Department of Civil Engineering                                                      viTüasßanCatiBhubec©keTskm<úCa

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T.Chhay                                                                                               NPIC

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Department of Civil Engineering                                                        viTüasßanCatiBhubec©keTskm<úCa
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T.Chhay                                                                                           NPIC

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Department of Civil Engineering                                             viTüasßanCatiBhubec©keTskm<úCa
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T.Chhay                                                                                                  NPIC

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Department of Civil Engineering                                                               viTüasßanCatiBhubec©keTskm<ú...
T.Chhay                                                                                        NPIC

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Department of Civil Engineering                                                            viTüasßanCatiBhubec©keTskm<úCa
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T.Chhay                                                                                                                   ...
Department of Civil Engineering                                                                                     viTüas...
T.Chhay                                                                                             NPIC

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Department of Civil Engineering                                              viTüasßanCatiBhubec©keTskm<úCa
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T.Chhay                                                                                                                   ...
Department of Civil Engineering                                                              viTüasßanCatiBhubec©keTskm<úC...
T.Chhay                                                                                                NPIC

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Department of Civil Engineering                                                             viTüasßanCatiBhubec©keTskm<úCa...
T.Chhay                                                                                                   NPIC
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Department of Civil Engineering                                                                viTüasßanCatiBhubec©keTskm<...
T.Chhay                                                                                                           NPIC

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Department of Civil Engineering                                                     viTüasßanCatiBhubec©keTskm<úCa
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T.Chhay                                                                            NPIC

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Department of Civil Engineering                                       viTüasßanCatiBhubec©keTskm<úCa




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T.Chhay                                                                                                NPIC

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Department of Civil Engineering                                                               viTüasßanCatiBhubec©keTskm<ú...
T.Chhay                                                                                          NPIC

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Department of Civil Engineering      viTüasßanCatiBhubec©keTskm<úCa




Members in Compression and Bending                ...
T.Chhay                                                                                     NPIC




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Department of Civil Engineering                                                   viTüasßanCatiBhubec©keTskm<úCa
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T.Chhay                                                                                                       NPIC

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Department of Civil Engineering                                           viTüasßanCatiBhubec©keTskm<úCa
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T.Chhay                            NPIC




eRKOgbgÁúMrgkarsgát; nigrgkarBt;    227
Department of Civil Engineering      viTüasßanCatiBhubec©keTskm<úCa




Members in Compression and Bending                ...
T.Chhay                            NPIC




eRKOgbgÁúMrgkarsgát; nigrgkarBt;    229
Department of Civil Engineering      viTüasßanCatiBhubec©keTskm<úCa




Members in Compression and Bending                ...
T.Chhay                                                                                       NPIC
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Department of Civil Engineering                                                viTüasßanCatiBhubec©keTskm<úCa
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T.Chhay                                                                                                         NPIC

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Department of Civil Engineering                                           viTüasßanCatiBhubec©keTskm<úCa
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T.Chhay                                                                                            NPIC

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Department of Civil Engineering                                                 viTüasßanCatiBhubec©keTskm<úCa
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T.Chhay                                                                                        NPIC

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Department of Civil Engineering                                                viTüasßanCatiBhubec©keTskm<úCa
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T.Chhay                                                                                                      NPIC

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Department of Civil Engineering                                                                      viTüasßanCatiBhubec©k...
Xi members in compression and bending
Xi members in compression and bending
Xi members in compression and bending
Xi members in compression and bending
Xi members in compression and bending
Xi members in compression and bending
Xi members in compression and bending
Xi members in compression and bending
Xi members in compression and bending
Xi members in compression and bending
Xi members in compression and bending
Xi members in compression and bending
Xi members in compression and bending
Xi members in compression and bending
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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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