Iv.flexural design of prestressed concrete elements

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Iv.flexural design of prestressed concrete elements

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 11. T.Chhay Flexural Design of Prestressed Concrete Elements 100
  12. 12. NPIC karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 101
  13. 13. T.Chhay Flexural Design of Prestressed Concrete Elements 102
  14. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 29. T.Chhay Flexural Design of Prestressed Concrete Elements 118
  30. 30. NPIC karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 119
  31. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 47. T.Chhay Flexural Design of Prestressed Concrete Elements 136
  48. 48. NPIC karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 137
  49. 49. T.Chhay Flexural Design of Prestressed Concrete Elements 138
  50. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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

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