• Share
  • Email
  • Embed
  • Like
  • Save
  • Private Content
Xv design for torsion
 

Xv design for torsion

on

  • 851 views

 

Statistics

Views

Total Views
851
Views on SlideShare
729
Embed Views
122

Actions

Likes
2
Downloads
0
Comments
0

3 Embeds 122

http://civilnpic.wordpress.com 60
http://civilnpic.wordpress.com 60
http://channakengineer.blogspot.com 2

Accessibility

Categories

Upload Details

Uploaded via as Adobe PDF

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

    Xv design for torsion Xv design for torsion Document Transcript

    • 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
    • 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
    • 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
    • T.Chhay NPIC karKNnasMrab;kMlaMgrmYl 349
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • T.Chhay NPIC karKNnasMrab;kMlaMgrmYl 377