Xv design for torsion

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Xv design for torsion

  1. 1. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa XV. karKNnasMrab;kMlaMgrmYl Design for Torsion 1> esckþIepþIm Introduction kugRtaMgrmYlekItmanenAkñúgmuxkat;FñwmenAeBlEdlm:Um:g;manGMeBIRsbeTAnwgmuxkat;enaH. m:Um:g;rmYleFVIeGayGgát;vil nigmansñameRbHenAelIépÞrbs;va CaTUeTAEtgekItmanenAelImuxkat;mUl. edIm,IbgðajkugRtaMgrmYl eKGnuvtþkMlaMgrmYl T Fñwm cantilever muxkat;mUlEdleFVIBI elastic homogenous material dUcbgðajkñúgrUbTI 15>1. kMlaMgrmYlnwgeFVIeGayFñwmvil. cMnuc B clt½eTA cMnuc B' enAxagcugrbs;Fñwm b:uEnþcugmçageTotrbs;FñwmRtUv)anbgáb;. mMu θ RtUv)aneKehAfa mMurmYl (angle of twist). bøg; AO' OB nwgdUrrageTACarag AO'OB' . edaysnμt;fa Ggát;enArkSaRbEvgrbs; vadEdl enaH shear strain KW BB' rθ γ = = L L Edl L CaRbEvgrbs;Fñwm nig r CakaMrbs;muxkat;rgVg;. enAkñμúgeRKOgbgÁúMebtugGarem: Ggát;nwgrgm:Um:g;rmYlenAeBlGgát;enaHekagenAkñúgbøg;/ RT cantilever slab/ mannaTICa spandrel beam (end beam)/ b¤CaEpñkrbs;CeNþIrvil. Design for Torsion 346
  2. 2. T.Chhay NPIC Ggát;eRKOgbgÁúMGacrgnUvEtkMlaMgrmYlsuT§ b¤enAkñúgkrNICaeRcIn vargCamYyKñakñúgeBlEtmYy nUvkMlaMgkat;TTwg nigm:Um:g;Bt;. ]TahrN_TI15>1 bgðajBIkMlaMgepSg²EdlGacGnuvtþmkelImuxkat; epSgKñaénFñwm cantilever. ]TahrN_TI15>1³ KNnakMlaMgEdlmanGMeBIenAmuxkat; !/ @ nig # énFñwm cantilever EdlbgðajenAkñúgrUbTI 15>2. Fñwm rgnUvkMlaMgbBaÄr P1 = 67kN / kMlaMgedk P2 = 53.5kN EdleFVIGMeBIenAcMnuc C nigbnÞúkedk P3 = 89kN EdlGnuvtþenAcMnuc B nigEkgeTAnwgTisedArbs;kMlaMg P2 . dMeNaHRsay³ yk N = kMlaMgEkg (normal force)/ V = kMlaMgkat; (shear force)/ M = m:Um:g;Bt; (bending moment)/ T =m:Um:g;rmYl (torsional moment). kMlaMgTaMgGs;RtUv)anbgðajenAkñúgtaragxageRkam³ muxkat; N (kN ) M x (kN .m) M y (kN .m) V x (kN ) V y (kN ) T (kN .m) ! 0 − 180.9 144.45 53.5 67 0 @ − 53.5 ¬sgát;¦ 0 144.45 89 67 180.9 # − 53.5 ¬sgát;¦ 241.2 464.85 89 67 180.9 karKNnasMrab;kMlaMgrmYl 347
  3. 3. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa RbsinebI P1 / P2 nig P3 CabnÞúkemKuN ¬ Pu = 1.2PD + 1.6PL ¦ enaHral;tMélenAkñúgtaragCakMlaMg KNnaemKuN. 2> m:Um:g;rmYlenAkñúgFñwm Torsional Moments in Beams dUcbgðajenAkñúgrUbTI 15>1 kMlaMgGacGnuvtþenAelIeRKagsMNg;GKar edayeFVIeGaymanm:Um:g; rmYl. RbsinebIkMlaMgcMcMnuc P GnuvtþenARtg;cMnuc C enAelIeRKag ABC dUcbgðajenAkñúgrUb 15>3 a vabegáItm:Um:g;rmYl T = PZ enAkñúgFñwm AB Rtg;cMnuc D . enAeBl D sßitenAkNþalElVgénFñwm AB enaHm:Um:g;rmYlKNnaenAkñúgkMNat; AD esμInwgm:Um:g;rmYlKNnaenAkñúgkMNat; DB b¤esμInwg 1 T . Rb 2 sinebIkMral cantilever slab RtUv)anRTedayFñwm AB ¬rUbTI 15>3 b¦ enaHkMralxNнbegáItm:Um:g;rmYl BRgayesμI mt tambeNþayFñwm AB . m:Um:g;rmYlBRgayesμIenH KWekItBIbnÞúkenAelIcMerokTTwkmYy Éktþarbs;kMralxNн. RbsinebI S CaTTwgén cantilever slab nig w CabnÞúkenAelIkMralxNн ¬ kN / m 2 ¦ enaH mt = wS 2 / 2 ¬ kN .m / m ¦énFñwm AB . m:Um:g;rmYlKNnaGtibrmaenAkñúgFñwm AB KW T = mt L / 2 EdlGnuvtþenARtg;cMnuc A nig B . krNIbnÞúkepSgeTotRtUv)anbgðajenAkñúgtarag 15>1. CaTUeTA düaRkamm:Um:g;rmYlenAkñúgFñwmmanrag nigmantMéldUcKñanwgdüaRkamkMlaMgkat;TTWgsM rab;FñwmEdlrgnUvkMlaMg mt nig T . Design for Torsion 348
  4. 4. T.Chhay NPIC karKNnasMrab;kMlaMgrmYl 349
  5. 5. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa 3> kugRtaMgrmYl Torsional Moments in Beams edayBicarNaelIFñwm cantilever Edlmanmuxkat;mUl ¬rUbTI15>1¦ enAeBlEdlm:Um:g;rmYl T manGMeBIelIFñwm vanwgbegáIteGaymankMlaMgkat;TTwg dV EkgeTAnwgkaMrbs;muxkat;. BIlkçxNÐl<nwg m:Um:g;rmYlxageRkARtUv)anTb;edaym:Um:g;rmYlxagkñúgEdlmantMél T esμIKñaEtTisedApÞúyKña . Rbsin ebI dV CakMlaMgkat;TTwgeFVIGMeBIelIépÞ dA ¬rUbTI 15>4¦ enaHGaMgtg;sIueténkMlaMgrmYlKW T = ∫ rdV edayyk v CakugRtaMgkMlaMgkat;TTWgenaH dV = vdA nig T = ∫ rvdA kMlaMgkat;TTwgeGLasÞicGtibrmaekItmanenAépÞxageRkArbs;muxkat;rgVg;Rtg;kaM r CamYynwgkM ras; dr enaHkMlaMgrmYlGacRtUv)ankMNt;edayKitm:Um:g;eFobnwgcMnuc 0 sMrab;RkLaépÞkg³ dT = (2πrdr )vr Edl 2πrdr CaRkLaépÞkg nig v CakugRtaMgkMlaMgkat;TTwgenAkñúgkg. dUcenH T = ∫ (2πrdr )vr = ∫ 2πr 2 dr R 0 R 0 ¬!%>!¦ sMrab;muxkat;RbehagEdlmankaMxagkñúg R1 / R T = ∫ 2πr 2 dr R1 ¬!%>@¦ sMrab;muxkat;tan; edayeRbIsmIkar ¬!%>!¦ nig v = vmax r / R R ⎛v r⎞ ⎛ 2π ⎞ R 3 T = ∫ 2πr 2 ⎜ max ⎟dr = ⎜ ⎟vmax ∫0 r dr 0 ⎝ R ⎠ ⎝ R ⎠ ⎛ 2π ⎞ ⎛π ⎞ 4 R =⎜ ⎟vmax = ⎜ ⎟vmax R 3 ⎝ R ⎠ 4 ⎝2⎠ Design for Torsion 350
  6. 6. T.Chhay NPIC vmax = 2T πR 3 ¬!%>#¦ m:Um:g;niclPaBb:UElrénmuxkat;rgVg;KW J = πR 4 / 2 . dUcenH kugRtaMgkMlaMgkat;GacRtUv)an sresrCaGnuKmn_énm:Um:g;niclPaBb:UElrdUcxageRkam³ vmax = TR J ¬!%>$¦ 4> m:Um:g;rmYlenAkñúgmuxkat;ctuekaN Torsional Moments in Rectangular Sections karKNnakugRtaMgenAkñúgGgát;manmuxkat;minmUlEdlrgbnÞúkrmYlminsamBaØdUckarKNnasM rab;muxkat;mUleT. b:uEnþ lT§plEdlTTYlBIRTwsþIeGLasÞic (theory of elasticity) bgðajfakugRtaMg kMlaMgkat;TTwgGtibrmasMrab;muxkat;ctuekaNEkgGacRtUv)ankMNt;dUcxageRkam³ vmax = 2 T αx y ¬!%>%¦ Edl T= kMlaMgrmYlEdlGnuvtþ x = RCugxøIrbs;muxkat;ctuekaN y = RCugEvgrbs;muxkat;ctuekaN α = emKuNEdlGaRs½ynwgpleFobén y / x tMélrbs;vaRtUv)aneGayenAkñúgtarag xageRkam. y/x 1 .0 1 .2 1 .5 2 .0 4 .0 10 α 0.208 0.219 0.231 0.246 0.282 0.312 kugRtaMgkMlaMgkat;TTwgGtibrmaekItmanenAtamGkS½énRCugEvg y ¬rUbTI 15>5¦. sMrab;Ggát;EdlekItBIkarpÁúMénmuxkat;ctuekaNEkg dUcCamuxkat;GkSr L / T nig I tMél α GacRtUv)ansnμt;faesμInwg 1/ 3 ehIymuxkat;GacRtUv)anEckecjCamuxkat;ctuekaNCaeRcInEdlman RCugEvg yi nigRCugxøI xi . kugRtaMgkMlaMgkat;TTwgGacRtUv)anKNnaBI vmax = 3T ¬!%>^¦ ∑ i ix2 y Edl ∑ xi2 y i CatMélEdl)anBIplbUkmuxkat;ctuekaNEkgtUc². enAeBlEdl y / x ≤ 10 eK GaceRbIsmIkarsMrYlxageRkam³ karKNnasMrab;kMlaMgrmYl 351
  7. 7. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa v max = 3T ¬!%>&¦ ⎛ x⎞ ∑ x 2 y⎜1 − 0.63 ⎜ ⎟ y⎟ ⎝ ⎠ 5> kMlaMgpÁÜbrvagkMlaMgkat; nigkMlaMgrmYl Combined Shear and Torsion enAkñúgkrNIGnuvtþn_CaeRcIn Ggát;eRKOgbgÁúMGacrgnUvTaMgkMlaMgkat; nigkMlaMgrmYlCamYyKña. kugRtaMgkMlaMgkat;GacnwgekItmanenAkñúgmuxkat;CamYynwgkugRtaMgkMlaMgkat;mFüm = v1 enAkñúgTis Design for Torsion 352
  8. 8. T.Chhay NPIC edAénkMlaMgkat; V ¬rUbTI 15>6 a¦. kMlaMgrmYl T begáItkugRtaMgrmYlenAelIRKb;RCugrbs;muxkat; ctuekaN ABCD ¬ rUbTI 15>6 a¦ CamYynig v3 > v2 . karBRgaykugRtaMgcugeRkayRtUv)anTTYlBI karbUkbBa©ÚlnUvT§iBlénkugRtaMgkMlaMgkat; nigkugRtaMgrmYl edIm,IbegáIttMélGtibrmaesμI v1 + v3 enA elIRCug CD b:uEnþRCug AB nwgmankugRtaMgcugeRkayesμI v1 − v3 . RCug AD nig BC nwgrgEtkugRtaMg rmYl v2 . muxkat;RtUvd)anKNnasMrab;kugRtaMgGtibrma v = (v1 + v3 ) . 6> RTwsþIkarrmYlsMrab;Ggát;ebtug Torsion Theories for Concrete Members eKmanviFICaeRcInsMrab;viPaKGgát;ebtugBRgwgedayEdkEdlrgkarrmYl b¤rgkarrmYl karBt; nigkarkat;kñúgeBlEtmYy. CaTUeTAviFIKNnasMGageTAelIRTwsþIeKalBIrKW³ the skew bending theory nig space truss analogy. 6>1> Skew Bending Theory viFIeKalrbs; skew bending theory EdlENnaMeday Hsu CaviFIEdlsikSakar)ak;énmuxkat; ctuekaNedaykarrmYlEdlekItedaykarBt;eFobGkS½RsbeTAnwgépÞénmuxkat; y FMCag nigeRTteday mMu 45o eTAnwgGkS½EvgénFñwm ¬rUbTI 15>7¦. QrelIviFIsaRsþenH m:Um:g;rmYlGb,brma Tn GacRtUv)an KNnadUcxageRkam³ ⎛ x2 y ⎞ Tn = ⎜ ⎜ 3 ⎟ r ⎟f ¬!%>*¦ ⎝ ⎠ Edl f r KWm:UDuldac;rbs;ebtug. f r RtUv)ansnμt;esμInwg 5 f 'c / 12 enAkñúgkrNIenH Edl RtUv)aneRbobeFobCamYy 7.5 f 'c /12 EdlTTYleday ACI Code sMrab;KNnaPaBdabenAkñúgFñwm. karKNnasMrab;kMlaMgrmYl 353
  9. 9. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa kMlaMgrmYlTb;edayebtugsMEdgdUcxageRkam³ ⎛ 1 ⎞ 2 Tc = ⎜ ⎟ x y f 'c ¬!%>(¦ ⎝ x⎠ nigkMlaMgTb;karrmYledayEdkTb;karrmYlKW α1 ( x1 y1 At f y ) Ts = s ¬!%>!0¦ dUcenH Tn = Tc + Ts Edl Tn lT§PaBTb;m:Um:g;rmYl nominal énmuxkat;. 6>2> Space Truss Analogy viFIsaRsþén space truss analogy KWQrelIkarsnμt;falT§PaBTb;Tl;karrmYlrbs;ebtugGar em:muxkat;ctuekaNRtUv)anKitecjEtBIEdknigebtugEdlBT§½CMuvijEdkb:ueNÑaH. kñúgkrNIenH muxkat; thin-wall RtUv)ansnμt;mannaTICa space truss ¬rUbTI 15>8¦. cMerokebtugvNнeRTtcenøaHsñameRbH Tb;kMlaMgsgát; b:uEnþEdkbeNþayenARCug nigEdkkgTb;nwgkMlaMgTajEdlekItedaym:Um:g;rmYl. kareFVIkarrbs;FñwmebtugGarem:EdlrgkarrmYlsuT§GacbgðajedayRkaPicénTMnak;TMngrvagkar rmYlnigmMurmYl dUcbgðajenAkñúgrUbTI15>9. eyIgemIleXIjfa muxnwgeRbH ebtugTb;nwgkugRtaMgrmYl nigEdkswgEtKμanrgkugRtaMg. eRkayeBleRbH kareFVIkarrbs;FñwmCalkçN³eGLasÞicminGacGnuvtþ)an dUcenHmMurmYlekIteLIgPøam² EdlekIneLIgrhUtdl;lT§PaBTb;Tl;m:Um:g;rmYlekItman. karkMNt;Edl manlkçN³Rbhak;RbEhlénlT§PaBTb;karrmYlsMrab;muxkat;eRbHGacnwgsMEdgdUcxageRkam³ ⎛A f ⎞ Tn = 2⎜ t s ⎟ x1 y1 ⎝ s ⎠ ¬!%>!!¦ Edl At = éneCIgmçagrbs;Edkkg Design for Torsion 354
  10. 10. T.Chhay NPIC s=KMlatEdkkg x1 nig y1 = RbEvgxøI nigRbEvgEvg KitBIGkS½eTAGkS½énEdkkgbiTCit b¤BIEdkenARCug. smIkarmunecalnUvlT§PaBTb;karrmYlrbs;ebtug. Mitchell nig Collins ENnaMnUvsmIkarxag eRkamedIm,IKNnamMurmYlkñúgmYyÉktþaRbEvg ψ ³ ⎛ P ⎞ ⎡⎛ ε ⎞ ⎛ P (ε tan α ) ⎞ 2ε d ⎤ ψ = ⎜ o ⎟ ⎢⎜ l ⎟ + ⎜ h h ⎜ 2 A ⎟ tan α ⎜ ⎟+ ⎟ sin α ⎥ ¬!%>!@¦ ⎝ o ⎠ ⎣⎝ P ⎠ ⎝ o ⎠ ⎦ Edl εl = bMErbMrYlrageFob (strain) enAkñúgEdkbeNþay (longitudinal reinforcing steel) ε h = bMErbMrYlrageFobenAkñúgEdkkg (hoop steel) ε d = bMErbMrYlrageFobebtugGgát;RTUgenARtg;TItaMgénkMlaMgpÁÜbénFarkMlaMgkat; (shear flow) Ph = brimaRtrbs;EdkkgKitRtwmGkS½Edk ⎡ ⎛ P ⎞⎤ α = mMuénkMlaMgsgát;Ggát;RTUg = (ε d + ε l ) / ⎢ε d + ε h ⎜ h ⎟⎥ ⎜P ⎟ ⎣ ⎝ o ⎠⎦ Ao =RkLaépÞEdlBT§½CMuvijedaykMlaMgkat; b¤ = torque / 2q ¬Edl q = FarkMlaMgkat;¦ Po = brimaRténKnøgFarkMlaMgkat; ¬brimaRtrbs; Ao ¦ smIkarmMurmYlxagelImanlkçN³RsedogKñanwgsmIkarmMukMeNagkñúgkarBt; ¬rUbTI 15>10¦ ε + εs φ = curvature = c d ¬!%>!#¦ karKNnasMrab;kMlaMgrmYl 355
  11. 11. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa Edl ε c nig ε s CabMErbMrYlrageFobenAkñúgebtug nigEdk erogKña. smIkard¾samBaØRtUv)anbk Rsayeday Solanki edIm,IkMNt;lT§PaBTb;nwgkarrmYlsuT§rbs;FñwmebtugGarem: edayQrelI space truss analogy dUcxageRkam³ 1 ⎡⎛ ∑ As f sy ⎞ ⎛ Ah f hy ⎞⎤ 2 Tu = (2 Ao )⎢⎜ ⎜ ⎟×⎜ ⎟ ⎜ ⎟⎥ ⎟ ¬!%>!$¦ ⎢⎝ Po ⎣ ⎠ ⎝ s ⎠⎥⎦ Edl / nig s RtUv)anBnül;BIxagelI Ao Po ∑ As f sy = kMlaMg yield énEdkbeNþayTaMgGs; Ah f hy = kMlaMg yield énEdkkg ACI Code )anTTYlykRTwsþIenHedIm,IKNnaGgát;eRKOgbgÁúMebtugEdlrgkarrmYl b¤karrmYl nigkarkat; enAkúñgviFIsaRsþd¾sMrYl. 7> ersIusþg;rmYlénGgát;ebtugsuT§ Torsional Strength of Plain Concrete Memgers Ggát;ebtugrgkarrmYlCaTUeTARtUv)anBRgwgedayEdkTb;nwgkarrmYlBiess. kñúgkrNIEdlkug RtaMgrmYlmantMéltUc nigRtUvkarKNnasMrab;Ggát;ebtugsuT§ kugRtaMgkMlaMgkat; vtc GacRtUv)ankMNt; edayeRbIsmIkar !%>^³ 3T f 'c vtc = ≤ φ∑ x y 2 2 Design for Torsion 356
  12. 12. T.Chhay NPIC nigmMurmYlKW θ = 3TL / x3 yG / Edl T Cam:Um:g;rmYlEdlGnuvtþmkelImuxkat; ¬tUcCagm:Um:g; rmYlEdleFVIeGayeRbH¦ nig G KWCam:UDulkMlaMgkat; nigGacRtUv)ansnμt;esμInwg 0.45 dgénm:UDuleG- LasÞicrbs;ebtug Ec Edl G = 2135 f 'c . kMlaMgkat;TTwgeFVIeGayeRbHedaysarkarrmYl (torsional cracking shear) vc enAkñúgebtugsuT§GacRtUv)ansnμt;esμI 0.5 f 'c . dUcenH sMrab;muxkat; ctuekaNebtugsuT§ φ 2 Tc = 12 x y f 'c ¬!%>!%¦ nigsMrab;muxkat;EdlpSMeLIgedayctuekaNEkgeRcIn φ Tc = 12 f 'c ∑ x 2 y ¬!%>!^¦ 8> karrmYlenAkñúgGgát;ebtugBRgwgedayEdk Torsion in Reinforced Concrete Memebers (ACI Code Procedure) 8>1> sBaØaNTUeTA General dMeNIrkarKNnasMrab;karrmYlmanlkçN³RsedogKñaeTAnwgkMlaMgkat;TTwgedaykarBt;. enA eBlEdlm:Um:g;rmYlemKuNGnuvtþenAelImuxkat;FMCaglT§PaBTb;m:Um:g;rmYlkñúgrbs;ebtugGacTb;)an enaHsñameRbHEdlekItedaykarrmYl (torsional crack) ekIteLIg dUcenHEdkTb;karrmYl (torsional reinforcement) kñúgTMrg;CaEdkkgbiTCit (closed stirrup or hoop reinforcement) RtUv)andak;. bEnßmBIelIEdkkgbiTCit EdkbeNþayk¾RtUv)andak;enAtamRCugrbs;Edkkg nigRtUv)anBRgayy:ag l¥enACMuvijmuxkat;. TaMgEdkkgbiTCit nigEdkbeNþaymansarsMxan;Nas;kñúgkarTb;nwgkMlaMgTaj Ggát;RTUgEdlbNþaymkBIkMlaMgrmYl EdkEtmYyRbePTnwgKμanRbsiT§PaBeTebIKμanEdkmYyRbePT eTot. EdkkgRtUvEtbiTCit edaysarkugRtaMgrmYlekItmanenARKb;RCugrbs;muxkat;. EdkcaM)ac;sMrab;karrmYlRtUv)anbEnßmelIEdkcaM)ac;sMrab;kMlaMgkat;/ sMrab;karBt; nigkMlaMg tamGkS½. EdkEdkcaM)ac;sMrab;karrmYlRtUv)andak;edIm,IeFVIeGayersIusþg;m:Um:g;rmYlrbs;muxkat; φTn FMCagb¤esμInwgm:Um:g;rmYlemKuN Tu EdlRtUv)anKNnaBIbnÞúkemKuN. φTn ≥ Tu ¬!%>!&¦ enAeBleKRtUvkarEdkTb;karrmYl ersIusþg;m:Um:g;rmYl φTn RtUv)anKNnaedaysnμt;kMlaMg rmYl Tu TaMgGs; RtUv)anTb;edayEdkkg nigEdkbeNþayCamYynwgersIusþg;Tb;karrmYlrbs;ebtug karKNnasMrab;kMlaMgrmYl 357
  13. 13. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa Tc = 0 . kñúgeBlCamYyKña ersIusþg;kMlaMgkat;EdlTb;edayebtug vc RtUv)ansnμt;enAdEdledayKμan karERbRbYledaysarvtþmanrbs;ersIusþg;rmYl. 8>2> )a:ra:Em:RtFrNImaRténkarrmYl Torsional Geometric Parameters enAkñúg ACI Code, Section 11.6 karKNnasMrab;karrmYlKWQrenAelI space truss analogy dUcbgðajenAkñúgrUbTI 15>8. eRkayeBlEdlsñameRbHedaykarrmYlekIteLIg karrmYlRtUv)anTb; edayEdkkgbiTCit EdkbeNþay nigersIusþg;kMlaMgsgát;Ggát;RTUgrbs;ebtug. sac;ebtugenAxageRkA EdkkgkøayeTACaKμanRbsiT§PaB nigRtUv)anecalenAkñúgkarKNna. RkLaépÞBT§½CMuvijedayGkS½énEdk Design for Torsion 358
  14. 14. T.Chhay NPIC kgbiTCitxageRkAbMput RtUv)ankMNt;eday Aoh ¬épÞqUtenAkñúgrUbTI 15>11¦. edaysarGgÁdéTeTot RtUv)aneRbIenAkñúgsmIkarKNna vak¾RtUv)anENnaMCadMbUgenATIenHedIm,ICYyeGaykaryl;nUvsmIkarman lkçN³gayRsYl. BIrUbTI 15>11 GgÁEdleGayRtUv)ankMNt;dUcxageRkam³ Acp = RkLaépÞmuxkat;ebtugEdlBT§½CMuvijedaybrimaRtxageRkAénmuxkat;ebtug Pcp = brimaRténmuxkat;ebtugTaMgmUl Acp Aoh = RkLaépÞEdlBT§½CMuvijedayGkS½énEdkrgkarrmYlTTwgbiTCitxageRkAbMput ¬épÞqUtkñúgrUbTI 15>11¦ Ao = RkLaépÞEdlBT§½CMuvijedayKnøgFarkMlaMgkat;TTwg nigGacykesμInwg 0.85 Aoh Ph = brimaRtebtugrbs;EdkrgkarrmYlTTwgbiTCitxageRkAbMput θ = mMuénkMlaMgsgát;Ggát;RTUgcenøaH 30 o eTA 60 o ¬b¤GacykesμInwg 45o sMrab;Ggát;ebtugGarem:¦ sMrab;muxkat;GkSr T nig L TTwgRbsiT§PaBénsøabmçag²RtUv)ankMNt;esμInwgkMBs;FñwmEdl sßitenABIelI b¤BIeRkamkMralxNн edayykmYyNaEdlFMCag b:uEnþminRtUvFMCag 4 dgkMras;kMralxNнeT ¬ACI Code, Sections 11.6.1 and 13.2.4¦. 8>3> m:Um:g;rmYleFVIeGayeRbH Tcr Cracking Torsional Moment Tcr m:Um:g;eFVIeGayeRbHeRkamm:Um:g;rmYlsuT§ Tcr GacRtUv)anTajecjedayCMnYsmuxkat;BitR)akd munnwgeRbH CamYynwg thin-walled tube smmUl t = 0.75 Acp / Pcp / CamYynwgRkLaépÞEdlBT§½CMuvij edayGkS½CBa¢aMg A0 = 2 Acp / 3 . enAeBlEdl kugRtaMgTajGtibrma ¬kugRtaMgem¦ mantMélesμI f 'c / 3 sñameRbHnwgekItman ehIyCaTUeTAm:Um:g;rmYl T esμInwg T = 2 Aoτt ¬!%>!*¦ Edl τ = kugRtaMgkMlaMgkat;edaykarrmYl = f 'c / 3 sMrab;sñameRbHedaykarrmYl. CMnYs τ eday f 'c / 3 f 'c ⎛ Acp ⎞ 2 Tcr = 3 ⎜ ⎜P ⎟ n ⎟ =T nig Tu = φTcr ¬!%>!(¦ ⎝ cp ⎠ edaysnμt;fam:Um:g;rmYltUcCagb¤esμInwg Tcr / 4 nwgmineFVIeGaymankarkat;bnßyersIusþg;Tb;karBt; b¤Tb;kMlaMgkat;enAkñúgGgát;énrcnasm<n§½ ACI Code, Section 11.6.1 GnuBaØateGayecalnUvT§iBlm:Um:g;rmYlenAkñúgGgát;ebtugGarem:enAeBlEdlm:Um:g;rmYlemKuN Tu ≤ φTcr / 4 b¤ karKNnasMrab;kMlaMgrmYl 359
  15. 15. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa f 'c ⎛ Acp ⎞ 2 Tu ≤ φ 12 ⎜ ⎟=T ⎜ Pcp ⎟ a ¬!%>@0¦ ⎝ ⎠ enAeBlEdl Tu FMCagtMélenAkñúgsmIkar !%>@0 Tu TaMgGs;RtUv)anTb;edayEdkkgbiTCit nigEdkbeNþay. m:Um:g;rmYl Tu RtUv)anKNnaBImuxkat;EdlmanTItaMgRtg;cMgay d BIépÞénTMr nig Tu = φTn Edl φ = 0.75 . ]TahrN_TI15>1³ sMrab;muxkat;bIEdlbgðajenAkñúgrUbTI 15>12 nigQrelIkarkMNt; ACI Code cUrkMNt; a. m:Um:g;eFVIeGayeRbH φTcr b. m:Um:g;rmYlemKuNGtibrma φTn EdlGacGnuvtþelImuxkat;nImYy²edaymineRbIEdkRTnugTb;kar rmYl. snμt; f 'c = 28MPa / f y = 400MPa / kMras;ebtugkarBarEdk 40mm nigeRbIEdkkg DB12 . dMeNaHRsay³ 1> muxkat; ! a. mU:m:g;eFVIeGayeRbH φTcr GacRtUv)anKNnaBIsmIkar !%>!( f 'c ⎛ Acp ⎞ 2 φTcr = φ ⎜ ⎟ 3 ⎜ Pcp ⎟ ⎝ ⎠ Design for Torsion 360
  16. 16. T.Chhay NPIC sMrab;muxkat;enH Acp = xo yo RkLaépÞmuxkat;TaMgmUl Edl xo = 400mm nig yo = 610mm Acp = 400 × 610 = 244000mm 2 Pcp = brimaRténmuxkat;ebtugTaMgmUl = 2( xo + yo ) = 2(400 + 610 ) = 2020mm 28 ⎛ 244000 2 ⎞ ⎜ ⎟ = 39kN .m φTcr = 0.75 3 ⎜ 2020 ⎟ ⎝ ⎠ b. φTn GnuBaØatEdlGacGnuvtþedaymineRbIEdkTb;karrmYlRtUv)anKNnaBIsmIkar !%>@0 φTcr 39 Ta = = = 9.75kN .m 4 4 2> muxkat; @ a. dMbUgKNna Acp nig Pcp sMrab;muxkat;enH nigGnuvtþsmIkarTI !%>!( edIm,IKNna φTcr . edaysnμt;søabRtUv)andak;CamYyEdkkgbiTCit enaHsøabRbsiT§PaBEdlRtUveRbIenA RCugmçag²énRTnugesμInwg $dgkMras;søab b¤ 4(100) = 400mm = hw = 400mm Acp = web area + area of effective flanges Acp = 500 × 350 + 2 ×100 × 400 = 255000mm 2 Pcp = 2(b + h ) = 2(350 + 2 × 400 + 500) = 3300mm 28 ⎛ 255000 2 ⎞ ⎜ ⎟ = 26kN .m φTcr = 0.75 3 ⎜ 3300 ⎟ ⎝ ⎠ cMNaM³ RbsinebIsøabRtUv)anecal ehIyEdkTb;karrmYlRtUv)andak;EtenAkñúgRTnug enaH Acp = 350 × 500 = 175000mm 2 Pcp = 2(350 + 500) = 1700mm φTcr = 23.8kN .m b. φTn GnuBaØatEdlGacGnuvtþedaymineRbIEdkTb;karrmYl φTcr 26 Ta = = = 6.5kN .m 4 4 3> muxkat; 3 karKNnasMrab;kMlaMgrmYl 361
  17. 17. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa a. snμt;søabRtUv)andak;EdkkgbiTCit RbEcgRbsiT§PaBesμInwg hw = 370mm < 4 ×150 = 600mm Acp = 350 × 520 + 370 ×150 = 237500mm 2 Pcp = 2(b + h) = 2(350 + 370 + 520) = 2480mm 28 ⎛ 237500 2 ⎞ ⎜ ⎟ = 30kN .m φTcr = 0.75 3 ⎜ 2480 ⎟ ⎝ ⎠ cMNaM³ RbsinebIsøabRtUv)anecal enaH Acp = 350 × 520 = 182000mm 2 Pcp = 2(350 + 520) = 1740mm φTcr = 25.2kN .m b. φTn GnuBaØat φTn = φTcr = 30 = 7.5kN.m 4 4 8>4> m:Um:g;rmYllMnwg nwgm:Um:g;rmYlRtUvKña Equilibrium Torsion and Compatibility Torsion kñúgkarviPaKeRKOgbgÁáúMGgát;ebtug kMlaMgepSg²EdlGnuvtþrYmman kMlaMgEkg (normal force)/ m:Um:g;Bt; (bending moment)/ kMlaMgkat; (shear force) nigm:Um:g;rmYl Edl)anBnül;enAkñúg]TahrN_ TI 15>1. karKNnaGgát;ebtugGarem:KWQrelIkar)ak;rbs;Ggát;GMeBIrbs;bnÞúkemKuN. sMrab;Ggát;sþa TicminkMNt; (statically indeterminate member) karEbgEckm:Um:g;mþgeTot (redistribution of moments) ekItmanmuneBl)ak; dUcenHm:Um:g;KNnaGacnwgRtUv)ankat;bnßy b:uEnþ sMrab;Ggát;sþaTickM Nt; (statically determinate member) dUcCaFñwmsamBaØ (simple beam) b¤Fñwm cantilever Kμankar EbgEckm:Um:g;mþgeTotekIteLIgeT. enAkñúgkarKNnaGgát;Edlrgm:Um:g;rmYl eKmanBIrkrNIEdlGacGnuvtþbnÞab;BIkareRbH. !> krNIm:Um:g;rmYllMnwg (equilibrium torsion case) ekItmanenAeBlm:Um:g;rmYlEdlRtUvkar sMrab;eRKOgbgÁúMsßitkñúgsßanPaBlMnwg ehIy Tu minGacRtUv)ankat;bnßyedaykarEbg EckeLIgvijrbs;m:Um:g;eT dUckrNIFñwmTMrsamBaØ. kñúgkrNIenHEdkTb;rmYlRtUv)andak; edIm,ITb;RKb; Tu . rUbTI 15>13 FñwmEdlenAEKmRTkMralxNн cantilever EdlKμankar EbgEckm:Um:g;mþgeTotekItman. @> krNIm:Um:g;rmYlRtUvKña (compatibility torsion case) ekItmanenAeBlm:Um:g;rmYl Tu Gac RtUv)ankat;bnßyedaykarEbgEckkMlaMgkñúgmþgeTotbnÞab;BIeRbH enAeBlEdlPaBRtUvKña Design for Torsion 362
  18. 18. T.Chhay NPIC énkMhUcRTg;RTay (compatibility of deformation) RtUv)anrkSa enAkñúgGgát;eRKOgbgÁúM. rUbTI 15>14 bgðajBI]TahrN_sMrab;krNIenH EdlFñwmTTwgBIrmanGMeBIelIFñwmEKmbegáIt m:Um:g;rmYl. mMurmYlFMekItman enAeBlsñameRbHedaykarrmYlelcecj Edlpþl;nUvkar bgEckbnÞúkd¾FMenAkñúgeRKOgbgÁúM. vanwgeTAdl;m:Um:g;rmYlEdleFVIeGayeRbH Tcr eRkamGM eBIénbnSM karBt; karkat; nigkarrmYl enAeBlEdlkugRtaMgem (principle stress) mantM élRbEhl f 'c / 3 . enAeBlEdl Tu > Tcr m:Um:g;rmYlesμInwg Tcr ¬smIkar !%>!(¦ EdlGacsnμt;ekItmanenARtg;muxkat;eRKaHfñak;enACitépÞénTMr. ACI Code kMNt;m:Um:g;rmYlKNnaesμInwgtMéltUcCageKén Tu Edl)anBIbnÞúkemKuN b¤ φTcr BIsmIkar !%>!(. 8>5> karkMNt;énersIusþg;m:Um:g;rmYl Limitation of Tortional Moment Strength ACI Code,Section 11.6.3 kMNt;TMhMmuxkat;edaysmIkarxageRkamBIr³ !> sMrab;muxkat;tan; karKNnasMrab;kMlaMgrmYl 363
  19. 19. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa 2 2 ⎛ Vu ⎞ ⎛ Tu Ph ⎞ ⎡⎛ ⎞ ⎤ ⎜ ⎟ +⎜ ⎟ ≤ φ ⎢⎜ Vc ⎟ + 2 ⎜ b d ⎟ ⎜ 1 .7 A 2 ⎟ ⎜ ⎟ f 'c ⎥ ¬!%>@!¦ ⎝ w ⎠ ⎝ oh ⎠ ⎣⎝ bw d ⎠ 3 ⎦ @> sMrab;muxkat;Rbehag ⎛ Vu ⎞ ⎛ Tu Ph ⎞ ⎡⎛ ⎞ ⎤ ⎜ ⎟+⎜ ⎜ b d ⎟ ⎜ 1.7 A2 ⎟ ⎟ ≤ φ ⎢⎜ Vc ⎟ + 2 ⎜b d ⎟ 3 f 'c ⎥ ¬!%>@@¦ ⎝ w ⎠ ⎝ oh ⎠ ⎣⎝ w ⎠ ⎦ Edl Vc = f 'c bwd / 6 = ersIusþg;kMlaMgkat;sMrab;ebtugTMgn;Fmμta. GgÁdéTeTotRtUv)ankM Nt;enAkñúgEpñk 8>2. karkMNt;enHKWQrelIPaBCak;EsþgEdlfaplbUkénkugRtaMgEdlbNþalBIkMlaMgkat; nigm:Um:g; rmYl ¬GgÁxageqVg¦ minRtUvFMCagkugRtaMgEdleFVIeGayeRbHbUknwg 2 f 'c / 3 . krNIdUcKñaRtUv)an GnuvtþedIm,IKNnakMlaMgkat;edayKμanm:Um:g;rmYlenAkñúgemeronTI 8. eKRtUvkarkarkMNt; (limitation) edIm,Ikat;bnßysñameRbH nigedIm,IkarBarEbképÞebtugEdlbNþalmkBIkugRtaMgkMlaMgkat;TTwgeRTt nigm:Um:g;rmYleRTt. 8>6> muxkat;Rbehag Hollow Section bnSMénkugRtaMgkMlaMgkat; nigkugRtaMgm:Um:g;rmYlenAkñúgmuxkat;RbehagRtUv)anbgðajenAkñúgrUb 15>6 EdlkMras;CBa¢aMgRtUvOansnμt;faefr. enAkñúgmuxkat;RbehagxøH kMras;CBa¢aMgGacERbRbYlCMuvij brimaRt. sMrab;krNIenH smIkar !%>@@ RtUv)ankMNt;enATItaMgEdlGgÁxageqVgmantMélGtibrma. cM NaMfa enAnwgsøabxagelI nigsøabxageRkam CaTUeTAkugRtaMgkMlaMgkat;RtUv)anecal. CaTUeTA Rbsin ebIkMras;CBa¢aMgénmuxkat;Rbehag t tUcCag Aoh / Ph enaHsmIkar !%>@@ køayCa ⎛ Vu ⎞ ⎛ Tu Ph ⎞ ⎡⎛ V ⎞ 2 ⎤ ⎜ ⎟+⎜⎜ ⎜ b d ⎟ 1.7 A t ⎟ ⎟ ≤ φ ⎢⎜ c ⎟ + ⎜b d ⎟ 3 f 'c ⎥ ¬!%>@#¦ ⎝ w ⎠ ⎝ oh ⎠ ⎣⎝ w ⎠ ⎦ (ACI Code, Section 11.6.3) . 8>7> EdkRTnug Web Reinforcement dUcEdl)anBnül;rYcehIy viFI ACI Code sMrab;KNnaGgát;Edlrgm:Um:g;rmYlKWQrelI space truss analogy enAkñúgrUbTI 15>8. bnÞab;BIkareRbHedaykarrmYl eKRtUvkarEdkBIrRbePTedIm,ITb;nwg m:Um:g;rmYlEdlGnuvtþ Tu KW EdkTTwg (transverse reinforcement) At enAkñúgTMrg;CaEdkkgbiTCit nig EdkbeNþay (longitudinal reinforcement) Al . ACI Code )anbgðajnUvsmIkarxageRkamedIm,I KNna At nig Al ³ !> EdkkgbiTCit At EdlGacKNnadUcxageRkam³ Design for Torsion 364
  20. 20. T.Chhay NPIC 2 Ao At f yt cot θ Tn = s ¬!%>@$¦ Edl Tn = Tφu nig φ = 0.75 At = RkLaépÞéneCIgmYyrbs;EdkkgbiTCit f yt = ersIusþg;yal (yield strength) rbs; At At ≤ 400MPa s = KMlatEdkkg Ao nig θ RtUv)ankMNt;enAkñúgEpñk 8>2. smIkar !%>@$ GacRtUv)ansresrdUcxageRkam At = Tn s 2 A f cot θ ¬!%>@%¦ o yt RbsinebI θ = 45o enaH cot θ = 1.0 nigRbsinebI enaHsmIkar !%>@% køayCa f yt = 400MPa At = s 800 Ao Tn ¬!%>@^¦ Edl Tn KitCa N .mm . KMlatEdkkg s minRtUvFMCagéntMéltUcCageKkñúgcMeNam Ph / 8 nig 300mm . sMrab;muxkat; RbehagrgkarrmYl cMgayEdlvas;BIGkS½énEdkkgeTAépÞxagkñúgrbs;CBa¢aMgminRtUvtUcCag 0.5 Aoh / Ph . @> EdkbeNþaybEnßm Al EdlcaM)ac;sMrab;karrmYlminKYrtUcCagtMélxageRkam³ ⎛ A ⎞ ⎛ f yt ⎞ 2 Al = ⎜ t ⎟ Ph ⎜ ⎜ f ⎟ ⎟ cot θ ¬!%>@&¦ s ⎝ ⎠ ⎝ y ⎠ Rbsin θ = 45 nig o f yt = f y = 400MPa sMrab;TaMgEdkkg nigEdkbeNþay enaHsmIkar !%>@& køayCa ⎛A ⎞ ⎛A ⎞ Al = ⎜ t ⎟ Ph = 2⎜ t ⎟( x1 + y1 ) ¬!%>@*¦ ⎝ s ⎠ ⎝ s ⎠ Ph RtUv)ankMNt;enAkñúgEpñk 8>2. cMNaMfa EdkEdlcaM)ac;sMrab;karrmYlKYrRtUv)anbEnßmBIelI EdlEdlcaM)ac;sMrab;kMlaMgkat; m:Um:g;Bt; nigkMlaMgtamGkS½EdleFVIGMeBIrYmKñaCamYykMlaMgrmYl. kar kMNt;epSgeTotsMrab;EdkbeNþay Al mandUcxageRkam³ a. Ggát;p©itEdktUcbMputsMrab;EdkbeNþayKW DB10 b¤KMlatEdkkgelI 24 b¤ s / 24 edayykmYyNaEdlmantMéltUcCageK. b. EdkbeNþayKYrRtUv)anBRgayCMuvijbrimaRtrrbs;EdkkgCamYyKMlatGtibrma 300mm . karKNnasMrab;kMlaMgrmYl 365
  21. 21. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa c. EdkbeNþayKYrEtdak;enAkñúgEdkkg y:agehacNas;k¾dak;EdkenARKb;mMurbs;Edkkg. EdkEdldak;enAnwgmMurbs;EdkkgRtUv)aneKrkeXIjfamanRbsiT§PaBkñúgkarbegáItersIu sþg;m:Um:g;rmYl nigkñúgkarkarBarsñameRbH. d. EdkTb;m:Um:g;rmYlRtUvdak;enAcMgay (bt + d ) BIcMnucEdlRTwsþIRtUvkar Edl bt CaTTwgén Epñkrbs;muxkat;EdlmanEdkkgTb;kMlaMgrmYl. 8>8> EdkTb;karrmYlGb,brma Minimum Torsional Reinforcement enAkEnøgNaEdlEdkTb;karrmYlGb,brmaRtUvkar EdkTb;karrmYlGb,brmaRtUv)ankMNt;dUc xageRkam (ACI Code, Section 11.6.5) ³ !> EdkkgbiTCitGb,brmasMrab;bnSMénkMlaMgkat;TTwg nigkarrmYl ¬emIlEpñk 8>6¦³ Av + 2 At ≥ 0.35bw s f ¬sMrab; f 'c < 31MPa ¦ yt ⎛b s⎞ Av + 2 At ≥ 0.063 f 'c ⎜ w ⎟ ⎜ f yt ⎟ ¬sMrab; f 'c ≤ 31MPa ¦ ¬!%>@(¦ ⎝ ⎠ Edl Av = RkLaépÞeCIgTaMgBIrrbs;EdkkgEdlkMNt;)anBIkMlaMgkat; At = RkLaépÞeCIgEtmYyrbs;EdkkgEdlkMNt;BIm:Um:g;rmYl s = KMlatEdkkg f yt = ersIusþg;yal (yield strength) rbs;Edkkg ≤ 400 MPa KMlatEdkkg s minKYrFMCagtMéltUcCagkñúgcMeNam Ph / 8 nig 300mm . KMlatenHRtUvkar edIm,IRKb;RKgsñameRbH. @> RkLaépÞEdksrubGb,brmarbs;EdkbeNþayTb;karrmYl³ 5 f 'c Acp ⎛ At ⎞ ⎛ f yt ⎞ Al min = − ⎜ ⎟ Ph ⎜ ⎜ f ⎟ ⎟ ¬!%>#0¦ yf ⎝ s ⎠ ⎝ y ⎠ Edl At / s minRtUvyktUcCag 173bw / f yt . Al Gb,brmaenAkñúgsmIkar !%>#0 RtUv)ankMNt;edIm,Ipþl;nUvGRtaGb,brmaénmaDEdkTb;kM laMgrmYlelImaDebtug mantMélRbEhl 1% sMrab;ebtugGarem:EdlrgkMlaMgrmYlsuT§. 9> segçbviFIsaRsþKNnaeday ACI Code Summary of ACI Code Procedures viFIsaRsþKNnasMrab;bnSMkMlaMgkat;TTwg nigkMlaMgrmYlGacRtUv)ansegçbdUcxageRkam³ Design for Torsion 366
  22. 22. T.Chhay NPIC !> KNnakMlaMgkat;TTwgemKuN Vu nigm:Um:g;rmYlemKuN Tu BIkMlaMgEdlGnuvtþmkelIeRKOg bgÁúM. tMéleRKaHfñak;sMrab;kMlaMgkat;TTwg nigkMlaMgrmYlKWsßitenARtg;muxkat;EdlmancM gay d BIépÞrbs;TMr. @> a. eKRtUvkarEdkkMlaMgkat;TTwgenAeBl Vu > φVc / 2 Edl Vc = f 'c bwd / 6 . b. EdkTb;karrmYlRtUvkarenAeBlEdl f 'c ⎛ Acp ⎞ 2 Tu > φ 12 ⎜ ⎜P ⎟ ⎟ ¬!%>@0¦ ⎝ cp ⎠ RbsinebIEdkRTnugRtUvkar GnuvtþviFIsaRsþxageRkam. #> KNnasMrab;kMlaMgkat;TTwg a. KNnaersIuisþg;kMlaMgkat; nominal Edlpþl;edayebtug Vc . kMNt;kMlaMgkat;TTwg EdlTb;edayEdkRTnug³ V − φVc Vu = φVc + Vs b¤ Vs = u φ b. eRbobeFob Vs Edl)anKNnaCamYynwgtMélGnuBaØatGtibrma 2 f 'c bw d / 3 eyag tam ACI Code. RbsinebI Vs tUcbnþkarKNna EtpÞúymkvijtMeLIgTMhMmuxkat;rbs; ebtug. c. EdkRTnugkMlaMgkat;TTwgRtUv)anKNnadUcxageRkam³ Vs s Av = f yt d Edl Av =RkLaépÞéneCIgTaMgBIrrbs;Edkkg s = KMlatEdkkg EdkkMlaMgkat;TTwgkñúgmYyÉktþaRbEvgKW Av V = s s f yt d d. RtYtBinitü Av / s Edl)anKNnaCamYynwg Av / s Gb,brma³ Av ⎛b ⎞ ⎛ ⎞ (min) = 0.063 f 'c ⎜ w ⎟ ≥ 0.35⎜ bw ⎟ s ⎜ f yt ⎟ ⎜ f yt ⎟ ⎝ ⎠ ⎝ ⎠ Av Gb,brma RtUv)ankMNt;edaybTdæaneRkambnSMénGMeBIrbs;kMlaMgkat;TTwg nigkM laMgrmYlRtUv)aneGayenAkñúgCMhanTI5 karKNnasMrab;kMlaMgrmYl 367
  23. 23. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa $> KNnasMrab;karrmYl³ a. RtYtBinitüfaetIm:Um:g;rmYlemKuN Tu begáItm:Um:g;rmYllMnwg (equilibrium torsion) b¤m:Um:g;rmYlRtUvKña (compatibility torsion). sMrab; equilibrium torsion eRbI Tu . sMrab; compatibility torsion m:Um:g;rmYlKNnaKWtMéltUcCageKén Tu BIbnÞúkemKuN nig f 'c ⎛ Acp ⎞ 2 Tu 2 = φ 3 ⎜ ⎜P ⎟ ⎟ ¬!%>!(¦ ⎝ cp ⎠ b. RtYtBinitüfaetITMhMénmuxkat;RKb;RKan;b¤Gt;. vaTTYl)anedayeRbIsmIkar !%>@! sMrab; muxkat;tan; b¤smIkar !%>@@ sMrab;muxkat;Rbehag. RbsinebItMélenAGgÁxageqVgFM Cag φ (Vc / bwd + 2 f 'c / 3) enaHbegáInmuxkat; pÞúymkvijKNnabnþ. sMrab;muxkat; Rbehag RtYtBinitüfaetIkMras;CBa¢aMg t tUcCag Aoh / Ph b¤Gt;. RbsinebIvatUcCageRbI smIkar !%>@# pÞúymkvijeRbIsmIkar !%>@@. c. kMNt;EdkkgbiTCitcaM)ac;BIsmIkar !%>@% At = Tn s 2 A f cot θ ¬!%>@%¦ o yt At / s minRtUvtUcCag 173bw / f yt . ehIy mMu θ Gacsnμt;esμI 45o / Tn = Tu / φ nig φ = 0.75 . snμt; Ao = 0.85 Aoh = 0.85(x1 y1 ) Edl x1 nig y1 CaTTwg nigkMBs;rbs;muxkat;KitBIGkS½eTAGkS½Edkkg ¬emIlrUb TI !%>!!¦. sMrab; θ = 45o nig f y = 400MPa At = s 800 Ao Tn ¬!%>@^¦ KMlatGnuBaØatGtibrma s KWtMéltUcCageKén 300mm b¤ Ph / 8 . d. kMNt;EdkbeNþaybEnßmBIsmIkar !%>@&³ ⎛ A ⎞ ⎛ f yt ⎞ 2 Al = ⎜ t ⎟ Ph ⎜ ⎜ f ⎟ ⎟ cot θ ¬!%>@& a ¦ ⎝ s ⎠ ⎝ y ⎠ EtminRtUvtUcCag ⎛ 5 f 'c Acp ⎞ ⎛ A ⎞ ⎛ f yt ⎞ Al min = ⎜ ⎜ 12 f y ⎟ − ⎜ t ⎟P ⎜ ⎟ ⎝ s ⎠ h⎜ fy ⎟ ⎟ ¬!%>@& b ¦ ⎝ ⎠ ⎝ ⎠ Design for Torsion 368
  24. 24. T.Chhay NPIC sMrab; θ = 45o nig f yt = 400MPa enaH Al = ( At / s )Ph ¬!%>@*¦ EdkbeNþayTb;karrmYlKYrmanGgát;p©ity:agticesμIKMlatEdkkgelI 24 b¤ s / 24 b:uEnþ minRtUvtUcCag DB10 . EdkbeNþayRtUvdak;enAkñúgEdkkgbiTCitCamYyKMlatGtibrma esμI 300mm . y:agehaceKRtUvdak;EdkmYyedImenARKb;mMurbs;Edkkg. CaTUeTAmYy PaKbIén Edk Al RtUv)anbEnßmeTAelIEdkTaj mYyPaKbIenABak;kNþalkMBs;rbs; mux kat; nigmYyPaKbIeTotenAEpñksgát;. %> kMNt;RkLaépÞsrubénEdkkgbiTCitEdlbNþalBI Vu nig Tu . Avt = ( Av + 2 At ) ≥ 0.35bw s f ¬!%>@(¦ yt eRCIserIsEdkkgbiTCitsmrmüCamYyKMlat s EdlmantMéltUcCageKkñúgcMeNam 300mm nig Ph / 8 . EdkkgKYrRtUv)andak;enAcMgay (bt + d ) eRkaycMnucEdlRTwsþIRtUvkar Edl bt = TTwgén muxkat;EdlTb;nwgkMlaMgrmYl. ]TahrN_15>3³ (Equilibrium Torsion) kMNt;brimaNEdkRTnugcaM)ac;sMrab;muxkat;ctuekaNEkgdUcbgðajenAkñúgrUbTI 15>15. muxkat;rgnUvkM laMgkat;emKuN Vu = 213.5kN nigkMlaMgrmYllMnwg (equilibrium torsion) Tu = 41kN .m enATItaMg EdlmancMgay d BIépÞénTMr. eKeGay f 'c = 28MPa nig f y = 400MPa . karKNnasMrab;kMlaMgrmYl 369
  25. 25. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ CMhanxageRkambgðajBIviFIsaRsþkñúgkarKNna 1> kMlaMgKNnaKW Vu = 213.5kN nig Tu = 41kN .m 2> a. EdkTb;kMlaMgkat;RtUvkarenAeBl Vu > φVc / 2 . φ 28 (400 )(520) ⋅10 − 3 = 137.6kN 0.75 φVc = f 'c bd = 6 6 φV Vu = 213.5kN > c = 68.8kN 2 eKRtUvkarEdkTb;kMlaMgkat;. b. eKRtUvkarEdkTb;karrmYlenAeBl f 'c ⎛ Acp ⎞ 2 Tu > φ ⎜ ⎟ =T 12 ⎜ Pcp ⎟ a ⎝ ⎠ Acp = xo yo = 400 × 580 = 232000mm 2 Pcp = 2( xo yo ) = 2(400 + 520) = 1840mm 0.75 28 (232000 )2 − 6 Ta = 10 = 9.7 kN .m 12 × 1840 Tu = 41kN .m > 9.7kN .m EdkTb;kMlaMgrmYlRtUvkarcaM)ac;. cMNaMfa RbsinebI Tu tUcCag 9.7kN.m enaHEdkTb;kar rmYlnwgminRtUvkar b:uEnþEdkTb;kMlaMgkat;RtUvkar. 3> KNnakMlaMgkat;TTwg³ a. Vu = φVc + φVs / Vs = 101.2kN 28 (400)(520 ) = 733.8kN > Vs 2 2 b. Vs (max) = f 'c bd = 3 3 101.2 ⋅10 3 c. Av s V = s = f y d 400 × 520 = 0.5mm 2 / m ¬eCIgBIr¦ Av 2s = 0.25mm 2 / m ¬eCIgmYy¦ 4> KNnasMrab;karrmYl a. kMlaMgrmYlKNna Tu = 41kN .m . KNnalkçN³muxkat; edaysnμt;kMras;ebtugkarBar Edk 40mm nigeRbIEdkkg DB12 ³ x1 = 400 − 2(40 + 6 ) = 308mm Design for Torsion 370
  26. 26. T.Chhay NPIC y1 = 580 − 2(40 + 6 ) = 488mm CakarGnuvtþn_ eKGacsnμt; x1 = b − 90mm nig y1 = h − 90mm Aoh = x1 y1 = 308 × 488 = 150304mm 2 Ao = 0.85 Aoh = 127758.4mm 2 Ph = 2(x1 + y1 ) = 2(308 + 488) = 1592mm θ = 45o nig cot θ = 1.0 b. RtYtBinitüPaBRKb;RKan;rbs;muxkat;edayeRbIsmIkar !%>@!³ 2 2 ⎛ Vu ⎞ ⎛ Tu Ph ⎞ ⎡⎛ ⎞ ⎤ ⎜ ⎜ ⎟ ≤ φ ⎢⎜ Vc ⎟ + 2 ⎜b d ⎟ +⎜ ⎟ 2 ⎟ ⎜b d ⎟ 3 f 'c ⎥ ⎝ w ⎠ ⎝ 1.7 Aoh ⎠ ⎣⎝ w ⎠ ⎦ φVc = 137.6kN nig Vc = 183.5kN 2 2 ⎛ 137600 ⎞ ⎛ 41000000 × 1592 ⎞ Left − hand side = ⎜ ⎟ +⎜ ⎟ = 1.82 MPa ⎝ 400 × 520 ⎠ ⎝ 1.7 × 150304 2 ⎠ ⎛ 183500 2 ⎞ Right − hand side = 0.75⎜ + 28 ⎟ = 3.3MPa > 1.82MPa ⎝ 400 × 520 3 ⎠ muxkat;RKb;RKan; c. kMNt;EdkkgbiTCitcaM)ac;EdlbNþalBIkarrmYlBIsmIkar !%>@%³ At Tn = s 2 Ao f yt cot θ Tu 41 Tn = = = 54.7 kN .m cot θ = 1.0 Ao = 127758.4mm 2 φ 0.75 54.7 ⋅10 6 At = s 2 × 127758.4 × 400 = 0.535mm 2 / m ¬eCIgmYy¦ d. kMNt;EdkbeNþaybEnßmBIsmIkarTI !%>@&³ ⎛A ⎞ ⎛ f yt ⎞ 2 Al = ⎜ t ⎟ Ph ⎜ ⎟ cot θ ⎝ s ⎠ ⎜ fy ⎝ ⎟ ⎠ At = 0.535 Ph = 1592mm f yt = f y = 400MPa cot θ = 1.0 s Al = 0.535 × 1592 = 851.72mm 2 5 f 'c Acp ⎛ A ⎞ ⎛ f yt ⎞ Al (min) = − ⎜ t ⎟ Ph ⎜ ⎟ 12 f y ⎝ s ⎠ ⎜ fy ⎟ ⎝ ⎠ At Acp = 232000mm 2 = 0.535 f yt = f y = 400MPa s karKNnasMrab;kMlaMgrmYl 371
  27. 27. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa 5 28 (232000) Al (min) = − (0.535)(1592) = 427mm 2 12 × 400 Al = 851.72mm 2 lb; 5> kMNt;RkLaépÞEdkkgsrub a. sMrab;eCIgmYyrbs;Edkkg Avt At Av = + s s 2s EdkkgEdlcaM)ac; Avt = 0.535 + 0.25 = 0.785mm 2 / m ¬eCIgmYy¦ eRbIEdk DB12 RkLaépÞmuxkat;rbs;EdkkgsMrab;eCIgmYyKW 113mm 2 = 144mm yk 140mm 113 spacing of stirrups = 0.785 b. KMlatGtibrma s = h = = 199mm b¤ 300mm mYyNaEdltUcCag. P 1592 8 8 KMlatEdleRbIKW 140mm < 199mm 0.35bw 0.35 × 400 c. Avt / s Gb,brma = = = 0.35mm 2 / m < 0.785mm 2 / m f yt 400 6> edIm,IrkkarBRgayEdkbeNþay cMNaMfa Al srub = 851.72mm 2 . eRbImYyPaKbIenAEpñkxag elI b¤ 851.72 / 3 = 283.9mm 2 edIm,IbEnßmenAkñúgEdkrgkarsgát; A's . eRbImYyPaKbIdak;enA EpñkxageRkam edIm,IbEnßmBIelIEdkrgkarTaj nigEdkmYyPaKbIeTotdak;enAkMBs;Bak;kNþal. a. RkLaépÞEdksrubenAEpñkxagelIesμI 226 + 283.9 = 509.9mm 2 . eRbI 3DB16 ¬ As = 603mm 2 ¦ b. RkLaépÞEdksrubenAEpñkxageRkamesμI 3078.8 + 283.9 = 3362.7 mm 2 . eRbI 3DB 28 nig 2DB32 enARCugmMu ¬ As srub = 3455.8mm 2 ¦ Al srubEdleRbI = (603 − 226 ) + (3455.8 − 3078.8) = 754mm 2 c. enAkMBs;Bak;kNþal eRbIEdk 2DB12 ¬ As = 226mm 2 ¦ bøg;srésEdklMGitRtUv)anbgðajenAkñúgrUbTI 15>15. KMlatEdkbeNþayesμInwg 230mm EdltUcCagKMlatEdkGtibrmaEdlRtUvkar 300mm 2 . Ggát;p©itEdkkg DB12 Edl eRbIFMCagGgát;p©itGb,brma DB10 b¤KMlatEdkkgelI 24 ¬ s / 24 = 5.8mm ¦. ]TahrN_15>4³ (Compatibility Torsion) edaHRsay]TahrN_TI 15>3 eLIgvij RbsinebIkMlaMgrmYlemKuNCa compatibility torsion. dMeNaHRsay³ Design for Torsion 372
  28. 28. T.Chhay NPIC eyagtamdMeNaHRsaykñúg]TahrNITI 15>3 !> kMlaMgKNnaKW V u = 213.5kN nig compatibility torsion Tu = 41kN .m @> CMhan (a) nig (b) dUcKñaenAkñúg]TahrN_TI 15>3. eKRtUvkarEdkRTnug. #> CMhan (c) KWdUcKña. $> KNnasMrab;kMlaMgrmYl³ edaysar compatibility torsion Tu = 41kN .m enaH Tu KNnaRtUvtUvCag 41kN .m b¤ φTcr RtUv)aneGayenAkñúgsmIkar !%>!( f 'c ⎛ Acp ⎞ 0.75 28 ⎛ 232000 2 ⎞ − 6 2 φTcr = φ ⎜ ⎟= ⎜ ⎟ ⋅10 = 38.7 kN .m 3 ⎜ Pcp ⎟ 3 ⎜ 1840 ⎟ ⎝ ⎠ ⎝ ⎠ edaysarEt φTcr < Tu / eRbI Tu = 38.7kN .m . GnuvtþeLIgvijRKb;CMhanenAkñúg]TahrN_TI 15>3 edayeRbI Tu = 38.7kN .m edIm,IkMNt;famuxkat;RKb;RKan;. At s = 0.5mm 2 / m ¬eCIgmYy¦ Al = 0.5 × 1592 = 796mm 2 eRbI Al = 852mm 2 > Al (min) %> Avt caM)ac; = 025 + 0.5 = 0.75mm 2 / m ¬eCIgmYy¦ . 113 s= = 150.6mm 0.75 eRbI 150mm . eRCIserIsEdkbeNþay nigEdkkgdUckñúg]TahrN_TI 15>3. ]TahrN_15>5³ (L-section with Equilibrium Torsion) FñwmxagénRbBn§½kMralxNнrbs;GKardUcbgðajenAkñúgrUbTI 15>16. muxkat;enAcMgay d BIépÞénTMrrg nUv Vu = 235kN nig equilibrium torque Tu = 27kN .m . KNnaEdkRTnugcaM)ac;edayeRbI f 'c = 28MPa nig f y = 400MPa sMrab;RKb;EdkEdleRbIenAkñúgFñwm. dMeNaHRsay³ 1> kMlaMgKNnaKW Vu = 235kN nig Tu = 27kN .m 2> a. EdkTb;kMlaMgkat;RtUvkarenAeBl Vu > φVc / 2 φ f 'c 0.75 28 φVc = bw d = 350 × 455 ⋅10 − 3 = 105.3kN 6 6 φVc Vu > = 52.65kN 2 karKNnasMrab;kMlaMgrmYl 373
  29. 29. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa eKRtUvkarEdkkMlaMgkat;TTwg b. RtYtBinitüfaetIEdkTb;karrmYlRtUvkarb¤Gt;. snμt;fasøabcUlrYmkñúgkarTb;karrmYl RbEvg søabRbsiT§PaBKW hw = 380mm < 150 × 4 = 600mm . xo = 350mm nig yo = 530mm Acp = (350 × 530) + (150 × 380 ) = 242500mm 2 Pcp = 2(730 + 530) = 2520mm ⎛ 242500 2 ⎞ − 6 BIsmIkar !%>@0 Ta = 0.75 12 28 ⎜ ⎜ 2520 ⎟ ⎟ ⋅10 = 7.7kN .m ⎝ ⎠ Tu > Ta muxkat;RtUvkarEdkTb;karrmYl. 3> KNnaEdkTb;kMlaMgkat;TTwg³ a. Vu = φVc + φVs 235 = 105.3 + 0.75Vs Vs = 173kN 2 b. Vs (max) = f 'c bw d = 561.8kN > Vs 3 c. Av s V = s = 173000 f y d 400 × 455 = 0.95mm 2 / m ¬eCIgBIr¦ Av 0.95 = = 0.475mm 2 / m 2s 2 4> KNnaEdkTb;karrmYl³ Tu = 27kN .m a. KNnalkçN³muxkat;edaysnμt; kMras;ebtugkarBarEdk 40mm nigEdkkg DB12 . RTnug x1 = 350 − (2 × 40) − 12 = 258mm y1 = 530 − (2 × 40) − 12 = 438mm søab x1 = 380mm ¬EdkkghYscUleTAkñúgRTnug¦ y1 = 150 − 92 = 58mm Aoh = (58 × 380) + (258 × 438) = 135044mm 2 Ao = 0.85 Aoh = 114787.4mm 2 Ph = 2(58 + 380) + 2(258 + 438) = 2268mm θ = 45o cot θ = 1.0 Design for Torsion 374
  30. 30. T.Chhay NPIC b. RtYtBinitüPaBRKb;RKan;rbs;muxkat;edayeRbIsmIkarTI !%>@!³ Vu = 235kN / φVc = 105.3kN / Vc = 140.4kN / Tu = 27kN .m ⎛ 235000 ⎞ ⎛ 27 ⋅10 × 2268 ⎞ 2 6 Left − hand side = ⎜ ⎟ +⎜ ⎟ = 2.47 MPa ⎝ 350 × 455 ⎠ ⎜ 1.7 × 135044 2 ⎟ ⎝ ⎠ ⎛ 140400 2 ⎞ Right − hand side = 0.75⎜ + 28 ⎟ = 3.3MPa ⎝ 350 × 455 3 ⎠ muxkat;manlkçN³RKb;RKan; c. kMNt;EdkkgbiTCitedIm,ITb;karrmYl At / s BIsmIkar !%>@%³ 27 ⋅ 10 6 At s = Tn = 2 Ao f yt cot θ 0.75 × 2 × 114787.4 × 400 = 0.392mm 2 / m ¬eCIgmYy¦ d. KNnaEdkbeNþaybEnßmBIsmIkar !%>@* ¬sMrab; f 'c = 400 MPa nig cot θ = 1.0 ¦ ⎛A ⎞ Al = ⎜ t ⎟ Ph = 0.392 × 2268 = 889mm 2 ⎝ s ⎠ Al min ¬BIsmIkar !%>#0¦ KW 5 28 × 242500 Al min = − 889 = 447.7mm 2 12 × 400 karcUlrYmrbs;søabRtUv)anecaledaysarTTYl)anlT§plxusKñatictYc nigtMélBlkmμ ticCag. 5> kMNt;RkLaépÞmuxkat;EdkkgbiTCit a. sMrab;eCIgmYy vt = t + v A A A s s 2s muxkat;cM)ac; Avt = 0.392 + 0.475 = 0.867mm 2 / m ¬eCIgmYy¦ eRCIserIsEdk DB12 ¬ As = 113mm 2 ¦ KMlatEdkkg = 0113 = 130mm eRbI 125mm .867 b. KMlatEdkGtibrma s max = h = = 283.5mm . eRbI s = 125mm dUckarKNna. P 2268 8 8 Avt 0.35bw 0.35 × 350 c. = = = 0.31mm 2 / m < 0.867mm 2 / m dUcenHeRbI s f yt 400 DB12 @125 6> kMNt;karBRgayrbs;EdkbeNþay. Al srubKW 889mm 2 . eRbImYyPaKbI b¤ 889 / 3 = 296.3mm 2 enAEpñkxagelI EpñkkNþal nigEpñkxageRkam. karKNnasMrab;kMlaMgrmYl 375
  31. 31. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa a. brimaNEdksrubenAEpñkxagelI = 628.3 + 296.3 = 924.6mm 2 eRbI 3DB20 ¬ As = 942.5mm 2 ¦ b. brimaNEdksrubenAEpñkxagelI = 2463 + 296.3 = 2759.3mm 2 eRbI 5DB28 ¬ As = 3078.8mm 2 ¦ Al srubEdleRbI = (942.5 − 628.3) + (3078.8 − 2463) = 930mm 2 c. eRbIEdk 2DB12 enABak;kNþalkMBs; ¬ As = 226mm 2 ¦. bøg;EdklMGitRtUv)anbgðajenA kñúgrUbTI 15>16. KMlatEdkbeNþayKW 190mm < 300mm . Ggát;p©itrbs;EdkkgKW 12mm EdlFMCagGgát;p©itEdk DB10 b¤KMlatEdkkgelI 24 ¬ s / 24 = 5.2mm ¦. bEnßmEdkbeNþay DB12 enARKb;mMurbs;EdkkgenAkñúgRTnugFñwm nigsøabFñwm. Design for Torsion 376
  32. 32. T.Chhay NPIC karKNnasMrab;kMlaMgrmYl 377

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