Iv.flexural design of prestressed concrete elements

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  • 1. T.Chhay IV. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; Flexural Design of Prestressed Concrete Elements 1> esckþIepþIm Introduction kugRtaMgBt;CalT§plénbnÞúkxageRkA nigm:Um:g;Bt;. kñúgkrNICaeRcIn vaCaGñkkMNt;kñúgkar eRCIserIsTMhMFrNImaRtrbs;ebtugeRbkugRtaMgedayminKitfavargkarTajCamun (pretensioned) b¤rg karTajCaeRkay (post-tensioned) eT. dMeNIrkarKNnacab;epþImCamYynwgkareRCIserIsmuxkat; bzm nigedaykarsakl,g nigkarEktMrUveKnwgTTYl)anmuxkat;cugeRkayCamYynwgTMhMlMGitrbs;muxkat; ehIynwgTMhM nigKnøgrbs;EdkeRbkugRtaMg. muxkat;RtUvbMeBjnUvkarkMNt;rbs;kugRtaMgBt;EdlRtUvkar rbs;ebtug nigEdk. bnÞab;BIenH vaRtUv)anviPaK nigbMeBjktþamYycMnYneTotdUcCa lT§PaBrgkarkat; lT§PaBrgkarrmYl PaBdab nigsñameRbH. edaysarTinñn½ysMrab;karviPaKxusKñaBITinñn½yEdlcaM)ac;sMrab;karKNna karKNnaTaMgGs;Ca karviPaK. dMbUgeKsnμt;lkçN³muxkat;FrNImaRtEdlRtUvrgeRbkugRtaMg nigbnÞab;mkeKcab;epþÍmkMNt; faetImuxkat;GacrgkMlaMgeRbkugRtaMg nigkMlaMgGnuvtþn_xageRkA)anedaysuvtßiPaBb¤k¾Gt;. dUcenHeyIg RtUvyl;BIeKalkarN_mUldæanénkarviPaK nigkarKNnamuxkat;EdlmanlkçN³sMrYly:agxøaMgEdl)an ENnaMkñúgemeronenH. dUc)aneXIjBICMBUkTI1 lkçN³emkanicmUldæanrbs;sMPar³ eKalkarN_lMnwgrbs; m:Um:g; couple xagkñúg nigeKalkarN_eGLasÞicéntMrYtpl (superposition) RtUv)aneRbIenARKb;dMNak; kalénkardak;bnÞúk. eKKNnamuxkat;ebtugGarem:rgkugRtaMgBt;EtkñúgsßanPaBkMNt;énkugRtaMgenAeBl)ak;sMrab; muxkat;EdleRCIserIs RbsinebIvabMeBjnUvtMrUvkard¾éTeTotdUcCa serviceability, lT§PaBkñúgkarkat;/ nigPaBs¥itrvagebtug nigEdk. b:uEnþ kñúgkarKNnaGgát;ebtugeRbkugRtaMg eKcaM)ac;RtUveFVIkarRtYtBinitü bEnßmeTotenAeBlepÞrkMlaMg nigsßanPaBkMNt;enAeBlrgbnÞúkeFVIkar k¾dUcCasßanPaBkMNt;enA eBl)ak;. karRtYtBinitüTaMgenHmansar³sMxan;sMrab;Fanafa sñameRbHedaysarbnÞúkeFVIkarGac ecal)an ehIyeKGacRKb;RKg)annUvT§iBlry³eBlyUrrbs;PaBdab nigPaBekag. eKeRbIsBaØadkedIm,IsMKal;kugRtaMgsgát; ehIyeKeRbIsBaØabUkedIm,IsMKal;kugRtaMgTajenAkñúg muxkat;ebtug. ragekag (convex or hogging shape) rbs;Ggát;bgðajm:Um:g;GviC¢man ehIyragpt (concave or sagging) bgðajmU:m:g;viC¢man dUcbgðajenAkñúgrUbTI 4>1. Flexural Design of Prestressed Concrete Elements 90
  • 2. NPIC mindUcKñaniwgkrNIGgát;ebtugGarem: kugRtaMgrbs;ebtugERbRbYleTAtamdMNak;kalepSg²én kardak;bnÞúkefr nigbnÞúkGefr. xageRkamCakarsegçbénkardak;bnÞúkTaMgenH³ eRkayeBlGnuvtþkMlaMgeRbkugRtaMgedIm Pi kMlaMgenHRtUv)anepÞrBIkabeRbkugRtaMgeTAebtug. TMgn;pÞal;TaMgGs; WD manGMeBIeTAelIGgát;rYmCamYynwgkMlaMgeRbkugRtaMgedIm RbsinebIGgát; enaHRTedayTMrsamBaØ ¬vaminmanTMrenAkNþalElVg¦. bnÞúkefrbEnßmTaMgGs; WSD edayrYmTaMg topping sMrab; composite action RtUv)anGnuvtþ eTAelIGgát;. kMhatbg;kMlaMgeRbkugRtaMgry³eBlxøIbMputekItman EdlnaMeGaymankarkat;bnßykMlaMg eRbkugRtaMg Peo . Ggát;rgnUvbnÞúkeFVIkareBjeljCamYynwgkMhatbg;ry³eBlyUrEdlbNþalmkBI creep, shrinkage nig stand relaxation EdlnaMeTAdl; net prestressing force Pe . bnÞúkelIsEdlmanGMeBIelIGgát;ekItmaneRkamlkçxNÐxøHEdlnaMdl;sßanPaBkMNt;enAeBl)ak;. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 91
  • 3. T.Chhay rUbTI 4>2 bgðajBICMhanénkardak;bnÞúk nigkarBRgaykugRtaMgelImuxkat;EdlRtUvnwgkardak; bnÞúktamCMhannImYy². ehIyrUbTI 4>3 bgðajBIdüaRkambnÞúk-kMhUcRTg;RTay ¬ekag b¤pt¦ sMrab; kardMNak;kalénkardak;bnÞúktaMgBIeBlTTYlT§iBlénTMgn;pÞal;rhUtdl;eBl)ak;. 2> kareRCIserIslkçN³FrNImaRténmuxkat; Selection of Geometrical Properties of Section Components k> eKalkarN_ENnaMTUeTA General Guideline eRkamlkçxNÐbnÞúkeFVIkar FñwmRtUv)ansnμt;famanlkçN³esμIsac; (homogenous) nigeGLasÞic. ehIyeKsnμt; ¬edaysarkarrMBwgTuk¦ fakMlaMgsgát;eRbkugRtaMgEdlbBa©ÚneTAebtugesÞIreFVIeGaysrés rgkarTajrbs;FñwmekItmansñameRbH dUcenHeKcat;Tukmuxkat;FñwmCamuxkat;KμansñameRbH (uncracked Flexural Design of Prestressed Concrete Elements 92
  • 4. NPIC section) . karviPaKkugRtaMgrbs;FñwmeRbkugRtaMgeRkamlkçxNÐTaMgenHminxusKñaBIkarviPaKkugRtaMgrbs; FñwmEdk ¬Edlkan;Etc,as;CagenH KW beam column¦. vaEtgEtmankMlaMgtamG½kSEdlbNþalBI kMlaMgeRbkugRtaMgeTaHbICaman b¤Kμanm:Um:g;Bt;EdlbNþalBIbnÞúkpÞal; b¤bnÞúkxageRkAd¾éTeTotk¾eday. dUc)aneXIjenAkñúgCMBUk1 vaCakarRbesIrEdlKnøgrbs;EdkeRbkugRtaMgcakp©itenARtg;muxkat; eRKaHfñak; dUcCamuxkat;kNþalElVgsMrab;FñwmTMrsamBaØ nigmuxkat;elITMrsMrab;FñwmCab;. RbsinebIeK eFVIkareRbobeFobrvagmuxkat;ctuekaN muxkat;EdlmansøabminsIuemRTImanRbsiT§PaBCagedaykareRbI R)as;ebtug nigkarRbmUlpþúMebtugenAkñúgtMbn;sgát;énmuxkat;EdleKRtUvkarCageK. x> m:UDulmuxkat;Gb,brma Minimum Section Modulus edIm,IKNna nigeRCIserIsmuxkat; CadMbUgeKRtUvkMNt;m:UDulmuxkat;EdlRtUvkar Sb nig S t . RbsinebI³ f ci = kugRtaMgsgát;GnuBaØatGtibrmaenAkñúgebtugeRkayeBlepÞrPøam² munnwgmankMhatbg; = 0.60 f 'ci kugRtaMgTajGnuBaØatGtibrmaenAkñúgebtugeRkayeBlepÞrPøam² munnwgmankMhatbg; f ti = = 3 f 'ci psi (0.25 f 'ci MPa ) ¬eKGacbegáIntMélenHdl; 6 f 'ci psi (0.5 f 'ci MPa ) enARtg;TMrsMrab;Ggát;TMrsmBaئ f c = kugRtaMgsgát;GnuBaØatGtibrmaenAkñúgebtugeRkayeBlepÞrenAeBlrgbnÞúkeFVIkar = 0.45 f 'c b¤ 0.60 f 'c enAeBlGnuBaØatedaykUd f t = kugRtaMgTajGnuBaØatGtibrmaenAkñúgebtugeRkayeBlepÞrenAeBlrgbnÞúkeFVIkar = 6 f 'ci psi (0.5 f 'ci MPa ) ¬eKGacbegáIntMélenHdl; 12 f 'ci psi ( f 'ci MPa ) enA kñúgRbB½n§mYyTis RbsinebIeKRtUvkarKNnaPaBdabry³eBlyUr¦. kugRtaMgsrésxageRkACak;EsþgenAkñúgebtugminGacFMCagkugRtaMgGnuBaØatEdl)anerobrab;xag elIeLIy. edayeRbImuxkat;minsIuemRTIGt;eRbH karsegçbénsmIkarkugRtaMgEdl)anBICMBUk 1EpñkTI 3 sM rab;dMNak;kalénkardak;bnÞúkepSg²mandUcxageRkam³ kugRtaMgenAeBlepÞr Stress at Transfer Pi ⎛ ect ⎞ M D ft =− ⎜1 − 2 ⎟ − t ≤ f ti (4.1a) Ac ⎝ r ⎠ S karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 93
  • 5. T.Chhay Pi ⎛ ecb ⎞ M D fb = − ⎜1 + 2 ⎟ + ≤ f ci (4.1b) Ac ⎝ r ⎠ Sb Edl Pi CakMlaMgeRbkugRtaMgedIm. eKKYreRbIbgÁúMkMlaMgedkrbs; Pi edIm,ITTYl)antMélkan;EtsuRkitCag. EtsMrab;karGnuvtþTaMgGs;eKmin)anKitdl;PaBRbesIrenHeT. kugRtaMgRbsiT§PaBeRkaykMhatbg; Effective Stress after Losses ⎛ ect ⎞ M D Pe ft =− ⎜1 − 2 ⎟ − t ≤ f t (4.2a) ⎝ Ac r ⎠ S P ⎛ ec ⎞ M f b = − e ⎜1 + 2b ⎟ + D ≤ f c (4.2b) Ac ⎝ r ⎠ Sb kugRtaMgénbnÞúkeFVIkarcugeRkay Service-load Final Stresses Pe ⎛ ect ⎞ M T ft =− ⎜1 − 2 ⎟ − t ≤ f c (4.3a) Ac ⎝ r ⎠ S P ⎛ ecb ⎞ M f b = − e ⎜1 + 2 ⎟ + T ≤ f t (4.3b) Ac ⎝ r ⎠ Sb Edl M T = M D + M SD + M L Pi = kMlaMgeRbkugRtaMgedIm Pe = kMlaMgeRbkugRtaMgRbsiT§PaBeRkayeBlxatbg;kMlaMgeRbkugRtaMg t bgðajfasrésxagelI nig b bgðajfasrésxageRkam e = cMNakp©itrbs; tendon BITIRbCMuTMgn;rbs;munkat;ebtug cgc (center of gravity of concrete section) q r 2 = kaer:énkaMniclPaB S t / Sb = m:UDulmuxkat;srésxagelI nigxageRkamrbs;muxkat;ebtug dMNak;kalénkacuHfykMlaMgsgát; (decompression) bgðajkarekIneLIgbMErbMrYlrageFob rbs;EdkEdlbNþalBIkarekIneLIgrbs;bnÞúk taMgBIdMNak;kalEdlkMlaMgeRbkugRtaMgRbsiT§PaB Pe eFVIGMeBIEtÉkÉgrhUtdl; dMNak;kalEdlbnÞúkbEnßmeFVIeGaykugRtaMgsgát;rbs;ebtugenARtg;nIv:U cgs kat;bnßydl;sUnü¬emIlrUb TI 4>3¦. enARtg;dMNak;kalenH bMErbMrYlkugRtaMgebtugEdlbNþalBI decompression KW Pe ⎛ e2 ⎞ f decomp = ⎜1 + ⎟ (4.3c) Ac ⎜ r2 ⎟ ⎝ ⎠ Flexural Design of Prestressed Concrete Elements 94
  • 6. NPIC TMnak;TMngenHQrelIkarsnμt;fabMErbMrYlrageFob (strain) rbs;ebtug nigEdkeRbkugRtaMgEdls¥itCab; eTAnwgebtugEk,reFVIeGaykarekIneLIgénkugRtaMgEdkesμInwgkarfycuHénkugRtaMgebtug. 1. FñwmEdlmancMNakp©itEdkeRbkugRtaMgERbRbYl Beam with Variable Tendon Eccentricity FñwmrgnUvkMlaMgeRbkugRtaMgCamYynwg tendon Edl harped b¤ draped. CaTUeTAcMNakp©itGti- brmaEtgEtsßitenARtg;muxkat;kNþalElVgsMrab;krNIFñwmTMrsamBaØ. edaysnμt;fakMlaMgeRbkugRtaMg RbsiT§PaBKW Pe = γPi Edl γ CapleFobkMlaMgeRbkugRtaMgEdlenAsl; (residual prestress ratio) kMhatbg;énkMlaMgeRb kugRtaMgKW Pi − Pe = (1 − γ )Pi (a) RbsinebIkugRtaMgsrésxageRkAbMputrbs;ebtugCak;EsþgsmmUleTAnwgkugRtaMgGnuBaØat BIsmIkar 4.1a nig b eyIgTTYl)anbMErbMrYlkugRtaMgenHeRkayeBlxatbg;kMlaMgeRbkugRtaMgdUcxageRkam³ ⎛ M ⎞ Δf t = (1 − γ )⎜ f ti + tD ⎟ (b) ⎝ S ⎠ ⎛ M ⎞ Δf b = (1 − γ )⎜ − f ci + D ⎟ ⎜ (c) ⎝ Sb ⎟ ⎠ BIrUb 4>4 (a) edaysarm:Um:g;bnÞúkefrbEnßm M SD nigm:Um:g;bnÞúkGefr M L manGMeBIeTAelIFñwm kugRtaMg suT§ (net stress) enAsrésxagelIKW f nt = f ti − Δf t − f c b¤ f nt = γf ti − (1 − γ ) tD − f c M S (d) Net stress enAsrésxageRkamKW f bn = f t − f ci − Δf b b¤ f bn = f t − γf ci − (1 − γ ) D M Sb (e) BIsmIkar (d) nig (e) muxkat;EdlRtUveRCIserIsmanm:UDulmuxkat;dUcxageRkam St ≥ (1 − γ )M D + M SD + M L (4.4a) γf ti − f c ehIy Sc ≥ (1 − γ )M D + M SD + ML (4.4b) f t − γf ci karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 95
  • 7. T.Chhay cMNakp©itEdlRtUvkarrbs;EdkeRbkugRtaMgenARtg;muxkat;eRKaHfñak; dUcCamuxkat;kNþalElVg KW ( )S t MD ec = f ti − f ci + (4.4c) P i Pi Edl f ci CakugRtaMgrbs;ebtugenAeBlepÞrRtg;nIv:UénTIRbCMuTMgn; cgc rbs;muxkat;ebtug ehIy Pi = f ci Ac dUcenH f ci = f ti − ct ( f ti − f ci ) (4.4d) h Flexural Design of Prestressed Concrete Elements 96
  • 8. NPIC 2. FñwmEdlmancMNakp©itEdkeRbkugRtaMgefr Beam with Constant Tendon Eccentricity FñwmEdlmancMNakp©itEdkeRbkugRtaMgefrCaFñwmEdlman tendon Rtg; dUckñúgkrNIFñwmeRbkug RtaMgTMrsamBaØcak;eRscEdlmantMéllμm. edaysar tendon mancMNakp©itFMenARtg;TMr vaeFVIeGay mankugRtaMgTajFMenAsrésxagelIedayminmankarkat;bnßyNamYyedaym:Um:g;bnÞúkbEnßm M D + M SD + M L eT. b¤eKGacniyaymü:ageTotfa sMrab;FñwmEbbenH muxkat;eRKaHfñak;KWsßitenARtg;TMr ehIykarBRgaykugRtaMgenARtg;TMrRtUv)anbgðajenAkñúgrUbTI 4>4 (b). dUcenH Δf t = (1 − γ )( f ti ) (a’) ehIy Δf b = (1 − γ )(− f ci ) (b’) Net stress enAsrésxagelI sMrab;lkçxNÐbnÞúkeFVIkareRkaykMhatbg;KW f nt = f ti − Δf t − f c b¤ f nt = γf ti − f cs (c’) karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 97
  • 9. T.Chhay Edl fcs CakugRtaMgbnÞúkeFVIkarCak;EsþgenAkñúgebtug. Net stress enAsrésxageRkamsMrab;lkçxNÐ bnÞúkeFVIkareRkaykMhatbg;KW f bn = f t − f ci − Δf b b¤ Δf bn = f t − γf ci (d’) BIsmIkar (c’) nig (d’) muxkat;EdlRtUveRCIserIsRtUvmanm:UDulmuxkat;dUcxageRkam³ M D + M SD + M L St ≥ (4.5a) γf ti − f c M + M SD + M L ehIy Sb ≥ D f t − γf ci (4.5b) cMNakp©itEdlRtUvkarenARtg;muxkat;eRKaHfñak; dUcCaRtg;TMrsMrab;muxkat;EdlmanlkçN³RsedogKñanwg GVIEdlRtUvkaredaysmIkar 4.5a nig b KW ( )S t ee = f ti − f ci (4.5c) P i RkaPictMNageGaym:UDulmuxkat;rbs; nominal section RtUv)anbgðajenAkñúg rUbTI 4>5. eKGaceRbIva kñúgkareRCIserIsmuxkat;sakl,gdMbUgkñúgdMeNIrkarKNna. Flexural Design of Prestressed Concrete Elements 98
  • 10. NPIC tarag 4>1 eGaynUvtMélm:UDulmuxkat;énmuxkat;ctuekaNEkg PCI sþg;dar. tarag 4>2 eGaynUvxñatxageRkAénmuxkat;GkSr T rbs; PCI sþg;dar nigmuxkat;GkSr I rbs; AASTHO erogKña k¾dUcCam:UDul muxkat;srésxagelIénmuxkat;TaMgenaHEdlRtUvkarkñúgkareRCIserIsmuxkat;bzmsMrab;kar viPaKeRkamlkçxNÐbnÞúkeFVIkar. tarag 4>4 (a) pþl;nUvxñatlMGiténragFrNImaRt “as built” én PCI sþg;dar nigmuxkat; AASTHO ehIytarag 4>4 (b) pþl;nUvlkçN³muxkat;rbs; girder EdleRbIenA kñúgrdæepSg². lkçN³ bulb section manenAkñúg]bsm<½n§ (appendix) C. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 99
  • 11. T.Chhay Flexural Design of Prestressed Concrete Elements 100
  • 12. NPIC karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 101
  • 13. T.Chhay Flexural Design of Prestressed Concrete Elements 102
  • 14. NPIC 3> ]TahrN_sMrab;karKNnaeRkamlkçxNÐbnÞúkeFVIkar Service-Load Design Examples k> cMNakp©itrbs;EdkeRbkugRtaMERbRbYl Variable Tendon Eccentricity ]TahrN_ 4>1³ KNnaFñwmeRbkugRtaMgmuxkat;GkSr T Dub sMrab;eFVIcMNtrfynþ. FñwmenHmanRbEvg 60 ft (18.3m ) nwgRtUv)anRTedayTMrsamBaØ. EdkeRbkugRtaMgEdleRbIenAkñúgFñwmenHRtUv)an harped. eKeRbIkugRtaMgGnuBaØatrbs; ACI 318 Building code. FñwmenHRtUvRTbnÞúkeFVIkarbEnßm 1,100 plf (16.1kN / m ) nigbnÞúkefrbEnßm 100 plf (1.5kN / m ) nigminman concrete topping eT. snμt;faeKeFVI FñwmenHedayeRbIebtugTMgn;Fmμta (normal-weight concrete) Edlman f 'c = 5,000 psi (34.5MPa ) ehIykugRtaMgebtugenAeBlepÞr f 'ci esμInwg 75% én f 'c . ehIysnμt;fakMhatbg;GaRs½ynwgeBl rbs;kMlaMgeRbkugRtaMgedImesμInwg 18% énkMlaMgeRbkugRtaMgedIm ehIy ultimate strength rbs;Edk eRbkugRtaMg f pu = 270,000 psi (1,862MPa ) sMrab; stress-relieved tendon nig f 't = 12 f 'c psi ( f 'c MPa ) . dMeNaHRsay³ γ = 100 − 18 = 82% f 'ci = 0.75 × 5,000 = −3,750 psi (25.9MPa ) eRbI f 't = 12 5,000 = 849 psi(5.9MPa ) CakugRtaMgrgkarTajGtibrma ehIysnμt;TMgn;xøÜn Rbhak;RbEhlnwg 1,000 plf (14.6kN / m). kMNt;m:Umg;Edl)anBITMgn;pÞal; wl 2 1,000(60 )2 MD = = × 12 = 5,400,000in. − lb(610kN .m ) 8 8 ehIym:Um:g;Edl)anBIbnÞúkbEnßmKW M SD + M L = (1,100 + 100)(60)2 × 12 = 6,480,000in. − lb(732kN .m ) 8 muxkat;eRKaHfñak;sßitenAEk,rkNþalElVg CakEnøgEdlm:Um:g;Edl)anBIbnÞúkefr nigbnÞúkefr bEnßmmantMélGtibram nigedaysar tendon RtUv)an harped dUcenHkñúgkrNIPaKeRcInmuxkat; eRKaHfñak;RtUv)anykenARtg; 0.40L BITMr Edl L CaElVgFñwm. BIsmIkar 4.4a nig b eyIg)an St ≥ (1 − γ )M D + M SD + M L γf ti − f c ≥ (1 − 0.82)5,400,000 + 6,480,000 = 3,104in3 (50,860cm3 ) 0.82 × 184 + 2,250 karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 103
  • 15. T.Chhay Sb ≥ (1 − γ )M D + M SD + M L f t − γf ci ≥ (1 − 0.82)5,400,000 + 6,480,000 = 2,766in3 45,330cm3 ( ) 849 + (0.82 × 2,250 ) BI eRCIserIs nontopped normal weight concrete double-T 12DT PCI design handbook 34 168-D1 edaysarvamantMélm:UDulmuxkat;srésxageRkamEk,rtMélEdlRtUvkarCageK. lkçN³muxkat;rbs;ebtugmandUcxageRkam³ Ac = 978in.2 ct = 8.23in. I c = 86,072in.4 cb = 25.77in. I r 2 = c = 88.0in.2 e c = 22 . 02 in . Ac S t = 10,458in.3 ee = 12.77in. Sb = 3,340in.3 WD = 1,019 plf V = 2.39in. S KNna strands nigRtYtBinitükugRtaMg BIrUbTI 4>7 TMgn;xøÜnEdlsnμt;mantMélEk,rTMgn;xøÜnCak;Esþg. KNnam:Um:g;Edl)anBITMgn;pÞal;Cak;EsþgBIm:Um:g;Edl)anBITMgn;pÞal;snμt; 1,019 MD = × 5,400,000 = 5,502,600in. − lb 1,000 f pi = 0.70 × 270,000 = 189,000 psi f pe = 0.82 f pi = 0.82 × 189,000 = 154,980 psi Flexural Design of Prestressed Concrete Elements 104
  • 16. NPIC (a) viPaKkugRtaMgenAeBlepÞr BIsmIkar 4.1a Pi ⎛ ect ⎞ M D ft =− ⎜1 − 2 ⎟ − t ≤ f ti = 184 psi Ac ⎝ r ⎠ S P ⎛ 22.02 × 8.23 ⎞ 5,502,600 bnÞab;mk 184 = − i ⎜1 − 978 ⎝ 88.0 ⎟− ⎠ 10,458 Pi = (184 + 526.16) 978 = 655,223lb 1.06 cMnYn tendon EdlRtUvkar = 655,223 189,000 × 0.153 = 22.66 edImtendon EdlmanGgát;p©it 1 / 2in. sakl,g tendon Ggát;p©it 1 / 2in. cMnYn 16 edIm sMrab;muxkat;sþg;dar Aps = 16 × 0.153 = 2.448in.2 ( .3cm 2 ) 15 Pi = 2.448 × 189,000 = 462,672lb(2,058kN ) Pe = 2.448 × 154,980 = 379,391lb(1,688kN ) (b) viPaKkugRtaMgeRkamGMeBIbnÞúkeFVIkarenAkNþalElVg Pe = 379,391lb(1,688kN ) 100(60 )212 M SD = = 540,000in. − lb(61kN .m ) 8 1,100(60 )212 ML = = 5,940,000in.lb(788kN .m ) 8 m:Um:g;srub M T = M D + M SD + M L = 5,502,600 + 6,480,000 = 11,982,600in. − lb(1,354kN .m ) BIsmIkar 4.3a Pe ⎛ ect ⎞ M T ft =− ⎜1 − 2 ⎟ − t Ac ⎝ r ⎠ S 379,391 ⎛ 22.02 × 8.23 ⎞ 11,982,600 =− ⎜1 − ⎟− 978 ⎝ 88.0 ⎠ 10,458 = 411 − 1146 = −735 psi < f c = −2250 psi O.K. (c) viPaKkugRtaMgRtg;muxkat;TMr ee = 12.77in.(324mm ) f ti = 6 f 'ci = 6 3,750 ≅ 367 psi f t = 12 f 'c = 12 5,000 = 849 pis (i) enAeBlepÞr karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 105
  • 17. T.Chhay 462,672 ⎛ 12.77 × 8.23 ⎞ ft =− ⎜1 − ⎟ − 0 = +92 psi (T ) 978 ⎝ 88.0 ⎠ 462,672 ⎛ 12.77 × 25.77 ⎞ fb = − ⎜1 + ⎟ + 0 = −2,240 psi (C ) 978 ⎝ 88.0 ⎠ < f ci = −2,250 psi O.K. RbsinebI fb > fci / eKRtUveFVIkarpøas;bþÚrcMNakp©it. (ii) eRkamGMeBIbnÞúkeFVIkar 379,391 ⎛ 12.77 × 8.23 ⎞ ft =− ⎜1 − ⎟ − 0 = +75 psi (T ) 978 ⎝ 88.0 ⎠ 379,391 ⎛ 12.77 × 25.77 ⎞ fb = − ⎜1 + ⎟ + 0 = −1.840 psi (C ) 978 ⎝ 88.0 ⎠ < f ci = −2,250 psi O.K. TTYlykmuxkat;sMrab;lkçxNÐbnÞúkeFVIkaredayeRbI strand Ggát;p©it 1 / 2in.(12.7mm) cMnYn 16 edImedaymancMNakp©itenAkNþalElVg ec = 22.02in.(560mm) nigcMNakp©itenAcugTMr ee = 12.77in. (324mm ) . x> cMNakp©itrbs;EdkeRbkugRtaMERbRbYledayminmankarkMNt;kMBs; Variable Tendon Eccentricity with No Height Limitation ]TahrN_ 4>2³ KNnamuxkat;GkSr I sMrab;FñwmEdlmanElVg 65 ft (19.8m) Edlmanm:UDulmuxkat;dUc xageRkam. cUreRbInUvkugRtaMgGnuBaØatdUcKñaEdl)aneGayenAkñúg]TahrN_ 4>1. S t EdlRtUvkar = 3,570in.3 (58,535cm3 ) Sb EdlRtUvkar = 3,780in.3 (61,940cm3 ) Flexural Design of Prestressed Concrete Elements 106
  • 18. NPIC dMeNaHRsay³ edaysarm:UDulmuxkat;enAsrésxagelI nigsrésxageRkamesÞIresμIKña eKGaceRCIserIsmux kat;sIuemRTI)an. bnÞab;mk viPaKmuxkat;enAkñúgrUbTI 4>8 EdleRCIserIsedaykarsakl,g nigEktMrUv. viPaKkugRtaMgenAeBlepÞr BIsmIkar 4.4d ct f ci = f ti − ( f ti − f ci ) h = +184 − 21.16 (+ 184 + 2,250) ≅ −1,104 psi(C )(7.6MPa ) 40 Pi = Ac f ci = 377 × 1,104 = 416,208lb(1,851kN ) 393(65)2 MD = × 12 = 2,490,638in. − lb(281kN .m ) 8 BIsmIkar 4.4c cMNakp©itEdlRtUvkarenARtg;muxkat;m:Um:g;GtibrmaenAkNþalElVgKW ( ec = f ti − f ci ) St M D Pi + Pi = (184 + 1,104 ) 3,572 2,490,638 + 416,208 416,208 = 11.05 + 5.98 = 17.04in.(433mm ) edaysar cb = 18.84in. nigedaysnμt;fakMras;ebtugkarBarEdk 3.75in. sakl,g ec = 18.84 − 3.75 ≅ 15.0in.(381mm ) RkLaépÞ tendon EdlRtUvkar P Ap = i = 416,208 f pi 189,000 ( = 2.2in 2 14.2cm 2 ) cMnYn tendon = 02153 = 14.38 edIm . .2 sakl,g tendon Ggát;p©it 1 / 2in.(12.7mm) cMnYn 13 edIm/ Ap = 1.99in.2 (12.8cm2 ) / ehIy kMlaMgeRbkugRtaMgedImCak;Esþg Pi = 189,000 × 1.99 = 376,110lb(1,673kN ) RtYtBinitükugRtaMgsrésxageRkArbs;ebtug BIsmIkar 4.1a Pi ⎛ ect ⎞ M D ft =− ⎜1 − 2 ⎟ − t Ac ⎝ r ⎠ S 376,110 ⎛ 15.0 × 21.16 ⎞ 2,490,638 =− ⎜1 − ⎟− 377 ⎝ 187.5 ⎠ 3,340 karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 107
  • 19. T.Chhay = +691.2 − 745.7 = −55 psi (C ) minmankugRtaMgTajenAeBlepÞr (O.K.) BIsmIkar 4.1b Pi ⎛ ecb ⎞ M D fb = − ⎜1 + 2 ⎟ + Ac ⎝ r ⎠ Sb 376,110 ⎛ 15 × 18.84 ⎞ 2,490,638 =− ⎜1 + ⎟+ 377 ⎝ 187.5 ⎠ 3,750 = −2,501.3 + 664.2 = −1,837 psi (C ) < f ci = 2,250 psi O.K. viPaKkugRtaMgenAeBlrgbnÞúkeFIVkar BIsmIkar 4.3a Pe ⎛ ect ⎞ M T ft =− ⎜1 − 2 ⎟ − t Ac ⎝ r ⎠ S Pe = 13 × 0.153 × 154,980 = 308,255lb(1,371kN ) m:Um:g;srub M T = M D + M SD + M L = 2,490,638 + 7,605,000 = 10,095,638in. − lb(1,141kN .m ) 308,255 ⎛ 15.0 × 21.16 ⎞ 10,095,638 ft =− ⎜1 − ⎟− 377 ⎝ 187.5 ⎠ 3,340 = +566.5 − 3,022.6 = −2,456 psi (C ) > f c = −2,250 psi dUcenH eKRtUvdMeLIgkMBs;rbs;muxkat; b¤eRbIebtugEdlmanersIusþg;FMCag. edayeRbI f 'c = 6,000 psi f c = 0.45 × 6,000 = −2,700 psi O.K. Pe ⎛ ecb ⎞ M T 308,255 ⎛ 15.0 × 18.84 ⎞ 10,095,638 fb = − ⎜1 + 2 ⎟ + =− ⎜1 + ⎟+ Ac ⎝ r ⎠ Sb 377 ⎝ 187.5 ⎠ 3,750 = −2,050 + 2,692.2 = 642 psi (T ) O.K. RtYtBinitümuxkat;Rtg;TMr kugRtaMgGnuBaØati f 'ci = 0.75 × 6,000 = 4,500 psi f ci = 0.60 × 4,500 = 2,700 psi f ti = 3 f 'ci = 201 psi sMrab;kNþalElVg f ti = 6 f 'ci = 402 psi sMrab;elITMr f c = 0.45 f 'c = 2,700 psi f t1 = 6 f 'c = 465 psi f t 2 = 12 f 'c = 930 psi (a) enAeBlepÞr Flexural Design of Prestressed Concrete Elements 108
  • 20. NPIC kugRtaMgsgát;srésxageRkArbs;muxkat;elITMr ⎛ ecb ⎞ pi fb = − ⎜1 + 2 ⎟ + 0 ⎝ Ac r ⎠ 376,110 ⎛ e × 18.84 ⎞ − 2,700 = − ⎜1 + ⎟ 377 ⎝ 187.5 ⎠ dUcenH e = 16.98in. dUcenHsakl,g ee = 12.49in. 376,110 ⎛ 12.49 × 21.16 ⎞ ft =− ⎜1 − ⎟−0 377 ⎝ 187.5 ⎠ = 409 psi (T ) > f ti = 402 psi 376,110 ⎛ 12.49 × 18.84 ⎞ fb = − ⎜1 + ⎟+0 377 ⎝ 187.5 ⎠ = 2,250 psi < f ci = 2,700 psi dUcenHeRbIEdkFmμtaenAsrésxagelIRtg;muxkat;elITMredIm,ITTYlykkugRtaMgTajkñúgebtugTaMg Gs; b¤eRbIebtugEdlmanersIusþg;FMCagsMrab;muxkat;enH b¤k¾kat;bnßycMNakp©it. (b) enAeBlrgbnÞúkeFVIkar 308,255 ⎛ 12.49 × 21.16 ⎞ ft =− ⎜1 − ⎟ − 0 = 335 psi (T ) < 930 psi O.K. 377 ⎝ 187.5 ⎠ 308,255 ⎛ 12.49 × 18.84 ⎞ fb = − ⎜1 + ⎟ + 0 = −1,844 psi (C ) < −2,700 psi O.K. 377 ⎝ 187.5 ⎠ dUcenH eKGacTTYlykFñwmebtugeRbkugRtaMgEdlmanmuxkat;GkSr I kMBs; 40in.(102cm) eRbIebtugTMgn;FmμtaEdlmanersIusþg; 6,000 psi(41.4MPa ) CamYynwg tendon Ggát;p©it 1 / 2in.(12.7 mm ) EdlmancMNakp©itenAkNþalElVg ec = 15.0in.(381mm ) nigcMNakp©itenARtg; muxkat;xagcug ee = 12.5in.(318mm) eKGaceRbIviFImü:ageTotsMrableFVIkaredaHRsay edaybnþeRbI f 'c = 5,000 psi b:uEnþeFVIkarpøas;bþÚrcMnYn EdkeRbkugRtaMg nigcMNakp©it. K> cMNakp©itrbs;EdkeRbkugRtaMefr Constant Tendon Eccentricity ]TahrN_ 4>2³ edaHRsay]TahrN_ 4>2 edaysnμt;fakabeRbkugRtaMgmancMNakp©itefr. eRbIebtug TMgn;FmμtaEdlmanersIusþg; f 'c = 5,000 psi(34.5MPa) ehIykugRtaMgTajGnuBaØatGtibrmarbs;eb tugKW ft = 12 f 'c = 849 psi . karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 109
  • 21. T.Chhay dMeNaHRsay³ edaysar tendon mancMNakp©itefr ehIym:Um:g;edaysarbnÞúkefr m:Um:g;edaysarbnÞúk efrbEnßm nigm:Um:g;edaysarm:Um:g;GefrRtg;muxkat;elITMrrbs;FñwmsamBaØesμIsUnü dUcenHeKRtUvKNna FñwmenHedayeRbImuxkat;Rtg;TMr. m:UDulmuxkat;EdlRtUvkarenARtg;TMrEdl)anBIsmIkar 4.5a KW M D + M SD + M L St ≥ γf ti − f c M + M SD + M L Sb ≥ D f t − γf ci snμt; WD = 425 plf . bnÞab;mk 425(65)2 MD = × 12 = 2,693,438in. − lb(304kN .m ) 8 M SD + M L = 7,605,000in. − lb(859kN .m ) dUcenH m:Um:g;srub M T = 10,298,438in. − lb(1,164kN .m ) ehIyeyIgk¾mankugRtaMgGnuBaØatdUcxageRkam f ci = −2,250 psi f 'ci = −3,750 psi f ti = 6 f 'ci = 367 psi sMrab;muxkat;elITMr f c = −2,250 psi (15.5MPa ) f t = 849 psi γ = 0.82 m:UDulmuxkat;EdlRtUvkar St = 10,298,438 0.82 × 367 + 2,250 ) = 4,035.8in.3 61,947cm3 ( Sb = 10,298,438 849 + 0.82 × 2,250 ) = 3,823.0in.3 62,713cm3 ( sakl,gelIkTI 1³ edaysar S EdlRtUvkar = 4,035.8 psi FMCag S rbs;muxkat;enA t t kñúg]TahrN_ 4>2 dUcenHeRCIserIsmuxkat;GkSr I Edlman h = 44in. dUcbgðajenAkñúgrUbTI 4>9. lkçN³muxkat;rbs;vamandUcxageRkam³ I c = 92,700in.4 r 2 = 228.9in.2 Ac = 405in.2 ct = 23.03in. Flexural Design of Prestressed Concrete Elements 110
  • 22. NPIC S t = 4,303in.3 cb = 20.97in. Sb = 4,420in.3 WD = 422 plf BIsmIkar 4.5c cMNakp©itEdlRtUvkarRtg;muxkat;elITMrEdlCamuxkat;eRKaHfñak;KW ( )S t ee = f ti − f ci P i Edl f ci = f ti − t ( f ti − f ci ) c h = 367 − 23.03 (367 + 2,250) = −1,002 psi(6.9MPa ) 44 nig Pi = Ac f ci = 405 × 1,002 = 405,810lb(1,805kN ) dUcenH ee = (367 + 1,002) 405030 = 13.60in.(346mm) 4, ,810 RkLaépÞEdkeRbkugRtaMgEdlRtUvkarKW = 2.15in.2 ( .4cm 2 ) P 405,810 Ap = i = 14 f 189,000 pi dUcenHeyIgsakl,geRbIEdkeRbkugRtaMgEdlmanGgát;p©it 1 / 2in. . cMnYn tendon EdlRtUvkarKW karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 111
  • 23. T.Chhay 2.15 / 0.153 = 14.05 dUcenHeRbI tendon Ggát;p©it 1 / 2in.(12.7mm) cMnYn 14 edIm. CalT§pl Pi = 14 × 0.153 × 189,000 = 404,838lb(1,801kN ) (a) viPaKkugRtaMgenAeBlepÞrenARtg;muxkat;xagcug BIsmIkar 4.1a pi ⎛ ect ⎞ M D 404,838 ⎛ 13.60 × 23.03 ⎞ ft =− ⎜1 − 2 ⎟ − t = − ⎜1 − ⎟−0 Ac ⎝ r ⎠ S 405 ⎝ 228.9 ⎠ = +368.2 psi (T ) ≅ f ti = 367 O.K. BIsmIkar 4.2b Pi ⎛ ecb ⎞ M D 404,838 ⎛ 13.6 × 20.97 ⎞ fb = − ⎜1 + 2 ⎟ + =− ⎜1 + ⎟+0 Ac ⎝ r ⎠ Sb 405 ⎝ 228.9 ⎠ = −2,245 psi (C ) ≅ f ci = −2,250 O.K. eKk¾GaceRbIvatMélTaMgenHsMrab;muxkat;kNþalElVgpgEdr edaysarcMNakp©it e efr. (b) viPaKkugRtaMgenAeBlrgbnÞúkeFVIkarcugeRkayenARtg;TMr Pe = 14 × 0.153 × 154,980 = 331,967lb(1,477kN ) m:Um:g;srub M T = M D + M SD + M L = 0 BIsmIkar 4.3a Pe⎛ ect ⎞ M T ft =− ⎜1 − 2 ⎟ − t Ac⎝ r ⎠ S 331,967 ⎛ 13.60 × 23.03 ⎞ =− ⎜1 − ⎟ − 0 = 302 psi (T ) < f t = 849 psi O.K. 405 ⎝ 228.9 ⎠ BIsmIkar 4.3b Pe ⎛ ecb ⎞ M T 331,967 ⎛ 13.6 × 20.97 ⎞ fb = − ⎜1 + 2 ⎟ + =− ⎜1 + ⎟+0 Ac ⎝ r ⎠ Sb 405 ⎝ 228.9 ⎠ = −1,841 psi (12.2MPa )(C ) < f c = −2,250 psi O.K. (c) viPaKkugRtaMgenAeBlrgbnÞúkeFVIkarcugeRkayenAkNþalElVg m:Um:g;srub M T = M D + M SD + M L = 10,298,438in. − lb dUcenHkugRtaMgsrésxageRkArbs;ebtugEdlbNþalBI M T KW = −2,555 psi (C )(17.6MPa ) MT 10,298,438 f1t = t =− S 4,030 = +2,330 psi (T )(16.1MPa ) M 10,298,438 f1b = T = Sb 4,030 dUcenH kugRtaMgsrésxageRkArbs;ebtugcugeRkayKW Flexural Design of Prestressed Concrete Elements 112
  • 24. NPIC f t = +302 − 2,555 = −2,253 psi (C ) ≅ f c = −2,250 psi TTYlyk)an f b = −1,841 + 2,330 = +489 psi (T ) < f t = 849 psi O.K. dUcenH TTYlykmuxkat;sakl,gEdlmancMNakp©itefr e = 13.6in.(345mm) sMrab; tendon Ggát;p©it 1 / 2in.(12.7mm) cMnYn 14 srés. 4> kareRCIserIsmuxkat; niglkçN³rbs;Fñwmd¾RtwmRtUv Proper Selection of Beam Sections and Properties k> eKalkarN_ENnaMTUeTA General Guidelines muxkat;ebtugeRbkugRtaMgmindUc steel-rolled section eT eRBaHvaminTan;manlkçN³sþg;dar eBjeljenAeLIy. kñúgkrNICaeRcIn visVkrKNnaeRKOgbgÁúMRtUvEteRCIserIsRbePTmuxkat;edIm,IeRbI R)as;enAkñúgKMeragenaH. enAkñúgkarKNnaFñwmTMrsamBaØPaKeRcIn cMgayBI cgc nigExS cgs EdleKsÁal; CacMNakp©it e smamaRteTAnwgkMlaMgeRbkugRtaMgEdlRtUvkar. CaTUeTA edaysarEteKKNnaeRcIneRbIm:Um:g;kNþalElVg eRBaHvamantMélFMCageK. cMNakp©it enAkNþalElVgkan;EtFM kMlaMgeRbkugRtaMgEdlRtUvkarkan;EttUc ehIyvapþl;nUvlkçNesdækic©kan;Et xøaMgkñúgkarKNna. sMrab;cMNakp©itFM eKRtUvkarRkLaépÞebtugenAsrésxagelIFMEdr. dUcenH muxkat; GkSr T nigmuxkat;GkSr I EdlmansøabFMCamuxkat;Edlsaksm. CaTUeTA muxkat;xagcugEtgCamux kat;tan;edIm,IeCosevogcMNakp©itFMenAelIbøg;m:Um:g;sUnü ehIyk¾edIm,IbegáInlT§PaBTb;kMlaMgkat;énmux kat;elITMr nigkarBar anchorage zone failure. muxkat;epSgeTotEdleKeRbIPaKeRcInEdrKW muxkat;GkSr T Dub. muxkat;enHbEnßmGtßRbeyaCn_ eTAmuxkat;GkSr T eTaledIm,IPaBgayRsYl nigesßrPaBkñúgkarelIkdak; nigdMeLIg. rUbTI 4>10 bgðaj BIRbePTmuxkat;EdleKeRcIneRbICaTUeTA. muxkat;d¾éTeTotdUcCakMralRbehagkñúg (hollow-core slab) muxkat;Gt;sIuemRTI k¾RtUv)aneRbICaTUeTApgEdr. cMNaMfa eKeRbImuxkat;mansøabCMnYseGaymuxkat; ctu- ekaNtan;EdlmankMBs;dUcKñaedayminman)at;bg;ersIusþg;rgkarBt;eT. b:uEnþ eKeRbImuxkat;ctuekaNCa girder EdlmanElVgxøI. eKeRbImuxkat;GkSr I CaRbePTFñwmkMralEdlmankMralxNÐsmascak;BIelIsMrab;eeRKOgbgÁMúcMNt rfynþEdlmanElVgEvg. CaTUeTA eKeRcIneRbImuxkat;GkSr T EdlmansøabxageRkamF¶n;dUcbgðajenA kñúgrUbTI 4>10 (d) enAkñúgeRKOgbgÁúMs<an. eKeRbImuxkat; T Duby:agTUlMTUlayenAkñúgRbB½n§kMralxNÐ karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 113
  • 25. T.Chhay rbs;GKar k¾dUcCageRKOgbgÁúMcMNtrfynþ edaysarRbeyaCn_énskmμPaBsmasrbs;søabFMxagelI EdlmanTTwgBI 10 ft eTA 15 ft . kMralRbehagkñúgCacMerokFñwmmYyTisRbehagkñúgEdlGacdMeLIgCakMralxNÐ)any:aggayRsYl. eKGaceRbIr:tRbehagragRbGb;Car:tFñwmsMrab;ElVgEvg EdleKsÁal;vaCaRbB½n§kMralkMNat;s<an (segmental bridge deck system). kMNat;r:t (segmental girder) enHmanlT§PaBTb;karrmYlFM ehIypleFoblT§PaBTb;karBt;elITMgn;xøÜnrbs;vaFMCagRbePTmuxkat;RbB½n§eRbkugRtaMgd¾éTeTot. x> RkLaépÞTaMgmUl muxkat;bMElg nigvtþmanrbs;bMBg; Gross Area, the Transformed Section, and the Presence of Ducts CaTUeTA RkLaépÞrbs;muxkat;TaMgmUlrbs;muxkat;ebtug (gross cross sectional area ) KWRKb; RKan;sMrab;eRbIenAkñúgkarKNna muxkat;ebtugeRbkugRtMgeRkamlkçxNÐbnÞúkeFVIkar. kñúgxN³EdlGñk KNnaxøHeBjcitþnwgkarKNna EdlmanlkçN³suRkitCagtamry³kareRbImuxkat;bMElg. PaBsuRkit Edl)anBIkarKitbBa©ÚlkarcUlrYm énmuxkat;rbs;EdkeTAkñúgPaBrwgRkaj (stiffness) rbs;ebtugmin Flexural Design of Prestressed Concrete Elements 114
  • 26. NPIC RtUv)anKitfaCakarcaM)ac;enaHeT. enA kúñgFñwmrgeRbkugRtaMgCaeRkay (post-tensioned beam) Edl bMBg;RtUv)ankMe)arebtug (grout), gross cross section enAEtRKb;RKan;sMrab;RKab;KNnaTaMgenH. man EtkñúgkrNIs<anElVgEvg nigFñwmeRbkugRtaMgEdlplitCalkçN³]sShkmμEdlmanRkLaépÞEdkeRbkug RtaMgFMeT EdleKRtUveRbImuxkat;bMElg b¤muxkat;ebtugsuT§ (net concrete area) EdlminKitbMBg;. K> Envelope sMrab;kardak;kabeRbkugRtaMg Envelopes for Tendon Placement kugRtaMgTajenAsrésxageRkAbMputrbs;ebtugeRkamlkçxNÐbnÞúkeFVIkarminGacFMCagkugRtaMg GnuBaØatEdleGayeday code dUcCa ACI, PCI, AASTHO b¤ CEB-FIP eT. dUcenH eKcaM)ac;RtUv begáItnUvtMbn;kMNt;mYyenAkññúgmuxkat;ebtugEdlCa envelope EdleKGacGnuvtþkMlaMgeRbkugRtaMgeday mineFVIeGaymankugRtaMgTajenAsrésxageRkAbMputrbs;ebtug. BIsmIkar 4.1a eyIgman Pi ⎛ ect ⎞ ft = 0 = − ⎜1 − 2 ⎟ Ac ⎝ r ⎠ 2 eK)an e= r ct dUcenH cMnucsñÚlxageRkam (lower kern point) r2 Kb = ct dUcKña BIsmIkar 4.1b RbsinebI fb = 0 enaHeK)an − e = r 2 / cb EdlsBaØadktMNageGaytMNag eGaykarvas;eLIgelIBIG½kSNWt ÉcMNakp©itviC¢manCakarvas;cuHeRkam. dUcenH upper kern point KW r2 Kt = cb BIkarkMNt;cMnucsñÚlxagelI nigxageRkammk eyIgeXIjy:agc,as;fa³ (a) RbsinebIkMlaMgeRbkugRtaMgmanGMeBIenAxageRkam lower kern point vanwgekItmankugRtaMgTaj enAsrésxagelIrbs;muxkat;ebtug. (b) RbsinebIkMlaMgeRbkugRtaMgmanGMeBIenAxagelI upper kern point vanwgekItmankugRtaMgTaj enAsrésxageRkamrbs;muxkat;ebtug. eKGackMNt;cMnucsñÚlxagsþaM nigxageqVgénExSsIuemRTIbBaÄrrbs;muxkat;tamlkçN³dUcKña dUc enHeKnwgTTYl)anépÞsñÚlsMrab;GnuvtþkMlaMgeRbkugRtaMgeTAelIEdkeRbkugRtaMg. rUbTI 4>11 bgðajBI sñÚlsMrab;muxkat;ctuekaN. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 115
  • 27. T.Chhay X> plRbeyaCn_énkardak;kabeRbkugRtaMgCa curved b¤ harped Advantages of Curved or Harped Tendons eTaHbICaeKeRbIEdkeRbkugRtaMgRtg;y:agTUlMTUlayenAkñúgFñwmRbEvglμmEdlcak;eRsck¾eday k¾CaTUeTAeKeRbIkabeRbkugRtaMgEdlmanTMrg;ekagenAkñúgGgát;rgkarTajCaeRkay (post-tensioned element) Edlcak;enAnwgkEnøgEdr. eKEck tendon EdlminRtg;CaBIrRbePT³ (a) Draped: manTMrg;ekagdUc)a:ra:bUl RtUv)aneKeRbIenAkñúgFñwmEdlrgbnÞúkxageRkABRgayesμICa bzm. (b) Harped: tendon eRTtEdlminCab; ¬tamn½yKNitviTüa¦ enARtg;bøg;rgbnÞúkcMcMnuc RtUv)aneK eRbIenAkñúgFñwmEdlrgbnÞúkcMcMnucTTwgG½kSCabzm. rUbTI 4>12/ 4>13 nig 4>14 bgðajBI alignment, m:m:g;Bt; nigkarBRgaykugRtaMgsMrab;Fñwm EdlrgkMlaMgeRbkugRtaMgedaykabeRbkugRtaMgRtg;/ draped/ nig harped erogKña. düaRkamTaMgenHcg; bgðajBIplcMeNjEpñkesdækic©rbs; draped nig harped tendon elIEdkeRbkugRtaMgRtg;. enAkñúgrUbTI 4>12 Rtg;muxkat; 1-1 kugRtaMgTajrbs;ebtugEdleKminR)afñacg;)an)anbgðajenAsrésxagelI. muxkat; 1-1 enAkñúgrUbTI 4>13 nig 4>14 bgðajfakugRtaMgsgát;rayesμIRbsinebI tendon eFVIGMeBIenA Rtg; cgc énmuxkat;enARtg;TMr. plRbeyaCn_epSgeTotrbs; draped nig harped tendon KWvaGnuBaØat eGayFñwmeRbkugRtaMgRTbnÞúkF¶n; edaysarT§iBllMnwgrbs;bgÁúMkMlaMgbBaÄrrbs;kabeRbkugRtaMgmin Rtg;. niyaymü:ageTot kMlaMgeRbkugRtaMgEdlRtUvkar Pp sMrab; parabolic tendon enAkñúgrUbTI 4>13 nig Ph sMrab; harped tendon enAkñúgrUbTI 4>14 mantMéltUcCagkMlaMgEdlRtUvkarenAkñúg straight Flexural Design of Prestressed Concrete Elements 116
  • 28. NPIC tendon enAkñúgrUbTI 4>14. dUcenH sMrab;kMritkugRtaMgdUcKña eKRtUvkarcMnYn strand ticCagsMrab;krNI draped b¤ harped tendon nigeBlxøHeKGaceRbImuxkat;ebtugtUcCagkñúgkarKNnaedayTTYl)annUv lT§plRbkbedayRbsiT§PaB ¬eRbobeFob]TahrN_ 4>2 nig 4>3 mþgeTot¦. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 117
  • 29. T.Chhay Flexural Design of Prestressed Concrete Elements 118
  • 30. NPIC karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 119
  • 31. T.Chhay g> Limiting-Eccentricity Envelopes eKcg;)ancMNakp©itKNnarbs; tendon tambeNþayElVgEdleFVIy:agNaminbegáItkugRtaMg TajenAsrésxageRkAbMputrbs;muxkat;FñwmEdleRKaHfñak;. RbsinebIeKmincg;)ankugRtaMgTajtam beNþayElVgrbs;FñwmenAkñúgrUbTI 4>15 EdleRbI draped tendon eKRtUvkMNt;cMNakp©itRtg;muxkat; tambeNþayFñwm. RbsinebI M D Cam:Um:g;TMgn;pÞal; ehIy M T Cam:Um:g;srubEdlekItBIbnÞúkTTwgG½kS TaMgGs; enaHédXñas;rbs;m:Um:g; couple EdlbegáIteday center-of-pressure line (C-line) nigG½kSTI RbCMuTMgn;rbs;EdkeRbkugRtaMg (cgs line) EdlekItBI M D nig M T KW amin nig amax erogKña dUc bgðajkñúgrUbTI 4>15. Lower cgs Envelop édXñas;Gb,brmarbs; tendon couple KW MD amin = (4.7a) Pi smIkarenHkMNt;cMgayGtibrmaenABIxageRkam bottom kern EdlCaTItaMgrbs;ExS cgs dUcenH C-line minFøak;enABIxageRkamExS bottom kern )aneT GBa©wgehIyvaGackarBarmineGaymankugRtaMgTajenA srésxagelIbMput)an. Flexural Design of Prestressed Concrete Elements 120
  • 32. NPIC dUcenH limiting bottom eccentricity KW eb = (amin + kb ) (4.7b) Upper cgs Envelop édXñas;Gtibrmarbs; tendon couple KW MT amax = (4.7c) Pe smIkarenHkMNt;cMgayGb,brmaenABIxageRkam top kern EdlCaTItaMgrbs;ExS cgs dUcenH C-line minsßitenABIxagelIExS top kern )aneT GBa©wgehIyvaGackarBarmineGaymankugRtaMgTajenAsrés xageRkambMput)an. dUcenH limiting top eccentricity KW et = (amax − kt ) (4.7d) kUdxøHGnuBaØateGayeRbIkugRtaMgTajkMNt;sMrab;enAeBlepÞr nigenAeBlrgbnÞúkeFVIkar. enAkñúgkrNI EbbenH eKGacGnuBaØateGayExS cgs GacsßitenAxageRkA limiting cgs envelop Edl)anbgðajenA kñúgsmIkar 4.7a nig c bnþicbnþÜc. RbsineKbEnßmcMNakp©itbEnßmenAelI cgs-line envelop enaHvanwgeFVIeGaymankugRtaMgTaj kMNt;enAelIsrésxagelI nigxageRkamrbs;ebtug. kugRtaMgxagelI nigxageRkambEnßmKW f (t ) = Pi e'b ct (4.8a) Ic nig P e' c f (b ) = e t b Ic (4.8b) Edl t nig b tMNageGaysrésxagelI nigxageRkam erogKña. BIsmIkar 4.6 cMNakp©itbEnßmEdl RtUvbEnßmeTAelIsmIkar 4.7b nig d KW f (t ) Ac kb e'b = (4.9a) Pi f (b ) Ac kt nig e't = Pe (4.9b) EnvelopEdlGnuBaØatkugRtaMgkMNt;RtUv)anbgðajenAkñúgrUbTI 4>16. eKKYrcMNaMfa enAeBl upper envelop enAxageRkAmuxkat; ehIykugRtaMgenAmantMélkMNt;GnuBaØat enaHbgðajfamuxkat;Kμan lkçN³esdækic©eT. bMErbMrYlcMNakp©it b¤kMlaMgeRbkugRtaMgeFIVeGaykarKNnakan;EtRbesI. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 121
  • 33. T.Chhay c> Envelopes EdkeRbkugRtaMg Prestressing Tendon Envelopes ]TahrN_ 4>4³ ]bmafaFñwmenAkñúg]TahrN_ 4>2 Ca post-tensioned bonded beam ehIyEdkeRbkug RtaMgmanrag)a:ra:bUl. kMNt; limiting envelop sMrab;TItaMgrbs; tendon EdlkMritkugRtaMgsrésrbs; ebtugminFMCagkugRtaMgGnuBaØat. Kitfamuxkat;Rtg;cMnuckNþalElVg mYyPaKbYnénElVg nigcugFñwmCa muxkat;EdlRtUvKNna. snμt;fatMélrbs;kMhatbg;eRbkugRtaMgdUcKñaenAkñúg]TahrN_ 4>2 b:uEnþ Pi = 549,423lb / Pe = 450,526lb / f 'c = 6,000 psi / ec = 13in nig ee = 6in . dMeNaHRsay³ BI]TahrN+_ 4>2 eyIgGacsegçbm:Um:g;KNnarbs;FñwmGkSr I niglkçN³muxkat;Edl RtUvkardUcxageRkam³ Pi = 549,423lb(2,431kN ) Pe = 450,526lb(2,004kN ) M D = 2,490,638in. − lb(281kN .m ) M SD + M L = 7,605,000in. − lb(859kN .m ) M T = M D + M SD + M L = 10,095,638in. − lb(1,141kN .m ) ( Ac = 377in.2 2,536cm 2 ) f 'c = 6,000 psi ( r 2 = 187.5in.2 1,210cm 2 ) ct = 21.16in.(537mm ) cb = 18.84in.(479mm ) Flexural Design of Prestressed Concrete Elements 122
  • 34. NPIC edaysarEtm:Um:g;Bt;enAkñúg]TahrN_enH)anmkBIbnÞúkBRgayesμI TMrg;rbs;düaRkamm:Um:g;man ragCa)a:ra:bUl CamYynwgm:Um:g;EdlmantMélsUnüenARtg;cugTMrrbs;FñwmsamBaØ. dUcenH m:Um:g;enARtg; mYyPaKbYnénRbEvgElVgKW M D = 0.75 × 2,490,638 = 1,867,979in. − lb(211kN .m ) M T = 0.75 × 10,095,638 = 7,571,729in. − lb(856kN .m ) BIsmIkar 4.6a nig b, kern point limit KW r 2 187.5 kt = = = 9.95in.(253mm ) cb 18.84 r 2 187.5 kb = = = 8.86in.(225mm ) ct 21.16 Lower envelop BIsmIkar 4.7a cMgayGtibrmaEdlExS cgs RtUv)andak;BIeRkam bottom kern edIm,IkarBarkug RtaMgTajenAsrésxagelIbMputRtUv)ankMNt;dUcxageRkam (i) kNþalElVg = 4.53in.(115mm ) M D 2,490,638 amin = = Pi 549,423 eyIgTTYl)an e1 = kb + amin = 8.86 + 4.53 = 13.39in.(340mm) (ii) mYyPaKbYnénElVg = 3.40in.(340mm ) 1,867,979 amin = 549,423 eyIgTTYl)an e2 = 8.86 + 3.40 = 12.26in.(311mm ) (iii) elITMr amin = 0 eyIgTTYl)an e3 = 8.86 + 0 = 8.86in.(225mm ) Upper envelop BIsmIkar 4.7b cMgayGtibrmaEdlExS cgs RtUv)andak;BIeRkam top kern edIm,IkarBarkugRtaMg TajenAsrésxageRkambMputRtUv)ankMNt;dUcxageRkam (i) kNþalElVg = 22.41in.(569mm ) M T 10,095,638 amin = = Pe 450,526 karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 123
  • 35. T.Chhay eyIgTTYl)an e1 = amax − kt = 22.41 − 9.95 = 12.46in.(316mm) kMras;ebtugkarBarEdk = 3.0in. cMNaMfa e1 minGacFMCag cb ebImindUecñaHeT tendon nwgenAxageRkAmuxkat;. (ii) mYyPaKbYnénElVg = 16.80in.(427mm ) 7,571,729 amin = 450,526 eyIgTTYl)an e2 = 16.80 − 9.95 = 6.85in.(174mm ) (iii) elITMr amin = 0 eyIgTTYl)an e3 = 0 − 9.95 = 9.95in.(− 253mm) ¬9.95in. sßitenABIelIExS cgs¦ sMrab;kargarGnuvtþn_ snμt;fakugRtaMgsrésTajGtibrmaeRkamlkçxNÐbnÞúkeFVIkarsMrab;kargarbegáIt cgs envelope minRtUvFMCag f t = 6 f 'c = 465 psi sMrab;srésxagelI nigxageRkam. BIsmIkar 4.9a cMNakp©itbEnßmEdlRtUvbEnßmeTAelI lower cgs envelope edIm,IGnuBaØateGaymankugRtaMgTajkMNt; enAsrésxagelIKW f (t ) Ac kb 465 × 377 × 8.86 e'b = = = 2.83in.(72mm ) Pi 549,423 dUcKña BIsmIkar 4.9b cMNakp©itEdlRtUvbEnßmeTAelI upper cgs envelop edIm,IGnuBaØateGayman kugRtagTajkMNt;enAsrésxageRkamKW f (b ) Ac kt 465 × 377 × 9.95 e't = = = 3.87in.(98mm ) Pe 450,526 dUcenH eyIgmantaragsegçbBI cgs envelope cMNkp©itdUcxageRkam³ Flexural Design of Prestressed Concrete Elements 124
  • 36. NPIC rUbTI 4>17 bgðajBI cgs envelope sMrab;kugRtaMgTajesμIsUnü nigkugRtaMgTajkMNt;enAkñúg ebtug. q> karkat;bnßykMlaMgeRbkugRtaMgenAEk,rTMr Reduction of Prestress Force near Support dUc)aneXIjBI]TahrN_ 4>3 nigEpñk K nig g xagelI straight tendon enAkñúg pretensioned member GacbNþaleGaymankugRtaMgTajFMenAsrésxageRkArbs;ebtugenARtg;TMr edaysarGvtþ- mankugRtaMgm:Um:g;Bt;Edl)anBITMgn;pÞal; nigbnÞúkbEnßm. eKmanviFIFmμtaBIrkñúgkarkat;bnßykugRtaMg enARtg;muxkat;TMrEdlbNþalmkBIkMlaMgeRbkugRtaMg. viFITaMgBIrenaHKW³ - pøas;bþÚrcMNakp©itrbs;kabxøHedayelIkBYkvaeLIgeTAkan;tMbn;TMrdUcbgðajenAkñúgrUbTI 4>18 (a). viFIenHkat;bnßytMélm:Um:g;. - eRsabkabxøHedaybMBg;)aøsÞiceTAkan;tMbn;TMr dUcbegðIjenAkñúgrUbTI 4>18(b). viFIenHkat;bnßy EpñkénkugRtaMgepÞrrbs;kabenAcMgayxøHBImuxkat;TMrénFñwmeRbkugRtaMgTMrsamBaØ. cMNaMfakabEdlelIkeLIgk¾RtUv)aneRbIenAkñúgFñwmeRbkugRtaMgElVgEvgEdlrgeRbkugRtaMgCa eRkaypgEdr. eKminRtUvkarEpñkminCab;rbs; tendon edaylkçN³RTwsþI edayelIkvaeLIgelI. kMhat bg;edaysarkMlaMgkkitbEnßmedaysarExSekagbBa©ÚleTAkñúgkarKNna b¤karviPaKmuxkat;. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 125
  • 37. T.Chhay 5> End Block at Support Anchorage Zones k> karEbgEckkugRtaMg Stress Distribution kugRtaMgsgát;cMcMnucd¾FMenAkñúgG½kSbeNþayekItmanenARtg;muxkat;TMrenAelIkMNat;d¾tUcénépÞ rbs;cugFñwm ¬TaMgenAkñúg pretensioned beam nig post-tensioned beam¦ EdlbNþalmkBIkMlaMg eRbkugRtaMgd¾FM. enAkñúg pretensioned beam bnÞúkepÞrcMcMnucrbs;kMlaMgeRbkugRtaMgeTAelIebtugEdl B½T§CMuvijekIteLIgbnþicmþg²rhUtdl;vakøayeTACamanlkçN³BRgayesμIelIRbEvg lt BIépÞénmuxkat;TMr. enAkñúg post-tensioned beam karEbgEck nigkarepÞrkMlaMgbnþicmþg²tamrebobenHminGaceFVI eTA)aneT edaysarkMlaMgmanGMeBIedaypÞal;eTAelIépÞrbs;cugFñwmtamry³ bearing plate nig anchors. ehIy tendon xøH b¤k¾TaMgGs;enAkñúg post-tensioned beam RtUv)anelIkeLIg b¤ draped eTAkan;srés xagelItamry³EpñkénRTnugrbs;muxkat;ebtug. edaysarkarpøas;bþÚrkugRtaMgsgát;tamG½kSBIcMcMnuceTABRgayesμIminsnSwm² vabegáIteGayman kugRtaMgTajTTwg (transverse tensile stress) FMkñúgTisbBaÄr dUcenHehIy longitudinal bursting cracks k¾ekItmanenA anchorage zone. enAeBlEdlkugRtaMgFMCagm:UDulkat;rbs;ebtug end block Flexural Design of Prestressed Concrete Elements 126
  • 38. NPIC nwgeRbHtambeNþay elIkElgEteKdak;EdkbBaÄrsmRsb. TItaMgrbs; concrete-bursting stress nig resulting bursting crack k¾dUcCa surface-spalling crack KWGaRs½ynwgTItaMg nigkarEbgEckkMlaMg cMcMnuctamTisedkEdlGnuvtþedayEdkeRbkugRtaMgeTAelI end bearing plate. eBlxøHeKcaM)ac;begáInRkLaépÞrbs;muxkat;eTArkTMredayeFVIkarBRgIkRTnugbnþicmþg²eGayesμI TTwgrbs;søabenARtg;TMr kñúgeKalbMNgedIm,IeFVIkarelIk tendon eLIgelI ¬emIlrUbTI 4>19(a)¦. b:uEnþ karekIneLIgRkLaépÞmuxkat;EbbenHmin)ancUlrYmkarBar bursting b¤ spalling crack eT ehIyvak¾min manT§iBlkñúgkarkat;bnßykMlaMgTajtamTTwgenAkñúgebtugEdr. tamBit TaMglT§plénkarBiesaF nigkarviPaKedayRTwsþIén three-dimension stress problem bgðajfakugRtaMgTajGacekIneLIg. dUcenH eKRtUvkardak; anchorage reinforcement caM)ac;enAkñúgtMbn;epÞrkMlaMgkñúgTMrg;Edkkg biTCit (closed ties b¤ stirrup) b¤]bkrN_ anchorage edaydak;B½T§CMuvijEdkeRbkugRtaMgemTaMgGs; nig EdkBRgwgFmμtatambeNþay. rUbTI 4>20 bgðajBIKnøgkugRtaMgTaj nigKnøgkugRtaMgsgát;. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 127
  • 39. T.Chhay x> RbEvgbgáb; nigRbEvgepÞrenAkñúgGgát;rgeRbkugRtaMgCamun nigkarKNna Anchorage Reinforcement Development and Transfer Length in Pretensioned Members and Design of Their Anchorage Reinforcement edaysarkMlaMgTaj (jacking force) RtUv)anRbElgeTAelIGgát;rgeRbkugRtaMgCamun enaH kMlaMgeRbkugRtaMgRtUv)anepÞredaylkçN³DINamictamry³épÞb:HrvagEdkeRbkugRtaMg nigebtugeTAeb tugEdlB½T§CMuvijEdkeRbkugRtaMg. PaBs¥itrvagEdkeRbkugRtaMg nigebtugelIRbEvgkMNt;rbs;Edk eRbkugRtaMgepÞrkMlaMgeRbkugRtaMgcMcMnucsnSwm²eTAmuxkat;TaMgmUlrbs;ebtugRtg;bøg;EdlecjBI end block eTAkan;kNþalElVg. RbEvgbgáb;kMNt;TMhMkMlaMgeRbkugRtaMgEdlGacekItmantambeNþayElVg. RbEvgbgáb;kan;EtEvg kMlaMgeRbkugRtaMgkan;EtFM. Ca]TahrN_ sMrab; 7-wire strand Ggát;p©it 1 / 2in. EdlmanRbEvgbgáb; 40in.(102cm) begáIt kugRtaMg 180,000 psi(1,241MPa ) b:uEnþCamYynwgRbEvgbgáb; 70in.(178cm) begáItkugRtaMg 206,000 psi (1,420MPa ) . BIrUbTI 4>21 vabgðajy:agc,as;faRbEvgbgáb; ld EdlbegáItkugRtaMgeBjeljCabnSM rvagRbEvgepÞr (transfer length) lt nigRbEvgs¥itedaykarBt; (flexural bond length) l f . 1 ⎛ f pe ⎞ lt = ⎜ ⎟d b (xñat US) (4.10a) 1,000 ⎜ 3 ⎟ ⎝ ⎠ Flexural Design of Prestressed Concrete Elements 128
  • 40. NPIC ⎛ f pe ⎞ lt = ⎜ ⎜ 20.7 ⎟d b ⎟ ( xñat SI) ⎝ ⎠ f pe b¤ lt = 3000 db (4.10b) nig lf = 1 1,000 ( f ps − f pe d b ) ( xñat US) (4.10c) lf = 1 6.9 ( f ps − f pe d b ) ( xñat SI) Edl kugRtaMgenAkñμúgEdkeRbkugRtaMgenAeBl nominal strength f ps = f pe = eRbkugRtaMgRbsiT§PaBeRkayeBlxatbg; d b = nominal diameter rbs;EdkeRbkugRtaMg edaybBa¢ÚlsmIkar 4.10b nig 4.10c eyIg)an 1 ⎛ ⎞ (xñat US) 2 ld min = ⎜ f ps − f pe ⎟d b (4.10d) 1,000 ⎝ 3 ⎠ 1 ⎛ ⎞ ld min = 6.9 ⎝ 2 ⎜ f ps − f pe ⎟d b 3 ⎠ ( xñat SI) karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 129
  • 41. T.Chhay smIkar 4.10d eGaynUvRbEvgbgáb;caM)ac;Gb,brmasMrab;EdkeRbkugRtaMg. RbsinebIeKeRsab EdkeRbkugRtaMgxøHeq<aHeTAkan;cugFñwmedIm,Ikat;bnßykugRtaMgs¥itenAEk,rxagcug enaHkugRtaMgepÞrenAkñúg tMbn;enaHRtUv)ankat;bnßy ehIyeKcaM)ac;RtUveFVIkarEksMrYledaybegáInRbEvgbgáb; ld . !> KNnaEdktMbn;epÞrenAkñúgFñwmrgeRbkugRtaMgCamun Design of Transfer Zone Reinforcement in Pretensioned Beams tamkarBiesaF Mattock et al. )anbegáItsmIkarEdl)anBIkarBiesaFsMrab;rkkMlaMgEdkkg srub F dUcxageRkam³ Pi h F = 0.0106 (4.11) lt Edl h CakMBs;rbs;FñwmrgeRbkugRtaMgCamun ehIy lt Ca transfer length. RbsinebIeKykkugRtaMg mFümenAkñúgEdkkgRtwmBak;kNþalkugRtaMgGnuBaØatGtibrma f s rbs;Edk enaH F = 1 / 2( At f s ) . edayCMnYsvacUleTAkñúgsmIkar 4.11 eyIgTTYl)an³ Ph At = 0.021 i f l ¬xñat Us¦ (4.12) s t At = 21,000 ¬xñat IS¦ Pi h f s lt Edl At CaRkLaépÞsrubrbs;Edkkg ehIy f s ≤ 20,000 psi(138MPa) sMrab;karRKb;RKgsñameRbH. @> kareRCIserIsEdkenAkñúgFñwmrgeRbkugRtaMgCamun Reinforcement Selection in Pretensioned Beams ]TahrN_ 4>5³ KNna anchorage reinforcement EdlRtUvkaredIm,IkarBar bursting crack b¤ spalling crack EdlekItmanenAkñúgFñwmén]TahrN_ 4>2. dMeNaHRsay³ Pi = 376,110lb(1,673kN ) BIsmIkar 4.12 At = 0.021 Pi lh fs t BIsmIkar 4.10b RbEvgepÞrKW lt = ( f pe / 3,000)db . dUcenH edaysar f pe = 154,980 psi nig d b = 1 / 2in. eyIgman × 0.5 = 25.83in.(66cm ) 154,980 lt = 3,000 dUcenH Ph At = 0.021 i f s lt Flexural Design of Prestressed Concrete Elements 130
  • 42. NPIC edaysar f s ≤ 20,000 psi / eyIgTTYl)an 376,110 × 40 At = 0.021 20,000 × 25.83 ( = 0.61in.2 3.9cm 2 ) sakl,gEdkkgbiTCit #3 2 × 0.11 = 0.22in.2 ¬Ggát;p©it 9.5mm ¦ cMnYnEdkkgGb,brma = 0..22 = 2.78 0 61 eRbIEdkkg #3 cMnYnbIkgedIm,Ih‘MuB½T§EdkembeNþay. cgh‘uMB½T§EdkeRbkugRtaMgCamYy helical steel wire elIRbEvgepÞr lt edIm,ITTYl)ankarepÞrEdlmanRbsiT§PaBl¥. K> Post-tensioned Anchorage Zones: Linear Elastic and Strut-and-Tie Theories eKGacKit anchorage zone CamaDebtugEdlkMlaMgeRbkugRtaMgcMcMnucenARtg; anchorage device BRgayCalkçN³smamaRttamTisTTwgeBjépÞTaMgmUlrbs;muxkat;ebtugtambeNþayElVg. RbEvgrbs;tMbn;enHGnuvtþtameKalkarN_ St. Venant EdlkugRtaMgkøayCaBRgayesμIenAcMgayRbhak; RbEhlmYyBImux anchorage device esμInwgkMBs; h rbs;muxkat;. RBIsTaMgmUlEdlman RbEvgepÞr h Ca anchorage zone srub. dUcenHtMbn;enHpSMeLIgedayBIrEpñk³ - General Zone: karraldalTUeTAéntMbn;enHRsedogKñanwg anchorage zone srub. dUcenH RbEvglatsn§wgtambeNþayFñwmesμInwgkMBs;muxkat; h enAkñúgkrNIsþg;dar. - Local Zone: tMbn;enHCaRBIsbEnßménebtugEdlB½T§CMuvij nigenABIxagmux anchorage device Pøam² nigBI confining reinforcement. emIlépÞqUtenAkñúgrUbTI 4>22 (c) nigTMhMrbs;vaenA kñúgrUbTI 4>22 (a). rUbenHbgðajBIkarEbgEckkugRtaMgTaj nigkugRtaMgsgát;enAkñúg local zone nig stress contour rbs;vaEdlTTYl)anBI finite element analysis rbs; Rutgers test. RbEvgrbs; tMbn;enHCatMélFMCageKkñúgcMeNamTTwgGtibrma b¤RbEvgrbs; anchorage device. eKeRCIserIs confining reinforcement eBj anchorage zone edIm,IkarBar bursting nig splitting EdlekItBIkMlaMgsgát;cMcMnucFMEdlbBa¢Úntamry³ anchorage device. elIsBIenH eKRtUvRtYt karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 131
  • 43. T.Chhay Binitü bearing stress enAelIebtugkñúg local zone EdlbNþalmkBIkMlaMgsgát;d¾FMenH edIm,IFana favaminFMCag allowable compressive bearing stress rbs;ebtug. Flexural Design of Prestressed Concrete Elements 132
  • 44. NPIC !> viFIKNnasMrab; General Zone Design Method for General Zone eKmanviFIbIEdlGacKNna anchorage zone - Linear Elastic Stress analysis approach Including Use of Finite Element: viFIenH Bak;B½n§nwgkarKNnasßanPaBlMGitrbs;kugRtaMgdUcCa linearly elastic. karGnuvtþén finite element method manPaBlM)akxøHkñúgkarbegáItKMrUrEdlmansñameRbHd¾RtwmRtUvenAkñúgebtug. Et CamYynwgkarsnμt;d¾smRsbmYyeKGacTTYl)annUvlT§plEdlGacTTYlyk)anmYy. - Equilibrium-Based Plasticity Approach dUcCa Strut-and Tie Method: viFI strut-and-tie pþl;nUvKnøgd¾l¥rbs;kMlaMgeRbkugRtaMgEdlmanTMrg;dUcCaeRKOgbgÁúM truss EdlkMlaMgkñúgrbs;va eKarBeTAtameKalkarN_lMnwgTUeTA. Ultimate load EdlBüakrN_edayviFIenHTTYlykBI kar)ak;énbgÁúM strut b¤ tie NamYy. viFIenHEtgEtpþl;nUvlT§plEdlmansuvtßiPaBsMrab;kargar Gnuvtþn_. - Approach Method: viFIenHGnuvtþsMrab;muxkat;ctuekaNEdlmindac;. @> viFIviPaK Linear Elastic sMrab;kMNt; Confining Reinforcement Linear Elastic Analysis Method for Confining Reinforcement Determination Anchorage zone rgnUvkugRtaMgbIkMritdUcbgðajenAkñúgrUbTI 4>22 (a) nig stress contour zone: - High bearing stress BImux anchorage device. eKRtUvkarebtugEdlmankarRtYtBinitüd¾Rtwm RtUvedIm,IkarBarkar)ak;edaykugRtaMgsgát;énkMNat;rgkarsgÁt;dUcbgðajenAkñúgRkLaépÞqñÚtén rUbTI 4>22(a) nig 4>22(c). - Extensive tensile-bursting stress enAkñúg tensile contour areas EdlEkgeTAnwgG½kSrbs; tendon dUcbgðajenAkñúgrUbTI 4>22(a) nig (b) nig enAkñúgrUbTI 4>23(b). - kugRtaMgsgát;FMenAkñúgEdnkugRtaMg (stress field) RkLaépÞ D nig E enAkñúgrUbTI 4>22(a). eKGaceRbI linear elastic stress analysis edIm,ITay)annUvTItaMgrbs;sñameRbH nigpþl;nUv kar)a:n;sμan Rbhak;RbEhlmYyEdlGacTTYlyk)anBIrMhUrkugRtaMgeRkayeBleRbH. RkLaépÞEdk TajRtUv)ankMNt;edIm,ITb;Tl;kMlaMgTajsrubEdlTTYl)anBIkarRbmUlpþMúkugRtaMgTajenAkñúgebtug. eKRtUvbEnßmEdkrgkarsgát;enAkñúgtMbn;sgát; RbsinebIkMlaMgsgát;FM. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 133
  • 45. T.Chhay EdlbgðajenAkñúgrUbTI 4>22 pþl;nUvkarKNnasßan Linearly elastic finite element analysis PaBrbs;kugRtaMgenAkñúg anchorage zone suRkitCag. b:uEnþ CMhanénkarKNnaRtUvkarefr³evlaeRcIn Cag nigcMNayeRcInCag. lT§plRtUv)ankMNt;edaysarPaBBi)akkñúgkarbegáItKMrUEdlmansñameRbH enAkñúgebtugd¾RtwmRtUv. eKGaceRbI nonlinear finite element analysis edIm,ITaynUv post-cracking response. Flexural Design of Prestressed Concrete Elements 134
  • 46. NPIC rUbTI 4>23 bgðajBI linearly elastic end block forces. vabgðajBIkMlaMg end-block nigkugRtaMgsrésEdlbNþalBIkMlaMgeRbkugRtaMg Pi k¾dUcCatMélm:Um:g;Bt;sMrab;kMBs;eRbH y EdlGac ekItmannImYy² BIelI)atFñwm CD . tMélm:Um:g;Gtibrma M max kMNt;TItaMgén horizontal bursting crack. m:Um:g;enHRtUv)anTb;Tl;edaym:Um:g; couple EdlekItBIkMlaMgTaj T én vertical anchorage zone reinforcement nigkMlaMgsgát; C Edlpþl;eday end-block concrete xN³EdlkMlaMgkat;tam Tisedk V enARtg; crack spite surface RtUv)anTb;Tl;eday aggregate interlock force. tamkarGegát Edkkg vertical anchorage zone Edlpþl;kMlaMg T RtUv)anEbgEckelItMbn;Edlman TTwg h / 2 BIépÞxagcugrbs;Fñwm EdldUcCa X enAkñúgrUbTI 4>23 GacERbRbYlBI h / 5 eTA h / 4 . BIsmIkarlMnwgrbs;m:Um:g; M max T= (4.13) h−x ehIyRkLaépÞrbs;EdkbBaÄrEdlRtUvkarsrubkøayCa T At = (4.14) fs EdlkugRtaMgEdk f s EdlRtUv)aneRbIenAkñúgkarKNnaenHminRtUvFMCag 20,000 psi(138.5MPa ) sMrab; karRKb;RKgTTwgsñameRbH. Casegçb nigCMnYseGay linear elastic finite element analysis eKGacTTYldMeNIrkar Edl)anENnaM eTaHbICaminminsUvsuRkitdUckarKNna anchorage y:aglMGitEdlnwgpþl;eGayenA kñúg]TahrN_ 4>6 Epñk (a) k¾eday. #> Strut-and-Tie Method for Confining End-Block Reinforcement Strut-and-tie concept KWQrelI plasticity approach Edl)a:n;RbmaNkMlaMgenAkñúg anchorage zone edayes‘rIén strut sgát;Rtg; nig tie TajRtg;EdlP¢ab;KñaRtg;cMnucmYyEdleKehAfa node edIm,IkøayeTACa truss Éktþa. kMlaMgsgát;RtUv)anTb;Tl;eday plastic compressive strut ehIykMlaMgTajRtUv)anTb;Tl;edayEdkminEmneRbkugRtaMg b¤edayEdkeRbkugRtaMg. Yield strength rbs; anchorage confining reinforcement RtUv)aneRbIedIm,IkMNt;RkLaépÞsrubrbs;EdkEdlcaM)ac; eRbIenAkñúg anchorage block. rUbTI 4>24 bgðajBIkMlaMgeRbkugRtaMgcMp©it nigcakp©it P BImuxcMnucén karGnuvtþkMlaMgTaMgenHtamry³ anchorage device eTAkan;cugén general zone EdlkugRtaMgkøayCa rayesμItameKalkarN_ St. Venatn. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 135
  • 47. T.Chhay Flexural Design of Prestressed Concrete Elements 136
  • 48. NPIC karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 137
  • 49. T.Chhay Flexural Design of Prestressed Concrete Elements 138
  • 50. NPIC eRkayBIekItmansñameRbHKYreGaycab;GarmμN_mk KnøgkugRtaMgsgát;enAkñúgebtug)anRbmUlpþúM KñaeTACaExSRtg;EdlGacKitdUcCa straight compressive strut rgkMlaMgsgát;tamG½kSmYy. Srut TaMg enHnwgkøayCacMENkrbs; truss ÉktþaEdkkugRtaMgTajemRtUv)anKitCa tension tie enAkñúg truss Éktþa EdlmanTItaMgrbs; node RtUv)ankMNt;edayTisedArbs; compression strut. rUbTI 4>25 (a) bgðajBI karbegáIt strut nigrUbTI 4>25(b) bgðajBI truss EdlekItBI strut-and-tie sMrab; multiple anchorage enAkñúgmuxkat;GkSr T. rUbTI 2>26 segçbBIKMnitén strut nig tie enAkñúg anchorage zone. rUbTI 2>27 bgðajBI standard strut-and-tie truss sMrab;krNIcMp©it nigcakp©iténmuxkat;tan; nigmuxkat;man søabEdleGayeday ACI 318-99 Code. eKsnμt; tension tie enAkñúg truss sib,nimitþmancMgay h / 2 BI anchorage device. karsnμt; enHGacGnuvtþeTA)anCamYynwgTItaMgrbs;kMlaMgTaj T enAkñúgrUbTI 4>23 én elastic stress-analysis karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 139
  • 51. T.Chhay approach. Epñk (b) én]TahrN_ 4>6 bgðajBIKnøgEdlsnμt;sMrab; anchorage zone enAkñúg I-beam EdleKBicarNa. ⎛ a⎞ Tburst = 0.25 ∑ Psu ⎜1 − ⎟ (4.15a) ⎝ h⎠ d burst = 0.5(h − 2e ) (4.15b) Edl ∑ Psu =plbUkénkMlaMg tendon emKuNsrub a = kMBs;rbs; anchorage device b¤RkumeTalén closely-spaced device e = cMNakp©itén anchorage device b¤Rkumén closely-spaced device BITIRbCMuTMgn;rbs;mux kat;Fñwm h = kMBs;rbs;muxkat; eKeRbI anchorage device Ca closely-spaced device RbsinebIKMlatBIG½kSeTAG½kSrbs;vamin FMCag 1.5 dgénTTwgrbs; anchorage device. 4. Allowable Bearing Stresses GnuBaØatGtibrmaenARtg; anchorage device seating minRtUvFMCagtMél Bearing stress RsedogKñaBIrEdlTTYl)anBIsmIkar 4.16a nig 4.16b dUcxageRkam³ f b ≤ 0.7φf 'ci A / Ag (4.16a) f b ≤ 2.25φf 'ci (4.16b) Edl kMlaMg tendon emKuNGtibrma Pu EckCamYynwg effective bearing area Ab fb = f 'ci = ersIusþg;sgát;rbs;ebtugenAeBlrgkugRtaMg A = RkLaépÞGtibrmaéncMENkrbs;épÞEdlRTEdlmanragFrNImaRtRsedogKñanwgRkLaépÞrg bnÞúk ehIyRtYtsIuKña. Ag = gross area rbs; bearing plate Ab = effective net area rbs; bearing plate EdlRtUv)anKNnaedaydkRkLaépÞ As BIRkLa épÞRbehagenAelI bearing plate. smIkar 4.16a nig 4.16b mann½yEtRbsinebIeKdak; general zone reinforcement nigRbsinebIRbEvg énkarlatsn§wgrbs;ebtugtambeNþayG½kSrbs; tendon BImux anchorage device esμIBIrdgRbEvgén local zone y:agtic. Flexural Design of Prestressed Concrete Elements 140
  • 52. NPIC X> KNnaEdk End Anchorage sMrab;FñwmeRbkugRtaMgrgkarTajCaeRkay Design of End Anchorage Reinforcement for Post-tensioned Beams ]TahrN_ 4>6³ KNna end anchorage reinforcement sMrab; post-tensioned beam enAkñúg]TahrN_ 4>2 EdleGayTMhM RbePT nigkarBRgayEdk. eRbIebtugTMgn;Fmμta f 'c = 5,000 psi(34.5MPa) . snμt;facugFñwmCabøúkctuekaNEdllUtcUleTAkñúgElVg 40in.(104cm) BIeRkay anchorage device bnÞab;mkkat;bnßykMras;RTnug 6in. . edaHRsaybBaðaedayeRbI (a) linear elastic stress analysis method, (b) plastic strut-and-tie method. KUrKMrU truss Edl)ankMNt;. dMeNaHRsay³ (a) edaHRsayeday linear elastic stress method³ !> begáItKMrUén tendon edaymancMNakp©it ee = 12.49in.(317mm) BI]TahrN_ 4>2. cb = 18.84in. dUcenHcMgayBIsrésxageRkamrbs;Fñwm = cb − ee = 6.35in.(161mm) sMrab;cMgayTIRbCMuTMgn;rbs; tendon Ggát;p©it 1 / 2in. cMnYn 13 edIm = 6.35in. BIsrésxageRkamFñwm sakl,gkartMerobCaCYredkdUcxageRkam CYredkTI 1³ 5 tendon enAcMgay 2.5in CYredkTI 2³ 5 tendon enAcMgay 7.0in. CYredkTI 3³ 3 tendon enAcMgay 11.5in. cMgayénTIRbCMuTMgn;rbs; tendon = 5 × 2.5 + 5 ×13.0 + 3 × 11.5 ≅ 6.35in. 7 O.K. @> Ultimate forces enAkñúgCYredkén tendon nig bearing capacity rbs;ebtug kMlaMg Pu1 CYredkTI 1³ 5 × 0.153 × 270,000 = 206,550lb(919kN ) kMlaMg Pu 2 CYredkTI 2³ 5 × 0.153 × 270,000 = 206,550lb(919kN ) kMlaMg Pu3 CYredkTI 3³ 3 × 0.153 × 270,000 = 123,930lb(551kN ) #> Elastic analysis énkMlaMg EckkMBs;FñwmCacMerokEdlmankMBs; 4in. dUcbgðajenAkñúgrUbTI 4>28(a) nigsnμt;fakugRtaMg ebtugrbs;cMeroknImYy²esμInwgkugRtaMgenARtg;G½kSrbs;cMerokenaH. bnÞab;mkKNnakMeNInm:Um:g;Edl bNþalBIkugRtaMgxagkñúg nigkMlaMgeRbkugRtaMgxageRkA Pi eFobnwgbøg;edknImYy²edIm,IkMNt; net moment enAelImuxkat;. Net moment GtibrmanwgkMNt;TItaMgrbs; potential horizontal bursting crack nigEdkEdlRtUvdak;edIm,IkarBarsñameRbHEdlnwgekItmanenaH. edayeRbIsBaØabUk (+) sMrab; karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 141
  • 53. T.Chhay m:Um:g;vilRsbRTnicnaLika. BI]TahrN_ 4>2 kMlaMgeRbkugRtaMgedImmuneBlxatbg;KW Pi = 376,110lb (1,673kN ) . BIrUbTI 4>28 m:Um:g;xagkñúgebtugenARtg;bøg; 4in. BIsrésxageRkamKW M c 4 = 2,117 × 4 × 18 × (2in.) = 304,848in. − lb = 0.3 ⋅ 10 6 in. − lb(34.4kN .m ) nigenARtg;bøg; 8in. BIsrésxageRkamKW 18 + 10 M c8 = 2,117 × 4 × 18 × (6in.) + 1,851 × 4 × × (2in.) 2 = 1,121,856in. − lb = 1.12 ⋅ 10 6 in. − lb(127 kN .m ) m:Um:g;kMlaMgeRbkugRtaMgenARtg;bøg; 8in. BIsrésxageRkamKW M c8 = 376,110 × (8 − 6.35) = −620,582in. − lb = −0.62 ⋅ 10 6 in. − lb(70.1kN .m ) Net moment KW = 1.12 ⋅106 − 0.62 ⋅106 = 0.50 ⋅106 in. − lb(56.6kN .m) tamrebobdUcKña eyIgGacrk net moment sMrab;cMerokd¾éTeTot ehIytMélrbs;vaRtUv)anerobdak;enA kñúgtarag 4>5. BItaragenH net moment GtibrmaKW + M max = +0.75 ⋅106 in. − lb(84.6kN .m) enARtg;bøg;edk 6.35in. BIsrésxageRkamrbs;Fñwm (bursting potential crack effect) ehIy Flexural Design of Prestressed Concrete Elements 142
  • 54. NPIC − M max = −0.20 ⋅ 106 in. − lb enARtg;bøg; 24in.(64cm) BIxagelIsrésxageRkamrbs;Fñwm (spalling potential crack effect) . $> KNna anchorage reinforcement BIsmIkar 4>11 nigedaysnμt;vaG½kSrbs;kMlaMgTajbBaÄr T KWenARtg;cMgay x ≈ 15in. eyIgTTYl)an M max 0.75 ⋅ 106 T= = = 30,000lb(133kN ) h−x 40 − 15 edayGnuBaØatkugRtaMgEdkGtibrma f s = 20,000 psi ¬kUdGnuBaØat 0.60 f y = 36,000 psi ¦ Bursting zone reinforcement KW At = Tb 30,000 = f s 20,000 ( = 1.50in 2 968mm 2 ) dUcenH sakl,gEdkkgbiTCit #3 ³ (As = 2 × 0.11 = 0.22in.2 ) cMnYnEdkkgEdlRtUvkar = 1..50 = 6.82 kg 0 22 eRbIEdkkg #3 cMnYn 6 kg bEnßmBIelIEdkkgsMrab;Tb;nwgkMlaMgkat;. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 143
  • 55. T.Chhay Spalling zone force − 0.2 × 106 Ts = = 8,000lb 40 − 15 dUcenH T As = s = 8,000 f s 20,000 ( ) = 0.40in.2 250mm 2 dUcenH eyIgman cMnYnEdkkg #3 EdlRtUvkar = 0..40 = 1.82 kg 0 22 eRbIEdkkg #3 cMnYnBIrkgbEnßmeTot. dUcenH cMnYnEdkkgsrub = 6.82 + 1.82 + 4 = 12.64 kg eRbIEdkkgbiTCit #3 cMnYn 12 kg. dak;EdkkgbBa©ÚleTAkñúgtMbn;sgát;enAkñúgrUbTI 4>23. dak;Edkkg #3 KMlatBIKña 3in. edayKitBIG½kSeTAG½kS edayEdkkgTImYycab;epþImCamYynwgKM lat 3in. BIcugFñwm. ehIy dak;Edk #3 RbEvg 10in. cMnYn 4 edImEdlmanKMlatBIKña 3in. KitBIG½kSeTA G½kS nigmanKMlat 2in. BIépÞxagcugRtg;TItaMg anchorage edaysarsñameRbHGacekItmantamTis bBaÄr nigTisedk. RbsinebImantMrUvkarrbs;plitkr eKRtUvbEnßm spiral reinforcement BIxageRkam anchor. (b) edaHRsayeday plastic Strut-and-tie method³ !> begáItKMrUén tendon EdlmancMNakp©it ee = 12.49in.(317mm) BI]TahrN_ 4>2 cb = 18.84in. dUcenHcMgayBIsrésrbs;Fñwm = cb − ee = 6.35in.(161mm) sMrab;cMgayTIRbCMuTMgn;rbs; strand Ggát;p©it 1 / 2in. cMnYn 13 edImEdlesμInwg 6.35in BIsrés xageRkamrbs;Fñwm sakl,gkartMerob tendon CaCYredkEdlmancMgayBIsrésxageRkamdUcteTA³ CYredkTI 1³ 5 tendon enARt;g 2.5in. CYredkTI 2³ 5 tendon enARt;g 7.0in. CYredkTI 3³ 3 tendon enARt;g 11.5in. cMgayénTIRbCMuTMgn;rbs; tendon = 5 × 2.5 + 5 ×13.0 + 3 ×11.5 ≅ 6.35in. O.K. 7 @> Ultimate force enAkñúgCYredkrbs; tendon nig bearing capacity rbs;ebtug kMlaMg Pu1 CYredkTI 1³ 5 × 0.153 × 270,000 = 206,550lb(919kN ) kMlaMg Pu 2 CYredkTI 2³ 5 × 0.153 × 270,000 = 206,550lb(919kN ) kMlaMg Pu3 CYredkTI 3³ 3 × 0.153 × 270,000 = 123,930lb(551kN ) Flexural Design of Prestressed Concrete Elements 144
  • 56. NPIC kMlaMgsgát; ultimate srub = 206,550 + 206,550 + 123,930 = 537,830lb(2,389kN ) RkLaépÞsrubrbs; rigid bearing plate EdlRT Supreme 13-chucks anchorage device = 14 × 11 + 6 × 4 = 178in.2 ( cm 2 ) 113 Bearing stress Cak;Esþg f b = = 3020 psi(20.8MPa ) 537,380 178 BIsmIkar 4.16(a) nig (b), bearing pressure GnuBaØatGtibrmaenAelIebtugKW f b ≤ 0.7φf 'ci A / Ag f b ≤ 2.25φf 'ci snμt;fa ersIusþg;ebtugdMbUgenAeBlEdlrgkugRtaMgKW f 'ci = 0.75 f 'c = 0.75 × 5,000 = 7,750 psi RkLaépÞcMp©it A rbs;ebtugEdlman bearing plate ≅ 18 ×14 + 10 × 7 = 322in.2 Bearing stress GnuBaØat f b = 0.70 × 0.90 × 3,750 322 = 3,178 psi > 3,020 psi O.K. 178 Bearing stress BIsmIkar 4.14(b) Gt;lub. #> KUr strut-and-tie model RbEvgcMgaysrub a enAkñúgrUbTI 4>25 rvagkMlaMg Pu1 nig Pu3 = 11.5 − 2.5 = 9.0in. dUcenHcMgay a / 2 BImux anchorage = 9.0 / 2 = 4.5in. sg; strut-and-tie edaysnμt;vadUcbgðajenAkñúgrUbTI 4>29. TMhMFrNImaRtsMrab;rkbgÁúMkMlaMgedkBI tie 1 − 2 nig 2 − 3 EdlmantMélkUtg;sg; 26.5 / 15.5 nig 13.0 / 15.5 erogKña. BIsþaTic viPaK truss enAkñúgrUbTI 4>29 edayTTYl)ankMlaMgGgát;dUcxag eRkam³ = 211,982lb(942kN ) rgkarTaj 26.5 tie 1 − 2 = 123,930 × 15.5 = 173,235lb(728kN ) rgkarTaj 13 tie 2 − 3 = 206,550 × 15.5 eRbItMélEdlFMCagkñúgcMeNamtMélTaMgBIredIm,IeRCIserIsEdkkgbiTCitEdlrgkarTaj. sakl,gEdkkg #3 Edlman tensile strength kñúgmYykg = φf y Av = 0.90 × 60,000 × 2(0.11) = 11,880lb cMnYnEdkkgEdlRtUvkar = 211,982 11,880 = 17.8 karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 145
  • 57. T.Chhay sMrab;EdkkgrgkarTaj a − b − c enAkñúgrUbTI 4>29 eRbIkMlaMg Pu = 173,235lb edIm,Idak;EdkkgbBaÄr #4 BIxagmux anchorage device. cab;epþImEdkkgTImYyenAcMgay 1 1 in. BIxagcug rigid steel plate 2 EdlepÞrkMlaMgBI anchorage device eTAebtug. cMnYnEdkkg = 0.9 × 60,000 × 2(0.20) = 8.0 173,235 eRbIEdkkg #4 cMnYn 8 kgEdlmancMgayBIKña 1 14 in. BIG½kSeTAG½kS ¬12.7mm @ 32mm ¦ Edl manEdkkgTImYycab;epþImenAcMgay 1 12 in. BIxagmux anchorage device. eKRtUvkarEdkkgEt 13 CMnYseGay 17.8 Edl)anBIkarKNna edaysarEpñkrbs;tMbn;RtUv)an Tb;Tl;edayEdkkg #4 . eRbIEdkkg #3 cMnYn 13 EdlmanKMlatBIKña 2 12 in. BIG½kSeTAG½kS ¬12.7mm @ 57 mm ¦ bnÞab;BIEdkkg #4 EdlmancMgaysrubTaMgGs; 40in.(104cm ) . cMNaMfaviFIenHRtUvkar confining tie eRcInCag elastic solution kñúgEpñk (a). rUbTI 4>30 bgðajBI anchorage zone confining reinforcement lMGitEdl)anBI strut-and-tie analysis. Flexural Design of Prestressed Concrete Elements 146
  • 58. NPIC 6> KNnaFñwmsmasrgkarBt; Flexural Design of Composite Beams muxkat;smas FmμtaCaeRKOgbgÁúMeRbkugRtaMgcak;Rsab;EdlenABIelIva kMralxNÐRtUv)ancak;enA kardæan ehIyvaeFVIkarCamYyKña ¬rUbTI 4>31¦. eBlxøH eKTl; prestressed element kñúgGMLúgeBlcak; nigEfTaM situ-cast top slab. kñúgkrNIEbbenH TMgn;kMralxNÐeFVIGMeBIEtelImuxkat;smas Edlmanm:U Dulmuxkat;FMCagmuxkat;cak;Rsab;. dUcenH karKNnakugRtaMgebtugRtUv)anykmkKitenAkñúgkarKNna. karEbgEckkugRtaMgebtugEdlbNþalBIGMeBIsmasRtUv)anbgðajenAkñugrUbTI 4>32. k> krNIkMralxNÐminmanTl; Unshored Slab Case BIsmIkar 4.2a nig b smIkarkugRtaMgsrésebtugxageRkAbMputmuncak;kMralxagelIKW Pe ⎛ ect ⎞ M D + M SD ft =− ⎜1 − 2 ⎟ − (4.17) Ac ⎝ r ⎠ St karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 147
  • 59. T.Chhay P ⎛ ec ⎞ M + M SD nig f b = − e ⎜1 + 2b ⎟ + D Ac ⎝ r ⎠ Sb (4.18) Edl S t nig Sb Cam:UDulmuxkat;rbs;muxkat;cak;Rsab;Etb:ueNÑaH ehIy M SD Cam:Um:g;dak;BIelIbEnßm dUcCaebtugkMral. eRkayeBlkMralcak;enAnwgkEnøgkkrwg ehIyvaGaceFVIkarlkçN³smasmk vaGacmanm:UDul muxkat; Sct nig Scb FMCagmun CamYynwgkarrMkileLIgelIeTArksrésxagelIrbs;ExS cgc. kugRtaMg srésebtugcUlrYmCamYynwgsmIkar 4.17 nig 4.18 sMrab;srésxagelI nigxageRkamrbs;Epñkcak; Rsab;rbs;muxkat;smas ¬nIv:U AA enAkñúgrUbTI 4>32(e)¦ KW ⎛ ect ⎞ M D + M SD M CSD + M L Pe ft =− ⎜1 − 2 ⎟ − − (4.19a) ⎝Ac r ⎠ St Sc t P ⎛ ec ⎞ M + M SD M CSD + M L nig f b = − e ⎜1 + 2t ⎟ + D Ac ⎝ r ⎠ Sb + S cb (4.19b) Edl M CSD CabnÞúkefrsmasdak;BIelIbEnßmeRkayeBldMeLIg dUcCaenAeBleFVIkar. ehIy Sct nig Scb Cam:UDulmuxkat;rbs;muxkat;smasenAnIv:UénsrésxagelI nigxageRkam erogKña rbs;muxkat;cak; Rsab;. kugRtaMgenAnIv:UsrésxagelI nigxageRkamrbs;kMralcak;enAnwgkEnøg ¬nIv:U BB nig AA rbs;mux kat; 4>32 (e)¦ KW M CSD + M L f ts = − t (4.20a) S cb Flexural Design of Prestressed Concrete Elements 148
  • 60. NPIC + ML nig M f bs = − CSD Sbcb (4.20b) Edl M CSD + M L Cam:Um:g;bEnßmEdlekIneLIgeRkayeBlekItmanskmμPaBsmas ehIy Scb nig Sbcb t Cam:UDulmuxkat;rbs;muxkat;smassMrab;srésxagelI nigxageRkam AA nig BB erogKña enAkñúgrUbTI 4>32(e). x> krNIkMralxNÐTl;eBj Fully Shored Slab Case kñúgkrNIkMralcak;enAkEnøgRtUv)anRTeBjrhUtdl;ekItmanskmμPaBsmas kugRtaMgsrés ebtugmuneBlRT nigmuneBlcak;ebtugkMralxagelIEdlkøayBIsmIkar 4.18 nig 4.19KW ⎛ ect ⎞ M D Pe ft =− ⎜1 − 2 ⎟ − t (4.21a) ⎝ Ac r ⎠ S P ⎛ ec ⎞ M nig f b = − e ⎜1 + 2b ⎟ + D Ac ⎝ r ⎠ Sb (4.21b) eRkayeBlkMralxagelIcak;rYc ehIyskmμPaBsmasekItmanenAeBlebtugkkrwg smIkar 4.19a nig b sMrab;FñwmEdlRtUv)anRTeRkayeBldMeLIgnwgkøayeTACa ⎛ ect ⎞ M D M SD + M CSD + M L Pe ft =− ⎜1 − 2 ⎟ − t − (4.22a) ⎝ Ac r ⎠ S t Sc P ⎛ ecb ⎞ M M + M CSD + M L nig f b = − e ⎜1 + 2 ⎟ + D + SD Ac ⎝ r ⎠ Sb S cb (4.22b) cMNaMfaeKRtUveFVIkarRtYtBinitüsMrab;kugRtaMgkat;tamTisedkEdlekItmanenARtg;épÞb:HrvagebtugEdl cak;enAnwgkEnøg CamYynwgFñwmcak;Rsab; ¬nwgbgðajenAkñúgCMBUk 5¦. K> TTwgsøabRbsiT§PaB Effective Flange Width karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 149
  • 61. T.Chhay edIm,IkMNt;skmμPaBsmastamRTwsþIEdlTb;Tl;kugRtaMgBt; eKRtUveFVIkarkMNt;TTwgkMralxNÐ EdlGaccUlrYmy:agmanRbsiT§PaBenAkñúgekIneLIgPaBrwgRkaj (stiffness) EdlTTYl)anBIskmμPaB smas. rUbTI 4>33 nigtarag 4>6 eGaynUvtMrUvkarrbs; ACI nig AASTHO sMrab;kMNt;TTwgsøabxag elIRbsiT§PaB (effective top slange width) rbs;muxkat;smas. RbsinebIersIusþg;rbs;ebtugEdlcak; BIxagelIxusBIersIusþg;rbs;muxkat;cak;eRsc eKRtUvEktMrUvTTwg b edayKitBIm:UDuleGLasÞicxusKñarbs; ebtugTaMgBIr edIm,IFanafabMErbMrYlrageFobrbs;sMPar³TaMgBIrenARtg;épÞb:HdUcKña. TTwgEksMrYlrbs; kMralxagelIsMrab;KNnam:Um:g;niclPaBsmas I cc KW bm = Ect (b ) = ncb (4.23) Ec Edl m:UDuleGLasÞicrbs;ebtugEdlcak;BIxagelI Ect = Ec = m:UDuleGLssÞicrbs;ebtugcak;Rsab; enAeBlEdlkMNt;TTwgEksMrYl bm rYcehIy eKRtUvBicarNaersIusþg;ebtugrbs;muxkat;smasTaMgmUlCa ersIusþg;EdlFMCag. 7> Summary of Step-by-Step Trial-and Adjustment Procedure for the Service-Load Design of Prestressed member !> eGaynUvGaMgtg;sIuetbnÞúkefrEdldak;BIelIbEnßm WSD / GaMgtg;sIuetbnÞúkGefr WL / RbEvg kMNt; nigkMBs;kMNt;/ ersIusþg;sMPar³ f pu / f 'c / RbePTebtug nigeBlxøHRbePTeRbkug RtaMg dUcCaTajCamun b¤CaeRkay. Flexural Design of Prestressed Concrete Elements 150
  • 62. NPIC @> snμt;TMhMrbs;bnÞúkpÞal; WD nigKNnam:Um:g; M D / M SD nig M L #> KNna f pi / fci / fti / ft nig fc Edl f pi = 0.70 f pu / fci = 0.60 f 'ci / fti = 6 f 'ci sMrab;muxkat;elITMr/ fc = 0.45 f 'c b¤ 0.60 f 'c tamkarGnuBaØat/ ft = 6 f 'c eTA12 f 'c $> KNnakMhatbg;kugRtaMg Δf pT = Δf pES + Δf pR + Δf pSH + Δf pCR + Δf pE + Δf pA + Δf B tamRbePTeRbkugRtaMgEdleRbI. kMNt;kugRtaMgsuT§ (net stress) f pe = f ps − Δf pT %> rkm:UDulmuxkat;tMrUvkarGb,brmaénmuxkat;Gb,brmaedIm,IKNnakugRtaMgebtugenAsrésxag elI nigxageRkam. (a) sMrab; harped b¤ draped tendon eRbImuxkat;kNþalElVg St ≥ (1 − γ )M D + M SD + M L γf ti − f c Sb ≥ (1 − γ )M D + M SD + M L f t − γf ci Edl γ = Pe Pi (b) sMrab; tendon Rtg;eRbImuxkat;elITMrxagcug M D + M DS + M L St ≥ γf ti − f c M + M SD + M L Sb ≥ D f t − γf ci ^> eRCIserIsmuxkat;sakl,gEdlmanm:UDulmuxkat;Ek,rnwgm:UDulmuxkat;EdlRtUvkar edIm,IRtYt BiinitüsMrab;tMrUvkarkugRtaMgsrésmuxkat;smasenAeBleRkay. &> sMrab; (a) muxkat;EdlsikSasßitenAkñúgElVg ¬CaTUeTAkNþalElVg b¤ 0.40 énElVg¦/ (b) muxkat;EdlsikSasßitenARtg;TMr nig (c) muxkat;d¾éTeTottambeNþayElVgsMrab;kareRbI tendon Rtg; nig draped tendon/ viPaKkugRtaMgsrésebtugrMBwgTukenAmuneBlepÞrkug RtaMgPøam² Pi ⎛ ect ⎞ M D ft =− ⎜1 − 2 ⎟ − t Ac ⎝ r ⎠ S P ⎛ ec ⎞ M f b = − i ⎜1 + 2b ⎟ + D Ac ⎝ r ⎠ Sb RbsinebIkugRtaMgFMCagtMélGnuBaØat eKRtUvBRgIkmuxkat; b¤eFVIkarpøas;bþÚrcMNakp©it ec b¤ ee b¤k¾TaMgBIr. *> viPaKkugRtaMgsrésebtugsMrab;lkçxNÐbnÞúkeFIVkar ¬dUckñúgCMhanCMBUk &¦ karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 151
  • 63. T.Chhay Pe ⎛ ect ⎞ M T ft =− ⎜1 − 2 ⎟ − t Ac ⎝ r ⎠ S P ⎛ ecb ⎞ M f b = − e ⎜1 + 2 ⎟ + T Ac ⎝ r ⎠ sb Edl M T = M D + M SD + M L / RbsinebIkugRtaMgFMCagtMélGnuBaØat eKRtUvBRgIkmuxkat; b¤eFVIkarpøas;bþÚrcMNakp©it ec b¤ ee b¤k¾TaMgBIr. (> sMrab;krNIEdleKeRbI strand eRcIn/ begáIt envelop EdlkMNt;cMNakp©itsMrab;kugRtaMgTaj sUnü eb = (kb − amin ) nig et = (amax − kt ) / Edl amin = M D / Pi nig amax = M T / Pe . RbsinebIeKeRbIeKeRbIkugRtaMgTajrbs;ebtugenAkñúgkarKNna bEnßm f (t ) Ac kb e'b = Pi f (b ) Ac kt nig e't = Pe eTAelI envelop xageRkam nigxagelI erogKña/ Edl f(t ) nig f(b) CakugRtaMgsrésxag eRkArbs;ebtugEdlKNnaedIm,IeRbIenAkñúgebtug !0> GegátkugRtaMg end-block anchorage zone/ nigKNnaEdkcaM)ac;edIm,IkarBar bursting b¤ spalling crack. kMNt;RbEvgbgáb;Gb,brma 1 ⎛ ⎞ ⎜ f ps − f pe ⎟d e ¬xñat US¦ 2 ld = 1,000 ⎝ 3 ⎠ 1 ⎛ ⎞ ld = 6.9 ⎝ 2 ⎜ f ps − f pe ⎟d b 3 ⎠ ( xñat SI) nigRbEvgepÞr lt = f pe 3,000 db (xñat US) f pe lt = 20.7 db (xñat SI) Post-tensioned anchrage KNnaEdl anchorage block. eRbI strut-and-tie plastic truss unit edim,IKNna kMlaMgTaj ultimate enAkñúg tie edIm,IeCIserIs confining reinforcement. Pretensional Prestress Transfer Zone Ph At = 0.021 i f s lt US ¬xñat ¦ At = 21,000 i Ph f s lt IS ¬xñat ¦ Flexural Design of Prestressed Concrete Elements 152
  • 64. NPIC !!> kMNt;kugRtaMgskmμPaBsmas nigeRCIserIsmuxkat;eLIgvij RbsinebIkugRtaMgFMCagkug RtaMgsrésebtugGnuBaØatGtibrmaTaMgenAkñúgmuxkat;cak;Rsab; nigkMralxagelIcak;enAnwg kEnøg. eRbITTwgRbsiT§PaBEksMrYl bm = (Ect / Ec )b sMrab;søabsmasxagelI enAeBl KNnam:UDUlmuxkat;rbs;muxkat;smas. (a) krNIkMralxNÐminTl; muneBlcak;ebtugkMralxagelIenAnwgkEnøg Pe ⎛ ect ⎞ M D + M SD ft =− ⎜1 − 2 ⎟ − Ac ⎝ r ⎠ St P ⎛ ecb ⎞ M + M SD M CSD + M L f b = − e ⎜1 + 2 ⎟ + D + Ac ⎝ r ⎠ Sb S cb eRkayeBlkMralxagelIRtUv)ancak; nigTTYlkarEfTaMedIm,IeGayekItmanskmμPaB smaseBj kugRtaMgenAsrésxagelI nigxageRkamrbs;Epñkcak;Rsab;énmuxkat; smasnwgkøayCa Pe ⎛ ect ⎞ M D + M SD M CSD + M L ft =− ⎜1 − 2 ⎟ − − Ac ⎝ r ⎠ St Sct P ⎛ ecb ⎞ M + M SD M CSD + M L f b = − e ⎜1 + 2 ⎟ + D + Ac ⎝ r ⎠ Sb S cb Edl Sct nig Scb Cam:UDulmuxkat;rbs;muxkat;smasenAnIv:UsrésxagelI nigxageRkam rbs;muxkat;cak;Rsab;. M L rYmbBa©ÚlTaMg M I RbsinebImankugRtaMgTgáic. kugRtaMgsrésenAnIv:UsrésxagelI nigxageRkaménkMralcak;enAnwgkEnøgKW M CSD + M L f ts = − t S cb M + ML f bs = − CSD Sbcb Edl Scb nig Sbcb Cam:UDulmuxkat;énmuxkat;smasenAnIv:UsrésxagelI nigxageRkam t rbs;kMralcak;enAnwgkEnøg. (b) krNIkMralxNÐTl;eBj muneBlTl; nigeRkayeBlcak;ebtugkMralxagelIenAnwgkEnøg ⎛ ect ⎞ M D Pe ft =− ⎜1 − 2 ⎟ − t ⎝Ac r ⎠ S P ⎛ ec ⎞ M f b = − e ⎜1 + 2b ⎟ + D Ac ⎝ r ⎠ Sb karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 153
  • 65. T.Chhay eRkayeBlkMralxNÐcak;enAkEnøgRtUv)anEfTaM nigskmμPaBsmaseBjekItman ⎛ ect ⎞ M D M SD + M CSD + M L Pe ft =− ⎜1 − 2 ⎟ − t − ⎝ Ac r ⎠ S t Sc P ⎛ ecb ⎞ M M + M CSD + M L f b = − e ⎜1 + 2 ⎟ + D + SD Ac ⎝ r ⎠ Sb S cb kMNt;TTwgRbsiT§PaBrbs;søabxagelIrbs;muxkat;smastam ACI b¤ AASTHO specification MD =m:Um:g;EdlbNþalBITMgn;pÞal;énGgát;cak;Rsab;/ M SD = m:Um:g;Edl bNþalBIkMralxNÐcak;Rsab; nigbnÞúksagsg;d¾éTeTot ehIy M CSD = m:Um:g;Edl bNþalBIbnÞúkdak;bEnßmBIelI. !@> bnþkMNt;ersIusþg;rbs;muxkat;sMrab;sßanPaBkMNt;enAeBl)ak; nigkMNt;ersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl. rUbTI 4>34 bgðajBI flowchart sMrab;karKNnaFñwmeRbkugRtaMgrgkarBt;eRkamlkçxNÐ bnÞúkeFVIkar. Flexural Design of Prestressed Concrete Elements 154
  • 66. NPIC karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 155
  • 67. T.Chhay Flexural Design of Prestressed Concrete Elements 156
  • 68. NPIC 8> KNnamuxkat;smasTMrsamBaØeRbkugRtaMgrgkarTajCaeRkay Design of Composite Post-Tensioned Prestressed Simply Supported Section ]TahrN_ 4>7³ Supported Section ³ s<anTMrsamBaØBIrpøÚv (two-lane) EdlmanElVgRbEvg 64 ft (19.5m) KitBIG½kSeTA G½kSén bearing. TTwgs<anEdlKitBIFñwmxageRkAmanRbEvg 28 ft (8.54m) EdlKitBIG½kSeTAG½kS. KMlatrbs;FñwmxagkñúgKW 7 ft KitBIG½kSeTAG½kS. KNnaFñwmTMrxagkñúgeRbkugRtaMgrgkarTajCaeRkay EdlmankMrals<anminTl;kñúgGMLúgeBlsagsg; edIm,IRTkardak;bnÞúk AASTHO HS20-44. kMNt; cMNakp©it nig tendon envelop. KNna anchorage block nig confining reiforcement. smμtikmμ³ ebtug ebtugcak;Rsab;man f 'c = 5,000 psi CaebtugTMgn;Fmμta ebtugcak;kMrals<ankMras; 5.75in man f 'c = 3,000 CaebtugTMgn;Fmμta f 'ci = 4,000 psi(27.6MPa ) f ci = 0.55 f 'ci = −2,200 psi (15.2MPa ) f c = 0.40 f 'c = −2,000 psi(13.8MPa ) f ti = 212 psi = 3 f 'c f t = 6 f 'c = 424 psi EdkeRbkugRtaMg f pu = 270,000 psi(1,862MPa ) f py = 0.90 f pu = 243,000 psi(1,675MPa ) f pi = 0.70 f pu = 189,000 psi(1,303MPa ) eRkaykMhatbg; = 0.80 f pi = 151,200 psi(1,043MPa) f pe kMNt; nigKUrnUvkarEbgEck tendon TaMgenAmuxkat;kNþalElVg nigmuxkat;xagcug. eRbItMélm:U m:g;bnÞúkGefrEdlrYmbBa©ÚlTaMgkMlaMgTgáicEdlbNþalBIkardak;bnÞúktam AASTHO HS20-44 sMrab; FñwmxagkñúgEdlmanElVg 64 ft (19.5m) = 9,300,000in. − lb(1,051kN .m) . dMeNaHRsay³ m:Um:g;Bt; nigkugRtaMgGnuBaØat ¬CMhan1-4¦ karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 157
  • 69. T.Chhay edaysareKsÁal;KMlatFñwm nigRbEvgElVg eKGackMNt;m:Um:g;Edl)anBIkMralcak;enAnwgkEnøg nig diaphragm. Clear distance rvagRTnugrbs;FñwmKW 7 ft − 6in = 6 ft 6in. . snμt;fakMrals<anEdl mankMras; 5.75in.(14.6cm) pSMBIkMralbnÞHcak;Rsab; 1.75in. CamYynwgkMralcak;BIelIkMras; 4in. nig diaphragm kMras; 8in.(20cm ) enAkNþalElVg nigkMBs; 45in.(122cm ) cak;rYmKñaCamYynwgkMrals<an. eyIgman³ TMgn; diaphragm = 12 × 12 × 6.5 ×150 = 2,500lb(11.6kN ) 8 45 TMgn;kMralcak;Rsab;kMras; 1.75in. = 112 × 7 ×150 = 153 plf (2.2kN / m) .75 TMgn;kMralcak;BIelIenAnwgkEnøgkMras; 4in. = 12 × 7 ×150 = 350 plf (5.1kN / m) 4 153(64 )2 M SD1 = × 12 = 940,032in. − lb 8 M CSD = bnÞúkefrdak;BIelIbEnøm = 0 PL WSD L2 M SD 2 = + 4 8 2,500 × 64 × 12 350(64 )212 = + 4 8 = 499,200 + 2,150,400 = 2,649,6500in. − lb srub = 940,032 + 2,649,000 = 3,589,632in. − lb(406kN .m) M SD M CSD = 0 enAkñúgkrNIenH m:UDulmuxkat;Gb,brma nigkareRCIserIsmuxkat;sakl,g ¬CMhan 1-5¦ 151,200 γ= = 0.80 189,00 St ≥ (1 − γ )M D + M SD + M L γfti − fc Sb ≥ (1 − γ )M D + M SD + M L ft − γf ci snμt;faFñwmcak;Rsab;manTMgn;pÞal;RbEhl 583 plf (8.5kN / m) . 583(64)2 × 12 MD = = 3,581,952in. − lb(405kN .m ) 8 S t min = (1 − 0.80)3,581,952 + 3,589,632 + 9,300,000 = 6,271in.3 103,222cm3 ( ) 0.80(212) − (− 2,000) Sb min = (1 − 0.80)3,581,952 + 3,589,632 + 9,300,000 = 6,229in.3 102,075cm3 ( ) 424 − 0.80(− 2,200) Flexural Design of Prestressed Concrete Elements 158
  • 70. NPIC CaTUeTA m:UDulmuxkat;Cak;EsþgEdlrMBwgTuksMrab;srésxagelIEtgEtFMCagm:UDulmuxkat;sMrab; srésxageRkamrbs;muxkat;smas. dUcenHeRCIserIsGgát;cak;Rsab;edayQrelI Sb = 6,229in.3 . edaysar AASTHO type IV manm:UDulmuxkat;FMCagm:UDulmuxkat;EdlRtUvkarxøaMgeBk dUcenH eyIgeRCIserIs AASTHO type III ¬rUbTI 4>35¦ Camuxkat;sakl,gEdlmantMélEk,r Sb = 6,229in.3 CageK. ( S t = 5,070in.3 83,082cm3 ) Sb = 6,186in.3 ( ,370cm3 ) 101 I c = 125,390in.4 (5.2 × 106 cm 4 ) Ac = 560in.2 (3,613cm 2 ) r = 223.9in ( ,445cm ) 2 1 2 2 ct = 24.73in.(62.8cm) cb = 20.27in.(51.5cm) WD = 583 plf (8.2kN / m ) sakl,g 7-wire stress-relieved tendon Ggát;p©it 0.5in.(12.7mm) cMnYn 22 edIm. kugRtaMgenAeBlepÞr M T = 3,581,952 + 3,589,632 + 9,300,000 = 16,471,584in. − lb(1,861kN .m ) ( Aps = 22 × 0.153 = 3.366in.2 21.7cm 2 ) karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 159
  • 71. T.Chhay Pi = Aps f pi = 3.366 × 0.70 × 270,000 = 636,174lb(2,830kN ) Pe = 0.80 Pi = 0.80 × 636,174 = 508,940lb(2,264kN ) r 2 223.91 kt = = = 11.05in.(28.1cm) cb 20.27 r 2 223.91 kb = = = 9.05in.(23.0cm) ct 24.73 M 3,581,952 amin = D = = 5.63in. Pi 636,174 M amax = T = Pe 16,471,584 508,940 ¬ = 32.36in. Upper envelop sßitenAxageRkAmuxkat; dUcenH eyIgRtUveFVIkarEksMrYlmuxkat;¦ eb = amin − kb = 5.63 + 9.05 = 14.68in. et = amax − kt = 32.36 − 11.05 = 21.31in. dUcenH eRbIcMNakp©itrbs; tendon kat;bnßy ec = 16.27in.(413mm ) ee = 10in.(254mm ) (a) muxkat;kNþalElVg Pi ⎛ ec ct ⎞ M D ft =− ⎜1 − 2 ⎟ − t Ac ⎝ r ⎠ S 636,174 ⎛ 16.27 × 24.73 ⎞ 3,581,952 =− ⎜1 − ⎟− 560 ⎝ 223.91 ⎠ 5,070 = 905.4 − 706.5 = 198.9 psi (T ) < 212 psi O.K. P ⎛ ec ⎞ M f b = − i ⎜1 + c 2b ⎟ + D Ac ⎝ r ⎠ Sb 636,174 ⎛ 16.27 × 20.27 ⎞ 3,581,952 =− ⎜1 + ⎟+ 560 ⎝ 223.91 ⎠ 6,186 = −2,809.3 + 579.0 = −2,230.3 psi(C )(15.4 MPa ) ≅ f ci = −2,200 psi O.K. (b) muxkat;Rtg;TMr Pi ⎛ ee ct ⎞ 636,174 ⎛ 10 × 24.73 ⎞ ft =− ⎜1 − 2 ⎟ = − ⎜1 − ⎟ Ac ⎝ r ⎠ 560 ⎝ 223.91 ⎠ = +118.7 psi (T ) < 212 psi O.K. P ⎛ ec ⎞ 636,174 ⎛ 10 × 20.27 ⎞ f b = − i ⎜1 + e 2b ⎟ = − ⎜1 + ⎟ Ac ⎝ r ⎠ 560 ⎝ 223.91 ⎠ = −2,164.4 psi (C ) < 2,200 psi O.K. lkçN³muxkat;smas Flexural Design of Prestressed Concrete Elements 160
  • 72. NPIC Ec (topping ) 57,000 3,000 = = 0.77 Ec (precast ) 57,000 5,000 TTwgsøabRbsiT§PaB = 7 ft = 84in.(213cm) TTwgsøabRbsiT§PaBEksMrYl = 0.77 × 84 = 65in.(165cm) c'b = (5.75 × 65)(47.875) + (560 × 20.27 ) = 31.32in. (5.75 × 65) + 560 65(5.75)3 I 'c = 125,390 + 560(31.32 − 20.27 )2 + + 65 × 5.75(16.56 )2 12 = 297,044in.4 r 2 = 318.12in.2 S cb = 9,490in.3 cak;Rsab; = 45 − 31.32 = 13.68in. ct = 21,714in.3 enARtg;srésxagelIrbs;muxkat;cak;Rsab; 297,044 Sc = t 13.68 c t kMralxagelI = 13.68 + 5.75 = 19.43in. = 15,288in.3 enARtg;srésxagelIrbs;kMral 297,044 S cs = t 19.43 = 19,251in.3 enARtg;srésxageRkamrbs;kMral 297,044 Sbcs = (19.43 − 4) kugRtaMgeRkayeBldMeLIgkMralcak;Rsab;kMras; 1.75in. EdlmanTMgn; WSD CakMral ¬CMhan 11a¦ muneBlcak; diaphragm nigkMral (a) muxkat;kNþalElVg Pe ⎛ ec ct ⎞ M D + M SD ft =− ⎜1 − 2 ⎟ − Ac ⎝ r ⎠ St M D + M SD = 3,581,952 + 940,032 = 4,521,984in. − lb 508,904 ⎛ 16.27 × 24.73 ⎞ 4,521,984 ft = ⎜1 − ⎟− 560 ⎝ 223.9 ⎠ 5,070 = +724.3 − 891.9 = −167.6 psi (C ) minEmnrgkarTaj O.K. Pe⎛ ec cb ⎞ M D + M SD fb = − ⎜1 + 2 ⎟ + Ac⎝ r ⎠ Sb 508,940 ⎛ 16.27 × 20.27 ⎞ 4,521,984 =− ⎜1 + ⎟+ 560 ⎝ 223.9 ⎠ 6,186 = −2,247.4 + 731.0 = −1,516.4 psi (C ) < 2,000 psi O.K. (b) muxkat;elITMr karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 161
  • 73. T.Chhay 508,904 ⎛ 10 × 24.73 ⎞ ft = ⎜1 − ⎟ = 94.9 psi(T ) < f t = 424 psi O.K. 560 ⎝ 223.9 ⎠ 508,940 ⎛ 10 × 20.27 ⎞ fb = − ⎜1 + ⎟ = −1,731.6 psi(11.9MPa )(C ) 560 ⎝ 223.9 ⎠ < −2,000 psi O.K. kugRtaMgeRkayeBlcak;ebtugkMralxagelIPøam² ¬ebtugkMralminTan;kkrwg¦ muxkat;kNþalElVg Pe ⎛ ec ct ⎞ M D + M SD ft =− ⎜1 − 2 ⎟ − Ac ⎝ r ⎠ St M D + M DS = 4,521,984 + 2,649,600 = 7,171,584in. − lb 508,940 ⎛ 16.27 × 24.73 ⎞ 7,171,584 ft =− ⎜1 − ⎟− 560 ⎝ 223.9 ⎠ 5,070 = 724.3 − 1,414.5 = −690.2 psi (C ) < −2,000 psi O.K. P ⎛ e c ⎞ M + M SD f b = − e ⎜1 + c 2b ⎟ + D Ac ⎝ r ⎠ Sb 508,940 ⎛ 16.27 × 20.27 ⎞ 7,171,584 =− ⎜1 + ⎟+ 560 ⎝ 223.9 ⎠ 6,186 = −2,247.4 + 1,159.3 = −1,088.1 psi (C ) < −2,000 psi (7.5MPa < 13.8MPa ) O.K. kugRtaMgenAeBlrgbnÞúkkargar ¬CMhan 11¦ edaybEnßmT§iBlrbs; M SD2 EdlbNþalBIkMral Gt;manTl; (a) muxkat;kNþalElVg Pe ⎛ ec ct ⎞ M D + M SD M CSD + M L ft =− ⎜1 − 2 ⎟ − − Ac ⎝ r ⎠ St Sct M D + M SD = 7,171,584in. − lb M CSD + M L = 0 + 9,300,000 = 9,300,000in. − lb BIdMNak;kalelIkmun f t = −690.2 psi(C )/ fb = −1088.1 psi(C ) . kugRtaMgenAsrésxagelIrbs;muxkat;cak;Rsab;énmuxkat;smasKW = −690.2 − 428.3 = −1,118.5 psi(C ) < −2,000 psi O.K. 9,300,000 f ct = −690.2 − 21,714 = −1088.1 + 979.8 = −108.3 psi(C ) 9,300,000 f bc = −1088.1 + 9,492 pleFobm:UDulEdl)anBImunmkKW n = 0.77 kugRtaMgenAsrésxagelIrbs;kMraleRkayeBlebtugkkrwgKW Flexural Design of Prestressed Concrete Elements 162
  • 74. NPIC × 0.77 = −468 psi(C ) 9,300,000 f cs = − t 15,288 kugRtaMgenAsrésxageRkamrbs;kMralKW × 0.77 = −372 psi(C ) 9,300,000 f bcs = − 19,251 (b) muxkat;elITMr karKNnadUcKñanwgkarKNnaenACMhanxagelI. lT§plKW f t = +94.9 psi (T ) nig f b = −1,731.6 psi(C ) . Tendon Envelope (a) muxkat;kNþalElVg amax = 32.36in. amin = 5.63in. (b) muxkat;Rtg;mYyPaKbYn M D = 2,686,464 M amin = D = 4.22in. Pi M T = 12,355,248 M amax = T = 24.28in. Pe emIlrUbTI 4>36 sMrab; tendon envelope nigrUbTI 4>37 sMrab;karEbgEckkugRtaMg. rUbTI 4>38 bgðajBIkugRtaMgtMbn; anchorage nig net moment tamkMBs;rbs;muxkat;Fñwm. KNna End-Block Anchorage (a) edaHRsaytam linear elastic method karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 163
  • 75. T.Chhay Pi = 636,174lb ee = 10in. f t = +119 psi (T ) f b = −2,164 psi(C ) Flexural Design of Prestressed Concrete Elements 164
  • 76. NPIC (i) Bursting Crack Reinforcement h = 45in. eRbI x = h / 3 = 15in. M max 2.15 × 106 Tb = = = 71,670lb(316kN ) h−x 45 − 15 T As = b = 71,670 f s 20,000 ( ) = 3.58in.2 23.1cm 2 (ii) Spalling Crack Reinforcement M 0.04 × 106 Tsp = max = = 1,330lb h−x 45 − 15 Tsp 1,330 As = = = 0.07in.2 20,000 20,000 Edksrub = 3.58 + 0.07 = 3.65in.2 (23.5cm2 ) sakl,gEdkbBaÄr #4 Ggát;p©it (12.7mm) cMnYnRtUvkar = 0.3.65 2 = 9.13 20 × kartMerob strand eRbIRkLa (grid) 2in. × 2in.(25mm × 25mm) sMrab;tMerobkarBRgay strand. cMNakp©itKW³ karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 165
  • 77. T.Chhay ec = 16.27in.(41.3cm) ee = 10.0in.(25.4cm) kartMerob strands edIm,ITTYlnUvcMNakp©it tendon EdlRtUvkarRtUv)anbgðaj enAkñúgrUbTI 4>39 (a). m:Um:g;tMbn; anchorage enAelIbøg;epSg²EdlmanKMlat 5in. tamkMBs;rbs;mux kat;RtUv)anbgðajenAkñúgtarag 4>7. Flexural Design of Prestressed Concrete Elements 166
  • 78. NPIC (b) edaHRsaytam Plastic Strut-and-tie Method (1) KNnacMgayTIRbCMuTMgn;rbs; tendon 4(2 ) + 6(4) + 4(6) + 3(5.17 ) + 2(20.27 ) + 3(27.77 ) = = 10.27in. 22 (2) KNna bearing capacity rbs;ebtugenARtg;bøg; anchorage device kMlaMgtamCYredk³ Pu1 / Pu3 = 4(0.153)(270,000) = 165,240lb Pu 2 = 6(0.153)(270,000) = 247,860lb / Pu 4 Pu 6 = 8(0.153)(270,000) = 123,930lb Pu 5 = 2(0.153)(270,000) = 82,620lb kMlaMgsgát; ultimate srub = 165,250(2) + 247,860 + 123,930(2) + 82,620 = 908,820lb RkLaépÞrbs; rigid bearing plates EdlRT anchorage chucks Ab = 16 × 12 + 8 × 16 = 320in.2 908,820 fb = = 2,840 psi 320 BIsmIkar 4.16 a nig b, bearing pressure GnuBaØatGtibrmaenAelIebtugKW f b ≤ 0.7φf 'ci A / Ag f b ≤ 2.25φf 'ci snμt;faersIusþg;edImrbs;ebtugenAeBlrgkugRtaMgKW f 'ci = 0.75 f 'c = 0.75 × 5,000 = 3,750 psi RkLaépÞcMp©it A rbs;ebtugEdlman bearing plate = 18 × 16 + 10 × 18 = 468in.2 Bearing stress GnuBaØat f b = 0.7 × 0.90 × 3,750 468 320 = 2,860 psi > 2,840 psi O.K. Bearing stress BIsmIkar 4.14b Gt;lub. (3) KUrKMrU strut-and tie nigeRCIserIs anchorage reinforcement eRCIserIscMgay a dUcenAkñúgrUbTI 4>39(b) cenøaHkMlaMgBIr Pu5 − Pu6 = 7.5in. dUcenH kMBs; a / 2 BIxagmux anchorage = 725 = 3.75in. yk 4in. . sg; strut-and-tie edaysnμt;vamanTMrg;dUckñúgrUbTI 4>39(b). Tension tie mankMlaMgcenøaH 33,495lb nig 102,024lb . edayeRCIserIs tie Edlman karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 167
  • 79. T.Chhay kMlaMgFMCageKKW 102,024lb nigedayeRbIEdkkg confining reinforcement GkSr U biTCit #3 enAkñúgtMbn; anchorage³ ersIusþg;TajkñúgmYykg = φf y Av = 0.90 × 60,000 × 2(0.11) = 11,880lb cMnYnEdkkgEdlRtUvkar = 102,024 11,880 = 8.59 eRbIEdkkgGkSr U biTCit #3 cMnYn 9 kg. edaysakl,gEdkkgGkSr U biTCit #4 enAkñúgtMbn;sgát;Ek,rbøg; anchorage device kñúgkrNIenH eKk¾Gacsnμt;eRbIkMlaMgRbhak;RbEhknwg 102,024lb . ersIusþg;Tajrbs; confining tie #4 mYy = 0.9 × 60,000 × 20(0.20) = 21,600lb Flexural Design of Prestressed Concrete Elements 168
  • 80. NPIC cMnYn tie = 102,,600 = 4.7 eRbIEdkkgGkSr U biTCit #4 cMnYn 5 21 024 edayeRbobeFobdMeNaHRsay (a) nig (b) eyIgTTYlyk confining reinforcement enAkñúgtMbn; anchorage elIRbEvg h = 45in. BIcugFñwmdUcxageRkam³ eRbIEdkkg #4 cMnYn 5 kgedaycab;epþImenARtg; 1.5in. BIbøg; anchorage device nigmanKMlatBIKña 1.5in. KitBIG½kSeTAG½kS. bnÞab;mkbnþCamYynwgEdkkg cMnYn 9 kg edaymanKMlatBIKña 5in. elIcMgay 40in. . cMNaMfa RbsinebIeKsnμt;cMnYn ExSKnøg compression strut kan;Ettic kMlaMg tenion tie nwgeFVIeGay confining reinforcement kan;EteRcIn. 9> KNnamuxkat;rgkarBt;eRkamlkçxNÐ Ultimate Strength Ultimate Strength Flexural Design k> m:Um:g;eRbH Cracking-Load Moment dUcEdl)anerobrab;enAkñúgCMBUk 1 PaBxusKñad¾sMxan;mYyrvagebtugeRbkugRtaMg nigebtugGarem: KW karcakq¶ayCabnþbnÞab;rbs; compressive C-line enAkñúgFñwmebtugeRbkugRtaMgBI tensile cgs line enA eBlEdlbnÞúkkan;EtekIneLIg. eKGacniyaymüa:geTotfa édXñas;rbs; couple xagkñúgekIneLIgeTA tamkarekIneLIgrbs;bnÞúkedaymineFVIkarpøas;bþÚrkugRtaMg f pe enAkñúgEdkeRbkugRtaMg. m:Um:g;Bt;enAEt bnþekIneLIgenAeBlEdlbnÞúkefr nigbnÞúkGefrEdldak;bEnßmBIelImanGMeBI ehUtdl;kugRtaMgsgát;rbs; ebtugenAsrésxageRkamRtg;nIv:UEdkrbs;FñwmTMrsamBaØesμInwgsUnü. eKehAdMNak;kalénkugRtaMgenHfa sßanPaBbnßykugRtaMgsgát; (limit of decompression). bnÞúkbEnßmxageRkA b¤bnÞúkelIseFVIeGayekIt mansñameRbHenAépÞxageRkam EdlkugRtaMgTajekItBIm:Um:g;eRbH (cracring moment) EdlbNþalmkBI bnÞúkeRbHdMbUg (first cracking load) eTAdl;m:UDuldac; (modulus of rupture) f r . enAdMNak;kalenH EdkekInkugRtaMgPøam² ehIykugRtaMgTajRtUv)anepÞrBIebtugeTAEdk. rUbTI 4>40 bgðajBIbnÞúk nigkugRtaMgEdkenAdMNak;kalepSg²énkardak;bnÞúk. vaminRtwmEt bgðajBIExSekagbnÞúk-kMhUcRTg;RTay EdlrYmmankarpøas;bþÚrCMraly:agrh½senAeBlrg first cracking load b:ueNÑaHeT vaEfmTaMgbgðajBIkarpøas;TItaMgrbs;ExSekageRkayBIrg decompression enAkñúg bonded prestressed beam. bnÞab;BIcMnucpøas;TIenH eKminGacKitfaFñwmeFVIkarCalkçN³eGLasÞiceT karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 169
  • 81. T.Chhay ehIykarrt;eLIgelIrbs; compressive C-line sßitenAnwg b¤efrEdleFIVeGaymuxkat;cab;epþImeFVIkarCa muxkat;ebtugGarem:EdlmanédXñas;m:Um:g;efr. eKcaM)ac;RtUvKNna first cracking load edaysarmankarkat;bnßyPaBrwgRkajrbs;muxkat; (section stiffness) dUcenHehIyeKRtUvBicarNaBIkarekIneLIgrbs;PaBdab. elIsBIenH eKRtUvRtYt BinitüBITMhMTTwgsñameRbHedIm,IkarBarERcHsIuEdk. kugRtaMgsrésrbs;ebtugenARtg;épÞrgkarTajKW Pe ⎛ ecb ⎞ M cr fb = − ⎜1 + 2 ⎟ + = fr (4.24) Ac ⎝ r ⎠ Sb Edl m:UDuldac; f r = 7.5 f 'c nigm:Um:g;eRbH M cr Cam:Um:g;EdlbNþalBIbnÞúkTaMgGs; (M D + M SD + M L ) . BIsmIkar 4.24 ⎛ r2 ⎞ M cr = f r Sb + Pe ⎜ e + ⎟ (4.25) ⎜ cb ⎟ ⎝ ⎠ cMNaMfatY r 2 / cb CatMél upper kern rbs; kt dUcenH Per 2 / cb Cam:Um:g;eGLasÞicEdlRtUvkaredIm,I dMeLIg C-line BInIv:UEdkeRbkugRtaMgeTAcMnuc upper kern edaymankugRtaMgTajesμIsUnüenAsrésxag Flexural Design of Prestressed Concrete Elements 170
  • 82. NPIC eRkam. dUcenHtY f r Sb Cam:Um:g;bEnßmEdlRtUvkaredIm,IeFVIeGaymansñameRbHdMbUgenAsrésrgkarTaj xageRkAedaysarGMeBIénbnÞúkelIs (overload) dUcCaenAsrésxageRkaménmuxkat;kNþalElVgrbs; FñwmTMrsamBaØ. x> eRbkugRtaMgedayEpñk Partial Prestressing Bakü partial prestressing CaBaküEdleKminTan;)anÉkPaBKñaenAeLIyeT edaysarEteKmin cg;ehAFñwmrgeRbkugRtaMgedayEpñkeT EdleKGaceFVIeGayeKRcLM. Partial prestressing BiBN’naBIFñwm eRbkugRtaMgEdleKeRbIEdkminEmneRbkugRtaMgbEnßmedIm,IRKb;RKgkarrIkral nigTMhMTTwgrbs;sñameRbH nigkarrYmcMENkrbs;vaeTAkñúg ultimate flexural moment strength. GtßRbeyaCn_d¾sMxan;BIrrbs; partial prestessting KWkareRbIR)as;sMPar³rYmpSMTaMgGs;y:agmanRbsiT§PaB nigkarRKb;RKgkMeNag elIslb;EdlbNþalBI long-term creep rbs;ebtugeRkamkugRtaMgsgát;. eKEtgEtKNnaFwñmebtug Garem:Ca underreinforced edIm,IFana ductile failure edayeGayEdkeFVIkardl; yield mun. eKGac KNnaFñwmeRbkugRtaMgCa uderreinforcement edayeRbIPaKryEdkFmμtaticEdlnaMeGayEdkrgkar Tajdac;enAeBl)ak; b¤k¾KNnaCa overreinforced edayeRbIPaKryEdkFmμtaeRcInEdleFVIeGayebtug EbkenARtg;srésrgkarsgát;xagelIEdlvamanlkçN³ ductile ticCag. RbePTénkar)ak;müa:geTotekItmanenARtg;nIv:U first cracking load Edl M cr Rbhak;RbEhl nwg nominal moment strength M n rbs;muxkat;. RbePTénkar)ak;enHGacekItmanenAkñúgGgát;Edl rgeRbkugRtaMg nigBRgwgedayEdkCamYynwgbrimaNEdktic b¤ekItmanenAkñúgGgát;EdlrgeRbkugRtaMgcM p©itCamYynwgbrimaNEdktic b¤enAkñúgGgát;muxkat;Rbehag. CaTUeTA eKENnaMeGayKNnabrimaNm:Um:g;eRbH M cr edIm,IkMNt;ersIusþg; nigEdnkMNt;rbs; overload Edlmuxkat;KNnaman. K> karkMNt;m:Um:g;eRbH Cracking Moment Evaluation ]TahrN_ 4>8³ KNnam:Um:g;eRbH M enAkñúgFñwmmuxkat;GkSr Ién]TahrN_ 4>2 ehIykMNt;TMhMrbs; cr overload moment EdlFñwmGacTb;Tl;)anenAeBlm:UDuldac;rbs;ebtug. ehIyKNnaemKuNsuvtßiPaB EdlFñwmGacRbqaMgnwgsñameRbHEdlbNþalBI overload. eKeGay f r = 7.5 f 'c = 7.5 5,000 = 530 psi (3.7 MPa ) . karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 171
  • 83. T.Chhay dMeNaHRsay³ BI]TahrN_ 4>2 Pe = 308,225lb(1,371kN ) r 2 / cb = 187.5 / 18.84 = 9.95in.(25.3cm) ( Sb = 3,750in.3 61,451cm3 ) ec = 14in.(35.6cm) M D + M SD = 2,490,638in. − lb(281kN .m ) M L = 7,605,000in. − lb(859kN .m ) BIsmIkar 4.25 ⎛ r2 ⎞ M cr = f r Sb + Pe ⎜ e + ⎟ = 530 × 3,750 + 308,255(14 + 9.95) ⎜ cb ⎟ ⎝ ⎠ = 9,370,207in. − lb M T = M D + M SD + M L = 2,490,638 + 7,605,000 = 10,095,638in. − lb(1,141kN .m ) overload moment = M T − M cr = 10,095,638 − 9,370,207 = 725,431in. − lb(83kN .m ) edaysarEt M T > M cr FñwmmansñameRbHedaysarkugRtaMgTajenAeBlrg service load dUckarsnμt;kñúgkarKNnaenAkñúg]TahrN_ 4>2. emKuNsuvtßiPaBRbqaMgnwgsñameRbHKW M cr 9,370,207 = = 0.93 M T 10,095,638 RbsinebIm:Um:g;edaysarbnÞúkeFVIkar M T tUcCag M cr emKuNsuvtßiPaBRbqaMgnwgsñameRbHnwgFM Cag 1 dUcenAkñúg nonpartially prestressed member. cMNaMfa enAeBleKeRbI nonprestressed member edIm,IbegáIt partially prestressed section m:Um:g;emKuNsrub φM n ≥ 1.2 M cr dUckarTamTareday ACI Code. 10> emKuNbnÞúk nigemKuNersIusþg; Load and Strength Factors k> PaBTukcitþ nigsuvtßiPaBrbs;eRKogbgÁúM<ebtug Reliability and structural safety of Concrete Components viFIEdl)anbegáIteLIgcMnYnbIenAkñúgTsvtSr_fμI²manT§iBly:agxøaMgeTAelIdMeNIrkarKNnana Flexural Design of Prestressed Concrete Elements 172
  • 84. NPIC eBlbc©úb,nñ k¾dUcCanaeBlGnaKtKW³ karekIneLIgy:ageRcInénkarKNnaGgát;ebtugtamkarBiesaF nig karviPaK/ viFIRbU)ab‘ÍlIetedIm,IBiBN’naBIkareFVIkarrbs;Ggát;ebtug/ nigkareRbIRbB½n§kMuBüÚT½rsMrab;viPaKBI suvtßiPaB nigPaBTukcitþénRbB½n§. rhUtdl;eBlfμI²enH emKuNsuvtßiPaBPaKeRcInkñúgkarKNnaKWQrelI karBiesaF. edaysarbTBiesaFn_bEnßmrYmnwgcMeNHdwgBIkar)ak;k¾dUcCakarsuaMCamYynwglkçN³rbs;eb tug emKuNsuvtßiPaBRtUv)anEksMrYl EdlPaKeRcInRtUv)ankat;bnßy. enAkñúgqñaM 1956/ Baker )anesñInUvviFId¾gayRsYlmYyénkarkMNt;emKuNsuvtßiPaB dUcbgðaj enAkñúgtarag 4>8 edayQrelIkarvaytMéltamRbU)ab‘ÍlIet. viFIenHKitfaTMgn;RbePTepSgKñaKYreRbIem KuNsuvtßiPaBxusKñaEdlmanT§iBldl;karKNna. T§iPBl)ak;lMeGog (weight failure effect) Wt sMrab;ktþaepSg²én workmanship, lkçxNÐénkardak;bnÞúk/ lT§plénkar)ak;/ niglT§PaBTb;Tl; RtUv)aneerobcMdak;enAkñúg tarag. emKuNsuvtßiPaBRbqaMgnwgkar)ak;KW ΣWt S .F . = 1.0 + (4.26) 10 EdltMéllMeGogsrubGtibrmarbs;)a:ra:Em:RtTaMgGs;EdlmanT§iBleTAelIkareFVIkaresμInwg 10.0 . niyaymüa:geTot emKuNsuvtßiPaBsMrab;karbnSMlkçxNÐEdlGaRkk;bMputEdlmanT§iBlelIkareFVIkar rbs;eRKOgbgÁúM S.F . = 2.0 . karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 173
  • 85. T.Chhay viFImüa:geTotCamYynwgcMnYn)a:ra:Em:RtRbU)ab‘ÍlIetEdledaHRsayCamYynwgbnÞúk nigersIusþg;tUc Cag. CaTUeTA viFIenHmanlkçN³RsedogKñasMrab;eRKOgbgÁúMEdk nigeRKOgbgÁúMebtug. TaMg load-and- resistance-factor-design method (LRFD) nig first-order second-moment method (FOSM) esñI nUvdMeNIrkarEdlTukcitþCaTUeTAsMrab;vaytMéllkçN³vinicä½yénkarKNnabnÞúkemKuNEdlQrelIRbU)a:- b‘ÍlIet (probability-based factored load design criteria). snμt;fa φi tMNageGayemKuNersIusþg;rbs;Ggát;ebtug ehIy γ i CaemKuNbnÞúksMrab;RbePT epSg²rbs;bnÞúk. RbsinebI Rn CaersIusþg;Fmμta (nominal resistance) rbs;Ggát;ebtug nig Wi Ca T§iBlbnÞúksMrab;RbePTepSg²énbnÞúkdak;BIelI φi Rn ≥ γ iWi (4.27) Edl i CabnÞúk dUcCabnÞúkefr/ bnÞúkGefr/ bnÞúkxül; b¤T§iBlGaRs½ynwgeBl>>> rUbTI 4>41(a) nig (b) bgðajBIkarEbgEck frequency dac;edayELkBIKñaénbnÞúkCak;Esþg W nigersIusþg; R EdlmantMélmFüm R nig W . rUbTI 4>41(c) bgðajBIkarRtYtelIKñaénkarEbgEck TaMgBIr ehIykat;KñaRtg;cMnuc C . eRKOgbgÁúMnwgmansuvtßiPaB nigGacTukcitþ)anenAeBlEdlbnÞúk W sßitenAxageqVgcMnuc C elI ExSekag W ehIyersIusþg; R sßitenAxagsþaMcMnuc C elIExSekag R . mü:agvijeTot eRKOgbgÁúMnwg)ak; RbsinebIbnÞúk b¤ersIusþg;sßitenAkñμúgépÞqUtenAkñúgrUbTI 4>41(c). RbsinebI β CasnÞsSn_suvtßiPaB (safety index) enaH R +W β= (4.28) σ R + σW 2 2 Edl σ R nig σW Ca standard deviation énersIusþg; nigbnÞúk erogKña. düaRkamrbs; safety index β sMrab;RbB½n§eRKOgbgÁúMEdlsnμt;RbqaMgnwgRbU)ab‘ÍlIeténkar)ak; rbs;RbB½n§RtUv)anbgðajenAkñúgrUbTI 4>42. eKGacsegáteXIjfaRbU)ab‘ÍlIetenHfycuHenAeBlEdl PaBxusKñaén R nig W ekIneLIg b¤PaBERbRbYlénersIusþg;Edlvas;eday standard deviation σ R nig σ W RtUv)ankat;bnßy EdleFIVeGayépÞqUtenAxageRkamcMnuc C enAkñúgrUbTI 4>41(c) rYmtUc. karbegáInPaBxusKñaén R − W b¤karkat;bnßy σ R b¤ σW RtUv)aneFVIeLIgedayBicarNaelI lkçN³esdækic©. eKGacTTYl)annUvlkçxNÐsuvtßiPaBedaykareRCIserIstMél safety index β tamry³ kareRCIserIs Rn nigWi d¾smRsbedayeRbIemKuNersIusþg; φi nig emKuNbnÞúk γ i smrmü enAkñúgsmIkar Flexural Design of Prestressed Concrete Elements 174
  • 86. NPIC 4.27. eKesñInUvtMél safety index β enAcenøaH 1.75 eTA 3.2 sMrab;eRKOgbgÁúMebtug EdltMéltUcbMput sMrab;karBarbnSMbnÞúkEdlmanxül; nigrBa¢ÜydI. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 175
  • 87. T.Chhay RbsinebI U i CabnÞúkxageRkAemKuN enaH Σγ iW = U i sMrab;karbnSMbnÞúkepSg². kñúgkrNIbnSM bnÞúkEdlminmanvtþmanRBwl nigsMBaFxag tMél U i EdlENnaMenAkñúg ASCE-7 Standard nig IBC 2000 sMrab; U i GtibrmaedIm,IeRbIenAkñúgsmIkar 4.2 Edl φi Rn ≥ γ iWi ≥ U i max KW U = φi Rn = max[1.2 D + 1.6 L ] (4.29) 11> emKuNbnÞúkACI nigkMritsuvtßiPaB ACI Load Factors and Safety Margins k> eKalkarN_TUeTA General Principles emKuNbnÞúk γ nigemKuNkat;bnßyersIusþg; φ eGaynUvemKuNsuvtßiPaBTaMGs;EdlQrelI RbePTbnÞúukEdl γ 1D + γ 2 L 1 S .F . = × (4.30) D+L φ Edl φ CaemKuNkat;bnßyersIusþg; ehIy γ 1 nig γ 2 CaemKuNbnÞúksMrab;bnÞúkefr D nigbnÞúkGefr L erogKña. CaTUeTA eKeRbIemKuNEtmYysMrab;bnÞúkefr nigemKuNmYyepSgeTotsMrab;bnÞúkGefr. emKuN Flexural Design of Prestressed Concrete Elements 176
  • 88. NPIC kat;bnßyersIusþg;ERbRbYleTAtamRbePTkugRtaMg. eKRtUvbEnßmemKuNeTAelIbnÞúkeFVIkarEdl)an)a:n; sμanedayKuNnwgemKuNbnÞúkdUcCa 1.2 sMrab;bnÞúkefr nig 1.6 sMrab;bnÞúkGefr EdlbnSMbnÞúkeKal bnÞúkTMnajCaplbUkrvagbnÞúkefr nigbnÞúkGefr. edaysarbnÞúkGefrRtUv)an)a:n;sμanedayeRbITMgn;én bnÞúkminzitezr dUcCamnusS nigeRKOgsgðarwm dUcenHeKBi)aknwg)a:n;sμantMélrbs;va)ansuRkitdUcbnÞúk efrNas;. edaysarmUlehtuenHehIyeTIbemKuNrbs;bnÞúkGefrFMCagemKuNbnÞúkefr. x> smIkaremKuNbnÞúk ACI ACI Load Factors Equations ACI 318 Building Code sMrab;eRKOgbgÁúMebtugCa international code. dUcenHvaGnueLameTA tam International Building Code, IBC 2000, IBC 2003 ehIyvaRsbeTAtam ASCE-7 Standard on Minimum Design Loads for Buildings and Other Structures. sþg;darTaMgBIrmantMélRbU)a:b‘Í- lIetdUcKñasMrab;emKuNersIusþg;suvtßiPaBEdlrMBwgTuk φi Rn Edl φ CaemKuNkat;bnßyersIusþg;Edl GaRs½ynwgRbePTrbs;kugRtaMgEdlBicarNaenAkñúgkarKNna EdlrYmmandUcCa karBt;begáag karkat; b¤karsgát;.l. dUcenH bnÞúkKNna b¤bnÞúkemKuN (design load or factored load) ACI fμI U RtUvmantMél y:agehacNas;esμInwgtMélEdlTTYl)anBIsmIkar 4.31(a) dl; (g). eRKOgbgÁúMTaMgGs;kMrnwgrgEtGMeBI bnÞúkefr nigGefrNas; dUcenHeKRtUveFVIkarGegátBIGMeBIrbs;T§iBlbnÞúkmYy b¤eRcIneFIVkarkñúgeBlEt mYy. smIkarxageRkambgðajBIbnSMbnÞúksMrab;sßanPaBEdlmanKitbBa©Úlxül; rBa¢ÜydI b¤sMBaFxag EdlbNþalBIdIbMeBj b¤Twk³ U = 1 .4 ( D + F ) (4.31a) U = 1.2(D + F + T ) + 1.6(L + H ) + 0.5(Lr or S or R ) (4.31b) U = 1.2 D + 1.6(Lr or S or R ) + (1.0 L or 0.8W ) (4.31c) U = 1.2 D + 1.6W + 0.5 L + 1.0(Lr or S or R ) (4.31d) U = 1 .2 D + 1 .0 E + 1 .0 L + 0 .2 S (4.31e) U = 0.9 D + 1.6W + 1.6 H (4.31f) U = 0 .9 D + 1 .0 E + 1 .6 H (4.31g) Edl D= bnÞúkefr/ E = bnÞúkrBa¢ÜydI/ F = bnÞúksMBaFxagrbs;Twk H = bnÞúkEdl)anBITMgn; nigsMBaFxagrbs;dI nigTwkenAkñúgdI L = bnÞúkGefr/ Lr = bnÞúkdMbUl/ R = bnÞúkTwkePøóg/ S = bnÞúkRBwl/ W = bnÞúkxül; karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 177
  • 89. T.Chhay krNIelIkElg (a) eKGnuBaØateGaykat;bnßyemKuNbnÞúkrbs; L enAkñúgsmIkar 4.31(c) eTAdl; 4.31(e) mkRtwm 0.5 elIkElgyandæan/ kEnøgsMrab;saFarN³ nigTIkEnøgEdlmanbnÞúkGefr FMCag 100lb / ft 2 (4.79kN / m2 ) (b) Code GnuBaØateGayeRbI 1.3W CMnYseGay 1.6W enAkñúgsmIkar 4.31(d) nig 4.31(f) sMrab;bnÞúkxül;Edlmin)ankat;bnßyedayemKuNTisedA (directionaly factor). (c) eKKYreRbI 1.4 E CMnYseGay 1.0 E enAkñúgsmIkar 4.31(e) nig4.31(g) enAeBlEdlbnÞúk TMnaj E RtUv)ankMNt;edayQrelI service-level seismic forces. (d) eKeGayemKuNrbs; H esμInwgsUnüenAkñúgsmIkar 4.31(f) nig 4.31(g) RbsinebIGMeBIEdl bNþalBI H RtUv)anRbqaMgedayGMeBIEdlbNþalBI W b¤ E . eKminRtUvKitbBa©ÚlsMBaF xagrbs;dIEdlpþl;ersIusþg;RbqaMgnwgGMeBIrbs;bnÞúkepSgeToteTAkñúg H eT EteKRtUvKit bBa©ÚlvaeTAkñúgersIusþg;KNna (design resistance). eKRtUveFVIbnSMbnÞúkeRcInRbePTedIm,IkMNt;lkçxNÐKNnaEdleRKaHfñak; CaBiessenAeBlEdl ersIusþg;GaRs½ynwgT§iBlbnÞúkeRcInCagmYy dUcCaersIusþg;sMrab;bnSMkarBt; nigbnÞúktamG½kS b¤ersIusþg; kat;enAkñúgGgát;EdlmanrgbnÞúktamG½kS. !> karkat;bnßybnÞúkGefr Reduction in Live Loads sMrab;épÞFM eKmanehtuplnwgsnμt;faGaMgtg;sIueténbnÞúkGefrTaMgGs;minGacRKbdNþb;elIépÞ eBj)aneT. dUcenH Ggát;EdlmanépÞrgT§iBl (influence area) cab;BI 400 ft 2 (37.2m2 ) eLIgeTAeK GaceRbIsmIkarxageRkamedIm,IKNnabnÞúkGefrkat;bnßy³ ⎛ 15 ⎞ L = Lo ⎜ 0.25 + ⎟ (xñat US) AI KitCa ft 2 (4.32a) ⎜ A ⎟ ⎝ I ⎠ ⎛ 4.57 ⎞ L = Lo ⎜ 0.25 + ⎜ ⎟ AI ⎟ ( xñat SI) AI KitCa m2 (4.32b) ⎝ ⎠ Edl bnÞúkGefrKNnakat;bnßy L= Lo = bnÞúkGefrKNnaEdlminkat;bnßy AI = Influence area . sMrab;eRKOgbgÁúMepSgeRkABI cantilevered construction AI CaépÞrg sMBaFsMrab;ssr. AI CaépÞrgsMBaFsMrab;Fñwm b¤RkLaépÞEdlesμInwgkMralxNÐBIrTis. Flexural Design of Prestressed Concrete Elements 178
  • 90. NPIC bnÞúkKNnakat;bnßyminGactUcCag 50% énbnÞúkGefr Lo sMrab;eRKOgbgÁúMEdlmanmYyCan; nigminRtUvtUcCag 40% énbnÞúkGefr Lo sMrab;eRKOgbgÁúMEdlmaneRcInCagmYyCan;. eKminRtUveFVIkar kat;bnßybnÞúkGefreT sMrab;bnÞúkGefrEdlmanTMhMtUcCagb¤esμInwg 100lb / ft 2 (4.79kN / m 2 ). sMrab; yandæanEdlmanrfynþBIreKGackat;bnßyy:ageRcInRtwm 20% . K> ersIusþg;KNna nigersIusþg;Fmμta³ emKuNkat;bnßyersIusþg; φ Design Strength vs Nominal Strength: Strength-Reduction Factor φ ersIusþg;rbs;Ggát;eRKOgbgÁúMEdlKNnaedayeRbIkarbnSMbnÞúkxagelICa nominal strength. Ca]TahrN_ sMrab;FñwmersIusþg;m:Um:g;énmuxkat;EdlKNnaedayeRbIsmIkarlMnwg niglkçN³rbs;ebtug nigEdkRtUv)aneKehAfa nominal strength moment M n edIm,IKitBIPaBminsuRkitkñúgkarsagsg; dUcCa TMhM b¤TItaMgrbs;Edk b¤bMErbMrYlrbs;lkçN³. ersIusþg;kat;bnßyrbs;Ggát;RtUv)ankMNt;Ca design strength rbs;Ggát;. sMrab;Fñwm ersIusþg;m:Um:g;KNna (design moment strength) φM n KYrmantMélesμI EtCakar RbesIrKYrmantMélFMCagm:Um:g;emKuNxageRkA (external factore moment) M u bnþicbnþÜc sMrab;lkç- xNÐEdlGaRkk;énbnÞúkemKuN U . emKuN φ ERbRbYleTAtamRbePTepSg²énkareFVIkar (behavior) rbs;Ggát;eRKOgbgÁúM nigERbRbYleTAtamRbePTrbs;eRKOgbgÁúM. ]TahrN_ sMrab;FñwmrgkarBt; φ = 0.9 . sMrab;ssrEdlmanEdkkgFmμtargbnÞúksgát;CaeKal emKuN φ = 0.65 . mUlehtuEdlersIu- sþg;kat;bnßyersIusþg;rbs;ssrmantMéltUcCagedaysarssrCaeRKOgbgÁúMsMxan;EdlRTeRKOgbgÁúMsrub karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 179
  • 91. T.Chhay nigedIm,IRbqaMgnwgkarrlMCabnþbnÞab; nigkar)ak;edayRsYyEdlminmanRbkasGasnñCamun. mü:ageTot FñwmRtUv)anKNnaedayeGayrgnUvPaBdabFMmunnwg)ak;. dUcenH lT§PaBrbs;FñwmsMrab;karRbkasGasnñ muneBl)ak;GnuBaØateGayeRbIemKuNkat;bnßyersIusþg;FMCag. tarag 4>9 segçbemKuNersIusþg; φ sMrab;Ggát;eRKOgbgÁúMepSg²Edlpþl;eGayeday ACI Code. emKuNkat;bnßyersIusþg;rbs; AASHTO AASTHO Strength-Reduction Factors karBt;begáag³ φ = 1.0 sMrab;Ggát;ebtugeRbkugRtaMgcak;Rsab;EdlplitenAeragcRk φ = 0.95 sMrab;Ggát;ebtugeRbkugRtaMgTajCaeRkayEdlcak;enAkardæan karkat; nigkarrmYl³ φ = 0.90 sMrab;Ggát;eRbkugRtaMg emIlemKuNepSgeTotrbs; LRFD nig Standard AASTHO enAkñúgCMBUkTI 12. 12> Limit State in Flexure at Ultimate Load in Bonded Members: Decompression to Ultimate Load k> esckþIepþIm Introduction FñwmebtugeRbkugRtaMgcab;epþImeFVIkardUcFñwmebtugGarem:enAeBlEdltMélrbs;m:Um:g;Bt;enAqøayBI m:Um:g;eRbH M cr nigm:Um:g;bnÞúkeFIVkarsrub M T . sñameRbHcab;epþImekItmanenAeBlkugRtaMgTajenAkñúg ebtugenARtg;srésxageRkAbMputrbs;muxkat;eRKaHfñak;FMCagkugRtaMgGtibrma f r ≅ 7.5 f 'c . munnwg eTAdl;kugRtaMgenH overload begáItkugRtaMgsgát;enAkñúgebtugenARtg;nIv:Urbs;EdkeRbkugRtaMgedIm,Ikat; bnßyCabnþbnÞab;rhUtdl;vaesμInwgsUnü EdleKeGayeQμaHfa decompression load. kMritkugRtaMgenA kñúg tendon RtUv)aneKeGayeQμaHfa decompression stress ¬emIlrUbTI 4>3 nig 4>43¦. GñkGegátxøHkMNt; decompression load CabnÞúkEdlsñameRbHTImYyelceLIgenARtg;srés xageRkAénmuxkat;eRKaHfñak; dUcCasrésxageRkamrbs;muxkat;kNþalElVgénFñwmTMrsamBaØ. PaB xusKñatictYcekItmanenAkñúgkarviPaKedayeRbIkarkMNt;niymn½yrbs; decompression load enH. edIm,IeFVItamsßanPaBénkardak;bnÞúkCaCMhan² snμt;faeRbkugRtaMgRbsiT§PaB f pe eRkamGMeBI bnÞúkeFVIkarEdlbNþalBIbnÞúkTaMgGs;begáIt)anCabMErbMrYlrageFob (strain) ε1 dUcxageRkam³ f pe ε1 = ε pe = (4.33a) E ps Flexural Design of Prestressed Concrete Elements 180
  • 92. NPIC eRkamGMeBI decompression, enAeBlEdlkugRtaMgsgát;enACMuvijebtugenARtg;nIv:UEdkeRbkugRtaMgman tMélesμIsUnüedaysarkugRtaMgTajEdlbNþalBI overload enaH decompression strain ε decomp = ε 2 pþl;lT§pldUcxageRkam³ Pe ⎛ e2 ⎞ ε 2 = ε decomp = ⎜1 + ⎟ (4.33b) Ac Ec ⎜ r2 ⎟ ⎝ ⎠ rUbTI 4>43 nigrUbTI 4>44 bgðajBIkarEbgEckkugRtaMgenAeBlrg nigeRkayeBlrg decompression EdlFñwmeRbkugRtaMgcab;epþImeFVIkarRsedognwgFñwmebtugGarem:. edaysarbnÞúkxiteTACitsßanPaBcugeRkay bMErbMrYlrageFobbEnßm ε 3 enAkñúgEdkbnþBIkarEbg EckkugRtaMgragRtIekaNdUcbgðajenAkñúgrUbTI 4>44(b) Edl compressive strain Gtibrma enARtg; srésrgkarsgát;xageRkAKW ε c = 0.003mm / mm . enAkñúgkrNIEbbenH karekIneLIgénbMErbMrYlrag eFobrbs;Edkedaysar overload EdlbEnßmBIelI decompression load KW ⎛d −c⎞ ε3 = εc ⎜ ⎟ (4.33c) ⎝ c ⎠ Edl c CakMBs;G½kSNWt. dUcenH bMErbMrYlrageFobsrubenAkñúgEdkeRbkugRtaMgenARtg;dMNak;kalenH køayCa ε s = ε1 + ε 2 + ε 3 (4.33d) eKGacTTYl)ankugRtaMg f ps enARtg; nominal strength y:aggayRsYlBIdüaRkamkugRtaMg-bMErbMrYl rageFobrbs;EdkEdlpþl;eGayedayplitkr. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 181
  • 93. T.Chhay Flexural Design of Prestressed Concrete Elements 182
  • 94. NPIC x> bøúkctuekaNsmmUl nigersIusþg;m:Um:g;Fmμta The Equivalent Retangular Block and Nominal Moment Strength vamansar³sMxan;Nas;EdleKGackMNt;ersIusþg;EdlbMrugTukenAkñúgFñwmeRbkugRtaMgmunnwg)ak; dUcEdl)anerobrab;enAkñúgCMBUkTI 1. dUcenH eKKYrbBa©ÚlersIusþg;m:Um:g;énmuxkat;eRbkugRtaMgeTAkñúgkar KNnasrubsMrab;karRtYtBinitüGMeBIrbs;bnÞúkeFVIkar Edlmanerobrab;lMGitenAkñúgEpñkTI1 dl;TI7 én CMBUkenH. karsnμt;xageRkamRtUv)aneFVIeLIgedIm,IkMNt;BIlkçN³eFIVkarrbs;muxkat;eRkamGMeBI ultimate load. !> snμt;fakarEbgEckbMErbMrYlrageFobmanlkçN³bnÞat; b¤smamaRt (linear). karsnμt;enHKW QrelI Bernoulli’s hypothesis Edlfamuxkat;erobesμIenAEterobesμImuneBlrgkarBt; ehIyEkgeTAnwgG½kSNWteRkayeBlrgkarBt;. @> bMErbMrYlrageFobrbs;Edk nigebtugEdlenACMuvijEdkmuneBlebtugeRbH b¤ Edk yield dUcKña eRkayeBlebtugeRbH b¤Edk yield. #> ebtugexSaykñúgkarTaj. vaeRbHenAdMNak;kaldMbUgénkardak;bnÞúkKWenARbEhlCa 10% én ersIusþg;rgkarsgát;kMNt;rbs;va. dUcenH eKminKitmuxkat;ebtugenAkñúgtMbn;TajsMrab;kar viPaK nigkarKNnamuxkat;rgkarBt; ehIyeKsnμt;EdkTajTTYlykkMlaMgTajTaMgGs;. edIm,IbMeBjlkçxNÐlMnwgénkMlaMgedk kMlaMgsgát; C enAkñúgebtugRtUvesμInwgkMlaMgTaj T enA kñúgEdk. Edl C =T (4.34) nimitþsBaØaenAkñúgrUbTI 4>44 RtUv)ankMNt;dUcxageRkam³ b = TTwgrbs;FñwmenARCugsgát; d = kMBs;rbs;FñwmEdlvas;BIsréssgát;xageRkAbMputeTAkan;TIRbCMuTMgn;rbs;RkLaépÞEdk h = kMBs;srubrbs;Fñwm K> viFIbMErbMrYlrageFobkMNt;sMrab;karviPaK nigkarKNnamuxkat; Strain Limits Method for Analysis and Design !> eKalkarN_TUeTA General Principles eBlxøHeKeGayeQμaHviFIenHfa unified method. muxkat;ebtugGaceTAdl;ersIusþg;Bt;Fmμta (nominal flexural strength) enAeBlEdlbMErbMrYlrageFobedaykarsgát;suT§ (net compressive karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 183
  • 95. T.Chhay strain) enAsréssgát;xageRkAbMput xiteTArkEdnkMNt;rbs; ACI Code KW 0.003mm / mm . enA eBlEdlbMErbMrYlrageFobedaykarTajsuT§ (net tensile strain) ε t mantMélFMRKb;RKan;EdltMélenaH FMCag b¤esμInwg 0.005mm / mm enaHkareFVIkarrbs;vamanlkçN³sVit (ductile). eKkMNt;lkçN³rbs; muxkat;FñwmCamuxkat;rgkarTaj (tension-controlled) CamYynwgkarRbkasGasnñBIkar)ak;edaybgðaj nUvsñameRbHeRcIn nigPaBdabFM. RbsinebI net tensile strain enAsrésrgkarTajxageRkAbMput ε t mantMéltUcdUcenAkñúgGgát; rgkarsgát; EdltUcCag b¤esμInwg compression-controlled strain limit enaHeKrMBwgfavanwg)ak;eday lkçN³RsYy (brittle) edaymankarRbkasGasnñtictYcbMputmunnwg)ak;. CaTUeTA Ggát;rgkarBt;Ca Ggát; tention-controlled ÉGgát;rgkarsgát;CaGgát; compression-controlled. b:uEnþ sMrab;muxkat;xøH dUcCamuxkat;EdlrgbnÞúktamG½kStUcEtrgm:Um:g;Bt;FM net tensile strain ε t enAelIsrésTajxageRkA nwgmantMélcenøaHsßanPaBkMNt;bMErbMrYlrageFobBIrKW compression-controlled strain limit ε t = f y / Es = 60,000 / 29 ⋅ 106 = 0.002 nig tension-controlled strain limit ε t = 0.005 . rUbTI 4>45 bgðajBItMbn;TaMgbIenH k¾dUcCabMErbMrYlénemKuNkat;bnßyersIusþg;EdlGacGnuvtþ)anenAkñúgcMNat;fñak; srubénkareFVIkarrbs;Ggát;. Flexural Design of Prestressed Concrete Elements 184
  • 96. NPIC sMrab; tension-controlled state, bMErbMrYlrageFobkMNt; (strain limit) ε t = 0.005 RtUvnwg pleFobEdk (reinforcement ratio) ρ / ρb = 0.63 Edl ρb CapleFobEdklMnwg (balanced rein- forcement ratio) sMrab;bMErbMrYlrageFoblMnwg (balanced strain) ε t = 0.005 . Net tensile strain ε t = 0.005 sMrab; tensioned-controlled state CatMéleTalEdlGacGnuvtþeTAkñúgRKb;RbePTEdk edayminKitfavaCaEdkFmμta b¤EdkeRbkugRtaMgeT. pleFobEdkx<s;EdlbegáIt net tensile strain tUc Cag 0.005 eFVIeGaytMélemKuN φ tUcCag 0.9 EdleFVIeGaymuxkat;minsUvmanlkçN³esdækic©. dUc enH eKKYrEtEfmEdkrgkarsgát; (compression reinforcement) RbsinebIcaM)ac; b¤dMeLIgkMBs;muxkat; edIm,IeFVIeGay strain enARtg; extreme tension reinforcement ε t ≥ 0.005 . bMErbMrYlrbs; φ CaGnuKmn_eTAnwgbMErbMrYlrageFob eKGackMNt;tMél φ sMrab; strain cenøaH ε t = 0.002 nig ε t = 0.005 tam linearly interpola- tion dUcbgðajenAkñúgsmIkarxageRkam³ sMrab;muxkat;EdleRbIEdkkgFmμta (tied sections) 0.65 ≤ [φ = 0.48 + 83ε t ] ≤ 0.90 (4.35a) sMrab;muxkat;EdleRbIEdkkgv½NÐ (spirally-reinforced sections) 0.70 ≤ [φ = 0.57 + 67ε t ] ≤ 0.90 (4.35b) bMErbMrYlrbs; φ CaGnuKmn_eTAnwgpleFobkMBs;G½kSNWtelIkMBs;RbsiT§PaB c / dt eKGacsMEdgsmIkar 4.35(a) nig 4.35(b) edayeRbIpleFobkMBs;G½kSNWt c elIkMBs;RbsiT§- PaB dt énRsTab;EdkEdlenACitépÞTajénmuxkat;CageKdUcxageRkam sMrab;muxkat;EdleRbIEdkkgFmμta (tied sections) ⎡ 0.25 ⎤ 0.65 ≤ ⎢φ = 0.23 + ⎥ ≤ 0.90 (4.36a) ⎣ c / dt ⎦ sMrab;muxkat;EdleRbIEdkkgv½NÐ (spirally-reinforced sections) ⎡ 0.20 ⎤ 0.70 ≤ ⎢φ = 0.37 + ⎥ ≤ 0.90 (4.36b) ⎣ c / dt ⎦ sMrab; balanced strain EdlEdkenARtg;xagTaj yield ehIykñúgeBlCamYyKñaebtugEbkenA Rtg;épÞsgát; ( f s = f y ) eKGackMNg;pleFobkMBs;G½kSNWt sMrab; ε t = 0.002 dUcxageRkam (xñat US) c 87,000 = (4.37) d t 87,000 + f y karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 185
  • 97. T.Chhay c = 600 d t 600 + f y ( xñat SI) Casegçb enAeBlEdl net tensile strain enAkñúg extreme tension reinforcement FMRKb;RKan; ¬FMCag b¤esμI 0.005 ¦ muxkat;RtUv)ankMNt;Ca tension-controlled EdlmankarRbkasGasnñénkar)ak; CamYynwgPaBdabFM. X> karEbgEckm:Um:g;GviC¢maneLIgvijenAkñúgFñwmCab; Negative Moment Distribution in Continuous Beams CodeGnuBaØatkat;bnßym:Um:g;eGLasÞicGviC¢man (negative elastic moment) enARtg;TMrrbs; FñwmCab;edayGRtatUcCag [1000ε t ] PaKry edaymantMélGtibrmaRtwm 20% . mUlehtuKWedaysar fa ductile members/ tMbn;snøak;)aøsÞic (plastic hinge region) ekItmanenARtg;cMnucm:Um:g;Gtibrma nigeFVIeGaymankarpøas;bþÚrdüaRkamm:Um:g;eGLasÞic. kñúgkrNICaeRcIn lT§plKWkarkat;bnßym:Um:g;GviC¢- man ehIyekIneLIgm:Um:g;viC¢manedaytMélRtUvKña. tamkarGnuBaØatrbs; code eKGaceFVIkarEbgEckm:U m:g;GviC¢maneLIgvij)anEt kñúgkrNI ε t ≥ 0.0075 Rtg;muxkat;Edlm:Um:g;RtUv)ankat;bnßy. eKminGac GnuvtþkarEbgEckm:Um:g;eLIgvijkñúgkrNIRbB½n§kMralEdlkMNt;smamaRteday direct design method (DDM) )aneT. rUbTI 4>46 bgðajBIkarEbgEckm:Um:g;eLIgvijGnuBaØatsMrab;ersIusþg;Gbrma. eKGacRbdUcbMEr bMrYlrageFobGb,brma 0.0075 enARtg;épÞrgkarTajeTAnwgkrNIEdlpleFobEdksMrab;bnSMEdkeRb Flexural Design of Prestressed Concrete Elements 186
  • 98. NPIC kugRtaMg nigEdkFmμtamansnÞsSn_Edk (reinforcement index) ω minFMCag 0.24β1 dUcEdnx<s;bMput sMrab;karKNnaedaylkçN³sVit (ductile design). eKRbdUcbMErbMrYlrageFobGtibrma 0.005 sMrab; tension-controlled state eTAnwg reinforcement index ω p = 0.32 β1 b¤ ωT = 0.36 β1 dUckarBN’na enAkñúg Code commentary. ACI 318 Code ykemKuNkat;bnßyersIusþg;Gtibrma φ = 0.90 sMrab; tension-controlled bending edIm,IeRbIenAkñúgkarKNnaersIusþg;rbs;Ggát;rgkarBt;. vaRtUvnwgpleFobkMBs;G½kSNWt c / d t = 0.375 sMrab;bMErbMrYlrageFob ε t = 0.005 eKENnaMpleFob c / d t tUcCagenH. sMrab;karEbg Eckm:Um:g;eLIgvijenAkñúgFñwmCab; eKENnaMeGayykpleFobkMBs;G½kSNWttUc. sMrab; net tensil strain EdlmantMél ε t = 0.0075 eKeGaykarEbgEckm:Um:g;eLIgvij 7.5% nigsMrab; ε t = 0.020 eKyk 20% sMrab;karEbgEckm:Um:g;eLIgvij dUcbgðajenAkñúgrUbTI 4>46. Tensile strain enARtg;Edk rgkarTajxageRkAbMputmantMél ⎛ dt ⎞ ε t = 0.003⎜ − 1⎟ (4.38a) ⎝ c ⎠ Ca]TahrN_ RbsinebI dt = 20in. ehIykMBs;G½kSNWt c = 5.1in. ⎛ dt ⎞ ⎛ 20 ⎞ ε t = 0.003⎜ − 1⎟ = 0.003⎜ − 1⎟ = 0.0088in. / in. > 0.0075in. / in. ⎝ c ⎠ ⎝ 5.1 ⎠ eKGnuvtþtMélGb,brmasMrab;karEbgEckm:Um:g;eLIgvijedaylkçN³ inelastic. enAkñúgkrNIenH karEbgEckm:Um:g;eLIgvijGnuBaØatGtibrma = 1000ε t = 8.8% tMélm:Um:g;GviC¢man = (100 − 8.8) = 91.2% karEbgEckm:Um:g;Gtibrma 20% mantMélRbhak;RbEhl = 0.24β1 dUcenAkñúg code BIxagedIm. eKGacbgðajEdnkMNt;enHedayTMnak;TMng reinforcement index sMrab; bonded prestressed concrete members dUcxageRkam³ ωp + d (ω − ω ') ≤ 0.24β1 (4.38b) dp eTaHbICa code GnuBaØatkarEbgEckm:Um:g;eLIgvijGnuBaØatGtibrma 20% b¤ [1000ε t ]% k¾ eday EteKKYrkMNt;PaKryEbgEcgm:Um:g;eLIgvijenHRtwmRbEhl 10% − 15% CakarRbesIr. Cakar segçb ACI 318-02 code ENnaMfa karEbgEckm:Um:g;eLIgvijenARtg;TMrrbs;FñwmCab;minRtUvFMCag [1000ε t ] PaKry CamYynwgtMélGtibrma 20% dUcbgðajenAkñúgrUbTI 4>46 ehIyeKRtUvdMeLIgm:Um:g; viC¢manenAkNþalElVg. b:uEnþ eKKYreFVIkarEbgEckm:Um:g;eLIgvijtamlkçN³ inelastic EtenAeBlEdl karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 187
  • 99. T.Chhay ε t ≥ 0.0075 Rtg;muxkat;Edlm:Um:g;RtUv)ankat;bnßy. karEktMrUvelIElVgmYyk¾GacGnuvtþeTAelIElVg déTepSgeTotTaMgGs;EdlrgkarBt; kMlaMgkat; nigmuxkat;EdlEdkRtUv)anbBaÄb;. eKKYrcMNaMfa brimaNsrubrbs;EdkeRbkugRtaMg nigEdkFmμtaRtUvmantMélRKb;RKan;edIm,IRT bnÞúkemKuNEdly:agehacesμInwg 1.2 dgénm:Um:g;EdleFVIeGayeRbH (cracking load) EdlKNnaeday QrelIm:UDuldac; f r . eKmineRbIkarpþl;eGayrbs; ACI 318 enHsMrab; (a) kMralxNÐBIrTis unboded post-tensioned slab nig (b) Ggát;rgkarBt;EdlmanersIusþg;kMlaMgkat; nigersIusþg;m:Um:g;Bt;y:agticesμI nwgm:Um:g;eRbHdMbUg (first cracking moment) M cr . g> ersIusþg;m:Um:g;Fmμtarbs;muxkat;ctuekaN Nominal Moment Strength of Rectangular Sections karEbgEckkugRtaMgsgát;Cak;EsþgenAkñúgmuxkat;enAeBl)ak;manTMrg;Ca)a:ra:bUlekIn dUcbgðaj enAkñúgrUbTI 4>44(c). eKRtUvkarcMNayeBleRcInedIm,IKNnamaDrbs;bøúkkugRtaMgsgát; RbsinebIvaman rag)a:ra:bUl. eKGaceRbIbøúkkugRtaMgctuekaNsmmUleday Whitney y:aggayRsYledayminman)at; bg;PaBsuRkitkñúgkarKNnakMlaMgsgát; k¾dUcersIusþg;m:Um:g;Bt;rbs;muxkat;. bøúkkugRtaMgsmmUlman kMBs; a nigersIusþg;sgát;mFüm 0.85 f 'c . dUcEdleXIjenAkñúgrUbTI 4>44(d) tMélrbs; a = β1c RtUv)ankMNt;edayeRbIemKuN β1 EdleFVIeGayRkLaépÞrbs;bøúkctuekaNsmmUlmantMélRbhak; RbEhlnwgbøúkkugRtaMg)a:ra:bUl Edlpþl;nUvkMlaMgsgát; C mantMélRsedogKñakñúgkrNITaMgBIr. tMél 0.85 f 'c sMrab;kugRtaMgmFüménbøúkkugRtaMgsmmUlQrelIlT§plBiesaFn_ebtugenAGa- yuy:agtic 28 éf¶. QrelIkarBiesaF bMErbMrYlrageFobGnuBaØatGtibrma 0.003 EdlTTYlykeday ACI CatMélkMNt;suvtßiPaB. eTaHbICamankaresñIeLIgnUvTMrg;bøúkkugRtaMgdUcCaragctuBñayk¾eday k¾eK TTYlykbøúkragctukaNsmmUlCaragbøúksþg;darkñúgkarviPaK nigKNnamuxkat;rbs;ebtugGarem:. Edk RtUv)ansnμt;faeFVIkarCalkçN³ elastoplastic. edayeRbInUvkarsnμt;BIxagedImTaMgGs; düaRkamEbgEckkugRtaMgRtUv)anbgðajenAkñúgrUbTI 4>44 (c) GacRtUv)anKUreLIgvijdUcbgðajenAkñúgrUbTI 4>44(d). eKGacsresrkMlaMgsgát; C = 0.85 f 'c ab EdlmaDénbøúksgát;enARtg; b¤enAEk,r ultimate enAeBlEdlEdkTaj yield (ε s > ε y ) . eKGacsresr kMlaMgTaj T = Aps f ps . dUcenH eyIgGacsresrsmIkarlMnwg 4.34 EdleGay C = T eLIgvijdUc xageRkam Aps f ps = 0.85 f 'c ab (4.39) Flexural Design of Prestressed Concrete Elements 188
  • 100. NPIC Aps f ps a = β1c = (4.40) 0.85 f 'c b eKTTYl)anersIusþg;m:Um:g;Fmμta (nominal moment strength) edayKuN C b¤ T CamYynwgéd Xñas;m:Um:g; (d p − a / 2) eyIgTTYl)an ⎛ a⎞ M n = A ps f ps ⎜ d p − ⎟ (4.41a) ⎝ 2⎠ Edl d p CacMgayBIsréssgát;eTATIRbCMuTMgn;rbs;EdkeRbkugRtaMg. PaKryEdk ρ p = Aps / bd p eGaysMrab;EtersIusþg;FmμtaénEdkeRbkugRtaMgb:ueNÑaH ⎛ f ps ⎞ M n = ρ p f ps bd 2 ⎜1 − 0.59 ρ p p⎜ ⎟ (4.41b) ⎝ f 'c ⎟ ⎠ RbsinebI ω p Ca reinforcement index = ρ p ( f ps / f 'c ) enaHsmIkar 4.41b køayCa ( M n = ρ p f ps bd p 1 − 0.59ω p 2 ) (4.41c) eKk¾RtUvGnuvtþdUcKñasMrab;karcUlrYmrbs;EdkTajFmμta dUcenHkMBs; a énbøúksgát;KW Aps f ps + As f y a= (4.42a) 0.85 f 'c b RbsinebI c = a / β1 enaHbMErbMrYlrageFobenARtg;nIv:UEdkFmμta ¬rUbTI 4>44¦ KW ⎛d −c⎞ ε3 = εc ⎜ ⎟ (4.42b) ⎝ c ⎠ sMrab;muxkat;ctuekaNEdlmanEdkTajFmμtaEtminKitbBa©ÚlEdksgát; smIkar 4.41b nwgkøayCa ⎛ f ps ⎞ ⎛ f ⎞ M n = ρ p f ps bd 2 ⎜1 − 0.59 ρ p p⎜ ⎟ + ρf y bd 2 ⎜1 − 0.59 y ⎟ (4.43a) ⎝ f 'c ⎟ ⎠ ⎜ ⎝ f 'c ⎟ ⎠ b¤eKGacsresrmü:ageTot ⎧ ⎪ ⎛ d ⎞⎫ ⎪ ⎧ ⎪ ⎛ dp ⎫ ⎞⎪ M n = Aps f ps ⎨1 − 0.59⎜ ω p + ω ⎟⎬ + As f y ⎨1 − 0.59⎜ ω p + ω ⎟⎬ ⎜ d ⎟⎪ (4.43b) ⎪ ⎜ d p ⎟⎪ ⎪ ⎩ ⎝ ⎠⎭ ⎩ ⎝ ⎠⎭ Edl ω = ρ ( f y / f 'c ) enaH ⎛ a⎞ ⎛ a⎞ M n = Aps f ps ⎜ d p − ⎟ + As f y ⎜ d − ⎟ (4.43c) ⎝ 2⎠ ⎝ 2⎠ eKGacKitbBa©ÚlkarcUlrYmrbs;Edksgát;RbsinebIva yield enaH Aps f ps + As f y − A's f y a= (4.44) 0.85 f 'c b Edl b CaTTwgmuxkat;énépÞsgát;rbs;Fñwm. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 189
  • 101. T.Chhay Kitm:Um:g;Rtg;TIRbCMuTMgn;rbs;bøúksgát;enAkñúgrUbTI 4>47 enaHersIusþg;m:Um:g;FmμtaenAkñúgsmIkar 4.43b nwgkøayCa ⎛ a⎞ ⎛ a⎞ ⎛a ⎞ M n = Aps f ps ⎜ d p − ⎟ + As f y ⎜ d − ⎟ + A's f y ⎜ − d ' ⎟ (4.45) ⎝ 2⎠ ⎝ 2⎠ ⎝2 ⎠ !> ersIusþg;m:Um:g;Fmμtarbs;muxkat;Edlmansøab Nominal Moment Strength of Flanged Sections enAeBlEdlkMras;søabsgát; h f tUcCagkMBs;G½kSNWt c nigkMBs;bøúkctuekaNsmmUl a enH eKGacKitmuxkat;enHCamuxkat;mansøabdUcenAkñúgrUbTI 4>48. Flexural Design of Prestressed Concrete Elements 190
  • 102. NPIC BIrUbeyIg)an T p + Ts = T pw + T pf (4.46) Edl Tp = kMlaMgeRbkugRtaMgsgát; = Aps f ps Ts = Ultimate force enAkñúgEdkFmμta = As f y T pw = EpñkénkMlaMgsrubenAkñúgEdkrgkarTajEdlRtUvkarsMrab;RTnug = Apw f ps Apw = RkLaépÞEdksrubEdlRtUvnwgkMlaMg T pw T pf = EpñkénkMlaMgsrubenAkñúgEdkTajEdlRtUvkarsMrab;søab C f = 0.85 f 'c (b − bw )h f C w = 0.85 f 'c bw a edayCMnYsvaeTAkñúgsmIkar 4.46 eyIgTTYl)an T pw = Aps f ps + As f y − 0.85 f 'c (b − bw )h f (4.47) eFVIplbUkkMlaMgTaMgGs;EdlmanenAkñúgrUbTI 4>48 (c) nig 4>48 (d) eyIg)an T pw + T pf = C w + C f A pw f ps enaH a= 0.85 f 'c bw (4.48a) Aps f ps + As f y − 0.85 f 'c (b − bw )h f b¤ a= 0.85 f 'c bw (4.48b) eKGacsresrsmIkar 4.45 sMrab;FñwmEdlmanEdksgát;eLIgvijedIm,ITTYl)anersIusþg;m:Um:g;FmμtasMrab; muxkat;mansøabEdlG½kSNWtsßitenAxageRkAsøab a > h f dUcxageRkam edayKitm:Um:g;eFobTIRbCMuTMgn; rbs;EdkeRbkugRtaMg³ ⎛ hf ⎞ ⎛ a⎞ ( ) M n = Apw f ps ⎜ d p − ⎟ + As f y d − d p + 0.85 f 'c (b − bw )h f ⎜ d p − ⎜ 2 ⎟ ⎟ (4.49a) ⎝ 2⎠ ⎝ ⎠ m:Um:g;KNna (design moment) enAkñúgkrNIenHKW M u = φM n (4.49b) Edl φ = 0.90 sMrab;karBt; edIm,IkMNt;fa G½kSNWtsßitenAxageRkAsøab EdlTamTarkarKNnamuxkat;mansøab eKRtUv kMNt;kMlaMgsgát;srub Cn nigkMlaMgTajsrub Tn ehIyeRbobeFobtMélrbs;va. RbsinebI Tp + Tn enAkñúgrUbTI 4>48 FMCag C f enaHG½kSNWtsßitenAxageRkAsøab ehIymuxkat;RtUv)anKitCamuxkat;man søab. RbsinebImindUecñaHeT vaRtUv)anKitCamuxkat;ctuekaNEdlmanTTwg b Casøabrgkarsgát;. viFImü:ageTotkñúgkarkMNt;famuxkat;GacKItCamuxkat;mansøabedayKNNatMélkMBs;bøúkctu- ekaNsmmUl a BIsmIkar 4.48b bnÞab;mkeyIgGackMNt;kMBs;G½kSNWt c = a / β1 . karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 191
  • 103. T.Chhay @> karkMNt;kugRtaMg)ak;Fmμtarbs;EdkeRbkugRtaMg f ps Determination of Prestressing Steel Nominal Failure Stress f ps eKGackMNt;kugRtaMg f ps rbs;EdkeRbkugRtaMgenAeBl)ak;edayeRbIbMErbMrYlrageFobRtUvKña (strain compatibility) tamry³dMNak;kalénkardak;bnÞúkepSg²rhUtdl;sßanPaBkMNt;enAeBl)ak;. eKRtUvkardMeNIrEbbenHRbsinebI Pe f pe = < 0.50 f pu (4.50a) Aps ACI 318 building code GnuBaØateGayeRbIkarKNnaEdlmantMélRbhak;RbEhl RbsinebI Pe f pe = ≥ 0.50 f pu (4.50b) Aps CamYynwgsmIkardac;edayELksMrab; bonded nig unbonded members. Bonded Tendons smIkarEdl)anBIkarBiesaFsMRab; bonded member KW ⎛ γp ⎡ f pu ⎤⎞ f ps = f pu ⎜1 − ⎜ ⎢ρ p + d (ω − ω ')⎥ ⎟ (4.51) ⎝ β1 ⎢ ⎣ f 'c d p ⎥⎟ ⎦⎠ Edl reinforcement index sMrab;EdkFmμtargkarsgát;KW ω ' = ρ ' ( f y / f 'c ) . RbsinebIeKKitEdksgát; enAeBlKNna f ps edaysmIkar 4.51 tY [ρ p ( f pu / f 'c ) + (d / d p )(ω − ω ')] minRtUvtUcCag 0.17 ehIy d ' minRtUvFMCag 0.15d p . ehIy γ p = 0.55 sMrab; f py / f pu EdlmintUcCag 0.80 sMrab; high-strength prestressing bar γ p = 0.40 sMrab; f py / f pu EdlmintUcCag 0.85 sMrab; stress-relieved strands γ p = 0.28 sMrab; f py / f pu EdlmintUcCag 0.90 sMrab; low-relaxation strands Bonded Tendons sMrab;pleFobElVgelIkMBs;tUcCag b¤esμI 35 f ps = f pe + 10,000 + f 'c 100 ρ p ¬xñat US¦ (4.52a) f ps = f pe + 70 + f 'c 100 ρ p ¬xñat SI¦ Edl f psminRtUvFMCag f py b¤ ( f pe + 60,000)psi b¤ ( f pe + 420)MPa . sMrab;pleFobElVgelIkMBs;FMCag 35 f ps = f pe + 10,000 + f 'c 300 ρ ¬xñat US¦ (4.52b) p Flexural Design of Prestressed Concrete Elements 192
  • 104. NPIC f ps = f pe + 70 + f 'c 300 ρ p ¬xñat SI¦ Edl minRtUvFMCag f py b¤ ( f pe + 30,000)psi b¤ ( f pe + 210)MPa . rUbTI 4>49 bgðaj seating f ps losses sMrab;RbePT unbonded tendons. cMNaMfa smIkar AASTHO sMrab;ersIusþg;KNna ultimate f ps xusBIsmIkar 4.51 nig 4.52. #> tMélkMNt;rbs;snÞsSn_Edk Limiting Values of the Reinforcement Index snÞsSn_Edk ω p CargVas;PaKryEdkenAkñúgmuxkat; EdlGackMNt;dUcxageRkam Aps f ps f ps ωp = = ρp (4.53) bd p f 'c f 'c EdkGb,brma RbsinebIPaKryEdktUcEmnETn muxkat;ebtugnwgexSaykñúgkarTb;Tl;nwgkugRtaMgTajeRkay eBlmansñameRbH ehIymuxkat;nwgeFVIkaresÞIrEtdUcCamuxkat;ebtugsuT§ (plain section). dUcenHeKRtUv BinitüPaKryEdkGb,brma ρ min CamYynwg ω p min enAkñúgkarKNnamuxkat;edIm,IkarBarkar)ak;EbbenH. brimaNsrubrbs;EdkeRbkugRtaMg nigEdkFmμtaEdlTamTareday ACI minRtUvtUcCagGVIEdlTamTar sMrab;begáItm:Um:g; M u = φM n eT dUcenH M u ≥ 1.2 M cr (4.54a) Edl M cr QrelIm:UDuldac; f r = 7.5 f 'c psi(0.623 f 'c MPa). EtelIkElgcMeBaHGgát;rgkarBt; EdlmanersIusþg;Bt; nigersIusþg; kat;y:agticesμInwgBIrdgénbnÞúkemKuNenAkñúgsmIkar 4.31. dUcKñaRk- LaépÞGb,brmarbs; bonded nonprestressing reinfocement enAkñúgFñwmEdleKarBtam ACI Code KW As min = 0.004 A (4.54b) Edl A CaEpñkrbs;muxkat;cenøaHépÞrgkarTajedaykarBt; nigTIRbCMuTMgn; cgc rbs; gross section. EdkenHRtUvEbgEckesμIkñúg precompressed tensile zone EdlsßitenAEk,rsrésrgkarTaj. enAkñúgkMralxNÐ flat plate BIrTis EdlkugRtaMgTajrbs;ebtugeRkamGMeBIbnÞúkeFVIkarFMCag 2 f 'c psi (0.17 f 'c MPa ) RkLaépÞ bonded nonprestressed steel EdlRtUvkarKW Nc As = (4.55a) 0.5 f y Edl f y ≤ 60,000 psi (420MPa ) Nc = kMlaMgTajenAkñúgebtugEdlbNþalBIplbUkbnÞúkefr nigbnÞúkGefr (D + L) karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 193
  • 105. T.Chhay Flexural Design of Prestressed Concrete Elements 194
  • 106. NPIC enAkñúgtMbn;m:Um:g;GviC¢manrbs;kMralxNÐenARtg;TMrssr RkLaépÞGb,brmarbs;EdkFmμtatam TisnImYy²KW As = 0.00075hl (4.55b) Edl h=kMras;kMralxNÐsrub l = RbEvgElVgtamTisRsbnwgEdkEdlkMBugKNna eKEbgEck As enAkñúgTTwgkMralxNÐenAcenøaHExSEdlsßitenAxageRkAépÞrbs;TMrssr 1.5h . eKRtUvdak;Edk (bars or wire) y:agticbYntamTisnImYy² nigmanKMlatminFMCag 12in.(30cm) . EdkGtibrma RbsinebIPaKryEdkFMeBk muxkat;ebtugnwgeFVIkarCalkçN³ everreinforced Edl nonductile failure nwgekIteLIgenAeBlebtugEbkdMbUgenARtg;sréssgát; EtEdkenAxagTajminTan;eFVIkardl; yield. eKEtKNnamuxkat;FñwmebtugGarem:CalkçN³ underreinforced CamYynwgbMErbMrYlrageFob ε t = 0.005 dUcEdl)anerobrab;rYcmkehIy. b:uEnþ sMrab;FñwmeRbkugRtaMg eKminGacGnuvtþEtkñúgkrNI underreinforced )aneT. kMlaMgeRbkug RtaMg Pi nig Pe enAeBlepÞr nigenAeBlrg service load RKb;RKgtMélénRkLaépÞEdkTajEdlcaM)ac; edayrYmbBa©ÚlTaMgEdkFmμtapg. elIsBIenH tMélrbs; yield strength nigtMélrbs; yield strain rbs;EdkeRbkugRtaMgminRtUv)ankMNt;eT. dUcenH eKKNnaFñwmeRbkugRtaMgedIm,IbMeBjRKb;tMrUvkarbnÞúk eFVIkarEdlGaceFVIkarCa underreinforcement b¤ overreinforcement enAsßanPaBkMNt;én ultimate- load design CaBiessRbsinebIvaCa partially prestressed beam. edIm,IFanakareFVIkarCalkçN³sVit (ductility), ACI code kMNt;PaKryrbs;EdkedaymineGay reinforcement index ω p minRtUvFMCag 0.36β1 cMNaMfaeKGacRbdUc 0.32β1 eTAnwg 0.005 . 0.05( f 'c −4,000) β1 = 0.85 − 1,000 ≥ 0.65 ¬xñat US¦ (4.56) 0.05( f 'c −28) β1 = 0.85 − 7 ≥ 0.65 ¬xñat SI¦ dUcerobrab;enAkñúg Code commentary sMrab;muxkat;EdlmanEtEdkeRbkugRtaMg f ps ωp = ρp ≤ 0.32β1 (4.57a) f 'c sMrab;muxkat;EdlmanEdkTaj nigEdksgát;Fmμta karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 195
  • 107. T.Chhay ⎡ ⎤ ⎢ω p + d (ω − ω ')⎥ ≤ 0.36β1 (4.57b) ⎢ ⎣ dp ⎥ ⎦ As f y Edl ω= bdf 'c A's f y nig ω' = bdf 'c sMrab;muxkat;mansøab ⎡ ⎤ ⎢ω pw + d (ωw − ω 'w )⎥ ≤ 0.36β1 (4.57c) ⎢ ⎣ dp ⎥ ⎦ Edl ω pw / ωw nig ω 'w RtUv)anKNnatamrebobdUcKñanwgsmIkar 4.57a, b EteKeRbITTwgRTnug bw enA PaKEbgénsmIkarTaMgenH. cMNaMfa tY ω p / [ω p + (d / d p )(ω − ω ')] nig [ω pw + (d / d p )(ωw − ω 'w )] KWesμInwg 0.85a / d p Edl a CakMBs;rbs;bøúkebtugsgát;ctuekaNsmmUl. (a) enAkñúgmuxkat;ctuekaN nigenAkñúgmuxkat;mansøabEdl a ≤ h f ⎡ ⎤ ⎡A f ⎛ f f ⎞⎤ ⎢ω p + d (ω − ω ')⎥ = ⎢ ps ⋅ ps + d ⎜ As ⋅ y − A's ⋅ y ⎟⎥ ⎜ ⎟ ⎢ ⎣ dp ⎥ ⎢ bd p f 'c d p ⎝ bd f 'c bd f 'c ⎠⎥ ⎦ ⎣ ⎦ A ps f ps + As f y − A's f y 0.85 f 'c ab 0.85a = = = bd p f 'c bd p f 'c dp (b) enAkñúgmuxkat;mansøabEdl a > h f / yk CF CakMlaMgsgát;pÁÜbrbs;ebtugenAkñúgsøab ⎡ d ⎤ ⎡ A f −C ( ) ⎛ (ωw − ω 'w )⎥ = ⎢ ps ps F + d ⎜ As ⋅ y − A's ⋅ y ⎟⎥ f f ⎞⎤ ⎢ω pw + ⎢ ⎣ dp ⎥ ⎢ bw d p f 'c ⎦ ⎣ d p ⎜ bw d f 'c bw d f 'c ⎟⎥ ⎝ ⎠⎦ A ps f ps + As f y − A's f y − C F = (4.57d) bw d p f 'c compression force in web = bwd pf ' c 0.85 f 'c bw a 0.85a = = bw d p f 'c dp smIkar 4.57a, b nig c RtUv)anelIkElg RbsinebIersIusþg;m:Um:g;KNnaminFMCagersIusþg;Edl QrelIEpñksgát;énm:Um:g; couple. mü:ageToteKGacniyayfa luHRtaEteKGnuvtþ strain compatibility analysis eTIbeKkMNt; overreinforced prestressed beam moment strength BIsmIkarEdl)anBIkar BiesaFdUcxageRkam sMrab;muxkat;ctuekaN Flexural Design of Prestressed Concrete Elements 196
  • 108. NPIC M n = 0.25 f 'c bd 2 (4.58a) sMrab;muxkat;mansøab ( M n = 0.25 f 'c bw d 2 + 0.85 f 'c (b − bw )h f d − 0.5h f ) (4.58b) eKGacEkERbsmIkarTaMgenHdUcxageRkam³ (a) sMrab; overreinforced rectangular section M n = f 'c bd 2 (0.68 β1 − 0.08 β12 ) p (4.59a) (b) sMrab; overreinforced flanged section M n = f 'c bw d 2 (0.36 β1 − 0.08 β12 ) + 0.85 f 'c (b − bw )h f (d p − 0.5h f ) (4.59b) p $> sßanPaBkMNt;rgkarBt;eRkamGMeBI ultimate Load enAkñúg Unbonded Tendons Limit State in Flexure at Ultimate Load in Nonbonded Tendons Post-tensioned tendon Edlmin grouted b¤k¾ asphalt coated Ca nonbonded tendons. Ca vi)ak enAeBlbnÞúkEdldak;BIelIbEnßmmanGMeBIeTAelIFñwm PaBrGilekItmanrvag tendon nigebtugEdl B½T§CMuvijva EdlGnuBaØateGayekItmankMhUcRTg;RTayesμItambeNþayRbEvgTaMgmUlrbs; tendon eRb kugRtaMg. edaysarsñameRbHekItmanenARtg;tMbn;Edlmanm:Um:g;FM karekIneLIgénkugRtaMgTajrbs; Edkmin)anRbmUlpþúMenARtg;kEnøgsñameRbHeT EtvaRtUv)anEbgEcgesμItambeNþay tendon EdlrGil edayesrI. CalT§pl karekIneLIg strain nigkugRtaMg enAkñúgkrNI nonbonded tUcCagenAkñúgkrNI bonded tendon dUcbnÞúkbnþekIneLIgrhUtdl; ultimate. dUcenH cMnYnsñameRbHticCag EtmanTMhMFM Cag ekItmanenAkñúg nonbonded prestressing. kugRtaMgcugeRkayenAkñúg prestressed tendons eRkam GMeBI ultimate load KYrEtmantMélFMCageRbkugRtaMgRbsiT§PaB f pe . edIm,IFanafaeRKOgbgÁúMEdlmankareFVIkar (serviceability perfomace) l¥ eKKYreRbIPaKryEdk Fmμtasmrmü. EdkFmμtaRKb;RKgsñameRbHEdlekItBIkarBt; nigRKb;RKgTMhMrbs;va ehIycUlrYmkar ekIneLIgénersIusþg;m:Um:g; M n rbs;muxkat;. vargnUvbMErbMrYlrageFobFMCag yield strain rbs;va eday sarkMhUcRTg;RTayrbs;vaenARtg;tMbn; postelastic RtUvEtRtUvKñanwgkMhUcRTg;RTayenAEk,r EdkeRbkug RtaMg. dUcenH kugRtaMgenAkñúgEdkFmμtaEtgEtFMCag yield strength rbs;vaeRkamGMeBI ultimate load. rUbTI 4>50 bgðajBIRbePTdüaRkamkugRtaMg-bMErbMrYlrageFobsMrab; 270k 7-wire 1 / 2in. prestressing strand cMENkÉrUbTI 4>51 bgðajBIdüaRkaménkugRtaMgrbs;EdkeRbkugRtaMg nigEdkFmμta CamYynwg seating losses. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 197
  • 109. T.Chhay tamkarerobrab;xagelI eyIgGacsnidæan)anfaeKGaceRbIsmIkarsMrab;KNna M n rbs;ersIusþg; FmμtasMrab; bonded beam dUcKñasMrab; nonbonded beam. cMNaMfa eBlxøHenAeBleKENnaMeGay grout the post-tensioned tendon, EtvaminEmnCakargayRsYlkñúgkareFVIy:agdUcenaHeT. ]TahrN_ enAkñúgRbB½n§kMralxNÐBIrTis b¤ shallow-box element EdlkMras;ebtugesþIg. dUcKña eKRtUvBicarNaBI tMélénsMBaFrbs; grouting kñúgkrNIEdlman tendon eRcIn. Flexural Design of Prestressed Concrete Elements 198
  • 110. NPIC 13> Preliminary Ultimate-Load Design RbsinebIkarKNnadMbUgcab;epþImenAeBlrg ultimate load m:Um:g;KNnaEdlRtUvkar M u = φM n y:agehacNas;k¾RtUvesμInwgm:Um:g;emKuN M u . kMBs;sakl,gdMbUgRtUvEtQrelIpleFobElVgelIkMBs; smrmü ehIyTTwgsøabxagelIrbs;muxkat;RtUv)ankMNt;eTAtamRbePTeRKOgbgÁúM dUcCaeKcUlcitþeRCIs erIs double-T-section b¤ hollow-box shaloow section sMrab;kMralGKar b¤cMNtrfynþ ehIyeKcUl citþ I-section sMrab;eRKOgbgÁúMs<an. kMBs;mFümrbs;FñwmeRbkugRtaMgKWRbEhl 75% énkMBs;rbs;FñwmebtugGarem:EdlRTbnÞúkesμIKña. eKalkarN_ENnaMepSgeTotsMrab;kareRCIserIsdMbUgKWeRbIkMBs; 0.6in(15mm) sMrab;ElVg 1 ft (300mm) . enAeBlEdleKeFVIkareRCIserIskMBs;sakl,grYcehIy eKGaccab;epþImkMNt;lkçN³FrNImaRtrbs;mux kat;. snμt;faTIRbCMuTMgn;rbs;EdkeRbkugRtaMgsßitenARbEhl 0.85 / h BIBak;kNþalkMBs;rbs;søab. ehIyédXñas;rbs;m:Um:g; couple KW jd ≅ 0.80h . dUcKña snμt;fa nominal strength rbs;EdkeRbkug RtaMgKW f ps ≅ 0.90 f pu . ehIyRkLaépÞrbs; Aps rbs;EdkeRbkugRtaMgKW Mu /φ Aps = (4.60a) 0.90 f pu (0.80h ) b¤ Aps = Mn 0.72 f pu h (4.60b) RbsinebIkMBs;bøúksgát; a esμInwgkMras;søab h f maDrbs;bøúksgát;énrUbTI 4>44(d) edayeRbItYénRkLa épÞ ba = A'c KW C = 0.85 f 'c A'c Mu T = 0.90 f pu Aps = 0.8h BIsmIkarlMnwgrbs;kMlaMg C = T . dUcenHRkLaépÞrbs;søabsgát;KW Mn Mn A'c = = (4.61) 0.85 f 'c (0.8h ) 0.68 f 'c h enAeBlEdleKeRCIserIsTTwgsøabsMrab;karsakl,gdMbUg ehIyeKsÁal;kMBs;Fñwm eKGaceRCIs erIskMras;RTnugEdlQrelItMrUvkarkMlaMgkat;Edlnwgerobrab;enAkñúgCMBUk 5. bnÞab;mk edaykarsak l,g nigEktMrUv eKGaceFVIkareRCIserIsmuxkat;d¾lsMrab;lkçxNÐtMrUvkarKNna nigviPaKbnþelIkugRtaMg ¥ sMrab;lkçxNÐbnÞúkeFVIkar. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 199
  • 111. T.Chhay 14> Summary Step-By-Step Procedure for Limit-State-At-Failure Design of the Prestressed Members !> eRCIserIs partial prestressing b¤k¾Gt; edaykareRbIPaKryRbsiT§PaBrbs;EdkFmμta. eRCIs erIskMBs;sakl,g h edayQrelIGRtapleFobkMBs;elIElVg 1 / 20 b¤k¾ 75% énkMBs;Edl RtUvkarsMrab;muxkat;ebtugGarem:eRkayeBlKNnaersIusþg;FmμtaEdlRtUvkar M n = M u / φ . @> eRCIserIskMras;søabsakl,gedayeGayRkLaépÞebtugsrubrbs;søab A'c ≅ M n / 0.68 f 'c h EdlQrelIkareRCIserIsTTwgsøabEdltMrUvedaykarerobcM nigKMlatrbs;Fñwm. eRCIserIsRkLa épÞdMbUgrbs;EdkeRbkugRtaMg Aps = M n / 0.72 f pu h . #> eRbItMélsmrmüsMrab;kugRtaMgEdk f ps enAeBl)ak;sMrab;karsakl,gdMbUg. RbsinebI f pe < 0.5 f pu enaHeKRtUvkar strain compatibility analysis. kMNt;faetI tondon Ca bonded b¤ unbonded. eRbItMéleRbkugRtaMgRbsiT§PaB f pe BI service-load analysis RbsinebI eK)aneFVIkarKNnarYcehIy. RbsinebI f pe > 0.5 f pu eRbItMélRbhak;RbEhldUcxageRkam (a) Bonded tendons ⎛ γp ⎧ ⎪ ⎫⎞ (ω − ω ')⎪ ⎟ f pu d f ps = f pu ⎜1 − ⎨ρ p + ⎬⎟ ⎜ β1 ⎪ f 'c d p ⎪⎠ ⎝ ⎩ ⎭ (b) Nonbonded tendons, pleFobElVgelIkMBs; ≤ 35 f ps = f pe + 10,000 + f 'c 100 ρ ¬xñat US¦ p f ps = f pe + 70 + f 'c 100 ρ p ¬xñat SI¦ (c) Nonbonded tendons, pleFobElVgelIkMBs; > 35 f ps = f pe + 10,000 + f 'c 300 ρ ¬xñat US¦ p f ps = f pe + 70 + f 'c 300 ρ p ¬xñat SI¦ cMNaMfa f ps EdleGayeday AASHTO xusBIsmIkarxagelI ehIyRtUv)anENnaMenACMBUkTI 12. Flexural Design of Prestressed Concrete Elements 200
  • 112. NPIC $> kMNt;fa etIeKKYrBicarNamuxkat;CaragctuekaN b¤muxkat;mansøabedaykMNt;TItaMgrbs; G½kSNWt c = a / β1 . RbsinebImuxkat;ctuekaN Aps f ps + As f y − A's f y a= 0.85 f 'c b RbsinebImuxkat;mansøab A pw f ps a= 0.85 f 'c bw Edl Apw f ps = Aps f ps + As f y − 0.85 f 'c (b − bw )h f %> RbsinebI h f FMCag c nig a viPaKGgát;Camuxkat;ctuekaNEdlmanEdkeTal (singly reinforced) b¤EdkDub (double reinforced). ^> kMNt;snÞsSn_Edk (reinforcement index) ω p / ω nig ω ' sMrab;krNI a < h f ¬G½kSNWt sßitenAkñúgsøab dUcenHeRbImuxkat;ctuekaN¦ (a) muxkat;ctuekaNEdlmanEtEdkeRbkugRtaMg f ps Aps f ps ωT = ω p = ρ p = ⋅ f 'c bd p f 'c (b) muxkat;ctuekaNEdlmanEdksgát;bEnßmBIelIEdkTajFmμta ωT = ω p + d (ω − ω ') dp RbsinebIsnÞsSn_srubenAkñúg (a) b¤ (b) tUcCag b¤esμInwg 0.36β1 enaHersIusþg;m:Um:g;KW ⎛ a⎞ ⎛ a⎞ ⎛a ⎞ M n = Aps f ps ⎜ d p − ⎟ + As f y ⎜ d − ⎟ + A's f y ⎜ − d ' ⎟ ⎝ 2⎠ ⎝ 2⎠ ⎝2 ⎠ &> kMNt;snÞsSn_Edk (reinforcement index) ω pw / ωw nig ω 'w sMrab;krNI a > h f ¬G½kS NWtsßitenAeRkAsøab¦ EdlmansnÞsSn_Edksrub ωT = ω pw + d (ωw − ω 'w ) dp eKKNnasnÞsSn_EdkedayQrelITTwgrbs;RTnug bw . karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 201
  • 113. T.Chhay RbsinebI snÞsSn_Edksrub ωT < 0.36β1 enaH ⎛ ⎞ ⎛ a⎞ ( ) M n = Apw f ps ⎜ d p − ⎟ + As f y d − d p + 0.85 f 'c (b − bw )h f ⎜ d p − ⎜ hf ⎟ ⎟ ⎝ 2⎠ ⎝ 2 ⎠ A pw f ps Edl a = 0.85 f ' b c w nig Apw f ps = Aps f ps + As f y − 0.85 f 'c (b − bw )h f RbsinebIsnÞsSn_Edksrub ωT > 0.36β1 / muxkat;Ca overreinforced ehIyersIusþg;Fmμta (nominal strength) KW M n = f 'c bw d 2 (0.36β1 − 0.08β12 ) + 0.85 f 'c (b − bw )h f (d p − 0.5h f ) p *> RtYtBinitüEdktMrUvkarGb,brma As > 0.004 A . dUcKña RtYtBinitüfaetI M u ≥ 1.2M cr b¤Gt; edIm,IFananUvPaBRKb;RKan;énkareRbIEdkFmμta CaBiessenAkñúg nonbonded tendon. (> eRCIserIsTMhM nigKMlatrbs;EdkTajFmμta (nonprestressed tension reinforcement) nigEdksgát; (compression reinforcement) sMrab;kEnøgNaEdlRtUvkarva. !0> epÞógpÞat;fa m:Um:g;KNna (design moment) M u = φM n FMCag b¤esμInwgm:Um:g;emKuN (factored moment) M u . RbsinebImindUecñaHeT eKRtUvEktMrUvkarKNnaeLIgvij. rUbTI 4>52 bgðayBI flowchart sMrab;sresrkmμviFIKNnaedaykMuBüÚT½redIm,IdMeNIrkarviPaK nominal flexural strength rbs;muxkat;eRbkugRtaMgmansøab nigmuxkat;ctuekaNedayyk d p Ca kMBs; cgs rbs; tendon mYyRsTab;. dUcKña rUbTI 4>53 bgðajBI flowchart sMrab;viPaK nominal flexural strength rbs;FñwmeRbkugRtaMgedayeRbI strain compatability analysis EdlmankMBs; strain eRcInRsTab; d p1 rhUtdl; d pn . Flowchart TaMgBIrRtUv)anGnuBaØateGayeRbIsMrab; fully prestressed beam EdlminmaneRbIEdkFmμta nigminmankugRtaMgTajenAkñúgebtug ehIy flowchart TaMgBIrenHk¾ RtUv)anGnuBaØateGayeRbIsMrab; partially prestressed beams EdlkugRtaMgTajkMNt;RtUv)anGnuBaØat eGayekItmanenAkñúgebtugrhUtdl;kareRbIEdkFmμta. kmμviFIkMuBüÚT½rEdlQrelI flowchart TaMgBIrenH GacRtUv)aneRbIdUcKñasMrab;kMBs;RbsiT§PaBeTal d p én cgs tendon profile. Flexural Design of Prestressed Concrete Elements 202
  • 114. NPIC karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 203
  • 115. T.Chhay Flexural Design of Prestressed Concrete Elements 204
  • 116. NPIC 15> Ultimate Strength Design of Prestressed Simply Supported Beam by Strain Compatibility ]TahrN_ 4>9³ sikSaKNna bonded beam enAkñúg]TahrN_ 4>2 eday ultimate-load theory eday eRbIEdkFmμtaedIm,IRTcMENkxøHrbs;bnÞúkemKuN. eRbI strain compatibility edIm,IkMNt; f ps Edlmux kat;EktMrUvRtUv)anbgðajenAkñúgrUbTI 4>54 CamYynwgkMralxNÐsmasxagelIEdlmankMras; 3in. (76mm) nig f pu = 270,000 psi(1,862MPa ) f py = 0.85 f pu sMrab; stress-relieved strands f y = 60,000 psi(414MPa ) f 'c = 5,000 psi(34.5MPa ) ebtugTMgn;Fmμta eRbI 7-wire 1 / 2in. dia tendon. EdkFmμtaRtUv)andak;edaymankMras;ebtugkarBarEdk (clear cover) 1.5in.(38mm ) ehIyeKminKitbBa©ÚlEdksgát;eT. elIsBIenHeKminKitxül; nigrBa¢ÜydIeT. dMeNaHRsay³ BI]TahrN_ 4>2 bnÞúkeFVIkar WL = 1,100 plf (16.1kN / m) bnÞúkeFVIkar W SD = 100 plf (1.46 kN / m ) karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 205
  • 117. T.Chhay bnÞúksnμt; WD = 393 plf (5.74kN / m) ElVgFñwm = 65 ft (19.8m) !> m:Um:g;emKuN ¬CMhanTI1¦ WU = 1.2(WD + WSD ) + 1.6WL = 1.2(100 + 393) + 1.6(1,100 ) = 2352 plf (34.4kN / m ) m:Um:g;emKuNRtUv)aneGayeday Wu l 2 2,352(65)212 Mu = = = 14,905,800in − lb(1684kN .m ) 8 8 ersIusþg;m:Um:g;FmμtaRtUvkar (requirement nominal moment strength) = 16,562,000in. − lb(1871kN .m ) Mu 14,905,800 Mn = = φ 0.9 @> CMerIsmuxkat;dMbUg ¬CMhanTI2¦ edaysnμt;kMBs;FñwmedayGaRta 1 / 20 énElVg × eyIgGacmankMBs;muxkat;sakl,g h = 652012 = 39in. yk 40in.(102cm) bnÞab;mksnμt;EdkFmμta 4#6 = 4 × 0.44 = 1.76in.2 (11.4cm2 ) BIsmIkar 4.61 = 121.8in.2 (786cm 2 ) Mn 16,562,000 A'c = = 0.68 f 'c h 0.68 × 5,000 × 40 snμt;TTwgsøabesμInwg 18in. kMras;søabmFüm = 121.8 / 18 ≅ 7.0in.(178mm) dUcenH ]bmakMras;RTnug bw = 6in.(152mm) edIm,IepÞógpÞat;tMrUvkarersIusþg;kMlaMgkat; BIsmIkar 4.60b = 2.13in.2 ( .3cm 2 ) Mn 16,562,000 Aps = = 13 0.72 f h 0.72 × 270,000 × 40 pu cMnYn 1 / 2in. stress-relieved wire strands = 2.13 / 0.153 = 13.9 dUcenHsakl,g 1 / 2in. tendon cMnYn 13 Aps = 13 × 0.153 = 1.99in.2 (12.8cm 2 ) #> KNnakugRtaMg f ps enAkñúg tendon eRbkugRtaMgEdlmanersIusþg;FmμtaedayeRbI strain- compatibility approach ¬CMhanTI 3¦ Flexural Design of Prestressed Concrete Elements 206
  • 118. NPIC lkçN³FrNImaRtrbs;muxkat;mantMélEk,rnwgTMhMsnμt;dUcCa kMBs; h nigTTwgsøab b . dUcenHeRbITinñn½yxageRkam Ac = 377in.2 ct = 21.16in. d p = 15 + ct = 15 + 21.16 = 36.16in. r 2 = 187.5in.2 e = 15in. enAkNþalElVg e 2 = 225in.2 e 2 / r 2 = 225 / 187.5 = 1.20 Ec = 57,000 5,000 = 4.03 ⋅ 106 psi 27.8 ⋅ 103 MPa ( ) ( ) E ps = 28 ⋅ 106 psi 193 ⋅ 103 MPa bMErbMrYlrageFobsgát;GnuBaØatGtibrma ε enAeBl)ak; = 0.003 c snμt;faeRbkugRtaMgRbsiT§PaBeRkamGMeBIbnÞúkeFVIkarKW f ≅ 155,000 psi(1,069MPa) pe f pe 155,000 (a) ε1 = ε pe = = = 0.0055 E ps 28 ⋅ 10 6 Pe = 13 × 0.153 × 155,000 = 308,295lb kMeNInbMErbMrYlrageFobEdkeRbkugRtaMgk¾dUcebtugRtUv)an decompressed edaykar ekIneLIgbnÞúkxageRkA ¬emIlrUbTI 4>3 nigsmIkar 4.3c¦ RtUv)aneGayeday Pe ⎛ e2 ⎞ ε 2 = ε decomp = ⎜1 + ⎟ Ac Ec ⎜ r2 ⎟ ⎝ ⎠ = 308,295 (1 + 1.20) = 0.0004 377 × 4.03 ⋅ 10 6 (b) snμt;fakugRtaMg f ps ≅ 205,000 psi Cakarsakl,gelIkTImYy. ]bmaG½kSNWtenAkñúg søabEdlmankMras; h f = 3 + 4.5 + 3.5 / 2 = 9.25in. . dUcenHBIsmIkar 4.41a Aps f ps + As f y 1.99 × 205,000 + 1.76 × 60,000 a= = 0.85 f 'c b 0.85 × 5,000 × 18 = 6.71in.(17cm ) < h f = 9.25in. dUcenH bøúksgát;smmUlsßitenAkñúgsøab ehIymuxkat;RtUv)anBicarNaCamuxkat;ctuekaN dUcenHsMrab;ebtug 5,000 psi β1 = 0.85 − 0.05 = 0.8 karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 207
  • 119. T.Chhay = 8.39in.(22.7cm ) a 6.71 c= = β1 0.80 ( ) d = 40 − 1.5 + 0.5in. for stirrup + 16 in. for bar ≅ 37.6in. 5 kMeNInén strain EdlbNþalBI overload rhUtdl; ultimate BIsmIkar 4.37c KW ⎛d −c⎞ ⎛ 37.6 − 8.39 ⎞ ε3 = εc ⎜ ⎟ = 0.003⎜ ⎟ = 0.0104 >> 0.005 O.K. ⎝ c ⎠ ⎝ 8.39 ⎠ ehIybMErbMrYlrageFobsrubKW ε ps = ε1 + ε 2 + ε 3 = 0.0055 + 0.004 + 0.0104 = 0.0163 BI stress-strain diagram enAkñúgrUbTI 4>50 eyIg)an f ps EdlRtUvnwg ε ps = 0.0163 KW 230,000 psi . sakl,gelIkTIBIrsMrab;tMél f ps snμt; f ps = 229,000 psi 1.99 × 229,000 + 1.76 × 60,000 a= 0.85 × 5,000 × 18 = 7.34in. BicarNamuxkat;CaragctuekaN 7.34 c= = 9.17in. 0.80 ⎛ 37.6 − 9.17 ⎞ ε 3 = 0.003⎜ ⎟ = 0.0093 ⎝ 9.17 ⎠ bMErbMrYlrageFobsrubKW ε ps = 0.0055 + 0.0004 + 0.0093 = 0.0152 BIrUbTI 4>50 f ps = 229,000 psi(1,579MPa) eRbI As = 4#6 = 1.76in.2 $> ersIusþg;m:Um:g;EdlGacman ¬CMhanTI6 dl;TI 10¦ BIsmIkar 4.43c RbsinebIG½kSNWtsßitenAkñúgsøab ⎛ 7.34 ⎞ ⎛ 7.34 ⎞ M n = 1.99 × 229,000⎜ 36.16 − ⎟ + 1.76 × 60,000⎜ 37.6 − ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ = 14,806,017 + 3,583,008 = 18,389,025in. − lb(2,078kN .m ) EdlRtUvkar = 16,562,000in. − lb O.K. > Mn karkat;bnßyRkLaépÞEdkFmμtaGaceFVIeGaymuxkat;kan;EtmanRbsiT§PaB edaysarersIu sþg;EdlGacmanFMCagersIusþg;m:Um:g;RtUvkarRbEhl 11% . %> RtYtBinitüEdkGb,brma nigEdkGtibrma ¬CMhanTI6 nigTI9¦ Flexural Design of Prestressed Concrete Elements 208
  • 120. NPIC (a) As min = 0.004 A Edl A CaRkLaépÞénEpñkrbs;muxkat;EdlsßitenAcenøaHmuxkat;rgkarTaj nig cgc. BImuxkat;énrUbTI 4>8 ⎛ 1.375 ⎞ A = 377 − 18⎜ 4.125 + ⎟ − 6(21.16 − 5.5) ≅ 201in. 2 ⎝ 2 ⎠ As min = 0.004 × 201 = 0.80in.2 < 1.76 EdleRbI O.K. (b) snÞsSn_EdkGtibrmaEdl)anBIsmIkar 4.57b KW ωp + d (ω − ω ') ≤ 0.36β1 < 0.29 sMrab; β1 = 0.80 d p ehIysnÞsSn_EdksrubCak;EsþgKW 1.99 × 229,000 37.6 ⎛ 1.76 × 60,000 ⎞ ωT = + ⎜ ⎟ 18 × 36.16 × 5,000 36.16 ⎝ 18 × 37.6 × 5,000 ⎠ = 0.14 + 0.03 = 0.17 < 0.29 O.K. ^> eRCIserIsmuxkat;sMrab; ultimate load ¬CMhanTI11¦ BICMhanTI1 dl;TI5 énkarKNna/ muxkat;enAkñúg]TahrN_4>2 CamYynwgmuxkat;EkERb EdlbgðajenAkñúgrUbTI 4>54 manersIusþg;m:Um:g;Fmμta M n EdlGacRTbnÞúkemKuN RbsinebIeK eRbIEdkFmμta #6 cMnYn 4 enAépÞrgkarTajCa partially prestressed section. dUcenH eKGacTTYlykmuxkat;enHkarBt; edaysarvak¾bMeBjnUvtMrUvkar service-load flexural stress TaMgenAkNþalElVg nigenARtg;TMr. cMNaMfa muxkat;GacbegáItEtersIusþg;FmμtatMrUvkar M n = 16,562,000in. − lb edaykarbEnßmEdkFmμtaenAépÞrgkarTajedIm,ITb;Tl;EpñkxøHénersIusþg; m:Um:g;RtUvkarsrub. cMNaMfa muxkat;enHRKb;RKan;sMrab;ebtug f 'c = 5,000 psi cMENkÉmuxkat;enA kñúg]TahrN_4>2 ebtugmanersIusþg; f 'c = 6,000 psi edIm,IkMueGayFMCag allowable service-load concrete stresses. dUcenH karKNnaeRkamGMeBI ultimate load KWmanlkçN³caM)ac;kñúgkarKNna ebtugeRbkugRtaMgedIm,IFanafa Ggát;GacRTRKb;bnÞúkemKuN nigCaEpñkTaMgmUlénkarKNnasrub. 16> Strength Design of Bonded Prestressed Beam Using Approximate Procedure ]TahrN_ 4>10³ sikSaKNnaFñwmenAkñúg]TahrN_ 4>9 Ca partially prestressed beam edayeRbI ACI approxiamate procedure RbsinebIeKGnuBaØat. eRbImuxkat;sþg;darEdleRbIenAkñúg]TahrN_ 4>2 karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 209
  • 121. T.Chhay CamYy (a) bonded prestressed streel nig (b) nonbonded prestressed steel. minKitBicarNakar cUlrYmrbs;Edksgát;Fmμta. dMeNaHRsay³ !> lkçN³muxkat; ¬CMhan1 nig2¦ TTwgsøabxagelIrbs;muxkat;enAkñúg]TahrN_ 4>2 KW b = 18in. ehIykMras;mFümrbs;va Edl)anBIrUbTI 4>8 KW 3.5 h f = 4.5 + = 6.25in. 2 sakl,gEdkTajFmμta #6(dia.12.7mm) cMnYn 4 bEnßmBIelIEdkeRbkugRtaMg. @> kugRtaMg f ps enAkúñgEdkeRbkugRtaMgeRkamersIusþg;Fmμta (nominal strength) ¬CMhanTI3¦ BI]TahrN_ 4>9 f pe ≅ 155,000 psi 0.5 f pu = 0.5 × 270,000 = 135,000 psi f pe > 0.5 f pu dUcenH eyIgGaceRbI ACI approximate procedure edIm,IkMNt; f ps . (A) Bonded case RbsinebIeKminsÁal;TItaMgG½kSNWt dMbUgeKRtUvviPaKvaCamuxkat;ctukaNsin. BIsmIkar 4.51 ⎛ γp ⎡ f pu ⎤⎞ f ps = f pu ⎜1 − ⎜ ⎢ρ p + d (ω − ω ')⎥ ⎟ ⎝ β1 ⎢ ⎣ f 'c d p ⎥⎟ ⎦⎠ f py 229,500 = f pu 270,000 = 0.85 eRbI γ p = 0.40 Aps = 13 × 0.153 = 1.99in.2 As = 4 × 0.44 = 1.76in.2 Aps 1.99 ρp = = = 0.0032 bd p 18 × 36.16 As f y 1.76 60,000 ω= × = × − 0.00132 bd f 'c 18 × 37.6 5,000 sMrab; ω ' = 0 / ⎛ 0.40 ⎡ ⎤⎞ ⎜ 0.80 ⎢0.0032 × 5,000 + 36.16 (0.0132 )⎥ ⎟ 270,000 37.6 f ps = 270,000⎜1 − ⎟ ⎝ ⎣ ⎦⎠ = 270,000 × 0.897 = 242,190 psi (1,670 MPa ) Flexural Design of Prestressed Concrete Elements 210
  • 122. NPIC 1.99 × 242,190 + 1.76 × 60,000 a= = 7.68in. > h f = 6.25in. 0.85 × 5,000 × 18 dUcenH G½kSNWtsßitenAxageRkAsøab ehIyeKRtUvviPaKvaCamuxkat;mansøab ¬GkSr T¦. eday eRbITTwgRTnug bw . Aps 1.99 ρp = = = 0.0092 bw d p 6 × 36.16 As fy 1.76 60,000 ωw = × = × = 0.0936 bw d f 'c 6 × 37.6 5,000 ⎛ 0.40 ⎡ ⎤⎞ ⎜ 0.80 ⎢0.0092 × 5,000 + 36.16 (0.0936 − 0 )⎥ ⎟ 270,000 37.6 f ps = 270,000⎜1 − ⎟ ⎝ ⎣ ⎦⎠ = 189,793 psi (1,309 MPa ) Apw f ps = Aps f ps + As f y − 0.85 f 'c (b − bw )h f = 1.99 × 189,793 + 1.76 × 60,000 − 0.85 × 5,000(18 − 6 ) × 6.25 = 377,688 + 105,600 − 318,750 = 164,538lb = 6.45in.(16.4cm ) 164,538 a= 0.85 × 5,000 × 6 #> ersIusþg;m:Um:g;FmμtaEdlGacman ¬CMhanTI4 dl;TI8¦ ⎛ hf ⎞ ⎛ a⎞ ( ) M n = Apw f ps ⎜ d p − ⎟ + As f y d − d p + 0.85 f 'c (b − bw )h f ⎜ d p − ⎜ 2 ⎟ ⎟ ⎝ 2⎠ ⎝ ⎠ ⎛ 6.45 ⎞ = 164,538⎜ 36.16 − ⎟ + 1.76(60,000)(37.6 − 36.16 ) ⎝ 2 ⎠ ⎛ 6.25 ⎞ + 0.85(5,000 )(18 − 6) × 6.25⎜ 36.16 − ⎟ = 16,071,226in. − lb(1,816kN .m ) ⎝ 2 ⎠ < Mn EdlRtUvkar = 16,562,000in. − lb(1,871kN .m) dUcmuxkat;enHminRKb;RKan;eT. bnþkarsakl,g nigEktMrUvmYyeTotedayeRbIEdkFmμtaeRcInCagmun. sakl,g #8(dia. 25mm) cMnYn 4 / As = 3.16in.2 (25cm2 ). eyIgman 3.16 60,000 ωw = × = 0.17 6 × 37.6 5,000 edayeGay f ps = 179,068 psi nig Apw f ps = 227,195lb(1010kN ) / dUcenH = 8.9in.(22.6cm ) 227,195 a= 0.85 × 5,000 × 6 karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 211
  • 123. T.Chhay ⎛ 8.9 ⎞ M n = 227,195⎜ 36.16 − ⎟ + 3.16(60,000 )(37.6 − 36.16) ⎝ 2 ⎠ ⎛ 6.25 ⎞ + 0.85(5,000)(18 − 6) × 6.25⎜ 36.16 − ⎟ ⎝ 2 ⎠ EdlRtUvkar = 16,562,000in. − lb O.K. = 18,007,283in. − lb(2035kN .m ) > M n dUcenH eRbIEdkFmμta 4#8 enAsrésxageRkam ehIyTTYlykkarKNnasMrab;krNI bonded. (b) Nonbonded Case × pleFobElVgelIkMBs; = 654012 = 19.5 < 35 dUcenH BIsmIkar 4.52a f 'c 5,000 f ps = f pe + 10,000 + = 155,000 + 10,000 + 100 ρ p 100 × 1.99 / (6 × 36.16 ) = 170,451 psi (1,175MPa ) cMNaMfa enATIenHeKeRbI bw = 6in. sMrab; ρ p edaysareKdwgehIyfamuxkat;eFVIkarCalkçN³ muxkat;mansøab ¬edaysarG½kSNWtsßitenAxageRkamsøab¦. dUcenH f ps = 170,451 psi (1,175MPa ) !> eRCIserIsEdkFmμta sakl,gEdkFmμta 4#8 edIm,ITb;Tl;EpñkxøHrbs;m:Um:g;emKuN³ As = 4 × 0.79 = 3.16in.2 ( .8cm 2 ) 19 Apw f ps = 1.99 × 170,451 + 3.16 × 60,000 − 0.85 × 5,000(18 − 6)6.25 = 210,047lb Apw f ps = 8.24in.(20.9cm ) 210,047 a= = 0.85 f 'c bw 0.85 × 5,000 × 6 @> ersIusþg;m:Um:g;EdlGacman ¬CMhanTI4 dl;TI8¦ BIsmIkar 4.48, ⎛ 8.24 ⎞ M n EdlGacman = 210,047⎜ 36.16 − ⎟ + 3.16 × 60,000(37.6 − 36.16) ⎝ 2 ⎠ ⎛ 6.25 ⎞ + 0.85 × 5,000(18 − 6) × 6.25⎜ 36.16 − ⎟ ⎝ 2 ⎠ = 17,537,057in. − lb(1981kN .m ) < Mn EdlRtUvkar = 16,562,000in. − lb O.K. Flexural Design of Prestressed Concrete Elements 212
  • 124. NPIC (c) RtYtBintükarkMNt;rbs;Edk !> EdkGb,brma BIsmIkar 4.25 m:Um:g;eRbH M cr RtUv)aneGayeday ⎛ 2⎞ M cr = f r Sb + Pe ⎜e + r ⎟ ⎜ cb ⎟ ⎝ ⎠ BI]TahrN_ 4>2/ f r = 7.5 5,000 = 530.3 psi(3.7MPa) . edaysar Sb = 3,750in.3 / e = 15in. / r 2 / cb = 187.5 / 18.84 = 9.95in. nig Pe = 308,255lb(1,371kN ) eyIgTTYl)an M cr = 530.3 × 3,750 + 308,255(15 + 9.95) = 9,680,585in. − lb(1,090kN .m ) 1.2 M cr = 1.2 × 9,680,585 = 11,616,702in. − lb(1,313kN .m ) M u = φM n = 0.90 × 18,026,667 = 16,224,000in. − lb(1,833kN .m ) cugeRkay BIsmIkar 4.54a M u > 1.2M cr dUcenH tMrUvkarEdkGb,brmaRtUv)anbMeBjsMrab;krNI bonded nig nonbonded. @> snÞsSn_EdkGnuBaØatGtibrma (maximum allowable reinforcement index) Max. Allow. ω p = 0.36 β1 = 0.36 × 0.80 = 0.288 ¬bMErbMrYlrageFobGb,brma ε t = 0.005 RtUvKñanwg 0.32β1 ¦ ω p Cak;Esþg = ρ p ( f ps / f 'c ) = 0.0032 × 170,451 = 0.109 5,000 ω Cak;Esþg = ρ ( f y / f 'c ) = 3.96 60,000 × = 0.0702 18 × 37.6 5,000 dUcenH ω p + ω = 0.109 + 0.0702 = 0.1862 < 0.288 O.K. dUcenH TTYlykkarKNnaEdleRbImuxkat;ebtugenAkñúg]TahrN_ 4>2 edaydak;bEnßmEdk Fmμta 4#8 enAEpñkrgkarTaj. cMNaMfa non-bonded section EdlmanRkLaépÞEdkFmμtadUcKñaman ersIusþg;m:Um:g;tUcCag bonded section EdldUckarrMBwgTuk. RbsinebI eKeRbI f 'c = 6,000 psi sMrab;karKNnaenAkñúg]TahrN_enH eKnwgRtUvkarEdkFmμta ticCag. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 213