V. shear and torsional strength design

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V. shear and torsional strength design

  1. 1. 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
  2. 2. 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
  3. 3. 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
  4. 4. 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
  5. 5. 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
  6. 6. 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
  7. 7. 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
  8. 8. 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
  9. 9. 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
  10. 10. 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
  11. 11. 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
  12. 12. 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
  13. 13. 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
  14. 14. 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
  15. 15. 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
  16. 16. 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
  17. 17. 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
  18. 18. 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
  19. 19. T.Chhay Shear and Torsion Strength Design 232
  20. 20. 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
  21. 21. 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
  22. 22. 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
  23. 23. 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
  24. 24. 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
  25. 25. 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
  26. 26. NPIC karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 239
  27. 27. 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
  28. 28. 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
  29. 29. 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
  30. 30. 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
  31. 31. T.Chhay Shear and Torsion Strength Design 244
  32. 32. 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
  33. 33. 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
  34. 34. 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
  35. 35. 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
  36. 36. 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
  37. 37. 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
  38. 38. 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
  39. 39. 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
  40. 40. 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
  41. 41. 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
  42. 42. 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
  43. 43. 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
  44. 44. 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
  45. 45. 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
  46. 46. 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
  47. 47. 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
  48. 48. 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
  49. 49. 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
  50. 50. 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

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