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V. shear and torsional strength design
 

V. shear and torsional strength design

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    V. shear and torsional strength design V. shear and torsional strength design Document Transcript

    • T.Chhay V. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl Shear and Torsion Strength Design 1> esckþIepþIm Introduction CMBUkenHnwgBN’naBIdMeNIrkarsikSaKNnamuxkat;ebtugeRbkugRtaMgedIm,ITb;Tl;kMlaMgkat; nigkM laMgrmYlEdlekItBIkMlaMgGnuvtþn_xageRkA. edaysarersIusþg;Tajrbs;ebtugexSayCagersIusþg;sgát; rbs;va karsikSaKNnasMrab;kMlaMgkat; nigkMlaMgrmYlkøayCaerOgsMxan;sMrab;RKb;RbePTeRKOgbgÁúM ebtugTaMgGs;. karEbgEckrvagkar)ak;rbs;FñwmebtugeRbkugRtaMgeRkamGMeBIkMlaMgkat; b¤bnSMkMlaMgkat; nigkM laMgrmYl KWxusBIkar)ak;eRkamGMeBIkMlaMgBt;begáag. va)ak;y:agelOnedaymin)anRbkasGsnñCamun RKb;RKan; ehIysñameRbHGgát;RTUgEdlekItmanmanTMhMFMCagsñameRbHedaysarGMeBIkMlaMgBt;begáag. TaMgkMlaMgkat; nigkMlaMgrmYlbegáItCakugRtaMgkat;. kugRtaMgenHGacegáItCakugRtaMgTajem (principal tensile stress) enARtg;muxkat;eRKaHfñak;EdlGacnwgmantMélFMCagersIusþg;Tajrbs;ebtug. cMNaMfa bøg;ekagrgkar)ak;edaykarrmYlKWbNþalmkBIm:Um:g;rmYlEdl)ak;bEnßmBIelI. kugRtaMgenAkñúgFñwmFmμta minRtwmEtekIteLIgedaysarkMlaMgkat;edaypÞal; (direct shear) b¤kMlaMgrmYlsuT§ (pure torsion) b:ueNÑaHeT b:uEnþvak¾ekIteLIgedaysarbnSMénkMlaMg nigm:Um:g;xageRkA. kMlaMgTaMgenaHbegáIteGayman kugRtaMgTajGgát;RTUg (diagonal tension stress) b¤kugRtaMgTajedaysarkarbegáag (flexural shear stress) enAkñúgGgát;. kugRtaMgkMlaMgkat;edaypÞal; b¤kugRtaMgrmYlsuT§ekItmanEtenAkñúgRbB½n§eRKOg bgÁúMxøHb:ueNÑaH dUcCakrNI corbel b¤ bracket EdlBak;B½n§nwgkMlaMgkat;edaypÞal; b¤ cantilever balcony EdlTak;TgCaBiessnwgkMlaMgrmYlsuT§enAelIFñwmTMr. 2> kareFVIkarrbs;Fñwm homogeneous eRkamGMeBIkMlaMgkat; Behavior of Homogeneous Beams in Shear BicarNaFatuGnnþtUcBIr A1 nig A2 rbs;FñwmctuekaNenAkñúgrUbTI 5>1 (a) EdlplitBIsMPar³ EdlmanlkçN³sac;mYy (homogeneous), lkçN³esμIsac; (isotropic) nig linearly elastic. rUbTI 5>2 (b) bgðajBIkarBRgaykugRtaMgBt; nigkarBRgaykugRtaMgkat;elIkMBs;rbs;muxkat;. kugRtaMg Shear and Torsion Strength Design 214
    • NPIC TajEkg (tensile normal stress) ft nigkugRtaMgkat; v CatMélenAkñúgFatu A1 enAelIkat;bøg; a − a Rtg;cMgay y BIG½kSNWt. BIeKalkarN_ classical emkanic eKGacsresrkugRtaMgEkg (normal stress) f nigkugRtaMg kat; v sMrab;Fatu A1 dUcxageRkam³ My f = (5.1) I nig v= VA y VQ Ib = Ib (5.2) Edl M nig V = m:Um:g;Bt; nigkMlaMgkat;enARtg;muxkat; a − a A = RkLaépÞrbs;muxkat;enARtg;bøg;Edlkat;tamTIRbCMuTMgn;rbs;Fatu A1 y = cMgayBIFatuGnnþtUceTAG½kSNWt y = cMgayBITIRbCMuTMgn;rbs; A eTAG½kSNWt karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 215
    • T.Chhay m:Um:g;niclPaBrbs;muxkat; I= Q = m:Um:g;sþaTicrbs;RkLaépÞmuxkat;EdlenABIxagelI b¤BIxageRkamG½kSNWt b = TTwgrbs;Fñwm rUbTI 5>2 bgðajBIkugRtaMgxagkñúgEdlmanGMeBIelIFatuGnnþtUc A1 nig A2 . edayeRbIrgVg;m: (Mohr’s cicle) enAkñúgrUbTI 5>2(b) kugRtaMgemsMrab;Fatu A1 enAkñúgtMbn;TajxageRkamG½kSNWtkøayCa 2 ⎛f ⎞ f t (max) f = t + ⎜ t ⎟ + v2 2 ⎝2⎠ kugRtaMgTajem (5.3a) 2 ⎛f ⎞ f c (max ) = ft 2 − ⎜ t ⎟ + v2 ⎝2⎠ kugRtaMgsgát;em (5.3b) nig tan 2θ max = v ft / 2 Shear and Torsion Strength Design 216
    • NPIC 3> kareFVIkarrbs;FñwmebtugGarem:CalkçN³minEmnsac;mYy Behavior of Concrete Beams as Nonhomogeneous Sections kareFVIkarrbs;FñwmebtugGarem: nigFñwmebtugeRbkugRtaMgxusBIFñwmEdk EdlersIusþg;Tajrbs; ebtugmantMélRbEhlmYyPaKdb;énersIusþg;sgát;rbs;ebtug. ersIusþg;sgát; fc enAkñúgFatu A2 énrUbTI 5>2(b) EdlenAxagelIG½kSNWtkarBareRbH edaysarkugRtaMgemGtibrmaenAkñúgFatuGnnþtUcCakugRtaMg sgát;. sMrab;Fatu A1 EdlenAxageRkamG½kSNWt kugRtaMgemGtibrmaCakugRtaMgTaj dUcenHnwgekItman karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 217
    • T.Chhay sñameRbH. edaysarm:Um:g;Bt; dUcKñanwgkugRtaMgTajfycuH ehIykugRtaMgkat;ekIneLIgkñúgTisedAeTArk TMr enaHkugRtaMgTajem ft (max ) nwgeFVIGMeBIelIbøg;RbEhl 45o Rtg;muxkat;Ek,rTMr dUceXIjenAkñúgrUbTI 5>3. edaysarersIusþg;Tajrbs;ebtugtUc sñameRbHGgát;RTUgekItmantambøg;Ekgnwgbøg;énkugRtaMg Tajem ehIysñameRbHenHeKeGayeQμaHfa sñameRbHGgát;RTUgTaj (diagonal tension crack). edIm,I karBarsñameRbHEbbenHBITIkEnøgcMh (opening) eKRtUvdak;EdkTajGgát;RTUgBiess. RbsinebIeKsnμt; ft EdlenAEk,rTMrénrUbTI 5>3 esμIsUnü enaHFatuGnnþtUcnwgesÞIrkøayeTACa sßanPaBénkugRtaMgkat;suT§ ehIykugRtaMgTajemEdleRbIsmIkar 5.3a nwgesμInwgkugRtaMgkMlaMgkat; v enAelIbøg; 45o . vaCakugRtaMgTajGgát;RTUgEdlbgáeGaymansñameRbHeRTt. karyl;y:agc,as;las;BI correct shear mechanism enAkñúgebtugGarem:enAminTan;RKb;RKan; enAeLIgeT. b:uEnþ viFIén ACI-ASCE Joint Committee 426 pþl;nUveKalkarN_mUldæanEdl)anBI lT§plénkarBiesaFy:ageRcInsn§wksn§ab;. 4> FñwmebtugEdlKμanEdkTajGgát;RTUg Concrete Beams without Diagonal Tension Reinforcement enAkñúgtMbn;énm:Um:g;Bt;FM sñameRbHekItmanesÞIrEtEkgnwgG½kSrbs;Fñwm. sñameRbHTaMgenHman eQμaHfa sñameRbHBt;begáag (flexural crack). enAkñúgtMbn;kMlaMgkat;FMedaysarkMlaMgTajGgát;RTUg sñameRbHeRTtekItmanbnþBI flexural crack ehIyRtUv)aneKehAfasñameRbHkMlaMgkat;begáag (flexural shear crack). rUbTI 5>4 bgðajBIRbePTsñameRbHEdlrMBwgnwgekItmanenAkñúgebtugGarem:edayman b¤ KμanEdkTajGgát;RTUgRKb;RKan;. Shear and Torsion Strength Design 218
    • NPIC enAkñúgFñwmeRbkugRtaMg muxkat;esÞIrEtrgkugRtaMgsgát;TaMgGs;eRkamGMeBIbnÞúkeFVIkar (service load). BIrUbTI 5>2 (c) nig (d) kugRtaMgemsMrab;Fatu A2 KW f t (max ) = − c + ( f c / 2)2 + v 2 f 2 kugRtaMgTajem (5.4a) f c (max ) = − c − ( f c / 2)2 + v 2 f 2 kugRtaMgsgát;em (5.4b) nig tan 2θ max = v f /2 c k> KMrU)ak;rbs;FñwmEdlKμanEdkTajGgát;RTUg Modes of Failure of Beams without Diagonal Tension Reinforcement pleFobElVgkat;elIkMBs; (slenderness ratio) rbs;FñwmkMNt;nUvKMrU)ak;rbs;Fñwm. rUbTI 5>5 bgðajBIkar)ak;sMrab;EdnkMNt;én slenderness ratio epSg². RbEvgElVgkMlaMgkat; (shear span) a sMrab;bnÞúkcMcMnucCacMgayrvagcMnucénkarGnuvtþbnÞúk nigépÞénTMr. sMrab;bnÞúkBRgay shear span lc Ca clear beam span. CaeKalkarN_ KMrUénkar)ak;manbIEbbKW kar)ak;edaykarbegáag (flexural failure), kar)ak;edaykMlaMgTajGgát;RTUg (diagonal tension failure) nigkar)ak;edaybnSMkarkat; nigkarsgát; (shear compression failure or web shear). Fñwmkan;EtRsav kareFVIkarrbs;vakan;EtxiteTArklkçN³ begáagdUckarerobrab;xageRkam. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 219
    • T.Chhay x> kar)ak;edaykarBt;begáag Flexural Failure [F] sMrab; flexural failure sñameRbHeRcInEtQrelIRbEvgmYyPaKbIEpñkkNþalrbs;Fñwm ehIyEkg eTAnwgExSénkugRtaMgem. sñameRbHenHekItBIkugRtaMgkMlaMgkat; v tUc nigkugRtaMgBt; f FM EdlbegáIt nUvkugRtaMgem ft (max ) esÞIrEtedk. enAkñúgKMrU)ak;EbbenHsñameRbHbBaÄrtUcb:unsréssk;cab;epþImekIt manenAtMbn;kNþalElVgeRkamGMeBIrbs;bnÞúkRtwm 50% én failure load. edaysarbnÞúkxageRkAekIn eLIg sñameRbHbEnßmekItmanenAtMbn;kNþalrbs;ElVg ehIysñameRbHdMbUgrIkFM ehIymanTisedAeq<aH eTArkG½kSNWt CamYynwgkarekIneLIgPaBdabrbs;FñwmKYreGaykt;sMKal;. RbsinebIFñwmenHCa under- reinforced member kar)ak;rbs;vaekIteLIgkñúglkçN³ ductile eday longitudinal flexural rein- forcement eFVIkareTAdl; yield. kareFVIkarRbePTenHpþl;nUvkarRbkasGasnñénkarrlMrbs;Fñwm. pleFobElVgkat;elIkMBs;sMrab;kareFVIkarenHmantMélFMCag 5.5 sMrab;krNIbnÞúkcMcMnuc nigFMCag 16 sMrab;bnÞúkBRgay. K> kar)ak;edaykMlaMgTajGgát;RTUg Diagonal Tension Failure [Flexure shear, FS] kar)ak;edaykMlaMgTajGgát;RTUgekItmanRbsinebIersIusþg;kMlaMgTajGgát;RTUgrbs;FñwmtUcCag ersIusþg;Bt;begáagrbs;va. pleFobElVgkat;elIkMBs;mantMélkNþalEdlERbRbYlcenøaH 2.5 eTA 5.5 sMrab;krNIbnÞúkcMcMnuc. eKKitFñwmEbbenHCa intermidate slenderness. sñameRbHcab;epþImCamYykar ekIteLIgén flexural crack bBaÄresþIg²enAkNþalElVg nigbnþedaykar)at;bg;PaBs¥itrvagEdkCamYy nwgebtugEdlB½T§CMuvijvaenARtg;TMr. bnÞab;mkeTot edayminmankarRbkasGsnñRKb;RKan;BIkarekItman kar)ak; sñameRbHGgát;RTUgBIr b¤bIekItmanenARtg;RbEhl 1.5d eTA 2d éncMgayBIépÞénTMrkñúgkrNIFñwm ebtugGarem: nigCaTUeTAekItmanRtg;RbEhlmYyPaKbYnénElVgkñúgkrNIFñwmeRbkugRtaMg. enAeBlEdl vaenAzitezr sñameRbHGgát;RTUgmYykñúgcMeNamenaHrIkhMkøayCasñameRbHTajGgát;RTUgem ehIylat sn§wgeTArksréssgát;xagelIrbs;Fñwm dUceXIjenAkñúgrUbTI 5>5(b) nig 5>5(c). cMNaMfa flexural crack minlatrIkraleTArkeTArkG½kSNWtenAkúñgkrNIKrU)ak;RsYyEbbenHeT. KMrU)ak;enHmanPaBdab M tUcCagKMrU)ak;elIkmunenAeBl)ak;. eTaHbICakMlaMgkat;xageRkAGtibrmaenARtg;TMrkþI TItaMgeRKaHfñak;rbs;kugRtaMgTajemGtibrma minsßitenARtg;TMreT. vaRtUv)ankat;bnßyenARtg;mxkat;enaH edaysarkMlaMgsgát;FMrbs;EdkeRbkug u RtaMgbEnßmBIelIkMlaMgsgát;bBaÄrénRbtikmμrbs;FñwmenARtg;TMr. vaCamUlehtuEdlsñameRbHGgát;RTUg Shear and Torsion Strength Design 220
    • NPIC EdlenAzitezrmanTItaMgq¶aycUlmkkñúgElVgRbEhlmYyPaKbYnénElVgsMrab;FñwmeRbkugRtaMgEdlman søab ehIyTItaMgenHGaRs½ynwgTMhMénkMlaMgeRbkugRtaMg nigPaBERbRbYlrbs;cMNakp©it. CarYm diagonal tension failure CalT§plénbnSMrbs;kugRtaMgBt; nigkugRtaMgkat; edayKitnUvkarcUlrYmrbs; bgÁúMbBaÄrrbs;kMlaMgeRbkugRtaMg nigRtUv)ankt;sMKal;edaysñameRbHbegáag nigsñameRbHGgát;RTUg. CakarRbesIr eKehAvafa flexure shear sMrab;FñwmeRbkugRtaMg ehIyvamanlkçN³gayRsYlBnül;Ca web shear Edlnwgerobrab;enAeBlbnÞab;. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 221
    • T.Chhay X> kar)ak;edaybnSMkarkat; nigkarsgát; Shear Compression Failure [Web Shear, WS] FñwmEdlRbQm shear compression failure EtgmanpleFobElVgkat;elIkMBs; 2.5 sMrab; krNIbnÞúkcMcMnuc nigtUcCag 5.0 sMrab;bnÞúkBRgay. dUckñúgkrNIkar)ak;edaykMlaMgTajGgát;RTUg sñameRbHedaysarkarBt;esþIg²mYycMnYncab;epþImekItmanenAkNþalElVg ehIybBaÄb;karrIkral edaysarkar)at;bg; PaBs¥itrvagEdkbeNþayCamYynwgebtugEdlB½T§CMuvijvaenARtg;TMr. bnÞab;mk sñameRbHeRTtEdlmanlkçN³ecaTCagsñameRbHkñμúgkrNIkar)ak;edaykMlaMgTajGgát;RTUgekIteLIg Pøam² nigbnþrIkraleq<aHeTArkG½kSNWt. GRtaénkarrIkralrbs;vaRtUv)ankat;bnßyedaysarkarEbk rbs;ebtugenAelIsréssgát;xagelI nigkarEbgEckkugRtaMgeLIgvijenAkñúgtMbn;xagelI. kar)ak;Pøam² ekIteLIgedaysarsñameRbHGgát;RTUgemrYmpSMngtMbn;ebtugEbk dUcbgðajenAkñúgrUbTI 5>5(c). eKKit w fakar)ak;RbePTenHmanlkçN³minsUvRsYydUckar)ak;edaysar diagonal tension failure edaysar karBRgaykugRtaMgeLIgvij. CakarBit vaCaRbePTénkar)ak;EdlmanlkçN³RsYyedaymankarRbkas GasnñEdlmanEdnkMNt; ehIyeKRtUveCosvagkarsikSaKNnaEbbenHdac;xat. FñwmebtugminmanlkçN³sac;mYy (homogeneous) ehIyCaTUeTAkarBRgayersIusþg;rbs;ebtug enAelIElVgTaMgmUlmanlkçN³minesμIKñaeT. dUcenH eKminGacrMBwgBIkarekIteLIgénsñameRbHGgát;RTUg enAelIcugTaMgsgçagrbs;Fñwm)aneT. ehIyedaysarlkçN³TaMgenH karbnSMCan;Kñaénkar)ak;edaykar Bt; nigkar)ak;edaykMlaMgTajGgát;RTUg CamYynwgkar)ak;edaysarkMlaMgTajGgát;RTUg nigkar)ak; edaybnSMkMlaMgkat; nigkMlaMgsgát;GacekItmanenAeBlpleFobElVgkat;elIkMBs;Can;Kña. RbsinebI eKdak;EdkkMlaMgkat; (shear reinforcement) smrmü eKGackat;bnßykar)ak;edaylkçN³RsYyrbs; Ggát;edkCamYynwgfvikarbEnßmtictYc. kar)ak;PaKeRcInEtgEt)ak;edaykMlaMgTajGgát;RTUg EdlCabnSMénT§iBlrbs;karBt;begáag nigkarkat;. RbePTénkar)ak;edaybnSMkMlaMgkat; nigkMlaMgsgát; ¬CalT§plénkarEbkénépÞsgát;xag elIrbs;ebtug nigkar)at;bg;lT§PaBTb;Tl;kMlaMgBt;¦ naMdl;karEbgEcksøabrgkarTajecjBIRTnug sMrab;muxkat;mansøabedaysarsñameRbHeRTtlatsn§wgeTArkTMr. karEbkénFñwmrbs;muxkat;eFVIeGay FñwmeFVIkardUcnwgFñÚ (tied arch). CakarRbesIr eKKYrehARbePTénkar)ak;enHCa kar)ak;kMlaMgRTnug (web-shear failure) sMrab;FñwmeRbkugRtaMg eKcaM)ac;kMNt;lT§PaBénbnSMkMlaMgBt; nigkMlaMgkat; nig lT§PaBkMlaMgkat;RTnugrbs;muxkat;Rtg;TItaMgeRKaHfñak;edIm,IkMNt;ersIusþg;kMlaMgkat;rbs;muxkat;eb- tug. Shear and Torsion Strength Design 222
    • NPIC karEbgEckkugRtaMgkat;tamTisedkGtibrmaenAkñúgmuxkat;eRbHrbs;muxkat;mansøabRtUv)an bgðajenAkñúgrUbTI 5>6. edaysarkarpøas;bþÚrTTwgmuxkat;y:agrh½senARtg;RCug A dUcenHeKcaM)ac; RtYtBinitülT§PaBrbs;muxkat;enARtg;TItaMgeRKaHfñak;tambeNþayElVg CaBiesssMrab;kar)ak;edaysar kMlaMgkat;RTnug (web-shear failure). 5> kugRtaMgkat; nigkugRtaMgemenAkñúgFñwmeRbkugRtaMg Shear and Principal Stresses in Prestressed Beams dUcEdl)anerobrab;enAkñúgEpñk 4/ flexure shear enAkñúgFñwmebtugeRbkugRtaMgrYmbBa©ÚlTaMgT§i- BlénkMlaMgeRbkugRtaMgsgát;EdlGnuvtþBIxageRkA. bgÁúMkMlaMgbBaÄrrbs;kMlaMgEdkeRbkugRtaMgkat; bnßykugRtaMgbBaÄrEdlekIteLIgedaybnÞúkTTwgG½kSxageRkA (external transverse load) ehIy net transverse load EdlFñwmRtUvTTYlmantMéltUcKYreGaykt;sMKal;sMrab;FñwmeRbkugRtaMgCagsMrab;Fñwm ebtugGarem:. elIsBIenH kMlaMgsgát;rbs;EdkeRbkugRtaMg ¬eTaHbIenAkñúgkrNI straight tendon¦ kat;bnßy T§iBlrbs; tensile flexural stresses y:ageRcIn dUcenHkarrIkraldalén flecural cracking nigTMhMén flexural cracking enAkñúgFñwmeRbkugRtaMgRtUv)ankat;bnßy. CalT§pl kMlaMgkat; nigkugRtaMgemenA kñúgFñwmeRbkugRtaMgmantMéltUcCagkMlaMgkat; nigkugRtaMgemenAkñúgFñwmebtugGarem:xøaMgNas;. dUcenH smIkarmUldæanEdlbegáIteLIgsMrab;kMlaMgkat;enAkñúgebtugeRbkugRtaMgmanlkçN³dUcKñanwgsmIkarmUl dæanEdlbegáIteLIgsMrab;ebtugeRbkugRtaMg. rUbTI 5>7 bgðajBIkarcUlrYmrbs;bgÁúMbBaÄrrbs;kMlaMgeRb kugRtaMgenAkñúgEpñkEdlminmanlMnwg b¤kMlaMgkat;bBaÄr V PaKeRcInEdlekItedaybnÞúkTTwgG½kSxag eRkA. kMlaMgkat;suT§ (net shearing force) Vc EdlTb;Tl;edayebtugKW Vc = V − V p (5.5) karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 223
    • T.Chhay BIsmIkar 5.2 kugRtaMgkat;ÉktþasuT§ v enARtg;kMBs;Nak¾edayrbs;muxkat;KW Vc Q vc = (5.6) Ib karEbgEckkugRtaMgsréssgát; fc EdlbNþalBIm:Um:g;Bt;xageRkAKW Pe Pe ec M T c fc = − ± m (5.7) Ac Ic Ic ehIyBIsmIkar 5.4a kugRtaMgTajemKW f 't = ( f c / 2)2 + vc2 − fc (5.8) 2 k> Flexural-Shear Strength edIm,IsikSaKNnasMrab;kMlaMgkat; eKcaM)ac;kMNt;faetI flexural shear b¤ web shear lubedIm,I eFVIkareRCIserIsersIusþg;kat; Vc rbs;ebtug. sñameRbHeRTtEdlmanlMnwg (inclined stabilized crack) enAcMgay d / 2 BI flexural crack EdlekItmanenAeBlrg first cracking load enAkñúgkrNI flexure shear RtUv)anbgðajenAkñúgrUbTI 5>8. RbsinebI kMBs;RbsiT§PaBCa d p ¬kMBs;BIsréssgát;eTATIRbCMu TMgn;rbs;EdkeRbkugRtaMgbeNþay¦ bMErbMrYlm:Um:g;rvagmuxkat; @ nig # KW Vd p M − M cr ≅ (5.9a) 2 b¤ V= M cr M /V − d p / 2 (5.9b) Shear and Torsion Strength Design 224
    • NPIC Edl V CakMlaMgkat;enARtg;muxkat;EdlBicarNa. lT§plénkarBiesFCaeRcInbgðajfaeKRtUvkar kMlaMgkat;bBaÄrbEnßmEdlmanTMhM 0.6bwd p f 'c sMrab;xñat US nig bwd p f 'c / 20 sMrab;xñat SI edIm,IeFVIeGaymansñameRbHeRTtenAkñúgrUbTI 5>8 eBjelj. dUcenH kMlaMgkat;bBaÄrsrubEdleFVIGMeBI enARtg;bøg;elx @ rbs;rUbTI 5>8 KW Vci = M cr M /V − d / 2 + 0.6bw d p f 'c + Vd ¬xñat US¦ (5.10) p Vci = M cr M /V − d p / 2 + bw d p f 'c 20 + Vd ¬xñat SI¦ karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 225
    • T.Chhay Edl Vd CakMlaMgkat;bBaÄrEdlbNþalBITMgn;pÞal;. bgÁúMbBaÄr V p rbs;kMlaMgeRbkugRtaMgminRtUv)an KitbBa©ÚlenAkñúgsmIkar 5.10 eT edaysarvamanTMhMtUctambeNþaymuxkat;ElVgEdlEdkeRbkugRtaMg minecatxøaMg. tMélrbs; V enAkñúgsmIkar 5.10 KWCakMlaMgkat;emKuN Vi enARtg;muxkat;EdlBicarNaEdl bNþalBIbnÞúkxageRkAEdlekIteLIgtMNalKñaCamYynwgm:Um:g;Gtibrma M max EdlekIteLIgenARtg;mux kat;enaH Vci = 0.6λ f 'c bw d p − Vd + Vi (M cr ) ≥ 1.7λ f 'c bwd p ¬xñat US¦ (5.11) M max λ f 'c bw d p λ f 'c bw d p Vci = 20 − Vd + Vi M max (M cr ) ≥ 7 ¬xñat SI¦ ≤ 5.0λ f 'c bw d p ¬xñat US¦ ≤ 0.42λ f 'c bw d p Edl sMrab;ebtugTMgn;Fmμta λ = 1 .0 = 0.85 sMrab; sand-lightweight concrete = 0.70 sMrab; all-lightweight concrete Vd = kMlaMgkat;Rtg;muxkat;EdlbNþalBIbnÞúkefrEdlKμanemKuN Vci = ersIusþg;kMlaMgkat;Fmμta (nominal shear strength) Edlpþl;eGayedayebtugenAeBl EdlsñameRbHkMlaMgTajekItBIbnSMénkMlaMgkat;bBaÄr nigm:U:m:g; Vi = kMlaMgkat;emKuNRtg;muxkat;EdlbnÞúkxageRkAekIteLIgenAeBlCamYyKñanwg M max sMrab;ebtugTMgn;Rsal λ = f ct / 6.7 f 'c sMrab;xñat US nig λ = fct / 0.556 f 'c sMrab;xñat SI RbsinebIeKsÁal; tensile splitting strength f ct . cMNaMfatMélrbs; f 'c minKYrFMCag 100 psi (0.69MPa ) . smIkarsMrab; M cr ¬m:Um:g;EdlbegáIt flexural cracking EdlbNþalBIbnÞúkxageRkA¦ RtUv)an eGayeday M cr = c (6 f 'c + f ce − f d ) ¬xñat US¦ I (5.12) y t M cr I ( = c 0.5 f 'c + f ce − f d yt ) ¬xñat IS¦ enAkñúg ACI Code eKeRbI f pe CMnYseGay f ce Shear and Torsion Strength Design 226
    • NPIC Edl ersIusþg;sgát;rbs;ebtugEdlbNþalBIeRbkugRtaMgRbsiT§PaBeRkaykMhatbg;enARtg; f ce = srésxageRkAbMputrbs;muxkat;EdlkugRtaMgTajRtUv)anbgáeLIgedaybnÞúkxageRkA. enARtg;TIRbCMuTMgn; fce = f c . f d = kugRtaMgEdlbNþalBIbnÞúkGefrKμanemKuNenARtg;srésxageRkAbMputrbs;muxkat;Edl bNþalEtBIbnÞúkpÞal;EdlkugRtaMgTajRtUv)anbgáedaybnÞúkxageRkA. yt = cMgayBIG½kSTIRbCMuTMgn;eTAsrésrgkarTajxageRkA ehIy M cr = EpñkxøHrbs;m:Um:g;énbnÞúkxageRkAEdlbgáeGaymansñameRbH. CakarsMrYl eKGacCMnYs Sb CMnYseGay I c / yt . rUbTI 5>9 bgðajBIdüaRkamrbs;smIkar 5.10 CamYynwgTinñn½yénkarBiesaF. cMNaMfa eKeRbIkarsikSaKNnadUcKñaEdlGnuvtþsMrab;muxkat;cak;Rsab; sMrab;karsikSaKNna kMlaMgkat;énmuxkat;smas. BIeRBaHkarsikSaKNnasMrab;kMlaMgkat;KWQrelIsßanPaBkMNt;edA eBl)ak;eRkamGMeBIbnÞúkemKuN. eTaHbICa muxkat;TaMgmUlrbs;muxkat;smasTb;Tl;nwgkMlaMgkat; smasdUcmuxkat;Edlcak;kñúgeBlEtmYy (monolithic section) k¾eday k¾karKNnaersIusþg;kMlaMg karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 227
    • T.Chhay kat; Vc KYrQrelIlkçN³rbs;muxkat;cak;Rsab; edaysarersIusþg;kMlaMgkat;PaKeRcInpþl;eGayeday RTnugrbs;muxkat;cak;Rsab;. dUcenH fce nig f d enAkñúgsmIkar 5.12 RtUv)anKNnaedayeRbIragFrNI maRtrbs;muxkat;cak;Rsab;. x> ersIusþg;kMlaMgkat;RTnug Web-Shear Strength sñameRbHkMlaMgkat;RTnug (web-shear crack) enAkñúgFñwmeRbkugRtaMgekIteLIgedaysarkugRtaMg EdlminGackMNt;)an (indeterminate stress) EdlCakarRbesIreKKYrKNnavaedaykugRtaMgTajemenA Rtg;bøg;eRKaHfñak;BIsmIkar 5.8. eKGaccat;TukkugRtaMgkat; vc CakugRtaMgkat;RTnug vcw nigmantMél GtibrmaenAEk,rTIRbCMuTMgn; cgc énmuxkat;EdlsñameRbHGgát;RTUgCak;EsþgekItman dUckarBiesaFeTA dl;kar)ak;CaeRcIn)anbgðaj. RbsinebIeKCMnYs vc sMrab; vcw nig fc sMrab; f c ¬EdlCakugRtaMgeb- tug fc EdlbNþalBIeRbkugRtaMgRbsiT§PaBenARtg;nIv:U cgc¦ enAkñúgsmIkar smIkarEdleGaykugRtaMg TajemenAkñúgebtugesμInwgersIusþg;TajpÞal; (direct tensile strength) køayCa ( f c / 2) + vcw − f2c f 't = 2 (5.13) Edl vcw = Vcw / (bwd p ) CakugRtaMgkat;enAkñúgebtugEdlbNþalBIbnÞúkTaMgGs;EdleFVIeGayman ersIusþg;kMlaMgkat;bBaÄrFmμta Vcw enAkñúgRTnug. edaHRsayrk vcw enAkñúgsmIkar 5.13 vcw = f 't 1 + f c / f 't (5.14a) edayeRbI f 't = 3.5 f 'c psi(0.3 f 'c MPa) CatMéld¾smrmüsMrab;kugRtaMgTajedayQrelIlT§pl énkarBiesaFCaeRcIn smIkar 5.14(a) køayCa vcw = 3.5 f 'c ⎛ 1 + f c / 3.5 f 'c ⎞ ⎜ ⎝ ⎟ ⎠ ¬xñat US¦ (5.14b) vcw = 0.3 f 'c ⎛ 1 + f c / 0.3 f 'c ⎞ ⎜ ⎝ ⎟ ⎠ ¬xñat SI¦ EdleyIgGacsMrYl)andUcxageRkam vcw = 3.5 f 'c + 0.3 f c ¬xñat US¦ (5.14c) vcw = 0.3( f 'c + f c ) ¬xñat SI¦ enAkñúg ACI Code eKeRbI f pc CMnYseGay f c . nimitþsBaØaEdleRbIenATIenHKWcg;bBa¢ak;favaCakugRtaMg enAkñúgebtugminEmnenAkñúgEdkeRbkugRtaMgeT. ersIusþg;kMlaMgkat;Fmμta Vcw EdleGayedayebtugenA eBlsñameRbHGgát;RTUgekItBIkugRtaMgTajemd¾FMenAkñúgRTnugkøayCa Vcw = (3.5λ f 'c + 0.3 f c )bw d p + V p ¬xñat US¦ (5.15) Shear and Torsion Strength Design 228
    • NPIC ( ) Vcw = 0.3 λ f 'c + f c bw d p + V p ¬xñat SI¦ Edl V p = bgÁúMbBaÄrénkMlaMgeRbkugRtaMgRbsiT§PaBenARtg;muxkat;BiessEdlcUlrYmnwgersIusþg; FmμtabEnßm λ = 1.0 sMrab;ebtugTMgn;Fmμta nigmantMéltUcCagenHsMrab;ebtugTMgn;Rsal d p = cMgayBIsréssgát;xageRkAeTATMRbCMuTMng;rbs;EdkeRbkugRtaMg b¤ 0.8h edayykmYy NaEdlFMCag ACI Code yktMél f c CakugRtaMgsgát;pÁÜbrbs;ebtugenARt;gTIRbCMuTMgn;rbs;muxkat; b¤Rtg; kEnøgEdlkat;KñarvagRTnug nigsøabenAeBlEdlTIRbCMuTMgn;sßitenAkñúgsøab. enAkñúgkrNImuxkat;smas eKKNna f c edayQrelIkugRtaMgEdlekIteLIgedaykMlaMgeRbkugRtaMg nigm:Um:g;EdlTb;Tl;eday Ggát;cak;Rsab;EdleFVIkarEtÉg. düaRkaménTMnak;TMngrvagkugRtaMgkat;RTnugFmμta (nominal web shear stress) vcw nigkugRtaMgsgát;rbs;ebtugRtg;TIRbCMuTMgn;RtUv)aneGayenAkñúgrUbTI 5>10. cMNaMfa PaBdUcKñarvagExSekagénsmIkar 5.14b nig c bgðajfasmIkar 5.14c RtUv)anEktMrUvBIsmIkar 5.14b edIm,IeGaymanlkçN³bnÞat;. Code GnuBaØateGayeRbIemKuN 1.0 CMnYseGayemKuN 0.3 sMrab;tYTIBIr énsmIkar 5.15. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 229
    • T.Chhay K> karRtYtBinitütMélrbs;V nigV sMrab;KNnaersIusþg;ebtugRTnugV ci cw c Controlling Values of Vci and Vcw for the Determination of the Web Concrete Strength Vc ACI Code mansmIkarbEnßmsMrab;kMNt;Vci nig Vcw edIm,IeRCIserIstMél Vc EdlRtUvkarenA kñúgkarKNna³ (a) enAkñúgGgát;eRbkugRtaMgEdlmuxkat;enAcMgay h / 2 BIépÞénTMrenAEk,rcugénGgát;CagRbEvgepÞr rbs;EdkeRbkugRtaMg enaHeKRtUvBicarNatMéleRbkugRtaMgkat;bnßyenAeBlKNna Vcw . tMél Vcw enHRtUv)anKitCaEdlGtibrmarbs; Vc enAkñúgsmIkar ⎛ Vu d p ⎞ Vcw = ⎜ 0.6λ f 'c + 700 ⎜ ⎟bw d p ≥ 2λ f 'c bw d p ⎝ Mu ⎟ ⎠ ≤ 5λ f 'c bw d p ¬xñat US¦ (5.16) ⎛ Vu d p ⎞ Vcw = ⎜ 0.05λ f 'c + 5 ⎜ ⎟bw d p ≥ 0.2λ f 'c bw d p ⎝ Mu ⎟ ⎠ ≤ 0.4λ f 'c bw d p ¬xñat SI¦ tMél Vu d p / M u minGacFMCag 1.0 eT. (b) enAkñúgGgát;eRbkugRtaMgEdl bonding rbs; tendon xøHmin)anBnøÚtdl;cugrbs;Ggát; enaHeKRtUv KiteRbkugRtaMgkat;bnßyenAeBlkMNt; Vc edayeRbIsmIkar 5.16 b¤eRbItMéltUcCageKkñúg cMeNamtMél Vc EdlTTYl)anBIsmIkar 5.11 nigBIsmIkar 5.15. dUcKña tMélrbs; Vcw Edl KNnaedayeRbIeRbkugRtaMgkat;bnßyRtUvyktMélGtibrmaénsmIkar 5.16. (c) eKGaceRbIsmIkar 5.16 kñúgkarkMNt; Vc sMrab;Ggát;EdlkMlaMgeRbkugRtaMgRbsiT§PaBmintUc Cag 40% énersIusþg;Tajrbs;EdkrgkarBt; (flexural reinforcement) ebImindUcenaHeT luH RtaEteKGnuvtþkarviPaKlMGitedayeRbIsmIkar 5.11 sMrab; Vci nigsmIkar 5.15 sMrab; Vcw ehIyedayeRCIserIsyktMéltUcCageKéntMélTaMgBIrCatMélkMNt; Vc edIm,IeRbICaersIusþg; rbs;RTnugkñúgkarKNnaEdkRTnug. (d) bøg;dMbUgsMrab;ersuIsþg;kat;FmμtaEdlRtUvkarsrub (total required nominal shear strength) Vn = Vu / φ EdlRtUv)aneRbIsMrab;KNnaEdkRTnugk¾sßitenARtg;cMgay h / 2 BIépÞrbs;TMr. Shear and Torsion Strength Design 230
    • NPIC 6> EdkkMlaMgkat;RTnug Web-Shear Reinforcement k> Web Steel Planar Truss Analogy edIm,IkarBarsñameRbHGgát;RTUgenAkñúgGgát;eRbkugRtaMg eTaHbIekIteLIgedaysar flexural- shear b¤ web-shear action k¾eday eKRtUvdak;EdkBRgwgtamTMrg;énExSditEdlBN’naBIKnøgrbs;kug RtaMgTajenAkñúgrUbTI 5>3. b:uEnþdMeNaHRsayEbbenHRtUv)anRcanecal ehIyTMrg;epSgeTotrbs;Edk RtUv)anécñRbDiteLIgedIm,ITb;Tl;nwgkugRtaMgTajenARtg;bøg;)ak;edaysarkMlaMgkat;eRKaHfñak; (critical shear failure plane). karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 231
    • T.Chhay Shear and Torsion Strength Design 232
    • NPIC KMrUénkar)ak;edaykMlaMgkat;eFVIeGayFñwmman stimulated arched section rgkarsgát;enAEpñk xagelI ehIyRtUv)ancgP¢ab;KñaenAEpñkxageRkamedayEdkTaj dUceXIjenAkñúgrUbTI 5>11(a). Rbsin ebIeKBinitüEtFatusgát;EdlbgðajenAkñúgrUbTI 5>11 (b) eKGacKitvaCaGgát;sgát; enAkñúg truss RtIekaN dUcbgðajenAkñúgrUbTI 5>11 (c) EdlmanBhuekaNénkMlaMg Cc / Tb nig Ts EdltMNageGaykMlaMg EdlmanGMeBIeTAelIGgát;rbs; truss dUcenHeKGacehAvafa truss analogy. kMlaMg Cc kMlaMgsgát;enA kñúg simulated concrete strut, kMlaMg Tb CakMeNInkMlaMgTajénEdkTaj beNþayem ehIy Ts CakMlaMg enAkñúgEdkBt; (bent bar). rUbTI 5>12 (a) bgðajBI analogy truss sMrab; krNIénkareRbIEdkkgbBaÄr (vertical stirrup) CMnYseGayEdkeRTt CamYynwgBhuekaNkMlaMgEdlmankMlaMgTajbBaÄr Ts CMnYs eGaykMlaMgTajeRTtenAkñúgrUbTI 5>11 (c). EdkkMlaMgkat;mantYnaTIsMxan;bYn³ - vaTb;Tl;EpñkxøHénkMlaMgkat;emKuNxageRkA Vu - vaRKb;RKgkarrIkFMénsñameRbHGgát;RTUg - vaeFVIeGayEdkbeNþayemsßitcMTItaMg dUcenHeKGacdak;Edk dowel EdlcaM)ac;edIm,IRT flexural load - vapþl;nUv confinement xøHdl;ebtugenAkñúgtMbn;sgát; RbsinebIeKeRbIEdkkgkñúgTMrg;biTCit x> Web Steel Planar Resistance RbsinebI Vc ¬ersIusþg;kMlaMgkat;FmμtaénRTnugebtugsuT§¦ mantMéltUcCagkMlaMgbBaÄrsrub Fmμta Vu / φ = Vn eKRtUvdak;EdkRTnugedIm,ITb;Tl;nUvPaBxusKñaénkMlaMgTaMgBIr. dUcenH Vs = Vn − Vc (5.17) enATIenH Vc CatMéltUcCageKén Vci nig Vcw . eKGackMNt; Vc BIsmIkar 5.11 b¤ 5.15 ehIyeKkMNt; Vs BIsmIkarlMnwgenAkñúg analogous triangular truss. BIsmIkar 5.11(c) Vs = Ts sin α = Cc sin β (5.18a) Edl Ts CakMlaMgpÁÜbénEdkkgRTnugTaMgGs;EdlEkgnwgbøg;sñameRbHGgát;RTUg ehIy n CacMnYnénKM lat s . RbsinebI s1 = ns enAkñúgGgát;rgkarTajxageRkamrbs; analogous truss enaH s1 = jd (cot α + cot β ) (5.18b) snμt;fa édXñas; jd ≅ d / kMlaMgEdkkgkñúgmYyÉktþaRbEvgBIsmIkar 5.18a nig b Edlman s1 = ns nwgkøayCa karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 233
    • T.Chhay Ts Ts V 1 = = s (5.18c) s1 ns sin α d (cos α + cos β ) RbsinebIeKmanEdkkgeRTtcMnYn n Edlman analoguos truss chord RbEvg s1 ehIyRbsin ebI Av CaRkLaépÞénEdkkgeRTtmYy enaH Ts = nAv f y (5.19a) dUcenH nAv = Vs ns d sin α (cot β + cot α ) f y (5.19b) b:uEnþGacsnμt;fa enAkñúgkrNI)ak;edaykMlaMgTajGgát;RTUg (diagonal tension failure) Ggát;RTUgrgkar sgát;manmMu β = 45o dUcenHsmIkar 5.19b nwgkøayCa Av f y d Vs = [sin α (1 + cot α )] s Av f y d b¤ Vs = s (sin α + cos α ) (5.20a) edaHRsayrk s edayeRbI Vs = Vu − Vc Av f y d s= (sin α + cos α ) (5.20b) Vu − Vc RbsinebIEdkRTnugeRTtpÁúMeLIgedayEdkeTal b¤RkuménEdkeTalEdlEdkTaMgenARtUv)anBt; nigRtUv)an dak;enAcMgaydUcKñaBIépÞénTMr enaH Vs = Av f y sin α ≤ 3.0 f 'c bw d ¬xñat US¦ Vs = Av f y sin α ≤ 0.25 f 'c bw d ¬xñat SI¦ RbsinebIeKeRbIEdkkgbBaÄr mMu α nwgesμInwg 90o enaHeK)an Av f y d Vs = (5.21a) s Av f y dAvφf y d b¤ s= = (Vu / φ ) − Vc Vu − φVc (5.21b) enAkñúgsmIkar 5.21a nig b/ d p CacMgayBIsréssgát;xageRkAbMputeTAkan;TIRbCMuTMgn;rbs;EdkeRbkug RtaMg ehIy d CacMgayBIsréssgát;xageRkAbMputeTATIRbCMuTMgn;rbs;EdkFmμta. tMélrbs; d p min RtUvtUcCag 0.80h eT. Shear and Torsion Strength Design 234
    • NPIC K> EdlkMNt;énTMhM nigKMlatrbs;Edkkg Limitation on Size and Spacing of Stirrups smIkar 5.20 nig 5.21 eGaynUvTMnak;TMngRcasKñarvagKMlatEdkkg nigkMlaMgkat; b¤kugRtaMg kat;EdlvaRtUvTb;Tl;. enAeBlEdl s fycuH (Vu − Vc ) nwgekIneLIg. edIm,IeGayEdkkgbBaÄrTb;Tl; sñameRbHGgát;RTUg dUcbgðajenAkñúgrUbTI 5>11 (c) eKRtUvGnuvtþEdnkMNt;KMlatGtibrmasMrab;Edkkg bBaÄrdUcxageRkam³ (a) smax ≤ 3 h ≤ 24in.(60cm) Edl h CakMBs;srubrbs;muxkat; 4 (b) RbsinebI Vs > 4λ f 'c bw d p ¬xñat US¦ Vs > λ f 'c bw d p / 3 ¬xñat SI¦ eKRtUvkat; bnßyKMlatGtibrmarbs; (a) Bak;kNþal ¬ smax ≤ 83 h ≤ 12in.(30cm) (c) RbsinebI Vs > 8λ f 'c bw d p ¬xñat US¦ Vs > 2λ f 'c bw d p / 3 ¬xñat SI¦ eKRtUvBRgIkmuxkat;. (d) RbsinebI Vu = φVn > φVc / 2 / eKRtUvdak;EdkkMlaMgkat;Gb,brma. eKKNnaRkLaépÞ EdkGb,brmaenHedaysmIkar Av = 0.75 f 'c w b s f b¤ Av = 50fbws edayykmYyNaEdlFMCag y y RbsinebIkMlaMgeRbkugRtaMgRbsiT§PaB Pe FMCag b¤esμInwg 40% énersIusþg;Tajrbs;EdkBt; (flexural reinforcement) enaH A ps f pu s dp Av = (5.22b) 80 f y d bw Edlvapþl;nUv Av Gb,brmaRtUvkartUcCag ehIyEdleKGaceRbIvaCMnYs)an. (e) edIm,IRbsiT§PaB EdkRTnugRtUvEtmanRbEvgbgáb; (development lengt) RtUvkareBjelj. enHmann½yfaEdkkgRtUvBnøÚtcUleTAkñúgEpñkrgkarsgát; nigEpñkrgkarTajrbs;muxkat;/ RtUvkarkMras;ebtugkarBarEdk (clear concrete cover) tUc nigeKGaceRbITMBk; 90o b¤ 135o enAkñúgtMbn;sgát;. rUbTI 5>13 bgðajBIdüaRkaménkardak;EdkkgRTnugeTAtamtMbn;énRbEvgElVgrbs;FñwmeRbkug RtaMgEdlrgGMeBIénbnÞúkBRgayesμI. épÞqUtCakMlaMgkat;elIs Vs EdlRtUvkarEdkRTnug. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 235
    • T.Chhay 7> ersIusþg;kMlaMgkat;edkenAkñúgeRKOgbgÁúMsmas Horizontal Shear Strength in Composite Construction snμt;fakMlaMgkat;tamTisedkepÞreBjeljRtg;épÞb:HénkEnøgEdlCYbKña. k> eRkamGMeBIbnÞúkeFVIkar Service-Load Level eKGackMNt;kugRtaMgkat;tamTisedkGtibrma vh BIeKalkarN_mUldæanrbs;emkanic VQ vh = (5.23) I c bv Edl V= kugRtaMgkat;KNnaKμanemKuN (unfactored design vertical shear) EdleFVIGMeBIelImux kat;smas Q = m:Um:g;RkLaépÞeFob cgc énkMNt;muxkat;EdlenABIxagelI b¤BIxageRkam cgc I c = m:Um:g;niclPaBénmuxkat;smasTaMgmUl bv = TTwgRtg;kEnøgb:Hrbs;muxkat;RTnugénGgát;cak;Rsab; b¤TTwgénmuxkat;EdleKKNna kMlaMgkat;edk Shear and Torsion Strength Design 236
    • NPIC eKGacsMrYlsmIkar 5.23 dUcxageRkam V vh = (5.24) bv d pc Edl d pc CakMBs;RbsiT§PaBBIsréssgát;xageRkAénmuxkat;smaseTATIRbCMuTMgn; cgc rbs;EdkeRbkug RtaMg. x> Ultimate-Load Level Direct Method: sMrab;kar)ak;enAkñúgsßanPaBkMNt; eKGacEkERbsmIkar 5.24 edayCMnYs V eday Vu dUcenHeyIgTTYl)an Vu vuh = (5.25a) bv d pc b¤ sMrab;ersIusþg;kMlaMgkat;bBaÄrFmμta Vn Vu / φ V vnh = = n (5.25b) bv d pc bv d pc Edl φ = 0.75 / RbsinebI Vnh CaersIusþg;kMlaMgkat;edkFmμta enaH Vu ≤ Vnh ehIyersIusþg;kMlaMgkat; FmμtasrubKW Vnh = vnhbv d pc (5.25c) ACI Code kMNt; vnh Rtwm 80 psi(0.55MPa ) RbsinebIeKmineRbIEdkEdkrgcaM (dowel) b¤EdkkgbBaÄr ehIyépÞ b:HmanlkçN³eRKIm b¤RbsinebIeKeRbIEdkkgbBaÄrGb,brma b:uEnþépÞb:HminmanlkçN³eRKIm. vnh GaceTAdl; 500 psi (3.45MPa ) EteKRtUveRbI friction theory CamYynwgkarsnμt;xageRkam (a) enAeBlEdlminmanEdkkgbBaÄr b:uEnþépÞb:HénGgát;cak;Rsab;manlkçN³eRKIm enaHeKeRbI Vnh ≤ 80 Ac ≤ 80bv d pc (5.26a) Edl Ac CaRkLaépÞrbs;ebtugEdlTb;Tl;kMlaMgkat; = bv d pc (b) enAeBlEdleKeRbIEdkkgGb,brma Edl Ac = 50(bws ) / f y b:uEnþépÞb:Hrbs;Ggát;cak; Rsab;minmanlkçN³eRKIm vnh ≤ 80bv d pc (c) RbsinebIépÞb:Hrbs;Ggát;cak;Rsab;manlkçN³eRKImEdlmankMBs; 1 / 4in.(6mm) ehIy EdkbBaÄrGb,brmaenAkñúg (b) RtUv)andak; enaHeKeRbI Vnh ≤ 500bv d pc (5.26b) karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 237
    • T.Chhay (d) RbsinebIkMlaMgkat;emKuN Vu > φ (500bv d pc )/ eKGaceRbI shear friction theory edIm,I KNnaEdk dowel. enAkñúgkrNIenH kMlaMgkat;edkTaMgGs;RtUv)anKitedaybøg;EkgEdl Vnh = μAvf f y (5.27) Edl Avf = RkLaépÞén shear-friction reinforcement/ in.2 f y = design yield strength/ minRtUvelIs 60,000 psi (414 MPa ) μ = emKuNkkit = 1.0λ sMrab;ebtugEdlcak;elIépÞebtugEdlmaneRKIm = 0.6λ sMrab;ebtugEdlcak;elIépÞebtugEdlminmaneRKIm λ = emKuNsMrab;RbePTebtug enAkñúgkrNITaMgGs; ersIusþg;kat;Fmμta Vn ≤ 0.20 f 'c Acc ≤ 800 Acc Edl Acc CaRkLaépÞb:Hrbs; ebtugEdlTb;Tl;nwgkMlaMgkat;epÞr (shear transfer). cMNaMfa enAkñúgkrNICaeRcIn kugRtaMgkat; vuh EdlTTYl)anBIkMlaMgkat;emKuNGt;FMCag 500 psi(3.45MPa ) eT. dUcenH eKmincaM)ac;RtUvkareRbIEt shear friction theory kñúgkarKNnaEdkrgcaM (dowel) sMrab;skmμPaBsmas (composite action) enaHeT. KMlatGnuBaØatGtibrmaénEdkrgcaM (dowel) b¤ tie sMrab;kMlaMgkat;edkKWtMéltUcCageKkñúg cMeNam 4 dgénTMhMtUcCageKénmuxkat;TMr nig 24in.(60cm) . Basic Method: ACI Code GnuBaØateGayeRbIviFIepSgeTotEdlkMlaMgkat;edkRtUv)anGegát edayKNnabMErbMrYlCak;EsþgénkMlaMgsgát; b¤kMlaMgTajenAkñúgbøg;NamYy nigedayepþrkMlaMgkat;edk enaHeTAGgát;EdlCaTMr. eKCMnYsRkLaépÞb:H Acc sMrab; bv d pc enAkñúgsmIkar 5.25b nig c enaHeK TTYl)an Vnh = vnh Acc (5.28) Edl Vnh ≥ Fh / kMlaMgkat;edk ehIyy:agehacNas;vaRtUvesμInwgkMlaMgsgát; C b¤kMlaMgTaj T enA kñúgrUbTI 5>14. ¬emIlsmIkar 5.30 sMrab;tMélrbs; Fh ¦ eKGackMNt;RkLaépÞb:H Acc dUcxageRkam Acc = bv lvh (5.29) Edl lvh CaRbEvgkMlaMgkat;edk (horizontal shear length) EdlkMNt;enAkñúgrUbTI 5.15(a) nig (b) sMrab;Ggát;TMrsamBaØ nigsMrab;Ggát;TMrCab; erogKña. Shear and Torsion Strength Design 238
    • NPIC karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 239
    • T.Chhay K> karKNnaEdkrgcaMskmμsmas Design of Composite-Action Dowel Reinforcement Edk tie sMrab;kMlaMgkat;edkGacpSMeLIgBIr)arEdkeTal (single bars or wires)/ BIEdkkgeCIg eRcIn (multiple leg stirrup) b¤BI vertical legs of welded wire fabric. KMlatrbs;vaminGacFMCagbYn dgénTMhMEdltUcCageKénGgát;TMr b¤ 24in.(60cm) edayykmYyNaEdltUcCageK. RbsinebI μ Ca emKuNkMlaMgkkit enaHeKGackMNt;kMlaMgkat;edkFmμta Fh enAkñúgrUbTI 5>14 dUcxageRkam Fh = μAvf f y ≤ Vnh (5.30) tMél ACI rbs; μ KWQrelIersIusþg;kkit-kMlaMgkat;kMNt; (limit shear-friction strength) 800 psi (5.5MPa ) ¬vaCatMélEdlmanlkçN³suvtßiPaBbnþicEdlbgðajedaykarBiesaF¦. viTüasßanebtugeRb kugRtaMg (Prestressed Concrete Institute) ENnaM μe = 2.9 CMnYseGay μ = 1.0λ sMrab;ebtug Edlcak;elIépÞebtugeRKIm ehIykMlaMgkat;KNnaGtibrma (maximum design shear force) Vu ≤ 0.25λ2 f 'c Ac ≤ 1,000λ2 Acc (5.31a) CamYynwgRkLaépÞcM)ac;rbs;Edkkkit-kMlaMgkat; (shear-friction steel) Vuh Avf = (5.31b) φf y μ e b¤ Avh = Vnh F = h μe f y μe f y (5.31c) edayeRbItMél PCI EdlminsUvsuvtßiPaB smIkar 5.31c køayCa Fh ≤ μ e Avf f y ≤ Vnh (5.32) λ 2 CamYynwg μe = 1,000F bv I vh ≤ 2.9 h Edl bvlvh = Acc / EdkGb,brmaKW 50bv s 50bv lvh Av = = (5.33) fy fy 8> CMhanKNnaEdkRTnugsMrab;kMlaMgkat; Web Reinforcement Design Procedure for Shear xageRkamCakarsegçbBICMhanénkarKNnaEdkRTnugsMrab;kMlaMgkat;³ !> kMNt;tMélersIusþg;kMlaMgkat;FmμtaEdlRtUvkar Vn = Vu / φ enARtg;cMgay h / 2 BIépÞénTMr Edl φ = 0.75 . Shear and Torsion Strength Design 240
    • NPIC @> KNnaersIusþg;kMlaMgkat;Fmμta (nominal shear strength) Vc EdlRTnugmanedayeRbIviFImYy kñúgcMeNamviFIBIrxageRkam³ (a) ACI conservative method RbsinebI f pe > 0.40 f pu ⎛ 700Vu d p ⎞ Vc = ⎜ 0.60λ f 'c + ⎜ ⎟bw d p ¬xñat US¦ ⎟ ⎝ M u ⎠ ⎛ λ f 'c V d⎞ Vc = ⎜ ⎜ 20 + 5 u ⎟bw d p Mu ⎟ ¬xñat SI¦ ⎝ ⎠ Edl 2λ f 'c bw d p ≤ Vc ≤ 5λ f 'c bw d p sMrab;xñat US λ f 'c bw d p / 5 ≤ Vc ≤ 0.4λ f 'c bw d p sMrab;xñat SI Vu d p ≤ 1.0 Mu ehIyeKKNna Vu enARtg;muxkat;dUcKñasMrab;karKNna M u . RbsinebIersIusþg;eRcokTajmFüm (average tensile splitting strength) fct sMrab; ebtugTMgn;Rsal enaH λ = fct / 6.7 f 'c sMrab;xñat US b¤ λ = fct / 0.556 f 'c sMrab;xñat SI CamYynwg f 'c Gt;FMCag 100 psi(0.67MPa ) . (b) Detailed analysis Edl Vc CatMéltUcCageKkñúgcMeNam Vci nig Vcw ¬xñat US¦ 1.7λ f 'c bw d p ≤ Vci = 0.60λ f 'c bw d p + Vd + Vi M max (M cr ) ≤ 5.0λ f 'c bwd p λ f 'c bw d p λ f 'c bw d p ¬xñat SI¦ 7 ≤ Vci = 20 + Vd + Vi M max (M cr ) ≤ 0.4λ f 'c bwd p ¬xñat US¦ ( ) Vcw = 3.5λ f 'c + 0.3 f c bw d p + V p ¬xñat SI¦ ( ) Vcw = 0.3 λ f 'c + f c bw d p + V p edayeRbItMélNaEdlFMCageKkñúgcMeNam d p nig 0.8h Edl M cr = (I c / yt )(6λ f 'c + fce − f d ) ¬xñat US¦ M cr = (I c / yt )(0.5λ f 'c + f ce − f d ) ¬xñat SI¦ b¤ M cr = Sb (6λ f 'c + fce − f d ) ¬xñat US¦ M cr = Sb (0.5λ f 'c + f ce − f d ) ¬xñat SI¦ Vi = kMlaMgkat;emKuNEdlbNþalBIbnÞúkGnuvtþn_BIxageRkAEdlekItmankñúg eBldMNalKñaCamYynwg M max karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 241
    • T.Chhay kugRtaMgsgát;enAkñúgebtugeRkayekItmankMhatbg;TaMgGs;enARtg; f ce = srésxageRkArbs;muxkat;EdlbnÞúkxageRkAbgáeGaymankugRtaMgTaj. f ce køayCa f c sMrab;kugRtaMgenARtg;TIRbCMuTMgn;rbs;muxkat;. #> RbsinebI Vu / φ ≤ Vc / 2 vaminRtUvkarEdkRTnugeT. RbsinebI Vc / 2 < Vu / φ < Vc vaRtUvkar EdkGb,brma. RbsinebI Vu / φ > Vc nig RbsinebI Vs = Vu / φ − Vc ≤ 8λ f 'c bw d p ¬xñat US¦ Vs = Vu / φ − Vc ≤ 2λ f 'c bw d p / 3 ¬xñat SI¦ eKRtUvKNnaEdkRTnug. RbsinebI Vs = Vu / φ − Vc > 8λ f 'c bw d p b¤ Vs > φ (Vc + 8λ f 'c bw d p ) ¬xñat US¦ Vs = Vu / φ − Vc > 2λ f 'c bw d p / 3 ¬xñat SI¦ eKRtUvtMeLIgmuxkat;. $> KNnaEdkRTnugGb,brmaEdlRtUvkar. KMlatKW s ≤ 0.75h b¤ 24in.(60cm) edayykmYyNa EdltUcCageK. Av min = 0.75 f 'c w b¤ Av min = ¬US¦ edayykmYyNaEdlFMCag b s 50bw s f y f y Av min = f 'c bw s 16 f y b¤ Av min = 50fbws ¬SI¦ edayykmYyNaEdlFMCag y RbsinebI / f pe ≥ 0.40 f pu Av min EdlmanlkçN³suvtßiPaBticCagCatMéltUcCageKkñúg cMeNam A ps f pu s dp Av = 80 f y d p bw Edl d p ≥ 0.80h CamYynwg Av min = 0.75 f 'c bw s fy b¤ Av min = 50fbws ¬US¦ y Av min = f 'c bw s 16 f y b¤ Av min = 50fbws ¬SI¦ y %> KNnaTMhM nigKMlatEdkRTnugEdlRtUvkar. RbsinebI ¬xñat US¦ Vs = (Vu / φ − Vc ) ≤ 4λ f 'c bw d p Vs = Vu / φ − Vc ≤ λ f 'c bw d p / 3 ¬xñat SI¦ enaHKMlatEdkkg s EdlRtUvkarKWesμInwgtMélEdlKNnaedaysmIkarEdleGayenAkñúgCMhan ^. Shear and Torsion Strength Design 242
    • NPIC EtRbsinebI ¬xñat US¦ Vs = (Vu / φ − Vc ) > 4λ f 'c bw d p Vs = Vu / φ − Vc > λ f 'c bw d p / 3 ¬xñat SI¦ enaHKMlatEdkkg s EdlRtUvkarKWesμInwgBak;kNþaléntMélEdlKNnaedaysmIkarEdleGay enAkñúgCMhan ^. ^> s = (VAvφf)y− V = VAv −yφdVp ≤ 0.75h ≤ 24in.(60cm) ≥ s Gb,brmaEdl)anBICMhan $ dp f u c u c &> sg; shear envelope enAelIElVgFñwm nigKUsbBa¢ak;tMbn;EdlRtUvkarEdkRTnug *> KUrBRgayEdkRTnugtambeNþayElVgedayeRbIEdkkgTMhM #3 b¤ #4 tamEdlcUlcitþ b:uEnþEdk kgminRtUvmanTMhMFMCag #6 eT. (> KNnaEdkrgcaM (dowel reinforcement) bBaÄrkñúgkrNImuxkat;smas (a) Vnh ≤ 80bv d pc sMrab;TaMgépÞb:HeRKImedayKμanEdkrgcaM b¤Edk tie bBaÄr nigsMrab;TaMgépÞb:H EdlmineRKImb:uEnþeRbIEdk tie bBaÄrGb,brma. eRbI 50bw s 50bv I vh Av = = fy fy (b) Vnh ≤ 500bw d pcsMrab;épÞeRKImEdlmanecjk,alfμkMBs; 1 / 4in.(6mm) (c) sMrab;krNIEdl Vnh > 500bw d pc / KNnaEdk tie bBaÄrsMrab; Vnh = Avf f y μ Edl Avf = RkLaépÞrbs;EdkrgcaMEdlmanlkçN³kkit (frictional steel dowel) μ = emKuNkkit = 1.0λ sMrab;épÞEdlmanlkçN³eRKIm Edl λ = 1.0 sMrab; ebtugTMgn;Fmμta. sMrab;RKb;krNITaMgGs; Vn ≤ Vnh ≤ 0.2 f 'c Acc ≤ 800 Acc Edl Acc = bv lvh . viFIepSgeToténkarKNnaRkLaépÞEdkrgcaM Avf KWedayKNnakMlaMgedk Fh enARtg;épÞkkit rbs;ebtugEdl Fh ≤ μ e Avf f y ≤ Vnh 1,000λ2bv lvh Edl μe = Fh ≤ 2.9 rUbTI 5>16 bgðajBICMhanKNnaEdl)anerobrab;xagelICaTMrg; flowchart. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 243
    • T.Chhay Shear and Torsion Strength Design 244
    • NPIC 9> kugRtaMgTajemenAkñúgmuxkat;mansøab nigKNnaEdkrgcaMbBaÄrenA kñúgmuxkat;smas Principal Tensile Stresses in Flanged Sections and Design of Dowel-Action Vertical Steel in Composite Sections ]TahrN_ 5>1³ FñwmeRbkugRtaMgmuxkat;GkSr T mankarBRgayénkugRtaMgeFVIkarsgát;dUcbgðajenAkñúg rUbTI 5>17. kMlaMgkat;bBaÄrxageRkAKNnaEdlKμanemKuN V = 120,000lb(554kN ) ehIykMlaMg kat;emKuN Vu = 190,000lb(845kN ) . (a) KNnakugRtaMgTajemenARtg;G½kSTIRbCMuTMgn; cgc nigenARtg;cMnucRCugEkg A énkEnøgkat;Kña rvagsøab nigRTnug nigKNnakugRtaMgkMlaMgkat;edkGtibrmaeRkamGMeBIbnÞúkeFVIkarsMrab;TI taMgTaMgenH. (b) KNnaersIusþg;kMlaMgkat;edkFmμtaEdlRtUvkarenARtg;épÞGnþrkmμ A − A rvagRTnugEdlcak; Rsab; nigsøabEdlcak;enAnwgkEnøg nigKNnaEdkrgcaM b¤Edk tie bBaÄrcaM)ac;edIm,IkarBarkar rGildac;enARtg; A − A EdlFanaskmμPaBsmaseBjelj. eRbI ACI direct method nig snμt;faépÞb:HmanlkçN³eRKIm. smμtikmμmandUcxageRkam³ f 'c sMrab;RTnug = 6,000 psi ebtugTMgn;Fmμta (normal-weigth concrete) f 'c sMrab;søab = 3,000 psi ebtugTMgn;Fmμta (normal-weigth concrete) TTwgsøabRbsiT§PaB bm = 60in.(152.4cm) Edl bm CaTTwgEktMrUvEdl)anKitbBa©ÚlPaBxusKñaénm:UDuleGLasÞicrbs;ebtugépñkEdlcak; Rsab; nigEpñkxagelIEdlcak;enAnwgkEnøg. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 245
    • T.Chhay dMeNaHRsay³ kugRtaMgkMlaMgkat;eFVIkaredk kugRtaMgkMlaMgkat;edkGtibrma VQ vh = Ibv Edl ( Q A = 60 × 12(19.32 − 6 ) = 9,590in.3 157,172cm3 ) 12 × (7.32 )2 Qcgc = 60 × 12(19.32 − 6 ) + 2 ( = 9,912in.3 162,429cm.3 ) dUcenHkugRtaMgkMlaMgkat;edkeRkamGMeBIbnÞúkeFVIkarKW enARtg;cMnuc A / vh = 120,000 × 9,12 = 235 psi(1.6MPa) 408,240 × 590 enARtg; cgc / vh = 120,000 × 9,12 = 243 psi(1.7MPa) 408,240 × 912 BIsmIkar 5.13 kugRtaMgTajemEdlRtUvKñaKW 2 ⎛f ⎞ enARtg; A / ⎝ 2 ⎠ 2 f f 't = ⎜ cA ⎟ + vh − cA 2 2 ⎛ 2,160 ⎞ ⎟ + (235) − = 25 psi (111Pa ) 2,160 = ⎜ 2 ⎝ 2 ⎠ 2 2 nigenARtg; cgc / f 't = ⎛ 1,831 ⎞ + (235)2 − 1,831 = 32 psi(221Pa ) ⎜ ⎝ 2 ⎠ ⎟ 2 dUcenH kugRtaMgTajemmantMéltUc nigminbgáeGaymansñameRbHeRkamGMeBIbnÞúkeFVIkareT. eKRtUvRtYtBinitükugRtaMgkMlaMgkat;edk vh = 235 psi enARtg;épÞb:H A − A edIm,IepÞóg pÞat;favasßitenAkñúgEdnkMNt;EdlGacTTYlyk)an. GnuelameTAtam AASTHO kugRtaMg GnuBaØatGtibrmaKW 160 psi(1.1MPa ) < 235 psi(1.6MPa ) dUcenHkarpþl;eGayCaBiesssMrab; EdkrgcaM b¤Edk tie bBaÄrbEnßmRtUv)aneFVIRbsinebIeKGnuvtþtamtMrUvkarrbs; AASTHO. KNnaEdkrgcaM (Dowel Reinforcement Design) Vu = 190,000lb Vnh EdlRtUvkar = Vφu = 190.,75 = 253,333lb(1126kN ) 0 000 bv = 12in.(30.5cm ) d pc = 57in.(145cm ) BIsmIkar 5.26b Shear and Torsion Strength Design 246
    • NPIC EdlGacman = 500bv d pc = 500 × 12 × 57 = 342,000lb(1520kN )253,333lb Vnh BIsmIkar 5.22 a/ 0.75 f 'c = 0.75 6,000 = 58 > 50 dUcenHeRbI 58 enAkñúgsmIkarenaH Av l ÉktþaGb,brma = 58bv = 60,× 12 = 0.0116in.2 / in tambeNþayElVg f 58 000 vh y dUcenHeRbIEdkkgbBaÄr #3 Edlman Av = 2 × 0.11 = 0.22in.2 ehIy s = 0.22 / 0.0116 = 18.9in.(48cm ) EdlKitBIG½kSeTAG½kS < 24in. dUcenHKWTTYlyk)an (O.K.). EdkRTnugbBaÄrsMrab; kMlaMgkat;enAkñúgRTnugKYrRtUvkarKMlattUcCagenH. dUcenH BnøÚtEdkkgRTnugTaMgGs;eTAkñúgkMralxNÐ xagelIEdlcak;enAnwgkEnøg. 10> KNnaEdkrgcaMsMrab;skmμPaBsmas Dowel Steel Design for Composite Action ]TahrN_ 5>2³ edayeRbI (a) emKuNkkit ACI nig (b) emKuNkkit PCI cUrKNnaEdkrgcaM (dowel reinforcement) én]TahrN_ 5>1 sMrab;skmμPaBsmaseBjelj (full composite action) edayviFI epSgeTot (alternative method). edaysnμt;ElVgRbsiT§PaBrbs;FñwmTMrsamBaØesμInwg 65 ft.(19.8m) . dMeNaHRsay³ BIrUbTI 5>14 nig 5>17 Atop = 60 × 12 = 720in.2 (4,645cm 2 ) Cc = 0.85 f 'c Atop = 0.85 × 3,000 × 720 = 1,836,000lb(8,167kN ) snμt; Aps f ps > Cc enAeBlEdleKmineGaykMlaMgeRbkugRtaMg. enaH Fh = 1,836,000lb 65 × 12 lvh = = 390in. 2 bv = 12in. 80bv lvh = 80 × 12 × 390 = 374,400lb(1,665kN ) < 1,836,000lb dUcenH eKRtUvkarEdk tie bBaÄr. sMrab;épÞeRKImEdlmanecjk,alfμkMBs; 1 / 4in.(6mm) nigmanEdkGb,brma Vnh = 500bv d = 50 × 12 × 57 = 342,000 Vu φ EdlRtUvkar = 190,000 0.75 = 253,333lb < Vnh = 342,000 < Fh EdlGacman = 1,836,000lb karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 247
    • T.Chhay eRbI Fh = 253,333lb sMrab;kMNt;EdkskmμPaBsmasEdlRtUvkar (required composite action reinforcement). (a) edayeRbItMél μ rbs; ACI BIsmIkar 5.27 nig μ = 1.0 CamYynwg lvh = 390in. Avf srub = = 4.2in.2 (26.3cm 2 ) 253,333 1.0 × 60,000 Avf Gb,brma = 50bfvlvh = 50 × 12000390 = 3.90in.2 (25.1cm2 ) 60, × y BI]TahrN_ 5>1 Avf Gb,brma = 0.0116in.2 / in. = 0.139in.2 / 12in. / lub. dUcenHsakl,g Edkkg #3 eyIgTTYl)an Av = 2 × 0.11 = 0.22in.2 enaH 390 × 0.22 = 20.42 BIG½kSeTAG½kS < 24in. < tMélGnuBaØatGtibrma 4 × 12 = 48in. l A s = vh v = A vf 4.2 dUcenHeRbIEdk tie GkSr U #3 EdlmanKMlat 20in. KitBIG½kSeTAG½kS. (b) edayeRbItMél μ e rbs; PCI λ = 1 .0 1,000λ2bv lvh μe = ≤ 2.9 Fh 1,000 × 1 × 12 × 390 = = 2.55 < 2.9 1,836,000 dUcenHeRbI μe = 2.55 . bnÞab;mk BIsmIkar 5.32 Avf tMrUvkar = = 1.66in.2 < Avf Gb,brma = 3.90in.2 lub 253,333 2.55 × 60,000 dUcenHeRbIEdk tie GkSr U #3 EdlmanKMlat 20in. EdlKitBIG½kSeTAG½kS ¬Ggát;p©it 9.5mm KMlat 55cm ¦ 11> Dowel Steel Design for Composite Action in an Inverted T Beam ]TahrN_ 5>3³ FñwmGkSr T bRBa©asEdlRTedayTMrsamBaØmanElVgRbsiT§PaB 24 ft (7.23m) . mux kat;rbs;FñwmenHRtUv)anbgðajenAkñúgrUbTI 5>18 edaymankMralxagelIEdlcak;enAnwgkEnøgkMras; 2in. (5.1cm) enAelIépÞGt;eRKIm. KNnaEdkkgrgcaMcaM)ac;edIm,IbegáItskmμPaBsmaseBjelj (full Shear and Torsion Strength Design 248
    • NPIC composite behavior) edaysnμt;fakMlaMgkat;emKuN Vu EdlFñwmRtUvrgenARtg;muxkat;eRKaHfñak;KW 160,000lb(712kN ) . smμtikmμmandUcxageRkam³ f 'c ¬cak;Rsab;¦ = 6,000 psi (41.4 MPa ) / ebtugTMgn;Fmμta f 'c ¬kMralxagelI¦ = 3,000 psi (20.7 MPa ) / ebtugTMgn;Fmμta EdkeRbkugRtaMg³ tendon 270k Ggát;p©it 1 / 2in.(12.7mm) cMnYn 12 f pu = 270,000 psi (1,862MPa ) f ps = 242,000 psi (1,669 MPa ) rbs;Edk tie = 60,000 psi(414MPa ) fy edayeRbITaMg ACI direct method nig alternative method CamYynwg μe RbsiT§PaBsMrab;karKNna. dMeNaHRsay³ d p = 2 + 2 + 10 + 12 − 3 = 23in. Aps = 12 × 0.153 = 1.836in.2 Tn = Aps f ps = 1.836 × 242,000 = 444,312lb(1,976kN ) bv = 12in. 24 × 12 lvh = = 144in. 2 Atop = 2 × 48 + 2 × 12 = 120in.2 Cc = 0.85 f 'cc Atop = 0.85 × 3,000 × 120 = 306,000lb(1,316kN ) < Tn = 444,312lb karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 249
    • T.Chhay dUcenH eRbI Fh = 306,000lb(1,361kN ) . bnÞab;mk sMrab;épÞmineRKIm Vnh EdlGacman = 80bv lvh = 80 × 12 × 144 = 138,240lb(615kN ) < Cc = 306,000lb dUcenH Edk tie EdlRtUvkarcaM)ac;sMrab;begáItskmμPaBsmaseBjeljedayeRbI λ = 1.0 . (a) ACI Direct Method tMrUvkar = Vφu = 160.,75 = 213,333lb(949kN ) Vnh 0 000 eRbI μ = 1.0 . bnÞab;mk BIsmIkar 5.26a CamYynwgEdkrgcaM (dowel reinforcement) eyIgman Vnh EdlGacman = 80bv d pc = 80 × 12 × 23 = 22,000lb << Vnh tMrUvkar BIsmIkar 5.27 sMrab;épÞEdlmineRKIm μ = 0.6λ = 0.60 . bnÞab;mk Avf srubtMrUvkar = n = V 213,333 = 5.93in.2 μf 0.60 × 60,000 y BIsmIkar 5.22(a)/ 0.75 f 'c = 0.75 6,000 = 58 > 50 dUcenHeRbI 58 enAkñúgsmIkar 58bv lvh 58 × 12 × 144 Avf Gb,brmatMrUvkar = = = 1.67in.2 < 5.93in.2 f y 60,000 dUcenHeRbI Avf = 5.23in.2 (33.7cm2 ) ehIysakl,gEdk tie GkSr U páab; #3 . bnÞab;mk Avf = 2 × 0.11 = 0.22in.2 ( .4cm 2 ) nigmanKMlat 1 lvh Av 144 × 0.22 s= = = 5.34in.(13.7cm ) Avf 5.93 KMlatGnuBaØatGtibrmaKW s = 4(2 + 2) = 16in. b¤ 0.75h = 0.75 × 26 = 19.5in. < 24in. . dUcenHeRbIEdk tie GkSr U páab; #3 KMlat 5in.(13cm) KitBIG½kSelIG½kSelIElVgTaMgmUl. (b) Alternativ Method edayeRbI μe Fh = 306,000lb 1,000λ2bv lvh 1,000 × 1.0 × 12 × 144 μe = = = 5.65 > 2.9 Fh 306,000 dUcenHeRbI μe = 2.9 bnÞab;mk BIsmIkar 5.31c eyIgTTYl)an Avf tMrUvkar = h = F 306,000 = 1.76in.2 μ f 2.9 × 60,000 e y Shear and Torsion Strength Design 250
    • NPIC Gb,brmatMrUvkarEdl)anBI (a) = 1.67in.2 < 1.76in.2 Avf dUcenHeRbI Avf = 1.76in.2 enaHKMlatKW l A 144 × 0.22 s = vh v = = 18in. BIG½kSeTAG½kS A vf 1.76 ehIyKMlatGnuBaØatGtibrmaKW s = 4(2 + 2 ) = 16in. < 24in. dUcenH eRbIEdk tie GkSr U páab; #3 manKMlat 16in. KitBIG½kSeTAG½kSelIElVgTaMgmUl. 12> Shear Strength and Web-Shear Steel Design in a Prestressed Beam ]TahrN_ 5>4³ KNna bonded beam én]TahrN_ 4>2 edIm,IeGaymansuvtßiPaBRbqaMgnwgkar)ak; edaykMlaMgkat; ehIyKNnaEdkRTnugtMrUvkar. dMeNaHRsay³ Tinñn½y nigkarkMNt;ersIusþg;kMlaMgkat;Fmμta (data and nominal shear strength determination) f pu = 270,000 psi (1,862MPa ) f y = 60,000 psi (414MPa ) f pe = 155,000 psi (1,069MPa ) f 'c = 5,000 psiebtugTMgn;Fmμta Aps = tendon Ggát;p©it 1 / 2in.(12.7 mm ) Edlman wire 7 cMnYn 13 = 1.99in.2 = (12.8cm 2 ) As = 4#6 = 1.76in.2 (11.4cm 2 ) RbEvgElVg = 65 ft (19.8m) bnÞúkeFVIkar WL = 1,100 plf (16.1kN / m) bnÞúkeFVIkar WSD = 100 plf (1.46kN / m) bnÞúkeFVIkar WD = 393 plf (5.7kN / m) h = 40in.(101.6cm ) d p = 36.16in(91.8cm ) d = 37.6in(95.5cm ) karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 251
    • T.Chhay bw = 6in.(15cm ) ec = 15in.(38cm ) ee = 12.5in.(32cm ) ( I c = 70,700in.4 18.09 × 106 cm 4 ) ( Ac = 377in.2 2,432cm 2 ) ( r 2 = 187.5in.2 1,210cm 2 ) cb = 18.84in.(48cm ) ct = 21.16in.(54cm ) Pe = 308,255lb(1.371kN ) bnÞúkemKuN Wu = 1.2D + 1.6L = 1.2(100 + 393) + 1.6 × 1,100 = 2,352 plf kMlaMgkat;enARtg;épÞTMr Vu = Wu L / 2 = (2,352 × 65) / 2 = 76,440lb Vn tMrUvkar = Vu / φ = 76,440 / 0.75 = 101,920lb enARtg;TMr bøg;enARtg; 12 d p BIépÞénTMr !> ersIusþg;kMlaMgkat;Fmμta (nominal shear strength) Vc rbs;RTnug ¬CMhanTI2 nigTI3¦ 1 36.16 dp = ≅ 1.5 ft 2 2 × 12 Vn = 101,920 × [(65 / 2) − 1.5] = 97,216lb 65 / 2 Vu enARtg;1 2 d p = 0.75 × 97,216 = 72,912lb f pe = 155,000 psi 0.40 f pu = 0.40 × 270,000 = 108,000 psi (745MPa ) < f pe = 155,000 psi (1,069 MPa ) eRbI ACI alternate method edaysar d p > 0.8h / eRbI d p = 36.16in. edaysnμt;faEdkeRbkugRtaMgxøHRtg;rhUtdl;TMr. BIsmIkar 5.16 Shear and Torsion Strength Design 252
    • NPIC ⎛ Vu d p ⎞ Vc = ⎜ 0.60λ f 'c + 700 ⎜ ⎟bw d p ≥ 2λ f 'c bw d p ≤ 5λ f 'c bw d p ⎝ Mu ⎟ ⎠ λ = 1 .0sMrab;ebtugTMgn;Rsal Wu (1.5)2 M u enARtg; d / 2 BIépÞ = Rbtikmμ × 1.5 − 2 2,352(1.5)2 = 76,440 × 1.5 − = 112,014 ft − lb 2 = 1,344,168in. − lb Vu d p 72,912 × 36.16 = = 1.96 > 1.0 Mu 1,344,168 dUcenHeRbI Vu d p / M u = 1.0 enaH Vc Gb,brma = 2λ f 'c bw d p = 2 × 1.0 5,000 × 6 × 36.16 = 30,683lb Vc Gtibrma = 5λ f 'c bw d p = 76,707lb(341kN ) Vc = (0.60 × 1.0 5,000 + 700 × 1.0 )6 × 36.16 = 161,077lb > Vc Gtibrma = 76,707lb bnÞab;mk Vc = 76,707lb / lub. dUcKña Vu / φ > Vc / 2 dUcenH eKRtUvkarEdkRTnug. Vu Vs = − Vc = 97,216 − 76,707 = 20,509lb φ 8λ f 'c bw d p = 8 × 1.0 5,000 × 6 × 36.16 = 122,713lb(546kN ) > Vs = 20,509lb dUcenHkMBs;rbs;muxkat;RKb;RKan;. @> EdkRTnugGb,brma ¬CMhanTI4¦ BIsmIkar 5.22b Av s Gb,brma = 80psf fdpu d p A b y p w 1.99 × 270,000 36.16 = = 0.0076in.2 / in. 80 × 60,000 × 36.16 6 #> EdkRTnugtMrUvkar ¬CMhanTI5 nigTI6¦ BIsmIkar 5.21b Av f y d p s= ≤ 0.75h ≤ 24in. Vu / φ − Vc b¤ Av s V = s = 20,509 f y d p 60,000 × 36.16 = 0.0095in.2 / in. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 253
    • T.Chhay ¬kMlaMgeRbkugRtaMgKWRtUvFMCag 0.4 × ersIusþg;Taj¦ EdkRTnugkMlaMgkat; (web-shear steel) EdlRtUvkarGb,brma Av / s = 0.0095in.2 / in. . dUcenHsakl,gEdkkgGkSr U #3 / Av = 2 × 0.11 = 0.22in.2 . enaHKMlatGtibrma = 23.2in.(59cm ) 0.22 s= 0.0095 nig 4λ f 'c bwd p = 4 × 1.0 5,000 × 6 × 36.16 = 61,366lb > Vs dUcenH eyIgminRtUvkareRbI 12 s . LÚv 0.75h = 0.75 × 40.0 = 30.0in. dUcenH eRbI web-shear reinforcement #3 KMlat 22in. KitBIG½kSeTAG½kS ¬EdkkgGgát;p©it 9.5mm KMlat 62cm KitBIG½kSeTAG½kS¦ bøg;EdlminRtUvkarEdkkg snμt;fabøg;enHsßitenAcMgay x BITMr. tamRtIekaNdUc 1 76,707 65 / 2 − x Vc = = 101,920 × 2 2 65 / 2 b¤ 65 2 −x= 76,707 65 101,920 4 × x = 20.3 ft (6.11m ) ≈ 244in. dUcenHeRbIEdkkgGkSr U #3 KMlat 22in KitBIG½kSeTAG½kSelIRbEvgRbEhl 244in. edayEdk kgTImYycab;epþImenARtg; 18in. BIépÞénTMr. BRgayEdkkgeTAdl;kNþalElVgRbsinebIeKRtUvkarskmμ- PaBsmas. 13> Web-Shear Steel Design by Detailed Procedures ]TahrN_ 5>5³ edaHRsay]TahrN_ 5>4 edaydMeNIrlMGitEdlkMNt;tMélrbs; V CatMéltUcCageK c én flexure shear Vci nig web shear Vcw . snμt;fa tendon RtUv)an harp enAkNþalElVg. ehIy snμt;fa f 'c = 6,000 psi . dMeNaHRsay³ Profile rbs;EdkeRbkugRtaMgRtuv)anbgðajenAkñúgrUbTI 5>19. bøgenARtg; d / 2 BIépÞrbs;TMr BI]TahrN_ 5>4/ Vn = 97,216lb !> Flexure-shear cracking, Vci ¬CMhanTI2¦ Shear and Torsion Strength Design 254
    • NPIC BIsmIkar 5.11 Vci = 0.60λ f 'c bw d p + Vd + Vi (M cr ) ≥ 1.7λ f 'c bwd p M max BIsmIkar 5.12 M cr = Ic yt ( 6λ f 'c + f ce − f d ) Edl I c / yt = Sb edaysar yt CacMgayBITMRbCMuTMgn;eTAsrésTajxageRkA. eyIgman I c = 70,700in.4 cb = 18.84in. Pe = 308,255lb Sb = 3,753in.3 r 2 = 187.5in.2 dUcenHBIsmIkar 4.3b kugRtaMgebtugenARtg;srésxageRkambMputEdlbNþalmkEtBIeRbkug RtaMgKW Pe ⎛ ecb ⎞ f ce = − ⎜1 + 2 ⎟ Ac ⎝ r ⎠ ehIycMNakp©itEdkeRbkugRtaMgenARtg; d p / 2 ≅ 1.5 ft BIépÞrbs;TMrKW e = 12.5 + (15 − 12.5) 1.5 = 12.62in. 65 / 2 308,255 ⎛ 12.62 × 18.84 ⎞ dUcenH f ce =− 377 ⎝ ⎜1 + 187.5 ⎟ ≅ −1,855 psi (12.8MPa ) ⎠ BI]TahrN_ 4>2/ bnÞúkefrKμanemKuNEdlbNþalBITMgn;pÞal; WD = 393 plf (5.7kN / m) KW WD x(l − x ) 393 × 1.5(65 − 1.5) × 12 Md /2 = = = 224,600in. − lb(25.4kN .m ) 2 2 karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 255
    • T.Chhay ehIykugRtaMgEdlbNþalBIbnÞúkGefrKμanemKuNenARtg;srésebtugxageRkAbMputEdlkugRtaMg TajRtUv)anbegáItedaybnÞúkxageRkAKW M d / 2cb 224,600 × 18.84 fd = = = 60 psi Ic 70,700 ehIy ( M cr = 3,753 6 × 1.0 × 6,000 + 1,855 − 60 ) = 8,480,872in. − lb(958kN .m ) ⎛l ⎞ ⎛ 65 ⎞ Vd = WD ⎜ − x ⎟ = 393⎜ − 1.5 ⎟ = 12,183lb(54.2kN ) ⎝2 ⎠ ⎝ 2 ⎠ WSD = 100 plf WL = 1,100 plf WU = 1.2 × 100 + 1.6 × 1,100 = 1,880 plf kMlaMgkat;emKuNenARtg;muxkat;EdlbNþalBIbnÞúkGnuvtþn_xageRkAEdlekIteLIgtMNalKñaCa mYynwg M max KW ⎛l ⎞ ⎛ 65 ⎞ Vi = WU ⎜ − x ⎟ = 1,880⎜ − 1.5 ⎟ = 58,280lb(259kN ) ⎝2 ⎠ ⎝ 2 ⎠ 1,880 × 1.5(65 − 1.5) nig M max = U (l − x ) = W x 2 2 × 12 = 1,074,420in. − lb(122kN .m ) dUcenH Vci = 0.6 × 1.0 6,000 × 6 × 36.16 + 12,183 + 58,280 1,074,420 (8,480,872) = 482,296lb(54.5kN .m ) 1.7λ f 'c bw d p = 1.7 × 1.0 6,000 × 6 × 36.16 = 28,569lb(127kN ) < Vci = 482,296lb dUcenH Vci = 482,296lb(214.5kN ) @> Web-shear cracking, Vcw ¬CMhanTI2¦ BIsmIkar 5.15 ( ) Vcw = 3.5 f 'c + 0.3 f c bw d p + V p f c = kugRtaMgsgát;enAkñúgebtugRtg; cgc ≅ 818 psi (5.6 MPa ) Pe 308,255 = = Ac 377 Vp = bgÁúMbBaÄrrbs;eRbkugRtaMgRbsiT§PaBenARtg;muxkat; = Pe tan θ Edl θ CamMurvag tendon eRTtCamYynwgbøg;edk. dUcenH Shear and Torsion Strength Design 256
    • NPIC V p = 308,255 (15 − 12.5) = 1,976lb(8.8kN ) 65 / 2 × 12 dUcenH Vcw = (3.5 6,000 + 0.3 × 818)× 6 × 36.16 + 1,976 = 114,038lb(507kN ) enAkñúgkrNIenH web-shear cracking manlkçN³lub ¬Edl Vc = Vcw = 114,038lb(507kN ) RtUv)aneRbIsMrab;KNnaEdkRTnug¦. eRbobeFobtMélenHCamYynwg Vc = 76,707lb(341kN ) EdlTTYl)anBIsmIkar 5.4 eday alternative method EdlmanlkçN³suvtßiPaBCag. BIsmIkar 5>4 − Vc = (97,216 − 114,038)lb Vu Vs = φ dUcenHeKminRtUvkarEdkkgeT elIkElgEt Vu / φ > 12 Vc . dUcenH eyIgKNnatYxageRkaydUc xageRkam = 57,019lb(254kN ) < 97,216lb(432kN ) 1 114,038 Vc = 2 2 edaysar Vu / φ > 12 Vc b:uEnþ < Vc dUcenHeRbIEdkkgGb,brmaenAkñúgkrNIenH #> EdkkgGb,brma ¬CMhanTI4¦ BI]TahrN_ 5>4 Av s tMrUvkar = 0.0077in.2 / in. dUcenH sakl,gEdkkg #3 / eyIgTTYl)an Av = 2 × 0.11 = 0.22in.2 ehIy = 28.94in.(73cm ) 0.22 s= 0.0077 bnÞab;mkeyIgRtYtBinitü Av Gb,brmaEdlCatMéltUcCageKkñúgcMeNamtMélTaMgBIrxageRkam 50bw s Av = fy nig Av = A ps f pu s 80 f y d p dp bw dUcenHKMlatGnuBaØatGtibrma ≤ 0.75h ≤ 24in. . eRbIEdkkgGkSr U #3 EdlmanKMlat 22in. KitBIG½kSeTAG½kSelIRbEvg 84in. BIépÞrbs;TMrdUcenAkñúg]TahrN_ 5>4. muxkat;sMrab;karerobEdk ¬CMhanTI8¦ RtUv)anbgðajenAkñúgrUbTI 5>20. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 257
    • T.Chhay 14> Design of Web Reinforcement for a PCI Double T-Beam ]TahrN_ 5>6³ Fñwm T DubTMrsamBaØ PCI 12 DT 34 manElVg 70 ft (21.3m) . varg service dead edayrYbbBa©ÚlTaMgkMralBIelIbEnßm nig service live load WL = 720 plf . load 200 plf (29.kN / m ) KNnaEdkkgEdlRtUvkaredIm,IkarBar shear cracking enARtg;muxkat;mYyPaKbYnénElVg 17 ft 6in. (5.3m ) BITMr edayKNna nominal web-shear strength Vc tam detailed design method. ehIy KNna dowel reinforcement RbsinebIcaM)ac; edaysnμt;faépÞxagelIrbs;Fñwm T cak;Rsab;Gt;eRKIm. lkçN³muxkat;RtUv)anbgðajenAkñúgrUbTI 5>21 CamYynwgTinñn½yxageRkam³ Shear and Torsion Strength Design 258
    • NPIC Tinñn½yepSgeTot³ f 'c ¬ebtugcak;Rsab;¦ = 5,000 psi (34.5MPa ) ebtugTMgn;Fmμta f 'cc ¬ebtugsMrab;cak;kMralxagelI¦ = 3,000 psi (20.7 MPa ) ebtugTMgn;Fmμta f 'ci = 4,000 psi (27.6MPa ) / f pu = 270,000 psi (1,862MPa ) low-relaxation steel f ps = 240,000 psi (1,655MPa ) f pe = 148,000 psi (1,020MPa ) ee = 11.38in.(28.3cm ) ec = 21.77in.(57.2cm ) Aps = strandGgát;p©it 1 / 2in.(12.7mm) cMnYn 18 f yv sMrab;Edkkg = 60,000 psi (414 MPa ) eRbItMélsMrab;RbEvgRbsiT§PaB d p sMrab;muxkat;kNþalElVgk¾dUcCamuxkat;epSgeTot. cMNaMfa bw sMrab;RTnugTaMgBIr = 2(4.75 + 7.75) / 2 = 12.5in.(32cm) . dMeNaHRsay³ Wu = 1.2(200 + 1,091) + 1.6 × 720 ≅ 2,615 plf (38.15kN / m ) 2,615 × 70 Vu enARtg;épÞrbs;TMr = 2 = 91,525lb enARtg; BIépÞrbs;TMr = 1 ⎛ (35 − 17.5) ⎞ Vn 17 ft 6in. ⎜ 91,525 × ⎟ 0.75 ⎝ 35 ⎠ = 61,017lb(271kN ) karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 259
    • T.Chhay !> Flexure-shear cracking Vci ¬CMhanTI2¦ d p = 34 − 25.77 + 21.77 = 30.0in.(76cm ) Pe = 18 × 0.153 × 148,000 = 407,592lb(18,176kN ) enARtg; e BITMr 17 ft 6in. = 11.38 + (21.77 − 11.38) 17.5 35 = 16.58in. eRbIlkçN³muxkat;cak;Rsab;sMrab;KNna f ce nig f d dUcerobrab;enAkñúgEpñkTI 5 Pe ⎛ ecb ⎞ 407,592 ⎛ 16.58 × 25.77 ⎞ f ce = − ⎜1 + 2 ⎟ = − ⎜1 + ⎟ Ac ⎝ r ⎠ 978 ⎝ 88.0 ⎠ = 2,440 psi (16.8MPa ) eRbIkugRtaMgsgát;srésxageRkAGnuBaØatdUcxageRkam³ (a) eRbkugRtaMg + bnÞúkGcié®nþy_ ³ fc = 0.45 f 'c (b) eRbkugRtaMg + bnÞúksrub ¬edayGnuBaØatedayekIneLIg 33% EdlbNþalBI transient load³ f c = 0.60 f 'c ¦. cMNaMfaeTaHbICa fce = 0.45 f 'c k¾eday k¾vaminCHT§iBldl;ersIusþg;kMlaMgkat;Edr edaysar f ce bNþalmkBIEtkMlaMgeRbkugRtaMgb:ueNÑaH ehIyedaysarkarrYmbBa©ÚlTaMgbnÞúkpÞal;)an kat;bnßyvaeGaytUcCag 0.45 f 'c . dUcenHeyIgman bnÞúkpÞal; WD = 1,019 plf WD x(l − x ) 1,019 × 17.5(70 − 17.5) M 17.5 = = × 12 2 2 = 5,617,238in. − lb(634kN .m ) Mcb 5,617,238 × 25.77 fd = = = 1,682 psi (11.6MPa ) Ic 86,072 ( M cr = Sb 6.0λ f 'c + f ce − f d ) ( = 3,340 6.0 × 1.0 5,000 + 2,440 − 1,682 ) = 3,948,762in. − lb(445kN .m ) eKKYrcMNaMfaemKuN 6.0 enAkñúgsmIkar cracking moment mantMéltUc edaysareKyk modulus of rupture 7.5 . RbsinebIeKeRbI 7.5 enAkñúgsmIkarxagelI enaH cracking moment nwgmantMél 4,303,022in. − lb GBa©wgvanwgkat;bnßycMnYnrbs;EdkkgenAkñúgkar KNnaenH. kMlaMgKμanemKuNEdlbNþalBIbnÞúlpÞal;KW Shear and Torsion Strength Design 260
    • NPIC ⎛l ⎞ ⎛ 70 ⎞ Vd = WD ⎜ − x ⎟ = 1,019⎜ − 17.5 ⎟ = 17,833lb ⎝2 ⎠ ⎝ 2 ⎠ WSD = 200 plf WL = 720 plf GaMgtg;sIuetbnÞúkxageRkAemKuNKW WU = 1.2 × 200 + 1.6 × 720 = 1,392 plf (20.4kN / m ) ⎛l ⎞ ⎛ 70 ⎞ Vi = WU ⎜ − x ⎟ = 1,392⎜ − 17.5 ⎟ = 24,360lb(108kN ) ⎝2 ⎠ ⎝ 2 ⎠ ⎛ l − x ⎞ 1,392 × 17.5(70 − 17.5) M max = WU x⎜ ⎟= × 12 ⎝ 2 ⎠ 2 = 7,673,400in. − lb(867 kN .m ) × (M cr ) ≥ 1.7λ f 'c bw d p Vi Vci = 0.6λ f 'c bw d p + V p + M max = 0.6 × 1.0 × 5,000 × 12.5 × 30.0 + 17,833 + 24,360 (3,948,762) 7,673,400 = 46,279lb(201kN ) 1.7λ f 'c bw d p = 1.7 × 1.0 5,000 × 12.5 × 30.0 = 45,078lb < 46,279lb dUcenH Vci = 46,279lb lub @> Web-shear cracking, Vcw ¬CMhanTI2¦ = 417 psi (2.9MPa ) Pe 407,592 fc = = Ac 978 sMrab;bgÁúMbBaÄrrbs;kMlaMgeRbkugRtaMg VP = Pe tan θ = 407,592 (21.77 − 11.38) = 10,083lb(44.0kN ) 70 / 2 × 12 ( ) Vcw = 3.5λ f 'c + 0.3 f c bw d p + V p = (3.5 ) 5,000 + 0.3 × 417 12.5 × 30.0 + 10,083 nig Vci = 46,279lb = 149,803lb Vc CatMéltUcCageKén Vci nig Vcw dUcenH Vc = Vci = 46,279lb #> KNnaEdkkg ¬CMhanTI3-8¦ eyIgman Vc = 46,279lb dUcenH 1 2 Vc = 23,140lb karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 261
    • T.Chhay enARtg;muxkat; 17.5 ft BITMr = 61,017lb > Vc > 12 Vc dUcenHeKcaM)ac;KNnaEdkkg. Vu / φ RbsinebI Vu / φ < Vc > 12 Vc eKRtUvkarRtwmEtEdkkgGb,brma. Av s tMrUvkar = Vnf − Vc = (Vu f/ φd) − Vc = 61,017 −×46,279 d 60,000 30.0 y p y p = 0.0082in.2 / in. edayeRbI d ≅ d p = 30.0in. nig bw = 12.5in. ⎛ Av ⎞ A ps f pu dp 18 × 0.153 270,000 30.0 ⎜ ⎟ = = × = 0.0080 ⎝ s ⎠ min 80 f y d p bw 80 60,000 × 30.0 12.5 0.75 f 'c = 0.75 5,000 = 53 ⎛ Av ⎞ 53bw 53 × 12.5 b¤ ⎜ ⎟ = = = 0.011in.2 / in. ⎝ s ⎠ min fy 60,000 dUcenH tMéltUcCageKéntMélGb,brmaTaMgBIrKW ⎛ Av ⎞ ⎜ ⎟ = 0.0080in.2 / in. = 0.010in.2 / ft sMrab;RTnugTaMgBIr b¤ 0.005in.2 / ft ⎝ s ⎠ min sMrab;RTnugmYy sakl,g D5 deformed welded wire fabric mYyCYredayKMlat 10in. KitBIG½kReTAG½kS. KMlatGnuBaØatGtibrmaKW 0.75h ≤ 24in. dUcenHeyIgman 0.75h = 0.75 × 34 = 25.5in. dUcenH TTYlykEdkkg D5 WWF mYyCYrkñúgmYyRsTab;edaymanKMlat 10in. KitBIG½kSeTA G½kSenARtg;muxkat;énmYyPaKbYnElVg. cMNaMfa edayeRbobeFobdMeNaHRsaysMrab; Vci nig Vcw enAkñúg]TahrN_ 5>6 tMélx<s;Cag eKrbs; Vci sßitenAEk,rTMr ehIyfycuHy:agelOneTArkkNþalElVg enAeBlEdlbMErbMrYlrbs; Vcw mantMéltUcCag dUcEdleXIjenAkñúgrUbTI 5>13. eKcaM)ac;KNna flexure shear Vci nig web shear Vcw enAeRcInmuxkat;tambeNþayElVgedIm,IkMNt;karrayEdkkgRbkbedayRbsiT§PaBbMput. kmμviFIkMu BüÚT½rsMrYlkarKNnatMélTaMgenHelIcenøaHefrNamYy ¬dUcCa 10 énElVg¦ ehIyeKGacsg;düaRkam 1 RsedogKñanwgdüaRkamenAkñúgrUbTI 5>13 EdlbgðajBIbMErbMrYlénersIusþg;kMlaMgkat;rbs;RTnugtam beNþayElVg. $> KNna dowel steel sMrab; full composite section Edlman topping bEnßm 2in. ¬CMhanTI9¦ RbsinebIeKbEnßm topping EbbenHeTAelI pretopped section enAeBleRkay. Shear and Torsion Strength Design 262
    • NPIC muxkat;enARtg; 12 d p BIépÞrbs;TMr d p = 30.0 + 2.0 = 32.0in. Vu enARtg;TMr = 91,525(408kN ) = 1.33 ft (40cm ) 1 32.0 dp = 2 2 × 12 h / 2 = 17in. = 1.33 ft ⎛ 35 − 1.33 ⎞ Vu = 91,525 × ⎜ ⎟ = 88,047lb(393kN ) ⎝ 35 ⎠ Vnh tMrUvkar V = u = φ 88,047 0.75 = 117,393lb(522kN ) bv = 12 ft htopping = 2in. BIrUbTI 5>14 Cc = 0.85 f 'cc Atop = 0.85 × 3,000 × 12 × 12 × 2 = 734,400lb(3,267kN ) Ts = Aps f ps = 18 × 0.153 × 240,000 = 660,960lb(2,940kN ) < Cc = 734,400lb dUcenH Fh = 660,960lb(2,178kN ) 70 × 12 lvh = = 420in.(1,067cm ) 2 bv = 144in.(366cm ) 80bv lvh = 80 × 144 × 420 = 4,838,400lb(21,520kN ) >> 660,960lb eKminRtUvkar dowel reinforcement edIm,ITak;enAkñúg topping EdlRtUvbEnßmenAeBleRkay 2in. edIm,I)an full composite action eT. eKTTYlykmuxkat;enH enAeBlvamanlkçN³RKb;RKan; sMrab; flexurl, PaBdab nigtMrUvkarsMrab;karRKb;RKgsñameRbH. 15> Brackets and Corbels Bracket nig corbel Ca short-haunched cantilever EdllyecjBIépÞxagrbs;ssr b¤ CBa¢aMgebtugedIm,IRTbnÞúkcMcMnucEdlF¶n; b¤Rbtikmμrbs;Fñwm. vaCaFatuy:agsMxan;rbs;eRKOgbgÁúMsMrab;RT Fñwmcak;Rsab;/ gantry girder nigTMrg;epSgeToténRbB½n§eRKOgbgÁúMcak;Rsab;. ebtugeRbkugRtaMg nigeb tugcak;Rsab;RtUv)aneRbIy:ageRcIn nigekIneLIgy:agrh½s ehIyElVgkan;EtEvgRtUv)ansagsg; Edl karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 263
    • T.Chhay eFVIeGaybnÞúkkMlaMgkat;TTwgFMenARtg;TMr. dUcenH karsikSaKNna bracket nig corbel køayCabBaða kan;EtsMxan;. suvtßiPaBrbs;eRKOgbgÁúMTaMgmUlGaRs½ynwgkarsikSaKNna nigkarsagsg;Ggát;TMrenH. sMrab; bracket nig corbel pleFobrbs; shear arm b¤ElVgelIkMBs;rbs; corbel CaerOy² mantMéltUcCag 1.0. pleFobtUcEbbenHEkERbsßanPaBkugRtaMgrbs;Ggát;eTACakugRtaMgBIrTMhM. dUcenH kMhUcRTg;RTayedaysarkMMlaMgkat; (shear deformation) nwgmanT§iBlelI nonlinear stress behavior rbs; bracket b¤ corbel enAkñúgsßanPaBeGLasÞic nigsßanPaBbnþ ehIyersIusþg;kMlaMgkat; køayeTACaktþasMxan;. Corbel k¾xusBIFñwmeRCA (deep beam) edaymankMlaMgedky:agFMEdlepÞrBIFñwm eTA corbel b¤ bracket. kMlaMgedkTaMgenH)anBI long-term shrinkage nig creep deformation rbs; Fñwm EdlkñúgkrNICaeRcInvaRtUv)anf<k;Cab;eTAnwg bracket. CaTUeTA sñameRbHPaKeRcInCasñameRbHbBaÄr b¤CasñameRbHkMlaMgkat;suT§eRTtecat (steeply inclined pure shear cracks). CaerOy² vacab;epþImBIcMnucEdlbnÞúkcMcMnucGnuvtþn_ ehIyraleq<aHeTA kan;RCugxageRkaméncMnuckat;Kñarvag bracket nigépÞssr dUcbgðajenAkñúgrUbTI 5>22(a). b¤mYyk¾va cab;epþImenARtg;kac;RCugxagelIrbs; bracket b¤corbel ehIybnþcuHeRkamesÞIEtbBaÄreTAkan;srésxag eRkamrbs;va dUcbgðajenAkñúgrUbTI 5>22(b). TMrg;énkar)ak;epSgeTotenAkñúgFatuTMrenHRtUv)anbgðaj enAkñúgrUbTI 5>22(c) nig (d). vak¾GacekIttamry³bnSMénrUbxagelI. Bearing failure k¾GacekIt eLIgBIkarEbkrbs;ebtugenABIxageRkam concentrated load-bearing plate RbsinebI bearing area minmansmamaRtRKb;RKan;. enAkñúgEpñkxageRkam eyIgnwgerobrab;BIEdksMxan;enAkñúg corbel b¤ bracket. kar)ak;rbs;va GacR)ab;BIkrNIEdlminman full anchorage development rbs;Edk. Shear and Torsion Strength Design 264
    • NPIC k> RTsþIkMlaMgkat;kkit sMrab;karepÞrkMlaMgkat;enAkñúg Corbel Shear Friction Hypothesis for Shear Transfer in Corbels Edlcak;enAeBlepSgKñaBIssrGacnwgman shear crack enARtg;épÞb:HedaysarekIt Corbel mankarepÞrkMlaMgkat;. pleFob a / d kan;EttUc karekItmankMlaMgkat;suT§ (pure shear) tambøg; bBaÄrkan;EtFM. kareFVIkarenHbgðajeGayeXIjkan;Etc,as;enAkñúgkrNIEdl corbel mansñameRbHenA Rtg;épÞb:HedaysarsMPar³BIrmindUcKña. viFIkMlaMgkat;kkit (shear friction approach) enAkñúgkrNIenHRtUv)anENnaMeday ACI dUc bgðajenAkñúgrUbTI 5>22(b). eKsnμt;fa bøg;bBaÄreRbH ¬a-a enAkñúgrUbTI 5>23¦ EdlekItmanenAelI corbel rGilenAeBlEdlvaeTAdl;sßanPaBkMNt;énkar)ak;rbs;va. eKeRbIemKuNkkit μ edIm,IbMElg kMlaMgTb;Tl;edkrbs; well-anchored closed ties eTACakMlaMgTb;Tl;FmμtabBaÄrEdlFMCagkMlaMg kat;emKuNxageRkA. dUcenH kMlaMgkat;Tb;Tl;bBaÄrFmμta (nominal vertical resisting shear force) Vn = Avf f y μ (5.34a) Vn Avf = (5.34b) f yμ Edl Avf CaRkLaépÞsrubrbs;Edkkgedk (horizontal anchored closed shear ties) kMlaMgkat;bBaÄremKuNxageRkA Vu ≤ φVn EdlsMrab;ebtugFmμta Vn ≤ 0.2 f 'c bw d (5.35a) b¤ Vn ≤ 800bw d ¬xñat US¦ (5.35b) karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 265
    • T.Chhay Vn ≤ 5.6bw d ¬xñat SI¦ edayykmYyNaEdlmantMéltUcCag. eKGackMNt;kMBs;RbsiT§PaBtMrUvkar d rbs; corbel BIsmIkar 5.35a b¤ b edayykmYyNaEdlmantMélFMCageK. sMrab; all-lightweight concrete b¤ sand-lightweight concretes ersIusþg;kMlaMgkat;Vn minRtUv mantMélFMCag (0.2 − 0.07a / d ) f 'c bwd b¤ (800 − 280a / d )bwd sMrab;xñat US (5.6 − 1.96)bwd sMrab;xñat SI. RbsinebI shear friction reinforcement eRTtedIm,IeFVIeGayekItmankMlaMgTajenAkñúg shear friction steel ( Vn = Avf f y μ sin α f + cos α f ) (5.35c) Edl α f CamMurvag shear friction reinforcement CamYynwgbøg;kMlaMgkat;. RkLaépÞrbs;EdkKW Vn Avf = ( f y μ sin α f + cos α f ) (5.35d) eKsnμt;faersIusþg;kMlaMgkat;TaMgGs;ekItmanedaysarersIusþg;enARtg;sñameRbHrvag corbel nigssr. emKuN ACI rbs;kMlaMgkat; μ mantMéldUcxageRkam³ ebtugcak;enAkñúgeBlCamYyKña (concrete cast monolithically) 1.4λ ebtugEdlcak;P¢ab;nwgebtugEdlmanépÞeRKIm 1 .0 λ ebtugEdlcak;P¢ab;nwgebtugEdlmanépÞGt;eRKIm 0.6λ ebtugEdlP¢ab;eTAnwgEdkeRKOgbgÁúM 0 .7 λ λ = 1.0 sMrab;ebtugTMgn;Fmμta/ 0.85 sMrab; sand-lightweight concrete nig 0.75 sMrab;ebtugTMgn; RsalTaMgGs;. tMél PCI manlkçN³suvtßiPaBticCagtMél ACI EdlQrelIkarBiesaF. RbsinebIeKeRbIebtugersIusþg;x<s; ¬polymer-modified concrete¦ enAkñúg corbel edIm,IP¢ab; CamYynwgebtugFmμtarbs;ssr enaHeKRtUveRbItMél μ FMCageKkñúgtaragxagelI. kargarenAkardæan RtUvkareRbItMélEdlFM. eKeGayEpñkxøHrbs;Edkedk Avf cUlrYmenAkñúgEdkTajxagelI ehIyeKBRgayEpñkEdlenA sl;rbs; Avf tamkMBs;rbs; corbel dUcbgðajenAkñúgrUbTI 5>24. karKNnaRsTab;EdkedkxagelI As nwgRtUv)anerobrab;enAEpñkbnþ. Shear and Torsion Strength Design 266
    • NPIC x> T§iBlkMlaMgedkxageRkA Horizontal External Force Effect enAeBlEdleKcak; corbel b¤ bracket kñúgeBlCamYyKñanwgssr b¤CBa¢aMg ehIyvargnUvkMlaMg TajedkFM Nuc Edl)anBIFñwmEdlRTeday corbel, eKeRbI modified apprach EdlCaerOy²eGayehA vafa strut theory approach. enARKb;krNITaMgGs; kMlaMgedkemKuN Nuc minGacFMCagkMlaMgkat;em KuNbBaÄr Vu eT. dUcEdl)aneXIjenAkñúgrUbTI 5>25 eKRtUvdak;Edk An eGayTb;nwgkMlaMg Nuc Edl N An = uc φf (5.36) y Vu a + N uc (h − d ) ehIy Af = φf y jd (5.37) eKk¾RtUvdak;Edk A f edIm,ITb;nwgm:Um:g;Bt;EdlbNþalBI Vu nig Nuc . tMélrbs; Nuc EdlBicarNaenAkñúgkarKNnaminKYrtUcCag 0.2Vu eT. eKGacTTYltMél Rbhak;RbEhlrbs;RkLaépÞEdkrgkarBt; A f (flexural steel area) edaysmIkarFmμtasMrab;sßan PaBkMNt;enAeBlFñwm)ak; Mu Af = (5.38) φf y jd Edl M u = Vu a + Nuc (h − d ) nig φ = 0.75 . G½kSrbs;muxkat;Edlsnμt;sßitenAtambeNþay compression strut eRTtpÁúM)anmMu β CamYynwg tension tie As dUcbgðajenAkñúgrUb. maDrbs;bøúksgát;KW karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 267
    • T.Chhay Ts As f y V Cc = 0.85 f 'c β1cb = = = u (5.39a) cos β cos β sin β BIsmIkarxagelI eKGacTTYl)ankMBs;rbs;bøúk β1c EdlEkgeTAnwgTisedArbs; compressive strut As f y β1c = (5.39b) 0.85 f 'c b cos β kMBs;RbsiT§PaB d − β1c / 2 cos β kñúgTisenAbBaÄreGayédXñas; jd rvagkMlaMg Ts nigbgÁúMkMlaMgedk Cc enAkñúgrUbTI 5>25. dUcenH β1c jd = d − (5.39c) 2 cos β RbsinebIeyIgCMnYstMél jd eTAkñúgsmIkar 5.38 enaH Mu Af = (5.40) φf y (d − β1c / 2 cos β ) Shear and Torsion Strength Design 268
    • NPIC edIm,Ikat;bnßykarsakl,g nwgEktMrUveRcIndg eKGaceRbItMélRbhak;RbEhlrbs;RbEvgéd jd Edl)anBIsmIkar 5.39c sMrab;krNICaeRcIndUcxageRkam jd ≅ 0.85d (5.41a) dUcenH Af = Mu 0.85φf yd (5.41b) eKGacKNnaEdkTaj As nigdak;dUcbgðajenAkñúgrUbTI 5>26 As ≥ 2 3 Avf + An (5.42) b¤ As ≥ A f + An (5.43) edayykmYyNaEdlmantMélFMCag. bnÞab;mk As f' ρ= ≥ 0.04 c bd fy RbsinebIeKsnμt; Ah CaRkLaépÞsrubrbs;EdkkgbiTCit b¤EdkkgEdlRsbeTAnwg As enaH Ah ≥ 0.5( As − An ) (5.44) RkLaépÞRTnab; (bearing) EdlenABIxageRkambnÞúkxageRkA Vu EdlenABIelI bracket minRtUvlyecj putBIEdkTajem As ehIyvak¾minRtUvlyecjBIépÞxagkñúgrbs; transverse welded anchor bar dUc bgðajenAkñúg rUbTI 5>26. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 269
    • T.Chhay K> CMhanénkarKNna corbel Sequence of Corbel Design Steps dUcEdl)anerobrab;kñúgEpñkxagedIm kMlaMgedkemKuN Nuc kMlaMgbBaÄremKuN Vu nigm:Um:g; Bt; [Vu a + Nuc (h − d )] manGMeBIelI corbel. edIm,IkarBarkar)ak; eKRtUvsikSaKNna corbel eGay Tb;Tl;nwg)a:ra:Em:RtTaMgbIenHkñúgeBlCamYyKña edayeRbIviFImYykñúgcMeNamviFIBIrxageRkamEdlGaRs½y eTAnwgRbePTénkarsagsg; corbel faetIeKsg;vakñúgeBlCamYyKñanwgssr b¤k¾Gt;. (a) sMrab; corbel Edlsg;kñúgeBlEtmYyCamYynwgssr KNnaRkLaépÞEdk Ah rbs;Edk kgbiTCitEdldak;enABIxageRkamEdkTajem As . EpñkxøHrbs;Edk Ah KWbNaþalmk BI An Edl)anBIsmIkar 5.36 EdlTb;Tl;nwgkMlaMgedk Nuc . (b) KNnaRkLaépÞEdk Avf eday shear friction hypothesis RbsinebI corbel nigssr minRtUv)ancak;kñúgeBlCamYyKña edayeRbIEpñkxøHrbs; Avf tamry³kMBs;rbs; corbel nigedaykarcUlrYmenAkñúglMnwgrbs;RkLaépÞrbs;RsTab;EdkxagelI. RkLaépÞEdkTajem As CaFatusMxan;énviFITaMgBIr. karKNna As GaRs½yeTAelIPaBlubrbs; smIkar 5.42 b¤ 5.43. RbsinebIsmIkar 5.42 lub eKeRbI As = 2 Avf + An ehIycMENkEdlenAsl; 3 1 A RtUv)anBRgayenAelIkMBs; 2 d enAEk,rnwg A . RbsinebIsmIkar 5.43 lub enaH A = A + A 3 vf 3 s s f n nigbEnßm 12 A f RtUv)andak;kñúgTMrg;EdkkgbiTCitRsbeTAnwg As ehIyRtUv)anBRgayenAelIRbEvg 2 d 3 enAEk,rnwg As . kñúgkrNITaMgBIr EdkTajembUknwgEdkkgbiTCitCabrimaNEdksrubEdl corbel RtUvkar. edaysaremkanicénkar)ak;mandWeRkkMNt;min)anx<s; ehIykarrBwgénkarsayPay shear crack manlkçN³minc,as;las; eBlxøHeKENnaMeGayeRCIserIsykRkLaépÞEdkTajem As NaEdlman tMélFM edayminKitBIkrNIénkarsg; corbel favaRtUv)ansg;kñúgeBlCamYynwgssrenaHeT. dUckarerobrab;BIxagedIm EdkkgbiTCitedkk¾CaFatusMxan;enAkñúg corbel Edr. dUcenH eKk¾RtUv kareRbIEdkkgbiTCiteRTtbEnßm. xageRkamCaCMhanénkarsikSaKNna corbel³ !> KNnakMlaMgbBaÄremKuN Vu nigkMlaMgTb;Tl;FmμtaVn rbs;muxkat;edayeGayVn ≥ Vu / φ Edl φ = 0.75 sMrab;RKb;karKNna. Vu / φ KYrEt ≤ 0.20 f 'c bwd b¤ ≤ 800bwd ¬xñat US¦ ≤ 5.6bw d ¬xñat SI¦ sMrab;ebtugTMgn;Fmμta. RbsinebImindUecñaHeT eKRtUvtMeLIgmuxkat;enA Rtg;TMr. Shear and Torsion Strength Design 270
    • NPIC @> KNna Avf = Vn / f y μ edIm,ITb;Tl; shear friction force nigeRbIvaenAkñúgkarKNnaEdk TajemxagelI As . #> KNnaRkLaépÞEdkrgkarBt; (flexural steel area) A f nigRkLaépÞEdkrgkarTajeday pÞal; (direct tension steel area) An Edl vu a + N uc (h − d ) Af = φf y jd Edl φ = 0.90 nig N uc An = φf y Edl φ = 0.75 $> KNnaRkLaépÞEdkem (a) As = 2 Avf + An nig (b) As = A f + An edayykmYyNaEdl 3 mantMélFMCageK. RbsinebI (a) lub enaHRkLaépÞEdkEdlenAsl; 13 Avf RtUv)aneRbICaEdk kgbiTCitRsbeTAnwg As ehIyRtUv)anBRgayelIkMBs; 2 d Ek,r As dUcbgðajenAkñúgrUbTI 3 5>24. RbsinebI (b) lub eRbI 12 A f bEnßmCaEdkkbiTCitEdlBRgayenAelIkMBs; 2 d Ek,r 3 nwg As dUcbgðajenAkñúgrUbTI 5>26. enaH Ah ≥ 0.5( As − An ) nig As f' ρ= ≥ 0.04 c bd fy b¤ f 'c As min = 0.04 bd fy %> eRCIserIsTMhM nigKMlatEdksMrab; corbel edayykcitþTikdak;CaBiesseTAelIkarerobcM EdklMGit edaysar corbel CaeRcIn)ak;edaysarkarlMGitEdkmin)anRtwmRtUv. rUbTI 5>27 bgðajBI flowchart sMrab;karKNnasmamaRtrbs; corbel. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 271
    • T.Chhay Shear and Torsion Strength Design 272
    • NPIC X> Design of a Bracket or Corbel ]TahrN_ 5>7³ sikSaKNna corbel edIm,IRTbnÞúkbBaÄremKuN V u EdlmanGMeBI = 80,00lb(160kN ) enAcMgay a = 5in(127mm) BIépÞrbs;ssr. Corbel manTTwg b = 10in.(254mm) kMras;srub h = 18in.(457 mm ) ehIykMBs;RbsiT§PaB d = 14in.(356mm ) . eKeGaynUvsmμtikmμdUcxageRkam³ f 'c = 5,000 psi (34.5MPa ) ebtugTMgn;Fmμta f y = 60,000 psi (414 MPa ) snμt;fa corbel RtUv)ansg;eRkayeBlssrsg;rYc b¤sg;kñúgeBlCamYyKñanwgssr. TMgn;pÞal;rbs; corbel RtUv)anecal. dMeNaHRsay³ CMhan! Vu 80,000 Vn ≥ = = 106,667lb φ 0.75 0.2 f 'c bw d = 0.2 × 5,000 × 10 × 14 = 140,000lb > Vn 800bw d = 800 × 10 × 14 = 112,000lb > Vn O.K. CMhan@ (a) karsg;kñúgeBlEtmYy (monolithic construction) sMrab;ebtugTMgn;Fmμta μ = 1.4λ = 1.270in.2 (819mm 2 ) V 106,667 Avf = u = φf μ 60,000 × 1.4 y (b) karcak;kñúgeBlepSgKña (nonmonolithic construction) μ = 1.0λ = 1.777in.2 (1110mm 2 ) 106,6667 Avf = 60,000 × 1.0 eRCIserIsyk Avf NaEdlmantMélFMCageK dUcenH Avf = 1.777in.2 CMhan# edaysareKmineGaytMélkMlaMgedkxageRkA Nuc EdlepÞrBIFñwmEdldak;BIelI enaHeyIgyk N uc Gb,brma = 0.20Vu = 0.2 × 80,000 = 16,000lb V a + N uc (h − d ) Af = Mu φf jd = u φf jd ¬Edl jd ≅ 0.85d ¦ y y 80,000 × 5 + 16,000(18 − 14 ) = 0.90 × 60,000(0.85 × 14) ( = 0.727in.2 469mm 2 ) N An = uc = 16,000 φf y 0.75 × 60,000 ( = 0.356in.2 280mm 2 ) CMhan$ karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 273
    • T.Chhay (a) As = (2 Avf + An ) = 2 × 1.777 + 0.356 = 1.541in.2 3 3 (b) As = A f + An = 0.727 + 0.356 = 1.083in.2 f 'c 5,000 As min = 0.04 bd = 0.04 × × 10 × 14 = 0.47in.2 fy 60,000 < 1.541in.2 O.K. yk As = 1.541in.2 (994mm2 ) EdkkgbiTCitedk edaykrNI (a) lub 1 3 Avf = 1 1.777 = 0.592in.2 3 Ah = 0.5( As − An ) = 0.5(1.541 − 0.356 ) = 0.593in.2 eday Ah > 13 Avf enaHeyIgyk Ah = 0.593in.2 CMhan% eRCIserIsTMhMEdk (a) As EdlRtUvkar = 1.541in.2 eRbI 3#7 = 1.80in.2 ¬EdkEdlmanGgát;p©it 22mm cMnYnbIedIm eGay As = 1.161mm2 ¦ (b) Ah EdlRtUvkar = 0.593in.2 eRbIEdkkgbiTCit 3#3 = 2 × 3 × 0.11 = 0.66in.2 BRgayelIkMBs; d = 9.33in. . dUcenH KMlatrbs;vaKW 3in. edayKitBIG½kSeTAG½kS. ehIyeRbI Edk framing 2 3 3#3 nigEdk welded 1#7 . bøg;lMGitsrésEdkrbs; bracket RtUv)anbgðajenAkñúgrUbTI 5>28. eKRtUvRtYtBinitü bearing area EdlenABIeRkambnÞúk nigKNna bearing pad Edl bearign stress eRkamGMeBI bnÞúkemKuN Vu minelIs φ (0.85 f 'c A1 ) Edl A1 CaRkLaépÞRTnab; (pad area). eyIgman emKuNkat;bnßyersIusþg; bearing φ = 0.70 Vu = 80,000lb = 0.70(0.85 × 5,000)A1 A1 = 80,000 0.70 × 0.85 × 5,000 ( = 26.9in.2 16,813mm 2 ) eRbI plate 5 12 in. × 5 12 in. . kMras;rbs;vaRtUv)anKNnaedayrkSa plate mineGayxUcRTg;RTay enAeBlrgbnÞúk Vu . Shear and Torsion Strength Design 274
    • NPIC 16> kugRtaMgrmYl nigersIusþg;rmYl Torsional Behavior and Strength k> esckþIepþIm Introduction kugRtaMgrmYlekItmanenAkñúg monolithic concrete enAeBlEdlbnÞúleFVIGMeBIenARtg;cMgaymYyBIG½kSbeNþayrbs;Ggát;eRKOgbgÁúM. cugFñwmrbs;kMralxNÐ/ spandrel beam EdlTTYlbnÞúkEtmçag/ canopy b¤dMbUlkEnøgcaMLanRkugEdllyecjBI monolithic beam enAelIssr/ peripheral beam EdlB½T§CMuvijkMralcMhr suT§EtCa]TahrN_énGgát;eRKOgbgÁúMEdl rgnUvm:Um:g;rmYl. m:Um:g;bgáeGaymankugRtaMgkat;FM. CalT§pl sñameRbHCaeRcInGacekItmanenAeBl EdlkugRtaMgrmYlFMCag allowable serviceability limits elIkElgEteKdak; special torsional reinforcement. enAkñúg spandrel beam Cak;EsþgénRbB½n§eRKOgbgÁúM TMhMénkarxUcxatedaysarkar rmYlminCasMxan;eT enHKWbNþalmkBIkarEbgEckkugRtaMgeLIgvijenAkñúgeRKOgbgÁúM. b:uEnþ eKKYrecosvag karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 275
    • T.Chhay kMhatbg;ersIusþg;edaysarkugRtaMgrmYledaykarKNnay:agsmrmünUv torsional reinforcement caM)ac;. esckþIepþImBIkarEbgEckkugRtaMgrmYlRtUvcab;epþImBI basic elastic behavior énmuxkat;samBaØ dUcCamuxkat;rgVg; b¤muxkat;ctuekaN. FñwmebtugPaKeRcInEdlrgnUvkarrmYlmanmuxkat;ctuekaN Ca]TahrN_ muxkat;mansøabdUcCaFñwmGkSr T nigFñwmGkSr L. eTaHbICaeKkMreRbImuxkat;rgVg;enAkñúg sMNg;ebtugk¾eday karerobrab;y:agsegçbBIkarrmYlenAkñúgmuxkat;rgVg;RtUv)aneRbICakarENnaMy:agl¥BI torsional behavior énmuxkat;déTeTot. kugRtaMgkMlaMgkat;esμInwgplKuNén shear strain CamYynwg shear modulus enARtg;tMbn;eGLa sÞicenAkñúgmuxkat;rgVg;. dUckñúgkrNI flexure kugRtaMgsmamaRteTAnwgcMgayrbs;vaBIG½kSNWt ¬G½kS kat;tamp©itrbs;rgVg;¦ ehIymantMélGtibrmaenAsrésxageRkA. RbsinebI r CakaMénmuxkat;rbs; Ggát;/ J = πr 4 / 2 Cam:Um:g;niclPaBb:UElr (polar moment of inertia) nig vte CakugRtaMgkat;eGLasÞic (elastic shearing stress) EdlbNþalmkBIm:Um:g;rmYleGLasÞic Te enaH Te r vte = (a) J enAeBlEdlGgát;muxkat;rgVg;ekItmankMhUcRTg;RTay G½kSrbs;sIuLaMghak;enArkSaPaBRtg; dEdl. kaMTaMgGs;enAkñúgmuxkat;k¾rkSaPaBRtg;Edr ¬minman warping¦ ehIyviledaymMudUcKñaeFobnwg G½kS. edaysarGgát;Edlmanmuxkat;mUlcab;epþImeFVIkarCalkçN³)aøsÞic kugRtaMgenAelIrgVg;)aøsÞicxag eRkAcab;epþImmantMélefr enAeBlEdlkugRtaMgbNþÚlxagkñúgenAmanlkçN³CaeGLasÞicenAeLIy dUc eXIjenAkñúgrUbTI 5>29. enAeBlEdlmuxkat;TaMgmUlkøayCa)aøsÞic/ b = 0 ehIykugRtaMgkMlaMgkat; 3 Tpr vtf = (b) 4 J Edl vtf CakugRtaMgkMlaMgkat;GsmamaRt (nonlinear shear stress) bNþalmkBI ultimate twisting moment T p . ¬GkSr f mann½y failure¦. enAkñúgmuxkat;ctuekaN bBaðarmYlmanlkçN³sμúKsμajCagqøayNas;. bøg;muxkat;edImrg warping edaysar applied torsional moment. m:Um:g;enHbegáItkugRtaMgtamG½kSk¾dUcCakugRtaMg circumferential shear stresses EdlmantMélsUnüenARtg;RCugEkgrbs;muxkat; nigRtg;TIRbCMuTMgn; rbs;ctuekaN ehIymantMélGtibrmaenARtg;EpñkkNþalénRCugxag dUcbgðajenAkñúgrUbTI 5>30. kugRtaMgkMlaMgkat;edaysarkarrmUlGtibrma (maximum torsional shearing stress) nwgekItmanenA Shear and Torsion Strength Design 276
    • NPIC Rtg;cMnuckNþal A nig B énTMhMFMCageKrbs;muxkat;. PaBsμúKsμajenHbEnßmelIPaBCak;EsþgEdlmux kat;ebtugGarem: nigebtugeRbkugRtaMgminEmnCasMPar³sac;mYy (homogeneous) b¤CasMPar³esμIsac; (isotropic) EdleFVIeGayvaBi)akbegáItrUbmnþKNitviTüaEdlmanlkçN³suRkitdUcmuxkat;rgVg;. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 277
    • T.Chhay kñúgry³eBlCag 70qñaM karviPaKm:Um:g;rmYl (torsional analysis) rbs;Ggát;ebtugQrelI (1) classical theory éneGLasÞicEdl)anbegáIteLIgtamry³rUbmnþKNitviTüarYmCamYynwg membrane analogy verification (St.-Venant’s) b¤ (2) RTwsþI)aøsÞic (theory of plasticity) EdlENnaMeday sand- heap analogy (Nadal’s). eKGnuvtþRTwsþITaMgBIrCaBiessenAkñúgsßanPaBénrmYlsuT§ (pure torsion). b:uEnþ eKrkeXIjtamry³karBiesaFfaRTwsþIeGLasÞicminbMeBjRKb;RKan;sMrab;karTsSTayy:agsuRkitnUv kugRtaMgenAkñúgebtugEdlrgkarrmYlsuT§. dUcenHeKrkeXIjfakareFVIkarrbs;ebtugeRkamGMeBIm:Um:g;rmYl RtUv)ansMEdg)anl¥CagedayviFI)aøsÞic. x> karrmYlsuT§enAkñúgGgát;ebtugsuT§ Pure Torsion in Plain Concrete Elements 1. karrmYlenAkñúgsMPar³eGLasÞic Torsion in Elastic Materials enAkñúgqñaM 1853 St.-Venant )anbgðajdMeNaHRsayrmYleGLasÞicEdlman warping Edl bNþalBIkarrmYlsuT§EdlekItmanenAkñúgminEmnrgVg;. enAqñaM 1903 Prantl )anbkRsaysar³sMxan;én rUbmnþKNitviTüaeday membran analogy model rbs;Kat;. KMrUenHbegáIteLIgedaymanTMnak;TMngrvag deflected surface rbs; loaded membran nig karEbgEckkugRtaMgenAkñgr)arEdlrgm:Um:g;Bt;. rUbTI ú 5>31 bgðajBI membrane analogy behavior sMrab;TMrg;ctuekaNk¾dUcCaTMrgGkSr L. sMrab;kMhUcRTg;RTaytUc smIkarDIepr:g;Esülrbs; deflected membrane surface manTMrg;dUc nwgsmIkarEdlkMNt;karBRgaykugRtaMgelImuxkat;rbs;r)arEdlrgm:Um:g;Bt;. dUcKña eyIgeXIjfa (1) bnÞat;b:HeTAnwg contour line enARKb;cMnucrbs; deflected membrane pþl;nUvTisedArbs;kugRtaMgkat; enARtg;muxkat;RtUvKñaén membrane Cak;EsþgEdlrgkarrmYl. (2) CMralGtibrmarbs; membran enARKb;cMnucTaMgGs;smamaRteTAnwgTMhMrbs;kugRtaMgkM;laMgkat; τ enARtg;cMnucRtUvKñaenAkñúgGgát;Cak; Esþg ehIy (3) m:Um:g;rmYlEdlGgát;Cak;EsþgrgKWsmamaRteTABIrdgénmaDenAxageRkam deflected membrane. eKGaceXIjBIrUbTI 5>30 nig 5>31(b) fa torsional shearing stress KWRcassmamaRteTAnwg cMgaycenøaH contour lines. ExSkan;EtCit kugRtaMgkan;EtFMEdlnaMkarsnñidæanEdl)anerobrab;BIxag edImfa torsional shearing stress GtibrmaekItmanenARtg;EpñkkNþalénRCugEvgrbs;ctuekaN. BI Shear and Torsion Strength Design 278
    • NPIC membrane analogy kugRtaMgGtibrmaRtUvEtsmamaRteTAnwgCMraleTrCageKénbnÞat;b:HRtg;cMnuc A nig B . RbsinebI δ CabMlas;TIGtibrmaén membrane BIbnÞat;b:HenARtg;cMnuc A enaHBIeKalkarN_ mUldæanénemkanic nigRTwsþIrbs; St.-Venant δ = b 2Gθ (5.45a) Edl G Ca shear modulus nig θ CamMurmYl (angle of twist). b:uEnþ vt (max ) smamaRteTAnwgCMralrbs; bnÞat;b:H dUcenH vt (max ) = k1bGθ (5.45b) Edl k1 CatMélefr. m:Um:g;rmYl (torsional moment) EdlRtUvKñasmamaRteTAnwgBIrdgénmaDEdlenA BIxageRkam membrane karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 279
    • T.Chhay ( ) Teα 2 2 δbh = k 2δbh 3 Edl k2 CatMélefr. b¤müa:geTot Te = k3b 3hGθ (5.45c) Edl k3 CatMélefr. BIsmIkar 5.45a nig b Teb Teb vt (max ) = 3 ≅ (5.45d) kb h J1 PaKEbg kb3h Cam:Um:g;niclPaBb:UElr (polar moment of inernertia) J1 rbs;muxkat;. edayeRbob eFobsmIkarenHeTAnwgsmIkar (a) sMrab;muxkat;rgVg;bgðajnUvPaBdUcKñaénsmIkarTaMgBIr elIkElgEtem KuN k sMrab;muxkat;ctuekaNEdlKitbBa©Úl shear strain EdlbNþalBI warping. eKGacsresr smIkar 5.45d dUcxageRkam Te vt (max ) = (5.46) kb 2 h eKk¾GacsresrCakugRtaMgenARtg;bøg;enAxagkñúgmuxkat; edayctuekaNcaritkñúgrgVg;EdlmanTMhM x nig y Edl x CaRCugxøI. dUcenH Te vt (max ) = (5.47) kx 2 y enAkñúgkareRbI membrane analogy approach eKRtUvcMNaMfa torsional shear stress pøas;bþÚrBIcMnucmYy eTAcMnucmYyeTottambeNþayG½kSEdldUcKñanwgG½kS AB enAkñúgrUbTI 5>31 edaysarkarpøas;bþÚrCMral rbs; analogous membrane nigkarEkERbRbEvgKNna torsional shear stress. 2. karrmYlenAkñúgsMPar³)aøsÞic Torsion in Plastic Materials dUcEdl)anerobrab;BIxagelI plastic sand-heap analogy pþl;karENnaMBIkareFVIkarrbs;sarFatu EdlmanlkçN³RsYy (brittle element) dUcCagFñwmebtugEdlrgkarrmYlsuT§kan;EtRbesICag elastic analogy. m:Um:g;rmYlk¾smamaRteTAnwgBIrdgénmaDenABIeRkamBMnUkxSac;. rUbTI 5>32bgðajBIKMrUkñúg bøg; nigKMrUkñúglMhrrbs;BMnUkxSac;. m:Um:g;rmYl Tp enAkñúgEpñk (d) rbs;rUbsmamaRteTAnwgBIrdgénmaD rbs;BMnUkxSac;ctuekaNEdlbgðajenAkñúgEpñk (b) niig (c). eyIgGaceXIjfaCMralrbs;RCugBMnUkxSac; EdlRtUv)anKitCa torsional shear stress mantMélefr enAkñúg sand-heap analogy approach Etva ERbRbYlenAkñúg membrane analogy approach. Shear and Torsion Strength Design 280
    • NPIC 3. Sand-Heap Analogy Applied to L-Beam Ggát;ebtugPaKeRcInEdlrgkarrmYleRcInCamuxkat;Edlmansøab ehIyCaFmμtaeRcInCaFñwmmux kat;GkSr L EdlpSMeLIgedayFñwmxageRkAénkMralxNÐ. Fñwmmuxkat;GkSr L enAkñúgrUbTI 5>33 RtUv)an erIssMrab;Gnuvtþ plastic sand-heap apperoach edIm,IkMNt;tMéllT§PaBm:Um:g;rmYl nigkugRtaMgkMlaMg kat;EdlvaRtUvrg. BMnUkxSac; (sand heap) RtUv)anbMEbkCamaDbI V1 = BIra:mItEdltMNageGayragkaer = y1bw / 3 2 V2 = EpñkRBIsénRTnugEdltMNageGayragctuekaNEkg = y1bw (h − bw ) / 2 V3 = RBIsEdltMNageGaysøabrbs;FñwmEdlsMedAelIEpñk PDI eTAdl; NQM = y2 h f (b − bw ) / 2 karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 281
    • T.Chhay m:Um:g;rmYlsmamaRteTAnwgBIrdgénmaDrbs;BMnUkxSac; dUcenH ⎡ y b 2 y b (h − b ) y 2 h f (b − bw ) ⎤ Tp ≅ ⎢ 1 w + 1 w w + ⎥2 (5.48) ⎢ ⎣ 3 2 2 ⎥ ⎦ ehIy torsional shear stress k¾smamaRteTAnwgCMralrbs;BMnUkxSac; dUcenH vt bw y1 = (5.49) 2 vt h f y2 = (5.50) 2 edayCMnYs y1 nig y2 BIsmIkar 5.49 nig 5.50 eTAkñúgsmIkar 5.48 eyIg)an Tp vt (max ) = (b2 / 6)(3h − b ) + (h2 / 2)(b − b ) w w f w (5.51) RbsinebIeyIgEckPaKyk nigPaKEbgrbs;smIkar 5.51 CamYynwg (b h)2 eyIg)anw Shear and Torsion Strength Design 282
    • NPIC T p h / (bw h )2 [1 (3 − bw / h)]+ 12 (h f / bw )2(b / h − bw / h) vt (max ) = 6 (5.52a) RbsinebIeKsnμt;fa Ct CaPaKEbgenAkñúgsmIkarenH nig J E = Ct /(bwh)2 enaHsmIkarxagelInwgkøayCa Tph vt (max ) = (5.52b) JE Edl J E Cam:Um:g;niclPaBb:UElrsmmUlEdlGnuKmn_eTAnwgragrbs;muxkat;Fñwm. cMNaMfasmIkar 5.52b manTMrg;RsedogKñaeTAnwgsmIkar 5.45d BI membrane analogy elIkElgtMélrbs;PaKEbgxusKña. eK k¾GacGnuvtþsmIkar 5.52a sMrab;muxkat;ctuekaNedayeGay h f = 0 . eKk¾RtUvTTYlsÁal;pgEdrfaebtugminCasMPar³)aøsÞictex©aHeT dUcenHersIusþg;rmYlCak;Esþg rbs;muxkat;ebtugsuT§mantMélsßitenAcenøaHrvagtMélén membrane analogy nigtMélén sand-heap analogy. eKk¾GacsresrsmIkar 5.52b edaykMNt;eGay Tp = Tc Ca nominal torsional strength rbs;muxkat;ebtugsuT§ ehIy vt (max) = vtc edayeRbInimitþsBaØaEdleRbIeday ACI Edl Tc = k 2b 2 hvtc (5.53a) b¤ Tc = k 2 x 2 yvtc Edl x CaTMhMtUcCageKrbs;muxkat;ctuekaN. tamry³kargarCamYyFñwmebtugGarem:CaeRcIn eKGacyk k2 = 1 / 3 . tMélenH)anmkBIkar BiesaFenAkñúg skew-bending theory rbs;ebtugsuT§. eK)ankMNt;yk 6 f 'c psi(0.5 f 'c MPa) CatMélkMNt;rbs;ersIusþg;rmYlsuT§rbs;Ggát;Edlminmandak;EdkrgrmYl. edayeRbIemKuNkat;bnßy 2.5 sMrab;sñameRbHdMbUg torsional load vtc = 2.4 f 'c psi (0.2 f 'c MPa ) nigedayeRbI k 2 = 1 / 3 enA kñúgsmIkar 5.53 eyIgTTYl)an Tc = 0.8 f 'c x 2 y ¬xñat US¦ (5.54a) Tc = 0.066 f 'c x 2 y ¬xñat IS¦ Edl x CaRCugxøICageKrbs;muxkat;ctuekaN. emKuNkat;bnßy 2.5 RtUv)aneRbIedIm,ITb;Tl;nwgT§iBl m:Um:g;Bt;EdlGacekItman. RbsinebImuxkat;manragGkSr T b¤ragGkSr L eKGacbMEbkRkLaépÞCabgÁúMragctuekaNdUcenA kñúgrUbTI 5>34 Edl Tc = 0.8 f 'c ∑ x 2 y ¬xñat US¦ (5.54b) Tc = 0.066 f 'c ∑ x 2 y ¬xñat IS¦ karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 283
    • T.Chhay 17> karrmYlenAkñúgGgát;ebtugGarem: nigebtugeRbkugRtaMg Torsion in Reinforced and Prestressed Concrete Elements karrmYlkMrnwgekItmanenAkñúgeRKOgbgÁúMebtugEdlminmanrgkarBt; nigkMlaMgkat;. karerobrab; kñúgEpñkxagelIpþl;nUvsBaØaNRKb;RKan;kñúgkarcUlrYmrbs;ebtugsuT§enAkñúgmuxkat;edIm,ITb;Tl;EpñkxøHén bnSMkugRtaMgEdlekItBIkMlaMgrmYl kMlaMgtamG½kS kMlaMgkat; nigkMlaMgBt;. kñúgkrNIPaKeRcIn lT§PaB rbs;ebtugsuT§EdlTb;Tl;karrmYlenAeBlEdlpSMCamYynwgbnÞúkdéTGacmantMéltUcCag enAeBlEdl vaTb;Tl;Etm:Um:g;rmYlxageRkAEdlmanemKuNdUcKña. dUcenH eKRtUvdak;EdkrgkarrmYl (torsional rein- forcement) edIm,ITb;Tl;nwgm:Um:g;rmYlFM. karrab;bBa©ÚlTaMgEdktambeNþay (longitudinal reinforcement) nigEdktamTTwg (tran- sverse reinforcement) edIm,IkarBarEpñkxøHrbs;m:Um:g;rmYlbgðajnUvFatufμIenAkñúgsMnMuénkMlaMg nigm:Um:g; enAkñúgmuxkat;. RbsinebI Tn = nominal torsional resistance srubcaM)ac;énmuxkat; edayrYmbBa©ÚlTaMgEdk Tc = nominal torsional resistance rbs;ebtugsuT§ Ts = torsional resistance rbs;Edk enaH Tn = Tc + Ts (5.55) Shear and Torsion Strength Design 284
    • NPIC eK)anesñIeLIgnUvRTwsþICaeRcInkalBIknøHstvtSmun. RTwsþITUeTAEdlnwgbgðajenATIenHepþateTA elI (a) shew bending theory, (b) space truss analogy theory, (c) compression field theory nig (d) plasticity equilibrium truss theory. elIkElg skew bending theory ecj RTwsþIepSgeTotBI carNarMhUrkMlaMgkat; (shear flow) enAkñúgmuxkat;RbGb;RbehagCaFatuemkñúgkarkMNt;ersIusþg;rmYl rbs;muxkat;tan; nigmuxkat;Rbehag. k> Skew-Bending Theory Skew-bending theory BicarNalMGitBIkMhUcRTg;RTayxagkñúgrbs;sMnMuén transverse warped surface tambeNþayFñwm. dMbUgRTwsþIenHRtUv)anesñIeLIgeday Lessig ehIyCabnþbnÞab;mankarcUlrYm BI Collin, Hsu, Zia, Gesund, Mattock nig Elfgren. Failure surface rbs;muxkat;FñwmEdlrgm:Um:g;Bt; (bending moment) M u enAEtrkSaPaBrab esμIeRkayeBlrgkarBt; dUcbgðajenAkñúgrUbTI 5>35(a). RbsinebIm:Um:g;rmYl (torsional moment) Tu FMCagersIusþg;rbs;muxkat; sñameRbHekItmanenAelIRCugbIrbs;muxkat;Fñwm ehIykugRtaMgsgát;ekItman enAelIEpñkxøHrbs;RCugTIbYntambeNþayFñwm. edaysarbnÞúkrmYlbnþrhUt dl;sßanPaBenAeBl)ak;/ skewed failure surface ekIteLIgedaysarbnSMén torsional moment Tu nig bending moment M u . G½kSNWtrbs; skewed surface nigépÞqUtenAkñúgrUbTI 535(b) EdlkMNt; nUvtMbn;sgát;ElgRtg;eTotehIy nigmanmMu θ CamYynwgmuxkat;bøg;edIm. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 285
    • T.Chhay munnwgeRbH TaMgEdktambeNþay nigEdkkgbiTCitmin)ancUlrYmy:agsMxan;eTAkñúg torsional stiffness rbs;muxkat;eT. enAdMNak;kalénkardak;bnÞúkeRkayeBleRbH PaBrwgRkaj (stiffness) rbs; muxkat;RtUv)ankat;bnßy b:uEnþ torsional resistance rbs;vaRtUv)anekIneLIgy:agxøaMgedayGaRs½yeTA nwgbrimaN nigkarEbgEbkénEdkbeNþay nigEdkkgbiTCit. eKGacTTYl)anersIusþg;rmYlbEnßmbnþic bnþÜcBIersIusþg;énebtugsuT§enAkñúgFñwmTal;EteKeRbIEdkbeNþay nigEdkkg. Skew-bending theory KittMbn;sgát;mankMBs;efr. eKsnμt;fasñameRbHenAelIépÞxagTaMgbI EdlenAsl;rbs;muxkat;ralesμI ehIyEdkkgenARtg;épÞenaHTTYlkMlaMgTajenAeBleRbH ehIyEdk beNþayCamYynwgebtugTb;Tl;nUvkMlaMgkat;tamry³ dowel action. rUbTI 5>36(a) bgðajBIkMlaMg EdleFVIGMeBIelI skewly bent plane. BhuekaNenAkñúgrUbTI 5>36(b) eGayersIusþg;kMlaMgkat; Fc rbs; ebtug/ kMlaMg T L rbs;EdkbeNþayskmμ (active longitudinal steel bar) enAkñúgtMbn;sgát; nig Cc kMlaMgbøúksgát;Fmμta. Shear and Torsion Strength Design 286
    • NPIC m:Um:g;rmYl Tc én resisting shearing force Fc RtUv)anbegáItedayépÞbøúksgát;qUtenAkñúgrUbTI 5>36 (a) KW × édXñas;rbs;vaeFobkMlaMg Fv enAkñúgrUb Fc Tc = cos 45o b¤ Tc = 2 Fc (0.8 x ) (5.56a) Edl x CaRCugxøICageKrbs;muxkat;Fñwm. karBiesaFCaeRcInedIm,IkMNt; Fc EdlCakugRtaMgkñúgrbs;ebtug esμInwg k1 f 'c nig geometrical torsional constant rbs;muxkat; k2 x 2 y eFVIeGaysmIkarxagelI køayCa 2.4 2 Tc = x y f 'c (5.56b) x x> Skew-Bending Theory Space truss analoguy theory RtUv)anbegáIteLIgdMbUgeday Rausch nigeRkaymkmankarcUl rYmeday Lampert, Collin, Hsu, Thulirman, Elfgren nigGñkdéTeTot. RTwsþIenHRtUv)aneFVIeGaykan; EtRbesIeLIgEfmeToteday Rabbat nig Collins enAelI variable angle space truss. Hse )anesñInUvkarpSMrvag equilibrium, compatibility nig softened constitutive law rbs;eb tugeGayeTACaRTwsþIEtmYyEdlGaceGayeK)a:n;sμan)annUvtMélkMlaMgkat; nigkMlaMgrmYlrbs;Fñwmkan; EtsuRkitEdlGacTTYlyk)an. eKeRbIeKalkarN_ shear flow edIm,IbMEbkTMnak;TMngsmIkarsMrab; lMnwgkMlaMgkat;. Space truss analogy CaKMrUEdlRtUv)aneRbIenAkñúgkarsikSaKNna shear-resisting stirruo EdlenAkñúgenaH diagonal tension crack RtUv)anTb;Tl;edayEdkkgenAeBlEdlvacab;epþIm ekIteLIg. edaysarrUbragminenAkñúgbøg; (nonplanar shape) rbs;muxkat;EdlbNþalBIm:Um:g;rmYl/ space truss EdlpSMeLIgedaysarEdkkgRtUv)aneRbICa diagonal tension members ehIy idealized concrete strips Edlman varable angle θ rvagsñameRbHRtUv)aneRbICa compression members (struts) dUcbgðajenAkñúgrUbTI 5>37. eKsnμt;enAkñúgRTwsþIenHfaFñwmebtugrgkugRtaMgrmYleFVIkarRsedogKñanwg thin-walled box Edlman shear flow efr enAkñúg wall cross section edaybegáItm:Um:g;rmYlefr. kareRbI hollow- walled section CagkareRbImuxkat;tan; edaysarEteKcg;bgðajBIkarTTYl)anCaBiessnUv ultimate torsional moment RsedogKña RbsinebIkMras;CBa¢aMgminesþIgeTenaH. karsniñdæanEbbenHekIteLIgenA eBlEdlkarBiesaFbgðajfa torsional strength rbs;muxkat;tan;pSMeLIgBIersIusþg;énRTugEdkkgbiTCit karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 287
    • T.Chhay EdlbegáIteLIgedayEdkbeNþay nigEdkTTwg idealized concrete inclined compression strut enAkñúgbøg;rbs;CBa¢aMgRTug. Compression strut CacMerokebtugeRTtcenøaHsñameRbHenAkñúgrUbTI 5>37. CEB-FIP code KWQrelI space truss model. enAkñúg code enH/ eKykkMras;CBa¢aMgRbsiT§- PaBrbs;FñwmRbehagesμInwg 1 Do Edl Do CaGgát;p©itrbs;rgVg;carwkkñúgctuekaNEdlP¢ab;EdkbeNþay 6 b¤ Do = xo enAkñúgrUbTI 5>37. Casegçb Gvtþmanrbs;sñÚlminmanT§iBldl;ersIusþg;rbs;Ggát;Edl rgkarrmYleT. dUcenH eKGacTTYlyk space truss analogy approach EdlQrelImuxkat;Rbehag. K> Compression Field Theory eKKitfa compression field theory CakrNIBiessrbs; general truss model theory. Elfgren )anesñIeLIgnUv compression field edIm,IBiBN’naBIbgÁúMén plasticity truss model EdlRtUv)an eRbIkñúgeBlbc©úb,nñenAkñúg European Code ehIy Collin nig Mitchell )anEklMGviFIenHedayesñIeLIg nUvBaküEdlnwgerobrab;CabnþbnÞab;. mMueRTt θ én diagonal crack enAkñúgrUbTI 5>37 b¤ compression Shear and Torsion Strength Design 288
    • NPIC strut cenøaH diagonal crack KWmin)anmMu 45o l¥eT b:uEnþeKcUlcitþeRbIEdnkMNt;EdlQrelIépÞrbs; longitudinal tension steel nig transverse toresional web steel (inclined or vertical closed strirrups or ties). rUbTI 5>38 bgðajBIkarBitEdlfakMlaMgrmYlRtUg)anTb;eday tengential component rbs; diagonal compression strut EdlbegáIt shear flow q enACMuvijbrimaRt. edaysnμt;fa ebtugminrgkMlaMgTajeRkayeBleRbH ehIy torsional shear RtUv)anTb;eday Ednén diagonal compression strut eKGackMNt;mMueRTt θ rbs; strut enHCa εl + εd tan 2 θ = (5.57) εt + εd Edl ε l = longitudinal tensile strain enAkñúgEdkem A ε t = transverse tensile strain enAkñúgr)arEdk B karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 289
    • T.Chhay ε d = diagonal compression strain eKGacTTYl)anRkLaépÞ Ao enAkñúgrUbEdlB½T§CMuvijeday shear flow q Ca ao Ao = Aoh − ph (5.58) 2 Edl Aoh = RkLaépÞEdlB½T§CMuvijedayG½kSrbs; hoop ph = brimaRtrbs; hoop edayKitykRtwmG½kSrbs; hoop ao = kMBs;bøúksgát; ¬dUcKñanwgkMBs; a rbs;bøúkctuekaNsmmUlEdlrg flexure¦ Shear and Torsion Strength Design 290
    • NPIC kMras;CBa¢aMgsmmUl td enAkñúgkarviPaKFñwmEdlrgkarrmYlRtUv)anbgðajenAkñúgrUbTI 5>39 ehIykMBs; ao rbs;bøúksgát;RtUv)ankMNt;enAkñúgsmIkar 5.61. Diagonal torsional crack k¾dUc exposed transver ties eRkayeBlkMras;ebtugkarBarEdk rebH (spall) eRkam torsion failure RtUv)anbgðajenAkñúgrUbTI 5>40. eKGackMNt; transverse strain nig longitudinal strain enAkñúgEdkenAeBl nominal torsional moment Tn dUcxageRkam ⎛ 0.85β1 f 'c Ao ⎞ εt = ⎜ ⎜ τ A tan θ − 1⎟0.003 ⎟ (5.59a) ⎝ n oh ⎠ ⎛ 0.85β1 f 'c Ao ⎞ εl = ⎜ ⎜ tan θ − 1⎟0.003 ⎟ (5.59b) ⎝ τ n Aoh ⎠ Edl nominal torsional shear stress KW Tn ph τn = 2 (5.60) Aoh eKGacTTYlRkLaépÞ Ao EdlB½T§eday shear flow BIsmIkar 5.60 nigsmIkarxageRkamsMrab;kMBs;bøúk sgát; ao EdlrgkarrmYledayGnuelamtam Collin nig Mitchell Aoh ⎡ Tn ph ⎛ 1 ⎞⎤ ao = ⎢1 − 1 − ⎜ tan θ + ⎟⎥ (5.61) ph ⎢⎣ 0.85 f 'c Aoh ⎝ 2 tan θ ⎠⎥ ⎦ karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 291
    • T.Chhay Edl Tn Ca nominal torsional moment strength enARtg;sßanPaBkMNt;enAeBl)ak;. eKKYr kt;cMNaMfasmIkar 5.58, 5.59, 5.60, 5.61 nig 5.63 KWEp¥kelIkarsnμt; (1) karrebH (spalling) rbs; RsTab;ebtugkarBarEdk nig(2) edayeRbI non-softened stress-strain curve rbs;ebtug. karsnμt; TaMgenHemIleTAminCaRtwmRtUveTedayBicarNakarrmYlCak;Esþgrbs;Ggát;ebtug. sMrab;bnSMkarrmYl nigkarkat;/ kugRtaMgkat;enARtg; nominal strength Tn nig Vn enAkñúg smIkar 5.61 køayCa Tn ph Vn − V p τn = 2 + (5.62) Aoh bv d v Edl Vp = bgÁúMkMlaMgkugRtaMgbBaÄr bv = TTwgRTnugRbsiT§PaBGb,brmaenAkñúgkMBs;kMlaMgkat; d v eRkayeBl spalling RsTab; karBar. dkGgát;p©itrbs; duct BITTwgRTnugRbsinebI ungrouted tendon b¤dkBak; kNþalGgát;p©itrbs; duct sMrab; grouted tendon. d v = kMBs;kMlaMgkat;RbsiT§PaB. eKGacKitvaCa flexural lever arm b:uEnþminRtUvtUcCagcMgay bBaÄrcenøaHG½kSrbs; bars b¤ prestressing tendon enARtg;kac;RCugrbs;Edkkg. tMél)a:n;sμanénmMueRTtrbs; compressive strut θ enAkñúgrUbTI 5>38 sßitenAcenøaH 24o sMrab; karrmYlsuT§ nig 90o sMrab;karBt;begáagsuT§. dUcenH eKeRCIserIsyktMél θ tUcCageKsMrab;kMlaMg rmYl eFVIeGayeKRtUvkar hoop steel kan;Ettic nignaMeGayeKRtUvkarRkLaépÞEdkbeNþaykan;EteRcIn. edaysarEdkkgéføCagEdkbeNþay enaHCMerIstMél θ kan;EttUcKWeKnwgTTYl)anlkçN³esdækic©kan; EtRbesIkñúgkarKNna. Compression field theory snμt;faeKeRCIserIslkçN³FrNImaRtrbs; designed section elImUldæanén yielding rbs;Edkkg nigEdkbeNþaymunnwgekItmankarEbktamGgát;RTUgrbs;ebtug. dUcenH transverse strain ε1 enAkñúgsmIkar 5.57 nig 5.58a KYrKitCa yield strain ε ty . eKGackMNt;mMurbs; compression strut θ EdlKitCadWeRkdUcxageRkam 35(τ n / f 'c ) 35(τ n / f 'c ) 10 + < θ < 80 − (5.63) 0.42 − 50ε l 0.42 − 65ε ty eKKYrcMNaMfaenAkñúg compression field theory eKsnμt;kugRtaMgTajemesμInwgsUnüeRkayeBl ebtugeRbH. Collin nig Mitchell Edl)anesñInUv modified compression field theory KitbBa©ÚlnUvkar cUlrYmrbs;EdkTajenAkñúgebtugcenøaHsñameRbH. ACI Code approach snμt;mMueRTt θ efr EdlenA kñúgenaH θ = 45o sMrab;ebtugGarm: nigθ = 37.5o sMrab;ebtugeRbkugRtaMg. Shear and Torsion Strength Design 292
    • NPIC X> Plasticity Equilibrium Truss Theory Hsu )anesñIeLIgnUv equilibrium, compatibility nig softene constitutive law rbs;ebtugenA kñúgRTwsþIEtmYyEdlGacTajnUvkugRtaMgkat; nigkugRtaMgrmYlrbs;FñwmEdlmanlkçN³suRkitGacTTYl yk)an. eKeRbI shear flow concept kñúgkarbMEbksmIkarEdlTak;TgsMrab; shear equilibrium. !> Equilibrium in Element Shear eKmanFatukaer:mYyEdlmankMras; t rgnUv shear flow q EdlbNþalmkBIkMlaMgkat;suT§Edl bgðajenAkñúgrUbTI 5>41. TaMgEdkenAkñúgTisbeNþay l TaMgEdkenAkñúgTisTTwg t KWsuT§EtrgkugRtaMg Ékta fl / sl nig fv / s erogKña EdleKGackMNt; shear flow q edaysmIkarlMnwg q = (Fl ) tan θ (5.64a) Edl Fl = Al f l / sl nig q = (Ft ) cot θ (5.64b) Edl Ft = At fv / s Al nig At CaRkLaépÞmuxkat;rbs;Edk ehIy sl nig st CaKMlatkñúgTis l nigTis t erogKña. BIragFrNImaRtrbs;RtIekaNenAkñúgrUbTI 5>41 eKGackMNt; shear flow eday q = ( f D t ) sin θ cos θ (5.65) karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 293
    • T.Chhay RbsinebIeKsnμt;EdkkñúgTisTaMgBIreFVIkardl; yield enaHsmIkar 5.64a nig 5.65 eGay Fty tan θ = (5.66a) Fly nig q y = Fly Fty (5.66b) EdlGkSr y mann½y yielding rbs;Edk. @> Equilibrium in Element Torsion rUbTI 5>42 bgðajBITIbRbehag nigmankMras;ERbRbYlEdlrgnUvkMlaMgrmYlsuT§. RTwsþI St.- Venant snμt;faragrbs;muxkat;enArkSadEdleRkamkMhUcRTg;RTayeGLasÞictUc ehIy warping deformation EdlEkgnwgmuxkat;KYrmantMéldUcKñatambeNþayG½kSrbs;Ggát;. dUcenH eKGacsnμt;fa manEtkugRtaMgkat;eTEdlekItmanenAkñúgCBa¢aMgRbGb;kñúgTMrg; shear flow q dUcbgðajenAkñúgrUbTI 5>42 a nigsnμt;fa kugRtaMgEkgkñúgbøg; (in-plane normal stresses) enAkñúgCBa¢aMgRtUv)anecal. Rb sinebIFatuGnnþtUcrbs;CBa¢aMg ABCD RtUv)anpþac;ecjdUcbgðajenAkñúgrUbTI 5>42 b, shear flow enAkñúg TisedA l RtUvesμInwg shear flow enAkñúgTisedA t b¤ τ l t1 = τ t t 2 (5.67) Shear and Torsion Strength Design 294
    • NPIC enAelIeKalkarN_enH eKKit shear flow q mantMélefrBaseBjmuxkat;. kMlaMgrmYlenAelIcMgay GnnþtUc dt tambeNþayKnøg shear flow KW qdt dUcenH torsional resistance rbs;m:Um:g;rmYlxag eRkA T enAkñúgrUbTI 5>42 a køayCa T = q ∫ rdt (5.68) BIrUbTI 5>42 a eyIgeXIjfa rdt enAkñúgGaMgetRkal;esIμnwgBIrdgénRkLaépÞrbs;RtIekaNEdlqUtEdl pÁúMeLIgeday r nig dt . RkLaépÞsrubCMuvijmuxkat;KW ∫ rdt = 2 Ao (5.69) Edl Ao = RkLaépÞmuxkat;EdlhMuB½T§eday shear flow center line. edayCMnYs 2 Ao eTAkñúgsmIkar 5.68 eKTTYl)an T q= (5.70) 2 Ao edayminKit warping, shear element EdlrgkarrmYlsuT§enAkñúgCBa¢aMgTIbrbs;rUbTI 5>42 a mantMél dUcKñaeTAnwg membrane shear element enAkñúgrUbTI 5>41 a. dUcenH edayCMnYs shear flow q Edl)anBIsmIkar 5.70 eTAkñúg 5.64 a, b nig 5.65 eK)ansmIkarlMnwgbIsMrab;karrmYldUcxageRkam T= Fl (2 Ao ) tan θ (5.71a) po Edl F l = Fl po ehIy po = brimaRtrbs;Knøg shear flow. F l CakMlaMgtambeNþaysrubEdl bNþalBIkarrmUl T = Ft (2 Ao ) cot θ (5.71b) T = ( f D t )(2 Ao ) sin θ cos θ (5.71c) eKGacsresrsmIkar 5.71 b enAeBl yield Ca 2 Ao At f yv Tn = cot θ (5.72) s Edl Tn CaersIusþg;m:Um:g;rmYlGtibrma. EdkrgkarrmYlcaM)ac;enAkñúgTisbeNþay nigTisTTwgkøayCa Tn s At = (5.73) 2 Ao f yv cot θ Al1 = At s ⎛ f yv ⎞ ⎜ ( ⎟ s cot 2 θ ⎜ f yl ⎟ l ) (5.74a) ⎝ ⎠ Edl Al = RkLaépÞsrubrbs;EdkrgkarrmYltambeNþayenAkñúgmuxkat; karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 295
    • T.Chhay #> Shear-Torsion-Bending Interaction BicarNaRbGb;ctuekaNenAkñúgrUbTI 5>37 nig 5>43. Shear flow q nwgmindUcKñaenAelICBa¢aMg TaMgbYnrbs;RbGb;enAeBlEdlrgbnSMkarkat; nigkarrmYldUceXIjenAkñúgrUbTI 5>43. kar)ak;Gacnwg ekIteLIgtamBIrEbb³ (a) EdkbeNþayrgkarTajxageRkam nigEdkkgTTwgeFVIkardl; yield (b) EdkbeNþayrgkarsgát;xagelI nigEdkkgTTwgeFVIkardl; yield EdkTajxageRkameFVIkardl; yield (a) RbsinebIkar)ak;enHekIteLIgedaysarEdkbeNþayxageRkam nigEdkkgeFVIkardl; yield edaysarbnSMénkugRtaMgkat; nigkugRtaMgrmYl eKGacbMEbksmIkarxageRkambIlkçxNÐlMnwg)andUcxag eRkam³ ( yo + xo ) 2 2 M ⎛ V ⎞ yo s ⎛ T ⎞ s ⎜ 2y ⎟ F A f + ⎜ 2A ⎟ +⎜ =1 (5.75) Fb yo ⎝ o ⎠ B t v ⎜ o ⎟ ⎟ ⎝ ⎠ FB At f v RbsinebI M o / Vo nig To Cam:Um:g; nigkMlaMgEdlmanGMeBIEtÉg eKGackMNt;BYkva)andUcxageRkam³ M o = FB yo (5.76a) ⎛F ⎞ At f v Vo = 2 yo ⎜ T ⎜y ⎟ ⎟ s sMrab;muxkat;RbGb; (5.76b) ⎝ o ⎠ ⎛ 2F ⎞ At f v To = 2 Ao ⎜ T ⎜ p ⎟ ⎟ s (5.76c) ⎝ o ⎠ Edl po = 2( yo + xo ) Shear and Torsion Strength Design 296
    • NPIC FT R= (5.76d) FB eKGacTTYl)an nondimensional interaction surface relationship edaybBa©ÚlsmIkar 5.76 eTAkñúg smIkar 5.75 2 2 ⎛ M ⎞ ⎛V ⎞ ⎛T ⎞ ⎜ ⎟ +⎜ ⎟ R +⎜ ⎟ R =1 ⎜ M ⎟ ⎜V ⎟ ⎜T ⎟ (5.77a) ⎝ o⎠ ⎝ o⎠ ⎝ o⎠ Edksgát;xagxagelIeFVIkardl; yield (b) RbsinebIkar)ak;enHekIteLIgedaysarEdkbeNþayxagelI ¬Edkrgkarsgát;¦ nigEdkkg eFVIkardl; yield enaHsmIkar 5.77 a nwgkøayCa 2 2 ⎛ M ⎞1 ⎛V ⎞ ⎛T ⎞ −⎜ ⎜ M ⎟ R + ⎜V ⎟ + ⎜T ⎟ = 1 ⎟ ⎜ ⎟ ⎜ ⎟ (5.77b) ⎝ o⎠ ⎝ o⎠ ⎝ o⎠ BIsmIkar 5.77a nig 5.77b Gnþrkmμrbs; V nig T manragCasmIkarrgVg;Edlmanm:Um:g;Bt; M efr sMrab;épÞ)ak;TaMgBIr. karRbsBVKñarvagépÞ)ak;TaMgBIrsMrab;kar)ak;TaMgBIrenHbegáIt)ancMnucx<s;bMput enAelIExSekagGnþrkmμrvag V nig T EdlsmIkar 5.77a nig 5.77b eGay 2 2 ⎛V ⎞ ⎛T ⎞ 1+ R ⎜ ⎟ +⎜ ⎟ = ⎜V ⎟ ⎜T ⎟ (5.78a) ⎝ o⎠ ⎝ o⎠ 2R smIkar 5.78 sMrab; R = 0.25 / 0.5 nig1 enAelIbøg;x<s;bMputeGaydüaRkamragrgVg;dUcbgðajenAkñúg rUbTI 5>44. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 297
    • T.Chhay kar)ak;tamrebObTIbIKWbNþalmkBIEdkxagelI EdkxageRkam nigEdkkgeFVIkardl; yield ehIyeKRtUvbEnßm shear flow EdlbNþalBIkMlaMgkat; nigm:Um:g;rmYlenAelIRKb;RCugTaMgGs;. smIkar 5.78a RtUv)anEksMrYldUcxageRkam 2 2 ⎛V ⎞ ⎛T ⎞ ⎛ VT ⎞ 1 + R ⎜ ⎟ + ⎜ ⎟ + 2⎜ ⎜V ⎟ ⎜ T ⎟ ⎜ V T ⎟ = 2R ⎟ (5.78b) ⎝ o⎠ ⎝ o⎠ ⎝ o o⎠ ersIusþg;m:Um:g;rmYlemKuN φTn RtUvFMCag b¤esμIm:Um:g;rmYlxageRkA Tu EdlbNþalmkBIbnÞúkemKuN. kñúg karKNna Tn (ACI 318-99) eKsnμt;eGaykMlaMgrmYlTaMgGs;RtUv)anTb;Tl;edayEdkkgbiTCit nig EdkbeNþay ehIysnμt;fa torsional moment Tc EdlTb;Tl;eday concrete compression strut man tMélesμIsUnü. enAkñúgeBlCamYyKña eKsnμt;fakMlaMgkat; Vc EdlTb;Tl;edayebtugminpøas;bþÚreday sarvtþmanrbs;m:Um:g;rmYleT. karsMrYlEbbenHkat;bnßyPaBlM)ak nigTMhMénsmIkarGnþrkmμsMrab; V / T nig M EdleRbIenAkñúgkUdBImun. Casegçb EdkkgsMrab;kMlaMgkat;RtUv)ankMNt;edaytMélrbs; Vs = Vn − Vc ÉEdkkgsMrab;m:Um:g;rmYlRtUv)ankMNt;edaytMélrbs; Tn Etb:ueNÑaH Edl Tn = Tu / φ ehIy φ = 0.85 . g> Design of Prestressed Concrete Beams Subjected to Combined Torsion, Shear, and Bending in Accordance with the ACI 319-99 Code edayeFVIkarEksMrYl equilibrium truss model éncMnuc X/ cMnucbnþbnÞab;Ca ACI 318 Code provision sMrab;karsikSaKNnaEdkbeNþay nigEdkTTwgenAkñúgGgát;eRbkugRtaMg. 1. Compatibility Torsion enAkñúgRbB½n§sþaTIcminkMNt; (statically indeterminate system) karsnμt;PaBrwgRkaj (stiff- ness), PaBRtUvKñμarbs;bMErbMrYlrageFob (compatibility of strain) enARtg;tMN nigkarEbgEckkugRtaMg eLIgvijGacCHT§iBldl;kugRtaMgpÁÜbEdlnaMeTAdl;karkat;bnßy torsional shearing stress. eK GnuBaØateGaymankarkat;bnßyenAkñúgtMélrbs;m:Umg;emKuNEdleRbIenAkñúgkarKNnaGgát; RbsinebI : Epñkénm:Um:g;enHGacRtUv)anEbgEckeTAkan;Ggát;EdlRbsBVKña. ACI Code GnuBaØat torsional moment emKuNGtibrmaenARtg;muxkat;eRKaHfñak; h / 2 BIépÞrbs;TMrsMrab;Ggát;ebtugeRbkugRtaMgxageRkam ⎛ Acp ⎞ 2 Tu = φ 4 f 'c ⎜ ⎜p ⎟ ⎟ 1+ f 4 f' ¬xñat US¦ (5.79) ⎝ pc ⎠ c Shear and Torsion Strength Design 298
    • NPIC φ f 'c ⎛ Acp ⎞ 2 Tu = 3 ⎜ ⎟ ⎜ pcp ⎟ 1+ 3f f 'c ¬xñat SI¦ ⎝ ⎠ Edl RkLaépÞEdlB½T§CMuvijedaybrimaRtxageRkArbs;muxkat;ebtug = xo yo Acp = pcp = brimaRtxageRkArbs;muxkat;ebtug Acp / = 2( xo + yo ) f = kugRtaMgsgát;mFümenAkñúgebtugenARtg;G½kSTIRbCMuTMgn;EdlbNþalEtBIkugRtaMgRbsiT§ PaBeRkayBIkMhatbg;. ACI Code kMNt; f c Ca f pc . karminKitT§iBleBjeljéntMélsrubrbs;m:Umg;rmYlxageRkAenAkñúgkrNIenH minnaMeTAdl; : kar)ak;rbs;eRKOgbgÁúMeT b:uEnþvaGacbgáeGaymansñameRbHFMRbsinebI 4φ f 'c (Acp / pcp ) sMrab;xñat 2 US b¤ φ f 'c (Acp / pcp )/ 3 sMrab;xñat SI mantMéltUcCagm:Um:g;rmYlemKuNCak;EsþgxøaMgeBkenaH. 2 RbsinebIkMlaMgrmYlemKuNCak;EsþgtUcCagGVIEdleGayenAkñúgsmIkar 5.79 enaHeKRtUvKNna FñwmsMrab;tMélkMlaMgrmYlEdltUcCag. b:uEnþsMrab;ebtugeRbkugRtaMg eKecal torsional moment Rbsin ebI ⎛ Acp ⎞ 2 Tu < φ f 'c ⎜ ⎜p ⎟ ⎟ 1+ f c 4 f' ¬xñat US¦ (5.80) ⎝ cp ⎠ c φ f 'c ⎛ Acp ⎞ 2 Tu < ⎜ 12 ⎜ pcp ⎟ ⎟ 1+ 3 f f 'c ¬xñat SI¦ ⎝ ⎠ 2. Torsional Moment Strength eKeRCIserIsTMhMmuxkat;edayQrelIeKalkarN_énkarkat;bnßysñameRbH nigkarBarkarpÞúHEbk épÞebtugEdlbgáedaykugRtaMgsgát;eRTtEdlbNþalBIkMlaMgkat; nigkMlaMgrmYlEdlkMNt;edayGgÁ xageqVgrbs;smIkar 5.81. TMhMFrNImaRtsMrab;ersIusþg;m:Um:g;rmYlenAkñúgebtugGarem: nigebtugeRbkug RtaMgRtUv)ankMNt;edaysmIkarxageRkam (a) muxkat;tan; 2 2 ⎛ Vu ⎞ ⎛ Tu ph ⎞ ⎛ Vc ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ b d ⎟ + ⎜ 1 .7 A 2 ⎟ ≤ φ ⎜ b d + 8 f ' c ⎟ ⎜ ⎟ ¬xñat US¦ (5.81) ⎝ w ⎠ ⎝ 0h ⎠ ⎝ w ⎠ 2 2 ⎛ Vu ⎞ ⎛ Tu ph ⎞ ⎛ Vc ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 2 ⎜ b d ⎟ + ⎜ 1 .7 A 2 ⎟ ≤ φ ⎜ b d + 3 f 'c ⎟ ⎟ ¬xñat SI¦ ⎝ w ⎠ ⎝ 0h ⎠ ⎝ w ⎠ (b) muxkat;Rbehag karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 299
    • T.Chhay ⎛ Vu ⎞ ⎛ Tu ph ⎞ ⎛ Vc ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ b d ⎟ + ⎜ 1.7 A2 ⎟ ≤ φ ⎜ b d + 8 f 'c ⎟ ⎜ ⎟ ¬xñat US¦ (5.82) ⎝ w ⎠ ⎝ 0h ⎠ ⎝ w ⎠ ⎛ Vu ⎞ ⎛ Tu ph ⎞ ⎛ Vc ⎞ ⎜ ⎜ ⎟ 2 ⎜ b d ⎟ + ⎜ 1.7 A2 ⎟ ≤ φ ⎜ b d + 3 f 'c ⎟ ⎟ ⎜ ⎟ ¬xñat SI¦ ⎝ w ⎠ ⎝ 0h ⎠ ⎝ w ⎠ E;dl A0h = RkLaépÞEdlB½T§CMuvijG½kSrbs; transverse torsional reinforcemet xageRkAeK bMput ph = brimaRtrbs; transcerse torsional reinforcement xageRkAeKbMput edayKitBI G½kSEdk RkLaépÞ A0h sMrab;muxkat;epSg²EdleGayenAkñúgrUbTI 5>45/ rUbTI 5>46 nigrUbTI 5>47 pþl; nUvkarENnaMedIm,IkMNt;RkLaépÞ A0h nigRkLaépÞ shear flow A0 ≅ 0.85 A0h enAkñúgsmIkar 5.84(a). Shear and Torsion Strength Design 300
    • NPIC plbUkénkugRtaMgenAGgÁxageqVgrbs;smIkar 5.82 minKYrFMCagkugRtaMgEdlbNþaleGayman shear cracking bUknwg 8 f 'c psi (0.664 f 'c MPa ) . vamanlkçN³RsedogKñanwg limiting strength Vs ≤ 8 f 'c psi (0.664 f 'c MPa ) sMrab;kMlaMgkat;EdlKμankMlaMgrmYl. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 301
    • T.Chhay 3. Hollow Section Wall Thickness kugRtaMgkat;EdlbNþalBIkMlaMgkat; nigkMlaMgrmYlekItmanenAkñúgCBa¢aMgrbs;muxkat;Rbehag dUceXIjenAkñúgrUbTI 5>48 a. cMNaMfaenAkñúgmuxkat;tan; kugRtaMgkMlaMgkat;EdlbNþalBIkarrmYlenA EtRbmUlpþúMenARtg;tMbn;xageRkArbs;muxkat;dUceXIjenAkñúgrUbTI 5>48b. RbsinebIkMras;CBa¢aMgrbs;muxkat;RbehagERbRbYl eKRtUvkMNt;muxkat;FrNImaRtenARtg;TItaMg EbbNaEdleFVIeGayGgÁxageqVgrbs;smIkar 5.82 mantMélGtibrma. dUcKña RbsinebIkMras;CBa¢aMg t < A0h / ph eKRtUvykGgÁxageqVgrbs;smIkar 5.82 esμInwg ⎛ Vu ⎞ ⎛ Tu ⎞ ⎜ b d ⎟ + ⎜ 1 .7 A t ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ w ⎠ ⎝ 0h ⎠ kMras;CBa¢aMg t CakMras;EdleKRtUvRtYtBinitükugRtaMg. ⎛ V d⎞ Vc = ⎜ 0.6λ f 'c + 700 u ⎟bw d ⎜ M ⎟ ¬xñat US¦ ⎝ u ⎠ ( Vc ≥ 1.7λ f 'c bw d ) ¬xñat US¦ Vc ≤ (5λ ) f 'c bw d ¬xñat US¦ (5.83) ⎛ V d⎞ Vc = ⎜ λ f 'c / 20 + 5 u ⎟ ⎜ Mu ⎟ ¬xñat SI¦ ⎝ ⎠ ( Vc ≥ 0.17λ f 'c bw d ) ¬xñat SI¦ Vc ≤ (0.4λ ) f 'c bw d ¬xñat SI¦ nig Vu d Mu ≤ 1.0 Shear and Torsion Strength Design 302
    • NPIC 4. Torsional Web Reinforcement eKGacTTYl)anersIusþg;rmYlbEnßmEdlbNþalBIEdkrgkarrmYlbEnßmedayeRbIEdkkg nigEdk beNþay. eKKYreRbIEdkkg nigEdkbeNþayEdlmanmaDesμIKñaedIm,ITb;Tl;nwgm:Um:g;rmYl. eKalkarN_ enHCaeKalkarN_rbs;smIkar ACI sMrab;kMNt; torsinal web steel. RbsinebI s CaKMlatEdkkg/ Al CaRkLaépÞEdkbeNþaysrub nig At CaRkLaépÞsMrab;eCIgmYyrbs;Edkkg. EdkTTwgsMrab;Tb;Tl;RtUv QrelIersIusþg;m:Um:g;rmYlxageRkAeBjelj Tn = (Tu / φ ) Edl 2 A0 At f yv Tn = cot θ (5.84a) s ¬emIlkarbMEbksmIkar 5.72¦ A0 = gross area B½T§CMuvijeday shear flow path At = RkLaépÞmuxkat;éneCIgmYyrbs;EdkkgbiTCit f yv = yield strength rbs; closed transverse torsional reinforcement ehIyvaminRtUvFMCag 60ksi(414 MPa ) θ= mMurbs; compression diagonal (struts) enAkñúg space truss analogy sMrab;karrmYl ¬emIlrUbTI 5>39¦ edaypøas;tYrbs;smIkar 5.84a/ RkLaépÞEdkkgkøayCa At Tn = (5.48b) s 2 A0 f yv cot θ eKKYrkMNt;RkLaépÞ A0 edaykarviPaK EtelIkElgfa ACI 318 Code GnuBaØateGayyk A0 = 0.85 A0h CMnYseGaykarviPaK. ersIusþg;rmYlemKuN φTn RtUvFMCag b¤esμInwgm:Um:g;rmYlemKuNxageRkA Tu . enAkñúg ACI 318- 99 code )ansnμt;fam:Um:g;rmYlTaMgGs;RtUv)anTb;Tl;edayEdkkgbiTCit nigEdkbeNþayedayminKit ersIusþg;rmYlrbs;ebtug Tc = 0 . kMlaMgkat; Vc RtUv)anTb;Tl;edayebtugedaysnμt;KμankarERbRbYl edaysarvtþmanrbs;m:Um:g;rmYl. eKminRtUvyktMélmMu θ EdlpÁúMeday concrete compression diagonal strut tUcCag 30o b¤FM Cag 60o eT. eKGacTTYl)anvaedaykarviPaKEdleFVIeLIgeday Hsu. EdkbeNþaybEnßmsMrab;m:Um:g; rmYlminKYrtUcCag At ⎛ f yv ⎞ 2 Al = ph ⎜ ⎟ cot θ (5.85) s ⎜ f yt ⎟ ⎝ ⎠ karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 303
    • T.Chhay Edl f yt = yield strengthrbs;EdkTb;karrmYlbeNþay vaminRtUvFMCag 60ksi = 414MPa . eKRtUveRbImMu θ dUcKñaTaMgenAkñúgsmIkar 5.84 TaMgenAkñúgsmIkar 5.85. eKKYrkt;cMNaMfa enA eBlEdleKyktMél θ kan;EttUc brimaNEdkkgEdlTamTaredaysmIkar 5.84 kan;EtfycuH. enA kñúgeBlCamYyKñaenaH brimaNEdkbeNþayEdlTamTaredaysmIkar 5.85 nwgekIneLIg. edIm,IeCosvagkarKNnamMu θ edaykarviPaK ACI Code GnuBaØateGayyktMélmMu θ esμInwg (i) 45o sMrab;Ggát;EdlminrgeRbkugRtaMg b¤Ggát;EdlmaneRbkugRtaMgtUcCagenAkñúg (ii) (ii) 37.5o sMrab;Ggát;eRbkugRtaMgEdlmankMlaMgeRbkugRtaMgRbsiT§PaBFMCag 40% énersIusþg; Tajrbs;EdkbeNþay. PCI )anENnaMeGayKNnatMélmMu θ BIsmIkar Tu / φ cot θ = (5.86) 1.7 A0h ( At / s ) f yv 5. Minimum Torsional Reinforcement eKcaM)ac;RtUvdak; torsional reinforcement Gb,brmaenARKb;tMbn;TaMgGs;Edl factored torsional moment θ FMCagtMélEdleGayedaysmIkar 5.80. enAkñúgkrNIEbbenH EdkkgTTwgbiT CitGb,brma EdlRtUvkarCatMéltUcCageKkñúgcMeNamKW Avt = Av + 2 At ≥ 50bw s f ¬xñat US¦ (5.87) yv Avt = Av + 2 At ≥ 0.35bw s f yv ¬xñat SI¦ Aps f pu nig Avt ≥ 80 f y d d bw KMlatGtibrmaminRtUvFMCagtMéltUcCageKén pn / 8 b¤12in.(30cm) . eKRtUvKNnaRkLaépÞsrubGtibrmarbs; longitudinal torsional reinforcement bEnßmeday 5 f 'c Acp ⎛ At ⎞ f yv Al , min = f − ⎜ ⎟ ph ⎝ s ⎠ f ¬xñat US¦ (5.88) yl yl 5 f 'c Acp ⎛A ⎞ f yv Al , min = 12 f yl − ⎜ t ⎟ ph ⎝ s ⎠ f yl ¬xñat SI¦ eKminRtUvyk At / s tUcCag 25bw / f yv eT. eKRtUvBRgayEdkbeNþaybEnßmEdlRtUvkarsMrab;Tb;Tl; karrmYlCMuvijbrimaRtrbs;EdkkgbiTCitCamYynwgKMlat 12in.(30cm) . eKRtUvdak;kabeRbkugRtaMgenA xagkñúgEdkkg ehIyEdkeRbkugRtaMgy:agticmYyRtUv)andak;enAkac;RCugnImYy²rbs;Edkkg. EdkeRb Shear and Torsion Strength Design 304
    • NPIC kugRtaMgRtUvmanGgát;p©ity:agtic 1 / 16 énKMlatEdkkg b:uEnþminRtUvtUcCagEdk #3 eT. eKRtUvbnøay EdkTb;Tl;karrmYlsMrab;cMgayGb,brma (bt + d ) ecjBIcMnucEdlRtUvkaredayRTwsþIsMrab;karrmYl BIeRBaH torsional diagonal crack ekItmankñúgTMrg; helical EdllanecjBIsñameRbHEdlekIteLIg edaysarkMlaMgkat; nigkarBt;. bt CaTTwgrbs;Epñkénmuxkat;EdlmanEdkkgsMrab;Tb;Tl;karrmYl. muxkat;eRKaHfñak;rbs;FñwmKWenARtg;cMgay d BIépÞrbs;TMrsMrab;Ggát;ebtugGarem: nigenARtg; h / 2 sMrab; Ggát;ebtugeRbkug Edl d CakMBs;RbsiT§PaB nig h CakMBs;srubrbs;muxkat;. 18> CMhankñúgkarsikSaKNnasMrab;bnSMénkarrmYl nigkarkat; Design Procedure for Combined Torsion and Shear xageRkamCakarsegçbénCMhansikSaKNnaEdl)anENnaMCalMdab;. rUbTI 5>49 bgðajBI flowchart EdlBN’naBIlMdab;énkarsikSaKNnatamTMrg;RkahVik. !> eFVIcMNat;fñak;rbs;m:Um:g;rmYlCa equilibrium torsion b¤ compatibility torsion. kMNt;muxkat; eRKaHfñak; nigkMNt;m:Um:g;rmYlemKuN Tu . muxkat;eRKaHfñak;RtUv)anKitenARtg; h / 2 BIépÞrbs; TMrsMrab;FñwmebtugeRbkugRtaMg. RbsinebI Tu tUcCag φ f 'c (Acp / pcp ) 1 + f c / 4 f 'c ¬xñat 2 US¦ b¤ ⎡φ f 'c (Acp / pcp ) 1 + 3 f c / f 'c ⎤ / 12 ¬xñat SI¦ sMrab;Ggát;eRbkugRtaMg eKminRtUv 2 ⎢ ⎣ ⎥ ⎦ KitT§iBlkMlaMgrmYleT. f c CakugRtaMgsgát;enAkñúgebtugeRkayeBlkMhatbg;enARtg;TIRbCMu TMgn;rbs;muxkat;EdlTb;Tl;nwgbnÞúkGnuvtþn_xageRkA ¬enAkñúg ACI Code eKeRbItY f pc CMnYs vij¦. @> RtYtBinitüfaetIm:Um:g;rmYlemKuN Tu bgáCa equilibrium torsion b¤Ca compatibility torsion. sMrab; compatibility torsion, design torsional moment RtUv)ankMNt;GaytUcCagm:Um:g; Tu Cak;Esþg b¤ Tu = φ 4 f 'c (Acp / pcp ) 1 + f c / 4 f 'c ¬xñat US¦ b¤ ¬xñat SI¦ Tu = 2 3φ f 'c (Acp / pcp ) 1 + 3 f c / f 'c sMrab;Ggát;ebtugeRbkugRtaMg. tMélrbs; design 2 nominal strength Tu RtUvsmmUleTAnwg Tu / φ Edl (a) sMrab;muxkat;tan; 2 2 ⎛ Vu ⎞ ⎛ Tu ph ⎞ ⎛ ⎞ ⎜ ⎟ +⎜ ⎟ ≤ φ ⎜ Vc + 8 f 'c ⎟ ⎜ b d ⎟ ⎜ 1 .7 A 2 ⎟ ⎜b d ⎟ ¬xñat US¦ ⎝ w ⎠ ⎝ 0h ⎠ ⎝ w ⎠ 2 2 ⎛ Vu ⎞ ⎛ Tu ph ⎞ ⎛ Vc ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ 2 ⎜ b d ⎟ + ⎜ 1 .7 A 2 ⎟ ≤ φ ⎜ b d + 3 f 'c ⎟ ⎟ ¬xñat SI¦ ⎝ w ⎠ ⎝ 0h ⎠ ⎝ w ⎠ karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 305
    • T.Chhay (b) muxkat;Rbehag ⎛ Vu ⎞ ⎛ Tu ph ⎞ ⎛ Vc ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ b d ⎟ + ⎜ 1.7 A2 ⎟ ≤ φ ⎜ b d + 8 f 'c ⎟ ⎜ ⎟ ¬xñat US¦ ⎝ w ⎠ ⎝ 0h ⎠ ⎝ w ⎠ ⎛ Vu ⎞ ⎛ Tu ph ⎞ ⎛ Vc ⎞ ⎜ ⎜ ⎟ 2 ⎜ b d ⎟ + ⎜ 1.7 A2 ⎟ ≤ φ ⎜ b d + 3 f 'c ⎟ ⎟ ⎜ ⎟ ¬xñat SI¦ ⎝ w ⎠ ⎝ 0h ⎠ ⎝ w ⎠ Shear and Torsion Strength Design 306
    • NPIC RbsinebIkMras;CBa¢aMgtUcCag A0h / ph enaHeKKYryktYTIBIrrbs;GgÁTImYyesμInwg Tu / 1.7 A0ht . ⎛ V d⎞ Vc = ⎜ 0.6λ f 'c + 700 u ⎟bw d ⎜ M ⎟ ¬xñat US¦ ⎝ u ⎠ ( Vc ≥ 1.7λ f 'c bw d ) ¬xñat US¦ Vc ≤ (5λ ) f 'c bw d ¬xñat US¦ ⎛ V d⎞ Vc = ⎜ λ f 'c / 20 + 5 u ⎟ ⎜ Mu ⎟ ¬xñat SI¦ ⎝ ⎠ karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 307
    • T.Chhay ( Vc ≥ 0.17λ f 'c bw d ) ¬xñat SI¦ Vc ≤ (0.4λ ) f 'c bw d ¬xñat SI¦ nig Vu d ≤ 1.0 Mu ehIy f pe ≥ 0.4 f pu #> eRCIserIsEdkkgbiTCitTb;karrmYlcaM)ac;EdlRtUveRbICaEdkTTwg edayeRbI yield strength Gtibrma 60ksi(414MPa) KW At Tn = 2 2 A0 f yv cot θ RbsinebIeKmineRbItMél A0 nig θ EdlTTYl)anBIkarviPaK b¤BIsmIkar 5.86 eKRtUveRbI A0 = 0.85 A0h nig θ = 45o sMrab;Ggát;minEmneRbkugRtaMg b¤ θ = 37.5o sMrab;Ggát;eRbkugRtaMg CamYynwgeRbkugRtaMgRbsiT§PaBEdlminRtUvtUcCagersIusþg;Tajrbs;EdkbeNþay. EdkbeNþay bEnßmRtUvesμInwg ⎛ A ⎞ ⎛ f yv ⎞ 2 Al = ⎜ t ⎟ ph ⎜ ⎟ cot θ ⎝ s ⎠ ⎜ f yl ⎟ ⎝ ⎠ b:uEnþvaminRtUvtUcCag ⎛A ⎞ Al , min = 5 f 'c Acp f yl − ⎜ t ⎟ ph ⎝ s ⎠ f yv f yl ¬xñat US¦ ⎛A ⎞ Al , min = 5 f 'c Acp 12 f yl − ⎜ t ⎟ ph ⎝ s ⎠ f yv f yl ¬xñat SI¦ Edl At / s minRtUvtUcCag 25bw / f yv . KMlatGnuBaØatGtibrmarbs;EdkkgTTwgCatMéltUcCageKkñúgcMeNam ph / 8 b¤ 12in.(30cm) ehIyGgát;p©itrbs;EdkbeNþay b¤EdkeRbkugRtaMgRtUvmanTMhMy:agticesμInwg 1 / 16 énKMlat rbs;Edkkg b:uEnþminRtUvtUcCagTMhMEdk #3 . $> KNnaEdkTb;kMlaMgkat; Av EdlcaM)ac;kñúgmYyKMlatÉktþaenAkñúgmuxkat;TTwg. Vu CakMlaMg kat;xageRkAemKuNenARtg;muxkat;eRKaHfñak;/ Vc CaersIusþg;kMlaMgkat;Fmμta (nominal shear resistance) rbs;ebtugenAkñúgRTnug ehIy Vs kMlaMgkat;EdlRtUv)anTb;Tl;edayEdkkg Av V = s s f yd Edl Vs = Vn − Vc ehIy Shear and Torsion Strength Design 308
    • NPIC ⎛ V d⎞ Vc = ⎜ 0.6λ f 'c + 700 u ⎟bw d ⎜ Mu ⎟ ¬xñat US¦ ⎝ ⎠ ( Vc ≥ 1.7λ f 'c bw d ) ¬xñat US¦ Vc ≤ (5λ ) f 'c bw d ¬xñat US¦ ⎛ V d⎞ Vc = ⎜ λ f 'c / 20 + 5 u ⎟ ⎜ Mu ⎟ ¬xñat SI¦ ⎝ ⎠ ( Vc ≥ 0.17λ f 'c bw d ) ¬xñat SI¦ Vc ≤ (0.4λ ) f 'c bw d ¬xñat SI¦ nig Vu d Mu ≤ 1.0 sMrab;ebtugTMgn;Fmμta λ = 1 .0 λ = 0.85 sMrab; sand lightweight concrete λ = 0.75 sMrab;ebtugTMgn;RsalTaMgGs; tMélrbs; Vn y:agehacNas;RtUvesμInwg Vu / φ . %> KNnamuxkat;EdkkgsMrab;Tb;nwgkarkat; nigkarrmYl Avt = 2 At + Av ⎧ bw s ⎪50 f sMrab; US 0.35 bw s sMrab;SI ⎪ yv f yv ≥ min ⎨ A f d ⎪ ps pu ⎪ 80 f y d ⎩ bw dak;EdkkgcMgay (bt + d ) ecjBIcMnucRTwsþIEdlRtUvkar kñúgenaH bt = TTwgrbs;muxkat;Edlman EdkkgTb;Tl;karrmYl. 19> Design of Web Reinforcement for Combined Torsion and Shear in Prestressed Beams ]TahrN_ 5>9³ kMralcMNtrfynþTMhMmFümRtUv)ansg;eLIgedayRbB½n§kMralebtugeRbkugRtaMgdUc bgðajenAkñúgrUbTI 5>50. kMralmanTMhM 36 ft × 54 ft (11m × 16.5m)KitBIG½kS ehIyFñwmGkSr T Dub Edlcak;Rsab;RbEvg 54 ft (16.5m) RtUv)anRTedayFñwmebtugeRbkugRtaMgcak;Rsab; spandrel L-beam EdlmanElVg 36 ftt (11m) KitBIG½kS ¬rUbTI 5>50 (a) nig (b)¦. Spandrel beam RtUv)anTb;karrmYl eday connection rbs;vaeTAnwgssrTMr. kMralrgnUv service superimposed dead load EdlbNþal karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 309
    • T.Chhay BIFñwm T Dub WSD = 77 psf (3,687 Pa ) nig service live load WL = 60 psf (2,873Pa ) . eKeRCIserIs ykkMBs;rbs; L-beam esμInwg 6'−30" (1.9m) EdleKGaceRbIvaCa parapet wall BIxagelIFñwmGkSr T Dub. sikSaKNnaEdkRTnugrbs; spandrel beam edIm,ITb;nwgbnSMkMlaMgrmYl nigkMlaMgkat;Edlva rg. xageRkamCaTinñn½yEdleKeGay lkçN³rbs;Fñwm Ac = 696in.2 (4,491cm 2 ) Shear and Torsion Strength Design 310
    • NPIC ( I c = 364,520in.4 93.3 × 106 cm 4 ) cb = 33.2in.(84.3cm ) ct = 41.8in.(106cm ) ( S t = 8,720in.3 142,895cm3 ) ( Sb = 10,990in.3 180,094cm3 ) WD = 725 plf (10.6kN / m ) ebtugTMgn;Fmμta f 'c = 5,000 psi (34.5MPa ) f y = 60,000 psi (418MPa ) sMrab;Edkkg lkçN³rbs;EdkeRbkugRtaMg Aps = 270K stress-relieved tendons Ggát;p©it 1 / 2in.(12.7 mm ) cMnYn 6 f pu = 270,000 psi (1,862MPa ) f ps = 255,000 psi (1,758MPa ) f pe = 155,000 psi (1,069MPa ) ( E ps = 28 × 106 psi 193 × 103 MPa ) d p = 71.5in.(190cm ) e = 71.5 − 41.8 = 29.7in.(75cm ) straight tendon minKitT§iBlkMlaMgxül; nigrBa¢ÜydI. dMeNaHRsay³ !> KNna Tu / Vu / M u / TSL / VSL EdlmanGMeBIelI L-beam ¬CMhanTI 1¦ (a) Service load WD = 725 plf (10.6kN / m ) 77 × 54 WSD = × 4 ft = 8,316lb / stem(37.0kN ) 2 60 × 54 WL = × 4 ft = 6,480lb / stem(28.0kN ) 2 PSL srubenAkñúgmYy stem= 8,316 + 6,480 = 14,796lb(65.7kN ) (b) Factored load WDu = 1.2 × 725 = 870 plf (12.7kN / m ) WSDu = 1.2 × 8,316 = 9,979lb / stem(44.4kN / m ) WLu = 1.6 × 6,480 = 10,368lb / stem(46.1kN ) karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 311
    • T.Chhay Pu srubenAkñúgmYy stem= 9,979 + 10,368 = 20,347lb(90.5kN ) Tu enARtg;épÞrbs;TMr = 1 Pu × arm × no. of stem 2 × × 9 = 61,041 ft − lb(82.8kN .m ) 20,347 8 = 2 12 TSL enARtg;épÞrbs;TMr = 14,796 20,347 × 61,041 = 44,388 ft − lb(60.2kN .m ) Vu enARtg;épÞrbs;TMr = 1 (Pu × no. of stem + factored WD × span ) 2 = 1 2 (20,347 × 9 + 870 × 34) = 106,352lb(474kN ) enARtg;épÞrbs;TMr = 12 (14,716 × 9 + 725 × 34) = 78,907lb(351kN ) VSL M u enARtg;épÞrbs;TMr = 0 dUcKña KNnatMélrbs; Tu / Vu / M u / tMél service load EdlRtUvKñaenARtg;cMnucb:Hén stem TTwgnImYy²tambeNþayElVgrbs;FñwmGkSr L nigsg;düaRkamm:Um:g;rmYl düaRkamkMlaMgkat; nigdüa Rkamm:Um:g;dUcbgðajenAkñúgrUbTI 5>51. Aps = 6 × 0.153 = 0.918in.2 Pe = Aps f pe = 0.918 × 155,000 = 142,290lb(633kN ) @> L-beam torsional geometrical details ¬CMhanTI 1¦ Acp = RkLaépÞEdlB½T§CMuvijedaybrimaRtxageRkArbs;muxkat;ebtug = 8 × 75 = 600in.2 (3871cm 2 ) pcp = brimaRtxageRkArbs;muxkat;ebtug = 2(8 + 75) = 166in.(422cm ) x1 = TMhMtUcCageKEdlKitBIG½kSEdkkgeTAG½kSEdkkg = 8 − 2(1.5 + 0.25) = 4.5in.(11.4cm ) y1 = 75 − 2(1.5 + 0.25) = 71.5in.(181.6cm ) h= kMBs;srub = 75in.(191cm) bw = TTwgRTnug = 8in.(20.3cm ) ph = brimaRtrbs;G½kSEdkkgxageRkAeK = 2( x1 + y1 ) = 2(4.5 + 71.5) = 152in. d p = kMBs;RbsiT§PaB = 7.5 − (1.5 + 0.5 + 0.5 + 1.0 ) = 71.5in.(182cm ) A0 h = épÞEdlB½T§CMuvijedayG½kSrbs;EdkkgTb;karrmYlxageRkAeK = x1 y1 = 4.5 × 71.5 = 322in.2 (2077cm 2 ) Ao = gross area EdlB½T§CuMvijeday shear flow path Shear and Torsion Strength Design 312
    • NPIC ( = 0.85 A0h = 0.85 × 322 = 274in.2 1766cm 2 ) θ= mMurbs;Ggát;RTUgrgkarsgát;rbs; struss analogy EdlrgkarrmYl = 37.5o sMrab;FñwmeRbkugRtaMg cot θ = 1.3 karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 313
    • T.Chhay #> ersIusþg;m:Um:g;eRbH (cracking moment capacity) ¬CMhanTI 1¦ f d = kugRtaMgbnÞúkefrKμanemKuN BIrUbTI 5>51 CamYynwg WD = 725lb / ft M D 725(36 )2 = 128 psi (0.9 MPa ) 1 fd = = × 12 × Sb 8 10,990 enAsrésxageRkAbMputrbs;muxkat; ⎛P Pe⎞ ⎛ 142,290 142,290 × 29.7 ⎞ f ce = −⎜ e + e ⎟ = −⎜ ⎜A ⎟ + ⎟ ⎝ c Sb ⎠ ⎝ 696 10,990 ⎠ = −(204.4 + 384.5) = 588.9 psi ≅ 589 psi (C )(4.1MPa ) enARtg;TIRbCMuTMgn;rbs;muxkat; f c = −204.4 psi M cr = Sb (6λ f 'c + f ce − f d ) = 10,990(6 × 1.0 5,000 + 589 − 128) = 9.73 × 106 in. − lb(1,100kN .m ) 1.2 M cr = 1.2 × 9.73 × 106 = 11.2 × 106 in. − lb Aps f ps 0.918 × 255,000 a= = = 6.9in.(175mm ) 0.85 f 'c b 0.85 × 5,000 × 8 ⎛ a⎞ ⎛ 6.9 ⎞ M n = Aps f ps ⎜ d p − ⎟ = 0.918 × 255,000 × ⎜ 71.5 − ⎟ ⎝ 2⎠ ⎝ 2 ⎠ = 15,930,000 = 15.9 × 106 in − lb(1,800kN .m ) > 1.2 M cr = 11.2 × 106 in. − lb dUcenH EdkrgkarBt;Gb,brmaRKb;RKan;sMrab;Tb;Tl;karBt;. $> epÞógpÞat;faetIeKRtUvkarEdkTb;Tl;karrmYlb¤Gt; ¬CMhanTI 2¦ BIsmIkar 5.80, kMlaMgrmYlGb,brmasMrab;karminKitkMlaMgrmYlkñúgkarsikSaKNna ⎛ Acp ⎞ 2 Tu ≤ φ f 'c ⎜ ⎟ 1+ f c ⎜ pcp ⎟ 4 f 'c ⎝ ⎠ ⎛ 600 2 ⎞ = 0.75 5,000 ⎜ ⎟ 1 + 204.4 ⎜ 166 ⎟ ⎝ ⎠ 4 5,000 = 150,953in. − lb(17.0kN .m ) edayBicarNamuxkat;enARtg; h / 2 BIépÞTMrenAkñúgrUbTI 5>51 b¤enARtg;cMgay 3 ft BIépÞrbs;TMr. Tu EdlRtUvkar = 1 (61,041 + 47,476 ) × 12 2 = 651,102in. − lb(73kN .m ) > 150,953in. − lb Shear and Torsion Strength Design 314
    • NPIC eKeRbItMélmFümCatMélEdlmansuvtßiPaBrbs;m:Um:g;rmYlCMnYseGay 47,476in. − lb . dUcenH eKRtUvBicarNam:Um:g;rmYl ehIyeKRtUvdak;EdkTb;m:Um:g;rmYlsmRsb. eRKOgbgÁúMrbs; cMNtrfynþsuT§Etcak;Rsab;. dUcenH snμt;eRbIlkçxNÐlMnwgrbs;kMlaMgrmYl nigminmankarEbgEck m:Um:g;eLIgvijedayeRbIkMlaMgrmYlemKuNGnuvtþn_srub Tu = 651,102in. − lb ¬dUcenH eKminGaceRbI smIkar 5.79 eT¦. %> RtYtBinitüPaBRKb;RKan;rbs;muxkat;sMrab;karrmYl (a) kMNt; Vc CatMéltUcCageKEdlTTYl)anBI Vci BIsmIkar 5.11 nig Vcw BIsmIkar 5.15. BIrUbTI 5>51 sMrab;muxkat;enARtg; h / 2 = 3 ft BIépÞrbs;TMr Vu EdlRtUvkar = 1 (105,482 + 81,655) = 93,569lb(417kN ) 2 M u EdlRtUvkar = 1 (105,900 + 439,500)12 = 3,272,400in. − lb(370kN .m ) 2 ⎡ M cr ⎤ Vci = ⎢0.6λ f 'c bw d + Vd + Vi ⎥ ⎣ M max ⎦ ≥ 1.7 f 'c bw d ≤ 5.0 f 'c bw d kMlaMgkat;EdlbNþalBIbnÞúkefrKμanemKuN Vd = Vd enARtg;épÞrbs;TMr = 1 (8,316 × 9 + 725 × 34) = 49,749lb 2 Vd enARtg;cMnuc A enAkñúgrUbTI 5>51 = 49,749 − 725 × 1 ft − 8,316 = 40,708lb Vd enARtg; 5 ft BITMr = 40,708 − 725 × 5 ft − 8,316 = 28,767lb enARtg;muxkat;EdlRtUvkarEdlenAcMgay h / 2 BIépÞrbs;TMr Vd = 1 2 (40,708 + 28,767 ) = 34,738lb(154.5kN ) Vi = kMlaMgkat;emKuNenARtg;muxkat;EdlbNþalBIbnÞúkGnuvtþn_xageRkAEdlmanGMeBI dMNalKñaCamYynwg M max 20,347 per stem = × 93,569 = 79,902lb 20,347 + 1,015 × 4 stem m:Um:g;emKuNGtibrmaenARtg;muxkat;EdlbNþalBIbnÞúkGnuvtþn_xageRkA M max = ¬bnÞúkGefr nigbnÞúkefrdak;bEnßm¦ m:Um:g;emKuN M u enARtg;cMnuc A enAkñúgrUbTI 5>51 EdlbNþalmkBIbnÞúkGefr nig SDL = 1 2 (20,347 × 9) = 91,562 ft − lb m:Um:g;emKuN M u enARtg;cMgay 5 ft BIépÞ karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 315
    • T.Chhay = 1 2 (20,347 × 9)5 − 20,347 × 4 = 376,420 ft − lb dUcenH M max enARtg; h / 2 = 3 ft BIépÞrbs;TMr = 1 2 (91,562 + 376,420)12 = 2.81 ⋅ 106 in. − lb M cr = 9.73 ⋅ 106 in. − lb ¬Edl)anBIxagelI¦ ⎡ 9.73 ⎤ dUcenH Vci = ⎢0.6 × 1.0 5,000 × 8 × 71.5 + 34,738 + 79,902 ⎣ 2.81⎥⎦ = 24,268 + 34,738 + 276,671 = 335,677lb = 587 psi (4.05MPa ) 335,677 vci = 8 × 71.5 1.7λ f 'c = 1.7 5,000 = 120 psi < 587 psi 5λ f 'c = 5.0 5,000 = 354 psi < 587 psi yk vci = 354 psi(2.4MPa ) BIsmIkar 5.15 Vcw = (3.5 f 'c + 0.3 f c )bw d + V p V p = 0 edaysar tendon Rtg; dUcenH ⎛ 142,290 ⎞ Vcw = 3.5 5,000 + 0.3⎜ ⎟ = 310 psi < vci = 354 psi ⎝ 696 ⎠ eRbI vc = 310 psi enAkñúgdMeNaHRsayenH Vc = vc bw d = 310 × 8 × 71.5 = 176,750lb(786kN ) (b) viFIepSgeTotsMrab;kMNt; Vc RbsinebI f pe > 0.4 f pu / ACI GnuBaØateGayeRbInUvsmIkar 5.16 EdlmanlkçN³suvtßiPaBCag ⎛ Vu d p ⎞ Vc = ⎜ 0.6λ f 'c + 700 ⎜ ⎟bw d p ⎝ Mu ⎟ ⎠ ≥ 2λ f 'c bw d ≤ 5.0λ f 'c bw d Vu d p 93,569 × 71.5 = = 2.04 > 1.0 Mu 3,272,400 eRbI VMd p = 1.0 u u vc = Vc bw d ( ) = 0.6 × 1.0 5,000 + 700 × 1.0 = 742 psi (5.1MPa ) > 5 f 'c = 354 psi Shear and Torsion Strength Design 316
    • NPIC vc = 354 psi (2.4MPa ) (c) RtYtBinitüPaBRKb;RKan; eyIgeRbI vc = 310 psi(2.1MPa) enAkñúgdMeNaHRsay (a) sMrab;karRtYtBinitü. BIsmIkar 5.81 sMrab;muxkat;tan; 2 2 2 ⎛ Vu ⎞ ⎛ Tu ph ⎞ 2 ⎛ ⎞ ⎜ ⎟ +⎜ ⎟ = ⎛ 93,569 ⎞ + ⎜ 651,102 × 152 ⎟ ⎜ ⎟ ⎜ ⎜ b d ⎟ ⎜ 1 .7 A 2 ⎟ ⎝ 8 × 71.5 ⎠ ⎝ 1.77 × (322 )2 ⎟ ⎝ w ⎠ ⎝ 0h ⎠ ⎠ 26,759 + 290,814 = 564 psi (3.9 MPa ) φ⎜ ⎜ ⎛ Vc ⎞ ( + 8 f 'c ⎟ = 0.75 310 + 8 5,000 ⎟ ) ⎝ bw d ⎠ = 656 psi (4.5MPa ) EdlGacman > 564 psi (3.8MPa ) dUcenHmuxkat;manlkçN³RKb;RKan;. ^> EdkTb;karrmYl (torsional reinforcment) ¬CMhanTI 3¦ Tn = Tu / φ = 651,102 / 0.75 = 868,137in. − lb(96kN .m ) BIsmIkar 5.38b At Tn 868,137 = = s 2 A0 f yv cot θ 2 × 274 × 60,000 × 1.3 ( ) = 0.0203in.2 / in. / one leg 0.046cm 2 / cm / one leg edayeRbI PCI method EdlenAkñúgviFIenHeKRtUvkarKNna cot θ Tu / φ 868,137 cot θ = = = 1.3 1.7 A0h ( At / s ) f yv 1.7(322)(0.0203) × 60,000 edaysnμt;tMélrbs; s rk ( At / s ) sMrab;TMhMEdkkgEdleRCIserIs nigbBa©ÚltMélrbs; ( At / s ) eTAkñúgsmIkar. &> EdkTb;kMlaMgkat; (shear reinforcment) ¬CMhanTI 4¦ Vc = 310 × 8 × 71.5 = 177,320lb(788kN ) = 124,757lb(550kN ) V 93,569 Vn = u = φ 0.75 Vs = (Vn − Vc ) b:uEnþ Vn < Vc eRbIEdkTb;kMlaMgkat;Gb,brma Av 50bw 50 × 8 = = s fy 60,000 karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 317
    • T.Chhay ( = 0.0067in.2 / in. / two legs 0.017cm 2 /cm/two legs ) Avi ⎛A ⎞ A = 2⎜ T ⎟ + v = 2 × 0.0203 + 0.0067 s ⎝ s ⎠ s ( = 0.0473in.2 / in. / two legs 0.110cm 2 / cm / two legs ) edassnμt;eRbIEdkkg #4 Ggát;p©it 12.7mm / Av = 2 × 0.20 = 0.40in.2 cross − sectional tie are 0.40 s= = = 8.5in. Avt / s 0.0473 KMlatGnuBaØatGtibrma smax = ph / 8 b¤ 12in. = 152 / 8 = 19in. b¤ 12in. ⎧ bw ⎪ 50 f ⎪ Av s Gb,brma = min ⎨ y Aps f pu d ⎪ ⎪ 80 f y d bw ⎩ 0.75 f 'c = 0.75 5,000 = 53 Av 53bw 53 × 8 = = s fy 60,000 ( = 0.0071in.2 / in. / two legs 0.017cm 2 / cm / two legs ) Av Aps f pu d 0.918 × 270,000 71.5 = = s 80 f y d bw 80 × 60,000 × 71.5 8 ( = 0.0022in.2 / in. / two legs 0.006cm 2 / cm / two legs ) s EdlGacman = 0.0473 > 0.0022 Av O.K. Ggát;p©itEdkGb,brma = s / 16 b¤ Edk #3 = 8.5 / 16 = 0.53 > 0.5in. sMrab;Edk #4 dUcenHeRbIEdkkg #5 / Av = 0.31× 2 = 0.62in.2 0.62 s= = 13.1in. > 12in. 0.0473 mü:agvijeToteKk¾GaceRbIEdkkg #4 )anEdr EteKRtUvkat;bnßyKMlatEdkkgBI 8.5in mkRtwm 8.5(0.5 / 0.53) ≅ 8.0in. . kareRbIEdkkg #4 KMlat 8in. KitBIG½kSeTAG½kSCMnYseGay #5 KMlat 8.5in. manPaBgayRsYl nigmankarniymeRcInCag edaysarEdk #4 gayBt;CagEdk #5 . *> EdkbeNþay (longitudinal reinforcement) ¬CMhanTI 5-6¦ BIsmIkar 5.85 Shear and Torsion Strength Design 318
    • NPIC At ⎛ f yv ⎞ Al = ph ⎜ ⎟ cot 2 θ s ⎜ f yl ⎟ ⎝ ⎠ ⎛ 60,000 ⎞ = 0.0203 × 152⎜ ( ⎟(1.3) = 5.21in.2 33.6cm 2 2 ) ⎝ 60,000 ⎠ BIsmIkar 5.86 5 f 'c Acp ⎛A ⎞ f yv Al , min = −⎜ t ⎟ ph f yl ⎝ s ⎠ f yl 5 5,000 × 600 60,000 = − 0.0203 × 152 × 60,000 60,000 ( ) = 0.50in 2 3.22cm 2 < Al = 5.21in.2 edayeRbIEdkbeNþay #4 = 0.2in.2 cMnYnEdk = 0..20 = 26.05 edIm 5 21 dak;Edk #4 cMnYn 12 edImenAépÞxagnImYy²CamYynwgKMlatesμIKña ¬EdkGgát;p©it 12.7mm cMnYn 12 sMrab;épÞnImYy²¦ nigbEnßmEdk 4 edImeTotedIm,IBRgwg dUcenHEdkbeNþaysrubman 28 edIm. cMNaM faKMlatGnuBaØatGtibrma 12in. . enAkñúgkrNIenH s ≅ 6.5in. KitBIG½kSeTAG½kS/ O.K.. karlMGitsrésEdk nigragFrNImaRtrbs;mxkat;RtUv)anbgðajenAkñúgrUbTI 5>52. u edIm,IeGaykarlMGitsrésEdkmanlkçN³eBjelj eKRtUvkarsikSaKNnaBIEdksMrab;Epñk Edllyecj (ledge reinforcement) nig hanger reinforcement k¾dUcCakarlMGitBI anchorage én EdkbeNþayenARtg;TMr. CMBUkTI10 EdlBN’naBIkarsikSaKNnatMN nwgpþl;nUvkarlMGitenH. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 319