 345 εy = = = 0.001725 200000 200000 1806 × 345 eday a= As1 f y = 0.85 f 'c b 0.85 × 20 × 250 = 146.6mm a 146.6 c= = = 172.47mm β1 0.85 172.47 − 60 ε 's = ( 172.47 ) × 0.003 = 0.00196 > ε y eFVIkardl;yar - RtYtBinitü ε t edayeyIgeRbI ρ nig R mkeRbIsMrab;edaHRsay edayeGaysac;lUteFobEdksuT§enAnIv:UTIRbCMuTMgn;Edk max u ε = 0.005 . dUcenHeyIgRtUvkMNt; ε sMrab;EdkenARsTab;eRkameKbMput. s t d t = 550 − 60 = 490mm d −c 490 − 172.47 εt = ( t c )0.003 = ( 172.47 )0.003 = 0.0055 > 0.005 RtwmRtUv dUcenH EdktMrUvkarsMrab;karTaj A = 24.75cm eRbI 5DB25 s 2 sMrab;karsgát; A' = 6.69cm eRbI 2DB22 s 2 ]TahrN_6³ eKmanFñwmmYymanmuxkat; b = 30cm nigkMBs; h = 50cm EdlRtUvRTnUvm:Um:g;Bt;KNna M = 400kN .m . KNnasésrEdktMrUvkar. smμtikmμ f ' = 28MPa nig f = 400MPa . u c y dMeNaHRsay³ manBIrviFIsaRsþ vIFITI1³ - kMNt;ersIusþg;m:Um:g;Gtibrmaénmuxkat;EdlmanEtEdkrgkarTajCaeKal sMrab; f ' = 28MPa / f = 400MPa / β = 0.85 c y 1 ρ = 0.030345 / ρ = 0.019 b max sMrab;muxkat;rgkarTaj φ = 0.9 karKNnaFñwmebtugGarem:rgkarkac;begáag 64
  13. 13. T.Chhay NPIC ρmax f y 0.019 × 400 ⇒ Ru (max) = φρmax f y (1 − ) = 0.9 × 0.019 × 400(1 − ) = 5.75MPa 1.7 f 'c 1.7 × 28 edaysnμt; d = h − 90 = 500 − 90 = 410mm EdkmanBIrRsTab; ⇒ M u1 = Ru bd 2 = 5.75 × 300 × 4102 = 290 × 106 N .mm < 290MPa dUcenHRtUvkarEdksgát; - KNna A , M , A nig A s1 u2 s2 s As1 = ρ maxbd = 0.019 × 30 × 41 = 23.37cm 2 M u 2 = M u − M u1 = 400 − 290 = 110kN .m M u 2 = φAs 2 f y (d − d ' ) edaysnμt; d '= 60mm 110 × 106 ⇒ As 2 = = 873mm 2 = 8.73cm 2 0.9 × 400 × (410 − 60) muxkat;EdkTajsrub A = A + A = 23.37 + 8.73 = 32.1cm s s1 s2 2 eRbIEdk 5DB30 - RtYtBinitüEdkrgkarsgát;eFVIkardl;yarb¤Gt; f 'c d ' 600 ρ − ρ ' ≥ K = 0.85β1 ( )( ) f y d 600 + f y 28 60 600 K = 0.852 ( )( ) = 0.00444 400 410 600 + 400 A 23.37 ρ − ρ ' = s1 = = 0.019 < K bd 30 × 41 Edkrgkarsgát;eFVIkarmindl;yareT dUcenH ⇒ f 's < f y - KNna f ' ³ s c − d' f 's = 600( ) ≤ fy c kMNt; c BI A s1 = 23.37cm2 As1 f y 2337 × 400 a= = = 130.92mm 0.85 f 'c b 0.85 × 28 × 300 a 130.92 c= = = 154.02mm β1 0.85 154.02 − 60 f 's = 600( ) = 366.3MPa ≤ f y 154.02 - KNna A' BI M s u2 = φA's f 's (d − d ' ) 110 × 106 A's = 0.9 × 366.3 × (410 − 60) = 953mm 2 = 9.53cm 2 eRbIEdk 2DB25 eyIgeXIjva eKarBtamlkçxNÐ Flexural Design on Reinforced Concrete Beams 65
  14. 14. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa f 's (ρ − ρ ' ) ≤ ρ max fy - RtYtBinitü sac;lUteFobEdksuT§ ε enAnIv:UEdkRsTab;xageRkam ¬eRBaHeyIgsnμt;fa sac;lUteFobEdksuT§ t ε enAnIv:UTIRbCMuTMgn;Edk¦ s d t = 500 − 60 = 440mm d −c 440 − 154.02 εt = ( t c )0.003 = ( 154.02 )0.003 = 0.0056 > 0.005 RtwmRtUv vIFITI2³ edayeRbIEdkrgkarTajBIrRsTab; nigsac;lUteFobEdksuT§enARsTab;eRkambMput ε t = 0.005 - KNna d = 500 − 60 = 440mm BIdüaRkamsac;lUteFob t c 0.003 0.003 = = = 0.375 dt 0.003 + ε t 0.008 ⇒ c = 0.375d t = 165mm ⇒ a = 0.85c = 140.25mm - kMlaMgsgát;kñúgebtug C1 = 0.85 f 'c ab = 0.85 × 28 × 140.25 × 300 = 1001385 N = 1001.385kN ¬sMrab;muxkat;manEdkrgkarTajCaeKal¦ eday C = T ⇒ A = T = 1001385 = 2503mm 1 1 f s1 400 1 2 = 25.03cm 2 y d = 500 − 90 = 410mm a 140.25 M u1 = φAs1 f y (d − ) = 0.9 × 2503 × 400 × (410 − ) = 306.25 × 106 N .mm = 306.25kN .m 2 2 A 25.03 ρ1 = s1 = = 0.0203 bd 30 × 41 - eday M u > M u1 dUcenHmuxkat;RtUvkarEdkrgkarsgát; karKNnaFñwmebtugGarem:rgkarkac;begáag 66
  15. 15. T.Chhay NPIC M u 2 = 400 − 306.25 = 93.75kN .m M u2 93.75 × 106 ⇒ As 2 = = = 744mm 2 = 7.44cm 2 0.9 f y (d − d ' ) 0.9 × 400 × (410 − 60) muxkat;EdkrgkarTajsrub A = A + A = 25.03 + 7.44 = 32.47cm eRbIEdk 5DB30 s s1 s2 2 - RtYtBinitü Edkrgkarsgát;eFVIkardl;yarb¤Gt; 28 60 600 K = 0.852 ( )( ) = 0.00444 400 410 600 + 400 ( ρ − ρ ' ) = ρ1 < K ⇒ Edkrgkarsgát;eFVIkarmindl;yareT dUcenH f 's < f y c − d' 165 − 60 f 's = 600( ) = 600( ) = 381.82MPa c 165 KNna A' BI M s u2 93.75 × 106 ⇒ A's = M u2 = 0.9 f 's (d − d ' ) 0.9 × 381.82 × (410 − 60) = 779mm 2 = 7.79cm 2 eRbIEdk 2DB25 - RtYtBinitüersIusþg;m:Um:g;kñúg As = 5 DB30 = 35.325cm 2 A's = 2 DB 25 = 9.81cm 2 As1 = As − A's = 25.515cm 2 a φM n = φ[ As1 f y (d − ) + A's f 's (d − d ' )] 2 140.25 φM n = 0.9[2551.5 × 400 × (410 − ) + 981× 381.82 × (410 − 60)] = 430.18 × 106 N .mm 2 d −c 410 − 165 eyIgeXIjfa εs = ( c )0.003 = ( 165 )0.003 = 0.0045 5> KNnamuxkat;GkSret T kñúgkarKNnamuxkat;GkSr T edaysÁal;m:Um:g;KNna M / kMras;søab t nigTTwgsøab b . kMras;rbs; u RTnug b ERbRbYlBI 20cm → 50cm . GBaØtiBIreTotEdlRtUvkarKNnaKW kMBs;RbsiT§iPaB d nigmuxkat;Edk w A . eKmanBIkrNIEdlCYbRbTH³ s - enAeBleKsÁal; d + RtYtBinitüfa muxkat;eFVIkarCaragctuekaNEkg b¤GkSret T edaysnμt; a = t KNnaersIusþg;m:Um:g;sMrab;søabTaMgmUl t φM nf = φ (0.85 f 'c )bt (d − ) 2 RbsinebI M > φM enaH a > t KNnaCaragGkSret T. RbsinebI M u nf u < φM nf enaH a < t KNna CaragctuekaNEkg. Flexural Design on Reinforced Concrete Beams 67
  16. 16. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa + RbsinebI a < t enaHKNna ρ = 0.85 f ' (1 − f c 1− 4M u 1.7φf 'c bd 2 ) / KNna A = ρbd . s y epÞógpÞat; ρ ≥ ρ . w min + RbsinebI a > t enaHKNna A sMrab;Epñksøabsgxag ¬>>>¦ sf Asf = 0.85 f 'c (b − bw )t / f y t M u 2 = φAsf f y (d − ) 2 m:Um:g;EdlTb;edayRTnugKW M u1 = M u − M u 2 KNna ρ edayeRbI M / b nig d 1 u1 w 0.85 f 'c 4 M u1 ρ1 = (1 − 1 − ) fy 1.7φf 'c bwd 2 nigKNna A = ρ b d s1 1 w muxkat;Edksrub A = A + A s s1 sf bnÞab;mkRtYtBinitü A ≤ A dUcKña RtUvRtYtBinitü ρ s s max w = A bw d ≥ ρ min + RbsinebI a = t enaH A = φ 0.85f f ' bt s c y - enAeBleKminsÁal; d nig A karKNnaRtUveFVItamviFIsaRsþxageRkam s + snμt; a = t nigKNnabrimaNEdksrub A EdlRtUvkarsMrab;Tb;nwgkMlaMgsgát;kñúgsøab sft TaMgmUl bt 0.85 f 'c bt Asft = fy + kMNt; d BI A nig a = t tamrUbmnþxageRkam³ sft t M u = φAsft f y (d − ) 2 RbsinebI eKyktam d KNnarkeXIjenaH A = A nig h = d + 65mm sMrab; × s sft EdkmanmYyRsTab; b¤ h = d + 90mm sMrab;EdkmanBIrRsTab;. × RbsinebI eKyk d fμIFMCag d KNnaenaHmuxkat;eFVIkarCaragctuekaNEkg. 1 ehIy ρ = 0.85 f ' (1 − 1 − 1.7φfM bd ) / KNna A = ρbd . f 4c ' u 2 s y c karKNnaFñwmebtugGarem:rgkarkac;begáag 68
  17. 17. T.Chhay NPIC RbsinebI eKyk d fμItUcCag d KNnaenaHmuxkat;eFVIkarCaragGkSr T. ehIy × 21 muxkat;EdkcugeRkay A FMCag A . kñúgkrNIenH eKRtUveFVIdUckrNIxagelI ¬>>>¦ edIm,IkM s sft Nt;muxkat; A . s ]TahrN_7³ eKmanFñwmmYymanGkSr T EdlmanRTnug b = 25cm RbEvgsøab b = 100cm kMras;søab w t = 10cm nigkMBs;RbsiT§iPaB d = 37cm . kMNt;muxkat;EdkcaM)ac; edIm,ITb;nwgm:Um:g; M = 375kN .m . u smμtikmμ f ' = 20MPa nig f = 400MPa . c y dMeNaHRsay³ - RtYtBinitüGkS½NWt edaysnμt; a = t = 10cm t 100 φM n = φ 0.85 f 'c bt (d − ) = 0.9 × 0.85 × 20 × 1000 × 100(370 − ) = 489.6 × 106 N .mm 2 2 eday φM > M ⇒ muxkat;manlkçN³CactuekaNEkg n u - kMNt;muxkat;EdkrgkarTaj 0.85 f 'c 4M u 0.85 × 20 4 × 375 × 106 ρ= (1 − 1 − )= (1 − 1 − ) = 0.00845 fy 1.7φf 'c bd 2 400 1.7 × 0.9 × 20 × 1000 × 3702 As = ρbd = 0.00845 × 100 × 37 = 31.265cm 2 edayeRbI 5DB30 = 35.325cm 2 - RtYtBinitü ρ = bAd = 25.× 37 = 0.0382 > ρ w 35 325 s min = 1.4 400 = 0.0035 w 0.85 f 'c As max = [(be − bw )t + 0.375β1bw d ] = 44.41cm 2 > As fy dUcKña a = 0.85 ff ' b = 035.325 ××400 = 8.31cm A s y .85 × 20 100 c 37 − 9.78 c= β1 a = 8.31 0.85 = 9.78cm nig ε s = 0.003( 9.78 ) = 0.00835 > 0.005 ]TahrN_8³ RbBn§½kMralxNÐdUcbgðajkñúgrUb EdlpÁúMeLIgedaykMralxNÐEdlmankMras; t = 8cm EdlRT edayFñwmRbEvg L = 430cm EdlmanKMlatBIKña l = 300cm KitBIGkS½mkGkS½. FñwmmanRTnug b = 35cm w Flexural Design on Reinforced Concrete Beams 69
  18. 18. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa nigkMBs; RbsiT§PaB d = 47cm . kMNt;muxkat;EdkcaM)ac;edIm,ITb;nwgm:Um:g;KNna M u = 575kN .m . smμtikmμ f ' = 20MPa nig f = 400MPa . c y dMeNaHRsay³ - kMNt;RbEvgsøab ⎧16t + bw ⎧16 × 80 + 350 = 1630mm ⎪ L ⎪ 4300 be = min ⎨ = min ⎨ = 1075mm ⎪ 4 ⎪ 4 ⎩ l ⎩ 3000mm dUcenH b = 1075mm e - RtYtBinitüTItaMgGkS½NWt edaysnμt; a = t t φM n = φ 0.85 f 'c bt (d − ) 2 80 φM n = 0.9 × 0.85 × 20 × 1075 × 80(470 − ) = 565.794 N .mm = 565.794kN .m < 575kN .m 2 eday φM < M ⇒ muxkat;manlkçN³CaragGkSr T dUcenH a > t . n u - kMNt;muxkat;EdksrubsMrab;Tb;Tl;CamYykMlaMgsgát;kñúgsøab 0.85 f 'c (b − bw )t 0.85 × 20 × (1075 − 350)80 Asf = = = 2456mm 2 fy 400 t 80 ⇒ M u 2 = φAsf f y (d − ) = 0.9 × 2456 × 400(470 − ) = 380.188 × 106 N .mm 2 2 0.85 f 'c 4 M u1 ⇒ ρ1 = (1 − 1 − ) fy 1.7φf 'c bwd 2 0.85 × 20 4 × 194.812 × 106 ρ1 = (1 − 1 − = 0.0077 400 1.7 × 0.9 × 20 × 350 × 4702 enaH A = ρ b d = 0.0077 × 35 × 47 = 12.67cm s1 1 w 2 ⇒ muxkat;EdkTajsrub A = 12.67 + 24.56 = 37.23cm s 2 eRbI 6DB30 BIrRsTab; - kMBs;srubrbs;muxkat; h = 470 + 90 = 560mm 0.85 f 'c As max = [(be − bw )t + 0.375β1bwd ] = 50.87cm 2 > As fy karKNnaFñwmebtugGarem:rgkarkac;begáag 70
  19. 19. T.Chhay NPIC 42.39 × 400 - RtYtBinitü ε / t a= As f y 0.85 f 'c b = 0.85 × 20 × 107.5 = 9.278cm a 9.278 c= = = 10.91cm β1 0.85 dt = 56 − 6 = 50cm 50 − 10.91 ⇒ ε t = 0.003( ) = 0.0107 > 0.005 10.91 ]TahrN_9³ sMrab;RbBn§½kMralxNÐmYy EdlmanTTwgsøabRbEvg b = 122cm TTwgRTnug b = 40 ehIykM e w ralxNÐmankMras; t = 10cm . KNnamuxkat;GkSr T edIm,IrgnUvm:Um:g;KNna M = 1100kN .m . smμtikmμ u f ' = 20MPa nig f = 400MPa . c y dMeNaHRsay³ - edaysarminsÁal;kMBs;RbsiT§PaB eyIgeGay a = t kMNt; muxkat;Edk A sMrab;søabTaMgmUl sft 0.85 f 'c bt 0.85 × 20 × 122 × 10 Asft = = = 51.85cm 2 fy 400 eGay M t = φAsft f y (d − ) u 2 d KNna Mu t 1100 × 106 100 ⇒d = + = + = 639.3mm φAsft f y 2 0.9 × 5185 × 400 2 eyIgeXIjfa RbsinebI d = 63.93cm enaH A = A s sft - RbsinebI d > 63.93cm / eyIgsnμt;yk d = 67cm enaH a < t muxkat;manlkçN³CaragctuekaNEkg PaKryEdk RtUv)anKNnatamrUbmnþxageRkam³ 0.85 f 'c 4M u ρ= (1 − 1 − ) fy 1.7φf 'c bw d 2 0.85 × 20 4 × 1100 × 106 ⇒ρ= (1 − 1 − = 0.006 400 1.7 × 0.9 × 20 × 1220 × 6702 enaHmuxkat;Edk A = ρbd = 0.006 × 122 × 67 = 49.04cm s 2 - RbsinebI d < 63.93cm / eyIgsnμt;yk d = 60cm enaH a > t muxkat;manlkçN³CaragGkSret T . 0.85 f 'c t (b − bw ) 0.85 × 20 × 10 × (122 − 40) Asf = = = 34.85cm 2 fy 400 Flexural Design on Reinforced Concrete Beams 71
  20. 20. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa t 100 M u 2 = φAsf f y (d − ) = 0.9 × 3485 × 400(600 − ) = 690 × 106 N .mm 2 2 M u1 = 1100 − 690 = 410kN .m sMrab;muxkat;EdkrgkMlaMgTajeKal b w / = 40cm d = 60cm nig M u1 = 410kN .m M u1 410 Ru = = = 2847.2 kN m 2 = 2.85MPa bw d 2 0.4 × 0.602 0.85 f 'c 4 Ru 0.85 × 20 4 × 2.85 ⇒ ρ1 = (1 − 1 − )= (1 − 1 − ) = 0.0088 fy 1.7φf 'c 400 1.7 × 0.9 × 20 As1 = ρ1bwd = 0.0088 × 40 × 60 = 21.12cm2 ⇒ As = As1 + Asf = 34.85 + 21.12 = 55.97cm2 eRbIEdk 7 DB32 = 56.27cm 2 - RtYtBinitü ε t As1 f y 21.12 × 400 a= = = 12.42cm 0.85 f 'c bw 0.85 × 20 × 40 a 12.42 c= = = 14.6cm 0.85 0.85 dt = 63cm ⎛d −c⎞ ⇒ ε t = 0.003⎜ t ⎝ c ⎠ ⎟ = 0.0099 > 0.005 muxkat;rgkarTaj - KNnamuxkat;EdksrubGtibrma f 'c 20 As max = 0.85 [(be − bw )t + 0.375β1bw d ] = 0.85 [(1220 − 400)100 + 0.375 × 0.85 × 400 × 600)] fy 400 As max = 6736mm 2 = 67.36cm 2 > 56.27cm 2 RtwmRtUv karKNnaFñwmebtugGarem:rgkarkac;begáag 72

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