5.beams

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  • 1. T.chhay V. Fñwm Beams 5>1> esckþIepþIm Introduction FñwmCaGgát;rbs;eRKOgbgÁúMEdlRTbnÞúkTTwg dUcenHehIy)aneFVIeGayvargnUvkarBt; (flexural or bending). RbsinebImanvtþmanbnÞúktamGkS½kñúgbrimaNmYyFMKYrsm vanwgRtUv)aneKehAvafa beam- column ¬EdlnwgRtUvbkRsayenAkñúgCMBUkTI6¦. enAkñúgGgát;eRKOgbgÁúMxøHEdlmanvtþman axial load kñúgtMéltictYc EtT§iBld¾sþÜcesþIgenHRtUv)aneKecalenAkñúgkarGnuvtþn_CaeRcIn ehIyeK)ancat; TukvaCa beam. CaTUeTAFñwmRtUv)aneKdak;kñúgTisedk nigrgnUvbnÞúkbBaÄr EtvamincaM)ac;EtkñúgkrNIEbb enHeT. Ggát;eRKOgbgÁúMEdlRtUv)aneKcat;TukCa beam RbsinebIvargnUvbnÞúky:agNaEdleFVIeGayva ekag (bending). rUbragmuxkat; (cross-sectional shape)EdlRtUv)aneKeRbICaTUeTArYmman W-, S- nig M- shapes. eBlxøH chanel shape k¾RtUv)aneRbIdUcCaFñwmEdlpSMeLIgBIEdkbnÞH kñúgTMrg; I-, H- b¤ box shape. Doubly symmetric shape dUcCa standard rolled W-, M- nig S-shape CarUbragEdlman RbsiT§PaBCaeK. CaTUeTA rUbragEdl)anBIkarpSMrbs;EdkbnÞHRtUv)aneKKitCa plate girder b:uEnþ AISC Specification EbgEck beam BI plate girder edayQrelIpleFobTTwgelIkMras; (width-thickness ratio) rbs;RTnug. rUbTI 5>1 bgðajTaMg hot-rolled shape nig built-up shapeCamYynwgTMhMEdlRtUv eRbIsMrab; width-thickness ratios. Rbsin t h 2555 ≤ F ¬xñat IS¦ th ≤ 970 ¬xñat US¦ F w y w y Ggát;eRKOgbgÁúMRtUv)aneKcat;TUkCa beam edayminKitfavaCa rolled shape b¤Ca built-up. EpñkenH RtUv)anerobrab;enAkñúg chapter F of the Specification, “Beams and Other Flexural Members” ehIyvak¾CaRbFanbTEdlRtUvykmkniyayenAkñúgCMBUkenH. RbsinebI 114 Fñwm
  • 2. T.chhay h 2555 tw > Fy ¬xñat IS¦ h tw ≤ 970 Fy ¬xñat US¦ enaHGgát;eRKOgbgÁúMRtUv)aneKcat;TukCa plate girder nwgRtUv)anerobrab;enAkñúg Chapter G of the specification, “Plate Girders”. enAkñúgesovePAenHeyIgnwgniyayBI plate girder kñúgCMBUkTI 10. edaysarEt slenderness rbs;RTnug plate girder RtUvkarBicarNaBiessenABIelI nigBIeRkamEdlcM)ac; sMrab;Fñwm. RKb; standard hot-rolled shape EdlGacrk)anenAkñúg Manual KWsßitenAkñúgRbePT beams. Built-up shape PaKeRcInRtUv)ancat;cMNat;fñak;Ca plate girder b:uEnþ built-up shape xøHRtUv)ancat; TukCaFñwmedaykarkMNt;rbs; AISC. sMrab; beams/ TMnak;TMngeKalrvagT§iBlbnÞúk (load effects) nig strength KW M u ≤ φb M n Edl Mu = bnSMénm:Um:g;emKuNEdlFMCageK φb = emKuNersIusþg;sMrab;Fñwm = 0.9 M n = nominal moment strength Design strength, φb M n enAeBlxøHRtUv)aneKehAfa design moment. 5>2> kugRtaMgBt; nigm:Um:g;)øasÞic Bending Stress and the Plastic Mement edIm,IGackMNt; nominal design strength M n dMbUgeyIgRtUvBinitüemIlkarRbRBwtþeTA (behavior) rbs;Fñwmtamry³énkardak;bnÞúkRKb;lkçxNÐ taMgBIbnÞúktUcrhUtdl;bnÞúkEdlGaceFVIeday Fñwm)ak;. BicarNaFñwmEdlbgðajenAkñúgrUbTI 5>2 a EdlRtUv)andak;edayeFVIy:agNaeGayvaekag eFobnwgGkS½em ¬GkS½ x − x sMrab; I- nig H-shape¦. sMrab; linear elastic material nigkMhUcRTg; RTaytUc karBRgaykugRtaMgBt;RtUv)anbgðajenAkñúg rUbTI 5>2 b CamYynwgkugRtaMgEdlRtUv)an snμt;faBRgayesμItamTTwgrbs;Fñwm. ¬kMlaMgkat;RtUv)anBicarNaedayELkenAkñúgEpñkTI 5>7¦. BI elementary mechanics of materials/ kugRtaMgRtg;cMNucNamYyGackMNt;)anBI flexural formula³ fb = My Ix ¬%>!¦ Edl M CamU:m:g;Bt;enAelImuxkat;EdlBicarNa/ y CacMgayEkgBIbøg;NWt ¬neutral plane) eTAcMnuc Edlcg;dwg nig I x Cam:Um:g;niclPaBénmuxkat;EdleFobnwgGkS½NWt. sMrab; homogeneous material 115 Fñwm
  • 3. T.chhay GkS½NWtRtYtsIuKñanwgGkS½TIRbCMuTMgn;. smIkar %>! KWQrenAelIkarsnμt;fa karBRgay strain man lkçN³CabnÞat;BIelIdl;eRkam Edlmüa:geToteyIgGacsnμt;fa muxkat;Edlrab (plane) munrgkarBt; enArkSarabdEdleRkaykarBt;. el;IsBIenH muxkat;FñwmRtUvEtmanGkS½sIuemRTIbBaÄr ehIybnÞúkRtUvEt sßitenAkñúgbøg;EdlmanGkS½sIemRTIenaH. FñwmEdlminbMeBjtamklçxNÐTaMgenHRtUv)anBicarNaenAkñúg EpñkTI 5>13. kugRtaMgGtibrmanwgekItenAsrésEpñkxageRkAbMput Edl y mantMélGtibrma. dUc enHvamantMélGtibrmaBIrKW kugRtaMgsgát;GtibrmarnAsrésEpñkxagelIbMput nigkugRtaMgTajGtibrma enAsrésEpñkxageRkambMput. RbsinebIGkS½NWtCaGkS½sIuemRTI kugRtaMgTaMgBIrenHnwgmantMélesμIKña. sMrab;kugRtaMgGtibrma smIkar %>! GacsresrkñúgTMrg; f max = Mc Ix = M = M Ix / c Sx ¬%>@¦ Edl c CacMNayEdkBIGkS½NWteTAsrésrEpñkxageRkAbMput ehIy S x Cam:UDulmuxkat;eGLasÞicénmux kat; (elastic section modulus) . sMrab;RKb;rUbragmuxkat; section modulus mantMélefr. sMrab;mux kat;minsIuemRTI S x nwgmantMélBIr³ mYysMrab;srésEpñkxagelIbMput nigmYyeTotsMrab;srésEpñkxag eRkambMput. tMélrbs; S x sMrab; standard rolled shape RtUv)andak;kñúg dimension and properties table enAkñúg Manual. 116 Fñwm
  • 4. T.chhay smIkar %>! nig %>@ mantMéleTA)ankñúgkrNIbnÞúktUclμmEdlsMPar³enAEtsßitenAkñúg linear elastic range. sMrab;eRKOgbgÁúMEdk vamann½yfakugRtaMg f max minRtUvFMCag f y ehIymann½yfa m:Um:g;minRtUvFMCag M y = Fy S x Edl M y Cam:Um:g;Bt;EdleFVIeGayFñwmeTAdl;cMnuc yielding. enAkñúgrUbTI 5>3 FñwmTMrsamBaØCamYynwgbnÞúkcMcMnucenAkNþalElVgRtUv)anbgðajnUvkardak; bnÞúktamdMNak;kalCabnþbnÞab;. enAeBl yielding cab;epþIm karBRgaykugRtaMgenAelImuxkat;Elg manlkçN³CabnÞat; ehIy yielding nwgrIkralBIsrésEpñkxageRkAeTAGkS½NWt. kñúgeBlCamYyKña 117 Fñwm
  • 5. T.chhay tMbn;Edlrg yield nwglatsn§wgtambeNþayFñwmBIGkS½kNþalrbs;FñwmEdlm:Um:g;Bt;mantMélesμInwg M y enATItaMgCaeRcIn. tMbn;Edlrg yield enHRtUv)angðajedayépÞBN’exμAenAkñúgrUbTI 5>3 c nig d. enAkñúgrUbTI 5>2 b yielding eTIbnwgcab;epþIm. enAkñúgrUbTI 5>2 c yielding )anrIkralcUleTAkñúgRTnug ehIyenAkñúgrUbTI 5>2 b muxkat;TaMgmUl)an yield. eKRtUvkarm:Um:g;bEnßmkñúgtMélCamFüm vaesμIRb Ehl 12% én yield moment edIm,InaMFñwmBIdMNak;kal (b) eTAdMNak;kal (d) sMrab; W-shape . enAeBleKeTAdl;dMNak;kal (d) RbsinebIenAEtbEnßmbnÞúkeTotFñwmnwg)ak; enAeBlEdlFatuTaMgGs; rbs;muxkat;)aneTAdl; yield plateau rbs; stress-strain curve ehIy unrestrict plastic flow nwg ekIteLIg. Plastic hing RtUv)aneLIgRtg;GkS½rbs;Fñwm ehIysnøak;enHCamYnnwgsnøak;BitR)akdenA xagcugrbs;FñwmbegáIt)anCa unstable machanism . kñúgeBl plastic collapse, mechanism motion RtUv)anbgðajenAkñúgrUbTI 5>4. Structural analysis EdlQrelIkarBicarNa collapse mechanism RtUv)aneKehAfa plastic analysis. karENnaMBI plastic analysis nig design RtUv)anerobrab;enAkñúg Appendix A kñugesovePAenH. lT§PaBm:Um:g;)aøsÞic EdlCam:Um:g;EdlRtUvkaredIm,IbegáItsnøak;)aøsÞic GacRtUv)anKNnay:ag gayRsYlBIkarBicarNakarBRgaykugRtaMgRtUvKña. enAkñúgrUbTI 5>5 ers‘ultg;kugRtaMgsgát; nigkug RtaMgTajRtUv)anbgðaj Edl Ac CaRkLaépÞmuxkat;Edlrgkarsgát; nig At CaRkLaépÞmuxkat;Edl rgkarTaj. RkLaépÞTaMgenHCaRkLaépÞEdlenABIxagelI nigBIxageRkamGkS½NWt)aøsÞic (plastic neutral axis) EdlmincaM)ac;dUcKñanwgGkS½NWteGLasÞic. BIsßanPaBlMnwgrbs;kMlaMg eyIg)an C =T Ac Fy = At Fy Ac = At dUcenHGkS½NWt)aøsÞicEckmuxkat;CaBIcMENkesμIKña. sMrab;rUbragEdlsIemRTIeFobnwgGkS½énkarBt; GkS½NWteGLasÞic nigGkS½NWt)aøsÞicKWdUcKña. m:Um:g;)aøsÞic M p Ca resisting couple EdlbegáIteLIg edaykMlaMgBIresμIKña nigmanTisedApÞúyKña b¤ ⎛ A⎞ M p = Fy ( Ac )a = Fy ( At )a = Fy ⎜ ⎟a = Fy Z ⎝2⎠ 118 Fñwm
  • 6. T.chhay Edl A= RkLaépÞmuxkat;srub a = cMgayrvagGkS½NWtrbs;RkLaépÞBak;kNþalTaMgBIr ⎛ A⎞ Z = ⎜ ⎟a = m:UDulmuxkat;)aøsÞic (plastic section modulus) ⎝2⎠ ]TahrN_ 5>1³ CamYynwg built-up shape EdlbgðajenAkñúgrUbTI 5>6 cUrkMNt; ¬k¦ elastic section modulus S nig yielding moment M y nig ¬x¦ plastic section modulus Z nig plastic moment M p . karekageFobnwgGkS½ x ehIyEdkEdleRbIKW A572 Grade 50 . dMeNaHRsay³ ¬k¦ edaysarvamanlkçN³sIuemRTI enaH elastic neutral axis ¬GkS½ x ¦ sßitenABak;kNþalmuxkat; ¬TItaMgrbs;TIRbCMuTMgn;¦. m:Um:g;niclPaBrbs;muxkat;GacRtUvkMNt;)anedayeRbIRTwsþIbTGkS½ Rsb (parallel axis theorem) ehIylT§plénkarKNnaRtUv)ansegçbenAkñúgtarag 5>1. tarag 5>1 Component I A d I + Ad 2 Flange 260417 5000 162.5 132291667 Flange 260417 5000 162.5 132291667 Web 28125000 - - 28125000 Sum 292.71×106 119 Fñwm
  • 7. T.chhay Elastic section modulus KW I 292.71 ⋅10 6 292.71 ⋅10 6 S= = = = 1.67 ⋅10 6 mm 3 c 25 + (300 / 2 ) 175 Yield moment KW M y = Fy S = 345 × 1.67 = 576.15kN .m cMeLIy³ S = 1.67 ⋅106 mm3 nig M y = 576.15kN .m ¬x¦ edaysarrUbragenHmanlkçN³sIuemRTIeFobnwgGkS½ x / enaHGkS½enHEckmuxkat;CaBIrcMEnkesμIKña ehIyGkS½enHk¾Ca plastic neutral axis Edr. TIRbCMuTMgn;rbs;épÞBak;kNþalxagelIRtUv)an kMNt;edayeRbI principle of moment. Kitm:Um:;g;eFobGkS½NWténmuxkat;TaMgmUl ¬rUbTI 5>6¦ ehIykarKNnaRtUv)anerobCatarag 5>2. tarag 5>2 Component A y Ay Flange 5000 162.5 812500 Web 1875 75 140625 Sum 6875 953125 y=∑ Ay 953125 = = 138.64mm ∑A 6875 rUbTI 5>7 bgðajfaédXñas;m:Um:g;rbs;m:Umg;KUrEdlekItmanenAxagkñúgKW : a = 2 y = 2(138.64) = 277.28mm ehIy plastic section modulus KW ⎛ A⎞ Z = ⎜ ⎟a = 6875 × 277.28 = 1.906 ⋅10 6 mm 3 ⎝2⎠ Plastic moment KW M p = Fy Z = 345 × 1.906 = 657.6kN .m 120 Fñwm
  • 8. T.chhay cMeLIy³ Z = 1.906 ⋅106 mm3 nig M p = 657.6kN .m ]TahrN_ 5>2³ KNna plastic moment, M p sMrab; W 10 × 60 rbs;Edk A36 . dMeNaHRsay³ BI dimensions and properties tables enAkñúg Part1 of the Manual A = 17.6in 2 A 17.6 = = 8.8in 2 2 TIRbCMuTMgn;sMrab;RkLaépÞBak;kNþalGacrk)anBIkñúgtaragsMrab; WT-shapes EdlRtUv)ankat; ecjBI W-shapes. rUbragEdlRtUvKñarbs;vaKW WT 5× 30 ehIycMgayBIépÞxageRkAbMputrbs;søab eTATIRbCMuTMgn;KW 0.884in dUcbgðajenAkñúgrUbTI 5>8. a = d − 2(0.884 ) = 10.22 − 2(0.884 ) = 8.452in ⎛ A⎞ Z = ⎜ ⎟a = 8.8(8.452) = 74.38in 3 ⎝2⎠ lT§plEdlTTYl)anenHmantMélRbhak;RbEhlnwgtMélEdleGayenAkñúg dimensions and properties tables ¬PaBxusKñabNþalmkBIkarKitcMnYnxÞg;eRkayex,ós¦ cMeLIy³ M p = Fy Z = 36(74.38) = 2678in. − kips = 223 ft − kips 5>3> lMnwg Stability RbsinebIFñwmGacrkSalMnwgrbs;va)anrhUtdl;vasßitkñúglkçxNÐ)aøsÞiceBjelj enaH nominal moment strength RtUv)aneKKitfamantMélesμInwg plastic moment capacity Edl Mn = M p pÞúymkvij M n < M p . 121 Fñwm
  • 9. T.chhay dUckrNIssrEdr PaBKμanlMnwgGacmann½yCalkçN³srub b¤Gacmann½yCalkçN³edaytMbn;. karekagrbs;Ggát;RtUv)anbgðajenAkñúgrUbTI 5>9 a. enAeBlFñwmekag tMbn;rgkarsgát; ¬EpñkxagelI GkS½NWt¦ manlkçN³ nigkareFVIkarRsedognwgssr ehIyvanwg buckle RbsinebIEpñkrbs;muxkat;man lkçN³RsavRKb;RKan;. EtvamindUcssr edaysartMbn;rgkarsgát;rb;muxkat;RtUv)anTb;edayEpñk EdlrgkarTaj ehIyPaBdabmkxageRkA (flexural buckling) RtUv)anbegáIteLIgeday twisting (torsion). karbegáItnUvPaBKμanlMnwgenHRtUv)aneKehAfa lateral-torsional buckling (LTB). eKGacbgáar Lateral-torsional buckling )aneday lateral bracing tMbn;rgkarsgát; CaBiesssøab Edlrgkarsgát; CamYynwgcenøaHRKb;RKan;. karBRgwgenHRtUv)anbgðajlkçN³nimitþsBaØaenAkñúgrUbTI 5>9 b. dUcGVIEdleyIg)aneXIj moment strength GaRs½yeTAnwgRbEvgEpñkEdlmin)anBRgwgEdlCa cMgayrvagcMnucénTMrxag (lateral support) . eTaHbICaFñwmGacTTYlm:Um:g;RKb;RKan;edIm,IeFVIeGayvaeTAdl;lkçxNÐ)aøsÞiceBjelj vak¾RtUv GaRs½yfaetIva)anrkSa cross-sectional integrity b¤Gt;. vanwg)at;bg; integrity RbsinebIEpñkrgkar sgát;NamYyrbs;muxkat; buckle. RbePT buckling GacCa compression flange buckling Edl eKehAfa flange local buckling(FLB) b¤ buckling énEpñkrgkarsgát;rbs;RTnug EdleKehAfa web local buckle (WLB). dUcEdl)anerobrab;enAkñúgCMBUk 4 RbePT local buckling epSgeTotekIteLIg edayGaRs½ynwg width-thickness ratio rbs;Epñkrgkarsgát;rbs;muxkatt;. rUbTI 5>10 bgðajBIT§iBlrbs; local and lateral-torsional buckling. FñwmR)aMdac;eday ELkRtUv)anbgðajenAkñúgRkaPicénbnÞúk-PaBdab. ExSekagTI ! CaExSekagbnÞúk-PaBdabrbs;FñwmEdl KμanlMnwg ¬edayviFINak¾eday¦ ehIy)at;bg;lT§PaBRTbnÞúkrbs;vamuneBlvaeTAdl; first yield ¬rUbTI 5>3 b¦. ExSekag @ nig # RtUvKñanwgFñwmEdlGacRTbnÞúkedayqøgkat; first yield bu:Enþmin)anyUrRKb; 122 Fñwm
  • 10. T.chhay RKan;edIm,IbegáItsnøak;)aøsÞic nigTTYl)an plastic collapse. RbsinebIvaGaceTAdl; plastic collapse enaHExSekagbnÞúk-PaBdabnwgmanlkçN³dUcExSekag $ b¤ %. ExSekag $ sMrab;krNIm:Um:g;esμIenAeBj RbEvgFñwmTaMgmUl ehIyExSekag % sMrab;FñwmEdlrgm:Um:g;ERbRbYl (moment gradient) . eKGac TTYl)ankarKNnaRbkbedaysuvtßiPaBCamYynwgFñwmEdlRtUvKñanwgExSekagNamYyénExSekagTaMgenH b:uEnþExSekag ! nig @ bgðajBIkareRbIsMPar³edayKμanlkμN³RbsiT§PaB. 5>4> cMNat;fñak;rbs;rUbrag Classification of Shapes AISC cat;cMNat;fñak;rUbragmuxkat;Ca compact, noncompact b¤ slender GaRs½ynwgtMél rbs; width-thickness ratios. sMrab; I- nig H-shapes pleFobsMrab;søab (unstiffened element) KW b f / 2t f ehIypleFobsMrab;RTnug (stiffened element) KW h / t w . eKGacrk)ankarcat;cMNat;fñak; rbs;muxkat;enAkñúg Section B5 of the specification, “Local Buckling” in Table B5.1. vanwg RtUv)ansegçbdUcxageRkam. edayyk λ = width-thickness ratio λ p = upper limit for compact category λr = upper limit for noncompact category enaH RbsinebI λ ≤ λ p ehIysøabP¢ab;eTAnwgRTnugCab;Kμandac; enaHrUbragmanlkçN³ compact. RbsinebI λ p < λ ≤ λr enaHrUbragmanlkçN³ uncompact. RbsinebI λ > λr enaHrUbragmanlkçN³ slender. cMNat;fñak;RtUvQrelI width-thickness ratio rbs;muxkat;EdlmantMélFMCag. ]TahrN_ RbsinebI RTnugCa compact ehIysøabCa noncompact enaHrUbragRtUv)ancat;cMNat;fñak;Ca noncompact . 123 Fñwm
  • 11. T.chhay tarag 5>3 RtUv)andkRsg;ecjBI AISC Table B5.1 nigman width-thickness ratio sMrabmuxkat; hot-rolled I- nig H-shape. tarag 5>3 Width-thickness parameters* λp λr Element λ IS US IS US bf 170 65 370 141 Flange 2t f Fy Fy Fy − 69 Fy − 10 h 1680 640 2550 970 Web tw Fy Fy Fy Fy * sMrab; hot-rolled I- nig H-shape rgkarBt; 5>5> Bending Strength of Compact Shapes FñwmGac)ak;edayvaTTYlm:Um:g; M p ehIyvakøayCa)aøsÞiceBjelj b¤k¾vaGac)ak;eday !> lateral-torsional buckling (LTB), eday elastically b¤ inelastically @> flange local buckling (FLB), eday elastically b¤ inelastically #> web local buckling (WLB), eday elastically b¤ inelastically RbsinebIkugRtaMgBt;Gtibrma (maximum bending stress) tUcCagEdnsmamaRt (proportional limit) enAeBlEdl buckling ekIteLIg failure enHRtUv)aneKehAfa elastic. RbsinebI minGBa©wgenH vaCa inelastic. ¬sUmemIlkarbkRsayEdlTak;TgenAkñúgEpñk 4>2 rbs;emeronTI 4 .¦ edIm,IgayRsYl CadMbUgeyIgcat;cMNat;fñak;FñwmCa compact, noncompact b¤ slender. kar erobrab;enAkñúgEpñkenHGnuvtþcMeBaHFñwmBIrRbePT³ ¬!¦ hot-rolled I-nig H-shape ekageFobGkS½xøaMg ehIyEdlbnÞúkenAkñúgbøg;énGkS½exSay ehIy ¬2¦ channels ekageFobGkS½xøaMg ehIybnÞúkdak;tam shear center b¤k¾RtUv)anTb;RbqaMgnwgkarrmYl. ¬ Shear center CacMnucenAelImuxkat; EdltamcMnuc enHbnÞúkTTwgRtUv)ankat;tam RbsinebIFñwmekagedayKμankarrmYl.¦ vanwgekItmancMeBaH I-nig H- Shapes. eKminBicarNaGMBI Hybrid beam ¬Edlsøab nigRTnugrbs;vamanersIusþg;epSgKña¦eT ehIy smIkar AISC xøHnwgRtUv)anEkERbbnþicbnþÜcedIm,IeqøIytbeTAnwgkarkMNt;enH edayeKCMnYs Fyf nig Fyw EdlCa yield strength rbs;søab nigRTnugeday Fy . 124 Fñwm
  • 12. T.chhay eyIgcab;epþImCamYynwg compact shape EdlRtUv)ankMNt;CarUbragEdlRTnugrbs;vaRtUv)an P¢ab;eTAsøabCab;tdac; ehIyEdlbMeBjnUvtMrUvkar width-thickness ratio xageRkamsMrab;søab nig RTnug³ bf 2t ≤ 170 F nig th ≤ 1680 ¬xñatCa IS ¦ 2btf ≤ 65 nig th ≤ 640 ¬xñatCa IS ¦ F F F f y w y f y w y sMrab;RKb; standard hot-rolled shape Edl)anrayeQμaHenAkñúg Manual )aneKarBlkçxNÐxag elI dUcenHeKRtUvkarBinitüEtpleFobsøabb:ueNÑaH. rUbragPaKeRcInk¾bMeBjtMrUvkarrbs;søabEdr dUcenH vaRtUv)ancat;cMNat;fñak;Ca compact. RbsinebIFñwmCa compact ehIymanTMrxagCab; b¤ unbraced length xøI enaH nomina’moment strength, M n Ca plastic moment capacity eBjrbs;rUbrag M p . sMrab;Ggát;EdlminmanTMrxagRKb;RKan; moment resistance RtUv)ankMNt;eday lateral-torsional buckling strength EdlmanlkçN³Ca elastic b¤ inelastic . RbePTTImYy (laterally supported compact beam) CakrNIEdlFmμta nigsamBaØCageK. AISC F1.1 eGay nominal strength Ca Mn = M p (AISC Equation F1.1) Edl M p = F y Z ≤ 1 .5 M y tMélkMNt;eday 1.5M y sMrab; M p KWedIm,IkarBarbnÞúkEdleFVIkarelIslb; nigRtUv)anbMeBj enAeBlEdl F y Z ≤ 1 .5 F y S b¤ Z ≤ 1.5 S sMrab; I- nig H-shape ekageFobGkS½xøaMg enaH Z / S EtgEttUcCag 1.5 Canic©. ¬b:uEnþsMrab; I- nig H- shape ekageFobGkS½exSay enaH Z / S nwgminEdltUcCag 1.5 eT.¦ ]TahrN_ 5>3³ FñwmEdlbgðajenAkñúgrUbTI 5>11 CaEdl A36 EdlmanrUbrag W 16 × 31 . vaRTkM ralxNнebtugGarem:Edlpþl;nUv continuous lateral support dl;søabrgkarsgát;. Service dead loadKW 450lb / ft . bnÞúkenHRtUv)andak;BIelIFñwm vaminRtUv)anKItbBa©ÚlbnÞúkpÞal;rbs;FñwmeT. Service live load KW 550lb / ft . etIFñwmenHman moment strength RKb;RKan;b¤eT? 125 Fñwm
  • 13. T.chhay dMeNaHRsay³ Service dead load srub edayrYmbBa©ÚlTaMgTMgn;rbs;FñwmKW wD = 450 + 31 = 481lb / ft sMrab;FñwmTMrsamBaØrgbnÞúkBRgayesμI m:Um:g;Bt;GtibrmaekItmanenAkNþalElVgesμInwg 1 M max = wL2 8 Edl w CabnÞúkEdlmanxñatkMlaMgelIÉktþaRbEvg ehIy L CaRbEvgElVg. enaH 1 2 0.481× 30 2 M D = wL = = 54.11 ft − kips 8 8 0.55 × 30 2 ML = = 61.88 ft − kips 8 edaysar dead load tUcCag live load min)an 8 dg enaHbnSMbnÞúk A4-2 nwgmantMélFMCageK³ M u = 1.2M D + 1.6M L = 1.2 × 54.11 + 1.6 × 61.88 = 164 ft − kips müa:gvijeTot bnÞúkGacRtUv)anKitemKuNmun wu = 1.2wD + 1.6wL = 1.2 × 0.431 + 1.6 × 0.550 = 1.457kips / ft 1 1.457 × 30 2 M u = wu L2 = = 164 ft − kips 8 8 RtYtBinitü compactness ³ bf 2t = 6.3 ¬BI Part 1 of the Manual ¦ f 65 Fy = 65 36 = 10.8 > 6.3 dUcenH søabCa compact . h tw < 640 Fy ¬sMrab;RKb;rUbragenAkñúg Manual ¦ dUcenH W 16 × 31 Ca compact sMrab;Edk A36 . edaysarFñwmCa compact ehIymanTMrxag M n = M p = F y Z x = 36(54.0 ) = 1944in − kips = 162 ft − kips RtYtBinitüsMrab; M p ≤ 1.5M y ³ Zx 54 = = 1.15 < 1.5 (OK) S x 47.2 φb M n = 0.90(162) = 146 ft − kips < 164 ft − kips (NG) cMeLIy³ Design moment tUcCagm:Um:g;emKuN dUcenH W 16 × 31 minRKb;RKan;. 126 Fñwm
  • 14. T.chhay eTaHbICakarRtYtBinitüsMrab; M p ≤ 1.5M y RtUv)aneFVIenAkñúg]TahrN_xagelI b:uEnþvamincaM)ac; sMrab; I- nig H-shape ekageFobGkS½xøaMg ehIyvaminRtUv)aneFVIdEdl²enAkñúgesobePAenHeT. rbs; compact shape CaGnuKmn_nwg unbraced length, Lb EdlRtUv)ankM Strength moment Nt;CacMgayrvagcMnucénTMrxag b¤karBRgwg. enAkñúgesovePAenH bgðajcMnucénTMrxageday “X” dUc bgðajenAkñúgrUbTI 5>12. TMnak;TMngrvag nominal strength M n nig unbraced length RtUv)an bgðajenAkñúgrUbTI 5>13 . RbsinebI unbraced length minFMCag L p FñwmRtUv)anBicarNamanTMr xageBj ehIy M n = M p . RbsinebI Lb FMCag L p b:uEnþtUcCag b¤esμI)a:ra:Em:Rt Lr enaHersIusþg;nwg QrelI inelastic LTB . RbsinebI Lb FMCag Lr enaHersIusþg;nwgQrelI elastic LTB . eKGacrksmIkarsMrab; enAkñúg theorical elastic lateral-torsional buckling strength Theory of Elastic Stability (Timoshenko and Gere, 1961) nigCamYykarpøas;bþÚrnimitþsBaØaxøH smIkarenHmanragdUcxageRkam³ 127 Fñwm
  • 15. T.chhay 2 π ⎛ πE ⎞ Mn = EI y GJ + ⎜ ⎟ I y C w ⎜L ⎟ ¬%>#¦ Lb ⎝ b⎠ Edl Lb = unbraced length G = shear modulus = 77225MPa b¤ = 11200ksi sMrab;eRKOgbgÁúMEdk J = torsional constant C w = warping constant ( mm 6 ) RbsinebIm:Um:g;enAeBlEdl lateral-torsional buckling ekIteLIgFMCagm:Um:g;EdlRtUvKñanwg first yield enaH strength QrenAelI inelastic behavior. m:Um:g;EdlRtUvKñanwg first yield KW M r = FL S x (AISC Equation F1-7) Edl FL CatMélEdltUcCageKkñúgcMeNam ( Fyf − Fr ) nig Fyw . enAkñúgsmIkarenH yield stress enA kñúgsøabRtUv)ankat;bnßyeday Fr kugRtaMgEdlenAsl; (residual stress) . sMrab; nonhybrid member, F yf = Fym = Fy ehIy FL EtgEtesμInwg F y − Fr . teTAmuxeTotenAkñúgCMBUkenH eyIg CMnYs FL eday Fy − Fr . Ca]TahrN_ eyIgsresr AISC Equation E1-7 Ca ( M r = Fy − Fr S x ) (AISC Equation F1-7) EdlkugRtaMgEdlenAsl; Fr = 10ksi = 69MPa sMrab; rolled-shapes nig Fr = 16.5ksi = 114MPa sMrab; welded built-up shapes. dUcbgðajenAkñúgrUbTI 5>13 RBMEdnrvag elastic behavior nig inelastic behavior KW unbraced length Lr EdltMélrbs; Lr RtUv)anTTYlBIsmIkar %># enAeBl Edl M n RtUv)andak;eGayesμI M r . eKTTYl)ansmIkarxageRkam³ Lr = ry X 1 (Fy − Fr ) ( ) 1 + 1 + X 2 Fy − Fr 2 (AISC Equation F1-6) Edl π EGJA X1 = Sx 2 2 (AISC Equation F1-8 and F1-9) 4C w ⎛ S x ⎞ X2 = ⎜ ⎟ I y ⎝ GJ ⎠ dUckrNIssrEdr inelastic behavior rbs;FñwmmanlkçN³sμúKsμajCag elastic behavior CaTUeTAeKeRcIneRbIrUbmnþEdl)anmkBIkarBiesaFn_ (empirical formulas). CamYynwgkarEktMrUvd¾tic tYc AISC )aneGayeRbIsmIkarxageRkam³ 128 Fñwm
  • 16. T.chhay ⎛ Lb − L p ⎞ ( Mn = M p − M p − Mr ⎜ ) ⎟ ⎜ Lr − L p ⎟ ¬%>$¦ ⎝ ⎠ 790ry 300ry Edl Lp = Fy ¬xñat ¦ IS Lp = Fy ¬xñat US¦ (AISC Equation F1-4) Nominal bending strength rbs; compact beam RtUv)anbgðajedaysmIkar %># nig %>$ rgnUv upper limit M p sMrab; inelastic beam RbsinebIm:Um;g;EdlGnuvtþBRgayesμIelI unbraced length Lb . RbsinebIdUcenaHeT vaman moment gradient ehIysmIkar %># nig %>$ RtUv)anEksMrYledayemKuN Cb . emKuNenHRtUv)aneGayeday AISC F1.2 kñúgTMrg; 12.5M max Cb = (AISC Equation F1-3) 2.5M max + 3M A + 4 M B + 3M C Edl M max = tMéldac;xatrbs;m:Um:g;GtibrmaenAkñúg unbraced length (including the end points) M A = tMéldac;xatrbs;m:Um:g;enAcMnucmYyPaKbYnén unbraced length M B = tMéldac;xatrbs;m:Um:g;enAcMnucBak;kNþalén unbraced length M C = tMéldac;xatrbs;m:Um:g;enAcMnucbIPaKbYnén unbraced length enAeBlm:Um:g;Bt;BRgayesμI tMél Cb esμInwg 12.5M Cb = = 1.0 2.5M + 3M + 4M + 3M ]TahrN_ 5>4³ kMNt; Cb sMrab;FñwmTMrsamBaØRTbnÞúkBRgayesμICamYyEtnwgkarTb;xagenAxagcug b:ueNÑaH. 129 Fñwm
  • 17. T.chhay dMeNaHRsay³ edaysarlkçN³suIemRTI m:Um:g;GtibrmasßitenAkNþalElVg dUcenH 1 M max = M B = wL2 8 dUcKña edaysarlkçN³sIuemRTI m:Um:g;enAcMnucmYyPaKbIesμIm:Um:g;enAcMnucbIPaKbYn. BIrUbTI 5>14 wL ⎛ L ⎞ wL ⎛ L ⎞ wL 2 wL2 3 M A = MC = ⎜ ⎟− ⎜ ⎟= − = wL2 2 ⎝4⎠ 4 ⎝8⎠ 8 32 32 12.5M max 12.5(1 / 8) Cb = = = 1.14 2.5M max + 3M A + 4 M B + 3M C 2.5(1 / 8) + 3(3 / 32) + 4(1 / 8) + 3(3 / 32) cMeLIy³ Cb = 1.14 rUbTI 5>15 bgðajBItMélrbs; Cb sMrab;krNIFmμtaCaeRcInénkardak;bnÞúk nigTMrxag. sMrab; unbraced cantilever beams, AISC kMNt;tMél Cb = 1.0 . tMél 1.0 CatMéltUc ¬edayminKitBIrrUbragrbs;Fñwm nigkardak;bnÞúk¦ b:uEnþkñúgkrNIxøHvaCatMélEdltUcEmnETn. karkMNt; TaMgGs;én nominal moment strength sMrab; compact shapes GacRtUv)ansegçbdUcxageRkam³ 130 Fñwm
  • 18. T.chhay sMrab; Lb ≤ L p / M n = M p ≤ 1.5 M y (AISC Equation F1-1) sMrab; L p < Lb ≤ Lr / ⎡ ⎛ −L ⎞⎤ ( M n = Cb ⎢ M p − M p − M r )⎜ Lb − L p ⎟⎥ ≤ M p ⎜L ⎟ (AISC Equation F1-2) ⎢ ⎣ ⎝ r p ⎠⎥ ⎦ sMrab; L p > Lr / M n = M cr ≤ M p (AISC Equation F1-12) 2 π ⎛ πE ⎞ Edl M cr = Cb EI y GJ + ⎜ ⎜ L ⎟ I y Cw ⎟ (AISC Equation F1-13) Lb ⎝ b⎠ 2 C S X 2 X1 X 2 = b x 1 1+ Lb / ry ( 2 Lb / ry 2 ) tMélefr X1 nig X 2 RtUv)ankMNt;BImun ehIyRtUv)anrayCataragenAkñúg dimensions and properties tables in the Manual. T§iBlrbs; Cb eTAelI nominal strength RtUv)anbgðajenAkñúgrUbTI5>16. eTaHbICa strength smamaRtedaypÞal;eTAnwg Cb k¾eday EtRkaPicenH)anbgðajy:agc,as;BIsar³sMxan;rbs; upper limit M p edayminKitBIsar³sMxan;rbs;smIkarEdlRtUveRbIsMrab; M n . ]TahrN_ 5>4³ kMNt; design strength φb M n sMrab; W 14 × 68 rbs;Edk A242 Edl³ k> TMrxagCab; x> unbraced length = 20 ft / Cb = 1.0 131 Fñwm
  • 19. T.chhay K> unbraced length = 20 ft / Cb = 1.75 dMeNaHRsay³ k> BI Part 1 of the Manual /W14 × 68 KWsßitenAkñúg shape group 2 /dUcenHvaGacman yield stress F y = 50ksi / kMNt;faetIrUbragenHCa compact, noncompact b¤ slender. bf 65 = 7.0 < 2t f 50 rUbragenHKW compact dUcenH M n = M p = Fy Z x = 50(115) = 5750in. − kips = 479.2 ft − kips cMeLIy³ φb M n = 0.9(479.2) = 431 ft − kips x> Lb = 20 ft nig Cb = 1.0 . KNna L p nig Lr ³ 300ry 300 × 2.46 Lp = = = 104.4in. = 8.7 ft Fy 50 BI torsion properties tables in Part 1 of the Manual, J = 3.02in 4 nig C w = 5380in 6 eTaHbICa X1 nig X 2 RtUv)anerobCataragenAkñúg dimensions and properties table in part 1 of the Manual eyIgnwgKNnavaenATIenHsMrab;bgðaj π EGJA π 29000(11200)(3.02)(20) X1 = = = 3021ksi Sx 2 103 2 2 2 C ⎛S ⎞ ⎛ 5380 ⎞⎛ 103 ⎞ −2 X 2 = 4 w ⎜ x ⎟ = 4⎜ ⎟⎜ ⎟ = 0.001649ksi I y ⎝ GJ ⎠ ⎝ 121 ⎠⎝ 11200 × 3.02 ⎠ ry X 1 Lr = 1 + 1 + X 2 ( Fy − Fr ) 2 ( Fy − Fr ) 2.46(3021) = 1 + 1 + 0.001649(50 − 10 )2 = 316.8in = 26.40 ft (50 − 10) edaysar L p < Lb < Lr strength QrelI inelastic LTB nig ( M r = Fy − Fr S x = ) (50 − 10)(103) = 343.3 ft − kips 12 ⎡ ⎛ Lb − L p ⎞⎤ M n = Cb ⎢ M p − M p − M r ⎜ ( ⎟⎥ ⎜ Lr − L p ⎟⎥ ) ⎢ ⎣ ⎝ ⎠⎦ ⎡ ⎛ 20 − 8.7 ⎞⎤ = 1.0⎢479.2 − (479.2 − 343.3)⎜ ⎟⎥ ⎣ ⎝ 26.4 − 8.7 ⎠⎦ 132 Fñwm
  • 20. T.chhay cMeLIy³ φb M n = 0.90(392.4) = 353 ft − kips K> Lb = 20 ft nig Cb = 1.75 . Design strength sMrab; Cb = 1.75 KWesμInwg 1.75 dgén Design strength sMrab; Cb = 1.0 . dUcenH M n = 1.75(392.4) = 686.7 ft − kips > M p = 479.2 ft − kips Nominal strength minGacFMCag M p / dUcenHeRbI nominal strength M n = 479.2 ft − kips cMeLIy³ φb M n = 0.90(479.2) = 431 ft − kips Part 4 of the Manual of Steel Construction, “Beam and Girder Design,” mantaragmanRbeyaCn_ CaeRcInsMrab;viPaK nigKNnaFñwm. Ca]TahrN_ Load Factor Design Selection Table raynUvrUbrag EdleRbICaTUeTAsMrab;Fñwm EdlRtUv)anerobCalMdab;én Z x . edaysar M p = Fy Z x rUbragk¾RtUv)an erobCalMdab;én design moment φb M p . tMélefrdéTeTotEdlmanRbeyaCn_k¾RtUv)anerobCatarag EdlrYmman L p nig Lr ¬EdlCaEpñkmYyEdlKYreGayFujRTan;kñúgkarKNna¦. Plastic Analysis enAkñúgkrNICaeRcIn m:Um:g;emKuNGtibrma M u nwgRtUv)anTTYlBI elastic structural analysis edayeRbIbnÞúkemKuN. eRkamlkçxNÐc,as;las; ersIusþg;EdlcaM)ac; (required strength) sMrab;rcna sm<n§½EdlminGackMNt;edaysþaTic (statically inderteminate structure) RtUv)anrkedayeRbI plastic analysis. AISC GnuBaØateGayeRbI plastic analysis RbsinebIrUbrag compact nigRbsinebI Lb ≤ L pd 24800 + 15200(M 1 / M 2 ) Edl L pd = Fy ry ¬xñat SI ¦ (AISC Equation F1-17) m:Um:g;EdltUcCageKkñúgcMeNamm:Um:g;cugTaMgBIrsMrab; unbraced segment M1 = M 2 = m:Um:g;EdlFMCageKkñúgcMeNamm:Um:g;cugTaMgBIrsMrab; unbraced segment pleFob M1 / M 2 viC¢manenAeBlEdlm:Um:g;begáIt reverse curvature enAkñúg unbraced segment. enAeBlenH Lb Ca unbraced length EdlenACab;nwgsnøak;)aøsÞicEdlCaEpñkmYyén failure mechanism. b:uEnþRbsinebIeKeRbI plastic analysis, nominal moment strength M n EdlenACab;nwg 133 Fñwm
  • 21. T.chhay snøak;cugeRkayEdlminenAEk,rsnøak;)aøsÞicRtUv)anKNnatamviFIdUcKñasMrab;FñwmEdlviPaKedayviFIeG LasÞic ehIyvaRtUvEttUcCag M p . 5>6> Bending Strength of Noncompact Shapes dUckarkt;cMNaMBImun standard W-, M-, nig S-shapes PaKeRcInCa compact sMrab; F y = 250 MPa nig F y = 350MPa . cMnYntictYcb:ueNÑaHCa noncompact edaysar width- thickness ratio rbs;søab b:uEnþKμanrUbragmYyNaCa slender eT. edaysarmUlehtuTaMgenH AISC Specification edaHRsay noncompact nig slender flexural member enAkñúg]bsm<n§½ (Appendix F). enAkñúgesovePAenH eyIgnwgBicarNa slender flexural member enAkñúgCMBUkTI10. CaTUeTA FñwmGac)ak;eday lateral-torsional buckling, flange local buckling b¤ web local buckling. RKb;RbePTénkar)ak;GacsßitenAkñúgEdneGLasÞic b¤ inelastic range. RTnugrbs;RKb; rolled shapes enAkñúg Manual Ca compact dUcenH noncompact shapes CaRbFanbTsMrab;Etsßan PaBkMNt; (limit states) én lateral-torsional buckling nig flange local buckling. ersIusþg;EdlRtUv nwgsßanPaBkMNt;TaMgBIrRtUv)anKNna ehIyeKyktMélEdltUcCageK. BI AISC Appendix F CamYy bf λ= 2t f RbsinebI λ p < λ ≤ λr / enaHsøabCa noncompact ehIy buckling Ca inelastic eyIgnwgTTYl)an ⎛ λ − λp ⎞ ( Mn = M p − M p − Mr ⎜ )⎟ ⎜ λr − λ p ⎟ (AISC Equation A-F1-3) ⎝ ⎠ Edl λp = 170 Fy IS ¬sMrab; ¦ λp = 65 Fy ¬sMrab; US ¦ λr = 370 F y − Fr ¬sMrab; IS ¦ λr = 141 F y − Fr ¬sMrab; US ¦ ( M r = F y − Fr S x ) kugRtaMgEdlenAesssl; = 69MPa = 10ksi sMrab; rolled shapes Fr = ¬GgÁenHRtUv)ankMNt;sMrab; nonhybrid beam¦ 134 Fñwm
  • 22. T.chhay ]TahrN_ 5>6³ FñwmTMrsamBaØmYymanRbEvg 40 feet RtUv)anTb;xagenAxagcugrbs;va ehIyvargnUv service load dUcxageRkam³ Dead load = 400lb / ft ¬edayrYmbBa©ÚlTaMgTMgn;Fñwm¦ Live load = 1000lb / ft RbsinebIeKeRbI AISC A572 Grade 50 etI W 14 × 90 RKb;RKan;b¤Gt;? dMeNaHRsay³ bnÞúkemKuN nigm:Um:g;emKuNKW wu = 1.2wD + 1.6 wL = 1.2(0.40) + 1.6(1.00) = 2.08kips / ft 1 2.08(40 )2 M u = wu L2 = = 416.0 ft − kips 8 8 kMNt;lkçN³rUbragmuxkat; ¬faetICa compact, noncompact b¤ slender¦³ bf λ= = 10.2 2t f 65 65 λp = = = 9.19 Fy 50 141 141 λr = = = 22.3 F y − Fr 50 − 10 eday λ p < λ < λr dUcenHrUbragenHKW noncompact. RtYtBinitülT§PaBRTRTg;edayQrelIsßanPaB kMNt;rbs; flange local buckling³ 50(157 ) M p = Fy Z x = = 654.2 ft − kips 12 Mr ( ) = F y − Fr S x = (50 − 10) 143 12 = 476.7 ft − kips ⎛ λ − λp ⎞ Mn ( = M p − M p − Mr ⎜ ) ⎟ = 652.4 − (654.2 − 476.7 )⎛ 10.2 − 9.19 ⎞ = 640.5 ft − kips ⎜ λr − λ p ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ 22.3 − 9.19 ⎠ Design strength EdlQrenAelI FLBdUcenH φb M n = 0.9(640.5) = 576 ft − kips RtYtBinitülT§PaBRTRTg;EdlQrelIsßanPaBkMNt;rbs; lateral-torsional buckling. BI Load Factor Design Selection Table³ L p = 15 ft nig Lr = 38.4 ft Lb = 40 ft > Lr dUcenHvanwg)ak;edayeGLasÞic LTB. 135 Fñwm
  • 23. T.chhay BI Part 1 of the Manual/ I y = 362in 4 J = 4.06in 4 C w = 16000in 6 sMrab;FñwmTMrsamBaØRTbnÞúkBRgayesμICamYynwgTMrxagenAxagcugsgçag Cb = 1.14 AISC Equation F1-13 eGay 2 π ⎛ πE ⎞ M n = Cb EI y GJ + ⎜ ⎟ ⎜ L ⎟ I yCw ≤ M p Lb ⎝ b⎠ ⎡ 2 ⎤ = 1.14 ⎢ π ⎛ π × 29000 ⎞ 29000(362)(11200)(4.06 ) + ⎜ ⎟ (362)(16000) ⎥ ⎢ 40(12 ) ⎝ 40 × 12 ⎠ ⎥ ⎣ ⎦ = 1.14(5412 ) = 6180in. − kips = 515.0 ft − kips M p = 654.2 ft − kips > 515.0 ft − kips edaysar 515.0 < 640.5 dUcenH LTB lub ehIy φb M n = (0.90)515.0 = 464 ft − kips > M u = 416 ft − kips (OK) / cMeLIy³ eday M u < φb M n enaHFñwmman moment strength RKb;RKan;. lkçN³kMNt;rbs; noncompact shapes RtUv)ansMrYleday Load Factor Design Selection Table. Noncompact shapes RtUv)ankMNt;sMKal;eday footnote farUbragCa noncompact sMrab; F y = 250 MPa = 36ksi b¤ F y = 350 MPa = 50ksi . Noncompact shapes k¾RtUv)anerobcMenAkñúg taragedaylkçN³xusEbøkKñadUcxageRkam³ !> sMrab; noncompact shapes tMélEdlmanenAkñúgtaragrbs; φb M p CatMélBitR)akdrbs; design strength EdlQrelI flange local buckling. enAkñúg]TahrN_TI 5>6 eyIg)an KNnatMélenHesμInwg 576 ft − kips b:uEnþtMélRtwmRtUvenAkñúgtarag φb M p KW 0.90(654.2 ) = 589 ft − kips @> tMél L p enAkñúgtaragCatMélrbs; unbraced length Edl nominal strength EdlQr elI inelastic lateral torsional buckling esμInwg nominal strength EdlQrelI flange 136 Fñwm
  • 24. T.chhay local buckling dUcenH nominal strength sMrab; unbraced length GtibrmaGacRtUv)an KitCaersIusþg;EdlQrelI web local buckling. ¬rMlwkfa L p sMrab; compact shapes Ca unbraced length GtibrmaEdl nominal strength GacRtUv)anKitesμInwg plastic moment¦. sMrab;rUbragenAkñúg]TahrN_5>6 karKNna nominal strength EdlQrelI FLB eTAersIusþg;EdlQrelI inelastic LTB (AISC Equation F1-2) CamYynwg Cb = 1.0 ³ ⎛ Lb − L p ⎞ M n = M p − (M p − M r )⎜ ⎟ ¬%>%¦ ⎜L −L ⎟ ⎝ r p⎠ tMélrbs; M r nig Lr RtUv)anTTYlBI]TahrN_ 5>6 ehIynwgminRtUv)anpøas;bþÚr. b:uEnþ tMélrbs; L p RtUvEt)anKNnaBI AISC Equation F1-4³ 300ry 300(3.70 ) Lp = = = 157.0in. = 13.08 ft. Fy 50 CMnYstMélxagelIkñúgsmIkar %>% eyIgTTYl)an ⎛ L − 13.08 ⎞ 640.5 = 654.2 − (654.2 − 476.7 )⎜ b ⎟ ⎝ 38.4 − 13.08 ⎠ Lb = 15.0 ft. enHCatMélbBa©ÚlkñúgtaragCa L p sMrab; W = 14 × 90 CamYynwg Fy = 50ksi . cMNaMfa 300ry Lp = Fy GaceRbIsMrab; noncompact shapes. RbsinebIeFVIEbbenH lT§plEdlTTYl)anenAkñúg smIkarsMrab; inelastic LTB EdlRtUv)aneRbIenAeBl Lb minmantMélFMRKb;RKan; enaH ersIusþg;EdlQrelI FLB nwglub. 5>7> Summary of Moment Strength viFIsaRsþkñúgkarKNna nominal moment strength sMrab; I- nig H-shaped sections Edl ekageFobnwgGkS½ x nwgRtUv)ansegçbenATIenH. GgÁTaMgGs;EdlmanenAkñúgsmIkarxageRkamRtUv)an kMNt;rYcehIyBImun ehIyelxsmIkarrbs; AISC minRtUv)anbgðajenATIenHeT. karsegçbenHsMrab;Et compact shapes nig noncompact shapes Etb:ueNÑaH ¬minmansMrab; slender shapes eT¦. 137 Fñwm
  • 25. T.chhay !> kMNt;faetIrUbrag compact b¤Gt; @> RbsinebIrUbrag compact, RtYtBinitüsMrab; lateral-torsional buckling dUcxageRkam³ RbsinebI Lb ≤ L p vaminEmn LTB ehIy M n = M p RbsinebI L p < Lb ≤ Lr / vaman inelastic LTB ehIy ⎡ ⎛ −L ⎞⎤ ( M n = Cb ⎢ M p M p − M r )⎜ Lb − L p ⎟⎥ ≤ M p ⎜L ⎟ ⎢ ⎣ ⎝ r p ⎠⎥ ⎦ RbsinebI Lb > br / vaman elastic LTB ehIy 2 π ⎛ πE ⎞ M n = Cb EI y GJ + ⎜ ⎟ I y C w ≤ M p ⎜L ⎟ Lb ⎝ b⎠ #> RbsinebIrUbrag noncompact edaysarsøab/ RTnug b¤TaMgBIr enaH nominal strength nwgCa tMéltUcCageKénersIusþg;EdlRtUvKñanwg flange local buckling, web local buckling nig lateral-torsional buckling. k> Flange local buckling³ RbsinebI λ ≤ λ p vaminman FLB. RbsinebI λ p < λ ≤ λr søabCa noncompact, ehIy ⎛ λ − λp ⎞ ( Mn = M p − M p − Mr ⎜ ⎜ λr − λ p ) ⎟≤Mp ⎟ ⎝ ⎠ x> Web local buckling³ RbsinebI λ ≤ λ p vaminman WLB. RbsinebI λ p < λ ≤ λr RTnugCa noncompact, ehIy ⎛ λ − λp ⎞ ( Mn = M p − M p − Mr ⎜ ⎜ λr − λ p ) ⎟≤Mp ⎟ ⎝ ⎠ K> Lateral-torsional buckling³ RbsinebI Lb ≤ L p vaminman LTB. RbsinebI L p < Lb ≤ Lr / vaman inelastic LTB ehIy ⎡ ⎛ −L ⎞⎤ ( M n = Cb ⎢ M p M p − M r )⎜ Lb − L p ⎟⎥ ≤ M p ⎜L ⎟ ⎢ ⎣ ⎝ r p ⎠⎥ ⎦ RbsinebI Lb > br / vaman elastic LTB ehIy 138 Fñwm
  • 26. T.chhay 2 π ⎛ πE ⎞ M n = Cb EI y GJ + ⎜ ⎟ I y C w ≤ M p ⎜L ⎟ Lb ⎝ b⎠ 5>8> ersIusþg;kMlaMgkat;TTwg Shear Strength ersIusþg;kMlaMgkat;rbs;FñwmRtUvEtRKb;RKan;edIm,IbMeBjTMnak;TMng Vu ≤ φvVn Edl Vu = kMlaMgkat;TTwgGtibrmaEdll)anBIkarbnSMbnÞúkemKuNFMCageK φv = emKuNersIusþg;sMrab;kMlaMgkat;TTwg = 0.9 Vn = nominal shear strength/ BicarNaFñwmsamBaØenAkñúgrUbTI 5>17. enAcMgay x BITMrxageqVgnigsßitenAelIGkS½NWtrbs; muxkat; sßanPaBrbs;kugRtaMgRtUv)anbgðajenAkñúgrUbTI 5>17 d . edaysarFatuenHsßitenAelIGkS½ NWt vaminrgnUvkugRtaMgBt;eT. BI elementary mechanics of materials/ kugRtaMgkMlaMgkat;TTwg (shearing stess) KW fv = VQ Ib ¬%>^¦ 139 Fñwm
  • 27. T.chhay Edl fv =kugRtaMgkMlaMgkat;TTwgbBaÄr nigedkenARtg;cMnucEdleyIgBicarNa V = kMlaMgkat;TTwgbBaÄrenARtg;muxkat;EdlBicarNa Q = m:Um:g;RkLaépÞTImYyeFobGkS½NWt rvagcMnucEdlBicarNanwgEpñkxagelIb¤EpñkxageRkam rbs;muxkat; I = m:Um:g;niclPaBeFobnwgGkS½NWt b = TTwgrbs;muxkat;enAcMnucEdlBicarNa smIkar %>^ KWQrelIkarsnμt;fakugRtaMgmantMélefreBjelITTwg b dUcenHvapþl;tMélsuRkit sMrab;Et b mantMéltUc. sMrab;muxkat;ctuekaNEkgEdlmankMBs; d nigTTwg b tMéllMeGogsMrab; d / b = 2 KWRbEhl 3% . sMrab; d / b = 1 tMéllMeGogKW 12% nigsMrab; d / b = 1 / 4 tMéllMeGogKW 100% (Higdon, Ohlsen, and Stiles, 1960). sMrab;mUlehtuenH smIkar %>^ minGacGnuvtþ)ansMrab; søabrbs; W-shape dUcKñasMrab;RTnugrbs;va. rUbTI 5>18 bgðajBIkarBRgaykugRtaMgkMlaMgkat;sMrab; W-shape. ExSdac;CakugRtaMgmFüm V / Aw EdlBRgayenAkñúgRTnug ehIytMélenHminxusKñaBIkugRtaMgGtibrmaenAkñúgRTnugeRcIneT. eyIg eXIjc,as;ehIyfa RTnugnwg yield y:agyUrmunnwgsøabc,ab;epþIm yield. edaysarbBaðaenH yielding rbs;RTnugsMEdgnUvsßanPaBlImItkMNt;mYy. edayyk shear yield stress esμInwg 60% én tensile yield stress eyIgGacsresrsmIkarsMrab;kugRtaMgenAkñúgRTnugenAeBl)ak;Ca V f v = n = 0.60 F y Aw Edl Aw = RkLaépÞmuxkat;rbs;RTnug. dUcenH nominal strength EdlRtUvKñanwgsßanPaBkMNt;enHKW Vn = 0.6 F y Aw 140 Fñwm
  • 28. T.chhay ehIyvaGacCa nominal strength in shear RbsinebIRTnugminman shear buckling. RbsinebIvaekIt eLIgvanwgGaRs½ynwgpleFob width-thickness ratio h / t w rbs;RTnug. pleFob h / t w rbs;RTnug EdlRsavxøaMgmantMélFMNas; enaHRTnugGacnwg buckle in shear eday inelastic b¤ elastic. TMnak;TM ngrvag shear strength nig width-thickness ration manlkçN³RsedogKñanwgTMnak;TMngrvag flexural strength nig width-thickness ratio ¬sMrab; FLB b¤ WLB¦ nigrvag flexural strength nig unbraced length ¬sMrab; LTB¦. TMnak;TMngRtUvbgðajenAkñúgrUbTI 5>19 nigRtUv)aneGayenAkñúg AISC F2.2 dUc xageRkam³ sMrab; h / t w < 418 / Fy ¬sMrab; US¦/ h / t w < 1100 / Fy ¬sMrab; IS¦ RTnugmanesßrPaB Vn = 0.6 F y Aw (AISC Equation F2-1) sMrab; 418 / Fy < h / t w ≤ 523 / Fy ¬sMrab; US¦/ 1100 / Fy ≤ h / t w < 1375 / Fy ¬sMrab; IS¦ enaH inelastic web buckling GacnwgekIteLIg 418 / Fy Vn = 0.6 Fy Aw h/t ¬sMrab; US¦ Vn = 0.6Fy Aw 1100//t Fy ¬sMrab; IS¦ h w w (AISC Equation F2-1) sMrab; 523 / Fy < h / t w ≤ 260 ¬sMrab; US¦/ 1375 / Fy ≤ h / t w < 260 ¬sMrab; IS¦ enaH sßanPaBkMNt;KW elastic web buckling Vn = 132000 Aw ¬sMrab; US¦ Vn = 910 Aw2 ¬sMrab; IS¦ (AISC Equation F2-1) (h / t w ) 2 (h / t w ) Edl Aw = RkLaépÞmuxkat;rbs;RTnug = dt w KitCa ¬ mm 2 ¦ d = kMBs;srubrbs;Fñwm Vn = nominal strength ¬KitCa KN ¦ RbsinebI h / t w > 260 enaHeKRtUvkar web stiffener ehIyvaRtUv)anbriyayenAkñúg Appendix F2 ¬b¤ Appendix G sMrab; plate girder ¦. AISC Equation F2-3 KWQrelI elastic stability theory, ehIy Equation F2-2 CasmIkar Edl)anBIkarBiesaFn_sMrab;tMbn; inelastic Edlpþl;nUvkarpøas;bþÚrrvagsßanPaBkMNt; web yielding nig elastic web buckling. kMlaMgkat;CabBaðaEdlkMrekItmansMrab; rolled steel beams karGnuvtþn_TUeTAKWbnÞab;BIKNna FñwmsMrab; flexural ehIyeyIgnwgRtYtBinitümuxkat;EdlTTYl)ansMrab;kMlaMgkat;TTwg. 141 Fñwm
  • 29. T.chhay ]TahrN_ 5>7³ RtYtBinitüFñwmenAkñúg]TahrN_ 5>6 sMrab;kMlaMgkat;TTwg. dMeNaHRsay³ BI]TahrN_ 5>6/ wu = 2.080kips / ft nig L = 40 ft . Edk W 14 × 90 CamYynwg F y = 50ksi RtUv)aneRbI. sMrab;FñwmTmrsamBaØRTbnÞúkBRgayesμI kMlaMgkat;GtibrmaekItmanenA elITMr ehIyesμInwgkMlaMgRbtikmμ w L 2.080(40) Vu = u = = 41.6kips 2 2 BI dimensions and properties tables in Part 1 of the Manual, web width-thickness ratio rbs; W 14 × 90 KW h = 25.9 tw 418 418 = = 59.11 Fy 50 edaysar h / t w < 418 / Fy enaHersIusþg;RtUv)anRKb;RKgeday shear yielding rbs;RTnug Vn = 0.6 Fy Aw = 0.6 Fy (dt w ) = 0.6(50 )(14.02 )(0.44 ) = 185.1kips φvVn = 0.90(185.1) = 167kips > 41.6kips (OK) cMeLIy³ Shear design strength FMCagkMlaMgkat;emKuN dUcenHFñwmmanlkçN³RKb;RKan;. tMél φvVn EdlRtUv)anerobCataragenAkñúg factored uniform load table enAkñúg part 4 of the Manual dUcnHkarKNnarbs;vaminmanRbeyaCn_sMrab; standard hot-rolled shapes. , 142 Fñwm
  • 30. T.chhay Block Shear Block shear Edl)anBicarNasMrab;tMNenAkñúgGgát;rgkarTaj k¾GacekItmanenAkñúgRbePTxøH rbs;tMNenAkñúgFñwmEdr. edIm,IsMrYlkñúgkartP¢ab;BIFñwmmYyeTAFñwmmYyeTot edayeGaynIv:UsøabxagelI esμIKña enaHRbEvgd¾xøIrbs;søabxagelIrbs;FñwmmYyRtUvEtkat;ecj b¤ coped. RbsinebI coped beam RtUv)antP¢ab;edayb‘ULúgdUckñúgrUbTI 5>20 kMNt; ABC cg;rEhkecj. bnÞúkEdlGnuvtþenAkñúgkrNI enHnwgCaRbtikmμbBaÄrrbs;Fñwm dUcenHkMlaMgkat;nwgekItenAtamExS AB ehIynwgekItmankMlaMgTaj tam BC . dUcenH block shear strength nwgCatMélEdlkMNt;rbs;Rbtikmμ. eyIg)anerobrab;BIkarKNna block shear strength enAkñúgCMBUkTI3rYcehIy b:uEnþeyIgnwgrMlwk vaeLIgvijenATIenH. kar)ak;GacekIteLIgedaybnSMén shear yielding nig tendion fracture b¤eday shear fracture nig tension yielding. AISC J4.3, “Block Shear Rupture Strength,” eGaysmIkar BIrsMrab; block shear design strength³ [ φRn = φ 0.6 Fy Agv + Fu Ant ] (AISC Equation J4.3a) φRn = φ [0.6 Fu Anv + F y Agt ] (AISC Equation J4.3b) Edl φ = 0.75 Agv = gross area rgkMlaMgkat; ¬enAkñúgrUbTI 5>20 RbEvg AB KuNnwgkMras;RTnug¦ Anv = net area rgkMlaMgkat; Agt = gross area rgkMlaMgTaj ¬enAkñúgrUbTI 5>20 RbEvg BC KuNnwgkMras;RTnug¦ Ant = net area rgkMlaMgTaj smIkarEdlmanlT§plFMCagKWCasmIkarEdlmantY fracture FMCag. ]TahrN_ 5>8³ kMNt;RbtikmμemKuNGtibrma EdlQrelI block shearEdlGacRTFñwmdUcbgðajkñúg rUbTI 5>21. 143 Fñwm
  • 31. T.chhay dMeNaHRsay³ Ggát;p©itRbehagRbsiT§PaBKW 3 / 4 + 1/ 8 = 7 / 8in. . gross nig net shear areas KW Agv = (2 + 3 + 3 + 3)t w = 11(0.300) = 3.300in.2 ⎛ 7⎞ Anv = ⎜11 − 3.5 × ⎟(0.300) = 2.381in.2 ⎝ 8⎠ gross nig net tension areas KW Agt = 1.25t w = 1.25(0.300) = 0.375in.2 ⎛ 7⎞ Ant = ⎜1.25 − 0.5 × ⎟(0.300 ) = 0.2438in.2 ⎝ 8⎠ AISC Equation J4.3a eGay [ ] φRn = φ 0.6 Fy Agv + Fu Ant = 0.75[0.6(36)(3.3) + 58(0.2438)] = 64.1kips AISC Equation J4.3b eGay [ ] φRn = φ 0.6 Fu Anv + Fy Agt = 0.75[0.6(58)(2.381) + 36(0.3750)] = 72.3kips tY fracture enAkñúg AISC Equation J4.3b mantMélFMCag ¬Edl 82.86>14.14¦ dUcenHsmIkarenH mantMélFMCag. cMelIy³ RbtikmμemKuNGtibrmaEdlQrelI block shear=72.3kips. 5>9> PaBdab Deflection bEnßmBIelIsuvtßiPaB eRKOgbgÁúMRtUvEt serviceable . eRKOgbgÁúMEdlman serviceable CaeRKOg bgÁúMEdleFVIkar)anl¥ minbNþaleGayGñkEdleRbIR)as;vamanGarmμN_favaKμansuvtßiPaB. sMrab;Fñwm edIm,ITTYl)an serviceable eKRtUvkMNt;bMlas;TIbBaÄr b¤PaBdab. PaBdabFMCaTUeTAekItmancMeBaH flexible beam EdlGacmanbBaðaCamYynwgrMjr½. PaBdabGacbgábBaðaeTAdl;Ggát;d¾éTeTotEdlP¢ab; 144 Fñwm
  • 32. T.chhay eTAnwgva edaybNþaleGaymankMhUcRTg;RTaytUc. elIsBIenH GñkeRbIR)as;sMNg;nwgeXIjPaB GviC¢manedaysarPaBdabFM ehIyeFVIkarsnidæanxusfasMNg;KμansuvtßiPaB. sMrab;krNITUeTArbs;FñwmTMrsamBaØEdlRTbnÞúkBRgayesμIdUckúñrUbTI 5>22 PaBdabbBaÄrGti- brmaKW³ 5 wL4 Δ= 384 EI eKGacrk)anrUbmnþPaBdabsMrab;FñwmeRcInRbePT niglkçxNÐdak;bnÞúkenAkñúg Part 4, “Beam and Girder Design,”of the Manual. sMrab;sßanPaBminFmμtaeKGaceRbI standard analytical method dUcCa method of virtual work CaedIm. PaBdabCa serviceability limit state minEmnCa sßanPaBkMNt;sMrab;ersIusþg;eT dUcenHCaTUeTAPaBdabRtUv)ankMNt;CamYy service loads. karkMNt;d¾smrmüsMrab;PaBdabGtibrmaGaRs½yeTAnwgtYnaTIrbs;Fñwm nwgkarRbmaNBIPaB xUcxatEdlekItBIPaBdab. AISC Specification pþl;nUvkarENnaMtictYcEdlmanEcgenAkñúg Chapter L, “Serviceability Design Consideration,” faeKRtUvEtRtYtBinitüPaBdab. eKGacrk)ankarkMNt; d¾smrmüsMrab;PaBdabBI governing building code. tMélxageRkamCaPaBdabGnuBaØatGtibrmasrub ¬service dead load bUknwg service live load¦. L Plastered construction: 360 L Unplastered floor construction: 240 L Unplastered roof construction: 180 Edl L CaRbEvgElVg. eBlxøHeKcaM)ac;eRbIkarkMNt;PaBdabCatMélwlx CagkareRbIPaBdabCatMélRbPaK. eBlxøH karkMNt;RtUv)anKitcMeBaHPaBdabEdlbNþalEtBI live load, edaysarCaerOy² dead load deflection RtUv)ankarBarkñúgeBlsagsg;. 145 Fñwm
  • 33. T.chhay ]TahrN_ 5>9³ RtYtBinitüPaBdab;rbs;FñwmEdlbgðajenAkñúg rUbTI 5>23. PaBdabGtibrmasrub GnuBaØatKW 240 . L dMeNaHRsay³ PaBdabGtibrmasrubGnuBaØat = 240 = 9100 = 38mm L 240 Total service load = 7.3 + 8 = 15.3kN / m 5 wL4 5 × 15.3 × 9100 4 Maximum total deflection = = = 32.2mm < 38mm (OK) 384 EI 384 × 2 ⋅105 × 212 ⋅10 6 cMeLIy³ FñwmbMeBjlkçxNÐPaBdab PondingCaPaBdabmYyEdlb:HBal;dl;suvtßiPaBrbs;eRKOgbgÁúM. vaeRKaHfñak;bMputsMrab;RbBn§½ kMralxNнrabesμIGaceFVIeGayTwkePøógdk;. RbsinebIRbBn§½bgðÚrTwksÞHkñúgGMLúgeBlePøóg TMgn;rbs;Twk Edldk;elIkMraleFVIeGaykMraldab EdlvabegáIt)anCaGagsMrab;sþúkTwkkan;EteRcIn. RbsinebIkrNI enHekIteLIgtQb;Qr enaHeRKOgbgÁúMGacnwg)ak;. AISC specification tMrUvfaRbBn§½dMbUlRtUvEtman PaBrwgRkajRKb;RKan;edIm,IkarBar ponding, elIsBIenH vaerobrab;BIkarkMNt;m:Um:g;niclPaB nig)a:ra:- Em:Rtd¾éTeTotenAkñúg Section K2, “Ponding”. 5>10> karKNnamuxkat; Design karKNnamuxkat;FñwmtMrUvkareRCIserIsrUbragmuxkat;EdlmanersIusþg;RKb;RKan; nigbMeBjtMrUvkar serviceability. enAeBleyIgKitBIersIusþg; flexure EtgEtmaneRKaHfñak;CagkMlaMgkat; dUcenHkar Gnuvtþn_TUeTAKWeKKNnamuxkat;sMrab; flexure rYcehIyRtYtBinitükMlaMgkat;tameRkay. viFIsaRsþkñúg karKNnamuxkat;RtUv)anerobrab;xageRkam³ !> kMNt;m:Um:g;emKuN/ M u . vadUcKñanwg required design strength, φb M n . TMgn;rbs;Fñwm CaEpñkrbs; desd load b:uEnþvaminRtUv)andwgenARtg;cMnucenH. eKGacsnμt;tMélenH b¤k¾eK ecalvasin bnÞab;mkeKnwgRtYtBinitüvaeLIgvijeRkayeBleKeRCIseIsrUbragehIy. 146 Fñwm
  • 34. T.chhay @> eRCIserIsrUbragEdlbMeBjnUvtMrUvkarersIusþg;enH. eKGacGnuvtþtamviFImYykñúgcMeNamviFIBIr xageRkam³ k> eRkayeBlsnμt;rUbragEdk KNna design strength rYcehIyeRbobeFobvaCamYy nwgm:Um:g;emKuN. epÞogpÞat;eLIgvijRbsinebIcaM)ac;. eKGaceRCIserIsrUbragsnμt; y:aggayRsYlEtenAkñúgsßanPaBkMNt;mYycMnYn ¬]TahrN_ 5>10¦. x> eRbI beam design charts in Part 4 of the Manual. eKcUlcitþviFIenH ehIyva RtUv)anBnül;enAkñúg]TahrN_ 5>10 xageRkam. #> RtYtBinitü shear strength. $> RtYtBinitüPaBdab. ]TahrN_ 5>10³ eRCIserIs standardhot-rolled shape of A36 sMrab;FñwmEdlbgðajenAkñúg rUbTI 5>24. FñwmenHmanTMrxagCab; ehIyRtUv)anRT uniform service live load 5kips / ft . PaBdab GtibrmaGnuBaØatsMrab;bnÞúkGefrKW L / 360 . dMeNaHRsay³ snμt;TMgn;FñwmesμI 100lb / ft . wu = 1.2 wD + 1.6 wL = 1.2(0.10) + 1.6(5.00) = 8.120kips / ft 1 8.12(30 )2 M u = wu L2 = = 913.5 ft − kips = requiredφb M n 8 8 snμt;farUbrag compact. sMrab;rUbrag compact ehIymanTMrxagCab; M n = M p = Z x Fy BI φb M n ≥ M u / φb F y Z x ≥ M u Mu 913.5(12) Zx ≥ = = 338.3in.3 φb Fy 0.90(36) CaFmμta Load Factor Design Selection Table erob rolled shapes EdlRtUv)aneRbICaFñwmedaytM él plastic section modulus fycuH. elIsBIenH RtUv)andak;CaRkumedayrUbragenAxagelIeKenAkñúg 147 Fñwm
  • 35. T.chhay Rkum ¬GkSrRkas;¦ rUbragEdlRsalCageKEdlman section modulus RKb;RKan;edIm,IbMeBj section modulus EdlfycuHenAkñúgRkum. kñúg]TahrN_enH rUbragEdlmantMélEk,rnwg section modulus requirement KW W 27 × 114 CamYynwg Z x = 343in.3 b:uEnþrUbragEdlRsalCageKKW W 30 × 108 Ca mYynwg Z x = 343in.3 . edaysar section modulusminsmamaRtedaypÞal;nwgRkLaépÞ karEdlman section modulus FMCamYynwgRkLaépÞtUc dUcenHTMgn;k¾GacRsaleTAtamRkLaépÞ. sakl,g W 30 ×108 . rUbrag compact dUcEdl)ansnμt; ¬noncompact shapesRtUv)ankM Nt;cMNaMenAkñúgtarag¦ dUcenH M n = M p dUcEdl)ansnμt;. TMgn;rbs;vaF¶n;Cagkarsnμt;bnþic dUcenHeKRtUvKNna required strength eLIgvij eTaHbICa W 30 × 108 manlT§PaBRTRTg;FMCaglT§PaBRTRTg;tMrUvkaredayrUbragsnμt;k¾eday EtvaPaKeRcInEtg EtmanlT§PaBRTRTg;FMCaglT§PaBRTRTg;tMrUvkaredayrUbragsnμt;. wu = 1.2(1.08) + 1.6(5.00) = 8.130kips / ft 8.130(30 )2 Mu = = 914.6 ft − kips 8 BI Load Factor Design Selection Table, φb M p = φb M n = 934 ft − kips > 914.6 ft − kips (OK) CMnYseGaykareRCIserIsrUbragEdlQrelI required section modulus, eKGaceRbI design strength φb M p edaysarvasmamaRtedaypÞal;nwg Z x ehIyvak¾RtUv)anrayenAkñúgtarag. bnÞab;mkeTot epÞógpÞat;kMlaMgkat; w L 8.13(30 ) Vu = u = = 122kips 2 2 BI factored uniform load tables / φvVn = 316kips > 122kips (OK) cugeRkaybMput epÞógpÞat;PaBdab. PaBdabGtibrmaGnuBaØatsMrab;bnÞúkGefrKW L / 360 L 30 × 12 = = 1in. 360 360 5 wL L4 5 (5.00 / 12 )(30 × 12 )4 Δ= = = 0.703in. < 1in. (OK) 384 EI x 384 29000(4470 ) cMeLIy³ eRbI W 30 × 108 . 148 Fñwm
  • 36. T.chhay Beam Design Charts eKmanRkaPic nigtaragCaeRcInsMrab;visVkrEdlGnuvtþn_ ehIyRkaPic nigtaragCMnYyTaMgenHCYy sMrYly:ageRcIndl;dMeNIrkarKNnamuxkat;. vaRtUv)aneKeRbIy:agTUlMTUlayenAkñúg design office b:uEnþ visVkrRtUvEteRbIvaedayRbytñ½. enAkñúgesovePAenHmin)anENnaMnUvRkaPic nigtaragTaMgGs;enaHlMGit Gs;eT b:uEnþRkaPic nigtaragxøHmansar³sMxan;kñúgkarENnaM CaBiessKW ExSekag design moment versus unbraced length EdleGayenAkñúg Part 4 of the Manual. ExSekagenHRtUv)anbgðajenAkñúgrUbTI 5>25 EdlbgðajBIRkaPic design moment φb M n Ca GnuKmn_én unbraced length Lb sMrab; particular compact shape. eKGacsg;RkaPicEbbenHsMrab; muxkat;epSg²CamYynwgtMélCak;lak;én Fy nig Cb edayeRbIsmikarsmRsbsMrab; moment strength. Manual chart rYmmanRKYsarénExSekagsMrab; rolled shapes CaeRcIn. ExSekagTaMgenHRtUv)an begáIteLIgCamYy Cb = 1.0 . sMrab;ExSekagepSgeTotrbs; Cb KuN design moment Edl)anBIta rageday Cb . RtUvcaMfa φb M n minGacFMCag φb M p ¬b¤ sMrab; noncompact shapes φb M n QrelI local buckling¦. bMerIbMras;rbs;RkaPicRtUv)anbgðajbgðajenAkñúgrUbTI 5>26 EdlExSekagEbbenHBIr RtUv)anbgðaj. cMNucNak¾edayenAelIRkaPicenH dUcCacMnucCYbKñaénExSdac;BIr bgðajBI design moment nig unbraced length. RbsinebIm:Um:g;Ca required moment capacity enaHExSekagEdlenABI elIcMnucenaHRtUvKñanwgFñwmEdlman moment capacity FMCag. ExSekagEdlenAxagsþaMKWsMrab;FñwmEdl man required moment capacity Cak;lak; eTaHbIsMrab; unbraced length FMCagk¾eday. dUcenH enA kñúgkarKNnamuxkat; RbsinebIeyIgdak; unbraced length nig required design strength cUleTAkñúg 149 Fñwm
  • 37. T.chhay RkaPic ExSekagenABIelI nigenABIsþaMcMnucenaH RtUvKñanwgFñwmEdlGacTTYlyk)an. RbsinebIeKKitTaMg ExSekagdac;² enaHExSekagsMrab;rUbragRsalCagsßitenABIelI nigBIxagsþaMExSekagdac;². cMNucenAelI ExSekagEdlRtUvnwg L p RtUv)anbgðajeday solid circle ehIy Lr RtUv)anbgðajeday open circle. eKmanExSekagBIrRbePT mYysMrab; Fy = 36ksi = 250MPa nigmYyeTotsMrab; Fy = 50ksi = 350 MPa . kñúg]TahrN_ 5>10 required design strength ¬EdlrYmbBa©ÚlTaMgTMgn;Fñwmsnμt;¦ KW 913.5 ft − kips ehIyvaman continuous lateral support. sMrab;TMrxagCab; eKGacyk Lb = 0 . BIRkaPic F y = 36ksi ExSekagRkas;TImYyenABIelI 913.5 ft − kips KW W 30 × 108 EdldUcKñanwgkareRCIserIs enAkñúg]TahrN_ 5>10. eTaHbICa Lb = 0 minRtUv)anbgðajenAkñúgRkaPicBiessk¾eday k¾tMéltUc bMputrbs; Lb EdlbgðajKWtUcCag L p sMrab;RKb;rUbragenAelITMBr½enaH. ExSekagFñwmEdlbgðajenAkñúgrUbTI 5>25 KWsMrab; compact shape dUcenHtMélrbs; φb M n sM rab;tMéltUcEdlRKb;RKan;rbs; Lb KW φb M p . dUcEdl)anerobrab;enAkñúgEpñk 5>6 RbsinebIrUbragCa noncompact tMélGtibrma φb M n nwgQrelI flange local buckling. vaCakarBitEdl maximum unbraced length sMrab; φb M n xagelInwgxusKñaBItMél L p EdlTTYlCamYynwg AISC Equation F1-4. The moment strength rbs; noncompact shapeRtUv)anbgðajCalkçN³RkaPicenAkñúgrUbTI 5>27 Edl maximum design strength RtUv)ankMNt;sMKal;eday φb M 'n ehIy maximum unbraced length EdlRtUvnwg φb M 'n xagelIRtUv)ansMKaleday L' p . 150 Fñwm
  • 38. T.chhay eTaHbICaRkaPicsMrab; compact nig noncompact shapes manlkçN³RsedogKñak¾eday k¾ φb M n nig Lb RtUv)aneRbIsMrab; compact shapes Et φb M 'n nig L' p RtUv)aneRbIsMrab; noncompact shapes. ]TahrN_ 5>11³ FñwmEdlbgðajenAkñúg rUbTI 5>28 RtUvRTbnÞúkcMcMnucGefrBIrEdlmYy²mantMél 20kips Rtg;cMnucmYyPaKbYn. PaBdabGtibrmaminRtUvFMCag L / 240 . Lateral support RtUv)anpþl; eGayenAcugrbs;Fñwm. eRbIEdk A572 Grade50 nigeRCIserIs rolled shape. 151 Fñwm
  • 39. T.chhay dMeNaHRsay³ RbsinebIeKecalTMgn;rbs;Fñwm enaHkMNat;FñwmcenøaHbnÞúkcMcMnucrgnUvm:Um:g;efr. M A = M B = M C = M max ehIy Cb = 1.0 eTaHRbsinCaeKKitTMgn;pÞal;rbs;Fñwmk¾eday k¾vaGacRtUv)anecaledayeFobnwgbnÞúkcMcMnuc ehIy Cb k¾enAEtmantMélesμI 1.0 EdlGnuBaØateGayeKGaceRbIRkaPicedayKμankarEkERb. edayminKitBITMgn;FñwmbeNþaHGasnñ eyIgTTYl)an M u = 6(1.6 × 20) = 192 ft − kips BIRkaPic CamYynwg Lb = 24 ft sakl,g W 15× 53 ³ φb M n = 219 ft − kips > 192 ft − kips (OK) LÚveyIgKitBITMgn;Fñwm M u = 192 + 1 (1.2 × 0.053)(24)2 = 197 ft − kips < 219 ft − kips (OK) 8 kMlaMgkat;TTwgKW 1.2(0.053)(24) Vu = 1.6(20) + = 32.8kips 2 BI factored uniform load tables/ φvVn = 112kips > 32.8kips (OK) PaBdabGtibrmaGnuBaØatKW L 24(12 ) = = 1.2in. 240 240 BI Beam Diagrams nig Formulas section in Part 4 of the Manual/ PaBdabGtibrma ¬enAkNþalElVg¦ sMrab;bnÞúkBIresμIKñaEdlRtUv)andak;sIuemRTIKñaKW Δ= Pa 24 EI ( 3L2 − 4a 2 . ) Edl P= GaMgtg;sIuetbnÞúkcMcMnuc a. =cMgayBITMreTAbnÞúk L = RbEvgElVg Δ= 20(6 × 12 ) 24 EI [ ] 3(24 × 12 )2 − 4(6 × 12 )2 = 13.69 × 10 6 EI sMrab;TMgn;pÞal;rbs;Fñwm PaBdabGtibrmak¾sßitenAkNþalElVgEd dUcenH 152 Fñwm
  • 40. T.chhay 5 wL4 5 (0.053 / 12 )(24 × 12 )4 0.04 × 10 6 Δ= = = 384 EI 384 EI EI PaBdabsrub 13.69 × 10 6 0.04 × 10 6 13.73 × 10 6 Δ= + = = 1.114in. < 1.2in. (OK) EI EI 29000(425) cMeLIy³ eRbI W12 × 53 . eTaHbICaRkaPicQrelI Cb = 1.0 k¾eday b:uEnþeKk¾GaceRbIvay:agRsYledIm,IKNnamuxkat;enA eBlEdl Cb minesIμnwg 1.0 edayEck required design strength eday Cb munnwgdak;vaeTAkñúgRka Pic. ]TahrN_ 5>12 nwgbgðajBIbec©keTsenH. ]TahrN_ 5>12³ eRbIEdk A36 ehIyeRCIserIs rolled shapes sMrab;FñwmenAkñúg rUbTI 5>29. bnÞúkcMcM nucCa service live load ehIybnÞúkBRgayesμIKW 30% CabnÞúkefr nig 70% CabnÞúkGefr. Lateral bracing RtUv)anpþl;eGayenAcug nigkNþalElVg. vaminmankarkMNt;sMrab;PaBdabeT. dMeNaHRsay³ edaysnμt;TMgn;FñwmesμI 100lb / ft. enaH wD = 0.30(3) + 0.10 = 1kips / ft. wL = 1.2(1.0 ) + 1.6(0.7 × 3) = 4.560kips / ft. Pu = 1.6(9) = 14.4kips bnÞúkemKuN nigRbtikmμRtUv)anbgðajenAkñúgrUbTI 5>30. m:Um:g;EdlcaM)ac;sMrab;KNna Cb ³ m:Um:g;Bt;enAcMgay x BIcugxageqVgKW ⎛ x⎞ M = 61.92 x − 4.590 x⎜ ⎟ = 61.92 x − 2.280 x 2 ⎝2⎠ ¬sMrab; x ≤ 12 ft ¦ sMrab; x = 3 ft / M A = 61.92(3) − 2.280(3)2 = 165.2 ft − kips 153 Fñwm
  • 41. T.chhay sMrab; x = 6 ft / M B = 61.92(6) − 2.280(6)2 = 289.4 ft − kips sMrab; x = 9 ft / M C = 61.92(9) − 2.280(9)2 = 372.6 ft − kips sMrab; x = 12 ft / M max = M u = 61.92(12) − 2.280(12)2 = 414.7 ft − kips 12.5M max Cb = 2.5M max + 3M A + 4 M B + 3M C 12.5(414.7 ) = = 1.36 2.5(414.7 ) + 3(165.2 ) + 4(289.4) + 3(372.6) bBa©ÚleTAkñúgRkaPicCamYynwg unbraced length Lb = 12 ft nigm:mU:g;Bt;KW M u 414.7 = = 305 ft − kips Cb 1.36 sakl,g W 21× 62 ³ φb M n = 343 ft − kips ¬sMrab; Cb = 1 ¦ edaysar Cb = 1.36 design strength BitR)akdKW φb M n = 1.36(343) = 466 ft − kips b:uEnþ design strength minRtUvelIs φb M p EdlesμIRtwmEt 389 ft − kips ¬TTYl)anBIRka Pic¦ dUcenH design strength BitR)akdRtUvEtesμInwg φb M n = 389 ft − kips < M u = 414.7 ft − kips (N.G.) sMrab;rUbragsakl,gbnÞab; eyIgRtUvrMkileLIgelIeTArkExSekagCab;Rkas;bnÞab;enAelIRkaPic eyIgTTYl)an W 21× 68 . sMrab; Lb = 12 ft design strength Edl)anBIRkaPicKW 385 ft − kips sMrab; Cb = 1.0 . ersIusþg;sMrab; Cb = 1.36 KW φb M n = 1.36(385) = 524 ft − kips > φb M p = 432 ft − kips dUcenH φb M n = φb M p = 432 ft − kips > M u = 414.7 ft − kips (OK) TMgn;FñwmKW 68lb / ft EdltUcCagTMgn;snμt; 100lb / ft . (OK) kMlaMgkat;TTwgKW Vu = 61.92kips ¬lT§plBitR)akdnwgtUcCagenHbnþic edaysarTMgn;pÞal;rbs;FñwmtUcCagbnÞúksnμt;¦ BI factored uniform load table φvVn = 177kips > 61.92kips (OK) cMeLIy³ eRbI W 21× 68 154 Fñwm
  • 42. T.chhay RbsinebItMrUvkarPaBdabRKb;RKgelIkarKNnamuxkat; eKRtUvkMNt;m:Um:g;niclPaBcaM)ac;Gb,- brma ehIyeKRtUvrkrUbragRsalCageKEdlRtUvnwgtMélenH. kargarenHRtUv)ansMrYly:ageRcIneday sar moment of inertia selection table in part 4 of the Manual. ]TahrN_ 5>13 nwgbgðajBIkar eRbIR)as;taragenH ehIynwgBnül;pgEdrBIviFIsaRsþkñúgkarKNnamuxkat;FñwmenAkñúgRbBn§½kMralxNн. ]TahrN_ 5>13³ EpñkénRbBn§½eRKagkMralRtUv)anbgðajenAkñúg rUbTI 5>31. kMralebtugBRgwgeday EdkmankMras; 4in. RtUv)anRTeday floor beams EdlmanKMlatBIKña 7 ft. . Floor beamsRtUv)anRT eday girders EdlRtUv)anbnþedayssr. ¬eBlxøH floor beamsRtUv)aneKehAfa filler beams¦. bEnßmBIelITMgn;rbs;rcnasm<n§½ bnÁÞúkrYmmanbnÞúkGefrBRgayesμI 80 psf nig movable partitions EdlRtUv)anKitCabnÞúkBRgayesμI 20 psf elIépÞkMral . PaBdabsrubGtibrmaminRtUvelIsBI 1/ 360 énRbEvgElVg. eRbIEdk A36 nigKNnamuxkat;rbs; floor beams. snμt;fakMralpþl;nUv continuous lateral support rbs; floor beams. 155 Fñwm
  • 43. T.chhay dMeNaHRsay³ eRbIebtugGarem:TMgn;FmμtaEdlmanTMgn; 150lb / ft 3 sMrab;KNnabnÞúkefr. TMgn;GacRtUv)anKitCabnÞúk kñúgmYyÉktþaépÞedayKuNTMgn;maDnwgkMras;kMralxNн. TMgn;kMralxNн = 150⎛⎜⎝ 12 ⎞⎟⎠ = 50 psf 4 snμt;faFñwmnImYy²RTnUvTTwgrgbnÞúk (tributary width) 7 ft. rbs;kMralxNн. kMralxNн³ 50(7) = 350lb / ft Partition³ 20(7 ) = 140lb / ft TMgn;Fñwm³ = 40lb / ft ¬)a:n;sμan¦ srub³ = 530lb / ft ¬ service dead load¦ eTaHbI partition Gacclt½)an b:uEnþ national model building codes KitvaCabnÞúkefr (BOCA, 1996; ICBO, 1997;nig SBCC, 1997). eyIgk¾KitvaCabnÞúkGefrEdrenATIenH. bnÞúkGefr³ 80(7) = 560lb / ft ehIybnÞúkemKuNsrubKW wu = 1.2wD + 1.6wL = 1.2(0.53) + 1.6(0.56) = 1.532kips / ft kartP¢ab;kMral-Fñwmpþl;nUv no moment restraint ehIyFñwmRtUv)anKitCaFñwmEdlRTedayTMrsamBaØ. 2 1.532(30 ) 2 1 M u = wu L = = 172.4 ft − kips 8 8 BI beam design chart CamYynwg Lb = 0 sakl,g W18× 35 ³ φb M u = 179.5 ft − kips > 172.4 ft − kips (OK) kMlaMgkat;TTwgKW 1532(30) Vu ≈ = 22.98kips 2 BI factored uniform load tables φvVn = 103kips > 22.98kips (OK) PaBdabGtibrmaGnuBaØatKW L 30(12) = = 1in. 360 360 5 wL4 5 (0.35 + 0.14 + 0.035 + 0.56)(30)4 (12)3 Δ= = = 1.3in. > 1in. (N.G.) 384 EI 384 29000(510) edayedaHRsaysmIkarPaBdabsMrab; required moment of inertia TTYl)an 156 Fñwm
  • 44. T.chhay 5wL4 384 5(1.085)(30)4 (12)3 I required = = = 682in.4 384 EΔ required 384(29000)(1) Moment of Inertia Selection Table RtUv)anerobcMeLIgkñúgviFIdUcKñanwg Load Factor Design Selection Table dUcenHkareRCIserIsrUbragEdlRsalCageKCamYynwgm:Um:g;niclPaBRKb;RKan;man lkçN³samBaØ. BI I x Table sakl,g W 21× 44 ³ I x = 843in.4 > 682in.4 (OK) φb M n = 257.5 ft − kips > 172.4 ft − kips (OK) TMgn;rbs;rUbragenHFMCagkarsnμt;dMbUgbnþic b:uEnþTMgn;EdlbEnßmenHminGaceRbobeFobnwg moment capacity d¾FMenaH)aneT. φvVn = 141kips > 22.98kips (OK) cMeLIy³ eRbI W 21× 44 . 5>11> rn§RbehagenAkñúgFñwm Holes in Beam RbsinebIkartP¢ab;FñwmRtUv)aneFVIeLIgCamYyb‘ULúg søab b¤RTnugrbs;FñwmRtUv)anecaHRbehag b¤xYg. elIsBIenH eBlxøHRTnugFñwmRtUv)anecaHrn§FM²edIm,Irt;eRKOgbrikçaepSg²dUcCa bMBugExSePøIg GKÁisnI bMBugxül;CaedIm. eKcUlcitþecaHrn§enAelIRTnugFñwmRtg;kEnøgNaEdlmankMlaMgkat;TTwgtUc ehIyrn§RbehagRtUv)anecaHenAelIsøabRtg;kEnøgNaEdlmanm:Um:g;tUc. b:uEnþeKminGaceFVIEbbenH)an rhUteT dUcenHeKRtUvKitBIT§iBlrbs;rn§Rbehag. sMrab;rn§RbehagtUc dUcsMrab;b‘ULúg T§iBlrbs;vanwgtUc CaBiesssMrab; flexure edaymUl ehtuBIr. TI1KW karkat;bnßymuxkat;tUc. TI2KW muxkat;EdlenAEk,rmin)ankat;bnßy ehIykarpøas; bþÚrmuxkat;énPaBminCab;tUcFMCag “weak link”. dUcenH AISC B10 GnuBaØateGayecalnUvT§iBlrbslrn§RbehagenAeBlEdl 0.75 Fu A fn ≥ 0.9 Fy A fg (AISC Equation B10-1) Edl A fg = gross flange are A fn = net flange are RbsinebIeKminCYbnUvlkçxNÐenHeT flexural properties RtUvEtQrelIRkLaépÞsøabrgkarTajRbsiT§ PaB 5 Fu A fe = A fn (AISC Equation B10-3) 6 Fy 157 Fñwm
  • 45. T.chhay ]TahrN_ 5>14³ KNna elastic section modulus EdlRtUv)ankat;bnßy S x sMrab;muxkat;Edl bgðajenAkñúgrUbTI 5>32. eKeRbIEdk A36 nigRbehagsMrab;b‘ULúgGgát;p©it 1in. . dMeNaHRsay³ A fg = b f t f = 7.635(0.81) = 6.184in 2 Ggát;p©itRbehagRbsiT§PaBKW 1 1 dh =1+ =1 in. 8 8 net flange area KW A fn = A fg − ∑ d h t f = 6.184 − 2(1.125)(0.810 ) = 4.362in.2 BI AISC Equation B10-1 0.75 Fu A fn = 0.75(58)(4.362 ) = 189.7kips nig 0.9Fy A fg = 0.9(36)(6.184) = 200.4kips edaysar 0.75Fu A fn < 0.9Fy A fg eyIgRtUvEtKitrn§Rbehag. edayeRbI AISC Equation B10-3 eGayRkLaépÞsøabRbsiT§PaB 5 Fu 5 ⎛ 58 ⎞ A fg = A fn = ⎜ ⎟4.362 = 5.856in.2 6 Fy 6 ⎝ 36 ⎠ RkLaépÞsøabenHRtUvKñanwgkarkat;bnßyeday 6.184 − 5.856 = 0.328in.2 . GkS½NWteGLasÞicsßitenA cMgay y BIkMBUlrbs;muxkat; 20.8(18.47 / 2 ) − 0.328(18.47 − 0.405) y= = 9.094in. 20.8 − 0.328 m:Um:g;niclPaBEdlRtUv)ankat;bnßyKW I x . = 1170 + 20.8(9.094 − 9.235)2 − 0.328(9.094 − 18.06)2 = 1144in.4 Sx sMrab;søabxagelIKW 158 Fñwm
  • 46. T.chhay I 1144 Sx = x = = 126in.3 y 9.094 Sx sMrab;søabxageRkamKW Ix 1144 Sx = = = 122in.3 d−y 18.47 − 9.094 cMeLIy³ The reduced elastic section modulus sMrab;EpñkxagelIKW 126in.3 nigsMrab;EpñkxageRkamKW 122in.3 . FñwmEdlmanrn§RbehagFMenAelIRTnug RtUvkarkarKNnaBiessEdlminmanerobrab;enAkñúgesov ePAenHeT. Design of Steel and Composite Beam with Web Openings KWCakarENnaMd¾manRb eyaCn_sMrab;RbFanbTenH (Darwin, 1990). 5>12> Open-Web Steel Joists Open-web steel joists CaRbePT truss EdlplitrYcCaeRscdUcbgðajenAkñúgrUbTI 5>33. Open-web steel joists xøHEdlmanTMhMtUc eRbIr)arEdkmUlCab;sMrab;eFVICaGgát;RTnug (web member) ehIyvaRtUv)aneKehA bar joists. vaRtUv)aneKeRbIenAkñúgkMral nigRbBn§½dMbUlsMrab;eRKOgbgÁúMCaeRcIn. sMrab;RbEvgElVgEdleGaydUcKña open-web steel joists manTMgn;RsalCag rolled shapes ehIyGvtþ manrbs;RTnugtan;GnuBaØateGayeKrt;RbBn§½brikçay:agRsYl. GaRs½yeTAnwgRbEvgElVg open-web steel joist manlkçN³esdækic©Cag rolled shapes eTaHbICavaKñaeKalkarN_ENnaMsMrab;karkMNt;vak¾ eday. eKGacrk open-web steel joists CamYynwgkMBs;sþg;dar niglT§PaBRTbnÞúkBIeragcRkCaeRcIn. Open-web steel joist xøHRtUv)anKNnaedIm,IeFVIkarCa floor b¤ roof joists ehIy open-web steel joists xøHeTotRtUv)anKNnaedIm,IeFVIkarCa girder EdlRTRbtikmμEdlRbmUlpþúMBI joists. AISC Specification min)anerobrab;BI open-web steel joists eT Etsßabn½mYyepSgeTotEdleKehAfa Steel Joist Institute (SJI) manBiBN’naBIva. ral;kareRbIR)as; steel joists rYmTaMgkarKNna nigkarplit RtUv)ane)aHBum<pSayenAkñúg Standard Specifications, Load Tables, nig Weight Table for Steel Joists and Joist Girders (SJI, 1994). 159 Fñwm
  • 47. T.chhay eKGaceRCIserIs open-web steel joists CamYynwg the aid of the standard load tables (SJI, 1994) ehIytaragmYyenAkñúgcMeNamenaHRtUv)anbgðajenAkñúgrUbTI 5>34 . CamYynwgkarpSMKñarvag ElVg nig joist eKnwgTTYl)antMélbnÞúkmYyKUr. elxxagelICa total service load capacity KitCa pounds kñúgmYy foot ehIyelxenAxageRkamCa service live load kñúgmYy foot EdlnwgbegáItPaBdab esμInwg 1/ 360 énRbEvgElVg. ¬eTaHbICabnÞúkenAkñúgtaragCa service load capacities k¾eday k¾eK GaceRbItaragenHy:aggayRsYlCamYynwgviFI LRFD EdleyIgnwgbgðajenATIenH¦. elxdMbUgénelx 160 Fñwm
  • 48. T.chhay sMKal;CakMBs;rbs; open-web steel joist EdlKitCa in. . taragk¾eGaypg EdrnUvTMgn;Rbhak;Rb EhlEdlKitCa pound kñúgmYy foot énRbEvg. eKGacrk open-web steel joists EdlRtUv)anKNnaedIm,ImannaTICa floor or roof joist ¬Edl pÞúyBImannaTICa girder¦ Ca open-web steel joist (K-series, both standard and KCS), longspan steel joists (LH-series), nig deep longspan steel joist (DLH-series). enAeBleyIgrMkilesrIeLIg kan;Etx<s; eyIgnwgTTYl)anRbEvgElVg niglT§PaBRTbnÞúkkan;EtFM. Ca]TahrN_ 8K1 manRbEvg ElVg 8 ft. niglT§PaBRTbnÞúk 550lb / ft. b:uEnþ 72DLH19 GacRTbnÞúk)an 497lb / ft. elIRbEvg 144 ft. . edayelIkElg KCS joists, open-web steel joists TaMgGs;RtUv)anKNnaCa trusses EdlRT edayTMrsamBaØ CamYynwgbnÞúkBRgayesμIenAelI top chord. kardak;bnÞúkenHeFVIeGay top chord rgnUv bending k¾dUc axial compression dUcenH top chord RtUv)anKNnaCa beam-column ¬emIlCMBUk 6¦. edIm,IFananUvesßrPaBrbs; top chord eKRtUvP¢ab; the floor or roof deck kñúgviFIEbbNaedIm,IeFVIeGay man continuous lateral support. TaMg top nig bottom chord members rbs; K-series joists RtUv)anplitedayEdkEdlman yield stress 50ksi . lT§PaBRTbnÞúkrbs; K-series joists RtUv)anepÞógpÞat;edaykarBiesaFn_ ehIy emKuNsuvtßiPaBGb,brmaRtUv)anbgðajeGayeXIjesμInwg 1.65 . viFIsaRsþd¾samBaØsMrab;eRbIR)as; standard load tables CamYynwg LRFD RtUv)anENnaMeday SJI (1994) ehIyRtUv)anbgðajenATIenH kñúgTMrg;EkERbbnþicbnþÜc. BicarNa TMnak;TMngeKal LRFD smIkar @>#³ ∑ γ i Qi ≤ φRn vaRtUv)ansresrsMrab;bnÞúkBRgayesμIkñúgTMrg;Ca wu ≤ φwn ¬%>&¦ Edl wu CabnÞúkBRgayesμIemKuN nig wn Ca nominal uniform load strength of the joist. Rbsin ebIeyIgeRbIpleFobmFümén nominal strength elI allowable strength esμInwg 1.65 eyIgGac * sresr nominal strength eday wn = 1.65wsji * cMNaMfaemKuNsuvtßiPaBsMrab; K-series joists RtUv)ankMNt;edaykarBesaFn_EdleFVIelIgedayplitkr. 161 Fñwm
  • 49. T.chhay Edl wsji Ca allowable strength (allowable load) EdleGayenAkñúg standard load tables. Design strength KW ( ) φwn = 0.9 1.65wsji = 1.485wsji ≈ 3 2 wsji LÚveyIgGacsresrsmIkar %>& Ca wu ≤ 3 2 wsji sMrab;eKalbMNgénkarKNna eyIgGacsresrTMnak;TMngenHCa required wsji = 2 3 wu ]TahrN_ 5>15³ eRbI load table EdleGayenAkñúg rUbTI 5>34 eRCIserIs open-web steel joist sMrab;RbBn§½kMral nigbnÞúkxageRkam. Joist spacing = 3 ft Span length = 20 ft bnÞúkKW³ kMralxNнkMras; 3in. bnÞúkefrepSgeTot = 20 psf bnÞúkGefr = 50 psf dMeNaHRsay³ sMrab;bnÞúkefr kMralxNн³ 50⎛⎝⎜ 12 ⎞⎟⎠ = 37.5 psf 3 bnÞúkefrepSgeTot = 20 psf TMgn;rbs; joist = 3 psf ¬]bma¦ srub = 60.5 psf wD = 60.5(3) = 181.5lb / ft sMrab;bnÞúkGefr 50 psf wL = 50(3) = 150lb / ft bnÞúkemKuNKW wu = 1.2 wD + 1.6 wL = 1.2(181.5) + 1.6(150 ) = 457.8lb' ft bMElgbnÞúkenHeTACa required service load³ wu = (457.8) = 305lb / ft 2 2 required wsji = 3 3 162 Fñwm
  • 50. T.chhay rUbTI 5>34 bgðajfa joist xageRkambMeBjnUvtMrUvkarénbnÞúkxagelI³ 12K 5 TMgn;RbEhl 7.1lb / ft / 14K 3 TMgn;RbEhl 6lb / ft nig 16K 2 TMgn;RbEhl 5.5lb / ft . edayminmankarkMNt;sMrab;kMBs; dUcenHeyIgerIsnUv joist NaEdlRsalCageK. cMeLIy³ eRbI 16K 2 . 5>13> bnÞHRTFñwm nigbnÞH)atssr Beam Bearing Plates and Column Base Plate viFIKNnabnÞHRTssrmanlkçN³RsedogKñanwgviFIKNnabnÞHRTFñwm ehIyedaysarmUlehtu enH eyIgnwgBicarNavaCamYyKña. elIsBI karkMNt;kMras;rbs;bnÞH)atssrtMrUveGaymankarBicarNa BI flexure dUcenHvaRtUv)anelIkykmkerobrab;enATIenH EdlminEmnenAkñúgCMBUk 4. kñúgkrNITaMgBIr tYnaTIrbs;bnÞHEdkKWEbgEckbnÞúkEdlRbmUlpþúM (concentrated load) eTAsMPar³EdlRTva. bnÞHRTFñwmmanBIrRbePTKW³ mYysMrab;bBa¢ÚnRbtikmμrbs;FñwmeTATMr dUcCaCBa¢aMgebtug nigmYy eTotsMrab;bBa¢ÚnbnÞúkeTAsøabxagelIrbs;Fñwm. dMbUg BicarNaTMrFñwmEdlbgðajenAkñúgrUbTI 5>35 . eTaHbICaFñwmCaeRcInRtUv)antP¢ab;eTAssrb¤eTAFñwmepSgeTotk¾eday EtRbePTénTMrEdlbgðajenATIenH RtUv)aneRbICaerOy² CaBiessenARtg; bridge abutments. karKNnaBIbnÞHRT rYmmanbICMhan³ !> kMNt;TMhM N EdleKGackarBar web yielding nig web crippling. @> kMNt;TMhM B EdlRkLaépÞ B × N manTMhMRKb;RKan;edIm,IkarBarsMPar³EdlRT ¬CaTUeTAKW ebtug¦ BIkarEbk. #> kMNt;kMras; t EdlbnÞHRTman bending strength RKb;RKan;. karBN’naBI Web yielding and web crippling manenAkñúg Chapter K of AISC Specifica- tion, “Strength Design Consideration”. ÉcMENk bearing strength rbs;ebtugRtUv)anniyayenA kñúg Chapter J, “Connections, Joints, and Fasteners”. 163 Fñwm
  • 51. T.chhay Web Yielding Web yielding KWCakarpÞúHEbkedaykarsgát; (compressive crushing) rbs;RTnugFñwmEdl bNþalBIkarGnuvtþn_kMlaMgsgát;edaypÞal;eTAsøabEdlenABIxagelI b¤BIxageRkamRTnug. kMlaMgenH GacCakMlaMgRbtikmμBITMrénRbePTdUcbgðajkñúg rUbTI 5>35 b¤vaGacCabnÞúkEdlbBa¢ÚneTAsøabeday ssr b¤FñwmepSgeTot. Yielding ekIteLIgenAeBlEdlkugRtaMgsgát;enAelImuxkat;edktamry³RTnug xiteTArkcMnuc yield. enAeBlbnÞúkRtUv)anbBa¢Úntamry³bnÞHEdk web yielding RtUv)ansnμt;faekIt manenAEk,rmuxkat;EdlmanTTwg t w . enAkñúg rolled shape muxkat;enARtg;cugénBitekag (toe of the fillet) EdlmancMgay k BIépÞxageRkArbs;søab ¬TMhMenHRtUv)anerobCatarag enAkñúg dimensions and properties tables in the Manual). RbsinebIbnÞúkRtUv)ansnμt;faEbgEckxøÜnvaeday slope 1 : 2.5 dUcbgðajenAkñúg rUbTI 5>36 RkLaépÞenARtg;TMrrgnUv yielding KW (2.5k + N )t w . edayKuN RkLaépÞenHnwg yield stress eGay nominal strength sMrab; web yielding enARtg;TMr³ Rn = (2.5k + N )Fy t w (AISC Equation K1-3) The bearing length N enARtg;TMrmikKYrtUcCag k . enARtg;bnÞúkxagkñúg beNþayrbs;muxkat;rgnUv yielding KW 2(2.5k ) + N = 5k + N The nominal strength KW Rn = (5k + N )Fy t w (AISC Equation K1-2) The design strength KW φRn , Edl φ = 1.0 Web Cripplimg Web cripplingCa buckling rbs;RTnugEdlbNþalBIkMlaMgsgát;EdlbBa¢Úntamry³søab. sMrab;bnÞúkxagkñúg nominal strength sMrab; web crippling KW³ 164 Fñwm
  • 52. T.chhay ⎡ 1.5 ⎤ ⎛N ⎞⎛ t w ⎞ ⎥ Fy t f Rn = 135t w ⎢1 + 3⎜ 2 ⎟⎜ ⎟ (AISC Equation K1-4) ⎢ ⎝ d ⎠⎜ t f ⎟ ⎝ ⎠ ⎥ tw ⎢ ⎣ ⎥ ⎦ sMrab;bnÞúkenARtg; b¤Ek,rTMr ¬minFMCagBak;kNþalkMBs;FñwmBIcug¦ nominal strength KW³ ⎡ 1.5 ⎤ ⎛N ⎞⎛ t w ⎞ ⎥ Fy t f Rn = 68t w ⎢1 + 3⎜ 2 ⎢ ⎟⎜ ⎟ ⎝ d ⎠⎜ t f ⎟ ⎥ tw sMrab; N ≤ 2 d (AISC Equation K1-5a) ⎢ ⎣ ⎝ ⎠ ⎥ ⎦ ⎡ 1.5 ⎤ 2⎢ ⎛ N ⎞⎛ t w ⎞ ⎥ Fy t f b¤ Rn = 68t w 1 + ⎜ 4 − 0.2 ⎟⎜ ⎟ ⎢ ⎝ d ⎠⎜ t f ⎟ ⎥ t w sMrab; N > 2 d (AISC Equation K1-5b) ⎢ ⎣ ⎝ ⎠ ⎥ ⎦ emKuNersIusþg;sMrab;sßanPaBkMNt;enHKW φ = 0.75 Concrete Bearing Strength sMPar³EdleRbIsMrab;RTFñwmGacCa ebtug dæ b¤sMPar³epSg²eTot b:uEnþCaTUeTAvaCaebtug. sMPar³enHRtUvEtTb;nwg bearing load EdlGnuvtþedaybnÞHEdk. The nominal bearing strength EdlbBa¢ak;enAkñúg AISC J9 dUcKñaenAkñúg American Concrete Institute’s Building Code (ACI, 1995). RbsinebI plate RKbeBjelIépÞrbs;TMr enaH nominal strength KW Pp = 0.85 f 'c A1 (AISC Equation J9-1) RbsinebI plate minRKbeBjelIépÞrbs;TMreT enaH nominal strength KW Pp = 0.85 f 'c A1 A2 / A1 (AISC Equation J9-2) 165 Fñwm
  • 53. T.chhay Edl ersIusþg;rgkarsgát; 28éf¶rbs;ebtug f 'c = A1 = bearing area R A2 = full area rbs;TMr RbsinebI A2 mincMCamYy A1 enaH A2 KYrmantMélFMCag A1 EdlvamanragFrNImaRtRsedog Kñanwg A1 dUcbgðajenAkñúgrUbTI 5>37. AISC tMrUveGay A2 / A1 ≤ 2 The design bearing strength KW φc Pp Edl φc = 0.60 . Plate Thickness enAeBlEdlbeNþay nigTTwgrbs;bnÞHTMrRtUv)ankMNt;ehIy bearing pressure mFümRtUv)an KitCabnÞúkBRgayesμIeTAelI)atén plate EdlRtUv)ansnμt;RTedayTTwg 2k EdlenAkNþalFñwmnig beNþay N dUcbgðajenAkñúgrUbTI 5>38. bnÞab;mkeTotbnÞHRtUv)anBicarNafaekageFobGkS½RsbeTA nwgElVgFñwm. dUcenH bnÞHRtUv)anKitCa cantilever EdlmanRbEvgElVg n = (B − 2k ) / 2 nigTTwg N . edIm,IgayRsYl TTwg 1in. RtUv)anBicarNa CamYynwgbnÞúkBRgayesμIKitCa lb / in. EdlesμInwg bearing pressure EdlKitCa lb / in.2 . BIrUbTI 5>38 m:Um:g;GtibrmaenAkñúgbnÞHKW Ru n R n2 Mu = ×n× = u BN 2 2 BN 166 Fñwm
  • 54. T.chhay Edl Ru / BN Ca bearing pressure mFümrvagbnÞHnigebtug. sMrab;muxkat;ctuekaNEkg EdlekageFobGkS½exSay (minor axis) enaH nominal moment strength M u esμInwg plastic moment capacity M p . dUcbgðajenAkñúgrUbTI 5>39 plastic moment sMrab;muxkat;ctuekaNEkg EdlmanTMhMTTwgmYyÉktþa nigkMras; t KW ⎛ t ⎞⎛ t ⎞ t2 M p = Fy ⎜1× ⎟⎜ ⎟ = Fy ⎝ 2 ⎠⎝ 2 ⎠ 4 edaysar φb M n RtUvEttUcCag M u φb M n ≥ M u t 2 Ru n 2 0 .9 F y ≥ 4 2 BN 2 Ru n 2 2.222 Ru n 2 t≥ 0.9 BNF y b¤ t≥ BNF y ¬%>* / %>(¦ ]TahrN_ 5>16³ KNna bearing plate edIm,IEbgEckRbtikmμrbs; W 21× 68 CamYynwgRbEvgElVg 15 ft. 10in. KitBIGkS½eTAGkS½rbs;TMr. Service load srub EdlKitbBa©ÚlTaMgTMgn;FñwmKW 9kips / ft EdlmanbnÞúkefr nigbnÞúkGefresμIKña. FñwmRtUv)anRTenABIelICBa¢aMgebtugGarem:Edlman f 'c = 3500 psi . TaMgbnÞHEdk nigFñwmCaEdk A36 . dMeNaHRsay³ bnÞúkemKuNKW wu = 1.2wD + 1.6wL = 1.2(4.5) + 1.6(4.5) = 12.6kips / ft. ehIyRbtikmμKW w L 12.6(15.83) Ru = u = = 99.73kips 2 2 kMNt;RbEvgrbs; bearing N EdlcaM)ac;edIm,IkarBar web yielding. BI AISC Equation K1-3, design strength sMrab;sßanPaBkMNt;enHKW Rn = (2.5k + N )Fy t w sMrab; φRn ≥ Ru / 1[2.5(1.438) + N ](36 )(0.430 ) ≥ 99.73 N ≥ 2.85in. 167 Fñwm
  • 55. T.chhay eRbI AISC Equation K1-5edIm,IkMNt;tMélrbs; N EdlcaM)ac;edIm,IkarBar web crippling. snμt; N / d ≥ 0.2 nigsakl,gTMrg;TIBIrrbs;smIkar. sMrab; φRn ≥ Ru / ⎡ 1.5 ⎤ 2⎢ ⎛ N ⎞⎛ t w ⎞ ⎥ Fy t f φ 68t w 1 + ⎜ 4 − 0.2 ⎟⎜ ⎟ ≥ Ru ⎢ ⎝ d ⎠⎜ t f ⎟ ⎥ ⎝ ⎠ tw ⎢ ⎣ ⎥ ⎦ ⎡ ⎛ 4N ⎞⎛ 0.43 ⎞ ⎤ 36(0.685) 1.5 0.75(68)(0.43) ⎢1 + ⎜ 2 − 0.2 ⎟⎜ ⎟ ⎥ ≥ 99.73 ⎢ ⎝ 21.13 ⎣ ⎠⎝ 0.685 ⎠ ⎥ ⎦ 0.43 N ≥ 5.27in. (controls) RtYtBinitükarsnμt; N 5.268 = = 0.25 > 0.2 (OK) d 21.13 sakl,g N = 6in. . kMNt;TMhM B BI bearing strength. karsnμt;EdlmansuvtßiPaBKWRkLaépÞeBj TaMgGs;rbs;TMrRtUv)aneRbI. φc (0.85) f 'c A1 ≥ Ru 0.6(0.85)(3.5)A1 ≥ 99.73 A1 ≥ 55.87in 2 tMélGb,brmarbs;TMhM B KW A 55.87 B= 1 = = 9.31in. N 6 TTwgsøabrbs; W 21× 68 KW 8.270in. EdleFVIeGaybnÞHEdkFMCagsøabbnþic EdleKcg;)an. sakl,g B = 10in. . kMNt;kMras;bnÞHEdkEdlcaM)ac; B − 2k 10 − 2(1.438) n= = = 3.562in. 2 2 BIsmIkar ¬%>(¦ 2.222 Ru n 2 2.222(99.73)(3.562 )2 t= = = 1.14in. BNF y 10(6 )(36 ) cMeLIy³ eRbI PL1 14 × 6 ×10 . 168 Fñwm
  • 56. T.chhay RbsinebIFñwmminRtUv)anBRgwgxagenARtg;cMnucrgbnÞúk ¬kñúgviFIEbbNaedIm,IkarBarbMlas;TIxag rvagsøabrgkMlaMgsgát; nigsøabrgkMlaMgTaj¦ eTenaH Specification tMrUveGayGegát sidesway web buckling (AISC K1.5). enAeBlbnÞúkGnuvtþeTAelIsøabTaMgBIr eKRtUvRtYtBinitü compression buckling (AISC K1.6). Column Base Plate dUcKñanwgkarKNna beam bearing plateEdr karKNna column base plate tMrUveGaymankar BicarNaBI bearing pressure eTAelIsMPar³EdlRT nig bending rbs;bnÞHEdk. PaBxusKñad¾FMbMputKW bending enAkñúg beam bearing plate KWmYyTis b:uEnþ column base plate rgnUv bending BIrTis. elIsBIenHeTot web crippling nig web yielding minEmnCabBaðaenAkñúgkarKNna column base plate eT. Column base plate GacRtUv)ancat;cMNat;fñak;CabnÞHFM b¤bnÞHtUc EdlbnÞHtUcmanTMhMRbhak; RbEhlTMhMssr. elIsBIenH bnÞHtUceFVIkarxusKña enAeBlvargbnÞúkRsal nigeBlvargbnÞúkF¶n;. kMras;rbs;bnÞHFMRtUv)ankMNt;BIkarBicarNaén bending rbs;EpñkénbnÞHEdllyecjBIssr. Bending RtUv)ansnμt;faekItmaneFobnwgGkS½enAkMBs;Bak;kNþalrbs;bnÞHEk,rRCugrbs;søabssr. GkS½BIrRsbeTAnwgRTnugmancMgayBIKña 0.80b f nigGkS½BIreTotRsbeTAnwgsøabmanKMlatBIKña 0.95d . 169 Fñwm
  • 57. T.chhay kñúgcMeNamcMerok cantilever 1in. BIrEdlsMKal;eday m nig n dUcenAkñúgrUbTI 5>40 tMélEdlFMCag eKRtUv)aneRbICMnYseGay n enAkñúgsmIikar %>* edIm,IKNnakMras;bnÞH b¤ t ≥l 2 Pu 0.9 BNF ¬%>!0¦ y Edl l CatMélFMCageKkñúgcMeNam m nig n . viFIenHsMedAdUceTAnwg cantilever method. bnÞH)attUcEdlRTTMgn;RsalRtUv)anKNnaedayeRbI Murray-Stockwell method (Murray, 1983). enAkñúgviFIenH EpñkénbnÞúkssrEdlGnuvtþenAkñúgRBMEdnrbs;muxkat;ssr ¬BIelIRkLaépÞ b f d ¦ RtUv)ansnμt;faEbgEckesμIenAelIRkLaépÞ H-shaped dUcbgðajkñúg rUbTI 5>41. dUcenH bearing pressure KWRbmUlpþúMenAEk,rExSRBMrbs;ssr. kMras;bnÞHRtUv)ankMNt;BI flexural analysis rbs;cMerok cantilever énTTwgÉktþa nigénRbEvg c . viFIenHpþl;lT§plCasmIkar t≥c 2 Po 0.9 A F ¬%>!!¦ H y Edl P Po = u × b f d BN = bnÞúkenAkñúgRkLaépÞ b f d = bnÞúkenAelIRklaépÞ H-shape AH = RklaépÞ H-shape c = TMhMEdlcaM)ac;edIm,IeGaykugRtaMg esμIeTAnwg design bearing stress rbs;sMPar³ Po AH EdlRT. cMNaMfasmIkar %>!! manTMrg;RsedogKñanwgsmIkar %>!0 edayeRbIkugRtaMg Pu / BN EdlCMnYs eday Po / AH . 170 Fñwm
  • 58. T.chhay sMrab;bnÞHEdlRTTMgn;F¶n; ¬RBMEdnrvagbnÞHRTTMgn;Rsal nigbnÞHRTTMgn;F¶n;minRtUv)ankMNt; Cak;lak;¦/ Thornton (1990a) EdlesñIrnUvkarviPaKedayQrelIkarBt;BIrTisrbs;EpñkénbnÞHrvagRT nug nigsøab. dUcEdl)anbgðajenAkñúg rUbTI 5>42 kMNat;énbnÞHenHRtUv)ancat;Tukfa fixed enAnwgRT nug/ TMrsamBaØenAnwgsøab nigTMenrenARCugmYyeTot. kMras;EdltMrUvkarKW 2 Pu t ≥ n' 0.9 BNF y Edl n'= 1 4 db f ¬%>!@¦ viFITaMgbIenHRtUv)anbBa©ÚlKñaeday Thornton (1990b) ehIykarsegçbmandUcxageRkam. kM ras;bnÞHEdlcaM)ac;KW t ≥l 2 Pu 0.9 BNF ¬%>!#¦ y Edl l = max(m, n, λn' ) 2 X λ= ≤1 1− 1− X ⎛ 4db ⎞ P X= ⎜ f ⎟ u ⎝ ( ⎜ d + b 2 ⎟φ P f ⎠ c p) 1 n'= db f 4 φc = 0.60 Pp = nominal bearing strength BI AISC Equation J9-1 b¤ J9-2. 171 Fñwm
  • 59. T.chhay CamYynwgsmIkarxagelIenH eKmincaM)ac;kMNt;fabnÞHFM b¤tUc rgbnÞúkRsal b¤F¶n;. λ Gacyk esμInwg 1.0 (Thornton, 1990b). viFIenHRsedogKñaeTAnwgGVIEdleGayenAkñúg Part 11 of the Manual (Volume II), “Connections for Tension and Compression”. ]TahrN_ 5>17³ eKeRbI W 10 × 49 CassrnwgRtUv)anRTeday concrete pierdUcbgðajkñúgrUbTI 5>43. épÞxagelIrbs; piermanTMhM 18in. ×18in. . KNnabnÞH A36 sMrab;bnÞúkefr 98kips nig bnÞúkGefr 145kips . ersIusþg;ebtugKW f 'c = 3000 psi . dMeNaHRsay³ bnÞúkemKuNKW Pu = 1.2 D + 1.6 L = 1.2(98) + 1.6(145) = 349.6kips KNna required bearing area φc Pp ≥ Pu φc (0.85) f 'c A1 A2 / A1 ≥ Pu 0.6(0.85)(3)A1 18(18) / A1 ≥ 349.6 A1 ≥ 161.1in.2 RtYtBinitü A2 / A1 = 18(18) / 161.1 = 1.41 < 2 (OK) mü:ageTot bnÞHRtUvEtmanTMhMFMCagTMhMssr dUcenH b f d = 10.00(9.98) = 99.8in.2 < 161.1in.2 (OK) sMrab; B = N = 13in. . A1 = 13(13) = 169in.2 TMhMrbs;cMerok cantilever m nig n GacRtUv)ankMNt;BI rUbTI 5>43 b¤ 172 Fñwm
  • 60. T.chhay N − 0.95d 13 − 9.48 m= = = 1.76in 2 2 N − 0.8b f 13 − 8 n= = = 2.5in. 2 2 BIsmIkar %>!@ 9.98(10) = 2.497in. 1 1 n' = db f = 4 4 edayyk λ = 1.0 eKTTYl)an l = max(m, n, n') = max (176,2.5,2.497 ) = 2.5in. BIsmIkar %>!#/ required plate thickness KW 2 Pu 2(349.6) t =l = 2.5 = 0.893in. 0.9 BNFy 0.9(13)(13)(36) cMeLIy³ eRbI PL1×13 ×13 . 5>14> Biaxial Bending Biaxial bending ekItmanenAeBlEdlFñwmrgnUvlkçxNÐbnÞúkEdlbegáIt bending tamTaMgGkS½ xøaMg (major or strong axis) nigGkS½exSay (minor or weak axis). dUckrNIbgðajenAkñúgrUbTI 5>44 EdlbnÞúkcMcMnuceTaleFVIGMeBIeTAelIGkS½beNþayrbs;Fñwm b:uEnþeRTteFobeTAnwgGkS½eKalnImYy²rbs; muxkat;. eTaHbICakardak;bnÞúkenHmanlkçN³TUeTACagkardak;bnÞúkBIelIkmunk¾eday k¾vaenAEtCakrNI 173 Fñwm
  • 61. T.chhay Biess edaysarbnÞúkkat;tam shear center rbs;muxkat;. The shear center KWCacMnucEdlbnÞúkeFVIGM eBIelIFñwmedaymineGayFñwmrgrmYl (no twisting nor torsion). TItaMgrbs; shear center GacRtUv)an kMNt;BI elementary mechanics of materials edayKNna internal resisting torsional moment EdlbMEbkBIrMhUrkMlaMgkat;enAkñúgmuxkat; eTA external torque. TItaMgrbs; shear center sMrab;muxkat;TUeTACaeRcInRtUv)anbgðajenAkñúg rUbTI5>45 a Edl shear center RtUv)ansMKal;eday “o”. tMélrbs; eo EdlkMNt;TItaMgrbs; shear center sMrab; channel shapes RtUv)anerobcMCataragenAkñúg Manual. CaTUeTA shear center EtgEtsßitenAelIGkS½ sIuemRTI dUcenH shear center nwgsßitenAelITIRbCMuTMgn;rbs;muxkat;EdlGkS½sIuemRTITaMgBIrkat;Kña. rUbTI 5>45 b bgðajTItaMgdabrbs;FñwmBIrepSgKña enAeBlbnÞúkGnuvtþkat;tam shear center nigminkat;tam shear center. krNITI1³ bnÞúkEdlGnuvtþkat;tam shear center RbsinebIbnÞúkeFVIGMeBIkat;tam shear center bBaðaFñwmrgnUvm:Um:g;Bt;FmμtakñúgTisedAEkgBIr. 174 Fñwm
  • 62. T.chhay dUcbgðajkñúg rUbTI 5>46 bnÞúkGacRtUv)anbMEbkCakMub:Usg;ctuekaNEkgkñúgTisedA x nigTisedA y EdlkMub:Usg;bnÞúknImYy²begáIt bending eFobGkS½epSgKña. edIm,IedaHRsayCamYybnÞúkpÁÜb mundMbUgeyIgsakl,gemIl chapter H of the Specification, “Manuals Under Combined Forces and Torsion” ¬ehIyemIleTACMBUkTI6 kñúgesovePAenH¦ sin. The Specification edaHRsaybnÞúkpÁÜbCadMbUgtamry³kareRbI interaction formulas EdlKitBIsar³sM xan;énT§iBlbnÞúknImYy²EdlmanTMnak;TMngeTAnwgersIusþg;EdlRtUvKñanwgT§iBlénbnÞúkenaH. ]Tahr- N_ RbsinebIman bending eFobEtnwgGkS½ x / M ux ≤ φb M nx b¤ φ M ux ≤ 1.0 b M nx Edl M ux = m:Um:g;Bt;emKuNeFobGkS½ x M nx = nominal moment strength eFobGkS½ x dUcKña RbsinmanEt bending eFobGkS½ y enaH M uy ≤ φb M ny b¤ φM uy ≤ 1.0M b ny Edl M uy = m:Um:g;Bt;emKuNeFobGkS½ y M ny = nominal moment strength eFobGkS½ y enAeBlmanRbePT bending TaMgBIr viFI interaction formula tMrUveGayplbUkpleFobTaMgBIrtUcCag b¤esμInwg 1.0 Edl M uy M ux φ M + φ M ≤ 1.0 ¬%>!$¦ b nx b ny 175 Fñwm
  • 63. T.chhay tamkarBit tMrUvkarenHGnuBaØateGay designer dak;bnÞúkkñúgTisedAmYyEdlminmandak;enAelITisedA mYyeTot. AISC Section H1 bBa©ÚlpleFobsMrab;bnÞúktamGkS½ nigeGay interaction formulas BIr EdlmYysMrab;bnÞúktamGkS½tUc nigmYyeTotsMrab;bnÞúktamGkS½FM ¬eyIgnwgsikSamUlehtusMrab;krNI enHenAkñúgCMBUk 6¦. CamYynwgm:Um:g;Bt;BIrTis ehIyKμanbnÞúktamGkS½ rUbmnþsMrab;bnÞúktamGkS½tUcKW Pu M ux M uy + + ≤ 1.0 (AISC Equation H1-1b) 2φPn φb M nx φb M ny RbsinebIbnÞúktamGkS½ Pu = 0 enaHsmIkarenHRtUvKñanwgsmIkar %>!$. mkdl;cMnucenH eKminBicarNaersIusþg;rbs;muxkat; I- nig H-shaped EdlekageFobGkS½ exSayeT. RbsinebIeFVIEbbenH vanwgmanlkçN³smBaØ. RKb;rUbragEdlekageFobnwgGkS½exSayrbs; vaminGac buckle kñúgTisedAepSgeToteT dUcenH lateral-torsional buckling minEmnCasßanPaBkM Nt;eT. RbsinebIrUbragmanlkçN³ compact enaH M ny = K py = F y Z y ≤ 1.5M yy Edl M yy = Fy S y = yield moment sMrab;GkS½ y . sMrab;muxkat; I- nig H-shaped EdlekageFob GkS½exSay Ednx<s;bMput 1.5M yy nwglubCanic© ¬ Z y / S y nwgFMCag 1.5 Canic©¦. RbsinebIrUbragCa noncompact ersIusþg;EdleGayeday AISC Equation A-F1-3 sMrab; flange local buckling b¤ web local buckling. ¬RKb; standard shapes EdlRtUv)anerobCataragenAkñúg Manual manRTnug compact dUcenHvaGacekItmanEt flange local buckling Etb:ueNÑaH.¦ ]TahrN_ 5>18³ W 21× 68 RtUv)aneRbICaFñwmTMrsamBaØEdlmanRbEvg 12 feet . søabrgkarsgát; RtUv)andak;TMrxagEtenAxagcug. bnÞúkeFVIGMeBItamry³ shear center CamYym:Um:g;emKuN M ux = 200 ft − kips ehIy M uy = 25 ft − kips . RbsinebIeKeRbI A36 etIFñwmenHbMeBjlkçxNÐ rbs; AISC Specification? snμt;fam:Um:g;TaMgBIrEbgEckesμIenAelIRbEvgrbs;Fñwm. dMeNaHRsay³ eKTTYl)anTinñn½yxageRkamsMrab; A36 BI Load Factor Design Selection Table. rUbragCa compact ehIy L p = 7.5 ft , Lr = 22.8 ft φb M p = 432 ft − kips, φb M r = 273 ft − kips The unbraced length Lb = 12 ft , dUcenH L p < Lb < Lr ehIysßanPaBkMNt;EdllubKWsßitenAkñúg 176 Fñwm
  • 64. T.chhay elastic lateral-torsion buckling enaH ⎡ ⎛ −L ⎞⎤ ( φb M nx = φb Cb ⎢ M p − M p − M r ⎜ )⎜ Lb − L p ⎟⎥ ⎟ ⎢ ⎣ L ⎝ r p ⎠⎥ ⎦ ⎡ ⎛ −L ⎞⎤ ( = Cb ⎢φb M p − φb M p − φb M r )⎜ Lb − L p ⎟⎥ ≤ φb M P ⎜L ⎟ ⎢ ⎣ ⎝ r p ⎠⎥ ⎦ edaysarm:Um:g;Bt;BRgayesμI/ Cb = 1.0 ehIy ⎡ ⎛ 12 − 7.5 ⎞⎤ φb M nx = 1.0⎢432 − (432 − 273)⎜ ⎟⎥ = 385.2 ft − kips ⎣ ⎝ 22.8 − 7.5 ⎠⎦ müa:gvijeTot eKGacTTYl φb M nx BI beam design charts edaysarrUbrag compact dUcenHvaKμan flange local bucklingehIy φb M ny = φb M py = φb Z y Fy = 0.90(24.4)(36 ) = 790.6in. − kips = 65.88 ft − kips RtYtBinitü Zy Sy = 24.4 15.7 = 1.55 > 1.5 dUcenHeRbI M ny = 1.5M yy = 1.5 F y S y = 1.5(36 )(15.7 ) = 847.8in. − kips = 70.75 ft − kips φb M ny = 0.9(70.75) = 63.59 ft − kips BIsmIkar %>!! M ux M uy 200 25 + = + = 0.912 < 1.0 (OK) φb M nx φb M ny 385.2 63.59 cMeLIy³ W 21× 68 RKb;RKan; krNITI2³ bnÞúkEdlGnuvtþminkat;tam shear center enAeBlEdlbnÞúkGnuvtþminkat;tam shear center rbs;muxkat; lT§plKWFñwmnwgrg flexure bUk nwg torsion. RbsinebIGaceFVIeTA)an rUbragFrNImaRtrbs;eRKOgbgÁúM nigtMNrKYrRtUvEkERbedIm,IbM)at;cM Nakp©it. bBaðarbs; torsion enAkñúg rolled shapes KWsμúKsμaj ehIyeyIgnwgedaHRsayvaCamYyviFIRb hak;RbEhl. eKGacrkkarerobrab;EdllkçN³lMGitsMrab;RbFanbT nig design aid RKb;RKan; enAkñúg Torsional Analysis of Structural Steel Members (AISC, 1997). lkçxNÐénkardak;bnÞúkEdleFVI eGayekItman torsion RtUv)anbgðajenAkñúg rUbTI 5>47 a. bnÞúkpÁÜbRtUv)andak;enAelIGkS½rbs;søab xagelI b:uEnþExSskmμrbs;vaminkat;tam shear center rbs;muxkat;eT. RbsinebIeyIgKitBIsßanPaBlM 177 Fñwm
  • 65. T.chhay nwg eyIgGacrMkilkMlaMgeTA shear center edaybEnßm couple. dUcenHeKTTYl)anRbBn§½lMnwgEdlpSM eLIgedaykMlaMgEdleGayeFVIGMeBIkat;tam shear center bUknwg twisting moment dUcEdl)anbgðaj. enAkñúg rUbTI 5>47 b eKmankMub:Usg;bnÞúkEtmYyEdlRtUvedaHRsay EtKMnitKWEtdUcKña. rUbTI 5>48 bgðajBIviFIEdlsMrYlkñúgkaredaHRsaykrNITaMgBIrenH. enAkñúgrUbTI5>48 a eK snμt;søabxagelIpþl;nUversIusþg;srubeTAnwgkMub:Usg;bnÞúkedk. enAkñúg rUbTI5>48 b m:Um:g;rmYl (twisting moment) RtUv)anTb;eday couple EdlpSMeLIgedaybnÞúkBIresμIKñaeFVIGMeBIelIsøabnImYy². tamviFIRbhak;RbEhl eKGacsnμt;fasøabnImYy²Tb;nwgkMlaMgdac;edayELkBIKña. dUcenH bBaða RtUv)ankat;bnßyeTACakrNIén bending rbs;rUbragBIr EdlrUbragnImYy²TTYlbnÞúktamry³ shear center. kñúgsßanPaBnImYy²Edl)anBiBN’naenAkñúg rUbTI 5>48 muxkat;EtRbEhlBak;kNþalb:ueNÑaH RtUv)anBicarNafamanRbsiT§PaBtamGkS½ y dUcenH enAeBlBicarNaersIusþg;rbs;rbs;søabeTal eRbItMélEtBak;kNþalrbs; Z y sMrab;muxkat;EdlmanenAkñúgtarag. Design of Roof Perlins édrENgdMbUl (roof purlin) CaEpñkénRbBn§½dMbUlCMral (sloping roof system) EdlrgnUvm:U m:g;Bt;BIrTis (biaxial bending) énRbePTEdleTIbnwgBN’na. sMrab; roof purlin EdlbgðajenAkñúg rUbTI 5>49 bnÞúkmanTisedAbBaÄr EtGkS½énkarBt;KWeRTt. lkçxNÐénkardak;bnÞúkenHRtUvnwgrUbTI 5>48 a. kMub:Usg;EkgeTAnwgdMbUlnwgbegáIt bending eFobGkS½ x ehIykMub:Usg;RsbBt;FñwmeFobGkS½ 178 Fñwm
  • 66. T.chhay y rbs;va. RbsinebI purlin RtUv)anRTedayTMrsamBaØenAnwg trusses ( b¤ rigid frame rafter) m:Um:g; Bt;GtibrmaeFobGkS½nImYy²KW wL2 / 8 Edl w CakMub:Usg;rbs;bnÞúk. RbsinebIeKeRbI sag rods vanwg pþl;nUv lateral support tamGkS½ x ehIynwgCaTMrsMrab;GkS½ y EdltMrUveGayKit purlin CaFñwmCab;. sMrab; sag rods EdlmanKMlatesμI eKGaceRbIrUbmnþsMrab;FñwmCab;enAkñúg Part 4 of the Manual. ]TahrN_ 5>19³ RbBn§½dMbUl trusses EdlbgðajenAkñúg rUbTI 5>50 EdlmanKMlatBIKña 15 ft. . éd rENgRtUv)andak;enAelItMNr nigenAelIcMnuckNþalrbs;Ggát;xagelI. eKdak; sag rods enAkNþal purlin nImYy². bnÞúk gravity srub EdlrYmbBa©ÚlTaMgTMgn;édrENgsnμt;KW 30 psf énépÞdMbUl CamYy nwgpleFobbnÞúkGefrelIbnÞúkefresμInwg 1.0 . edaysnμt;favaCalkçxNÐdak;bnÞúkeRKaHfñak; cUreRbIEdk A36 nigeRCIserIs W-shape EdlmankMBs; 6in. sMrab;édrENg. dMeNaHRsay³ sMrab;lkçxNÐbnÞúkenH bnÞúkefr bUknwg roof live load edayKμanxül; nigRBil bnSMbnÞúk (A4-3) nwgmantMélFMCageK³ wu 1.2wD + 1.6 Lr = 1.2(15) + 1.6(15) = 42 psf TTwgénépÞrgsMBaFEdlmanGMeBIelIédrENgKW 15 10 = 7.906 ft. 2 3 179 Fñwm
  • 67. T.chhay enaH bnÞúkelIédrENg = 42(7.906) = 332.1lb / ft kMub:Usg;Ekg = 10 (332.1) = 315.1lb / ft 3 kMub:Usg;Rsb = 1 (332.1) = 105.0lb / ft 10 nig M ux = 1 (0.3151)(15)2 = 8.862 ft − kips 8 CamYynwg sag rods Edldak;enAcMnuckNþalédrENgnImYy² enaHédrENgCaFñwmCab;BIrElVgtamTis exSay. BI “Beam Diagrams and Formulas” section in Part 4 of the Manual, m:Um:g;Bt;enAelI TrMxagkñúgCamYynwgkarrgbnÞúkEtmYyElVgKW 1 M= wL2 16 Edl w=bnÞúkBRgayesμI L = RbEvgElVg ¬ElVgBIresμIKña¦ CamYynwgbnÞúkenAelIElVgTaMgBIr m:Um:g;GacTTYleday superposition³ wL2 (2) = wL2 1 1 M = M max = 16 8 dUcenH M uy = 8 (0.105)(15 / 2)2 = 0.7382 ft − kips 1 edIm,IeRCIserIsrUbragsakl,g eRbI beam design charts nigeRCIserIsrUbragCamYynwgersIusþg;tam GkS½xøaMgFM. sMrab; unbraced length 15 / 2 = 7.5 ft / sakl,g W 6 × 9 . sMrab; Cb = 1.0 / φb M nx = 14.0 ft − kips . BI rUbTI 5>15 b Cb = 1.3 sMrab;lkçxNÐbbnÞúk nig lkçxNÐTMrxagénFñwmenH. dUcenH φb M nx = 1.30(14.0 ) = 18.20 ft − kips b:uEnþ φb M px = 16.8 ft − kips < 18.20 ft − kips dUcenHeRbI φb M nx = 16.8 ft − kips rUbragenH compact dUcenH φb M ny = φb M py = φb Z y Fy = 0.9(1.72 )36 = 55.73in. − kips = 4.644 ft − kips Zy b:uEnþ Sy = 1.72 1.11 = 1.55 > 1.5 180 Fñwm
  • 68. T.chhay dUcenH φb M ny = φb (1.5M yy ) = φb (1.5F y S y ) = 0.9(1.5)(36 )(1.11) = 53.95in. − kips = 4.496 ft − kips edaysarbnÞúkRtUv)anGnuvtþenAelIsøabxagelI eRbIlT§PaBenHEtBak;kNþaledIm,ITb;Tl;nwgT§iBl rmYl. BIsmIkar %>!$ M ux M uy 8.862 0.7382 + = + = 0.856 < 1.0 (OK) φb M nx φb M ny 16.8 4.496 / 2 kMlaMgkat;TTwgKW 0.3151(15) Vu = = 2.363kips 2 BI factored uniform load table³ φvVn = 19.5kips > 2.363kips (OK) cMeLIy³ eRbI W 6 × 9 . 5>15> ersIusþg;m:Um:g;Bt;rbs;rUbragepSg² Bending Strength of Various Shape W-, S- nig M-shapes Ca hot-rolled shapes EdleKeRbICaTUeTAsMrab;Fñwm ehIy bending strength rbs;vaRtUv)anerobrab;BIxagedIm. b:uEnþeBlxøHeKk¾eRbIrUbragepSg²eTotCa flexural members Edr ehIykñúgEpñkenHnwgniyaysegçbBIkarpþl;eGayrbs; AISC. smIkarTaMgGs;)anBI Chapter F b¤ Appendix F of the Specification. eKeGay Nominal strength sMrab; compact nig noncompact hot-rolled shapes b:uEnþminEmnsMrab; slender shapes b¤rUbragEdlpSMeLIgBIEdkbnÞHeT. kñúgEpñkenHmin)anpþl;nUv]TahrN_CatMélelxeT Et]TahrN_ 6>11 bBa©ÚlnUvkarKNnaBI flexural strength rbs; structural tee-shape. dUcEdl)anerobrab;BIxagedIm smIkarKWsMrab; nonhybrid section ( Fyw = Fyf = Fy ) nig sMrab;krNIBt;Etb:ueNÑaH ¬minmanbnÞúktamGkS½eT¦. I. Channels A. Width-thickness parameters for flexure 1. søab bf λ= tf / λp = 65 Fy nig λr = 141 F y − 10 ¬sMrab; US¦ bf λ= tf / λp = 170 Fy nig λr = 370 F y − 69 ¬sMrab; IS¦ 181 Fñwm
  • 69. T.chhay 2. RTnug λ= h tw / λp = 640 Fy nig λr = 970 F y − 10 ¬sMrab; US¦ λ= h tw / λp = 1680 Fy nig λr = 2550 Fy ¬sMrab; IS¦ B. Bending eFobGkS½xøaMg [CamYy ¬!¦ bnÞúkGnuvtþkat;tam shear center ehIysßitenA kñúgbøg;RsbnwgRTnug b¤ ¬@¦ karTb;RbqaMgnwgkarrmYlenAcMnucbnÞúkGnuvtþ nigenARtg;TMr] ³ M n dUcKñasMrab; I-shapes ¬emIlEpñk 5>5 nig 5>6¦. C. Bending eFobGkS½exSay³ M n dUcKñasMrab; I-shapes ¬emIlEpñk 5>14¦. II. Rectangular Structural Tubes A. Width-thickness parameters ¬emIlrUb 5>51¦ 1. søab λ= b t / λp = 190 Fy nig λr = 238 Fy ¬sMrab; US¦ λ= b t / λp = 500 Fy nig λr = 625 Fy ¬sMrab; IS¦ 2. RTnug λ= h t / λp = 640 Fy nig λr = 970 Fy ¬sMrab; US¦ λ= h t / λp = 1680 Fy nig λr = 2550 Fy ¬sMrab; IS¦ RbsinebIeKmindwgTMhMBitR)akd b nig h EdlbgðajenAkñúg rUbTI 5>51 eKGac)a:n; sμanedayykTTwgsrub b¤kMBs;srubdknwgbIdgkMras; ¬lkçN³rbs;EdkTIbRCug EdlmanenAkñúg Manual KWQrelIkaMxageRkAEdlesμInwg 2t ¦. 182 Fñwm
  • 70. T.chhay B. Bending eFobGkS½xøaMg ¬bnÞúkenAkñúgbøg;sIuemRTI¦ 1. rUbrag compact sMrab;rUbrag compact ersIusþg;nwgQrelIsßanPaBkMNt;én lateral-torsional buckling (LTB). sMrab; Lb ≤ L p M n = M p ≤ 1 .5 M y (AISC Equation F1-1) sMrab; L p < Lb ≤ Lr ⎡ ⎛ −L ⎞⎤ ( M n = Cb ⎢ M p − M p − M r )⎜ Lb − L p ⎟⎥ ≤ M p (AISC Equation F1-2) ⎜L ⎟ ⎢ ⎣ ⎝ r p ⎠⎥ ⎦ sMrab; Lb > Lr M n = M cr ≤ M p (AISC Equation F1-12) Edl M cr = 57000Cb JA Lb / ry xñat US (AISC Equation F1-14) M cr = 393000Cb JA Lb / ry xñat IS 3750ry JA Lp = Mp xñat US (AISC Equation F1-5) 25855ry JA Lp = Mp xñat IS 57000ry JA Lr = Mp xñat US (AISC Equation F1-10) 393000ry JA Lr = Mp xñat IS M r = Fy S x (AISC Equation F1-11) 2. rUbrag noncompact ³ The nominal strength esμInwgtMélEdltUcCageKéntMél Edl)anKNnasMrab;sßanPaBkMNt; lateral torsional buckling (LTB), flange local buckling (FLB) b¤ web local buckling (WLB). sMrab;sßanPaBnImYy² én local buckling TaMgBIr ersIusþg;RtUv)ankMNt;BIsmIkarxageRkam³ ⎛ λ − λp ⎞ ( Mn = M p − M p − Mr ⎜ ) ⎜ λr − λ p ⎟ (AISC Equation A-F1-3) ⎟ ⎝ ⎠ 183 Fñwm
  • 71. T.chhay C. Bending eFobGkS½exSay³ vaminmansßanPaBkMNt; LTB sMrab;RKb;rUbragEdlrgkar Bt;eFobGkS½exSayrbs;va. 1. rUbrag compact M n = M p ≤ 1 .5 M y (AISC Equation F1-1) 2. rUbrag nonompact³ RtYtBinitü FLB nig WLB CamYynwg AISC Equation A-F- 1-3. III. Square Structural Tubes A. Width-thickness parameters λ= b t / λp = 190 Fy nig λr = 238 Fy ¬sMrab; US¦ λ= b t / λp = 500 Fy nig λr = 625 Fy ¬sMrab; IS¦ B. Nominal bending strength vaminmansßanPaBkMNt; LTB sMrab;rUbragkaer: ¬b¤ctuekaNEkgeT¦. 1. rUbrag compact M n = M p ≤ 1 .5 M y (AISC Equation F1-1) 2. rUbrag nonompact³ ersIusþg;RtUv)ankMNt;eday local buckling eday WLB b¤ FLB edayykmYyNaEdl M n tUcCageK. ⎛ λ − λp ⎞ ( Mn = M p − M p − Mr ⎜ ⎜ λr − λ p ) ⎟ ⎟ (AISC Equation A-F1-3) ⎝ ⎠ IV. Solid Rectangular Bars sMrab; rectangular bars sßanPaBkMNt;EdlGacGnuvtþ)anKW LTB sMrab;GkS½Bt;xøaMg local buckling minEmnCasßanPaBkMNt;sMrab;GkS½Bt;xøaMg b¤k¾exSay. A. Bending eFobGkS½xøaMg sMrab; Lb ≤ L p M n = M p ≤ 1 .5 M y (AISC Equation F1-1) sMrab; L p < Lb ≤ Lr ⎡ ⎛ −L ⎞⎤ ( M n = Cb ⎢ M p − M p − M r )⎜ Lb − L p ⎟⎥ ≤ M p (AISC Equation F1-2) ⎜L ⎟ ⎢ ⎣ ⎝ r p ⎠⎥ ⎦ sMrab; Lb > Lr M n = M cr ≤ M p (AISC Equation F1-12) 184 Fñwm
  • 72. T.chhay Edl M cr = 57000Cb JA Lb / ry xñat US (AISC Equation F1-14) M cr = 393000Cb JA Lb / ry xñat IS 3750ry JA Lp = Mp xñat US (AISC Equation F1-5) 25855ry JA Lp = Mp xñat IS 57000ry JA Lr = Mp xñat US (AISC Equation F1-10) 393000ry JA Lr = Mp xñat IS M r = Fy S x (AISC Equation F1-11) B. Bending eFobGkS½exSay³ M n = M p ≤ 1 .5 M y (AISC Equation F1-1) V. Tees and double-anfle Shapes A. Width-thickness parameters 1. Tees a. søab bf λ= 2t f nig λr = 95 Fy eKmineRbI λ p ¬sMrab; US¦ bf λ= 2t f nig λr = 250 Fy eKmineRbI λ p ¬sMrab; IS¦ b. RTnug λ= d tw nig λr = 127 Fy eKmineRbI λ p ¬sMrab; US¦ λ= d tw nig λr = 333 Fy eKmineRbI λ p ¬sMrab; IS¦ 2. Double angles with separators, either leg λ= b t λr = nig76 Fy eKmineRbI λ p ¬sMrab; US¦ λ= b t nig λr = 200 Fy eKmineRbI λ p ¬sMrab; IS¦ 185 Fñwm
  • 73. T.chhay 3. Double angles in continuous contact, outstanding leg λ= b t / λr = 95 Fy λp eKmineRbIUS ¬sMrab; ¦ λ= b t / λr = 250 Fy eKmineRbI λ p ¬sMrab; IS¦ B. CamYybnÞúkenAkñúgbøg;sIuemRTI π EI y GJ ⎡ 2⎤ M n = M cr = ⎢B + 1 + B ⎥ (AISC Equation F1-15) Lb ⎣ ⎦ Edl M n ≤ 1 .5 M ysMrab; stem rgkarTaj M n ≤ 1.0 M y sMrab; stem rgkarsgát; ⎛ d ⎞ Iy B = ±2.3⎜ ⎟ ⎜L ⎟ j (AISC Equation F1-16) ⎝ b⎠ M y = Fy S x eKeRbIsBaØabUksMrab; B enAeBlEdl stem rgkarTaj ehIysBaØadkenAeBlEdl stem rgkarsgát; enARKb;kEnøgTaMgGs;tambeNþay unbraced length. C. Bending eFobGkS½exSay³ sMrab; nonslender shapes (λ ≤ λr ) M n = M p ≤ 1 .5 M y VI. Solid circular and square shapes M n = M p ≤ 1 .5 M y VII. Hollow circular shapes A. Width-thickness parameters λ= D t λp = / 2070 Fy nig λr = 8970 Fy ¬sMrab; US¦ λ= D t / λp = 14270 Fy nig λr = 61850 Fy ¬sMrab; IS¦ Edl D CaGgát;p©itxageRkA B. Norminal bending strength: vaminmansßanPaBkMNt; LTB sMrab;rUbragmUl ¬b¤kaer:¦. ersIusþg;RtUv)ankNt;eday local buckling. sMrab; λ ≤ λ p M n = M p ≤ 1 .5 M y 186 Fñwm
  • 74. T.chhay sMrab; λ p < λ ≤ λr ⎛ 600 ⎞ Mn = ⎜ + Fy ⎟ S (AISC Appendix F, Table A-F1.1) ⎝ D/t ⎠ 187 Fñwm