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4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
4.compression members
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4.compression members

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  • 1. T.chhay IV. eRKOgbgÁúMrgkarsgát; Compression Members 4>1> esckþIepþIm Introduction eRKOgbgÁúMrgkarsgát; CaeRKOgbgÁúMsMNg;EdlrgEtkMlaMgsgát;tamGkS½. bnÞúkEdlGnuvtþtam GkS½beNþaykat;tamTIRbCMuTMgn;rbs;muxkat;Ggát; ehIykugRtaMg (stress) GacesμInwg f a = P A Edl f a RtUv)anKitfamantMélesμIKñaelImuxkat;TaMgmUl. b:uEnþCak;EsþgeKminEdlTTYl)ansßanPaBl¥Ebb enHeT eKminGaceCosputBIkMlaMgcakp©itxøH)aneLIy. CalT§pleKnwgTTYl)ankarBt; b:uEnþvaGac RtUv)aneKKitkMlaMgrg (secondary load) nigGacRtUv)anecalRbsinebIlkçxNÐénkardak;bnÞúkesÞIrEt dUcKñanwgRTwsþI. karBt;minGacRtUvecaleT RbsinebIvaCam:Um:g;Bt;Edl)anBIkarKNna. eyIgnwgKit sßanPaBenHenAkñúgCMBUkTI6. CaTUeTA Ggát;rgkarsgát;EdlekItmanenAkñúgGKar nigs<anKW ssr ¬CaGgát;bBaÄrEdlman tYnaTIcMbgKWRTnUvbnÞúkbBaÄr¦. Ggát;rgkarsgát;k¾RtUv)aneRbIenAkñúgeRKOgbgÁúM truss nigCaeRKOgbgáúMén RbBn§½BRgwgpgEdr. Ggát;rgkarsgát;EdlmanRbEvgxøIminRtUv)aneKcat;cMNat;fñak;Ca column eT Etva RtUv)aneKehAfa strut. 4>2> RTwsþIssr Column Theory edayBicarNaGgát;rgkarsgát;Evg ehIyRsavdUcbgðajenAkñúgrUbTI 4>1 a . RbsinebIbnÞúktam GkS½ P RtUv)andak;yWt² enAeBlmYybnÞúkenaHnwgmantMélRKb;RKan;edIm,IeFVIeGayGgát;KμanesßrPaB ehIyragrbs;Ggát;nwgekagdUcbgðajedayExSdac;. bnÞúkEdleFVIeGayGgát;ekagRtUv)aneKehAfa 68 eRKOgbgÁúMrgkarsgát;
  • 2. T.chhay critical buckling load . RbsinebI Ggát;manRbEvgxøI ehIyFat;dUcbgðajenAkñúgrUbTI 4>2 b enaHeKRtUv karbnÞúkEdlmantMélFMCagmunedIm,IeFVIeGayGgát;enaHsßitkñúgsßanPaBKμanesßrPaB. RbsinebIGgát; enaH kan;EtxøI kar)ak;nwgekIteLIgeday compressive yielding Cageday buckling. munnwg)ak; kugRtaMgsgát; P A nwgrayesμIenAelImuxkat;RKb;cMnucTaMgGs;énbeNþayRbEvgrbs;ssr eTaHCa)ak; eday yielding b¤k¾)ak;eday buckling. bnÞúkEdleFVIeGay buckling ekItman CaGnuKmn_eTAnwg slendernesss ehIysMrab;Ggát;EdlRsavxøaMg bnÞúkenHnwgmantMéltUcNas;. RbsinebIGgát;manlkçN³RsavxøaMg EdlkugRtaMgmunnwg buckling EdltUcCagEdnsmamaRt (proportional limit) ¬EdlGgát;sßitenAkñúglkçN³eGLasÞic¦ critical buckling load RtUv)aneGay dUcxageRkam³ π 2 EI Pcr = 2 L ¬$>!¦ Edl E Cam:UDuleGLasÞic (modulus of elasticity), I Cam:Um:g;niclPaBénRkLaépÞmuxkat; (moment of inertia of the cross-sectional are) EdleFobnwgGkS½emEdltUc (minor principal axis), L CaRb Evgrbs;Ggát;cenøaHTMr. edIm,IeGaysmIkar ¬$>!¦ mann½y luHRtaEtGgát;sßitkñúgsßanPaBeGLasÞic ehIycugrbs;vaGacviledayesrI EtminRtUvrMkileTAxageT. cugTMrenHbMeBjlkçxNÐedayTMrsnøak; (hinge) b¤ pinned dUcbgðajkñúgrUbTI 4>2 . TMnak;TMngd¾KYreGaycab;GarmμN_enHRtUv)anrkeXIjdMbUgbMput edayGñkR)aCJKNitviTüaCnCatisVIseQμaH Leonhard Euler Edle)aHBum<enAkñúgqñaM 1759. bnÞúkeRKaH fñak; (critical load) enH enAeBlxøHRtUv)aneKehAfa Euler load b¤ Euler buckling load . smIkarTI $>! RtUv)aneKbgðajedIm,IeFVIeGayeCOedaykarBiesaFn_y:ageRcIn. karsMraybBa¢ak;rbs;smIkarenH RtUv)aneGayedIm,IbgðajBIPaBsMxan;rbs;lkçxNÐcugTMr. 69 eRKOgbgÁúMrgkarsgát;
  • 3. T.chhay edIm,IgayRsYlkñúgkarbkRsay Ggát;RtUv)andak;eGayedkelIGkS½ x dUcEdleGaykñúgrUbTI 4>3. TMr roller Edldak;enATIenHedIm,ITb;Ggát;mineGaycl½teTAelI b¤cuHeRkam. bnÞúksgát;tamGkS½ RtUv)anGnuvtþ ehIyekIneLIgsnSwm². bnÞúkxagbeNþaHGasnñRtUv)andak;edIm,IeFVIeGayGgát;dabdUcrUb ragEdlbgðajedayExSdac; ehIyGgát;nwgRtLb;eTArkrUbragedImvijenAeBlEdlbnÞúkbeNþaHGsnñ enaHRtUv)aneKdkecjRbsinebIbnÞúktamGkS½mantMéltUcCag critical buckling load . Critical buckling load, Pcr RtUv)ankMNt;CabnÞúkEdlmantMélFMRKb;RKan;edIm,IrkSarUbragdabrbs;Ggát;enA eBlEdlbnÞúkxagbeNþaHGasnñRtUv)aneKdakecj. smIkarDIepr:g;Esül (differential equation) sMrab;rUbragdabrbs;Ggát;eGLasÞicEdlrgkar Bt;KW³ d2y dx 2 =− M IE ¬$>@¦ Edl x CacMgayrbs;cMnucEdlsßitenAelIGkS½beNþayrbs;Ggát;/ y CaPaBdabrbs;Ggát;enARtg;cMnuc enaH/ nig M Cam:Um:g;Bt;enARtg;cMnucenaH. E nig I RtUv)anbgðajBIxagelI b:uEnþm:Um:g;niclPaB I enA TIenHKWeFobnwgGkS½énkarBt;. smIkarenHRtUv)anTajeday Jacob Bernoulli ehIyRtUv)anbMEbk eday Euler EdleRbIR)as;vasMrab;bBaðaekagrbs;ssr. BI rUbTI 4>3 eyIgeXIjfaenAeBlEdlGgát; ekagedaysarbnÞúktamGkS½ Pcr enAcMgay x BITMrxageqVgeyIgmanPaBdab y ehIym:Um:g;Bt;enARtg; cMnucenaHKW Pcr y . enaHsmIkar $>@ GacsresrdUcxageRkam³ Pcr y"+ y=0 EI Edl RBIm KWCaDIepr:g;Esültam x . smIkarenHCa second order, linear, ordinary differential equation CamYynwgemKuNefr ehIymandMeNaHRsay y = A cos(cx) + B sin(cx) 70 eRKOgbgÁúMrgkarsgát;
  • 4. T.chhay Edl c = Pcr EI ehIy A nig B CatMélefr. tMélefrTaMgenH RtUv)ankMNt;edayGnuvtþnUvlkçxNÐRBMEdndUcxageRkam³ Rtg; x = 0 / y = 0 ³ 0 = A cos(0) + B sin(0) enaH A = 0 Rtg; x = L / y = 0 ³ 0 = B sin(cL) lkçxNÐcugeRkayenHtMrUveGay sin(cL) = 0 RbsinebI B ≠ 0 ¬cMeLIyminsMxan; EdlRtUvKñanwg P = 0 ¦. sMrab; sin(cL) = 0 / cL = 0, π , 2π , 3π , ... = nπ , n = 0, 1, 2, 3, ... BI c = Pcr EI ⎛ ⎞ ehIy Pcr = n πL2 EI 2 2 eyIgTTYl)an cL = ⎜ ⎜ Pcr ⎟ L = nπ , ⎟ Pcr 2 EI L = n 2π 2 ⎝ EI ⎠ tMélCaeRcInrbs; n RtUvKñanwgrUbragekag (buckling mode) epSg². n = 1 bgðajnUvrUbragekagTImYy (first mode). n = 2 KWrUbragekagTIBIr (second mode).l. tMél n = 0 CakrNIKμanbnÞúk EdlCa krNIminsMxan;. rUbragénkarekagTaMgenHRtUv)anbgðajenAkñúgrUbTI 4>4. tMél n minGacFMCagmYy elIkElgEtGgát;rgkarsgát;RtUv)anTb;BIkardabenARtg;cMnucEdleFVIeGaykMeNagbt;Ebn. dUcenHdMeNaHRsayrbs;smIkarDIepr:g;EsülKW ⎛ nπx ⎞ y = B sin ⎜ ⎟ ⎝ L ⎠ ehIyemKuN B CatMélminkMNt;. lT§plenHRtUv)aneRbICacMeLIy linear kñúgsmIkarDIepr:g;EsültM Nag)atuPUt nonlinear. 71 eRKOgbgÁúMrgkarsgát;
  • 5. T.chhay sMrab;krNIFmμtarbs;Ggát;rgkarsgát;EdkKμanTMrenAcenøaHcugsgçagrbs;va n =1 enaHsmIkar Euler RtUv)ansresrCa π 2 EI Pcr = L2 ¬$>#¦ vamanlkçN³gayRsYlCagkñúgkarsresrsmIkar $># kñúgTMrg;dUcxageRkam π 2 EI π 2 EAr 2 π 2 EA Pcr = = = L2 L2 (L / r )2 Edl A CaRkLaépÞmuxkat; nig r CakaMniclPaB (radius of gyration) EdleFobnwgGkS½Edlekag. pleFob L / r Ca slenderness ratio. Ggát;EdlmanlkçN³kan;EtRsav tMél slenderness ration kan;EtFM. RbsinebI critical load RtUv)anEckedayRkLaépÞmuxkat; enaHeKnwgTTYl)an critical buckling stress dUcxageRkam³ π 2E P Fcr = cr = ¬$>$¦ A 2 (L / r ) sMrab;kugRtaMgrgkarsgát; karekagnwgekIteLIgtamGkS½EdlRtUvKñanwg r . karekagnwgekIteLIgPøam² enAeBlEdlbnÞúkEdlGnuvtþmkelIGgát;esμInwgbnÞúkEdleGaykñúgsmIkar $># ehIyssrnwgKμanesßr PaBeFobGkS½em (principal axis) EdleFVIeGay slenderness ratio mantMélFMCageK. CaTUeTAvaCa GkS½Edlmanm:Um:g;niclPaBtUcCageK ¬eyIgnwgBinitükrNIenHenAeBleRkay¦. dUcenHm:Um:g;niclPaB Gb,brma nigkaMniclPaBGb,brmaRtUv)aneRbIenAkñúgsmIkar $># nig $>$. ]TahrN_4>1³ ssrEdlmanmuxkat W 300 × 0.73 RtUv)aneRbIedIm,IRTbnÞúksgát;tamGkS½ 645kN . ssrenHmanRbEvg 6m nigmanTMr pinned enAcugsgçag. edaymikKitBIemKuNbnÞúk nigemKuNersIusþg; cUreFVIkarGegátBIesßrPaBrbs;Ggát;enH. ¬eKminRtUvkardwgBIm:akrbs;EdkeT edaysar critical buckling load CaGnuKmn_énm:UDuleGLasÞic minEmn yield stress b¤ ultimate tensile strength¦. dMeNaHRsay³ sMrab; W 300 × 0.73 tMélGb,brmarbs; r = r = 49.8mm y tMélGtibrmarbs; L = 6000 = 120.5 r 49.8 72 eRKOgbgÁúMrgkarsgát;
  • 6. T.chhay π 2 EA π 2 × 200 ⋅ 103 × 9.48 ⋅ 103 Pcr = = ⋅ 10 − 3 = 1288.7 kN (L r ) 2 120.5 2 cMeLIy³ edaysarbnÞúkGnuvtþKW 645kN tUcCag P enaHssrrkSaesßrPaB ehIymanemKuNsuvtßiPaB cr RbqaMgnwg bucklingesμInwg 1288.7 / 645 = 2.0 . eRkaymkeK)anrkeXIjfa smIkarrbs; Euler minGaceRbICamYyGgát;rgkarsgát;EdlFat; xøI nigminRsav. mUlehtuKWfa slenderness ratio tUcrbs;Ggát;bNþaleGayman buckling stress FM ¬BIsmIkar $>$¦. RbsinebI buckling stress FMCag proportional limit rbs;sMPar³ enaHTMnak;TMng rvag stress nig strain nwgmin linear ehIym:UDuleGLasÞic E nwgminGacykmkeRbI)aneT. ¬kñúg]TahrN_ 4>1 buckling stress KW Pcr / A = 1288.7 / 9.48 = 136MPa EdltUcCag proportional limit sMrab;RKb;eRKOgbgÁúMEdkTaMgGs;. enAkñúgqñaM 1889 Friedrich Engesser )anesñI eLIgdMbUgkñúgkareRbIR)as; tangent modulus Et enAkñúgsmIkar $>#. sMrab;sMPar³EdlmanExSekag stress-strain dUckñúgrUbTI 4>5/ E ElgCatMélefrsMrab;kugRtaMgEdlFMCag proportional limit Fpl . Tangent modulus Et RtUv)ankMNt;Ca slope énbnÞat;b:HeTAnwgExSekag stress-strain sMrab;tMélrbs; f EdlsßitenAcenøaH Fpl nig Fy . RbsinebI buckling stress Pcr / A sßitenAkñúgtMbn;enH vaRtUv)an bgðajdUcxageRkam³ π 2 Et I Pcr = L2 ¬$>%¦ smIkar $>% dUcKñanwgsmIkar Euler RKan;EtCMnYs E eday Et . 73 eRKOgbgÁúMrgkarsgát;
  • 7. T.chhay ExSekag stress-strain EdlbgðajenAkñúgrUbTI 4>5 manlkçN³xusKñaBIrUbEdl)anbgðajBImun sMrab; ductile steel ¬enAkñúgrUbTI 1>3 nig1>4¦ edaysarEtvamantMbn; nonlinear . ExSekagenHCaRb ePTénkarBiesaFn_karsgát;rbs;Edk W-shape RbEvgxøI EdleKehAfa stub column. Nonlinearity CalT§pldMbUgénvtþmanrbs; residual stress enAkñúg W-shape. enAeBlEdlEdk hot-rolled shape TukeGayRtCak; muxkat;TaMgmUlrbs;EdkminRtUv)anRtCak;edayGRtadUcKñaeT. ]TahrN_ enAcugsøab rbs;EdkRtCak;elOnCagkEnøgCYbKñarvagsøab nigRTnug. karRtCak;minRBmKñaEbbenHbegáIteGayman kugRtaMgenACab;kñúgEdkrhUt. ktþaepSgeTotdUcCa karpSar nigkarBt;RtCak;edIm,IbegáItFñwmekag GacCa ktþabNþaleGayman residual stress b:uEnþdMeNIrkareFVIeGayRtCak;CaktþacMbg. cMNaMfa Et mantMéltUcCag E ehIysMrab; L / r EdlmantMéldUcKñaRtUvKña eKnwgTTYl)an critical load Pcr tUc. edaysarEtPaBERbRbYlrbs; Et karkMNt;tMél Pcr enAkñúg inelastic range edayeRbIsmIkarTI $>% BitCamankarBi)ak. CaTUeTA trial-and-error approach RtUv)aneRbICamYynwg ExSekag stress-strain dUcbgðajkñúgrUbTI 4>5 edIm,IkMNt; Et sMrab;tMélsakl,grbs;tMél Pcr . sMrab;mUlehtuenH design specification CaeRcIn rYmTaMg AISC Specification manrUbmnþEdl)anBIkar BiesaFn_ (empirical formulas) sMrab; inelastic column. sMrab;RKb;sMPar³TaMgGs; critical buckling stress RtUv)ansg;CadüaRkamCaGnuKmn_eTAnwg slenderness dUcbgðajenAkñúgrUbTI 4>6 . ExSekag tangent modulus b:HeTAnwgExSekag Euler Rtg;cMnucEdlRtUvKñanwg proportional limit rbs;sMPar³. bnSMExSekagenH RtUv)aneKehAfa column strength curve EdlBN’naBIesßrPaBrbs;RKb;ssrTaMgGs;. eRkABI Fy , E nig Et EdlCalkçN³ rbs;sMPar³ ersIusþg;CaGnuKmn_nwg slenderness ratio. 74 eRKOgbgÁúMrgkarsgát;
  • 8. T.chhay RbEvgRbsiT§PaB (effective Length) TaMgsmIkar Euler nigsmIkar tangent modulusQrelIkarsnμt;dUcxageRkam³ !> ssrmanlkçN³Rtg;l¥ @> bnÞúkGnuvtþtamGkS½ KμancMNakp©it #> ssrmanTMr pinned enAcugsgçag lkçxNÐBIrdMbUgmann½yfa Kμanm:Um:g;Bt;enAkñúgGgát;mugeBlekag (buckling). dUc)anerobrab; BIxagedIm m:Um:g;écdnüxøHnwgekItman b:uEnþvaRtUv)anecalkñúgkrNICaeRcIn. tMrUvkarsMrab;TMr pinned Cakar kMNt;mYyEdlBi)ak Edlkarpþl;eGayRtUv)aneFVIsMrab;lkçxNÐTMrepSg²eTot. lkçxNÐTMr pinned tMrUv eGayTb;Ggát;BIkarrMkilxag b:uEnþminTb;nwgkarvilCMuvijTMreT. CakarBit karbegáIttMN pinned EdlKμan kkitKWminGaceFVIeTA)anl¥enaHeT dUcenHlkçxNÐTMrenHRKan;EtmanlkçN³Rbhak;RbEhlb:ueNÑaH. Cak;EsþgssrTaMgGs;RtUvEtxUcRTg;RTaytamGkS½edayesrI. lkçxNÐcugepSgeTotGacRtUv)anBnül;enAkñúgsmIkarTI $>#. CaTUeTA m:Um:g;Bt;GacCaGnuKmn_ én x EdlCalT§plenAkñúg nonhomogeneous differential equation. vamanlkçxNÐRBMEdnxusBI smIkaredIm EtviFIsaRsþKNnadUcKñaTaMgRsug. smIkarEdlCacMelIysMrab; Pcr manTMrg;dUcKña. ]Ta- hrN_ edayBicarNaGgát;rgkarsgát;EdlmanTMrmYyCa pinned nigmYyeTotCa fixed Tb;nwgkarvil nigkar rMkil dUcbgðajenAkñúgrUbTI 4>7 . smIkar Euler sMrab;krNIenH EdlRtUv)anbkRsaytamrebobdUc smIkar $># eKTTYl)an 75 eRKOgbgÁúMrgkarsgát;
  • 9. T.chhay 2.05π 2 EI Pcr = L2 2.05π 2 EA π 2 EA b¤ Pcr = (L / r)2 = (0.70 L / r ) 2 dUcenHGgát;rgkarsgát;enHmanlT§PaBRTbnÞúkesμInwgGgát;EdlmanTMr pinned sgçagEdr Et RbEvgrbs;vaRtUv)anKitRtwm 70% bueNÑaH. eKnwgTTYl)ansmIkarkñúgTMrg;RsedogKñaenHsMrab;ssr EdlmanlkçxNÐTMrepSg². Column-buckling problem GacRtUv)anbegáItCarUbmnþkñúgTMrgCa forth-order differential equation CMnYseGaysmIkar $>@. kareFVIEbbenHedIm,IgayRsYlkñúgkaredaHRsayCamYylkçxNÐRBM EdneRkABITMr pinned . edIm,IPaBgayRsYl smIkarsMrab; critical buckling load nwgRtUv)ansresrkñúgTMrg;dUcxageRkam³ π 2 EA b¤ Pcr = π Et A2 2 Pcr = ¬$>^ a/ $>^ b¦ (KL / r ) 2 (KL / r ) Edl KL CaRbEvgRbsiT§PaB (effective length) nig K CaemKuNRbEvgRbsiT§PaB (effective length factor). emKuNRbEvgRbsiT§PaBsMrab;Ggát;rgkarsgát; fixed-pinned KW 0.70 . sMrab;cugsgçagman TMr fixed Tb;nwgkarvil nigrMkil enaH K = 0.50 . tMélrbs; K sMrab;krNITaMgenH nigkrNIepSg eTotmanenAkñúgtarag C_C2.1 enAkúñg Commentary to the AISC Specification. enAkñúgtaragenaH eKeGaytMélrbs; K cMnYnBIr³ mYyCatMéltamRTwsþI nigmYyeTotCatMélsMrab;karKNna (recommended design value) EdlRtUv)anykmkeRbIenAeBlEdleKmanlkçxNÐTMresÞIrl¥tex©aH. dUcenH luHRtaEtTMr fixed KWbgáb;tex©aHeTIbtMélKNnaEdlmanlkçN³snSMsMécCagRtUv)anykmk eRbI. EtcMNaMfa tMéltamRTwsþI nigtMélsMrab;karKNnamantMéldUcKñasMrab;lkçxNÐ (d)nig (f) enAkñúg Commentary Table C-C2.1. mUlehtuKWfaPaBEdlminGaceFVI)anrbs;TMrsnøak;KμankkitEdl l¥tex©aH b¤rbs;TMr pinned )anbegáIteGaymankarTb;nwgkarvil nigeFVIeGaytMél K fycuH. dUcenHkareRbItMéltamRTwsþIkñúg krNITaMgBIrKWmantMéltUc. kareRbIRbEvgRbsiT§PaB KL CMnYseGayRbEvg L min)aneFVIeGaymankarpøas;bþÚrTMnak;TMng Edl)anerobrab;knøgmkeT. ExSekagersIusþg;ssr (column strength curve) Edl)anbgðajenAkñúg rUbTI 4>6 minmankarpøas;bþÚreT ebIRKan;EteFVIkarpøas;bþÚreQμaHGkS½Gab;sIusmk KL enaH. Critical 76 eRKOgbgÁúMrgkarsgát;
  • 10. T.chhay buckling stress EdlRtUvKñanwgRbEvgEdleGay eTaHCaRbEvgBitR)akd b¤RbEvgRbsiT§PaBk¾eday k¾ eQμaHrbs;vaelIGkS½GredaenenArkSadEdl. 4>3> tMrUvkarrbs; AISC AISC Requirements tMrUvkarCamUldæansMrab;Ggát;rgkarsgát;RtUv)anerobrab;enAkñúg Chapter E of the AISC Specification. TMnak;TMngrvagbnÞúk nigersIusþg; ¬smIkar @>#¦ manTMrg; Pu ≤ φc Pn Edl Pu = plbUkbnÞúkemKuN Pn = nominal compressive strength = Ag Fcr Fcr = critical buckling stress emKuNersIusþg;sMrab;Ggát;rgkarsgát; = 0.85 φc = CMnYseGaykareRbIsmIkar critical buckling stress Fcr CaGnuKmn_én slenderness ration KL / r specification eRbInUv slenderness parameter KL Fy λc = (AISC Equation E2-4) rπ E vaCa)a:ra:Em:RtKμanxñat ebIeTaHCasmIkarmanlkçN³sMPar³cUlrYmk¾eday. sMrab;ssreGLasÞic (elastic column) smIkar $>$ GacRtUv)ansresrCa π 2E 1 Fcr = = Fy (LK / r ) 2 λ2 c edIm,IKitbBa¢ÚlnUvT§iBlrbs;PaBminRtg;dMbUg (initial crookedness) smIkarxagelIRtUv)ankat;bnßy dUcxageRkam 0.877 Fcr = Fy λ2 c sMrab; inelastic column EdleRbI tangent modulus equation ¬smIkar $>^ b¦ RtUv)anCMnYseday ( Fcr = 0.658λc Fy 2 ) Edl)anKitpgEdrnUv initial crookedness. dUcenHdMeNaHRsayedaypÞal;GacTTYl)an edayeCos vagnUv trial-and error approach EdlmanCab;CamYynwgkareRbIR)as; tangent modulus equation. RbsinebIeKyk λc = 1.5 CaRBMEdnrvagssreGLasÞic nigssrminEmneGLasÞic enaH AISC equation sMrab; critical buckling stressGacRtUv)ansegçbdUcxageRkam³ sMrab; λc ≤ 1.5 77 eRKOgbgÁúMrgkarsgát;
  • 11. T.chhay ( Fcr = 0.658λc Fy 2 ) (AISC Equation E2-2) sMrab; λc > 1.5 0.877 Fcr = Fy (AISC Equation E2-3) λ2 c tMrUvkarTaMgenHRtUv)anbgðajCalkçN³RkaPicenAkñúgrUbTI 4>8. AISC Equation E2-2 nig E2-3 RtUv)ansegçbBIsmIkarcMnYn 5 Edlman λc 5 lMdab; (Galambos, 1988). smIkarTaMgenHQrelIkarBiesaFn_ nigRTwsþIEdlKitbBa¢ÚlnUv residual stress nig initial out-of straightness esμInwg L / 1500 / Edl L CaRbEvgGgát;. AISC esñInUvpleFobPaBrlas;Gtibrma (maximum slenderness ration) KL / r esμInwg 200 sMrab;Ggát;rgkarsgát;. eTaHbICamankarkMNt;EtmYyk¾eday EtenAkñúgkarGnuvtþn_eKGacykpl eFobkMNt;FMCagenH edaysarssrEdlmanlkçN³RsavCag nwgmanersIusþg;tUc ehIyvanwgminman lkçN³esdækic©. ]TahrN_4>2³ kMNt;ersIusþg;sgát;KNnarbs; W 360 ×1.08 EdlmanRbEvg 6m nigmanTMr pinned. eRbIEdk A36 . dMeNaHRsay³ Slenderness ratio³ tMélGtibrmarbs; KL = KL = 1.0(63 ) = 95.24 < 200 r r 6000 (OK) y KL Fy 95.24 250 λc = = = 1.072 rπ E π 200000 sMrab; λ c < 1.5 78 eRKOgbgÁúMrgkarsgát;
  • 12. T.chhay Fcr = (0.658)λc Fy = (0.658)1.072 (250 ) = 154.5MPa 2 2 Pn = Ag Fcr = 14100 × 154.5 × 10 −3 = 2177kN φc Pn = 0.85 × 2177 = 1850kN cMeLIy³ ersIusþg;sgát;KNna (design compressive strength) = 1850kN . enAkñúg]TahrN_ 4>2/ eday ry < rx enaHvanwgmanersIusþg;FMCagtamTis x . EdkTIbRCugmux kat;kaer: Camuxkat;EdlmanRbsiT§PaBCageKsMrab;Ggát;rgkarsgát; edaysar ry = rx enaHersIusþg; rbs;vanwgesμIKñaTaMgBIrTis. eBlxøHEdkTIbmUlRbehagk¾RtUv)aneRbICaGgát;egkarsgát;sMrab;mUlehtu dUcKña. rUbragénkar)ak;Edl)anBicarNayUrmkehIyKWsMedAeTAelIkarekagedaykarBt; (flexural buckling) dUcGgát;rgkarBt; enAeBlEdlvaKμanesßrPaB. sMrab;muxkat;xøH Ggát;nwg)ak;edayrmYl (twisting) KWekagedayrmYl (torsional buckling)b¤edaybnSMén twisting nig bending (flexural- torsional buckling). eyIgnwgBicarNavaenAkñúgEpñkTI 4>6. esßrPaBedaytMbn; Local Stability ersIusþg;EdlRtUvKñanwg buckling mode minGacnwgekIteLIg)aneT RbsinebIEpñkrbs;muxkat; manlkçN³esþIgeBkEdlnwgekItman local buckling. GesßrPaBRbePTenHKWCakarekagedaytMbn; b¤ wrinkle enAtMbn;epSg²Kña. RbsinebIvaekIteLIg muxkat;KμanRbsiT§PaBeBj)anyUr hIyGgát;nwg)ak;. muxkat;rUbragGkSr I nig H Edlmansøab b¤RTnugesþIgnwggayrg)atuPUtenH ehIyeKKYrEteCogvagkñúg kareRbIR)as;va. RbsinebImindUecñaHeT ersIusþg;sgát;EdleGayeday AISC Equation E2-2 nig E2-3 RtUvEtkat;bnßy. karvas;EvgnUvPaBgayrgnUv)atuPUtenHKWKNnapleFobTTwgelIkMras; (width- thickness ratio) rbs;Epñkénmuxkat;nImYy². EpñkBIrRbePTRtUv)anBicarNa ³ unstiffened element EdlRCug mYytambeNþayTisedAbnÞúkminRtUv)an support, nig stiffened element EdlRCugTaMg sgçagrbs;vaRtUv)an support. tMélkMNt;rbs; width-thickness ratio RtUv)aneGayenAkñúg AISC B5, “Local Buckling” EdlrUbragrbs;muxkat;RtUv)ancat;cMNat;fñak;Ca compact, noncompact b¤ slender GaRs½yeTAtam tMélrbs;pleFob. sMrab;Epñkrgkarsgát;esμI dUcCaGgát;rgkMlaMgsgát;tamGkS½ ersIusþg;RtUv)ankat; 79 eRKOgbgÁúMrgkarsgát;
  • 13. T.chhay bnßyRbsinebIrUbragman slender element. Width-thickness ratio RtUv)aneGayeQμaHsMKal;CaTU eTAfa λ . GaRs½yeTAnwgEpñkrbs;muxkat; λ GacCapleFob b / t b¤ h / tw EdlnwgRtUv)anbgðajenA TIenH. RbsinebI λ FMCagtMélkMNt; λr rUbragKW slender ehIyeKrkviFIedIm,IkarBar local buckling. ¬sMrab;rUbrag compact nig uncompact nwgRtUvykmkniyaykñúgCMBUkTI5¦ sMrab;rUbragGkSr I nig H søabrbs;vaRtUv)ancat;TukCa unstiffened element ehIyTTwgrbs;søabGacRtUv)anKitEtBak;kNþal. edayeRbI AISC notation eyIg)an³ b bf / 2 bf λ= = = t tf 2t f Edl b f nig t f CaTTwg nigkMras;rbs;søab. lImItx<s;KW 250 λr = fy RTnugrbs;rUbragGkSr I nig H Ca stiffened element ehIy stiffened width KWCacMgaycenøaH root rbs;søab. Width-thickness parameter KW h λ= tw Edl h CacMgaycenøaH root rbs;søab ehIy tw CaTTwgsøab. lImItx<s;bMputKW 665 λr = fy tMélrbs;pleFob b f / 2t f nig h / tw RtUv)anerobcMdak;enAkñúg dimension and properties tables in Part 1 of the manual. Stiffened element nig unstiffened element rbs;rUbragmuxkat;CaeRcInRtUv)anbgðajenAkñúg rUbTI 4>9. EdnkMNt; λr Edl)anmkBI AISC B5 RtUv)aneGaysMrab;krNInImYy². 80 eRKOgbgÁúMrgkarsgát;
  • 14. T.chhay ]TahrN_³ Gegát;ssrenAkñúg]TahrN_ 4>2 sMrab; local buckling. dMeNaHRsay³ sMrab; W 360 × 1.08 / b = 256mm / t = 19.9mm / nig f f bf 256 = = 6.43 2t f 2 × 19.9 tMélén b f / 2t f k¾RtUv)andak;enAkñúg properties table. 250 = 15.8 > 6.43 (OK) 250 h tw = 25.3 ¬BI properties table ¦ 81 eRKOgbgÁúMrgkarsgát;
  • 15. T.chhay 665 665 = = 42 > 25.3 (OK) fy 250 cMeLIy³ Local instability minmanbBaða. eKk¾GnuBaØateGayeRbIrUbragmuxkat;EdlminbMeBjtMrUvkar width-thickness ration pgEdr k¾ b:uEnþGgát;EbbenaHminRtUv)anGnuBaØateGayRTbnÞúkF¶n;²dUcrUbragmuxkat;EdlbMeBjlkçxNÐeT. müa:g vijeTot design strength k¾GacRtUv)ankat;bnßyedaysarEt local buckling. dMeNIrkarTUeTAkñúgkar GegátmandUcxageRkam. - RbsinebI width-thickness ration λ FMCag eyagtam Appendix B of λr the Specification nigKNnaemKuNkat;bnßy (reduction factor) Q . - KNna λc dUcFmμta³ λc = KL Fy rπ E - RbsinebI Qλc ≤ 1.5 / Fcr = Q⎛ 0.658Qλc2 ⎞Fy ⎜ ⎟ (AISC Eq. A-B5-15) ⎝ ⎠ - RbsinebI Qλc > 1.5 / Fcr = ⎡ 0.877 ⎤ Fy ⎢ 2 ⎥ (AISC Eq. A-B5-16) ⎢ λc ⎥ ⎣ ⎦ - Design strength KW φc Pn = 0.85 Ag Fcr kñúgkrNICaeRcIneKGacrk rolled shape EdlbMeBjtMrUvkar width-thickness ratio dUcenHeK mincaM)ac;eFVInUvdMeNIrénkarKNnaenHeT. enAkñúgesovePAenH eyIgBicarNaEtGgát;rgkarsgát;Edlman λ < λr bu:eNÑaH. taragsMrab;Ggát;rgkarsgát; Tables for Compression Members Manual mantaragEdlmanRbeyaCn_CaeRcInsMrab;karviPaK nigkarKNna. sMrab;Ggát;rgkar sgát;Edl strength rbs;valubeday flexural buckling ¬RbePTEdl)anBicarNaknøgmk¦/ tarag 3- 36, 3-50 nig 4 enAkñúg Numerical Value section rbs; Specification nig column load table enAkñúg part 3 rbs; Manual, “Column Design,” manRbeyaCn_CageK. tarag 3-36 eGaynUvtMél φc Fcr Ca GnuKmn_én KL / r sMrab; Fy = 36ksi = 250MPa . tarag 3-50 sMrab; Fy = 50ksi = 350MPa nig tarag 4 eGay φc Fcr / Fy CaGnuKmn_én λc . ¬RKb; Manual table TaMgGs;sMrab; Fy = 50ksi 82 eRKOgbgÁúMrgkarsgát;
  • 16. T.chhay = 350 MPa xusBItaragsMrab; Fy = 36ksi = 250MPa edaykarpat;BN’RbepH¦. Column load table eGay design strength rbs;rUbragEdleRCIserIssMrab;tMélRbEvgRbsiT§PaB (effective length) CaeRcIn. tarag 3-36 nig 3-50 bBa©b;edaylImItx<s;bMput KL / r = 200 ehIy column load table rYm bBa©ÚltMél KL EdlRtUvKñanwg KL / r = 200 . kareRbIR)as;nUvtaragnImYy²RtUv)anbgðajenA kñúg]TahrN_xageRkam. ]TahrN_ 4>4³ KNna design strength rbs;Ggát;rgkarsgát;rbs; W 14 × 74 EdlmanRbEvg 20 ft nigmanTMr pinned renAcugsgçag edayeRbI ¬!¦ Table 3-36 ¬@¦ Table 4 nig ¬#¦ column load table . eRbIEdk A36 . dMeNaHRsay³ Slenderness ratio: tMélGtibrma KL = KL = 1.0(2.4812) = 96.77 < 200 r r 20 × (OK) y KL Fy 96.77 36 λc = = = 1.085 rπ E π 29000 ¬!¦ sMrab; Fy = 36ksi / eyIgeRbI Table 3-36 . tMélrbs; φc Fcr RtUv)aneGaysMrab;tMél KL / r Kt;/ sMrab;tMél KL / r TsSPaK eyIgGaceFVIkarrMkil ex,óseLIg (rounded up) b¤eFVI linear interpolation. enAkñúgesovePAenHeyIgnwgeRbI linear interpolation sMrab;RKb;taragTaMgGs;elIkElgEtmankarbgðajR)ab;. sMrab; KL / r = 96.77 φc Fcr = 18.69ksi φc Pn = φc Ag Fcr = Ag (φc Fcr ) = 21.8(18.69) = 407 kips ¬@¦ BI Table 4 sMrab; λc = 1.085 eyIg)an Fcr φc = 0.519 Fy ⎛ ⎞ φc Pn = Ag ⎜ φc Fcr ⎟ Fy = 21.8(0.519)(36) = 407kips ⎜ Fy ⎟ ⎝ ⎠ ¬#¦ Column load table in Part 3 of the Manual eGay design strength sMrab;muxkat;rUbrag W, HP, pipe, tube, double-angle, WT nig single-angle. tMélenAkñúgtaragsMrab;rUbragsIuemRTI (W, 83 eRKOgbgÁúMrgkarsgát;
  • 17. T.chhay HP, pipe nig tube)RtUv)anKNnaedayeRbI radius of gyrationsMrab;rUbragnImYy². sMrab;]TahrN_ enH k = 1.0 dUcenH KL = 1.0(20 ) = 20 ft sMrab; W 14 × 74 / Edk A36 nig KL = 20 ft eyIgTTYl)an φc Pn = 407kips . tMélEdl)anBI Table 3-36, 3-50 nig 4 KWQrelI flexural buckling nig AISC Equation E2-2 nig E2-3. dUcenH local stability RtUv)ansnμt; ehIy width-thickness ratio nwgminFMCagtMél kMNt;eLIy. Design strength enAkñúg column load table )anKitbBa¢ÚlenAkarkat;bnßycaM)ac; enAeBlEdl width-thickness ratio FMCagtMélkMNt;. 4>4> karKNnamuxkat; Design kareRCIserIsnUv rolled shape EdlmanlkçN³esdækic© edIm,ITb;nwgbnÞúksgát;EdleGayman lkçN³samBaØCamYynwgkareRbIR)as; column load tables. emIltaragCamYynwg effective length ehIyrMkiltamTisedk rhUtdl;eyIgrkeXIjnUv design strength EdleyIgcg;)an ¬b¤mantMélFMCag bnþicbnþÜc¦. kñúgkrNIxøH eyIgRtUvbnþrkrhUtdl;eyIgGacrk)anrUbragEdlmanTMgn;RsalCageK. CaTU eTArUbrag (W, WT, etc) RtUv)aneKeFVIkarsMercmuneK. CaerOy² TMhM nigrUbragrbs;muxkat; RtUv)andwgmun edaytMrUvkarsßabtükmμ nigtMrUvkard¾éTeTot. dUcEdl)anbgðajBIxagedIm RKb;tMél EdlmanenAkñúgtaragRtUvKñanwg slenderness ratio tUcCagb¤esμInwg 200 . rUbragGt;sIuemRTI (structural tees and the single and double-angles) RtUvkarnUvkarBicarNaBiessEdlnwgmanbk RsayenAkñúgEpñk 4>6. ]TahrN_ 4>5³ Ggát;rgkarsgát;RTnUv service dead load 165kips = 734kN nig service live load 535kips = 2380kN . Ggát;enHmanTbEvg 26 ft = 7925mm ehIymanTMr pinned sgçag. eRbIEdk A36 nigerIsRubrag W 14 . dMeNaHRsay³ KNnabnÞúkemKuN (factored load)³ Pu = 1.2 × 165 + 1.6 × 535 = 1054kips b¤ 4689kN dUcenH required design strength φc Pn = 1054kips 84 eRKOgbgÁúMrgkarsgát;
  • 18. T.chhay BI column load table sMrab; KL = 26 ft / W 14 × 176 man design strength φc Pn = 1150kips cMeLIy³ eRbI W 14 × 176 . ]TahrN_ 4>6³ eRCIserIsrUbrag W EdlmanTMgn;RsalCageKbMputEdlGacRTbnÞúksgát;emKuN Pu = 190kips = 845kN . RbEvgRbsiT§PaBKW 24 ft = 7315m . eRbIEdk ASTM A572 Grade 50. dMeNaHRsay³ viFId¾smrmüenATIenHKWdMbUgeyIgerIsrUbragEdlRsalCageKenAkñúg nominal size nI mYy² ehIybnÞab;mkeTIberIsrUbragEdlRsalCageKelIrUbragTaMgGs;. CMerIsmandUcxageRkam³ W 4 / W 5 nig W 6 ³ KμanrUbragNamYyenAkñúgtaragEdlGacyk)an W8 ³ W 8 × 58 / φc Pn = 194kips W 10 ³ W 10 × 49 / φc Pn = 239kips W 12 ³ W 12 × 53 / φc Pn = 247kips W 14 ³ W 14 × 61 / φc Pn = 276kips cMNaMfa load capacity minsmamaRtnwgTMgn;eT ¬b¤RkLaépÞmuxkat;eT¦. eTaHbICa W 8 × 58 man design strength tUcCageKkñúgcMeNamCMerIsTaMgbYn EtvamanTMgn;F¶n;CageKbnÞab; W 14 × 61 . cMeLIy³ eRbI W 10 × 49 . sMrab;rUbragEdlKμanenAkñúg column load table, eKRtUveRbI trial-and-error approach. dMeNIr karTUeTAKWsnμt;rUbrag bnÞab;mkKNna design strength rbs;va. RbsinebIersIusþg;tUceBk ¬Kμansuvtßi PaB¦ b¤FMeBk ¬KμanlkçN³esd©kic©¦ eKRtUveFVIkarsakl,gepSgeTot. viFIsaRsþkñúgkareFVI trial selection mandUcxageRkam³ !> snμt;tMélsMrab; critical buckling stress Fcr . karBinitü AISC equation E2-2 nig E2-3 bgðajfatMél Fcr GtibrmatamRTwsþICa yield stress Fy . @> BItMrUvkarKW φc Pn ≥ Pu / yk φc Ag Fcr ≥ Pu enaH Ag ≥ φ PF u c cr #> eRCIserIsrUbragEdlRtUvKñanwgRkLaépÞcaM)ac;. $> KNna Fcr nig φc Pn sMrab;rUbragsakl,g. 85 eRKOgbgÁúMrgkarsgát;
  • 19. T.chhay %> eFVIkarEktMrUveLIgvijRbsinebIcaM)ac;. RbsinebI design strength mantMélEk,rtMélRtUvkar TMhMEdlmanenAkñúgtaragbnÞab;GacRtUv)ansakl,g. RbsinebImindUecñaHeT eFVIkarKNna eLIgvijTaMgRsug. eRbItMél Fcr EdlrkeXIjsMrab;tMélsakl,gCatMélsMrab;CMhanTI !>. ^> RtYtBinitü local stability ¬RtYtBinitü width-thickness ration). EktMrUveLIgvijRbsin ebIcaM)ac;. ]TahrN_ 4>7³ eRCIserIsrUbrag W 460 rbs;Edk A36 EdlGacRTbnÞúkemKuN (factored load) 4688kN . RbEvgRbsiT§PaBKW 7925mm . dMeNaHRsay³ sakl,g Fcr = 165.5kN ¬BIrPaKbIén Fy ¦³ Pu 4688 ⋅103 Required Ag = = = 33.325 ⋅10 − 3 m 2 φc Fcr 0.85 × 165.5 sakl,g W 460 × 2.8 Ag = 36.39 ⋅ 10 −3 m 2 > 33.325 ⋅ 10 −3 m 2 KL 7925 = = 111.8 < 200 (OK) rmin 70.9 KL Fy 111.8 250 λc = = = 1.258 < 1.5 rπ E π 200000 eRbI AISC Equation E2-2 ⎛ 2⎞ Fcr = ⎜ 0.658λc ⎟ Fy = (0.658)(1.258) (250) = 128.9MPa 2 ⎝ ⎠ φc Pn = 0.85 Ag Fcr = 0.85 × 36.39 ⋅10 −3 ×128.9 ⋅103 = 3987kN < 4688kN (N.G) sakl,g Fcr = 128.9 MPa ¬tMélEdleTIbnwg)anBIkarKNnasMrab; W 460 × 2.8 ¦ Pu 4688 ⋅103 Required Ag = = = 42.787 ⋅10 − 3 m 2 φc Fcr 0.85 ×128.9 sakl,g W 460 × 3.41 Ag = 44.39 ⋅10 −3 m 2 > 42.787 ⋅10 −3 m 2 KL 7925 = = 109.5 < 200 (OK) rmin 72.4 KL Fy 109.5 250 λc = = = 1.232 < 1.5 rπ E π 200000 eRbI AISC Equation E2-2 86 eRKOgbgÁúMrgkarsgát;
  • 20. T.chhay ⎛ 2⎞ Fcr = ⎜ 0.658λc ⎟ Fy = (0.658)(1.232 ) (250) = 132.45MPa 2 ⎝ ⎠ φc Pn = 0.85 Ag Fcr = 0.85 × 44.39 ⋅10 −3 ×132.45 ⋅103 = 4997.5kN > 4688kN (O.K) edaysarrUbragenHminmanenAkñúg column load table dUcenHeKRtUvkarRtYtBinitü width-thickness ration bf 250 = 2 .8 < = 15.8 (O.K) 2t f 250 h 665 = 13.8 < = 42.2 (O.K) tw 250 cMeLIy³ eRbIEdk W 460 × 3.41 RbsinebIeKeRbI table 3-36 b¤ table 3-50 tMélsakl,grbs; φc Fcr manlkçN³gayRsYlkñúg kareRbIenAkñúgsmIkar Pu Required Ag = φc Fcr 4>5> esckþIbEnßmsMrab;RbEvgRbsiT§PaB More on Effective Length enAkñúgEpñk 4>2 “column theory” )anENnaMBIRbEvgRbsiT§PaB. RKb;gGát;rgkarsgát;TaMg RtUv)anKitCaTMr pinned edayminKitBIlkçxNÐcugTMrBitR)akd EdlnegeFVIeGayRbEvgRbsiT§PaB KL mantMélxusBIRbEvgBitR)akd. CamYynwgkarEkERbenH load capacity rbs;Ggát;rgkarsgát;Ca GnuKmn_Etnwg slenderness parameter λc . enAeBlEdleKsÁal;lkçN³rbs;sMPar³ vaCaGnuKmn_eTA nwg slenderness ration KL . RbsinebIGgát;rgkarsgát;manTMrepSgKñaenAelIGkS½emrbs;va enaHvanwgmanRbEvgRbsiT§PaB epSgKñaenAelIGkS½TaMgBIr. enAkñúgrUbTI 4>10 W -shape RtUv)aneRbICassr ehIyenAEpñkxagelIva RtUv)anBRgwgedayGgát;edkenAelITisTaMgBIrEdlEkgKña. Ggát;TaMgenHkarBarkarrMkilrbs;ssrRKb; TisedA EtkarlMGitrbs;ssrminRtUv)anbgðajEdlGnuBaØateGaykarviltictYcekItman. eRkamlkç- xNÐenH Ggát;GacnwgRtUv)anKitCaTMr pinned enAEpñkxagelI. sMrab;mUlehtudUcKña tMNedIm,IRTTMrenA xageRkamk¾GacKitCatMN pinned Edr. CaTUeTA eKBi)aknwgTTYl lkçxNÐ rigid b¤ fixed Nas; luHRtaEteKdak; lkçxNÐBiess. tMNFmμta CaTUeTAxiteTArktMN hinge b¤ pinned . enARtg;Bak; kNþalkMBs;ssrRtUv)anBRgwgEttamTismYy. tMNkarBarEtkarrMkil EtvaminTb;karvileT. kar BRgwgenHkarBarkarrMkiltamGkS½exSayrbs;muxkat; b:uEnþmin)anTb;karrMkilTisxøaMgeT. dUc)anbgðaj 87 eRKOgbgÁúMrgkarsgát;
  • 21. T.chhay enAkñúgrUbTI 4>10 RbsinebIGgát;ekagtamGkS½xøaMg RbEvgRbsiT§PaBrbs;vaKW 7.9m b:uEnþkarekagtam TisexSayGacekagkñúgrUbrag second buckling mode RtUvKñanwgRbEvgRbsiT§PaB 3.95m . edaysarersIusþg;rbs;vaRcassmamaRteTAnwgkaer:én slenderness ratio ssrnwgekagkñúgTisedA Edlman slenderness ration FMCageK dUcenHeKRtUveRbobeFob K x L / rx CamYynwg K y L / ry . enA kñúgrUbTI 4>10 pleFob 7900 / rx RtUv)aneRbobeFobCamYynwg 3950 / ry ¬Edl rx nig ry KitCa mm ¦ ehIypleFobEdl mantMélFMCageKRtUv)aneRbIsMrab;kMNt; nominal axial compressive strength Pn . ]TahrN_4>8³ Edk W 300 × 0.95 manRbEvg 7.2m RtUv)anRTedayTMr pinned sgçag ehIyTb;tamTis exSayRtg;cMnucmYyPaKbI dUcbgðajkñúgrUbTI 4>11. eRbIEdk A36 kMNt; design compressive strength . 88 eRKOgbgÁúMrgkarsgát;
  • 22. T.chhay dMeNaHRsay³ K x L 7200 = = 53.7 rx 134.1 K y L 2400 = = 31.3 ry 76.7 K x L / rx mantMélFMCag dUcenHvamanlkçN³lub. BI table 3-36 CamYynwg KL / r = 53.7 φc Fcr = 26.29ksi = 26.29 × 6.895 = 181.3MPa φc Pn = Ag (φc Fcr ) = 12.32 ⋅ 103 × 181.3 ⋅ 10 −3 = 2233.6kN cMeLIy³ Design strength = 2233.6kN Design strength EdleGayenAkñúg column load table KWQrelIRbEvgRbsiT§PaBtamGkS½ y . dMeNIrkarsMrab;eRbIR)as;taragenHCamYynwg K x L GaceFVIeTA)anedaydwgBIedImehtuEdleKTTYl)an tMélenAkñúgtaragenH. edaycab;epþImCamYynwgtMél KL eKnwgTTYl)an φc Pn edaydMeNIrkarRsedog KñanwgdMeNIrkarxageRkam³ - KL RtUv)anEckeday ry edIm,ITTYl)an KL / ry . - KNna slenderness parameter λc = rKL Fyπ E y - KNna Fcr - KNna design strength φc Pn = 0.85 Ag Fcr dUcenHersIusþg;Edl)anerobCataragKWQrelItMélrbs; KL EdlesμInwg K yL . RbsinebIlT§PaBRT RTg;eFobnwgTisedA x eKGaceRbItaragedayCMnYs 89 eRKOgbgÁúMrgkarsgát;
  • 23. T.chhay KxL KL = rx / ry enaHbnÞúkEdlenAkñúgtaragnwgQrelI KL K x L /( rx / ry ) K x L = = ry ry rx pleFob rrx RtUv)aneGayenAkñúg column load table sMrab;rUbragnImYy². y ]TahrN_ 4>9³ Ggát;rgkarsgát;dUcbgðajenAkñúgrUbTI 4>12 manTMr pinned sgçagehIyenARtUv)anTb; tamTisenABak;kNþalkMBs;ssr. Service load KW 400Kips EdlbnÞúkefr nigbnÞúkGefrmantMél esμIKña. eRCIserIs W-Shape EdlmanTMgn;RsalCageK. dMeNaHRsay³ Factored load = Pu = 1.2 × 200 + 1.6 × 200 = 560kips edaysnμt;faTisedAexSaylub ehIyBinitüemIlkñúg column load table CamYynwg KL = 9 feet . cab;epþImCamYynwgrUbragtUcCageK dMbUgeyIgrk)anrUbrag W 10 × 77 CamYynwg design strength 632kips . RtYtBinitüGkS½xøaMg KxL 18 = = 10.40 ft > 9 ft rx / ry 1.73 KxLmanlkçN³lubsMrab;rUbragenH emIltaragCamYy KL = 10.4 feet . W 10 × 77 enAEtCarUbragRsalCageKsMrab; W 10 CamYynwg design strength 612kips ¬eRkayeBleFVI interpolation¦. bnþGegátelI W 12 × 72 ³ KxL 18 = = 10.3 ft > 9 ft rx / ry 1.75 KxL enAEtlub ehIyman design strength 592kips . kMNt;rUbragEdkRsalCageKsMrab; W 14 . rUbragEdlRsalCageKKW W 14 × 74 EtvaF¶n;CagrUbragEdl)anrkBIelIkmun. cMeLIy³ eRbIEdk W 12 × 72 90 eRKOgbgÁúMrgkarsgát;
  • 24. T.chhay RKb;eBlTaMgGs;EdlGaceFVIeTA)an GñkKNnaKYrEtbEnßmTMrsMrab;TisedAexSayrbs;ssr. RbsinebImindUcenaHeT Ggát;nwgKμanRbsiT§PaB³ vamanersIusþg;FMEtmYyTis. enAeBl K x L mantMél xusKñaBI K y L enaH K y L nwglub elIkElgEt rx / ry tUcCag K x L / K y L . enAeBlpleFobTaMgBIr esμIKña ssrnwgmanersIusþg;esμIKñakñúgTisedATaMgBIr. sMrab; W-shape enAkñúg column load table rx / ry sßitenAcenøaH 1.6 nig 1.8 elIkElgsMrab;rUbragEdlRsalCagxøH. ]TahrN_ 4>10³ ssrEdlbgðajenAkñúgrUbTI 4>13 RTnUv factored axial load 840 Kips . eRbIEdk A36 ehIyeRCIserIs W-Shape. dMeNaHRsay³ K x L = 20 ft nigtMélGtibrmarbs; K y L = 8 ft RbEvgRbsiT§PaB K x L manlkçN³lubenAeBlEdl KxL > K yL rx / ry b¤k¾enAeBlEdl ⎛r ⎞ KxL rx / ry > K yL KxL > ⎜ x ⎜ ry ( ⎟ KyL ⎟ ) ⎝ ⎠ kñúgkrNIenH K x L 20 kyL = 8 = 2.5 b¤ k x L = 2.5K y L edaysar K x L mantMélFMCag K y L q¶ay enaH K x L RbEhlCanwglub. mUlehtuKWfatMél rx / ry EdlmanenAkñúgtaragPaKeRcInmantMéltUcCag 2.5 dUcenH k x L = 2.5 K y L TMngCanwgFMCag 91 eRKOgbgÁúMrgkarsgát;
  • 25. T.chhay (rx / ry )K y L . sakl,g rx / ry = 1.7 ³ K x L 20 = = 11.76 > K y L rx / ry 1.7 rMkillT§pleGayeTACa KL = 12 ft ehIyBinitüemIlenAkñúg column load table . sakl,g W 10× 112 ¬ φc Pn = 865kips ¦³ tMélBitR)akd rK/xrL = 120 = 11.5 ft < 12 ft .74 x y φc Pn > 840kips EdlRtUvkar ¬edayeFVI interpolation φc Pn = 876kips RtYtBinitü W 12×106 KxL 20 = = 11.4 ft rx / ry 1.76 sMrab; KL = 12 ft φc Pn = 853kips > 840 ft (OK) GegátrUbrag W 14 . sMrab; rx / ry = 1.7 ¬pleFobRbEhlsMrab;RKb;krNIEdlGacekItman¦ K x L 20 = = 11.76 ft > K y L = 8 ft rx / ry 1.7 sMrab; KL = 12 ft / W 14 ×109 EdlmanlT§PaBRTRTg; 905kips CarUbragEdlRsalCagsMrab; W 14 . RbEvg 12 ft CatMélEdlmanlkçN³snSMsMécénRbEvgRbsiT§PaBBitR)akd rUbragenHKWRKb;RKan;. cMeLIy³ eRbI W 12 ×106 ¬RsalCageKkñúgcMeNambIrUbragEdl)ansikSa¦ 92 eRKOgbgÁúMrgkarsgát;
  • 26. T.chhay sMrab;ssrdac;edayELk (isolated column) EdlminEmnCaEpñkrbs;eRKagCab; (continuous frame), Table C-C2.1 enAkñúg Commentary to the specification manlkçN³RKb;RKan;CaTUeTA. EtsMrab;eRKagrwg (rigid frame) enAkñúgrUbTI 4>14. ssrenAkñúgeRKagenHminmanlkçN³ÉkraC EtvaCa Epñkrbs;rcnasm<n§½Cab;. elIkElgsMrab;ssrEdlenACan;eRkam ssrRtUv)anTb;enAcugsgçagrbs;va edayFñwmnwgssrd¾éTeTot. eRKagenHk¾Ca unbraced frame mann½yfaeRKagGacmanbMlas;TItamTis edk ehIyssrTaMgGs;rgnUv sidesway. RbsinebIeKeRbI Table C-C2.1 sMrab;eRKagenH ssrCan; eRkameKmanlkçxNÐRbhak;RbEhlnwglkçxNÐ (f) ehIytMélrbs; K = 0 GacRtUv)aneRbI. sMrab; ssrEdldUcssr AB tMélrbs; K = 1.2 EdlRtUvKñanwglkçxNÐ (c) GacRtUv)aneRCIserIs. EtdM eNIrkarEdlsmRsbCag nwgKitGMBIkMriténkarTb;Edlpþl;eGayedaytMNrbs;Ggát;. karTb;nwgkarvilEdlpþl;eGayedayFñwm b¤rtenAxagcugssrCaGnuKmn_eTAnwg rotational stiffness rbs;Ggát;EdlRbsBVKñaenARtg;cMnucenaH. Rotational stiffness rbs;Ggát;CasmamaRteTA nwg EI / L / Edl I Ca moment of inertia rbs;muxkat;eFobnwgGkS½énkarBt;. Gaylord nig 93 eRKOgbgÁúMrgkarsgát;
  • 27. T.chhay Stallmeyer (1992) )anbgðajfaemKuNRbEvgRbsiT§PaB K GaRs½ynwgpleFobrbs; column stiffness elI girder stiffness enAxagcugrbs;Ggát;nImYy² EdlGacsMEdgCa G= ∑ Ec I c / Lc = ∑ I c / Lc ¬$>&¦ ∑E I /L g g g ∑I /L g g Edl ∑ Ec I c / Lc = plbUk stiffness rbs;ssrTaMgGs;EdlenAcugrbs;ssrEdlBicarNa ∑ E g I g / Lg = plbUk stiffness rbs;rtTaMgGs;EdlenAcugrbs;ssrEdlBicarNa m:UDuleGLasÞicrbs;eRKOgbgÁúMEdk Ec = E g = E = RbsinebIssrEdlRsavxøaMg (very slender column) RtUv)anP¢ab;eTAnwgrtEdlmanmuxkat;FM enaHrtnwgkarBarkarvilrbs;ssry:agmanRbsiT§PaB. cugrbs;ssrmanlkçN³ approximately fixed enaH K nwgmantMéltUc. lkçxNÐenHRtUvKñanwgtMéltUcbMputrbs; G EdleGayedaysmikar $>&. b:uEnþ cugrbs;ssrmaM (stiff column) EdlP¢ab;eTAnwg flexible beam Gacnwgpþl;karvileday esrIdl;ssr EdlRtUvKñanwglkçxNÐTMr pinned EdleGaytMél G nig K FM. TMnak;TMngrvag G nig K RtUv)andak;enAkñúg Jackson-Mooreland Alignment Chart (Johnston, 1976) EdlRtUv)anpliteLIgvijenAkñúg Figure C-C2.2 enAkñúg Commentary . edIm,I TTYl)antMél K BIr nomogram mYykñúgcMeNamTaMgBIr dMbUgKNnatMél G enAcugnImYy²rbs;ssr edayeGaymYyCa G A nigmYyeTotCa GB . P¢ab; G A nig GB edaybnÞat;Rtg; ehIyGantMél K enAelIbnÞat;kNþal. emKuNRbsiT§PaBEdlTTYl)anCatMélEdleFobTAnwgGkS½énkarBt; EdlCaGkS½ EkgeTAnwgbøg;rbs;eRKag. karviPaKdac;edayELkGaceFVIeLIgsMrab;karekagEdleFobnwgGkS½mYy eTot. CaFmμta beam-to-column connection enAkñúgTisedAenHnwgminbBa¢Únm:Um:g; ¬ sidesway RtUv)ankarBareday bracing ¦ ehIy K GacnwgykesμI 1.0 . ]TahrN_ 4>11³ eRKagrwgEdlbgðajenAkñúgrUbTI 4>15 CaeRKag unbraced frame . Ggát;nImYy² RtUv)andak;edayeGayRTnugrbs;vasßitenAkñúgbøg;rbs;eRKag. kMNt;emKuNRbEvgRbsiT§PaB K x sMrab;ssr AB nig BC . 94 eRKOgbgÁúMrgkarsgát;
  • 28. T.chhay dMeNaHRsay³ ssr AB ³ sMrab;tMN A G= ∑ I c / Lc = 347 / 3.6 + 445 / 3.6 = 220 = 0.95 ∑ I g / Lg 560 / 6 + 762 / 5.5 231.9 sMrab;tMN B G= ∑ I c / Lc = 445 / 3.6 + 445 / 4.6 = 220.3 = 0.95 ∑ I g / Lg 560 / 6 + 762 / 5.5 231.9 BI alignment chart sMrab; sidesway uninhibited CamYynwg G A = 0.95 nig GB = 0.95 / K = 1.3 sMrab;ssr AB . ssr BC sMrab;tMN B KNnadUcmun G = 0.95 sMrab;tMN C Rtg;TMr pinned . sßanPaBrbs;vamanlkçN³dUceTAnwgssrEdlmaMxøaMgP¢ab;eTAnwg infinity flexible girder Edlrtman stiffness esμIsUnü. dUcenHpleFobPaBrwgRkajrbs;ssr (column stiffness) elIPaBrwgRkajrbs;rt (girder stiffness) mantMélesμIGnnþsMrab;snøak;Kμan kkiteBjelj (perfectly frictionless hinge). lkçxNÐcugenHGaceFVIeTA)ansMrab;karsμanenAkñúgkarGnuvtþn_ dUcenH eyIgGacyk G = 10 sMrab;tMN enH. BI alignment chart CamYynwg G A = 0.95 nig GB = 10 / K = 1.85 sMrab;ssr BC . 95 eRKOgbgÁúMrgkarsgát;
  • 29. T.chhay dUcEdl)anKUsbgðajenAkñúg]TahrN_4>11 sMrab;TMr pinned G KYrRtUv)anykesμInwg 10.0 sMrab; TMr fixed G KYrRtUv)anykesμInwg 1.0 . lkçxNÐTMr fixed RtUvKñanwgrtEdlrwgmaMxøaMg (infinitely stiff girder) nig flexible column EdlRtUvKñanwgtMéltamRTwsþI G = 0 . kñúgkareRbIR)as; alignment chart enAkñúg Commentary )anENnaMeGayeRbI G = 1.0 edaysarEteKBi)annwgTTYl)anTMr fixed eBjelj. Unbraced frame manlT§PaBTb;nUvkMlaMgxagedaysartMNEdlTb;nwgm:Um:g;rbs;va. Ca erOy²eRKagbEnßmedayRbBn§½BRgwgtamrUbragepSg² eRKagEbbenHRtUv)aneKehAfa braced frame. karTb;kMlaMgxagbEnßmGaceFVIeLIgkñúgTMrg;Ca diagonal bracing dUcbgðajenAkñúgrUbTI 4>16 b¤ rigid shear wall. kñúgkrNIepSgeTot ssrRtUv)anTb;eday panel b¤ bay sMrab;kMBs;TaMgmUlrbs;eRKag. TMrenHbegáItCa cantilever structure EdlTb;nwgbMlas;TItamTisedk ehIyk¾pþl;nUvTMrtamTisedksM rab;ElVgd¾éTeTot. GaRs½yeTAnwgTMhMrbs;eRKOgbgÁúM ElVgeRcInCagmYyGacRtUv)anBRgwg. ssrEdl CaGgát;rbs; braced frame RtUv)ankarBarBI sidesway nigmankarTb;karvilenAxagcugrbs;vaxøH. dUc enHvaCaRbePTGgát;sßitenAkúñgcenøaHkrNI (a) nig (d) enAkñúg Table C-C2.1 rbs; Commentary ehIy K sßitenAcenøaH 0.5 nig 1.0 . dUcenH 1.0 CatMélEdltUcsMrab;Ggát; én braced frame ehIy 96 eRKOgbgÁúMrgkarsgát;
  • 30. T.chhay CatMélEdl AISC C2.1 ENnaMeGayeRbI elIkElgEtmankarviPaKNamYyeFVIeLIg. karviPaKGaceFVI eTA)anedayeRbI alignment chart sMrab; braced frame . kareRbI nomogram nwgpþl;lT§plCa effective length factor EdltUcCag 1.0 bnþicbnþÜc ehIyeKnwgTTYl)ankarsnSMsMécxøH . * CamYynwg design aid xøH eKeRbI alignment chart kñúglkçxNÐEdleKbegáItvaeLIg. lkçxNÐ TaMgenHmanenAkñúg Section C2 of the Commentary to the Specification ehIyminRtUv)anerobrab; enATIenHeT. RKb;lkçxNÐTaMgGs;nwgRtUv)anbMeBjesÞIrEtTaMgGs;CaTUeTA RbsinebIdUcenaHeT PaBxus KñaenaHCaEpñkmYyKYreGayRbugRbytñ½. lkçxNÐmYyEdlminRtUv)anbMeBjCaTUeTAenaH KWtMrUvkarEdlfa RKb;karRbRBwtþeTArbs;Ggát;sßitkñúglkçN³eGLasÞic. RbsinebI slenderness parameter λc tUcCag 1.5 ssrnwgekageday inelastic ehIyemKuNRbEvgRbsiT§PaBEdlTTYl)anBI alignment chart nwg mantMéltUcEmnETn. ssrPaKeRcInsßitenAkñúgRkumenH. dMeNIrkargayRsYlkñúgkarkMNt; K sMrab; inelastic column GnuBaØateGayeRbI alignment chart (Yura, 1971 and Dique, 1973). edIm,Ibk RsaydMeNIrkarenH eyIgcab;epþImCamYynwg critical buckling load sMrab; inelastic column Edl eGayedaysmIkar $>^ b. edayEckvanwgRkLaépÞmuxkat;eKTTYl)an buckling stress³ π 2 Et Fcr = (KL / r )2 Rotational stiffnessrbs;ssrenAkñúgkrNIenHCasmamaRtnwg Et I c / Lc ehIytMélEdlsmRsb rbs; G sMrab;eRbIenAkñúg alignment chart KW Ginelastic = ∑ Et I c / Lc = Et G elastic ∑ EI g / Lg E eday Et tUcCag E enaH Ginelastic nwgtUcCag Gelastic ehIy effective length factor K nwgRtUv)an kat;bnßy CalT§pleKTTYl)ankarKNnamYyEdlmanlkçN³esdækic©Cag. edIm,IkMNt; Et / E Edl eKeGayeQμaHfa emKuNkat;bnßyPaBrwgRkaj (stiffness reduction factor SRF)/ BicarNaTMnak;TMng xageRkamsMrab;ssrEdlmanTMrcug pinned ³ Fcr (inelastic) π 2 Et / (L / r )2 Et = = ¬$>*¦ F cr (elastic) 2 π E / (L / r ) 2 E * RbsinebIeRKagRtUv)anBRgwgTb;nwg sidesway tMN beam-to-column minRtUvkar moment resisting ehIyRbBn§½BRgwgGac RtUv)anKNnaedIm,ITb;nUvRKb; sidesway tendency . b:uEnþRbsinebItMNminEmnCa moment resisting vanwgminmanPaBCab; rvagssr nigrt ehIyeKminGaceRbI alignment chart . sMrab; braced frame RbePTenH K x KYrRtUvykesμInwg 1.0 . 97 eRKOgbgÁúMrgkarsgát;
  • 31. T.chhay AISC eRbItMélRbhak;RbEhlsMrab;Epñk inelastic én column strength curve dUcenHsmIkar $>* Ca tMélRbhak;RbEhlenAeBlEdl AISC Equation E2-2 nig E2-3 RtUv)aneRbIsMrab; Fcr . RbsinebI eyIgeGay P P /φ Fcr = cr ≈ u c A A enAeBlEdl Fcr (inelastic) / Fcr (elastic) CaGnuKmn_én Pu /(φc A) . ]TahrN_ sMrab; Pu / (φc A) = 180MPa nig F y = 250 MPa Fcr (inelastic) ≈ 180MPa = 0.658λc Fy = 0.658λc (250) 2 2 λc = 0.785 2 0.877 0.877 Fcr (elastic) = Fy = 250 = 279.3MPa λc 2 0.785 dUcenHemKuNkat;bnßyPaBrwgRkajKW Fcr (inelastic) 180 SRF = = = 0.644 Fcr (elastic) 279.3 edaysar φc efr enaH SRF k¾CaGnuKmn_én Pu / A . tMélrbs; SRF EdlCaGnuKmn_én Pu / A RtUv)aneGayenAkúñg Table 3-1 in Part 3 of the Manual. ]TahrN_ 4>12³ rUb 4>17 bgðajBI rigid unbraced frame. Ggát;TaMgGs;RtUv)andak;edayeFVIy:ag NaeGaykarBt; eFobnwgGkS½xøaMg. TMrxagRtUv)andak;enAtMNnImYy²edaytMNFmμtaEdlBRgwgkñúgTis edAEkgeTAnwg eRKag. kMNt;emKuNRbEvgRbsiT§PaBedayeFobnwgGkS½nImYy²sMrab;Ggát; AB . bnÞúktamGkS½em KuNenAelIGgát;enHKW 180kips ehIyeKeRbIEdk A36 . dMeNaHRsay³ KNnaemKuNeGLasÞicrbs; G sMrab;tMN A / ∑ (I c / Lc ) = 170 / 12 = 14.17 = 1.52 ∑ (I g / Lg ) 88.6 / 20 + 88.6 / 18 9.35 sMrab;tMN B 98 eRKOgbgÁúMrgkarsgát;
  • 32. T.chhay ∑ (I c / Lc ) = 2(170 / 12) = 28.3 = 1.35 ∑ (I g / Lg ) 190 / 20 + 190 / 18 21.0 BI alignment chart sMrab; unbraced frames, K x = 1.43 / edayQrelI elastic behavior dUcenH KxL Fy 1.43(12 )(12 ) 36 λc = = = 0.5512 rxπ E 4.19 ⋅ π 29000 edaysar λc tUcCag 1.5 enaHeKRtUveRbIemKuN K inelastic EdleGay Pu 180 = = 18.5ksi A 9.71 BI Table 3-1in Part 3 of the Manual emKuNkat;bnßyPaBrwgRkaj SRF = 0.83 sMrab;tMN A Ginelastic = SRF × Gelastic = 0.83 × 1.52 = 1.26 sMrab;tMN B Ginelastic = SRF × Gelastic = 0.83 × 1.35 = 1.12 cMeLIy³ BI alignment chart K x = 1.37 . edaysarlkçxNÐTMrFmμtasMrab;eRKag enaH K y esμInwg 1. RbsinebIcugssrCaTMr fixed (G=1.0) b¤ pinned (G=10.0) tMélrbs; G minRtUv)anKuNnwg SRF eT. 99 eRKOgbgÁúMrgkarsgát;
  • 33. T.chhay 4>6>karekagedayrmYl nigedayBt;-rmYl Torsional and Flexural-Torsional Buckling enAeBlEdlGgát;rgkarsgát;edaybnÞúkcMGkS½ køayCaKμanesßrPaB ¬minEmn locally unstable¦ vaGacekagkñúgrUbragmYykñúgcMeNamrUbragbI dUcbgðajenAkñúgrUbTI 4>18. !> karekagedaykarBt; (flexural buckling) eyIg)anBicarNakarekagRbePTenHtaMgBImunrhUtmk dl;eBlenH. vaCaPaBdabEdlekIteLIgedaykarBt; (bending or flexure) CMuvijGkS½EdlRtUv nwgpleFobPaBrwgRkaj (slenderness ratio) FMCageK ¬rUbTI 4>18 a¦. CaTUeTAvaCa minor principle axis EdlmankaMniclPaB (radius of gyration) tUcCageK. Ggát;rgkarsgát;Edlman muxkat;RKb;rUbragGac)ak;tamTMrg;enH. @> karekagedayrmYl (torsional buckling) kar)ak; (failure) edayRbePTenHKWbNþaledaykarmYl (twisting) tamGkS½beNþayrbs;Ggát;. vaGacekIteLIgEtCamYynwgGgát;EdlmanlkçN³Rsav xøaMg ehIymanmuxkat;sIuemRTIDub (double symmetrical cross section) ¬rUbTI 4>18 b¦. Standard hot-rolled shapes mingaynwgrgnUvkarekagedayrmYlenHNas; b:uEnþGgát; built-up BI bnÞHesþIggayeRKaH nigKYreFVIkarGegát. rUbragExVgbgðajnUvPaBgayrgeRKaHBiesssMrab;RbePT énkarekagenH. rUbragenHGac)anmkBIkarpÁúMBIbnÞHdUcbgðajenAkñúgrUb b¤ built-up BImMubYnTl;xñgKña. 100 eRKOgbgÁúMrgkarsgát;
  • 34. T.chhay #> karekagedaykarBt;-rmYl (flexural-torsional buckling) kar)ak;RbePTenHbegáIteLIgedaybnSM énkarekagedaykarBt; nigkarekagedayrmYl. Ggát;ekag nigrmYlkñúgeBlEtmYy¬rUbTI4>18 c¦. karekagRbePTenHGacekIteLIgEtCamYymuxkat;EdlmanrUbragminsIuemRTI TaMgrUbragEdlman GkS½sIuemRTImYyTis dUcCa channel, structural tee, double-angle shape nig equal-leg sigle angles nigrUbragEdlKμanGkS½sIuemRTI dUcCa unequal-leg single angle. AISC Specification tMrUvnUvkarviPaKBI torsional b¤ flexural-torsional buckling enAeBl smrmü. Section E3 of the Specification erobrab;BIGgát; double angle nig tee-shaped ehIy Appendix E3 pþl;nUvviFITUeTAEdlGaceRbIsMrab;RKb;rUbragminsIuemRTI. dMbUgeyIgerobrab;BIviFIEdlmanenAkñúg Appendix E3. vaQrenAelIkareRbIR)as; slenderness parameter λe CMnYseGay λc . eyIgTTYl λe dUcxageRkam. BI Euler buckling stress/ π 2E Fe = (KL / r )2 Slenderness ratio GacsresrCa KL π 2E = r Fe RbsinebI Fe RtUv)ankMNt;Ca elastic buckling stress EdlRtUvnwgrUbragénkar)ak;Edllub eTaHeday flexural, torsional b¤ flexural-torsional enaH slenderness ratio EdlRtUvKñaKW ⎛ KL ⎞ π 2E ⎜ ⎟ = ⎝ r ⎠e Fe ehIy slenderness parameter EdlRtUvKñaKW (KL / r )e Fy 2 F y ( KL / r ) e Fy λe = = = π E π 2E Fe dMeNIrkarKNnamandUcxageRkam³ !> kNt; Fe sMrab; torsional elastic buckling b¤ flexural-torsional elastic buckling BIsmIkarEdleGayenAkñúg Appendix E3. @> KNna effective slenderness parameter, λe . #> KNna critical stress Fcr BIsmIkarFmμta (AISC Equations E2-2 and E2-3) b:uEnþeRbI λe CMnYs eGay λc . bnÞab;mk design strength KW 101 eRKOgbgÁúMrgkarsgát;
  • 35. T.chhay φc Pn = φc Ag Fcr Edl φc = 0.85 dUcKñasMrab; flexural buckling. smIkarsMrab; Fe EdleGayenAkñúg AISC Appendix E3KWQrelI well-established theory Edlmankñúg Theory of Elastic Stabality (Timoshenko and Gere, 1961). elIkElgsMrab;karpøas; bþÚrxøHenAkñúg notation vamansmIkardUcKñaenAkñúgesovePAenaH edayKμankarsMrYl. sMrab; doubly symmetrical shapes (torsional buckling)/ ⎡ π 2 EC w ⎤ 1 Fe = ⎢ + GJ ⎥ (AISC Equation A-E3-5) ⎢ (K z L )2 ⎣ ⎥ Ix + I y ⎦ sMrab; singly symmetrical shape (flexural-torsional buckling)/ Fey + Fez ⎛ ⎜ 4 Fey Fez H ⎞ ⎟ Fe = 1− 1− ( ) ⎜ ⎟ (AISC Equation A-E3-6) 2H ⎜ Fey + Fez 2 ⎟ ⎝ ⎠ sMrab;rUbragEdlKμanGkS½sIuemRTI (flexural-torsional buckling)/ (Fe − Fex )(Fe − Fey )(Fe − Fez ) − Fe2 ( Fe − Fey )(xo / r o )2 (AISC Equation A-E3-7) − Fe2 (Fe ( − Fex ) yo / r o ) 2 =0 smIkarcugeRkayCasmIkardWeRkTI3 dUcenHrbs; Fe KWtUcNas;. CasMNagl¥ PaBcaM)ac;kñúgkaredaH RsaysmIkarenHKWticbMput edaysareKkMreRbIrUbragminsIuemRTICaGgát;rgkarsgát;Nas;. GgÁEdl min)ankMNt;BImunEdleRbIenAkñúgsmIkarTaMgbIenHRtUv)ankMNt;dUcteTA³ C w = warping constant Kz = emKuNRbEvgRbsiT§PaBsMrab;karekagedayrmYl EdlQrelIbrimaNénkarTb;cug RbqaMgnwgkarrmYltamGkS½beNþay. G = shear modulus J = torsional constant ¬esμIeTAnwg polar moment of inertia sMrab;Etmuxkat;mUl¦ π 2E Fex = (AISC Equation A-E3-10) (K x L / rx )2 π 2E Fey = (K y L / ry )2 (AISC Equation A-E3-11) Edl y CaGkS½sIuemRTIsMrab; singly symmetric shapes. ⎡ π 2 EC w ⎤ 1 Fez = ⎢ + GJ ⎥ (AISC Equation A-E3-12) ⎢ (K z L )2 ⎣ 2 ⎥ Ar o ⎦ 102 eRKOgbgÁúMrgkarsgát;
  • 36. T.chhay ⎛ x2 + y2 ⎞ H = 1− ⎜ o o ⎟ (AISC Equation A-E3-9) ⎜ 2 ⎟ ⎝ ro ⎠ Edl xo nig yo CakUGredaenén shear center rbs;muxkat;edayeFobnwgTIRbCMuTMgn;. Shear center CacMnucenAelImuxkat;EdlbnÞúkeFVIeGayGgát;ekagedayminrmYl. Shear center RtUv)anniyay lMGitenAkñúgCMBUk 5. 2 2 2 Ix + Iy r o = xo + y o + (AISC Equation A-E3-8) A eKGacrktMélefrEdleRbIenAkñúgsmIkarTaMgbIsMrab; Fe enAkñúgtarag torsion properties nig flexural-torsional properties enAkñúg part 1 of the Manual . sMrab; W, M, S nig HP shapes, J nig C w RtUv)aneGay. eKeGaytMél J / C w / r o nig H RtUv)aneGaysMrab; channel, single angle nig structural tee. taragsMrab; double angle eGaytMél r o nig H ¬ J nig Cw esμInwgBIrdgén tMélEdleGaysMrab; single angle¦. dUc)anbgðajBIxagelI eKkMrnwgviPaKkarekagedayrmYlsMrab;muxkat;sIuemRTIDub. dUcKña eKkMr eRbIrUbragKμanGkS½sIuemRTICaGgát;rgkarsgát; ehIyeKkMrnwgviPaK flexural-tensional buckling én Ggát;RbePTenHEdr RbsinebIman eKcaM)ac;RtUvEtviPaKva. sMrab;ehtuplTaMgenH eyIgkMNt;kar BicarNaelIrUbrag flexural-torsional buckling CamYynwgGkS½sIuemRTImYy. elIsBIenH double angle EdlCa built-up shape CaRbePTrUbragEdleKniymeRbIeRcIn. sMrab; singly symmetrical shape, flexural-torsional buckling stress Fe TTYl)anBI AISC Equation A-E3-6. enAkñúgsmIkarenH y RtUv)ankMNt;CaGkS½sIuemRTI ¬edayminKitBITisedArbs; Ggát;¦ehIy flexural-torsional buckling RtUv)anKitEttamGkS½mYyenH ¬flexural bucklingtamTis enHnwgminekItman¦. GkS½ x RbQmEtnwg flexural buckling. dUcenH sMrab; singly symmetrical shape eKGacmanersIusþg;BIrKW flexural-torsional buckling tamGkS½ y ¬GkS½sIuemRTI¦ b¤ flexural buckling eFobGkS½ x . edIm,IkMNt;mYyNamanlkçN³lub KNnaersIusþg;EdlRtUvnwgGkS½nImYy² ehIyeRbItMélNaEdltUcCag. ]TahrN_ 4>13³ KNna design compressive strength rbs; WT13.5 × 80.5 . RbEvgRbsiT§PaB tamGkS½ x KW 25 feet 6inches RbEvgRbsiT§PaBtamGkS½ y KW 20 feet ehIyRbEvgRbsiT§PaBtam GkS½ z KW 20 feet . eRbIEdk A36 . 103 eRKOgbgÁúMrgkarsgát;
  • 37. T.chhay dMeNaHRsay³ KNna design compressive strength sMrab;GkS½ x ³ K x L 25.5(12 ) = = 77.27 rx 3.96 KL Fy 77.27 36 λc = = = 0.8666 < 1.5 rπ E π 29000 eRbI AISC Equation E2-2 Fcr = (0.658)λc Fy = (0.658)0.8666 (36) = 26.29ksi 2 2 φc Pn = φc Ag Fcr = 0.85(23.7 )(26.29) = 530kips KNna flexural-torsional buckling strength CMuvijGkS½ y π 2E π 2 (29000) Fey = = = 52.17ksi (K y L / ry )2 (74.07) 2 ⎡ π 2 EC w ⎤ 1 Fez = ⎢ + GJ ⎥ ⎢ (K z L )2 ⎣ 2 ⎥ Ar o ⎦ ⎡ π 2 (29000)(42.7 ) ⎤ + 11200(7.31)⎥ 1 =⎢ = 107.7ksi ⎢ (20 × 12) ⎥ 23.7(5.67 ) 2 2 ⎣ ⎦ Fey + Fez = 52.17 + 107.7 = 159.9ksi Fey + Fez ⎡ 4 Fey Fez H ⎤ Fe = ⎢1 − 1 − ⎥ 2H ⎢ ⎣ ( Fey + Fex 2 ⎥ ⎦ ) 159.9 ⎡ 4(52.17 )(107.7 )(0.813) ⎤ = ⎢1 − 1 − ⎥ = 45.81ksi 2(0.813) ⎢ (159.9)2 ⎥ ⎣ ⎦ Fy 36 λe = = = 0.8865 Fe 45.81 edaysartMélenHtUcCag 1.5 eRbI AISC Equation E2-2 CamYynwg λe CMnYseGay λc ³ ⎛ 2 ⎞ Fcr = ⎜ 0.658λe ⎟ Fy = (0.658)(0.8865) (36) = 25.91ksi 2 ⎝ ⎠ φc Pn = φc Ag Fcr = 0.85(23.7 )(25.91) = 522kips (controls) cMeLIy³ Design strength = 522kips cMNaMfa enAeBlEdl Fcr nig Fe RtUv)anKNna karKNnasMrab; flexural buckling CMuvij GkS½ x nig flexural-torsional buckling CMuvijGkS½ y manlkçN³dUcKña. dUcenHbnÞab;BI Fcr nig Fe 104 eRKOgbgÁúMrgkarsgát;
  • 38. T.chhay RtUv)anKNna TaMg λc nig λe GacRtUv)anKNna ehIytMélEdltUcCagRtUv)aneRbIedIm,IKNna strength . kareFVIEbbenHedIm,Ikat;bnßykarcaM)ac;kñúgkarKNna strength sMrab;GkS½TaMgBIr. dMeNIrkarviPaK flexural-torsional buckling elI double-angle nig tee EdleGayenAkñúg AISC Section E3 CakarEksMrYldMeNIrkarviPaKEdleGayenAkñúg AISC Appendix E3. vak¾mankar EkERbkMNt;cMNaMxøHdUcCa³ BI Fe eTACa Fcrft / Fey eTACa Fcry nig Fez eTACa Fcrz . kugRtaMg Fcry RtUv)anrkBI AISC E2 nigQrelI flexural buckling eFobGkS½ y . edIm,ITTYl)an Fcrz eyIgGacecalGkS½TImYyrbs; AISC Equation A-E3-12 enaH GJ Fcrz = 2 Ar o karlubecalenHGacGnuBaØat)an BIeRBaHsMrab; double-angle nig tee GgÁTImYymantMéltUc Gacecal)anebIeFobnwgGgÁTIBIr. Flexural buckling stress Fcry RtUv)anKNnaCamYynwgsmIkarFmμtarbs; AISC Chapter E edayeRbI KL / r EdlRtUvKñanwgGkS½ y ¬GkS½sIuemRTI¦. bnÞab;mkeTot nominal strength GacRtUv)anKNnadUcxageRkam Pn = Ag Fcrft ⎛ Fcry + Fcrz ⎞⎡ 4 Fcry Fcrz H ⎤ Edl Fcrft = ⎜ ⎟ ⎢1 − 1 − ⎥ ( ) ⎜ ⎟⎢ (AISC Equation E3-1) ⎝ 2H ⎠⎣ Fcry + Fcrz 2 ⎥ ⎦ RKb;GgÁTaMgGs;Edl)anmkBI Appendix E3 rkSadEdl. dMeNIrkarenH RtUv)aneRbIsMrab;Et double-angle nig tee eRBaHvapþl;nUvcMeLIysuRkitCagkareRbIdMeNIrkarEdleGayenAkñúg Appendix E3. ]TahrN_ 4>14³ KNna design strength rbs;rUbragenAkñúg]TahrN_TI 4>13 edayeRbIsmIkarrbs; AISC Equation E3. dMeNaHRsay³ BI]TahrN_ 4>13 flexural buckling strength sMrab;GkS½ x KW 530kips ehIy K y L / ry = 74.07 . BI AISC E2-4, slenderness parameter KW KL Fy 74.07 36 λc = = = 0.8307 < 1.5 rπ E π 29000 BI AISC Equation E2-2, 105 eRKOgbgÁúMrgkarsgát;
  • 39. T.chhay ⎛ 2 ⎞ Fcr = Fcry = ⎜ 0.658λc ⎟ F y = (0.658)(0.8307 ) (36) = 26.97ksi 2 ⎝ ⎠ BI AISC E3, GJ 11200(7.31) Fcrz = = = 107.5ksi 2 Ar o 23.7(5.67 )2 Fcry + Fcrz = 26.97 + 107.5 = 134.5ksi ⎛ Fcry + Fcrz ⎞⎡ 4 Fcry Fcrz H ⎤ Fcrft = ⎜ ⎟ ⎢1 − 1 − ⎥ ⎜ ⎝ 2H ⎟⎢ ⎠⎣ ( ) Fcry + Fcrz 2 ⎥ ⎦ 134.5 ⎡ 4(26.97 )(107.5)(0.813) ⎤ = ⎢1 − 1 − ⎥ = 25.48ksi 2(0.813) ⎢ (134.5)2 ⎥ ⎣ ⎦ φc Pn = φc Ag Fcrft = 0.85(23.7 )(25.48) = 513 (Control) cMeLIy³ Design strength = 513kips lT§plenAkñúg]TahrN_ 4>13 nig 4>14 bgðajBIkMhuskñúgkareRbIR)as; Appendix E3 sMrab; rUbragenHmanlkçN³minsnSMsMéc. viFIsaRsþEdleRbIenAkñúg]TahrN_ 4>14 EdlQrelI AISC Specification E3 EtgEtRtUv)aneRbIsMrab; double angle nig tee. b:uEnþkñúgkarGnuvtþn_ ersIusþg;rbs; double angle nig tee PaKeRcInGacrk)anenAkñúg column load table. taragTaMgenaHKWQrelIviFI saRsþEdlesñIeLIgeday AISC E3 ehIyk¾GaceRbIedIm,IepÞógpÞal;lT§plrbs;]TahrN_ 4>14. taragpþl;nUvtMél design strength BI EdlmYyCa flexural buckling eFobGkS½ x nigmYyeTotCa flexural-torsional buckling eGobGkS½ y . taragTaMgenaHk¾pþl;pgEdrsMrab;Ggát;rgkarsgát; single-angle. Design strength EdleGay edayminQrelIRTwsþI flexural-torsional buckling RtUv)aneGayenAkñúg specification dac;edayELk sMrab; single-angle member enAkñúg Part 6 of the Manual, Specification and Codes. 4>7> Built-up Member RbsinebIeKsÁal;lkçN³muxkat; (cross-sectional properties)rbs;Ggát;rgkarsgát; built-up karviPaKrbs;vamanlkçN³RsedogKñasMrab;Ggát;rgkarsgát;epSgeTot RbsinebIEpñkpÁúMrbs;muxkat;t P¢ab;)anl¥. AISC E4 mankarlMGitCaeRcInEdlTak;TgeTAnwgkartP¢ab;enH CamYynwgtMrUvkardac;eday ELksMrab;Ggát;EdlpÁúMeday rolled shape mYy b¤eRcIn nigGgát;EdlpÁúMeday plate b¤bnSMén plate nig 106 eRKOgbgÁúMrgkarsgát;
  • 40. T.chhay Edkrag (shape). munnwgBicarNaBIbBaðatP¢ab; eyIgnwgrMlwkBIkarKNnalkçN³muxkat;rbs;rUbrag built-up. Design strength rbs;Ggát;rgkarsgát; built-up CaGnuKmn_eTAnwg slenderness parameter λc . dUcenHeKRtUvkMNt;GkS½em nigkaMniclPaBEdlRtUvKñanwgGkS½TaMgenaH. sMrab;muxkat; homogenous GkS½emRtYtsIunwgGkS½TIRbCMuTMgn;. viFIsaRsþkñúgkarKNnaRtUv)anbgðajenAkñúg]TahrN_ 4>15. EpñkpÁúMrbs;muxkat;RtUv)ansnμt;fatP¢ab;)anl¥. ]TahrN_ 4>15³ ssrEdlbgðajenAkñúgrUbTI 4>19 RtUv)anplitedaykarpSarEdkbnÞH 4"× 38 " BIelI søabrbs;Edk W 18× 35 . EdkpÁúMTaMgBIrCaEdk A36 . RbEvgRbsiT§PaBeFobGkS½TaMgBIrKW 15 feet . edaysnμt;EdkpÁúMTaMgBIrRtUv)antP¢ab;edayeFVIy:agNaeGayGgát;manRbsiT§PaBeBj ehIyKNna design strength edayQrelI flexural buckling. dMeNaHRsay³ CamYynwgkarbEnßmEdkBIelI rUbragmanlkçN³minsIuemRTIbnþic b:uEnþT§iBl flexural- torsionalRtUv)anecal. GkS½sIuemRTIbBaÄrCaGkS½emmYyEdleKminRtUvkarKNna. eKnwgrkGkS½emedkedayeRbI principle of moment³ plbUkm:Um:g;RkLaépÞrbs;FatupSMnImYy²eFobnwgGkS½NamYy ¬enAkñúg]Ta- hrN_enH GkS½edksßitenAEpñkxagelIrbs; plateRtUv)aneRbI¦ RtUvEtesμInwgm:Um:g;RkLaépÞsrub. eyIgeRbItarag 4>1 edIm,IsMrYldl;karKNna. tarag 4>1 Component A y Ay Plate 1.500 0.1875 0.2812 W 10.3 9.225 95.02 ∑ 11.8 95.30 y= ∑ Ay = 95.30 = 8.076in ∑ A 11.8 107 eRKOgbgÁúMrgkarsgát;
  • 41. T.chhay CamYynwgTItaMgrbs;GkS½TIRbCMuTMgn;edkEdl)anKNnaxagelI eyIgGacKNnam:Um:g;niclPaB eFobnwgGkS½enHedayeRbI parallel-axis theorem³ I = I + Ad 2 Edl I=m:Um:g;niclPaBeFobGkS½TIRbCMuTMgn;rbs;RkLaépÞpSM A = RkLaépÞrbs;EpñkpSM I = m:Um:g;niclPaBeFobGkS½RsbeTAnwgGkS½TIRbCMuTMgn;rbs;RkLaépÞpSM d = cMgayEkgrvagGkS½BIr karcUlrYmBIRkLaépÞpSMnImYy²RtUv)anKNna nigRtUv)anbUkedIm,ITTYl)anm:Um:g;niclaPaBrbs; composite area. tarag 4>2 CataragEdlbEnßmBIelItarag 4>1 edayrYmbBa©ÚlkarKNnaenH. sMrab;GkS½Qr y = (3 8)(4)3 + 15.3 = 17.30in 4 1 12 ¬ controle¦ Iy 17.30 ry = = = 1.211in A 11.8 tarag 4>2 Component A y Ay I d I + Ad 2 Plate 1.500 0.1875 0.2812 0.01758 7.889 93.37` W 10.3 9.225 95.02 510 1.149 523.6 ∑ 11.8 95.30 617.0= I x KL F y 15(12 ) 36 λc = = = 1.667 > 1.5 rπ E 1.211π 29000 ⎡ 0.877 ⎤ ⎡ 0.877 ⎤ Fcr = ⎢ ⎥ Fy = ⎢ ⎥ (36) = 11.36ksi ⎢ λ2 ⎥ ⎣ (1.667 ) ⎥ 2 ⎣ c ⎦ ⎢ ⎦ φc Pn = φc Ag Fcr = 0.85(11.8)(11.36) = 114kips cMeLIy³ Design strength = 114kips . tMrUvkarkartP¢ab;sMrab; Built-up Members EdlpSMeLIgeday Rolled Shapes rUbrag built-up EdleKniymCageKKWrUbragEdlpÁúMeLIgeday rolled shap EdleKeGayeQμaH fa double-angle shape. Ggát;RbePTenHnwgRtUv)aneRbIedIm,IbgðajBItMrUvkarsMrab;Ggát; built-up Rb ePTenH. rUbTI 4>20 bgðajGgát;rgkarsgát;rbs; truss EdlP¢ab;eTAnwg gusset plate enAxagcugnImYy 108 eRKOgbgÁúMrgkarsgát;
  • 42. T.chhay ²rbs;va. edIm,IrkSa back-to-back seperation rbs; angle tambeNþayRbEvg fillers nig spacers EdlmankMras;esμInwg gusset plate RtUv)andak;enA angles edayKMlatesμI²Kña. KMlatRtUvEtmantMél tUcRKb;RKan;edIm,IeFVIeGay built-up member enHeFVIkarCalkçN³EtmYy. RbsinebIGgát;ekageFob GkS½ x ¬flexural buckling¦ eRKOgP¢ab; (connector) minrgnUvbnÞúkKNnaNamYyeT ehIybBaðaénkar tP¢ab;KWsamBaØedayrkSaTItaMgrbs;Ggát;TaMgBIr. edImI,Fanafa built-up member eFVIkarCalkçN³Et mYy AISC tMrUvfa stiffness rbs;FatupSMnImYy²minRtUvFMCagbIPaKbYnén stiffness rbs; built-up member eT. a 3 KL ≤ ri 4 r Edl a= KMlatrbs;eRKOgP¢ab; ri = kaMniclPaBGb,brmarbs;FatupÁúM KL / r = maximum slenderness ratio rbs; built-up member RbsinebIGgátekageFobGkS½sIuemeRTI ¬EdlvargnUv flexural-torsional buckling eFobGkS½ y ¦ eRKOgP¢ab;rgnUvkMlaMgkat;. lkçxNÐenHGacRtUv)anemIleXIjedayBicarNa planks BIrEdleRbICa Fñwm RtUv)anbgðajenAkñúgrUbTI 4>21. RbsinebI plank minRtUv)anP¢ab; vanwgrGiltamépÞb:H enAeBl EdlvargbnÞúk ehIyvanwgeFVIkarCaFñwmBIrdac;edayELkBIKña. enAeBlEdlvaRtUv)anP¢ab;edayb‘ULúg ¬b¤eRKOgP¢ab;epSgeTotdUcCa EdkeKal¦ plank TaMgBIrnwgeFVIkarEtmYy ehIyersIusþg;Tb;nwgkarrGil nwgbegáItCakMlaMgkat;enAkñúgb‘ULúg. kareFVIkarEbbenHekItmanenAkñúg double-angle shape enAeBl 109 eRKOgbgÁúMrgkarsgát;
  • 43. T.chhay karekageFobnwgGkS½ y . RbsinebIFñwm plank RtUv)andak;edayeGaykarekagekItmaneFobnwgGkS½ epSgeTot ¬GkS½ b ¦ enaH plank TaMgBIrnwgekagkñúglkçN³dUcKña ehIyKμankarrGil nigKμankMlaMgkat;. kareFVIkarenHmanlkçN³RsedogKñanwgkarekageFobGkS½ x rbs; double-angle shape . enAeBlEdl eRKOgP¢ab;rgnUvkMlaMgkat; eKRtUvkar modified slenderness ratio EdlmantMélFMCagtMélBitR)akd. ASIC E4 BicarNaeRKOgP¢ab;BIrRbePT³ ¬!¦ snug-tight bolt nig ¬@¦ pSar b¤ fully tightned bolt . karbriyaylMGitBIkartP¢ab;manenAkñúgCMBUkTI7. Column load table sMrab; double-angle KWQrelIkarpSar b¤ fully tightened bolt. sMrab;krNIenH³ 2 2 ⎛ KL ⎞ ⎛ KL ⎞ α 2 ⎛ a ⎞ ( ) ⎜ ⎟ = ⎜ ⎟ + 0.82 ⎜ ⎜r ⎟ ⎟ (AISC Equation E4-2) ⎝ r ⎠m ⎝ r ⎠o 1+α 2 ⎝ ib ⎠ Edl (KL / r )o = original unmodified slenderness ratio (KL / r )m = modified slenderness ratio rib = kaMniclPaBrbs;FatupSMeFobGkS½RsbeTAGkS½énkarekagrbs;Ggát; h α = separation ratio = 2rib h= cMgayrvagTIRbCMuTMgn;rbs;FatupSM ¬EkgeTAnwgGkS½énkarekagrbs;Ggát;¦ enAeBlEdleRKOgP¢ab;Ca snug-tight bolts 2 2 ⎛ KL ⎞ ⎛ KL ⎞ ⎛ a ⎞ ⎜ ⎟ = ⎜ ⎟ +⎜ ⎟ (AISC Equation E4-1) ⎝ r ⎠m ⎝ r ⎠ o ⎜ rib ⎟ ⎝ ⎠ Column load tablesMrab; double-angle shape bgðajBIcMnYneRKOgP¢ab;caM)ac;sMrab; flexural- torsional buckling strength EdleGaytamGkS½ y . cMnYneRKOgP¢ab;sMrab; flexural buckling strength tamGkS½ x RtUv)ankMNt;tamPaBcaM)ac;Edlfa slenderness rbs; angle mYyminRtUvFMCagbI PaKbYnén slenderness rbs; double-angle shape TaMgmUleT. ]TahrN_ 4>16³ KNna design strengthrbs;Ggát;rgkarsgát;EdlbgðajenAkñúgrUbTI 4>22. Edl rag angle BI 5 × 3 × 12 RtUv)andak;edayeGayeCIgEvgTl;xñgKña ehIyXøatBIKña 38 inch . RbEvg RbsiT§PaB KL = 16 feet nigman fully tightened intermediate connectors cMnYn 3 . eRbIEdk A36 . 110 eRKOgbgÁúMrgkarsgát;
  • 44. T.chhay dMeNaHRsay³ KNna flexural buckling strength sMrab;GkS½ x KL 16(12) = = 120.8 rx 1.59 KL Fy 120.8 36 λc = = = 1.355 < 1.5 rπ E π 29000 eRbI AISC Equation E2-2 Fcr = (0.658)λc Fy = (0.658)(1.355) (36 ) = 16.69ksi 2 2 φc Pn = φc Ag Fcr = 0.85(7.5)(16.69 ) = 106kips sMrab;GkS½ y KL 16(12 = = 153.6 ry 1.25 edIm,IkMNt; flexuaral-torsional buckling strength sMrab;GkS½ y eRbI modified slenderness ratio edayQrelIKMlatrbs;eRKOgP¢ab;. KMlatrbs;eRKOgP¢ab;KW 16(12) a= = 48in 4 bnÞab;mk a a = = 48 ri rz 0.648 = 74.07 < 0.75(153.6) = 115.2 (OK) rib = ry = 0.829in h = 2(0.75) + = 1.875in 3 8 h 1.875 α= = = 1.131 2rib 2(0.829) BI AISC Equation E4-2, modified slenderness ration KW 2 2 ⎛ KL ⎞ ⎛ KL ⎞ α2 ⎛ a ⎞ ⎜ ⎟ = ⎜ ⎝ r ⎠m ⎟ + 0.82 ⎝ r ⎠o (⎜ ⎟ ⎜ ⎟ ) 1 + α 2 ⎝ rib ⎠ 111 eRKOgbgÁúMrgkarsgát;
  • 45. T.chhay (153.6) + 0.82 (1.131) 2 2 2 ⎛ 48 ⎞ [ ] 2 = ⎜ ⎟ = 158.5 1 + (1.313) ⎝ 0.829 ⎠ tMélenHRtUv)aneRbICMnYseGay KL / ry sMrab;KNna Fcry KL Fy 158.5 36 λc = = = 1.778 > 1.5 rπ E π 29000 eRbI AISC Equation E2-3 ⎡ 0.877 ⎤ ⎡ 0.877 ⎤ Fcry = ⎢ ⎥ Fy = ⎢ ⎥ (36 ) = 9.987 ksi ⎢ λc ⎥ ⎣ 2 ⎦ ⎢ (1.778)2 ⎥ ⎣ ⎦ GJ 11200(2 × 0.322) Fcrz = = = 151.4ksi Ar o 2 7.5(2.52)2 Fcry + Fcrz = 9.987 + 151.4 = 161.4ksi ⎛ Fcry + Fcrz ⎞⎡ 4 Fcry Fcrz H ⎤ Fcrft = ⎜ ⎟ ⎢1 − 1 − ⎥ ⎜ ⎝ 2H ⎟⎢ ⎠⎣ ( ) Fcry + Fcrz 2 ⎥ ⎦ 161.4 ⎡ 4(9.987 )(151.4 )(0.645) ⎤ = ⎢1 − 1 − ⎥ = 9.748ksi 2(0.645) ⎢ (161.4)2 ⎥ ⎣ ⎦ φc Pn = φc Ag Fcrft = 0.85(7.50)(9.748) = 62.1kips (control) cMNaMfa lT§plenAkñúg]TahrN_enHmantMélRsedogKñanwgtMélEdleGayenAkñúg column load table cMeLIy³ Design strength KW 62kips ]TahrN_ 4>17³ KNnaGgát;rgkarsgát;EbEvg 14 feet edIm,IRTbnÞúkemKuN 50kips . eRbIEdkrUbrag double angle EdlmaneCIgxøITl;xñgKña nigmanKMlatBIKña 3 8 inch . Ggát;RtUv)anTl;BRgwgenARtg; Bak;kNþalRbEvgedIm,ITb;nwgkarekageFobGkS½ x ¬GkS½EdlRsbeTAnwgeCIgEvg¦. kMNt;cMnYneRKOg P¢ab;enAkNþalEdlRtUvkar ¬EdkEdlBRgwgenABak;kNþalRbEvgRtUv)anpþl;eRKOgP¢ab;mYy¦. eRbIEdk A36 . dMeNaHRsay³ BI column load table eRCiserIs 2L3 12 × 3 × 14 EdlmanTMgn; 10.8lb / ft . smtßPaBrbs;muxkat;enHKW 51kips edayQrelIkarekageFobGkS½ y CamYynwgRbEvgRbsiT§PaB 14 feet . ¬ersIusþg;EdlRtUvKñanwg flexural buckling eFobGkS½ x KW 60kips EdlQrelIRbEvg RbsiT§PaB 7 feet ¦. 112 eRKOgbgÁúMrgkarsgát;
  • 46. T.chhay karekageFobGkS½ y eFVIeGayeRKOgP¢ab;rgkMlaMgkat; dUcenHcMnYneRKOgP¢ab;RKb;RKan; RtUv)andak;edIm,ITb;Tl;nwgkMlaMgenH. taragbgðajfa vaRtUvkareRKOgP¢ab;cMnYn 3 . cMeLIy³ eRbI 2L3 12 × 3 × 14 CamYynwgeRKOgP¢ab;cMnYn 3 sMrab;RbEvg 14 feet . tMrUvkarcaM)ac;énkartP¢ab;sMrab; built-up member EdlpSMeLIgeday plate b¤ both plate CamYynwg shapes enAeBlEdl built-up member EdlpSMeLIgeday rolled shapes BIr b¤eRcInedaymanKMlat dac;BIKña plate RtUv)aneRbIedIm,ItP¢ab; shape. AISC E4 mankarlMGitCaeRcInGMBItMrUvkarcaM)ac;sMrab; kartP¢ab; nigTMhMrbs; plate . tMrUvkarcaM)ac;énkartP¢ab;RtUv)aneGaybEnßmsMrab; built-up compression member d¾éTeTotEdlpSMeLIgeday plate b¤ plate CamYynwg shape . 113 eRKOgbgÁúMrgkarsgát;

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