Xi members in compression and bending

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Xi members in compression and bending

  1. 1. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa XI. eRKOgbgÁúMrgkarsgát; nigrgkarBt; 1> esckþIepþIm eRKOgbgÁúMbBaÄrCaEpñkmYyrbs;eRKagsMNg; EdlrgkMlaMgsgát; nigm:Um:g;. kMlaMgTaMgenH )anBIkMlaMg xageRkAdUcCa bnÞúkefr bnÞúkGefr nigbnÞúkxül;. kMlaMgRtUv)ankMNt;eday karKNnaedayéd b¤edaykMuBüÚT½r EdlQrelIeKalkarN¾sþaTic nigviPaKeRKOgbgÁMú (structural analysis). Ca]TahrN¾ sMrab;rUb (1) bgðajfa eRKagQrelITMr hinged BIr EdlrgbnÞúkemKuNBRgayesμI enAelIGgát; BC. düaRkamm:Um:g;Bt;RtUv)anKUrenA EpñkxagTaj. kMNat;ssr AB nig CD rgnUvkMlaMgsgát; nigm:Um:g;Bt;. pleFobrvagm:Um:g;Bt; nigkMlaMg sgát; RtUv)aneGayeQμaHfa cMgaycakp©it e Edl e = M . e CacMgayBITIRbCMuTMgn;)aøsÞic(plastic centroid) n P n énmuxkat;eTA cMnucénbnÞúkmanGMeBI. TIRbCMuTMgn;)aøsÞic (plastic centroid) RtUv)anTTYledaykarkMNt;TItaMgkM laMgpÁÜbbegáItedaysésEdk nigebtug edaysnμt;kugRtaMgsgát;sMrab;Edk f nigkugRtaMgsgát;sMrab;ebtug y 0.85 f ' . sMrab;muxkat;sIuemRTI TIRbCMuTMgn;)aøsÞic (plastic centroid) RtYtsIuKñaCamYyTIRbCMuTMgn;rbs;muxkat;. c sMrab;muxkat;minsIuemRTI TIRbCMuTMgn;)aøsÞic (plastic centroid) RtUv)ankMNt;edayeRbIm:Um:g; eFobnwgGkS½ arbitrary axis. rUbTI1³ eRKagTMr pin BIrCamYynwgdüaRkamm:Um:g; ]TahrN_1³ kMNt;TIRbCMuTMgn;)aøsÞic (plastic centroid) énmuxkat;dUcbgðajkñúgrUbTI2. smμtikmμ³ f 'c = 28MPa nig f = 400MPa . y dMeNaHRsay³ !> kugRtaMgsgát;sMrab;ebtugRtUv)ankMNt;yk 0.85 f ' rUbTI2³ TIRbCuMTMgn;)øasÞic (P.C)énmuxkat; c F = kMlaMgkñúgrbs;ebtug = 0.85 f ' A c c g Members in Compression and Bending 196
  2. 2. T.Chhay NPIC = (0.85 × 28) × 350 × 500 = 4165kN Fc sßitenAelITIRbCMuTMgn;énmuxkat;ebtug ¬enAcMgay 250mm BIGkS½ A − A ¦ @> kMlaMgenAkñúgsésEdk π × 282 Fs1 = As1 f y = 4 × × 400 = 985.2kN 4 π × 282 Fs 2 = As 2 f y = 2 × × 400 = 492.6kN 4 #> kMNt;m:Um:g;eFob A − A (4165 × 250) + (985.2 × 65) + (492.6 × 435) x= = 233.85mm 4165 + 985.2 + 492.6 dUcenH TIRbCMuTMgn;)aøsÞic (plastic centroid) RtUvsßitenAelIcMgay 233.85mm BIGkS½ A − A $> RbsinebI A = A ¬muxkat;sIuemRTI¦ dUcenH x = 250mm BIGkS½ A − A . s1 s2 2> karsnμt;sMrab;KNnassr GaRs½ytam ACI Code EdnkMNt;sMrab;karKNnassrkMNt;dUcxageRkam³ !> kMhUcRTg;RTayeFob strain enAkñúgebtug nigEdk RtUvsmamaRteTAnwgcMgayBIGkS½NWt. @> RtUvEtbMeBjlkçxNÐ smIkarlMnwgénkMlaMg nigPaBRtUvKñaénkMhUcRTg;RTayeFob strain compatibility. #> kMhUcRTg;RTayeFobrbs;ebtugrgkarsgát;EdleRbIR)as;GtibrmaKW 0.003 . $> ersIusþg;rbs;ebtugrgkarTajGacRtUvecal. %> kugRtaMgenAkñúgEdkKW f = εE ≤ f . s s y ^> bøúkkugRtaMgGackMNt;manragctuekaNCamYykugRtaMg 0.85 f ' BRgayBIRbEvg a = β c . Edl c 1 c CacMgayBIGkS½NWt nig ⎧ 0.85 f 'c ≤ 28MPa ⎪ f ' −28 β1 = ⎨0.85 − 0.05( c 7 ) sMrab;ebtugEdlmanersIusþg; 28MPa < f 'c ≤ 56 MPa ⎪ f 'c > 56MPa ⎩ 0.65 3> düaRkamGnþrkmμbnÞúk-m:Um:g; Load-moment interaction diagram enAeBlEdlbnÞúktamGkS½RtUv)anGnuvtþmkelIssrxøI krNIdUcxageRkamGacekIteLIg edayGaRs½y eTAnwgTItaMgGnuvtþbnÞúkedayeFobeTAnwg TIRbCMuTMgn;)aøsÞic (plastic centroid). rUb eRKOgbgÁúMrgkarsgát; nigrgkarBt; 197
  3. 3. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa kMlaMgsgát;tamGkS½ P CakMlaMgsgát;tamGkS½EdlmantMélFMGnuvtþenAelITIRbCMuTMgn;)aøsÞic (plastic o centroid) e = 0 nig M = 0 . kar)ak;rbs;ssr ekIteLIgedayebtugEbk nigEdkeFVIkardl; yielding. n vaRtUv)ansMEdgeday P enAelIExSekag. o rUbTI3 a³ düaRkamGnþrkmμbnÞúk-m:Um:g; !> Maximum nominal axial load P : CakrNIEdlkMlaMgtamGkS½GnuvtþeTAelImuxkat;CamYy cM n max gaycakp©it eccentricity Gb,rma. tam ACI Code, P = 0.80P sMrab;ssrEdkkgdac;² tie n max o column nig P = 0.85P sMrab;ssrEdlmanEdkkgdUcrWus½r spirally reinforced column . kar)ak; n max o ekIteLIgedayebtugEbk nigEdkeFVIkardl; yielding. @> Compression failure: CakrNIEdlbnÞúktamGkS½FMGnuvtþenAcMgaycakp©ittUc. bnÞúktamGkS½kñúg krNIenHmantMélERbRbYlBI tMélGtibrma P = P eTAtMélGb,rma P = P (balanced load). s n n max n b r)ak;edayebtugEbkenAEpñkrgkarsgát;CamYYynwgbMErbMrYlrageFob strain = 0.003 ÉcMEnkkugRtaMgkñúgEdk ¬EpñkrgkarTaj¦ KWtUcCag yield strength f < f . kñúgkrNIenH P > P nig e < e . s y n e n #> Balanced condition P : lkçxNÐenHekItmaneLIgenAeBlEdl bMErbMrYlrageFobrgkarsgát; b compression strain enAkñúgebtugmantMélesμI 0.003 ehIybMErbMrYlrageFobrbs;EdkrgkarTajmantMél Members in Compression and Bending 198
  4. 4. T.Chhay NPIC εy = fy Es . kar)ak;rbs;ebtugekIteLIgtMNalKñanwgEdk yield. m:Um:g;EdlekItedaysarbnÞúkenHRtUv)an eKehAfa balanced moment M cMgaycakp©itRtUv)aneKehAfa balanced eccentricity e = M . b P b b b $> Tension failure: CakrNIekItmanenAxN³Edl bnÞúktamGkS½tUc nigcMNakp©itFM ehIyEdlmanm:U m:g;FM. muneBl)ak; kugRtaMgTajekItmanenAelIEpñkd¾FMénmuxkat; bNþaleGayEdkrgkarTaj yield muneBl rUbTI3 b³ muxkat;ssrEdlbgðajBITItaMgbnÞk P sMrab;lkçxNÐbnÞúkepSg² ú n eRKOgbgÁúMrgkarsgát; nigrgkarBt; 199
  5. 5. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa ebtugEbk. enAeBl)ak; bMErbMrYlrageFobrbs;EdkrgkarTajmantMélFMCagbMErbMrYlrageFob yield ε y ehIybMErbMrYlrageFobenAkñúgebtugesIμ 0.003 . krNIenHekItmanBI Balanced condition eTAdl; pure flexure P < P nig e > e . n b e %> Pure flexure: muxkat;kñúgkrNIenHrgm:Um:g;Bt; M Edl P = 0 . kar)ak;dUcKñanwg kar)ak;rbs; n n FñwmrgkarBt;. cMNakp©itRtUv)ansnμt;fa Gnñn. 4> karpþl;nUvsuvtßiPaB Safety provisions !> emKuNbnÞúksMrab; bnÞúk gravity nigbnÞúkxül; U = 1 .4 D U = 1 .2 D + 1 .6 L U = 1.2 D + 1.6 L + 0.8W U = 1.2 D + 1.0 L + 1.6W U = 0.9 D + 1.6W bnSMbnÞúkemKuNEdlmantMélsFMCageKRtUv)anykmkeRbIR)as;sMrab;karKNna. @> emKuNkat;bnßyersIusþg; φ eRbIsMrab;KNnassrGaRs½yeTAnwgkrNIxageRkam³ k> enAeBl P = φP ≥ 0.1 f ' A eBlenaH φ = 0.65 sMrab;ssrEdkkgdac;² tie column u n c g ehIy φ = 0.7 sMrab;muxkat;ssr EdlmanEdkkgdUcrWus½r spirally reinforced column . krNIenHssrRtUv )anrMBwg fa)ak;edaykarrsgát;. rUbTI4³ tMélemKuN φ Members in Compression and Bending 200
  6. 6. T.Chhay NPIC x> muxkat;EdlbMErbMrYlrageFobrgkarTajsuT§ net tensile strain ε sMrab;ersIusþg;Fmμta t nominal strength enAkñúgEdkrgkarTajeRkAeKbMput KWsißtenAcenøaH 0.005 nig 0.002 (transition region) φ ERbRbYlCalkçN³bnÞat;cenøaH 0.9 nig 0.67 b¤ ¬ 0.7 ¦. sMrab;muxkat;EdlmanEdkkgdUcrWus½r spiral section ⎡ 1 5⎤ ) b¤ φ = 0.7 + 0.2 ⎢ 200 φ = 0.7 + (ε − 0.002)( t − ⎥ (-1) 3 c/d 3 ⎣ t ⎦ sMrab;muxkat;epSgeTot φ = 0.65 + (ε t − 0.002)( 250 ) b¤ φ = 0.65 + 0.25⎡ c /1d ⎢ 5⎤ − ⎥ (-2) 3 ⎣ t 3⎦ K> enAeBl P = 0 kñúgkrNIrgkarBt;suT§ φ = 0.9 sMrab; muxkat; tension-control section u nig ERbRbYlBI 0.9 nig 0.65 b¤ ¬ 0.7 ¦ enAkñúgtMbn; transion region. 5> Balanced condition – muxkat;ctuekaN Balanced condition ekItmanenAkñúgmuxkat;ssrenAeBlEdl bnÞúkEdlGnuvtþmkelImuxkat;ssr Edlman nominal strength begáItbMErbMrYlrageFobesμI 0.003 enAkñúgsésrgkarsgát;rbs;ebtug nigbMEr bMrYlrageFobesμI ε = E enAkñúgr)arEdkrgkarTajkñúgeBldMNalKña. enHKWCakrNIBiessEdl GkS½NWt f y y s GacRtUv)ankMNt;BI strain diagram edaysÁal;tMélFMbMput. enAeBlEdlbnÞúkcakp©itmantMél FMCag P b enaHeKehAmuxkat;enaHfa compression control. pÞúymkvijeKehAfa tension control . karviPaK balanced column section GacRtUv)anBnül;dUcxageRkam³ !> yk c CacMgayBIsésrgkarsgát;q¶aybMputmkGkS½NWt. BI strain diagram b cb (balanced) 0.003 = (-3) d fy 0.003 + Es eRKOgbgÁúMrgkarsgát; nigrgkarBt; 201
  7. 7. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa eday E s = 200000MPa 600d ⇒ Cb = 600 + f y kMBs;bøúkrgkarsgát;smmUl equivalent compressive block ⎛ 600 ⎞ ab = β1cb = ⎜ ⎟β d (-4) ⎜ 600 + f ⎟ 1 ⎝ y ⎠ ⎧ 0.85 f 'c ≤ 28MPa ⎪ f ' −28 Edl β1 = ⎨0.85 − 0.05( c 7 ) sMrab;ebtugEdlmanersIusþg; 28MPa < f 'c ≤ 56 MPa ⎪ f 'c > 56MPa ⎩ 0.65 @> BIsmIkarlMnwg plbUkkMlaMgtamTisedkesIμsUnü Pb − Cc − C s + T = 0 Edl Cc = 0.85 f 'c ab nig T = A f s y (-5) Cs = A' ( f 's −0.85 f 'c ) enAeBlEdlEdkrgkarsgát;eFVIkardl; yield f 's = f y ⎛ c − d'⎞ f ' = 600⎜ s ⎟ ≤ f pÞúymkvij y ⎝ c ⎠ ⇒ Pb = 0.85 f 'c ab + A's ( f 's −0.85 f 'c ) − As f y (-6) #> cMNakp©it e RtUv)anvas;BI plastic centroid nig e' RtUv)anvas;BITIRbCMuTMgn;énEdkrgkarTaj. b e' = e + d " ¬sMrab;krNIrnH e' = e + d " ¦ Edl d " CacMgayBITIRbCMuTMgn;)øasÞiceTATIRbCMuTMgn;Edkrgkar b Taj. e RtUv)anKNnaedayKitm:Um:g;Rtg; plastic centroid b a Pb eb = Cc (d − − d " ) + C s (d − d '− d " ) + Td " (-7) 2 b¤ a Pb eb = 0.85 f 'c ab(d − − d " ) + A' ( f 's −0.85 f 'c )(d − d '−d " ) + As f y d " 2 (-8) cMNakp©it balanced eccentricity Mb eb = (-9) Pb sMrab;muxkat;minEmnctuekaN eyIgeRbIviFIsaRsþdUcKñakñúgkarviPaK edayKitRkLaépÞBitR)akdrbs;ebtug rgkarsgát;. emKuNkat;bnßyersIusþg; φ sMrab; balanced condition CamYy f = 400MPa RtUv)ansnμt;yk 0.65 y b¤ 0.7 . enHedaysar ε = ε = E = 0.002 . f s t y s ]TahrN_2³ kMNt;kMlaMgsgát; balanced compressive force P rYckMNt; e nig M sMrab;muxkat; b b b bgðajkñúgrUb. eKeGay f ' = 27MPa nig f = 400MPa . c y dMeNaHRsay³ Members in Compression and Bending 202
  8. 8. T.Chhay NPIC !> sMrab; balanced condition bMErbMrYlrageFobenAkñúgebtugKW 0.003 ehIy bMErbMrYlrageFobenAkñúgEdk 400 εy = = 0.002 200000 @> TItaMgGkS½NWt 600 cb = d = 0.6 × 500 = 300mm 600 + f y rUbTI6³ balanced condition ab = β1cb = 0.85 × 300 = 255mm @> RtYtBinitü Edkrgkarsgát;/ BI strain diagram ε 's c − d ' 300 − 50 = = 0.003 c 300 ⇒ ε 's = 0.0025 > ε y dUcenHEdkrgkarsgát; yield ⎛ c − d" ⎞ b¤RtYtBinitütam f 's = 600⎜ ⎝ c ⎠ ⎟ ≤ fy ⎛ 300 − 50 ⎞ f 's = 600⎜ ⎟ = 500MPa > 400 MPa ⎝ 300 ⎠ dUcenH f ' = f = 400MPa s y $> KNnakMlaMgmanGMeBImkelImuxkat; Cc = 0.85 f 'c ab = 0.85 × 27 × 255 × 350 = 2048.3kN Ts = As f y = 28 2 × π × 400 = 985.2kN C s = A' s ( f y − 0.85 f 'c ) = 28 2 π (400 − 0.85 × 27) = 928.7 kN %> KNna P nig e e b Pb = Cc + Cs − T = 2048.3 + 928.7 − 985.2 = 1991.8kN a M b = Pb eb = Cc (d − − d " ) + C s (d − d '− d " ) + Td " 2 eRKOgbgÁúMrgkarsgát; nigrgkarBt; 203
  9. 9. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa 255 M b = 1991.8eb = 2048.3(500 − − 225) + 928.7(500 − 50 − 225) + 985.2 × 225 = 732.8kN .m 2 732.8 ⇒ eb = = 0.368m 1991.8 ^> sMrab; balanced condition φ = 0.65 φPb = 0.65 × 1991.8 = 1294.67kN φM b = 0.65 × 732.8 = 476.32kN .m 6> muxkat;ssreRkamGMeBIbnÞúkcakp©it Column sections under eccentric loading sMrab;krNIBIr enAeBlEdlmuxkat;ssr)ak;edaykarsgát;b¤karTaj smIkarlMnwgmUldæanBIrGac RtUv)aneRbIsMrab;viPaKssrEdlrgbnÞúkcakp©it. rUbTI7³ krNITUeTA muxkat;ctuekaNEkg !> plbUkkMlaMgtamGkS½edk b¤tamGkS½QRtUvesμIsUnü @> plbUkm:Um:g;eFobnwgGkS½NamYyRtUvesμIsUnü eyagtamrUb eKGacsresrsmIkarTaMgBIrxagelI dYcxageRkam !> P − C − C + T = 0 n c s (-10) Edl C = 0.85 f ' ab c c C = A' ( f ' −0.85 f ' ) s s s ¬RbsinebIEdkrgkarsgát; yield enaH f ' = f ¦ c s y T=A f s s ¬RbsinebIEdkrgkarTaj yield enaH f = f ¦ s y @> Kitm:Um:g;Rtg;cMnuc A s a Pn e'−Cc (d − ) − Cs (d − d ' ) = 0 (-11) 2 Edl e' = e + d " ¬ d " CacMgayBITIRbCMuTMgn;)øasÞiceTATIRbCMuTMgn;rbs;EdkrgkarTaj¦ b¤ e' = e + d − h 2 sMrab;muxkat;ssrEdlmanEdksIuemRTI Members in Compression and Bending 204
  10. 10. T.Chhay NPIC 1⎡ a ⎤ Pn = ⎢Cc (d − 2 ) + C s (d − d ' )⎥ (-12) e' ⎣ ⎦ Kitm:Um:g;Rtg; C c ⎡ a ⎤ a a Pn ⎢e'−(d − )⎥ − T (d − ) − C s ( − d ' ) = 0 (-13) ⎣ 2 ⎦ 2 2 a a T (d − ) + Cs ( − d ' ) Pn = 2 2 (-14) a (e'+ − d ) 2 RbsidnebI A = A' ehIy s s f s = f 's = f y enaH As f y (d − d ' ) As f y (d − d ' ) Pn = = (-15) a h a e'+ − d e− + 2 2 2 h a Pn (e − + ) As = A's = 2 2 (-16) f y (d − d ' ) 7> ersIusþg;rbs;ssrsMrab;kar)ak;edaykarTaj Stregth of columns for tension failure enAeBlEdlssrrgbnÞúkcakp©itCamYynwgcMNakp©it e FM enaHeKrMBwgfassrnwg)ak;edaykarTaj. ssr)ak;edayEdkeFIVkardl; yield ebtugEbkenAeBlEdl strain rbs;EdkFMCag ε (ε = f E ) . Kñúg y y y s krNIenH nominal strength P nwgmantMéltUcCag P b¤k¾ cMNakp©it e = M P FMCag balanced n b n n eccentricity e . edaysarkñúgkrNIxøHeKmankarBi)akkñúgkarTsSn_TayfavaCamuxkat; tension control b b¤ compression control enaHeKGacsnμt;fa tension failure GacekIteLIgenAeBl e > d . Karsnμt;enH GaceFVIeLIgenAeBleRkay. smIkarlMnwgTUeTA P − C − C + T = 0 nig n c s P e'−C (d − ) − C (d − d ' ) = 0 GacRtUv)aneRbIR)as;sMrab;KNna nominal strength rbs;ssr. a n c s 2 !> sMrab;kar)ak;edaykarTaj EdkrgkarTaj yield f = f . snμt;fakugRtaMgEdkrgkarsgát; s y f' = f . s y @> KNna P = C + C − T n c s Edl C = 0.85 f ' ab c c Cs = A's ( f y − 0.85 f 'c ) T = As f y #> KNna P edayKitm:Um:g;Rtg; A n s eRKOgbgÁúMrgkarsgát; nigrgkarBt; 205
  11. 11. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa a Pn e' = Cc (d − ) + C s (d − d ' ) 2 Edl e' = e + d " b¤ e' = e + d − h enAeBl A = A' 2 s s $> BICMhan @ nig # eyIg)an 1⎡ a ⎤ Cc + C s − T = ⎢Cc (d − 2 ) + C s (d − d ' )⎥ e' ⎣ ⎦ vaCasmIkarTIdWeRkTI2 EdlmanGBaØti a . CMnYstMél C / C nig T ehIyedaHRsayrk a . c s %> eRkayBICMnYs C / C nig T smIkardWeRkTI2 Gacsresry:agsMrYldUcxageRkam c s Aa 2 + Ba + C = 0 Edl A = 0.425 f 'c b B = 0.85 f 'c b(e'− d ) = 2 A(e'− d ) C = A's ( f 's −0.85 f 'c )(e'− d + d ' ) − As f y e' − B ± B 2 − 4 AC ⇒a= 2A RbsinebI f ' −0.85 f ' < 0 RtUvykvaesμI 0 . s c ^> CMnYs a eTAkñúsmIkarCMhan @edIm,ITTYl P . m:Um:g; M kMNt;tam n n M n = Pn e &> RtYtBinitüemIlfaetIEdkrgkarsgát; yield dUckarsnμt;b¤Gt;. RbsinebI ε ' ≥ ε enaH Edkrgkar s y sgát; yield . pÞúymkvij f ' = E ε ' . Gnuvtþn_CMhan @ dl;% mþgeTot. ε ' = [(c − d ' ) / c]0.003 / s s s s ε = y E f y nig c = a / β . 1 s *> RtYtBinitüfamuxkat;Ca tension control . Tension control enAeBlNa e > e b¤ P < P . b n b (> Net tensile strain ε enAkñúgmuxkat; CaFmμtaFMCag limit strain sMrab;muxkat; compression- t controlled section 0.002 . dUcenHtMélénemKuNkat;bnßyersIusþg; φ ERbRbYlcenøaHBI 0.65 ¬b¤ 0.70 ¦ nig 0.90 . smIkar φ = 0.7 + (ε − 0.002)( 200 ) b¤ φ = 0.7 + 0.2⎡ c /1d − 5 ⎤ sMrab;muxkat; EdlmanEdk t ⎢ ⎥ 3 3 ⎣ t ⎦ kgdUcrWus½r spiral section nig φ = 0.65 + (ε t − 0.002)( 250 ) b¤ φ = 0.65 + 0.25⎡ c /1d ⎢ 5⎤ − ⎥ sMrab;muxkat; 3 ⎣ t 3⎦ epSgeTot RtUv)aneRbIsMrab;KNnark emKuNkat;bnßyersIusþg; φ . ]TahrN_3³ kMNt; nominal compressive strength P sMrab;muxkat;EdleGaydUckñúgrUbxageRkam Rbsin n ebI e = 500mm . dMeNaHRsay³ Members in Compression and Bending 206
  12. 12. T.Chhay NPIC !> eday e = 500mm > d = 485mm snμt;famuxkat;)ak;kñúglkçxNÐ tension failure condion controls ¬EdlRtUveFVIkarRtYtBinitüenAeBleRkayeTot¦. Strain rbs;EdkrgkarTaj ε GacFMCag ε dUcenHeyIg s y ykkugRtaMg f . edaysnμt;faEdkrgkarsgát; yield f ' = f EdlRtUvRtYtBinitüenAeBleRkay. y s y rUbTI8³ ]TahrN_TI3 kar)ak;edaykarTaj Tension failure @> BIsmIkarlMnwg P = C + C − T n c s Edl C = 0.85 f ' ab = 0.85 × 27 × 350a = 8.03akN c c 282 π C s = A's ( f y − 0.85 f 'c ) = 4 (400 − 0.85 × 27) = 928.68kN 4 T = As f y = 28 2 π 400 = 985.2kN ⇒ Pn = 8.03a + 928.68 − 985.2 = 8.03a − 56.52 ¬1¦ #> Kitm:Um:g;Rtg; A s 1⎡ a ⎤ Pn = ⎢Cc (d − 2 ) + C s (d − d ' )⎥ e' ⎣ ⎦ edayTIRbCMuTMgn;)øasÞic plastic centroid sßitenAelITIRbCMuTMgn;énmuxkat; d "= 210mm . e' = e + d " = 500 + 210 = 710mm 1 ⎡ ⎤ Pn = 710 ⎣ a ⎢8.03a(485 − 2 ) + 928.68(485 − 65)⎥ = −0.0056a + 5.49a + 549.36 ⎦ 2 ¬2¦ $> pÁúMsmIkar ¬1¦ nig ¬2¦ eyIg)an 0.0056a 2 + 2.54a − 605.88 = 0 ⇒ a = 172.74mm %> Pn = 8.03 × 172.74 − 56.52 = 1330.58kN eRKOgbgÁúMrgkarsgát; nigrgkarBt; 207
  13. 13. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa M n = 1330.58 × 0.5 = 665.29kN .m ^> RtYtBinitüfa Edkrgkarsgát; yield b¤Gt; 172.74 c= = 203.22mm 0.85 203.22 − 65 ε 's = 203.22 0.003 = 0.00204 > ε y = 400 200000 = 0.002 dUcenHEdksgát; yield RtYtBinitü strain enAkñúgEdkTaj ⎛ 485 − 203.22 ⎞ εs = ⎜ ⎟0.003 = 0.00416 > ε y ⎝ 203.22 ⎠ RbsinebIEdksgát;Gt; yield eRbI f ' = ε ' E rYceFVIkarKNnaeLIgvij. s s s &> KNna φ ³ eday ε = 0.00416 muxkat;sßitenAkñúgtMbn; transition region t ⎛ 250 ⎞ φ = 0.65 + (ε t − 0.002 )⎜ ⎟ = 0.83 ⎝ 3 ⎠ φPn = 0.83 × 1330.58 = 1104.38kN φM n = 0.83 × 665.29 = 552.19kN .m 8> ersIusþg;rbs;ssrsMrab;kar)ak;edaykarsgát; Stregth of columns for compression failure RbsinebIbnÞúkGnuvtþn_sgát; P FMCagbnÞúk balanced force P b¤cMNakp©it e = M tUcCag e enaH n b P n b n ssrnwgrMBwgfaRtUv)ak;edaykarsgát;. kñúgkrNI compression controls ehIy strain rbs;ebtugnwgmantMél 0.003 Edl strain rbs;EdkmantMéltUcCag ε . PaKeRcInrbs;muxkat;ssrnwgrgkarsgát;. GkS½Nwtxit y eTArkEdkrgkarTaj edaybegáInmuxkat;sgát; dUcenHcMgayeTAGkS½NWt c > c . b rUbTI9³ düaRkam strain enAeBl compression controls Members in Compression and Bending 208
  14. 14. T.Chhay NPIC edaysareKBi)akkñúgkarTsSn_TaynUvmuxkat;ssrfa tension failure b¤ compression failure eK)ansnμt;fa enAeBl e < 2d enaHssr)ak;eday compression failure EdlRtUvepÞógpÞat;enAeBleRkay. 3 edIm,IKNna nominal load strength P eKeRbIeKalkarN_sþaTic. karviPaKmuxkat;ssrsMrab;kar)ak;eday n karsgát; compression failure eKGaceRbIsmIkar P − C − C + T = 0 nigsmIkar n c s P e'−C (d − ) − C (d − d ' ) = 0 nigdMeNaHRsaymYykñúgcMeNamdMeNaHRsayxageRkam. a n c s 2 8>1> dMeNaHRsay Trial solution dMeNaHRsayenHRtUv)ansegçbdUcCMhanxageRkam³ !> KNnacMgayeTAGkS½NwtsMrab;muxkat; balanced section c b 600d t cb = (-3) 600 + f y @> kMNt; P edayeRbIlkçxNÐlMnwg n Pn = Cc + Cs − T (-10) #> KNna P edayKitm:Um:g;Rtg;EdkrgkarTaj A n s a Pn e' = Cc (d − ) + Cs (d − d ' ) (-11) 2 Edl e' = e + d "kñúgkrNITUeTA b¤ e' = e + d − enAeBl A = A' h 2 s s Cc = 0.85 f 'c ab Cs = A's ( f 's −0.85 f 'c ) T = As f s $> edaysnμt;tMél c > c KNna a = β c . snμt; b 1 f 's = f y %> KNna f s ⎛d −c⎞ f s = ε s Es = 600⎜ t ⎟ ≤ fy ⎝ c ⎠ ^> CMnYstMélEdlrkeXIjeTAkñúgsmIkarCMhan @ nigCMhan # edIm,Irk P nig P . RbsinebI n1 n2 P ≈ P eRCIsyktMéltUcCageK b¤mFümPaKén P nig P . EtebI P mantMélxusKñaq¶ayBI P eK n1 n2 n1 n2 n1 n2 RtUvsnμt; c b¤ a fμI ehIyeFVIkarKNnaeLIgvijcab;epþImBICMhan $ rhUtdl; P ≈ P . ¬eKGacTTYlyk)an n1 n2 ebI P nig P xusKña 1% ¦. n1 n2 eRKOgbgÁúMrgkarsgát; nigrgkarBt; 209
  15. 15. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa &> epÞógpÞat;fa Edkrgkarsgát; yield edayKNna ε ' = 0.003[(c − d ') / c] ehIyeRbobeFobCamYy s εy = fy E . enAeBlEdl ε ' ≥ ε Edkrgkarsgát; yield RbsinebImindUcenaHeT f ' = ε ' E b¤ s s y s s s ⎛ c − d' ⎞ f s = 600⎜ ⎟ ≤ fy ⎝ c ⎠ *> epÞógpÞat;fa e < e b¤ P > P sMrab; compression failure. b n b (> sMrab;muxkat; compression controlled section CaTUeTA net tensile strain ε enAkñúgmuxkat;tUc t Cag 0.002 . dUcenH emKuNkat;bnßyersIusþg; φ = 0.65 ¬b¤ 0.70 sMrab;ssrEdleRbIEdkkgCab;¦. ]TahrN_4³ kMNt; nominal compressive strength P sMrab;muxkat;EdleGay RbsinebI e = 254mm . n rUbTI10³ ]TahrN_TI4 Compression controls dMeNaHRsay³ !> edaysar e = 254mm < 23d = 333.33mm . snμt; compression failure. karsnμt;enHRtUvepÞógpÞat;enA eBleRkay. KNnacMgayeTAGkS½NwtsMrab; balanced section c : b 600d t 600 × 500 cb = = = 300mm 600 + f y 600 + 400 @> BIsmIkarlMnwg Pn = Cc + Cs − T (-10) Edl Cc = 0.85 f 'c ab = 0.85 × 27 × a × 350 = 8.03akN C s = A' s ( f y − 0.85 f 'c ) = 282 π (400 − 0.85 × 27) = 928.68kN edaysnμt;Edkrgkarsgát; yield ¬karsnμt;enHRtUvepÞógpÞat;enAeBleRkay¦ T = A f = 28 πf = 2.46 f kN ¬ f < f ¦ s s 2 s s s y Pn = 8.03a + 928.68 − 2.46 f s #> Kitm:Um:g;Rtg; A s Members in Compression and Bending 210
  16. 16. T.Chhay NPIC 1⎡ a ⎤ Pn = ⎢Cc (d − 2 ) + Cs (d − d ' )⎥ (-11) e' ⎣ ⎦ TIRbCMuTMgn;)øasÞicsßitenAelITIRbCMuTMgn;rbs;muxkat; d "= 225mm e' = e + d " = 254 + 225 = 479mm 1 ⎡ ⎛ a⎞ ⎤ Pn = ⎢8.03a⎜ 500 − 2 ⎟ + 928.8(500 − 50)⎥ = 8.38a − 0.0084a + 872.57 2 479 ⎣ ⎝ ⎠ ⎦ $> edaysnμt; c = 342mm EdlmantMélFMCag c b = 300mm a = 0.85 × 342 = 290.7 mm CMnYstMél a eTAkñúgsmIkarkñúgCMhanTIBIrxagelIeyIg)an Pn1 = 8.38 × 290.7 − 0.0084 × 290.7 2 + 872.57 = 2598.78kN %> KNna f BIdüaRkam strain enAeBlEdl c = 340mm s 500 − 342 fs = 600 = 277.19 MPa 342 f 277.19 ε s = εt = s = = 0.00139 Es 200000 ^> edayCMnYs a = 290.7mm nig f s = 277.19MPa eTAkñúgsmIkarCMhanTImYyedIm,IKNna P n2 Pn 2 = 8.03 × 290.7 + 928.68 − 2.46 × 277.19 = 2581.11kN eday P nig P mantMélxusKñamindl; 1% dUcenHeyIgyk P = 2581.11kN n1 n2 n M n = Pn e = 2581.11 × 0.254 = 655.6kN .m &> epÞógpÞat;fa Edkrgkarsgát; yield BIdüaRkam strain 342 − 50 ε 's = 0.003 = 0.00256 > ε y = 0.002 342 dUcenH Edkrgkarsgát; yield dUckarsnμt;. *> P = 2581.11kN > P = 1991.8kN ehIy e = 254mm < e = 368mm bgðajfavaCamuxkat; n b b compression control dUckarsnμt;. cMNaM³ eKGaceFVIkarsakl,gKNnaedIm,IeGay P nig P mantMél n1 n2 kan;EtesμIkña. (> KNna φ d t = d = 500mm c = 342mm 500 − 342 εt ¬enAnIv:UedkrgkarTaj¦ = 0.003 342 = 0.00139 < 0.002 enaH φ = 0.65 φPn = 0.65 × 2581.11 = 1677.72kN φM n = 0.65 × 655.6 = 426.14kN .m eRKOgbgÁúMrgkarsgát; nigrgkarBt; 211
  17. 17. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa 8>2> dMeNaHRsayviPaKcMnYn Numerical Analysis Solution enAeBl compression control karviPaKssrGaceFVIeTA)anedaykat;bnßykarKNnamkRtwmsmIkar dWeRkTI3 EdlmanTMrg; Aa + Ba + Ca + D = 0 rYcedaHRsayrktMél a edayviFIcMnYn numerical method 3 2 b¤ a GacTTYl)anBIm:asIunKitelx. BIsmIkarlMnwg Pn = Cc + C s − T = (0.85 f 'c ab) + A's ( f y − 0.85 f 'c ) − As f s (-10) Kitm:Um:g;Rtg;EdkTaj A s 1⎡ a ⎤ 1⎡ a ⎤ Pn = ⎢Cc (d − 2 ) + Cs (d − d ' )⎥ = e' ⎢0.85 f 'c ab(d − 2 ) + A's ( f y − 0.85 f 'c )(d − d ' )⎥ (-11) e' ⎣ ⎦ ⎣ ⎦ BIdüaRkam strain a (d − ) ⎛d −c⎞ β1 εs = ⎜ t ⎟0.003 = 0.003 ⎝ c ⎠ a β1 kugRtaMgenAkñúgEdkTajKW 600 f s = ε s E s = 200000ε s = ( β1 d − a ) a edayCMnYstMél f eTAkñúgsmIkar (-10) nigedaHRsaysmIkar (-10) nig (-11) eRkayBIsMrYlrYceyIg s TTYl)an ⎛ 0.85 f 'c b ⎞ 3 ⎜ ⎟a + [0.85 f 'c b(e'− d )]a + [ A's ( f y − 0.85 f 'c )(e'−d + d ' ) + 600 As e' ]a − 600 As e' β1d = 0 2 ⎝ c ⎠ enHCasmIkardWeRkTI3 EdlmanTMrg; Aa 3 + Ba 2 + Ca + D = 0 Edl A = 0.852f ' b c B = 0.85 f 'c b(e'− d ) C = A's ( f y − 0.85 f 'c )(e'− d + d ' ) + 600 As e' D = −600 As e' β1d enAeBlEdleKKNna)antMél A / B / C nig D enaH a GacRtUv)anKNnaedayviFIsakl,g b¤ TTYl)anedaypÞal;Bim:asIunKitelx. dMeNaHRsaysmIkardWeRkTI3 GacTTYl)anedayeRbIviFI Newton- Raphson . viFIenHmanRbsiT§PaBkñúgkaredaHRsayrkb¤srbs; f ( x ) = 0 . vaTak;Tgnwgbec©keTssamBaØ ehIyeKqab;TTYl)ancMelIyedayeFVItamCMhanxageRkam³ !> eGay f (a) = Aa + Ba + Ca + D nigKNna A / B / C nig D 3 2 @> KNnaedrIevTImYyén f (a) ³ f ' (a) = 3 Aa + 2Ba + C 2 #> edaysnμt;tMéldMbUg a KNnatMélbnÞab; o Members in Compression and Bending 212
  18. 18. T.Chhay NPIC f ( ao ) a1 = ao − f ' ( ao ) $> edayeRbItMél a KNna a dUcsmIkarxagelI 1 2 f (a1 ) a2 = a1 − f ' (a1 ) %> Gnuvtþn_nUvviFIenHrhUtdl;)antMélsuRkitmYy a ≅ a . kñúgkrNIviPaKssrenAeBl compression n n −1 control tMél a EtgEtFMCag a . dUcenH eKcab;epþImCamYy a = a ehIyGnuvtþsmIkarxagelIBIrdgedIm,I b o b TTYl)ancMelIy. ]TahrN_5³ eFVI]TahrN_TI4eLIgvijedayeRbIviFI numerical analysis dMeNaHRsay³ !> KNna A / B / C nig D nigkMNt; f (a) 0.85 f 'c b 0.85 × 27 × 350 A= = = 4016.25 2 2 B = 0.85 f 'c b(e'− d ) = 0.85 × 27 × 350(479 − 500) = −168682.5 C = A' s ( f y − 0.85 f 'c )(e'−d + d ' ) + 600 As e' C = 282 π (400 − 0.85 × 27)(479 − 500 + 50) + 600 × 282 π × 479 C = 734800328.08 D = −600 As e' β1d = −600 × 282 π × 479 × 0.85 × 500 = −300844190383.4 f (a) = 4016.25a 3 − 168682.5a 2 + 734800328.08a − 300844190383.4 @> KNnaedrIevTI1 f ' (a) = 12048.75a 2 − 337365a + 734800328.08 #> eGay a o = ab = 255mm sMrab;muxkat; balanced section c b = 300mm nig a b = 255mm f (255) a1 = 255 − = 295.39 f ' (255) $> nigKNna a ³ 2 f (295.39) a2 = 295.39 − = 292.4mm f ' (295.39) tMélrbs; a mantMélRsedognwg a enAkñúg]TahrN_TI3. CMnYstMél a eTAkñúgsmIkar (-10) b¤ (-11) eKTTYl)an P = 2594.66kN n 8>3> dMeNaHRsayRbEhl Approximate Solution smIkar approximate RtUv)anesñIreLIgeday Whitney edIm,IedaHRsayrk nominal compressive strength sMrab;ssrxøI enAeBl compression control. eRKOgbgÁúMrgkarsgát; nigrgkarBt; 213
  19. 19. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa bhf 'c A's f y Pn = + (-17) 3he e + 1.18 + 0.5 d 2 (d − d ' ) smIkarxagelIenHGaceRbIeTA)ansMrab;EtssrEdlmansésEdksIuemRTItMerobEtmYyRsTab; ehIyRsbeTA nwgGkS½énkarBt;. smIkar approximate TI2 RtUv)anesñIeLIgeday Hsu 1.5 Pn − Pb ⎛ M n ⎞ +⎜ ⎟ = 1.0 (-18) Po − Pb ⎜ M o ⎟ ⎝ ⎠ Edl = Pn nominal axial strength énmuxkat;ssr = Pb , M b nominal load nig nominal moment énmuxkat; balanced section M n = nominal bending moment = Pn ⋅ e enAeBl e=0 P = 0.85 f ' ( A − A ) + A f Po = nominal axial load o c g st st y A = gross area énmuxkat; = bh g A = muxkat;EdkbeNþayminEmnkugRtaMgsrub st ]TahrN_6³ kMNt; nominal compressive strength P sMrab;muxkat;EdleGaydUckñúg]TahrN_TI4 n edaysmIkar ¬*>#>!¦ nig ¬*>#>@¦edayeRbInUvcMNakp©itdUcKña e = 254mm rYceRbobeFobcMelIy. dMeNaHRsay³ !> dMeNaHRsaytamsmIkar Whitney k> lkçN³énmuxkat; b = 350mm / h = 550mm / d = 500mm / d ' = 50mm / A' = 2463mm 2 nig (d − d ' ) = 450mm x> GnuvtþsmIkar Whitney 350 × 550 × 27 2463 × 400 Pn = + = 2745.15kN 3 × 550 × 254 254 + 1.18 + 0.5 500 2 450 φPn = 0.65 × 2745.15 = 1784.35kN K> P EdlKNnaedaysmIkar Whitney CatMélEdlminsnSMsMécenAkñúg]TahrN_enH ehIytMél n P = 2745.15kN KWFMCagtMélsuRkit P = 2581.11kN EdlKNnaedaysmIkarsþaTickñúg]TahrN_TI4. n n @> dMeNaHRsaytamsmIkar Hsu k> sMrab; balanced condition P = 1991.8kN nig M = 732.8kN ¬]TahrN_TI2¦ b b x> P = 0.85 f ' ( A − A ) + A f = 0.85 × 27 × (550 × 350 − 2 × 2463) + 2 × 2463 × 400 o c g st st y Po = 6275.22kN Members in Compression and Bending 214
  20. 20. T.Chhay NPIC Pn − 1991 .8 1 .5 ⎛ 0254 Pn ⎞ K> +⎜ 6275 .22 − 1991 .8 ⎝ 732 .8 ⎠ ⎟ =1 edayKuNnwg 1000 ehIyedaHRsayrk P n 0.23346 Pn + 0.00654 Pn1.5 = 1465 Edaykarsakl,gtMél P = 2587.65kN EdlmantMélRbEhl 2581.11kN EdlKNnaedaysþa n Tic. 9> ]TahrN_sMrab;düaRkamGnþrkmμ Interaction Diagram Example enAkñúg]TahrN_TI2 bnÞúk balanced load P , M nig e RtUv)anKNnasMrab;muxkat;dUckñúgrUbTI6 b b b ¬ e = 368mm ¦. dUcKña enAkñúg]TahrN_TI3 nigTI4 load capacity sMrab;muxkat;dUcKñaRtUv)anKNnasMrab; b krNIenAeBl e = 500mm ¬tension failure¦ nigenAeBl e = 254mm ¬compression failure¦. tMélTaMg enHnwgRtUvbgðajenAkñúgtaragTI1. edIm,IKUrdüaRkamGnþrkmμbnÞúk-m:Um:g; tMélepSg²én φP nig φM RtUv)anKNnasMrab;tMél e epSg² n n Edl e ERbRbYlBI e = 0 eTA e = Gtibrma sMrab;krNIm:Um:g;Bt;suT§ pure moment enAeBl P = 0 . düa n Rkamnþr kmμbnÞúk-m:Um:g;RtUv)anbgðajkñúgrUbTI11. bnÞÞúk φP = 4078.90kN CabnÞúkcMGkS½tamRTwsþI enAeBl no e = 0 . Et ACI Code GnuBaØatibnÞúkGb,brmaRtwmEt 0.8φP = 3263.12kN EdlRtUvKñanwg cMNakp©wtGb, no brma. cMNaMfa sMrab;kar)ak;edaykarsgát; compression failure e < e nig P > P ehIysMrab;kar)ak; b n b taragTI1³taragKNnasegçb rUbTI11³düaRkamGnþrkmμbnÞúk-m:Um:g; eRKOgbgÁúMrgkarsgát; nigrgkarBt; 215
  21. 21. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa edaykarTaj tension failure e > e nig P < P . krNI e = Gtibrma ssrrgnUvm:Um:g;Bt;suT§dUckrNIFñwm. b n b 10> ssrmuxkat;ctuekaNCamYyEdkxag Rectangular columns with side bars enAkñúgmuxkat;ssrxøH EdkRtUv)aneKdak;BRgaytamRCugTaMgGs;. EdkxagRtUv)andak;tamkMBs;énmux kat;edaybEnßmeTAelIEdkTaj nigEdksgát; A nig A' ehIyRtUv)aneKkMNt;eGayeQμaH A ¬rUbTI12¦. s s ss kñúgkrNIenH viFIsaRsþkñúgkarKNnaEdl)anBnül;rYcmkehIyGacRtUv)anGnuvtþ edayKitBicarNabEnßmkar pøas;bþÚr strain tamkMBs;énmuxkat; nigTMnak;TMngkMlaMgenAkñúgEdkxagnImYy²eTAkñúgtMbn;sgát; b¤tMbn;Taj énmuxkat;. kMlaMgTaMgenHRtUv)anbUkbEnßmeTAelI C C nig T edIm,IkMNt; P smIkarmanragdUcxag c s n eRkam³ Pn = Cc + ∑ C s − ∑ T (-10a) ]TahrN_TI7 Bnül;BIkarKNnaenH. cMNaMfa RbsinebIEdkxagsßitenAEk,rGkS½NWt ¬rUb12 b¦ strain nigkMlaMg enAkñúgEdkmantMéltUcNas;EdleKGacecal)an. cMENkEdkEdlsßitenAEk,r A nig A' mantM s s élFMKYrsm nigCYybegáInlT§PaBRTRTg;bnÞúkénmuxkat;. rUbTI12³ EdkxagenAkñúgmuxkat;ctuekaNEkg ]TahrN_7³ kMNt;bnÞúk m:Um:g; M nigcMNakp©it e sMrab;muxkat;bgðajkñúgrUbTI13. edayeRbI Pb b b f ' = 28MPa nig f = 400 MPa . c y dMeNaHRsay³ muxkat; balanced section RtUv)anKNnadUcKñanwg]TahrN_TI2Edr. eKeGay π 32 b = h = 550mm / d = 485mm nig d ' = 65mm . A = A' = 5 2 = 4021.24mm (5DB32 ) / s s 2 4 nigEdkxag 6DB32 ¬ 3DB32 sMrab;mçag¦. 1> KNnacMgayeTAGkS½NWt Members in Compression and Bending 216
  22. 22. T.Chhay NPIC ⎛ 600 ⎞ cb = ⎜ ⎟d = ⎛ 600 ⎞485 = 291mm ⎜ 600 + f ⎟ t ⎜ 600 + 400 ⎟ ⎝ ⎠ ⎝ y ⎠ ab = 0.85cb = 0.85 × 291 = 247.35mm 2> KNnakMlaMgenAkñúgebtug nigEdk tamry³rUb 13 a . enAtMbn;sgát; Cc = 0.85 f 'c ab = 0.85 × 28 × 247.35 × 550 = 3237.81kN C s = A' s ( f ' s −0.85 f 'c ) enAnIv:U − 65mm ⎛ c − d'⎞ ⎛ 291 − 65 ⎞ f ' s1 = 600⎜ ⎟ = 600⎜ ⎟ = 466 MPa > 400MPa ⎝ c ⎠ ⎝ 291 ⎠ dUcenH f 's = 400MPa Cs1 = 4021.24(400 − 0.85 × 28) = 1512.79kN eRKOgbgÁúMrgkarsgát; nigrgkarBt; 217
  23. 23. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa enAnIv:U − 170mm ⎛ c − d'⎞ ⎛ 291 − 170 ⎞ f 's 2 = 600⎜ ⎟ = 600⎜ ⎟ = 249.48MPa ⎝ c ⎠ ⎝ 291 ⎠ π 32 2 Cs 2 = 2 × (249.48 − 0.85 × 28) = 363kN 4 enAnIv:U − 275mm ⎛ c − d' ⎞ ⎛ 291 − 275 ⎞ f 's 3 = 600⎜ ⎟ = 600⎜ ⎟ = 32.99MPa ⎝ c ⎠ ⎝ 291 ⎠ π 32 2 Cs 3 = 2 × (32.99 − 0.85 × 28) = 14.78kN 4 enAkñúgtMbn;Taj ¬nIv:U − 380mm ¦ Members in Compression and Bending 218
  24. 24. T.Chhay NPIC 380 − 291 ε s4 = × 0.003 = 917.53 ⋅ 10 −6 291 f s 4 = 200000 × 917.53 ⋅ 10 −6 = 183.5MPa π 32 2 T1 = 2 × (183.5) = 295.16kN 4 T2 = 4021.24 × 400 = 1608.5kN 3> KNna Pb = Cc + ∑ C s − ∑ T Pb = 3237.81 + (1512.79 + 363 + 14.78) − (295.16 + 1608.5) = 3224.72kN 4> KNnam:Um:g;Rtg;TIRbCMuTMgn;)øasÞic M b = 3237.81× 151.325 + 1512.79 × 210 + 363 × 105 + 295.16 × 105 + 1608.5 × 210 M b = 1214.54kN .m M 1214.54 eb = b = = 0.377m Pb 3224.72 5> KNna φ sMrab; balanced section ε t / = ε y = 0.002 φ = 0.65 φPn = 0.65 × 3224.72 = 2096.07kN niig φM b = 0.65 ×1214.54 = 749.45kN .m ]TahrN_8³ edaHRsay]TahrN_TI7 eLIgvijenAeBlEdl e = 152mm . dMeNaHRsay³ 1> edaysar enaHvaCalkçxNÐ)ak;edaykarsgát; compression failure e = 152mm < eb = 326mm condition. snμt; c = 399.5mm ¬edaykarsakl,g¦ nig a = 399.5 × 0.85 a = 339.58mm ¬ rUbTI13 b ¦. 2> KNnakMlaMgenAkñúgebtug nigEdk Cc = 0.85 × 28 × 339.58 × 550 = 4445.1kN dUcKñanwgkrNI balanced f s1 = 400MPa nig C s1 = 1512.79kN f s 2 = 344.68MPa nig Cs 2 = 516.13kN f s3 = 186.98MPa nig Cs 3 = 262.48kN f s 4 = 29.29MPa nig Cs 4 = 8.83kN f s 5 = 128.41MPa nig T = 516.37 kN 3> KNna Pn = Cc + ∑ C s − ∑ T = 6228.96kN M n = Pn ⋅ e = 6228.96 × 152 = 946.8kN .m eRKOgbgÁúMrgkarsgát; nigrgkarBt; 219
  25. 25. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa 4> KNna Pn edayKitm:Um:g;Rtg; As 1⎡ a ⎤ Pn = ⎢Cc (d − 2 ) + Cs1 (d − d ' ) + C s 2 (d − d '− s ) + C s 3 (d − d '−2s ) + C s 4 (d − d '−3s )⎥ e' ⎣ ⎦ h 550 e' = e + d − = 152 + 485 − = 362mm 2 2 s = 105mm KMlatrvagEdkxag ¬efr sMrab;]TahrN_enH¦ ⎡ ⎛ 339.58 ⎞ ⎤ 1 ⎢4445.1⎜ 485 − ⎟ + 1512.79(485 − 65) + 516.13(485 − 65 − 105)⎥ Pn = ⎝ 2 ⎠ 362 ⎢ ⎥ ⎢+ 262.48(485 − 65 − 2 × 105) + 8.83(485 − 65 − 3 × 105) ⎣ ⎥ ⎦ Pn = 6230kN 5> KNna φ d t = d = 485mm c = 399.5mm εt ¬enAnIv:UEdkTaj¦ = 0.03(dt − c ) / c = 0.03(485 − 399.5) / 399.5 = 0.00064 eday ε t < 0.002 enaH φ = 0.65 φPn = 0.65 × 6228.96 = 4048.8kN φM n = 0.65 × 946.8 = 615.42kN .m cMNaM³ RbsinebIEdkxagminRtUv)anKit enaH Pb = 3142.1kN Pn ¬enA e = 152mm ¦ = 4592.23 + 1512.79 − 422.48 = 5682.54kN RbsinebIeKKitEdkxagenaH Pb ekIneLIgRbEhl 2.6% nig Pn ekIneLIgRbEhl 9.6% . 11> lT§PaBRTbnÞúkrbs;ssrmuxkat;mUl Load Capacity of Circular Columns 11>1 lkçxNÐ Balanced Condition tMélénbnÞúk balanced load Pn nig m:Um:g; balanced moment M n sMrab;muxkat;mUlGacRtUv)ankM Nt;edayeRbIsmIkarlMnwgdUckrNImuxkat;cuekaNpgEdr. sésrEdkenAkñúgmuxkat;rgVg;EdlRtUv)antMerobeTA tamcMNayBIGkS½TIRbCMuTMgn;)øasÞicERbRbYl KWGaRs½yeTAnwgcMnYnEdkenAkñúgmuxkat;. bBaðacMbgKWrkkMBs;bøúk sgát; a nigkugRtaMgenAkñúgsésrEdk. ]TahrN_xageRkamBnül;BIkarviPaKmuxkat;eRkamlkçxNÐ balanced condition. nitiviFIdUcKñaGacRtUv)aneRbIedIm,IviPaKmuxkat;sMrab; tension control b¤ compression control. ]TahrN_9³ kMNt;bnÞúk balanced load Pn nig m:Um:g; balanced moment M n sMrab;ssrmuxkat;rgVg;Edkkg vNÐGgát;p©it 400mm CamYynig 8DB28 dUcbgðajkñúgrUbTI14. eKeGay f 'c = 28MPa nig Fy = 400MPa . dMeNaHRsay³ Members in Compression and Bending 220
  26. 26. T.Chhay NPIC 1> edaysarEtEdksIuemRTInwgGkS½ A − A Edlkat;tamTIRbCMuTMgn;rgVg; enaHTIRbCMuTMgn;)øasÞicsßitenAelI GkS½enaH. 2> kMNt;TItaMgTIRbCMuTMgn;GkS½NWt fy d t = 329.34mm εy = Es cb 0.003 600 = = d t 0.003 + ε y 600 + f y 600 cb = 329.34 = 197.6mm 600 + 400 ab = 167.96mm 3> kMNt;lkçN³rbs;cMerokrgVg; circular segment ¬rgVg;TI15¦ RkLaépÞcMerokrgVg; = r 2 (α − sin α cos α ) (-19) TItaMgTIRbCMuTMgn; x ¬BITIRbCMuTMgn; 0¦ 2 (r sin 3 α ) x= (-20) 3 α − sin α cos α Z =r−x (-21) ⎛ a⎞ r cos α = (r − a ) b¤ cos α = ⎜1 − ⎟ ⎝ r⎠ (-22) ⎛ 167.96 ⎞ cos α = ⎜1 − ⎟ = 0.16 ⎝ 200 ⎠ / ⇒ α = 80.79o sin α = 0.987 nig α = 1.41rad RkLaépÞcMerokrgVg; = 2002 (1.41 − 0.16 × 0.987) = 50083.2mm 2 ⎛2⎞ 200 × 0.987 3 x =⎜ ⎟ = 102.39mm ⎝ 3 ⎠ (1.41 − 0.987 × 0.16) Z = 200 − 102.39 = 97.61mm 4> kMNt;kMlaMgsgát; Cc Cc = 0.85 f 'c × RkLaépÞcMerokrgVg; = 0.85 × 28 × 50083.2 = 1192kN vaeFVIGMeBIenA 102.39mm BITIRbCMuTMgn;ssr eRKOgbgÁúMrgkarsgát; nigrgkarBt; 221
  27. 27. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa Members in Compression and Bending 222
  28. 28. T.Chhay NPIC 5> KNna strain, stress nig kMlaMgenAkñúgEdkrgkarTaj nigEdkrgkarsgát;. kMNt;bMErbMrYlrageFob strain BIdüaRkambMErbMrYlrageFob. sMrab; T1 ε = ε y = 0.002 f s = f y = 400MPa π 282 T1 = 2 × 400 = 492.6kN 4 sMrab; T2 55.98 ε s3 = ε y = 8.5 ⋅ 10 − 4 131.74 f s 3 = 8.5 ⋅ 10 −4 × 200000 = 170MPa π 282 T2 = 2 × 170 = 209.36kN 4 sMrab; Cs1 126.94 ε s1 = × 0.003 = 1.93 ⋅10−3 197.6 f s1 = 1.93 ⋅ 10 −3 × 200000 = 386MPa < 400 MPa 282 π C s1 = 2 (386 − 0.85 × 28) = 446.05kN 4 sMrab; Cs2 51.18 ε s2 = × 0.003 = 7.77 ⋅10 − 4 197.6 f s 2 = 7.77 ⋅10−4 × 200000 = 155.4 MPa 282 π cs 2 = 2 (155.4 − 0.85 × 28) = 162.07kN 4 6> kugRtaMgenAkñúgEdkrgkarsgát;RtUv)ankat;bnßy edIm,IKitenAkñúgebtugEdlCMnYsedayEdk. eRKOgbgÁúMrgkarsgát; nigrgkarBt; 223
  29. 29. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa kMlaMg balanced KW Pb = Cc + ∑ Cs − ∑ T Pb = 1192 + (446.05 + 162.07) − (492.6 + 209.36) = 1098.16kN sMrab;muxkat; balanced ε t = 0.002 nig φ = 0.65 φPb = 713.8kN 7> Kitm:Um:g;Rtg;TIRbCMuTMgn;)øasÞic ¬GkS½ A − A kat;tamGkS½TIRbCMuTMgn;¦ sMrab;kMlaMgTaMgGs; M b = Pb eb = [Cc × 102.39 + Cs1 × 129.34 + Cs 2 × 53.58 + T1 × 129.34 + T2 × 53.58) M b = 263.36kN .m φM b = 171.18kN .m 263.36 eb = = 239.8mm 1098.16 11>2 ersIusþg;rbs;muxkat;mUlsMrab;kar)ak;edaykarsgát; Strength of circular column for compression failure muxkat;ssreRkamkMlaMgcMNakp©it GacRtUv)anviPaKtamCMhandUcmuxkat; balanced Edr. va TTYl)anedaykarsnμt; C > Cb b¤ a > ab nigKNnakMlaMgenAkñúgebtug nig EdkenATItaMgepSgKñaedIm,I kMNt; Pn1 = Cc + ∑ Cs − ∑ T . dUcKña M n GacRtUv)anKNnaedayKitm:Um:g;Rtg;TIRbCMuTMgn;)øasÞic ¬TIRbCuMTMgn;rbs; muxkat;¦ ehIykMNt; Pn2 = Men . RbsinebItMél Pn1 nig Pn2 minRbhak;RbEhl KñaeTenaH snμt; C b¤ a fμI ehIyeFVIkarKNnaeLIgvij ¬emIlcMnucTI8¦. tMélxusKñarvag Pn1 nig Pn2 sßit enArgVg; 1% . muxkat; Camuxkat; compression controls enAeBl e < eb b¤ Pn > Pb . sMrab;]TahrN_ RbsinebIvaTamTarkMNt;ersIusþg;rbs;muxkat;ssrenAkñúg]TahrN_TI9 enAeBlEdl e = 150mm Pn GacRtUv)anKNnaedayCMhandUcnwg]TahrN_TI 9. 1> eday e = 150mm tUcCag eb = 239.8mm lkçxNÐ)ak;edaykarsgát; compression failure condion ekIteLIg. 2> snμt; c = 225mm ¬edaykarsμan¦ > Cb = 197.6 nig a = 191.25mm 3> KNna x = 89.63mm / Z = 110.37mm RkLaépÞcMerokrgVg; = 59332.97mm2 4> -5> KNnakMlaMg³ Cc = 1412.125kN / Cs1 = 463.29kN Cs 2 = 228.73kN / T1 = 342.66kN / T2 = 93.84kN 6> KNna Pn1 = Cc + ∑ Cs − ∑ T = 1667.64kN 7> Kitm:Um:g;Rtg;GkS½ssr ¬TIRbCMuTMgn;)øasÞic¦³ M n = 248.1kN .m Members in Compression and Bending 224
  30. 30. T.Chhay NPIC Pn 2 = Mn e = 1653.97 mm EdlmantMélRbEhl Pn1 ¬tMélxusKñaRbEhl 1% ¦. dUcenH Pn = 1653.97 kN cMNaM³ RbsinebIEdkkgrbs;ssrCaEdkkgvNÐdUcrWus½renaH φ = 0.70 . smIkartMélRbEhl approximate equation sMrab;karKNna Pn sMrab;muxkat;mUl enAeBl compression controls RtUv)anesñIeLIgedayelak Whitney Ag f ' c Ast f y Pn = + (-23) ⎡ 9.6he ⎤ ⎛ 3e ⎞ ⎢ + 1.18⎥ ⎜ ⎜ D + 1⎟ ⎟ ⎢ (0.8h + 0.67 Ds ) ⎝ s ⎠ 2 ⎣ ⎥ ⎦ Edl RkLaépÞmuxkat; gross area Ag = h = Ggát;p©itmuxkat; Ds = Ggát;p©itmuxkat;ssrEdlvas;BITIRbCMuTMgn;EdkmçageTATIRbCMuTMgn;EdkmçageTot Ast = RkLaépÞmuxkat;EdkbBaÄr e = cMNakp©itEdlvas;BITIRbCMuTMgn;)øasÞic ]TahrN_10³ KNnaersIusþg;kMlaMgsgát; nominal Pn sMrab;muxkat;sMrab;]TahrN_TI9 edayeRbIsmI kar Whitney RbsinebIcMNakp©it e = 150mm dMeNaHRsay³ 1> e = 150mm tUcCag eb = 239.8mm . tamkarKNnadUceBlmun bgðajfamuxkat;ssrCamuxkat; compression controls. 2> edayeRbIsmIkar Whitney h = 400mm π π Ag = 4 h2 = 4 400 2 = 125663.7mm 2 / D s = 400 − 120 = 280mm π × 28 2 As = 8 × = 4926mm 2 4 125663.7 × 28 4926 × 400 Pn = + = 1785.94kN ⎡ 9.6 × 400 × 150 ⎤ ⎛ 3 × 150 ⎞ ⎢ + 1.18⎥ ⎜ + 1⎟ ⎢ (0.8 × 400 + 0.67 × 280 ) ⎝ 280 ⎠ 2 ⎣ ⎥ ⎦ 3> M n = Pn e = 1785.94 × 0.15 = 267.89kN .m tMél Pn enAeBlenHFMCagtMél Pn = 1653.97kN EdlKNnaenAeBlmunedaysþaTic. eRKOgbgÁúMrgkarsgát; nigrgkarBt; 225
  31. 31. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa 11>3 ersIusþg;rbs;muxkat;mUlsMrab;kar)ak;edaykarTaj Strength of circular column for tension failure kar)ak;edaykarTajsMrab;ssrmUlenAeBlbnÞúkRtUv)anGnuvtþn_enARtg;cMNakp©it e > eb b¤ Pn < Pb . enAkñúgkrNIenH muxkat;ssrGacRtUv)anviPaKtamCMhandUckarviPaKmuxkat; balanced nigdUc kñúg]TahrN_TI8. karviPaKRtUv)aneFVIeLIgedaysnμt; C < Cb b¤ a < ab rYcehIyGnuvtþn_tamCMhanBnül; kñúgEpñkTI11>1. cMNaMfa edaysarEtsésrEdkRtUv)antMerobedaymancenøaHefrtambrimaRtmuxkat;rgVg; enaHEdkTaj As Edlpþl;eGayGacmantMéltUc ehIylT§PaBRTbnÞúkk¾køayCamantMéltUc. dUcenH eK)anENnaMeGayeCosvagkareRbIR)as;muxkat;mUlsMrab;krNIkar)ak;edaykarTaj tension failure. 12> karviPaK nigkarKNnassredayeRbIdüaRkam Analysis and Design of Column Using Charts karviPaKmuxkat;ssrEdl)anBnül;BIxagedImKWQrelIeKalkarN_sþaTic. sMrab;karviPaK b¤kar KNnassrCaCMhandMbUg düaRkamb¤taragBiessGacRtUv)aneRbIedIm,IkMNt; φPn nig φM n sMrab;muxkat; EdleGay nigkar KNnamuxkat;EdkcaM)ac;sMrab; Pu nig M u EdleGay. düaRkam nigtaragenHRtUv)ane)aH Bum<pSayeday viTüasßanebtugGaemric American Concrete Institute (ACI) viTüasßanebtugBRgwgedayEdk Concrete Reinforcing Steel Institute (CRSI) nigsmaKmsIum:g;t_Br½Eln Porland Cement Association (PCA). karKNnassrcugeRkayRtUvEteFVIeLIgedayQrelIsmIkarsþaTic edaykarKNnaedayéd b¤eday kmμviFIkMuBüÚTr½. kareRbIdüaRkam ACI RtUv)anbgðajenAkñúg]TahrN_xageRkam. düaRkamRtUv)anbgðajkñúgrUb TI16 nigrUbTI17. Tinñn½yTaMgenHRtUv)ankMnt;sMrab;mxkat;ssrdUcbgðajenARCugxagelIEpñkxagsþaMéntarag. u ]TahrN_11³ kMnt;sésrEdkcaM)ac;sMrab;ssr short tied column dUcbgðajenAkñúgrUbTI 18 a edIm,IRTnUvbnÞúkemKuN 2150kN nigm:Um:g;emKuN 440kN.m . ssrmanTTwg 350mm nigbeNþaysrub h = 500mm . eRbI f 'c = 28MPa / f y = 400MPa . dMeNaHRsay³ 1> cMNakp©it e = M u = 2150 = 204.65mm Pu 440 yk d = 500 − 60 = 440mm γh = 500 − 120 = 380mm enaH γ = 380 = 0.76 500 2> eday e = 204.65 < d snμt;famuxkat;)ak;edaykarsgát; compression-controlled section CamYynwg φ = 0.65 Members in Compression and Bending 226
  32. 32. T.Chhay NPIC eRKOgbgÁúMrgkarsgát; nigrgkarBt; 227
  33. 33. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa Members in Compression and Bending 228
  34. 34. T.Chhay NPIC eRKOgbgÁúMrgkarsgát; nigrgkarBt; 229
  35. 35. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa Members in Compression and Bending 230
  36. 36. T.Chhay NPIC 2150 Pn = = 3307.7kN 0.65 nig Mn = 440 0.65 = 676.9kN .m Pn 3307.7 ⋅ 10 3 Kn = = = 0.675 f 'c Ag 28 × 350 × 500 ⎛e⎞ ⎛ 204.65 ⎞ Rn = K n ⎜ ⎟ = 0.675⎜ ⎟ = 0.276 ⎝h⎠ ⎝ 500 ⎠ 3> BItaragkñúgrUbTI16 sMrab; γ = 0.7 / ρ = 0.048 dUcKñasMrab; γ = 0.8 / ρ = 0.043 eday interpolation sMrab; γ = 0.76 / ρ = 0.045 As = 0.045 × 500 × 350 = 7875mm 2 eRbI 10DB32 (As = 8042.48mm 2 )/ R)aMedImenAtamRCugxøI. eRbIEdkkg DB10 @ 350mm ¬rUbTI18 a¦ ]TahrN_12³ eRbItaragedIm,IkMNt;bnÞúkersIusþg; φP rbs;ssrxøIdUcbgðajkñúgrUbTI 18 b EdlGnuvtþ n enAcMgaycMNakp©it e = 305mm . eRbI f 'c = 35MPa nig f y = 400MPa . dMeNaHRsay³ A. lkçN³rbs;muxkat;³ H = 600mm / γh = 600 − 120 = 480mm ¬cMgayrvagEdlTaj 32 2 × π 8× nigEdk sgát;¦. γ = 600 = 0.8 ehIy 480 ρ= 4 600 × 350 = 0.03 B. eday e < d / snμt;vaCamuxkat; compression-controlled section. eRKOgbgÁúMrgkarsgát; nigrgkarBt; 231
  37. 37. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa yk ε t = 0.002 / fs fy = 1.0 ehIy φ = 0.65 BItaragkñúgrUbTI 17 eKTTYl)an Pn K n = 0.36 = 35 × 600 × 350 dUcenH Pn = 2646kN C. EpÞógpÞat;karsnμt;sMrab;muxkat; compression controlled³ sMrab; K n = 0.36 e Rn = K n = 0.183 h BIdüaRkameyIgTTYl)an ρ = 0.019 < 0.03 / dUcenH Pn > 2646kN ¬edIm,I)an ρ = 0.03 ¦ D. karsakl,gelIkTI2³ yk ε t = 0.0015 / f s = 0.0015 × 200000 = 300MPa f s 300 = = 0.75 ρ = 0.03 K n = 0.43 f y 400 Pn = 0.43 × 35 × 600 × 350 = 3160.5kN E. EpÞógpÞat;karsnμt;³ sMrab; K n = 0.43 / Rn = K n h = 0.219 e BItarag ρ = 0.03 dUcGVIEdleGay dUcenH Pn = 3160.5kN φPn = 0.65 × 3160.5 = 2054.3kN nig φM n = 626.6kN .m tamkarviPaK φPn = 2027kN ¬mantMélRbEhlKñanwgkarKNnaedayeRbItarag¦. 13> karKNnassreRkambnÞúkcakp©it Design of Columns Under Eccentric Loading karKNnassrmanlkçN³sμúKsμajCagkarviPaKssr edaysarEtbnÞúkxageRkA Pu nigm:Um:g; M u Casmμtikmμ ehIyeKRtUvkarkMNt;nUvGBaØtiCaeRcIndUcCa b / h / As / A's CamYynwgkarkMNt;rbs; ACI Code. vaCakarGnuvtþn_TueTAedaysnμt;dMbUgnUvmuxkat;ssr ehIykMnt;brimaNmuxkat;EdkRtUvkar. RbsinebIGñk KNnaRtUvkardUrmuxkat;EdkKNna enaHmuxkat;ssrk¾RtUv)anEkERbeTAtamenaHEdr. ]TahrN_xageRkam bgðajBIkarKNnassr. 13>1 KNnassrsMrab;kar)ak;edaykarsgát; Design of Column for Compression Failure sMrab; compression failure eKniymeRbI As = A's sMrab;muxkat;ctuekaN. cMNakp©it e = M u . Pu edayQrenAelItMélrbs; e eKman2krNIRtUv)anbegáIteLIg 1> enAeBlEdl e ≤ 100mm krNIcMNakp©itGb,brmaGacekItman EdlGaceKNnaedayeRbIrUbmnþ Members in Compression and Bending 232
  38. 38. T.Chhay NPIC [ ( Pu = φPn = φK 0.85 f 'c Ag + Ast f y − 0.85 f 'c )] Edl φ = 0.65 nig K = 0.80 sMrab;ssrEdl man dkkgdac;² nig φ = 0.70 nig K = 0.85 sMrab;ssrEdlmanEdkvNн ¬sUmemIl]TahrN_kñúg emeronssrrgkMlaMgcMGkS½¦. sMrab;krNIepSgBIenH GñkKNnaGacGnuvtþtamkrNITI2. krNIbnÞúk enHRbRBwtþeTAsMrab;ssrGKarCan;eRkaménGKareRcInCan; Edlm:Um:g; M u )anmkBIRbBn§½mYyCan; nig Pu )anmkBIbnÞúkRKb;Can;EdlmanGMeBIenABIelIva. 2> tMbn; compression failure KWtMbn;EdlsßitenAcenøaHGkS½QreTAbnÞat; balanced load dUcbgðaj kñúgrUbTI 3 nigrUbTI11. kñúgkrNIenH muxkat; bh GacRtUv)ansnμt; ehIybnÞab;mkmuxkat;EdkRtUv)an KNnasMrab; Pu nig M u EdleGay. CMhanénkarKNnaRtUv)ansegçbdUcxageRkam³ k> snμt;muxkat;kaer b¤ctuekaN bh rYckMNt; d / d ' nig e = M u P u A' s f y x> edaysnμt; As = A's KNna A's BIsmIkar Pn = 3he 'c bhf + e edayeRbI + 1.18 + 0.5 d2 (d − d ' ) TMhMmuxkat;Edl)ansnμt; nig φ = 0.65 sMrab;ssrEdleRbIEdkkgFmμta. yk As = A's rYc eRCIserIsmuxkat;RKb;RKan;. kMNt;muxkat;BitR)akdEdleRbIsMrab; As nig A's . mü:agvij eToteKGaceRbIdüaRkam ACI. + K> epÞógpÞat;fa 1% ≤ ρ g = As bhA's ≤ 8% . RbsinebI ρ g mantMéltUc kat;bnßymuxkat; snμt; b:uEnþBRgIkmuxkat;RbsinebIeKcg;)anmuxkat;EdktUc. X> epÞógpÞat;PaBRKb;RKan;rbs;muxkat;cugeRkayedayKNna φPn BIsmIkarsþaTic dUcBnül;kñúg]TahrN_xagedIm. φPn ≥ Pu . g> kMNt;EdkkgcaM)ac;. rUbmnþRbhak;RbEhl approximate formula y:agsamBaØ sMrab;kMNt;muxkat;ssrdMbUg bh b¤PaK ryEdksrub total steel retio ρ g KW Pn = K c bh 2 b¤ Pu = φPn = φK c bh 2 ¬-24¦ Edl K c mantMéldUcbgðajkñúgtaragTI2 nigbgðajkñúgrUbTI19 sMrab;Edk f y = 400MPa nig As = A' s . xñatrbs; K c KW kN / m 3 . taragTI2 tMélrbs; K ¬ f c y = 400MPa ¦ Kc ρ g (% ) f 'c (28MPa) f 'c (35MPa) f 'c (42MPa) 1% 24817 30246 35286 4% 37574 43003 48044 eRKOgbgÁúMrgkarsgát; nigrgkarBt; 233
  39. 39. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa 8% 54675 60103 65144 3 K c ( kN/m ) 69000 64000 59000 54000 49000 44000 39000 f'c=28MPa 34000 f'c=35MPa 29000 f'c=42MPa 24000 1 2 3 4 5 6 7 8 ρ g (%) rUbTI19 tMél K c nig ρ g tMél K c RtUv)anbgðajkñúgtaragTI2 CatMélRbhak;RbEhl niggayRsYleRbI edaysar K c ekIn eLIgmþg 5429 sMrab;karekIneLIgrbs; f 'c mþg 7 . sMrab;muxkat;dUcKña enAeBlEdlcMNakp©it e = M u Pu ekIneLIg Pn fycuH dUcenH K c fycuH. dUcenH tMél K c sMEdgbnÞúk Pn enAelIdüaRkamGnþrkmμcenøaH 0.8Pno nig Pb dUcbgðajkñúgrUbTI 3 nigTI 11. Linear interpolation GacRtUv)aneRbI. ]TahrN_ K c = 46124.5 sMrab; ρ g = 6% nig f 'c = 28MPa . CMhankñúgkarKNnamuxkat;ssrGacRtUv)ansegçbdUcxageRkam³ 1> snμt;muxkat;dMbUgsMrab;muxkat;ssr bh 2> KNna K c = (φbh 2 ) Pu 3> kMNt; ρ g BItaragTI 2 sMrab; f 'c EdleGay 4> kMNt; As = A's = ρ g2bh rYceRCIserIsEdkbBaÄr nigEdkkg. 5> kMNt; φPn énmuxkat;cugeRkaytamsmIkarsþaTic ¬dMeNaHRsayCak;lak;¦. tMélén φPn KYrEt mantMélFMCagb¤esμI Pu . RbsinebImindUecñaHeT EktMrUv bh b¤ ρ g . Members in Compression and Bending 234
  40. 40. T.Chhay NPIC mü:agvijeTot RbsinebIeKcg;)anPaKryEdksrubCak;lak; ]TahrN_ ρ g = 6% bnÞab;mkGnuvtþdUc xageRkam³ 1> snμt; ρ g dUcEdlTamTar nigbnÞab;mkKNna e = M u Pu 2> edayQrelI f 'c nig ρ g EdleGay/ kMNt; K c BItaragTI2 3> KNna bh 2 = φPu bnÞab;mkeRCIserIs b nig h . GnuvtþCMhan 4 nig 5 eLIgvij. Kc eKKYrEtepÞógpÞat;fa 1% ≤ ρ g ≤ 8% . dUcKña epÞógpÞat;fa c Edl)anmkBIkarKNnatamsþaTicmantMélFMCag cb = 600 +dft sMrab; compression failure . 600 y ]TahrN_13³ kMNt;muxkat;EdkTaj nigmuxkat;Edksgát;sMrab;ssrEdleRbIEdkkgFmμtamanmuxkat; 400 × 600 edIm,IRTbnÞúk Pu = 3470kN nig M u = 530kN .m . edayeRbI f 'c = 28kN nig f y = 400MPa . dMeNaHRsay³ 1> KNna e = M u = 3470 = 152.74mm . eyIgman h = 600mm yk d = 550mm nig d ' = 50mm P 530 u edaysar e < 2 d = 366.67mm snμt;fa compression failure. 3 2> snμt; As = A's . kMNt;tMéldMbUgrbs; A's tamrUbmnþ bhf 'c A' s f y Pn = + (-17) 3he e + 1.18 + 0.5 d 2 (d − d ' ) sMrab; P 3470 Pn = u = φ 0.65 = 5338.5kN A' s = 4271.8mm 2 = As eRKOgbgÁúMrgkarsgát; nigrgkarBt; 235
  41. 41. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa eRbIEdk DB32 ⇒ n = 6 edIm 6 DB32 = 4825.5mm 2 sMrab; As nig A' s ¬rUbTI 20¦ 3> ρ g = 2 × 4825.5 = 4% EdlvaRtUvEttUcCag 0.08 nig FMCag 0.01 400 × 600 4> epÞógpÞat;muxkat;edaysmIkarsþaTictamCMhanénkarKNnakñúg]TahrN_TI4 eKTTYl)an a = 430.18mm / c = 506.09mm / Cc = 4095.32kN C s = 4825.5(400 − 0.85 × 28) = 1815.35kN ⎛d −c⎞ ⎛ 550 − 506.09 ⎞ f s = 600⎜ ⎟ = 600⎜ ⎟ = 52.06MPa ⎝ c ⎠ ⎝ 506.09 ⎠ T = As f s = 4825.5 × 52.06 = 251.21kN Pn = Cc + C s − T = 5659.5kN > 5338.5kN cMNaMfa RbsinebI φPn < Pu cUrdMeLIg As nig A's rYceFVIkarKNnaeLIgvij. 5> epÞógpÞat; Pn edayeRbIsmIkar Pn = e' ⎡Cc ⎛ d − a ⎞ + Cs (d − d ' )⎤ Edl e' = e + d − h 1 ⎢ ⎜ ⎟ ⎥ ⎣ ⎝ 2⎠ ⎦ 2 eyIgTTYl)an Pn = 5659kN 6> sMrab; muxkat; balanced section ⎛ 600 ⎞ cb = ⎜ ⎟d t = 600 550 = 330mm ⎜ 600 + f y ⎟ 1000 ⎝ ⎠ edaysarEt c = 506.09mm > cb = 330mm vaCakrNI compression failure dUckarsnμt;. 7> edayeRbIEdkkgmanGgát;p©it 10mm KMlatEdkkg ⎧48φ ⎧48 ×10 ⎧480 ⎪ ⎪ ⎪ min ⎨16d = min ⎨16 × 32 = min ⎨512 = 400 ⎪ b ⎪ 400 ⎪400 ⎩ ⎩ ⎩ dUcenHeRbIEdkkg DB10 @ 400 . ]TahrN_14³ eFVI]TahrN_TI13 eLIgvijedayeRbIsmIkar ¬-24¦ dMeNaHRsay³ 1> muxkat;ssrEdleGay 400 × 600 2> kMNt; K c BIsmIkar ¬-24¦ 3> K c = φbh 2 = 0.65 ×3470× 0.6 2 = 37073 m3 Pu 0 .4 kN Members in Compression and Bending 236
  42. 42. T.Chhay NPIC 4> BItaragTI 2 b¤rUbTI19 sMrab; K c = 37073 mm3 f 'c = 28MPa eday interpolation kN 4 −1 eyIgTTYl)an ρ g = 1 + (37073 − 24817) 37574 − 24817 = 3.88% 5> KNna As = A's = ρbh / 2 = 0.0388(400)(600)/ 2 = 4656mm 2 eRbIEdk DB32 ⇒ n = 6 edIm 6> 6DB32 = 4825.5mm 2 7> kMNt; φPu edayeFVItamCMhan 4-7 sMrab;]TahrN_TI13. Pn = 5659kN > Pn = 5338.5kN dUcenHmuxkat;EdkRKb;RKan; 8> RbsinebImuxkat;minRKb;RKan; b¤ φPn < Pn tMeLIgmuxkat; As nig A's rYceFVIkarepÞógpÞat;eLIgvij edIm,ITTYl)antMélEk,r. ]TahrN_15³ KNnamuxkat;ssrctuekaNEkgedIm,IRTbnÞúk Pu = 3150kN nig M u = 630kN .m CamYynwgPaKryEdksrub ρ g RbEhl 4% . eRbI f ' = 28MPa / f = 400MPa nig b = 450mm . c y dMeNaHRsay³ 1> KNna e = M u = 3150 = 0.2m . snμt; compression failure ( φ = 0.65 ) ¬RtUvepÞógpÞat;enA Pu 630 eBleRkay¦ ehIy As = A's 2> sMrab; ρ = 4% nig f 'c = 28MPa enaH K c = 37574 ¬taragTI2¦ 3> KNna bh 2 BIsmIkar (-24): Pu = φK cbh 2 rW 3150 = 0.65(37574)(0.45)h 2 dUcenH h = 0.535m dUcenHyk h = 550mm . KNna As = A's = 0.04(450 × 550) = 4950mm 2 . eRCIserIs 5DB36 2 ( As = 5089.4mm ) dUcbgðajkñúgrUbTI 21. eRbIEdkkg DB12 @ 450 . 2 4> epÞógpÞat; muxkat;cugeRkayedaykarviPaK RsedogKñanwg]TahrN_TI4 eyIgTTYl)an a = 327.8mm / c = 385.65mm / Cc = 0.85 f 'c ab = 3510.7kN / f ' s = 400 MPa / ⎛d −c⎞ C s = A' s ( f y − 0.85 f 'c ) = 1914.6kN / f s = 600⎜ ⎟ = 146.79 MPa / nig T = 747 kN ⎝ c ⎠ dUcenH Pn = Cc + Cs − T = 4678.3kN ehIy Pu = φPn = 3041kN < 3150kN edaysarmuxkat;minRKb;RKan; eyIgRtUvtMeLIgmuxkat;Edk b¤muxkat;ebtug rYceFVIkarepÞógpÞat;eLIg vij. ¬yk h = 600mm ¦ rUbTI21. 5> sMrab;muxkat; balanced section eRKOgbgÁúMrgkarsgát; nigrgkarBt; 237
  43. 43. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa ⎛ 600 ⎞ cb = ⎜ ⎟d t = 318mm < c = 436.32mm ⎝ 600 + 400 ⎠ ¬ d = 530mm ¦ dUcenH vaCa compression failure dUckarsnμt;. 13>2 KNnassrsMrab;kar)ak;edaykarTaj Design of Column for tension Failure Kar)ak;edaykarTaj tension failure ekItmanenAeBl Pn < Pu b¤k¾cMNakp©it e > eb dUckarBnül; enAkñúgEpñkTI7. kñúgkarKNnassr Pu nig M u RtUv)aneGay ehIyvaTamTarnUvkarkMNt;muxkat;ssr nigmux kat;Edk. vaGacRtUv)ansnμt; ¬dUckarENnaM¦ favaCa tension control enAeBlNa ⎧530 sMrab; h < 600mm M u >⎨ 600 sMrab; h ≥ 600mm . kñúgkrNIenH muxkat;ssrGacRtUv)ansnμt; ehIybnÞab;mk As nig Pu ⎩ A' s RtUv)ankMNt;. düaRkam ACI GacRtUv)aneRbIedIm,IKNna ρ g sMrab;muxkat;EdleGayCamYynwg As = A's . cMNaMfa φ ERbRbYlcenøaH 0.65(0.7) nig 0.9 dUckarBnül;kñúgEpñkTI 4. enAeBl tension controls EdkTaj yields b:uEnþEdksgát;Gac yields nigmin yields. karsnμt;dMbUg f ' s = f y nig As = A' s . smIkar (-16) ¬kñúgEpñkTI 6¦ GacRtUv)aneRbIedIm,IKNnatMéldMbUgrbs; As nig A' s . ⎛ h a⎞ Pn ⎜ e − + ⎟ As = A' s = ⎝ 2 2⎠ (-16) f y (d − d ') edaysar a minRtUv)andwgenAeLIy snμt; a = 0.4d nig Pu = φPn bnÞab;mk Pu (e − 0.5h + 0.2d ) As = A' s = (-25) φf y (d − d ' ) muxkat;ssrcugeRkayKYrRtUv)anepÞógpÞat;edaysmIkarsþaTicedIm,Ibgðajfa φP n ≥ Pu . ]TahrN_ TI16 Bnül;BIviFIsaRsþkñúgkarKNnaenH. Members in Compression and Bending 238
  44. 44. T.Chhay NPIC enAeBlbnÞúk P mantMéltUcNas;ebIeRbobeFobCamYynwgm:Um:g; M TMhMrbs;muxkat;GacRtUv)ankM u u Nt;edayeRbIEt M EtmYy)anehIy edaysnμt;fa P = 0 . muxkat;cugeRkayKYrRtUv)anepÞógpÞat;eday u u smIkarsþaTic. krNIenHekIteLIgsMrab;eRKagGKarmYyCan; b¤BIrCan; EdlGKarenaHRtUv)aneKeRbIsMrab;eFVICa saltaMgBiBN’ b¤k¾GKarTaMgLayNaEdlmanlkçN³dUcKñaenaH. sMrab;krNIenH A' GacRtUv)ansnμt;eGay s mantMéltUcCag A . karKNnay:aglMGitsMrab;saltaMgBiBN’kMBs;mYyCan;Edlmansnøak;BIrRtUv)anBnül; s enAkñúgCMBUkTI 16 FñwmCab; nigeRKag. ]TahrN_16³ kMNt;sésrEdkcaM)ac;sMrab;ssrragctuekaNEkg 400 × 560 EdlmanEdkkg FmμtaRTbnÞúk P = 1140kN nig M = 850kN .m . eRbI f ' = 28MPa nig f = 400MPa u u c y dMeNaHRsay³ 1> KNna e = M P u = 850 1140 = 0.7456m . yk d = 560 − 60 = 500mm . edaysar u Mu Pu = 745.6mm > 530mm b¤edaysar e > d snμt;fassrenH)ak;edaykrNI tension failure enaH φ = 0.9 ¬RtUvepÞógpÞat;enAeBleRkay¦. 2> snμt; A = A' nig f ' = f nigeRbIsmIkar (-25) edIm,IkMnt; A nig A' . eday P = 1140kN / s s s y s s u e = 745.6mm / h = 560mm / d = 500mm / nig d ' = 60mm 1140 ⋅ 10 3 (745.6 − 0.5 × 560 + 0.2 × 500 ) As = A' s = = 4070.71mm 2 0.9 × 400(500 − 60 ) eRbI 5DB32 (4021.24mm ) sMrab; A nig A' . ¬rUbTI22¦ 2 s s 3> epÞógpÞat; ρ = 2400 × 560) = 0.0359 EdltUcCag 0.08 nigFMCag 0.01 . (4021.24 g 4> epÞógpÞat;kareRCIserIsmuxkat;edaysmIkarsþaTic EdlkarKNnaRsedogKñanwg]TahrN_TI3 eRKOgbgÁúMrgkarsgát; nigrgkarBt; 239
  45. 45. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa a. kMNt;tMélrbs; a edayeRbIsmIkarTUeTA Aa + Ba + C = 0 CamYynwg 2 e' = e + d − = 965.6mm / A = 0.425 f ' b = 4760 / B = 2 A(e'− d ) = 4432512 / h c 2 C = A' ( f − 0.85 f ' )(e'− d + d ') − A f e' = −758040793 . eKTTYl)an a = 147.62mm s y c s y nig c = a / 0.85 = 173.67mm . b. epÞógpÞat; f ' ³ f ' = 600⎛ c −c d ' ⎞ = 600⎛ 173.67.6760 ⎞ = 392.71MPa s ⎜ ⎝ s ⎟ ⎠ ⎜ ⎝ 173 − ⎟ ⎠ c. KNna a eLIgvij C = A' s ( f ' s −0.85 f ' c )(e'−d + d ') − As f ' s e' = −773454123.3 eKTTYl)an a = 150.25mm nig c = 176.77mm d. epÞógpÞat; f ' ³ f ' = 600⎛ c −c d ' ⎞ = 600⎛ 176.77.7760 ⎞ = 396.34MPa s ⎜ ⎝ s ⎟ ⎠ ⎜ ⎝ 176 − ⎟ ⎠ KNna C = 0.85 × 28 × 150.25 × 400 = 1430.38kN c C s = A' s ( f ' s −0.85 f ' c ) = 4021.24(396.34 − 0.85 × 28) = 1498.07kN T = As f y = 4021.24 × 400 = 1608.5kN e. Pn = C c + C s − T = 1319.95kN 5> KNna φ ³ ε = 0.003⎛ d c− c ⎞ = 0.0055 edaysarEt ε = 0.0055 > 0.005 enaH φ = 0.9 ⎜ t ⎟ t t ⎝ ⎠ 6> φP = 0.9 × 1319.95 = 1187.95kN > 1140kN muxkat;RKb;RKan; n 14> karBt;tamBIrTis Biaxial Bending karviPaK nigkarKNnassreRkamGMeBIbnÞúkcakp©itEdl)anBiPakSaknøgmk CakrNIkarBt;mYyTis. enHmann½yfa P GnuvtþenAelIGkS½ y ¬rUbTI23¦ begáIt)anbnSMénkMlaMgcMGkS½ P nigm:Um:g;Bt;CMuvijGkS½ x n n esμInwg M nx = Pn e y b¤ P GnuvtþenAelIGkS½ x ¬rUbTI24¦ CamYynwgcMNakp©it e begáIt)anbnSMénkMlaMgcM n x GkS½ P nigm:Um:g;Bt; M ny = Pn e x . n Members in Compression and Bending 240

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