Your SlideShare is downloading. ×
4.compression members
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

4.compression members

671

Published on

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
671
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
0
Comments
0
Likes
0
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 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;

×