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Vii. camber, deflection, and crack control

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Vii. camber, deflection, and crack control

  1. 1. Department of Civil Engineering NPIC VII. PaBekag PaBdab nigkarRKb;RKgsñameRbH Camber, Deflection and Crack Control 1> esckþIepþIm Introduction PaBdab nigsñameRbHrbs;Ggát;ebtugeRbkugRtaMgk¾sMxan;dUckarKNnaPaBdab nigsñameRbH rbs;Ggát;ebtugGarem:Edr. Ggát;ebtugeRbkugRtaMgmanlkçN³Rsav (slender) CagGgát;ebtugGarem: ehIykareFVIkarrbs;vargT§iBleday flexural cracking eFVIeGayvaeKkan;EtRby½tñkñúgkarRKb;RKg PaBdab nigsñameRbH. karKNnadMbUgBak;B½n§nwgkarKNnasmamaRtmuxkat;rbs;Ggát;eRKOgbgÁúMsMrab; sßanPaBkMNt;én flexural stresses eRkamGMeBI service load nigsMrab;sßanPaBkMNt;énkar)ak; Edlrg karBt;begáag kMlaMgkat; nigkMlaMgrmYl edayrYmbBa©ÚlTaMg anchorage development strength. kar KNnaEdlmanlkçN³eBjeljluHRtaEtmankarkMNt;TMhMén long-term deflection, camber nigTMhM sñameRbH ehIytMélTaMgenHsßitenAkñúgkMrit allowable serviceability. Ggát;ebtugeRbkugRtaMgrgkMlaMgsgát;cakp©itCaGcié®nþy_EdlbNþalBIkMlaMgeRbkugRtaMgCH T§iBly:agxøaMgdl; long-term creep deformation rbs;va. karbraC½ykñúgkarTajTukCamun nigkar RKb;RKgkMhUcRTg;RTayEbbenHGacnaMeGayman camber FM EdlGacbgáeGaymanépÞe)a:g nignaMeGay karbgðÚrTwkBIdMbUlGKarminmanlkçN³smRsb/ bgáeGaykarebIkbrelIs<anminmanpasuxPaB nigbgá eGaymansñameRbHenAelItYGKar EdlrYmbBa©ÚlTaMgkarBi)akkñúgkareFVIbg¥Üc nigTVarrt;Rtg;Kña. PaBBi)akkñúgkarTajTukCamunnUvkMhatbg; long-term prestress EdlmanlkçN³suRkiteFVI eGayeKkan;EtBi)akkñúgkar)a:n;RbmanTMhMén camber EdlrMBwgTukeGaysuRkitEdr. PaBsuRkitkan;Et Bi)akTTYl)ansMrab; partially prestressed concrete system EdlsñameRbHkMNt;RtUv)anGnuBaØattam ry³kareRbIEdkFmμtabEnßm. Creep strain enAkñúgebtugbegáIn camber dUcEdlvabgáeGaymankarekIn eLIgnUvkMeNagCalkçN³GviC¢manEdlCaTUeTAvamantMélFMCagkarfycuHEdlbegáItedaykarfycuHénkM hatbg;eRbkugRtaMgedaysar creep, shrinkage nig stress relaxation. kar)a:n;RbmanEdll¥bMput énkarekIneLIgén camber KYrEp¥kelIbTBiesaFn_/ EdnkMNt;énpleFobElVgelIkMBs;Fñwm nigkareRCIs erIsm:UDul Ec rbs;ebtugd_RtwmRtUv. karKNna moment-curvature relationship eRkamdMNak;kalén kardak;bnÞúkCabnþbnÞab;rhUtdl;sßanPaBkMNt;énkar)ak;k¾GacCYykñúgkarkMNt;PaBdabrbs;Ggát; eGaymanlkçN³kan;EtsuRkit. edaysarkugRtaMgFMenAkñúgEdkeRbkugRtaMg ERcHsIuEdlbNþalBIsñameRbHGaceFVIeGayeRKOg PaBekag PaBdab nigkarRKb;RKgsñameRbH 407
  2. 2. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa bgÁúM)at;bg;lT§PaBRTRTg;. dUcenH EdnkMNt;énTMhMrbs;sñameRbH nigKMlatrbs;vaRtUv)ankMNt; ehIy dMeNIrkarénkarkMNt;TMhMsñameRbHsmRsbRtUv)aneRbI. ACI 318 Code )ancat;cMNat;fñak;eGay Ggát;rgkarBt;begágebtugeRbkugRtaMgCabIfñak;KW³ (a) Class U: f t ≤ 7.5 f 'c psi (0.623 f 'c MPa ) (7.1a) enAkñgkrNIenH eKeRbI gross section sMrab;lkçN³muxkat;enAeBlkMNt;eRbkugRtaMgeRkamGMeBI service load nigkMNt;PaBdab. eKminRtUvkareRbI skin reinforcement enAépÞbBaÄreT. (b) Class T: 7.5 f 'c ≤ f t ≤ 12 f 'c psi ( f 'c MPa ) (7.1b) cMNat;fñak;enHenAcenøaHmuxkat;eRbH nigmuxkat;Gt;eRbH. eKeRbI gross section kñúgkarKNna stress. eKeRbI cracked bi-liner section sMrab;KNnaPaBdab. eKminRtUvkareRbI skin reinforcement enAépÞbBaÄreT. (C) Class C: f t > 12 f 'c (7.1c) cMNat;fñak;enHsMrab;muxkat;eRbH. dUcenH eKeRbImuxkat;eRbHsMrab;kMNt;kugRtaMg nigPaBdab eRkamGMeBI service load. eKcaM)ac;RtUvKNna Δf ps b¤ f s sMrab;RKb;RKgsñameRbH Edl Δ ps = kugRtaMg EdlekIneLIgbnÞab;BIsßanPaBdkkMlaMgsgát; (decompression) ehIy f s = kugRtaMgenAkñúgEdkFmμta enAeBlEdlEdkFmμtaRtUv)aneRbIEdr. RbB½n§kMralxNÐeRbkugRtaMgBIrTisRtUv)ansikSaKNnaCa Class U. 2> karsnμt;kñúgkarKNnaPaBdab Basic Assumptions in Deflection Calculations eKGackMNt;PaBdabBIdüaRkamm:Um:g;énkMlaMgeRbkugRtaMgCamYynwgbnÞúkTTwgG½kSxageRkA (external transverse loading) b¤BITMnak;TMngm:Um:g; nigkMeNag (moment-curvature relationships). enAkñúgkrNINak¾eday eKRtUveFVIkarsnμt;dUcxageRkam³ - RkLaépÞmuxkat;rbs;ebtugRtUvEtsuRkitRKb;RKan;edIm,IKNnam:Um:g;niclPaB elIkElgenA eBlEdleKRtUvkarcaM)ac;karKNnaEdlmanlkçN³kan;EtRbesIr. - m:UDulrbs;ebtug Ec = 33w1.5 f 'c psi(0.043w1.5 f 'c MPa) EdltMélrbs; f 'c RtUvKña nwgersIusþg;sgát;rbs;sMNakKMrUragsIuLaMgrbs;ebtugenAGayuEdleKRtUvkarkMNt; Ec . Camber, Deflection and Crack Control 408
  3. 3. Department of Civil Engineering NPIC - GnuvtþeKalkarN_ superposition kñúgkarKNnaPaBdabEdlbNþalBIbnÞúkTTwgG½kS nig camber EdlbNþalBIkMlaMgeRbkugRtaMg. - eKGaceFVIkarKNnaPaBdabTaMgGs;edayQrelIG½kSTIRbCMuTMgn;rbs;EdkeRbkugRtaMg (cgs) Edl strand RtUv)anKitCa single tendon. - PaBdabEdlbNþalBI shear deformation minRtUv)anKit - eKGacKitmuxkat;Ca totally elastic rhUtdl; decompression load. bnÞab;mk m:Um:g;niclPaB énmuxkat;EdleRbH I cr Gacpþl;nUvkarkMNt;PaBdab nig camber kan;EtsuRkit. 3> PaBdabry³eBlxøI¬xN³¦ rbs;Ggát;eRbH nigGgát;EdlKμaneRbH Short-Term (Instantaneous) Deflection of Uncracked and Cracked Members k> TMnak;TMngrvagbnÞúk nigPaBdab Load-Deflection Relationship PaBdabry³eBlxøIenAkñúgGgát;ebtugeRbkugRtaMgRtUv)anKNnaedaysnμt;vamuxkat;manlkçN³ esμIsac; (homogeneous), lkçN³sac;mYy (isotropic) nigeGLasÞic. karsnμt;EbbenHCaviFIénkareFVI karCak;Esþg Edlm:UDul Ec rbs;ebtugERbRbYleTAtamGayurbs;ebtug ehIym:Um:g;niclPaBERbRbYleTA tamdMNak;kalénkardak;bnÞúk eTaHbImuxkat;eRbH b¤mineRbHk¾eday. PaBekag PaBdab nigkarRKb;RKgsñameRbH 409
  4. 4. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa Cak;Esþg TMnak;TMngrvagbnÞúk nigPaBEdkCa trilinear dUcEdlbgðajenAkñúgrUbTI 7>1. tMbn;bI munkar)ak;KW³ tMbn;TI I dMNak;kalmuneBleRbH (precracking stage) EdlGgát;minmansñameRbHeT. tMbn;TI II dMNak;kaleRkayeBleRbH (postcracking stage) EdlGgát;eRKOgbgÁúMbegáIt acceptable controlled cracking TaMgkarBRgay nigTMhM. tMbn;TI III dMNak;kaleRbHeRkayrgbnÞúk (postserviceability cracking stage) EdlkugRtaMg enAkñúgEdkTajeFVIkardl;sßanPaBkMNt;én yielding. !> tMbn;TI1 Precracking stage kMNat;Ggát;muneBleRbHrbs;ExSekagrvagbnúÞk nigPaBdabKWCaExSRtg;EdlkMNt;kareFVIkarCa lkçN³eGLasÞiceBjelj dUcenAkñúgrUbTI 7>1. kugRtaMgTajGtibrmaenAkñúgFñwmenAkñúgtMbn;enHtUc CagersIusþg;TajkñúgkarBt;begáag EdlvatUcCagm:UDuldac; ft rbs;ebtug. eKGacPaBrwgRkajkñúgkarBt; begáag EI rbs;FñwmedayeRbIm:UDulyuaMg Ec rbs;ebtug ehIym:Um:g;niclPaBrbs;muxkat;ebtugEdlGt; eRbH. kareFVIkarrvagbnÞúk nigPaBdabGaRs½yy:agxøageTAnwgTMnak;TMngrvagkugRtaMg nigbMErbMrYlrag M eFobrbs;ebtug. düaRkamTMnak;TMngrvagkugRtaMg nigbMErbMrYlrageFobKMrUrbs;ebtugRtUv)anbgðajenAkñúgrUbTI 7>2. eKGac)a:n;RbmaNtMé;lrbs; Ec EdleRbIsmIkarEdl)anBIkarBiesaFrbs; ACI EdleGayenAkñúgem eronTI 2. Camber, Deflection and Crack Control 410
  5. 5. Department of Civil Engineering NPIC ( Ec = 33w1.5 f 'c psi 0.043w1.5 f 'c MPa ) (7.2a) b¤ ( ) Ec = 57,000 f 'c psi 4780 f 'c MPa sMrab;ebtugTMgn;Fmμta tMbn;muneBleRbHcb;enAeBlEdlsñameRbHedaykarBt;begáagdMbUgcab;epþImekItman enAeBl EdlkugRtaMgebtugeFVIkareTAdl;ersIusþg;énm:UDuldac; f r . RsedogKñaeTAnwgersIusþg;TajedaykarbMEbk edaypÞal; (direct tensile splitting strength) m:UDuldac;rbs;ebtugKWsmamaRteTAnwgb¤skaer:énersIu- sþg; sgát;rbs;va. sMrab;eKalbMNgénkarsikSaKNna eKGacyktMélrbs;m:UDuldac;sMrab;ebtugesμInwg f r = 7.5λ f 'c psi (0.623λ f 'c MPa ) (7.2b) Edl λ = 1.0 sMrab;ebtugTMgn;Fmμta (normal-weight concrete). RbsinebIeKeRbI all-lightweight concrete enaHeKyk λ = 0.75 ehIyRbsinebIeKeRbI sand-lightweight concrete enaH λ = 0.85 . RbsinebIeKeGaym:UDuldac; f r esIμnwgkugRtaMgEdlekIteLIgeday cracking moment M cr (decompression moment) enaH Pc ⎛ ecb ⎞ M cr fb = ft = − ⎜1 + 2 ⎟ − (7.3a) Ac ⎝ r ⎠ Sb EdlGkSr b tMNageGaysrésxageRkamenARtg;kNþalElVgénFñwmTMrsamBaØ. RbsinebIcMgayén srésrgkarTajxageRkAbMputrbs;ebtugBITMRbCMuTMgn;rbs;muxkat;ebtugCa yt enaH cracking moment RtUv)aneGayeday I g ⎡ Pe ⎛ ecb ⎞ ⎤ M cr = ⎢ ⎜1 + 2 ⎟ + 7.5λ f 'c ⎥ (7.3b) yt ⎣ Ac ⎝ r ⎠ ⎦ ⎡ P ⎛ ecb ⎞⎤ b¤ M cr = S b ⎢7.5λ f 'c + e ⎜1 + 2 ⎟⎥ Ac ⎝ r ⎠⎦ (7.3c) ⎣ Edl Sb = m:UDulmuxkat;enAsrésxageRkam. BIsmIkar 5.12, cracking moment EdlbNþalBIEpñkén bnÞúkGefrEdleFVIeGaymansñameRbHKW M cr = Sb [6.0λ f 'c + f ce − f d ] ¬xñat US¦ (7.4a) M cr = Sb [0.5λ f 'c + f ce − f d ] ¬xñat SI¦ Edl f cr = kugRtaMgsgát;enARtg;TIRbCMuTMgn;rbs;muxkat;ebtugEdlbNþalEtBIkMlaMgeRbkugRtaMg RbsiT§PaBeRkayeBlxatbg; enAeBlbnÞúkxageRkAeFVIeGaymankugRtaMgTaj f d = kugRtaMgebtugenARtg;srésTajxageRkAEdlbNþalBIbnÞúkGefrKμanemKuN enAeBl EdlbnÞúkxageRkAbgáeGaymankugRtaMgTaj nigsñameRbH PaBekag PaBdab nigkarRKb;RKgsñameRbH 411
  6. 6. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa eKk¾GaceRbIemKuN 7.5 CMnYseGayemKuN 6.0 ¬xñat US¦b¤ 0.623 CMnYseGay 0.5 ¬xñat SI¦ sMrab;kMNt;PaBdabrbs;Fñwm. eKGacbMElgsmIkar 7.3a eGayeTACaTMrg; PCI EdleGaynUvlT§pl dUcKña M cr ⎛ f − fr ⎞ = 1 − ⎜ tl ⎜ f ⎟ ⎟ (7.4b) Ma ⎝ L ⎠ Edl Ma = m:Um:g;EdlekItBIbnÞúkGefrKμanemKuNGtibrma f tl = kugRtaMgrbs;ebtugeRkamGMeBI service load srubcugeRkayenAkñúgGgát; f r = m:UDuldac; f L = kugRtaMgrbs;ebtugeRkamGMeBI service live load enAkñúgGgát; @> karKNnam:Um:g;eRbH M Calculation of Cracking Moment M cr cr ]TahrN_ 7>1³ KNna cracking moment M sMrab;muxkat;FñwmctuekaNEkgEdlmanTTwg b = 12in. cr (305mm) ehIykMBs;srub h = 34in.(610mm ) nigman . kugRtaMgebtug f 'c = 4,000 psi(27.6MPa ) f b EdlbNþalBIkMlaMgeRbkugRtaMgcakp©itKW 1,850 psi (12.8MPa ) kñúgkarsgát;. ykm:UDuldac;esμInwg 7.5 f 'c . dMeNaHRsay³ m:UDuldac; f r = 7.5 f 'c = 7.5 4,000 = 474 psi(3.27MPa) . ehIy I g = bh3 / 12 = 12(24 )3 / 12 = 12 = 13,824in.4 (575,400cm 4 )/ yt = 24 / 2 = 12in.(305mm ) eTAsrésrgkarTaj ehIy Sb = I g / yt = 13,824 / 12 = 1,125in.3 (18,878cm3 ). ⎡ P ⎛ ecb ⎞⎤ M cr = S b ⎢7.5λ f 'c + e ⎜1 + 2 ⎟⎥ = 1.152[474 + 1850] ⎣ Ac ⎝ r ⎠⎦ = 2.68 ⋅ 10 6 in. − lb(302.9kN .m ) RbsinebIFñwmenHminrgeRbkugRtaMg enaH cracking moment KW M cr = f r I g / yt = 474 × 13,824 / 12 = 0.546 ⋅ 106 in. − lb(61.7kN .m ) #> tMbn;TI2 Postcracking service-load stage tMbn;muneRbHcb;enAeBlsñameRbHTImYycab;epþm ehIycl½tcUltMbn;TI2 rbs;düaRkamTMnak; I TMngrvagbnÞúk nigPaBdabénrUbTI 7>1. FñwmPaKeRcInsßitenAkñúgtMbn;enHeRkamT§iBl service load. FñwmrgnUvdWeRkénsñameRbHEdlERbRbYltambeNþayElVgEdlRtUvKñanwgkugRtaMg nigPaBdabenARtg;mux Camber, Deflection and Crack Control 412
  7. 7. Department of Civil Engineering NPIC kat;nImYy². dUcenH sñameRbHnwgrIkFM nigeRCAenAkNþalElVg EdlsñameRbHEdlmanTMhMtUc²ekItman enAEk,rTMrrbs;FñwmsamBaØ. enAeBlEdl flexural cracking ekItman karcUlrYmrbs;ebtugenAkñúgtMbn;TajnwgfycuHy:ag xøaMg. dUcenH flexural rigidity rbs;muxkat;RtUv)ankat;bnßyEdleFVIeGayExSekagbnÞúk-PaBdab (load- deflection curve) enAkúñgtMbn;enHecattUcCagenAkñúgdMNak;kalmuneRbH (precracking stage). eday sarTMhMrbs;sñameRbHekIneLIg PaBrwgRkajnwgfycuH EdleFVIeGayPaBs¥itrbs;EdkmantMélTabEdl vaRtUvKñanwg karfycuHénm:Um:g;niclPaBrbs;muxkat;eRbH. eKGacKNnam:Um:g;niclPaB I cr énmuxkat; EdleRbH (cracked section) BIeKalkarN_rbs;emkanic. $> tMbn;TI2 Postserviceability cracking stage and limit state of deflection behavior at failure düaRkaménTMnak;TMngrvagbnÞúk nigPaBdabénrUbTI 7>1 enAkñúgtMbn;TI3manlkçN³rabesμICag enAkñúgtMbn;mun² EdlenHKWbNþalmkBIkMhatbg;énPaBrwgRkajrbs;muxkat;y:ageRcIn edaysarsñam eRbHFM² nigkarrIkFMrbs; stabilized cracks BaseBjElVg. edaysarbnÞúkbnþekIneLIg enaHbMErbMrYl rageFob ε s enAkñúgEdkenAkñúgtMbn;TajbnþekIneLIgtameRkay yield strain ε y edayminmankugRtaMg bEnßm. FñwmRtUv)anBicarNafa)ak;eday yielding dMbUgrbs;Edk TajenAkñúgdMNak;kalenH. vabnþdab edayKμankardak;bnÞúkbEnßm nigsñameRbHbnþcMhr ehIy G½kSNWtbnþeLIgelIeTArksréssgát;xageRkA bMput. cugeRkay secondary compression failure ekIteLIg EdlnaMeTAdl;karpÞúHEbkrbs;ebtugenA kñúgtMbn;m:Um:g;GtibrmaEdlbnþedaykar)ak;. x> muxkat;Gt;eRbH Uncracked Sections !> karKNnaPaBdab Deflection calculation eKmanbMNgcg;KNnaPaBdabsMrab;muxkat;ebtugeRbkugRtaMgGt;eRbHeGaykan;EtsuRkitCag karKNnaPaBdabsMrab;muxkat;EdleRbHedaysarkarsnμt;énkareFVIkarCalkçN³eGLasÞicmanlkçN³ RbesIrCag. kareRbIR)as;m:Um:g;niclPaBrbs; gross section minCHT§iBldl;suRkitPaBkñúgkarKNna dUc transformed section eT. PaBekag PaBdab nigkarRKb;RKgsñameRbH 413
  8. 8. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa ]bmafaFñwmrgeRbkugRtaMgCamYynwgcMNakp©itrbs;EdkeRbkugRtaMgefrdUcbgðajenAkñúgrUbTI 7>3. eRbIkarkMNt;sBaØaéndüaRkam primary moment enAelIépÞrgkarTajrbs;Fñwm ehIyGnuvtþ elastic weight method edaybMElgdüaRkamm:Um:g;FmμtaeGayeTACa elastic weight M 1 / (Ec I c ) enAelIElVgFñwm l . bnÞab;mkm:Um:g;rbs; weight intensity (Pe) /(Ec I c )énkNþalElVg AC enAkúñgrUbTI 7>3(c) BIelIcMnuckNþalElVg C eGay Pel ⎛l⎞ Pe ⎛ l l ⎞ Pel 2 δc = ⎜ ⎟− ⎜ × ⎟= (7.5) 2 Ec I c ⎝ 2 ⎠ Ec I c ⎝ 2 4 ⎠ 8 Ec I c Camber, Deflection and Crack Control 414
  9. 9. Department of Civil Engineering NPIC cMNaMfa eKKUrdüaRkamPaBdabenAkñúgrUbTI 7>3 (d) BIelIExSeKal (base line) dUcEdlFñwmekageLIgelI edaysarkMlaMgeRbkugRtaMg. eKGaceFVIkarKNnaRsedogKñasMrab; tendon profile NamYy nigsMrab;RbePTbnÞúkTTwgG½kS (transverse loading) NamYyEdlminKitfaragFrNImaRtrbs;EdkeRbkugRtaMg b¤kardak;bnÞúkman lkçN³sIuemRTIk¾Gt;. PaBdab b¤ camber cugeRkayKWCa superposition énPaBdabEdlbNþalBI kMlaMgeRbkugRtaMgCamYynwgPaBdabEdlbNþalBIbnÞúkxageRkA. @> karKNnabMErbMrYlrageFob nigkMeNag Strain and Curvature Evaluation karEbgEckbMErbMrYlrageFobtamkMBs;rbs;muxkat;enAdMNak;kalrgbnÞúkmanragCabnÞat; dUc bgðajenAkñúgrUbTI 7>4 EdlmanmMurbs;kMeNagGaRs½ynwgbMErbMrYlrageFobrbs;srésxagelI ε ct nigbMErbMrYlrageFobrbs;srésxageRkam ε cb rbs;ebtug. BIkarEbgEckbMErbMrYlrageFob (strain distribution) smIkarkMeNagenAdMNak;kalénkardak;bnÞúkepSg²mandUcxageRkam³ (I) dMNak;kalrgkMlaMgeRbkugRtaMgdMbUg (initial prestress) ε cbi − ε cti φi = (7.6a) h (II) dMNak;kalrgeRbkugRtaMgRbsiT§PaBeRkayeBlxatbg; (effective prestress after losses) ε cbe − ε cte φe = (7.6b) h PaBekag PaBdab nigkarRKb;RKgsñameRbH 415
  10. 10. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa (III) dMNak;kalrgbnÞúkeFVIkar (service load) ε ct − ε cb φ= (7.6c) h (IV) dMNak;kal)ak; (failure) εu φu = (7.6d) c eRbIsBaØabUksMrab; tensile strain nigsBaØadksMrab; compressive strain. rUbTI 7>4 c bgðajBI karEbgEckkugRtaMg (stress distribution) sMrab;muxkat;Gt;eRbH. vaRtUv)anEkERbedIm,Ibgðajfakug RtaMgTajenAsrésxageRkamRbsinebImuxkat;enaHmansñameRbH. kMeNagRbsiT§PaB (effective curvature) φe enAkñúgsmIkar 7.4 (b) eRkaykMhatbg;CaplbUk EdleRbIsBaØasmRsbrvagkMeNagedIm (initial curvature) φi CamYynwgbMErbMrYlrbs;kMeNag dφl Edl bNþalBIkMhatbg;eRbkugRtaMgedaysar creep/ relaxation nig shrinkage nigbMErbMrYlrbs;kMeNag dφ2 EdlbNþalmkBI creep énebtugeRkamGMeBIkMlaMgeRbkugRtaMg. φe = φi + dφ1 + dφ2 (7.7) EdlBImUldæanénemkanicrbs;sMPar³ (basic mechanics of materials) M φ= (7.8a) Ec I c sMrab; primary moment M1 = Pee dUcenHeyIg)an Pe e φ= (7.8b) Ec I c edayCMnYsvaeTAkñúgsmIkar 7.5 sMrab;FñwmTMrsamBaØEdlmancMNakp©itebs;EdkeRbkugRtaMgefr eK)an φl 2 δc = (7.9a) 8 smIkarTUeTAsMrab;PaBdabEdleRbIkMeNagRtUv)anesñIeLIgeday Tadros manrag l2 2 δ = φc − (φe − φc ) a (7.9b) 8 6 Edl φc = kMeNagRtg;kNþalElVg φe = kMeNagRtg;TMr a = )a:ra:Em:RtRbEvgCaGnuKmn_én tendon profile Camber, Deflection and Crack Control 416
  11. 11. Department of Civil Engineering NPIC #> PaBdabPøam²énFñwmTMrsamBaØEdlrgeRbkugRtaMgedayEdkeRbkugRtaMgrag)a:ra:bUl Immediate Deflection of Simply Supported Beam Prestressed with Parabolic Tendon ]TahrN_ 7>2³ kMNt;PaBdabkNþalElVgPøam²rbs;FñwmEdlbgðajenAkñúgrUbTI 7>5 EdlrgeRbkug RtaMgedayEdkeRbkugRtaMgrag)a:ra:bUlEdlmancMNakp©itGtibrma e enAkNþalElVg nigkMlaMgeRbkug RtaMgRbsiT§PaB Pe . eRbI elastic weight method nig equivalent weight method. ElVgrbs;FñwmKW l nigPaBrwgRkajrbs;vaKW Ec I c . dMeNaHRsay³ Elastic weight method BIsmIkar 7.5 (b) 1 ⎛ P el 2 ⎞ P el R 'e = ⎜ e × ⎟ = e 2 ⎜ Ec I c 3 ⎟ 3Ec I c ⎝ ⎠ m:Um:g;EdlbNþalBI elastic weight We eFobcMNuc C kNþalElVgKW ⎛ l ⎞ ⎡ P el 2 ⎛ 3 l ⎞⎤ M c = δ c = R 'e ⎜ ⎟ − ⎢ e × ⎜ × ⎟ ⎥ ⎝ 2 ⎠ ⎣ Ec I c 6 ⎝ 8 2 ⎠ ⎦ PaBekag PaBdab nigkarRKb;RKgsñameRbH 417
  12. 12. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa 1 ⎛ Pe el 2 3Pe el 2 ⎞ 5Pe el 2 = ⎜ − ⎟= Ec I c ⎜ 6 48 ⎟ 48Ec I c ⎝ ⎠ 5 Pe el 2 enaH δc = 48 Ec I c (a) Equivalent weight method BIemeronTI1 equivalent balancing load intensity W Edl)anBIsMBaFén parabolic tendon eTAelIebtugKW 8 Pe e W = l2 BImUldæanénemkanicrbs;sMPar³ PaBdabkNþalElVgrbs;TMrsmBaØEdlrgbnÞúkBRgayesμIKW 5 wl 4 δc = (b) 384 Ec I c edayCMnYsGaMgtg;sIuetbnÞúk W eTAkñúgsmIkarxagelI eyIg)an 5 Pe el 2 δc = (c) 48 Ec I c dUckarrMBwgTuk eyIgTTYl)ansmIkar (c) nigsmIkar (a) sMrab;PaBdabkNþalElVgrbs;Fñwm. rUbTI 7>6 bgðajBIsmIkarPaBdabkNþalElVgsMrab;FñwmTMrsamBaØ Edlb®gÁb;elIsmIkar kMlaMgkat; nigsmIkarm:Um:g;sMrab;FñwmCab;EdleGayenAkñúgrUbTI 6>12. K> muxkat;eRbH Cracked Sections !> viFIKNnam:Um:g;niclPaBRbsiT§PaB Effective-moment-of-inertia Computation Method enAeBlEdlGgát;eRbkugRtaMgrgbnÞúkelIs (overload) b¤enAkñúgkrNIGgát;eRbkugRtaMgedayEpñk EdleKGnuBaØateGayman limited controlled cracking enaHkareRbI gross moment of inertia I g nwg pþl;nUvkar)a:n;sμan camber b¤PaBdabrbs;FñwmeRbkugRtaMgmanlkçN³esÞIrminRtwmRtUvtamPaBCak; Esþg. tamlkçN³RTwsþI eKKYreRbIm:Um:g;niclPaBrbs;muxkat;EdleRbH (cracked moment of inertia) I cr sMrab; muxkat;EdlekItmansñameRbH enAxN³EdleKeRbI gross moment of inertia I g sMrab;muxkat;FñwmenA cenøaHmuxkat;mansñameRbH. b:uEnþ eBlxøHeKminRtUvkarPaBeFVIeGayRbesIreLIgtamry³kareFVIplbUk énkMeNInPaBdabtambeNþayFñwmeT edaysareKBi)akkñúgkarkMNt;PaBdabeGay)ansuRkit. dUcenH eKGacykm:Um:g;niclPaBRbsiT§PaB I e CatMélmFümtambeNþayElVgrbs; simply supported bonded tendon beam/ vaCaviFIEdlbegáIteLIgeday Branson. eyagtamviFIenHeyIg)an³ Camber, Deflection and Crack Control 418
  13. 13. Department of Civil Engineering NPIC 3 ⎛M ⎞ I e = I cr + ⎜ cr ⎜M ⎟ ( ) ⎟ I g − I cr ≤ I g (7.10a) ⎝ a ⎠ eKGacsresrsmIkar 7.10a kñúgTMrg; PaBekag PaBdab nigkarRKb;RKgsñameRbH 419
  14. 14. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa ⎛M ⎞ 3 ⎡ ⎛M ⎞ 3⎤ I e = ⎜ cr ⎜M ⎟ I g + ⎢1 − ⎜ cr ⎟ ⎟ ⎥ I cr ≤ I g (7.10b) ⎝ a ⎠ ⎢ ⎜ Ma ⎟ ⎥ ⎣ ⎝ ⎠ ⎦ eKGacCMnYspleFob (M cr / M a ) BIsmIkar 7.4b eTAkñúgsmIkar 7.10 a nig b edIm,ITTYl)an m:Um:g;niclPaBRbsiT§PaB M cr ⎛ f − fr ⎞ = 1 − ⎜ tl ⎜ f ⎟ ⎟ (7.11) Ma ⎝ L ⎠ Edl m:Um:g;niclPaBrbs;muxkat;EdleRbH BIsmIkar 7.13 xageRkam I cr = I g = m:Um:g;niclPaBrbs;muxkat;TaMgmUl (gross moment of inertia) cMNaMfa TaMg M cr nig M a Cam:Um:g;KμanemKuNEdlbNþalmkEtBIbnÞúkGefrb:ueNÑaH EdleKyk M cr CacMENkénm:Um:g;EdlekItBIbnÞúkGefrEdlbgáeGaymansñameRbH. dUcenH m:Um:g;niclPaBRbsiT§- PaB I e enAkñúgsmIkar 7.10a nig b GaRs½ynwgm:Um:g;Gtibrma M a tambeNþayElVgEdlCab;Tak;Tg nwglT§PaBTb;m:Um:g;eRbH M cr rbs;muxkat;. enAkñúgkrNIFñwmCab;Gt;eRbHEdlmancugsgçagCab; I e mFüm = 0.70 I m + 0.15(I e1 + I e 2 ) (7.12a) sMrab;FñwmCab;Gt;eRbHEdlmancugmçagCab; I e mFüm = 0.85I m + 0.15(I cont.end ) (7.12b) Edl I m Cam:Um;g;niclPaBénmuxkat;kNþalElVg ehIy I e1 nig I e2 Cam:Um:g;niclPaBénmuxkat;cug. @> Bilinear Computation Method kñúgTMrg;RkaPic/ bilinear moment-deflection relationship sMrab;tMbn;TI I niigtMbn;TI II Edl manerobrab;enAkñúgcMnuc 3>k EdlGnuelameTAtam ACI Code. düaRkamsMrab;tMbn; I g nig I cr RtUv)anbgðajenAkñúgrUbTI 7>7. m:Um:g;niclPaBRbsiT§PaB I e rbs; Branson eGaynUvPaBdabPøam² srubmFüm δ tot = δ e + δ cr EdlBIxagedIm. ACI Code TamTarnUvkarKNnaPaBdabenAtMbn;EdleRbHenAkñúg bonded tendon beam KWEp¥k elI transformed section enARKb;eBlEdlkugRtaMgTaj ft enAkñúgebtugFMCag 6 f 'c . dUcenH eKGac kMNt; δ cr enAkñúgrUbTI 7>7 edayeRbI I cr transformed EdleRbIkarcUlrYmrbs;EdkBRgwgenAkñúg bilinear method kñúgkarKNnaPaBdab. eKGacKNnam:Um:g;niclPaBrbs;muxkat;EdleRbHeday PCI approach sMrab;Ggát;rgeRbkugRtaMgeBjtamsmIkarxageRkam Camber, Deflection and Crack Control 420
  15. 15. Department of Civil Engineering NPIC ( ) I cr = n p A ps d 2 1 − 1.6 n p ρ p p (7.13a) Edl n p = E ps / Ec . RbsinebIeKeRbIEdkFmμtaeGayrgkugRtaMgTaj ¬enAkñúgGgát;eRbkugRtaMgeday Epñk¦ eKGacEkERbsmIkar 7.13 eGayeTACa I cr = (n p A ps d 2 + ns As d 2 )(1 − 1.6 n p ρ p + ns ρ ) p (7.13b) Edl ns = Es / Ec sMrab;EdkFmμta/ d = kMBs;RbsiT§PaBeTAdl;TIRbCMuTMgn;rbs;EdkFmμta b¤Edkminrg eRbkugRtaMg (nonprestressed strand steel). #> viFIkMeNInm:Um:g;-kMeNag Incremental Moment-Curvature Method eKGacKNnam:Um:g;niclPaBrbs;muxkat;EdleRbHkan;EtsuRkitBITMnak;TMngrvagm:Um:g;nigkMeNag (moment-curvature relationship) tambeNþayElVgFñwm nigBIkarEbgEckkugRtaMg nigbMErbMrYlrag eFobelIkMBs;énmuxkat;eRKaHfñak;. dUcbgðajenAkñúgrUbTI 7>4(d) sMrab; strain ε cr enAeBlmansñam eRbHdMbUg ε cr M φcr = = (7.14) c Ec I cr Edl ε cr Ca strain enARtg;srésrgkarsgát;rbs;ebtugxageRkAbMput nig M Cam:Um:g;srubEdlrYmbBa©Úl TaMg prestressing primary moment M1 eFobnwgTIRbCMuTMgn;rbs;muxkat;EdlBicarNa. eKGac sresrsmIkar 7.14 eLIgvij enaHeyIg)an PaBekag PaBdab nigkarRKb;RKgsñameRbH 421
  16. 16. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa Mc Mc I cr = = (7.15) Ec ε cr f Edl f CakugRtaMgrbs;ebtugenARtg;srésrgkarsgát;rbs;muxkat;. Flowchart sMrab;KNnaPaBdabPøam² nigsMrab;sg;düaRkamTMnak;TMngrvagm:Um:g; nigkMeNag manbgðajenAkñúgrUbTI 7>8. Camber, Deflection and Crack Control 422
  17. 17. Department of Civil Engineering NPIC PaBekag PaBdab nigkarRKb;RKgsñameRbH 423
  18. 18. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa Camber, Deflection and Crack Control 424
  19. 19. Department of Civil Engineering NPIC 4> PaBdabry³eBlxøIeRkamGMeBIbnÞúkeFVIkar Short-Term Deflection at Service Load k> ]TahrN_ 7>3 Non-Composite Uncracked Double T-Beam Deflection kMNt;PaBdabeGLasÞicPøam² ¬ry³eBlxøI¦ srubén 12 DT 34 Beam enAkñúg]TahrN_ 4>1 EdleRbI (a) viFIm:Um:g;niclPaBEdlGacGnuvtþ)an I g b¤ I e / (b) viFIkMeNInm:Um:g;-kMeNag (incremental moment-curvature method). FñwmrgnUv superimposed service load 1,100 plf (16.1kN / m ) nig superimposed dead load 100 plf (1.5kN / m ) . FñwmenHrgnUv bonded pretensioned CamYynwg stress- relieved strands 7-wire-270ksi ¬ f pu = 270ksi = 1,862MPa ¦ Ggát;p©it 1 / 2in.(12.7 mm ) cMnYn 16 ¬ Aps = 2.448in 2 ¦. enAkñúg]TahrN_enHminKitBIkarcUlrYmrbs;EdkminrgeRbkugRtaMgenAkñúgkarKNna m:Um:g;niclPaBeT. snμt;faeKTaj (jack) strand rhUtdl; 0.70 f pu Edl)anBIkMlaMgeRbkugRtaMgedIm Pi = 462,672lb . eRbkugRtaMgRbsiT§PaB Pe = 379,391lb ekItmanenAeBlrgkarGnuvtþbnÞúkelIkdMbUg Kw 30éf¶eRkayeBldMeLIg nigminKitbBa©ÚlkMhatbg;GaRs½ynwgeBlTaMgGs;. PaBekag PaBdab nigkarRKb;RKgsñameRbH 425
  20. 20. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa Tiinñn½y³ (a) lkçxN³FrNImaRt (geometrical properties) ¬rUbTI 7>9¦ Ac = 978in.2 (6,310cm 2 ) I c = 86,072in.4 (3.59 ⋅10 6 cm 4 ) S b = 3,340in.3 (5.47 ⋅10 6 cm 3 ) S t = 10,458in.3 WD = 1,019 plf bnÞúkpÞal; WSD = 100 plf (1.46kN / m ) WL = 1,100 plf (16.05kN / m ) ec = 22.02in. ee = 12.77in. Camber, Deflection and Crack Control 426
  21. 21. Department of Civil Engineering NPIC cb = 25.77in. ct = 8.23in. ( A ps = 16 × 0.153 = 2.448in.2 15.3cm 2 ) Pi = 462,672(2,058kN ) enAeBlepÞr Pe = 379,391lb(1.688kN ) (b) lkçN³sMPar³ (material properties) V / S = 2.39in. RH = 70% f 'c = 5,000 psi f 'ci = 3,750 psi f pu = 270,000 psi (1,862MPa ) f pi = 189,000 psi (1,303MPa ) f pe = 154,980 psi (1,067 MPa ) f py = 230,000 psi E ps = 28.5 ⋅10 6 psi (196GPa ) (c) kugRtaMgGnuBaØat (allowable stresses) f ci = 2,250 psi f c = 2,250 psi f ti = 184 psi ¬kNþalElVg¦ f t = 849 psi ¬kNþalElVg¦ dMeNaHRsay (a) !> kugRtaMgenARtg;muxkat;kNþalElVg eyIgmancMNakp©itkNþalElVg ec = 22.02in.(559mm ) m:Um:g;Bt;ekIteLIgedaysarbnÞúkpÞal;xøÜnGtibrma 1,019(60 )2 MD = × 12 = 5,502,600in. − lb 8 (a) enAeBlepÞr (at transfer) PaBekag PaBdab nigkarRKb;RKgsñameRbH 427
  22. 22. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa kugRtaMgEdlRtUv)anKNnaKW BIsmIkar 4.1a Pi ⎛ ec ct ⎞ M D ft =− ⎜1 − 2 ⎟ − t Ac ⎝ r ⎠ S 462,672 ⎛ 22.02 × 8.73 ⎞ 5,502,600 =− ⎜1 − ⎟− 978 ⎝ 88.0 ⎠ 10,458 = +501 − 526 = −25 psi (C ) < f t = +184 psi(T ) / O.K. Pi ⎛ ec cb ⎞ M D fb = − ⎜1 + 2 ⎟ + Ac ⎝ r ⎠ Sb 462,672 ⎛ 22.02 × 25.77 ⎞ 5,502,600 =− ⎜1 + ⎟+ 978 ⎝ 88.0 ⎠ 3,340 = −3,524 + 1,647 = −1,877 psi (C ) < −2,250 psi / O.K. (b) enAeBlrgbnÞúkeFVIkar (service load) 100(60 )2 12 M SD = = 540,000in. − lb(61kN .m ) 8 1,100(60 )2 12 ML = = 5,940,000in. − lb(672kN .m ) 8 edaysarbnÞúkGefr ft = 5,940,000 10,458 = −568 psi (C ) edaysarbnÞúkGefr fb = 5,940,000 3,340 = 1,778 psi (T ) 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 ⎛ ec ct ⎞ M T Pe 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 = −2,250 psi O.K. BIsmIkar 4.3b Pi ⎛ ec cb ⎞ M T fb = − ⎜1 + 2 ⎟ + Ac ⎝ r ⎠ Sb 379,391 ⎛ 22.02 × 25.77 ⎞ 11,982,600 =− ⎜1 − ⎟+ 978 ⎝ 88.0 ⎠ 3,340 = −2,689 + 3,587 = +698 pis (T ) < 849 psi O.K. Camber, Deflection and Crack Control 428
  23. 23. Department of Civil Engineering NPIC eKGnuBaØateGayeRbI gross moment of inertia I g sMrab;karKNnaPaBdab. kñúgkrNIEbbenH eKGacyk effective moment of inertia I e esμInwg I g . RbsinebIeRbobeFobCamYy modules of rupture f r = 7.5 f 'c = 7.5 5,000 = 530 psi eKrMBwgfanwgmansñameRbHtUc² (minor cracking) ehIyedIm,IlkçN³suvtßiPaB (conservative) eKGnuBaØateGayRbIemKuN 7.5 . @> kugRtaMgenARtg;muxkat;TMr BIsmIkar 4.1 f ti = 6 f 'ci = 6 3,750 = 367 psi f t = 12 f 'c = 12 5,000 = 849 psi ee = 12.77in. eFVIdUcKñaenAkñúgCMhanénkarKNnakugRtaMgRtg;muxkat;kNþalElVg edayeRbI M = 0 CMnYskñúg smIkarkñúgral;CMhanxagelI. karRtYtBinitükugRtaMgmuxkat;TMrenAeBlepÞreGaynUvkugRtaMgEdlman tMéltUcCagkugRtaMgGnuBaØat O.K.. taragsegçbénkugRtaMgsrés ( psi ) #> KNnaPaBdab nigPaBekag (camber) enAeBlepÞr BI basic mechanics of materials b¤BIsmIkar 7>6 sMrab; a = l / 2 camber enAkNþalElVg EdlbNþalBI single harp b¤ depression énEdkeRbkugRtaMgKW Pec l 2 P(ee − ec )l 2 δ ↑= + 8EI 24 EI dUcenH Eci = 57,000 f 'ci = 57,000 3,750 = 3.49 ⋅10 6 psi (24.1MPa ) Ec = 57,000 f 'c = 57,000 5,000 = 4.03 ⋅10 6 psi (27.8MPa ) 462,672 × 22.02 × (60 × 12 )2 462,672 × (12.77 − 22.02)(60 × 12)2 δ pi ↑= + 8 × 3.49 ⋅10 6 × 86,702 24 × 3.49 ⋅10 6 × 86,072 PaBekag PaBdab nigkarRKb;RKgsñameRbH 429
  24. 24. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa = −2.20 + 0.31 = −1.89in.(48mm ) ↑ PaBdabeLIgelIenH (camber) KWbNþalEtmkBIkMlaMgeRbkugRtaMgb:ueNÑaH. bnÞúkpÞal;enAkñúg 1in. KW 1,019 / 12 = 84.9lb / in. ehIyPaBdabEdlbNþalBIbnÞúkpÞal;KW δ D ↓= 5wl 4 / 384 EI 5 × 84.9(60 × 12)4 δD = = 0.99in. ↓ 384 × 3.49 ⋅10 6 × 86,072 dUcenH net camber enAeBlepÞrKW − 1.89 ↑ +0.99 ↓= −0.90in. ↑ (25mm) $> KNnaPaBdabPøam²srubeRkamGMeBI service load énmuxkat;Gt;eRbH (a) PaBdabedaysar superimposed dead load edayeRbI Ec = 4.03 ⋅106 psi Eci ⎛ 100 ⎞ ⎛ 3.49 ⎞⎛ 100 ⎞ δ SD = 0.99 ⎜ ⎟ = 0.99⎜ ⎟⎜ ⎟ = 0.08in.(2.0mm ) ↓ Ec ⎝ 1,019 ⎠ ⎝ 4.03 ⎠⎝ 1,019 ⎠ (b) PaBdabedaysarbnÞúkGefr 5wl 4 5(1100 )(60 × 12)4 1 δL = = × = 0.93in. ↓ 384 Ec I c 384 × 4.03 ⋅10 × 86,072 12 6 esckþIsegçbén camber nigPaBdabry³eBlxøIeRkamGMeBI service load mandUcxageRkam³ camber edaysarkMlaMgeRbkugRtaMgdMbUg = 1.89in.(48mm ) ↑ PaBdabedaysarbnÞúkpÞal; = 0.99in.(25mm) ↓ PaBdabedaysar superimposed dead load = 0.08in.(2mm) ↓ net deflection enAeBlepÞr = −1.89 + 0.99 = −0.90in. ↑ RbsinebIeKBicarNaPaBdabedaysarkMhatbg;BIdMNak;epÞrrhUtdl;ry³eBl 30éf¶ enaH camber RtUv)ankat;bnßy)an ⎛ 462,672 − 379,391 ⎞ ⎛ 0.34 ⎞ = 1.89⎜ ⎟ = 1.89⎜ ⎟ = 0.34in. ↓ ⎝ 462,672 ⎠ ⎝ 462,672 ⎠ dMeNaHRsay (b) dMeNaHRsaytamviFIkMeNInm:Um:g; nigkMeNag (incremental moment curvature method) ΔP = Pi − Pe = 462,672 − 379,391 = 83,281lb(370kN ) bMErbMrYlrageFobedaysarkMlaMgeRbkugRtaMgenAeBlepÞr enAry³eBl 7éf¶ Eci = 3.49 ⋅106 psi (i) edaysarkMlaMgeRbkugRtaMg Pi kNþalElVg³ Camber, Deflection and Crack Control 430
  25. 25. Department of Civil Engineering NPIC f t = +501 psi f b = −3,524 psi 501 εc = t = +144 ⋅10 − 6 in. / in. 3.49 ⋅10 6 ε cb = −1,010 ⋅10 −6 in. / in. elITMr³ f t = +92 psi f b = −2,242 psi ε e = 26 ⋅10 −6 in. / in. t ε et = −642 ⋅10 −6 in. / in. ¬1 psi = 6.895kPa ¦ (ii) edaysarkMlaMgeRbkugRtaMg nigbnÞúkpÞal; Pi + WD kNþalElVg³ f t = −25 psi ε c = −7.2 ⋅10 −6 in. / in. t f b = −1,877 psi ε cb = −537.8 ⋅10 −6 in. / in. TMr³ dUcKñanwgkrNI (i) bMErbMrYl strain EdlbNþalBIkMhatbg;eRbkugRtaMg − ΔP = 83,281lb Eci = 3.49 ⋅10 −6 psi muxkat;kNþalElVg Δf t = − (− ΔP ) ⎛1 − ect ⎞ = + 83,281 ⎛1 − 22.02 × 8.23 ⎞ = −90 psi(C ) ⎜ ⎟ ⎜ ⎟ Ac ⎝ r2 ⎠ 978 ⎝ 88.0 ⎠ − 90 Δε c = t = −26 ⋅10 − 6 in. / in. 3.49 ⋅10 6 Δf b = − (− ΔP ) ⎛1 + ecb ⎞ = 83,281 ⎛1 + 22.02 × 25.77 ⎞ = +634 psi(T ) ⎜ ⎟ ⎜ ⎟ Ac ⎝ r2 ⎠ 978 ⎝ 88.0 ⎠ 634 Δε cb = = +182 ⋅10 − 6 in. / in. 3.49 ⋅10 6 muxkat;Rtg;TMr Δf t = − (− ΔP ) ⎛1 − ect ⎞ = 83,281 ⎛1 − 12.77 × 8.23 ⎞ = −16.5 psi(C ) ⎜ ⎟ ⎜ ⎟ Ac ⎝ r2 ⎠ 978 ⎝ 88.0 ⎠ PaBekag PaBdab nigkarRKb;RKgsñameRbH 431
  26. 26. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa − 16.5 Δε e = t = −5 ⋅10 − 6 in. / in. 3.49 ⋅10 6 Δf b = − (− ΔP ) ⎛1 + ecb ⎞ = + 83,281 ⎛1 + 12.77 × 25.77 ⎞ = 404 psi(T ) ⎜ ⎟ ⎜ ⎟ Ac ⎝ r2 ⎠ 978 ⎝ 88.0 ⎠ + 404 ΔEbe = = +116 ⋅10 − 6 in. / in. 3.49 ⋅10 6 edaybUk strain enAeBlepÞrbEnßmBIelI strain EdlbNþalBIkMhatbg;eRbkugRtaMgeGaykar EbgEck strain eRkamGMeBI service load eRkayeBlrgEtkMlaMgeRbkugRtaMg dUcbgðajenAkñúgrUbTI 7>10. BIrUbTI 7>10 kMeNagenAkNþalElVg − 828 − 118 φc = × 10 − 6 = −27.82 ⋅10 − 6 rad / in. 34 kMeNagenARtg;TMr − 526 − 21 φe = × 10 − 6 = −16.09 ⋅10 − 6 rad / in. 34 BIrUbTI 7>6/ sMrab; a = l / 2 / camber rbs;FñwmEdlbNþalEtBI Pe KW Camber, Deflection and Crack Control 432
  27. 27. Department of Civil Engineering NPIC ⎛ l2 ⎞ 2 δ e ↑= φc ⎜ ⎟ + (φe − φc ) l ⎜ ⎟ ⎝8⎠ 24 = −27.82 ⋅ 10 −6 (60 × 12)2 + (− 16.09 + 27.82) ⋅10 −6 (60 × 12)2 8 24 = −1.80 + 0.25 = −1.55in. ↑ (39mm ) (camber) EdlRsedogKñaeTAnwg (− 1.89 + 0.34) = −1.55in. ↑ eRkayeBlxatbg;enAkñúgdMeNaHRsay elIkmun. PaBdabEdlbNþalmkBIbnÞúkpÞal; WD / superimposed dead load WSD nigbnÞúkGefr WL KWRsedogKñanwgdMeNaHRsayelIkmun. cMNaMfatMélPaBEdl)anBIkarKNnaxusBItMélPaBdabCak;EsþgcenøaHBI 20% eTA 40% eday sar)a:ra:Em:RtCaeRcInEdlCHT§iBldl;m:UDulrbs;ebtug. dUcenH eKKYryktMélEdlKNnaenARKb;CM- hanTaMgGs;rbs;dMeNaHRsaybIxÞg;eRkayek,ósedIm,IkMurGayvaCHT§iBlxøaMgdl;lT§plcugeRkay. 5> PaBdabry³eBlxøIrbs;FñwmeRbkugRtaMgEdleRbH Short-Term Deflection of Cracked Prestressed Beams k> PaBdabry³eBlxøIrbs;FñwmenAkñúg]TahrN_ 7>3 RbsinebImuxkat;maneRbH Short-Term Deflection of Cracked Prestressed Beam in Example 7.3 if cracked ]TahrN_ 7>4³ edaHRsay]TahrN_ 7>3 eday (a) bilinear method, (b) viFIm:Um:g;RbsiT§PaBsMrab; lkçxNÐkugRtaMgTaj fb = 750 psi ¬EdlkugRtaMgTajmantMélFMCagm:UDuldac; f r = 7.5 f 'c = 530 psi ¦ eRkamGMeBI service load enAkNþalElVgRtg;srésxageRkamCMnYseGay f b = −56 psi(C ) enAkñúg]TahrN_elIkmun. snμt;fa net beam camber EdlbNþalBIkMlaMgeRbkugRtaMg nigbnÞúkpÞal;KW δ = 0.95in. . dMeNaHRsay³ Net tensile stressbnÞab;BI first cracking load Rtg;m:UDuldac;KW f net = fb − f r = 750 − 530 = +220 psi (T ) . BIrUbTI 7>3/ kugRtaMgTajEdlbNþaledaysarEtbnÞúkGefrenARtg;srésxageRkamKW + 1,778 psi . enAeBlenH edaysar WL = 1,100 plf cMENkénbnÞúkEdlmin)aneFVIeGaymankugRtaMg TajenARtg;srésxageRkamKW w1 = (1,778 − 220) ×1,100 = 964 plf 1,778 964 = = 80lb / in. 12 PaBekag PaBdab nigkarRKb;RKgsñameRbH 433
  28. 28. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa PaBdabEdlkMNt;eday I g énmuxkat;EdlGt;eRbHKW 5w1l 4 5 × 80(60 × 12)4 δg = = = 0.8in. ↓ (20mm ) 384 Ec I g 384 × 4.03 ⋅10 6 × 86,072 (a) bilinear method ( I cr = n p A ps d p 1 − 1.6 n p ρ p 2 ) E ps 28.5 ⋅ 106 np = = = 7.07 Ec 4.03 ⋅ 10 6 d p = ec + ct = 22.02 + 8.23 = 30.25in. > 0.8h = 27.2in. dp EdleRbI = 30.25in. nig Aps = 2.448in.2 enaH A ps 2.448 ρp = = = 0.0006 bd p 144 × 30.25 ( I cr = 7.07 × 2.448(30.25)2 1 − 1.6 7.07 × 0.0006 ) ( ) = 14,187in.4 5.9 ⋅ 105 cm 4 tulüPaBénbnÞúksrubEdleFVIeGaymuxkat;eRbHKW 1,100 − 964 w2 = = 11.3lb / in. 1,100 × 12 5w2l 4 5 × 11.3(60 × 12 )4 δ cr = = 384 Ec I cr 384 × 4.03 ⋅ 10 6 × 14,187 = 0.69in. ↓ (17mm ) dUcenH PaBdabsrubEdlbNþalBIbnÞúkGefr δ L = 0.80 + 0.69 = +1.49in. ↓ (38mm ) (b) viFIm:Um:g;niclPaBRbsiT§PaB (effective moment inertia moment) I e BIsmIkar 7.10b ⎛M ⎞ 3 ⎡ ⎛M ⎞ 3⎤ I e = ⎜ cr ⎜M ⎟ I g + ⎢1 − ⎜ cr ⎟ ⎟ ⎥ I cr ≤ I g ⎝ a ⎠ ⎢ ⎜ Ma ⎟ ⎥ ⎣ ⎝ ⎠ ⎦ BIsmIkar 7.11 ⎛ M cr ⎞ ⎛ f − ft ⎞ ⎜ ⎜M ⎟ = 1 − ⎜ tl ⎟ ⎜ f ⎟ ⎟ ⎝ a ⎠ ⎝ L ⎠ f tl =kugRtaMgsrubcugeRkay = +750 psi(T ) f r = m:UDuldac; = 530 psi )anBIelIkmun f L = kugRtaMgbnÞúkGefr = 1,778 psi Camber, Deflection and Crack Control 434
  29. 29. Department of Civil Engineering NPIC ⎛ M cr ⎞ ⎛ 750 − 530 ⎞ ⎜ ⎜M ⎟ = 1− ⎜ ⎟ ⎟ = 1 − 0.124 = 0.876 ⎝ a ⎠ ⎝ 1,778 ⎠ 3 ⎛ M cr ⎞ ⎜ ⎜M ⎟ = 0.67 ⎟ ⎝ a ⎠ I e = 0.67 × 86,072 + (1 − 0.67 )14,187 = 62,350in.4 GaMgtg;sIuetbnÞúkGefrsrub = 1,100 / 12 = 92lb / in. PaBdabEdlbNþalBIbnÞúkGefr 5 × 92(60 × 12 )4 δL = = 1.28in. ↓ (33mm ) 384 × 4.03 ⋅ 10 6 × 62,350 edayeRbobeFobCamYynwg 1.49in. enAkñúgdMeNaHRsay (a) eyIgyk δ L = +1.49in. ↓ . eRbI tMélenHsMrab; final net long-term deflection eRkayeBlxatbg;dUcGIVEdl)anerobCataragenA kúñg]TahrN_ 7>6. 6> karsg;düaRkamTMnak;TMngrvagm:Um:g; nigkMeNag Construction of Moment-Curvature Diagram ]TahrN_ 7>5³ cUrsg;düaRkamTMnak;TMngm:Um:g; nigkMeNagsMrab;muxkat;kNþalElVgrbs; bonded double-T beam enAkñúg]TahrN_ 7>3 sMrab;CMhanénkarekIneLIgnUvbMErbMrYlrageFobdUcxageRkam³ !> bMErbMrYlrageFobenAeBlepÞr f pi = 189,000 psi EdlbNþalEtBI Pi @> bMErbMrYlrageFobenAeBl f pe = 154,980 psi muneBlrgbnÞúkTMnaj #> enAeBldkkMlaMg (decompression) enARtg;TIRbCMuTMgn;rbs;EdkeRbkugRtaMg $> enAeBlkugRtaMgeFVIkardl;m:UDuldac; (modulus of rupture) %> muxkat;EdlmaneRbH. bMErbMrYlrageFob ε c1 enAsrésxagelI = 0.001in. / in. ^> muxkat;EdlmaneRbH. bMErbMrYlrageFob ε c1 enAsrésxagelI = 0.003in. / in. dMeNaHRsay³ !> dMNak;kalepÞrkMlaMgeRbkugRtaMg BITinñn½ysMrab;]TahrN_ 7>3 kugRtaMgkNþalElVgEdlbNþalmkEtBIkMlaMgeRbkugRtaMgKWman dUcxageRkam³ f t = +501 psi PaBekag PaBdab nigkarRKb;RKgsñameRbH 435
  30. 30. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa f b = −3,524 psi + 501 εc = t = +144 ⋅10 − 6 in. / in. 3.49 ⋅10 6 − 3,524 ε cb = = −1,010 ⋅10 − 6 in. / in. 3.49 ⋅10 6 φi = (ε cb −εc t = )(− 1,010 − 144) ×10 − 6 = −33.94 ⋅10 − 6 rad / in. h 34 BI]TahrN_ 7>3 m:Um:g;EdlbNþalmkBI Pi + M D KW M i = −462,672 × 22.02 + 5,502,600 = −4.69 ⋅10 6 in. − lb @> dMNak;kaleRkayeBlxagbg; enAkñúgdMNak;kaldkbnÞúkCabnþbnÞab; tMélrbs;m:Um:g; M g EdlbNþalmkBIbnÞúkTMnajRtUv)an rkedaykarkat;bnßykugRtaMgenAkñúgEdkeRbkugRtaMgrhUtdl;sUnü. BI]TahrN_ 4>1/ Pe = 379,391lb . dUcenH Pe 379,391 = = 0.82 Pi 462.672 kugRtaMg nigbMErbMrYlrageFobenAkNþalElVgeBlepÞrkMlaMgeRbkugRtaMg Pi KW f ct = +501 psi f cb = −3,524 psi ε c = +144 ⋅10 −6 in. / in. t ε cb = −1,010 ⋅10 −6 in. / in. kat;bnßybMErbMrYlrageFobrhUtdl;dMNak;kal Pe dUcxageRkam³ ε c = 0.82(144 ⋅10 −6 ) = 118 ⋅10 −6 in. / in. t ε cb = 0.82(− 1,010 ⋅10 −6 ) = −828 ⋅10 −6 in. / in. karBRgaybMErbMrYlrageFobnwgkøaydUcGVIEdlbgðajenAkñúgrUbTI 7>11 φ2 = (ε cb − ε ct ) = (− 828 − 118)10− 6 = −27.82 ⋅10− 6 rad / in. h 34 m:Um:g;EdlbNþalBIbnÞúkTMnaj M g = 0 cMNaMfakarEbgEckbMErbMrYlrageFobenAkñúgrUbTI 7>11 KWbNþalBIkMlaMgeRbkugRtaMg Pe . eRbI düaRkamTMnak;TMngkugRtaMg nigbMErbMrYlrageFobkñúgrUbTI 7>12 sMrab;EdkeRbkugRtaMg nigeRbIdüaRkam kñúgrUbTI 7>13 sMrab;ebtugedIm,IkMNt;kugRtaMgCak;Esþgtamry³ strain compatibility. Camber, Deflection and Crack Control 436
  31. 31. Department of Civil Engineering NPIC PaBekag PaBdab nigkarRKb;RKgsñameRbH 437
  32. 32. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa #> dMNak;kaleRkaydkbnÞúkCamYynwgkugRtaMgebtugsUnüenARtg;TIRbCMuTMgn;rbs;EdkeRbkugRtaMg BIrUbTI 7>12 bMErbMrYlrageFobénkardkbnÞúkenARtg;nIv:UTIRbCMuTMgn;EdkeRbkugRtaMgKW 26.01 ε decomp = −828 ⋅10 − 6 × = 723 ⋅10 − 6 in. / in. 26.01 + 3.75 f nig ε pe = Epe = 27.5,⋅9806 = 5,636 ⋅10 − 6 in. / in. 154 10 ps PaBRtUvKña (compatibility) rbs;bMErbMrYlrageFobTamTareGayEdkeRbkugRtaMgenAkñúg bonded beam manbMErbMrYlrageFobdUcKña dUcEdlkugRtaMgTajrbs;ebtugEdlB½T§CMuvijvaekIneLIgedIm,Ikat; bnßykugRtaMgsgát;enARtg;nIv:UTIRbCMuTMgn;rbs;EdkeRbkugRtaMgrhUtdl;esμIsUnü. dUcenH bMErbMrYlrageFobsrub ε pe = 5,636 ⋅10−6 + 723 ⋅10−6 = 6,359 ⋅10−6 in. / in. BIdüaRkamTMnak;TMngkugRtaMg nigbMErbMrYlrageFobenAkñúgrUbTI 7>12 kugRtaMg f pe = 177,00 psi dUcenH eyIg)an Pe EdlEksMrYl = 177,000 × 0.153 × 16 = 433,296 433,296 ⎛ 22.02 × 8.23 ⎞ f t EdlEksMrYl = − ⎜1 − ⎟ ≅ +469 psi (T ) 978 ⎝ 88.0 ⎠ + 469 εc = − t = 116 ⋅10 − 6 in. / in. 4.03 ⋅10 6 fb EdlEksMrYl = − 433,296 ⎛1 + 22.02 ×.0 .77 ⎞ ≅ −3,300 psi(C ) 978 ⎝ ⎜ 88 25 ⎟ ⎠ − 3,300 ε cb = = −819 ⋅10 − 6 in. / in. 4.03 ⋅10 6 M decomp × y M decomp × 22.02 f decomp = = = 2,884 psi Ic 86,072 M decomp = 2,884 × 86,072 22.02 ( = 11.27 ⋅10 6 in. − lb 1.27 ⋅10 6 N .m ) M decomp 11.27 ⋅10 6 ft = = = −1,078 psi (C ) St 10,458 net stress f t = −1,078 + 469 = −609 psi (C )(4.16 MPa ) − 609 εc = t = −151.1 ⋅10 − 6 in. / in. 4.03 ⋅10 6 11.27 ⋅10 6 11.27 ⋅10 6 fb = = = +3,374 psi (T ) Sb 3,340 net stress f b = +3,374 − 3,300 = +74 psi (T ) 74 ε cb = = +18.4 ⋅10 − 6 in. / in. 4.03 ⋅10 6 Camber, Deflection and Crack Control 438
  33. 33. Department of Civil Engineering NPIC φ decomp = (ε cb −εc t = ) (18.4 + 151.1) × 10 − 6 = +4.99 ⋅10 − 6 rad / in. h 34 M = 11.27 ⋅10 6 in. − lb rUbTI 7>14 eGaynUvkarBRgaykugRtaMg nigbMErbMrYlrageFobenAkúñgFñwmenHenAkñúgsßanPaBénkar dkbnÞúk. $> dMNak;kalm:UDuldac; f r = 7.5λ f 'c = 7.5 5,000 = 530 psi ⎡ P ⎛ ec ⎞⎤ M cr = S b ⎢7.5λ f 'c + e ⎜1 + 2b ⎟⎥ ⎣ Ac ⎝ r ⎠⎦ BIelIkmun GgÁTIBIrénsmIkarxagelIsMrab;m:Um:g;eGaykugRtaMg 3,300 psi . dUcenH M cr = 3,340(530 + 3,300) = 12.8 ⋅10 6 in. − lb net bottom concrete stress = m:UDuldac; f r sMrab;krNIenH = +530 psi(T ) + 530 ε cb = = +132 ⋅10 − 6 in. / in. 4.03 ⋅10 6 12.8 ⋅10 6 ft = = −1,224 psi (C ) 10,458 net stress f t = −1,224 + 469 = −755 psi (C ) − 755 εc = t = −187 ⋅10 − 6 in. / in. 4.03 ⋅10 6 φs = (ε cb −εc t = ) (132 + 187 ) ×10 − 6 h 34 = +9.38 ⋅10 −6 rad / in. PaBekag PaBdab nigkarRKb;RKgsñameRbH 439
  34. 34. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa %> dMNak;kalmuxkat;mansñameRbH/ ε c = 0.001in. / in. BIelIkmun/ ε pe = 6,359 ⋅10 −6 = 0.0064in. / in. . tamkarsakl,g nigEktMrUv snμt;kMBs;G½kS NWt c = 1.5in. BIxageRkamsrésxagelIbMputrbs;søab. ehIy Δε ps CabMErbMrYlrageFobbEnßmenAkñúg bonded prestressing strand EdlbNþalBI ε c = 0.001in. / in. enAsrésxagelIbMput ehIyBIRtIekaN dUc (similar triangle) enAkñúgrUbTI 7>15 Δε ps = (30.25 − 1.5) × 0.001 = 0.0192in. / in. 1.5 dUcenH srub = 0.0192 + 0.0064 = 0.0256in. / in. ε ps BIdüaRkamTMnak;TMngrvagkugRtaMg nigbMErbMrYlrageFobrbs;EdkeRbkugRtaMgenAkñúgrUbTI 7>12 kugRtaMgEdlRtUvnwgtMélbMErbMrYlrageFob ε ps srubKW f ps ≅ 260,000 psi nig A ps = 16 × 0.153 = 2.448in.2 dUcenH kMlaMgTaj T p = 260,000 × 2.448 = 636,480lb BIrUbTI 7>13/ f c = 3,000 psi RtUvKñanwg ε c = 0.001in. / in. . enaH kMlaMgsgát; Cc = (12 × 12 × 1.5)3,000 = 648,000 > T = 636,480lb dUcenH eKKYrkat;bnßykMBs;G½kSNWt. sakl,gelIkTIBIr snμt; c = 1.45in. . enaH Δε ps = (30.25 − 1.45) × 0.001 = 0.0199in. / in. 1.45 nig ε ps srub = 0.0199 + 0.0064 = 0.0263in. / in. Camber, Deflection and Crack Control 440
  35. 35. Department of Civil Engineering NPIC BIrUbTI 7>13/ f ps ≅ 255,000 psi / T p = 255,000 × 2.448 = 624,240lb nig Cc = (12 × 12 × 1.45)3000 = 624,400lb ≅ T p . dUcenH c Edlsnμt; = 1.45in. KW O.K. ⎛ 1.45 ⎞ M n = 624,240⎜ 30.25 − ⎟ = 18.4 ⋅10 in. − lb 6 ⎝ 2 ⎠ nigBIsmIkar 7.5d εu 0.001 φu = = = 690 ⋅ 10 − 6 rad / in. c 1.45 ^> dMNak;kalmuxkat;mansñameRbHeBj/ ε c = 0.003in. / in. (ultimate load) ε c = 0.003in. / in. CabMErbMrYlrageFobGtibrmaEdlGnuBaØateday ACI Code eRkamGMeBI ultimate load. snμt; f ps = 263,000 psi . enaH A ps f ps 2.448 × 263,000 a= = = 1.1in. 0.85 f 'c b 0.85 × 5,000 × 144 a 1.1 c= = = 1.38in. β1 0.8 BIrUbTI 7>15 30.25 − 1.38 ε ps = × 0.003 = 0.0628in. / in. 1.38 ε ps srub = 0.0628 + 0.0064 = 0.0692in. / in. BIdüaRkamTMnak;TMngrvagkugRtaMg nigbMErbMrYlrageFobenAkúñgrUbTI 7>13/ f ps ≅ f pu = 270,000 psi . dUcenH eRbI a ≅ 1.1in. EdleGay ⎛ a⎞ ⎛ 1.1 ⎞ M n = A ps f ps ⎜ d p − ⎟ = 2.448 × 270,000⎜ 30.25 − ⎟ ⎝ 2⎠ ⎝ 2 ⎠ PaBekag PaBdab nigkarRKb;RKgsñameRbH 441
  36. 36. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa = 19.6 ⋅10 6 in. − lb yk c ≅ 1.4in. εu 0.003 φu = = = 2,143 ⋅10 − 6 rad / in. c 1 .4 düaRkaménTMnak;TMngrvagm:Um:g; nigkMeNagRtUv)anbgðajenAkñúgrUbTI 7>16. düaRkamTMnak;TMng rvagbnÞúk nigPaBdabmanTMrg;RsedogKña ehIyeyIgGacsnñidæanvaecjBIdüaRkamTMnak;TMngrvagm:Um:g; nig kMeNag. 7> T§iBlénry³eBlEvgeTAelIPaBdab nigPaBekag Long-Term Effects on Deflection and Camber k> viFIemKuN PCI PCI Multipliers Method ACI Codepþl;nUvsmIkarxageRkamsMrab;)a:n;RbmaNemKuNGaRs½ynwgeBlsMrab;PaBdabén Ggát;ebtugeRbkugRtaMg³ ξ λ= (7.16) 1 + 50 ρ ' Edl ξ= emKuNGaRs½yeBlsMrab;bnÞúkGcié®nþy_ (sustained load) ρ ' = pleFobEdkrgkarsgát; λ = emKuNsMrab;PaBdabry³eBlEvgbEnßm kñúgTMrg;RsedogKña/ PCI multipliers method pþl;nUvemKuN C1 EdlKitT§iBlénry³eBlEvgenAkñúg Ggát;ebtugeRbkugRtaMg. Et C1 xusBI λ enAkñúgsmIkar 7.16 edaysarkarkMNt;PaBdab nig camber ry³eBlEvgenAkñúgGgát;eRbkugRtaMgmanlkçN³sμúKsμajCagedaysarktþadUcxageRkam³ !> T§iBlry³eBlEvgénkMlaMgeRbkugRtaMg nigkMhateRbkugRtaMg. @> karekIneLIgénersIusþg;rbs;ebtugeRkayeBlkMlaMgeRbkugRtaMgfycuHedaysarkMhatbg;. #> T§iBlénPaBdab nig camber kñúgGMLúgeBldMeLIg. edaysarktþaTaMgenH eKminGaceRbIsmIkar 7.16 eT. tarag 7>1 pþl;nUvemKuNénPaBdab nig camber Pøam²d¾smrmü RbsinebI camber nigPaB dabEdl)anKNnaBIdMbUgRtUv)anKitdac;edayELkBIKñaedIm,IKitBIT§iBlénkMhatbg;kMlaMgeRbkugRtaMg eTAelI camber. Camber, Deflection and Crack Control 442
  37. 37. Department of Civil Engineering NPIC nig Brason ENnaMfaeKGacTTYl)annUvkarkat;bnßyCaGcié®nþy_nUv camber ry³eBl Shaikh EvgedaykarbEnßmEdkminrgeRbkugRtaMg. enAkñúgkrNIenH eKGaceRbIemKuNEdlkat;bnßy C2 Edl eGayeday C1 + As / A ps C2 = (7.17) 1 + As / A ps Edl C1 = emKuNEdl)anBItarag 7>1 As = RkLaépÞrbs;EdkminrgeRbkugRtaMg A ps = RkLaépÞrbs;EdkrgeRbkugRtaMg x> viFIkMeNIntameBl Incremental Time-Steps Method viFIkMeNIntameBl (incremental time-steps method) KWQrelIbnSMénkarKNnaPaBdabCa- mYynwgkarKNnakMhatbg;edaysar creep, shrinkage nig relaxation EdlGaRs½ynwgeBl. kar KNnaBICIvitrbs;eRKOgbgÁúMEbgEckCaeRcIncenøaHeBlEdleRCIserIsedayQrelIeKalkarN_énEdn kMNt;rbs;bMErbMrYlrageFobebtugCak;lak; (specific concrete strain limits) dUcCabMErbMrYlrageFob Éktþa ε c1 = 0.001 nig ε c1 = 0.002in. / in. nig ultimate allowable strain ε c1 = 0.003in. / in. . eK PaBekag PaBdab nigkarRKb;RKgsñameRbH 443
  38. 38. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa KNnakarBRgaybMErbMrYlrageFob/ kMeNag nigkMlaMgeRbkugRtaMgsMrab;cenøaHeBlnImYy²CamYynwgkM- eNInénkMhatbg;edaysarbMErbMrYlrbs;karrYmmaD/ creep nig relaxation EdlekItmankñúgcenøaHeBl enaH. eKRtUveFVIkarKNnaenHCadEdl²sMrab;cenøaHkMeNInbnþbnÞab; nigkareFVIplbUkénkarKNnaenH pþl;eGayeyIgnUvPaBdabGaRs½ynwgeBlcugeRkaysMrab;muxkat;Cak;lak;NamYyenAtambeNþayElVg rbs;Fñwm. eKRtUveFIVkarKNnaenHsMrab;cMnYncMnucenAelIbeNþayElVgFñwmRKb;RKan; dUcCakNþalElVg nigcM- nucmYyPaKbYnedIm,IGackMNt;düaRkamTMnak;TMngrvagPaBdab nigkMeNageGaymanlkçN³suRkit. eKGacsmIkarTUeTAsMrab;mMuvilsrub (total rotation) enAcugbBa©b;éncenøaHeBldUcxageRkam³ t t Pi e x ex e φt = − + ∑ (Pn −1 − Pn ) − ∑ (C n − C n−1 )Pn −1 x (7.18a) Ec I c 0 Ec I c 0 Ec I c Edl Pi = kMlaMgeRbkugRtaMgedImmuneBlxatbg; e x = cMNakp©itrbs; tendon enARtg;muxkat;NamYytambeNþayElVg n −1 = cMnuccab;epþIméncenøaHeBl (time-step) n = cugbBa©b;én time-step Edl)anniyayBIxagelI C n−1 / C n = emKuN creep enAcMnuccab;epþIm nigcMnucbBa©b; erogKña én time-step NamYy Pn − Pn−1 = kMhatbg;eRbkugRtaMgenARtg;cenøaHeBlNamYyEdlekItBIktþaTaMgGs; Cak;Esþg eKeFVIkarKNnay:agl¥itl¥n;EbbenHEtenAkñúgkarkMNt;rkPaBdab nigPaBekagrbs; RbB½n§s<anEdlmanElVgEvg² dUcCas<anEdlsg;CakMNat;² (segmental bridge) EdlkardMeLIg nigkar pÁúMkMNat;s<anenaHTamTarnUvkar)a:n;RbmaNPaBdabeGaymanlkçN³suRkit. BIsmIkar 7.18a PaBdab srubenARtg;muxkat;NamYyKW δ x = φt kl 2 (7.18b) ]bmafaeKeRbIbMErbMrYlrageFobxageRkamBI]TahrN_ 7>7 xageRkamedIm,IbgðajBIkarKNna kMeNInénmMuvil (incremental rotation) nigmMuvilsrub (total rotation)³ ε ' n−1 = gross strain EdlbNþalEtmkBIkMlaMgeRbkugRtaMgenAsrésxagelIbMput Edl ε c = 144 ⋅ 10 −6 in. / in. ¬rUbTI 7>19¦ t ε b,n−1 = gross strain EdlbNþalEtBIkMlaMgeRbkugRtaMgenAsrésxageRkambMput Edl ε cb = −1,010 ⋅ 10 −6 in. / in. ¬rUbTI 7>19¦ Camber, Deflection and Crack Control 444
  39. 39. Department of Civil Engineering NPIC Δε CR ,n = t kMeNInénbMErbMrYlrageFobedaysar gross creep enAsrésxagelIbMput Edl Δε CRc = 127 ⋅10 −6 in. / in. ¬rUbTI 7>20¦ t Δε CRb, n = kMeNInénbMErbMrYlrageFobedaysar gross creep enAsrésxageRkambMput Edl Δε CRcb = −895 ⋅10 −6 in. / in. ¬rUbTI 7>20¦ Δε ps , n = karkat;bnßybMErbMrYlrageFobedaysarkMhatbg;eRbkugRtaMgEdlbgáedaykMlaMg creep ΔP, n ¬dUcCa 169 ⋅10 −6 in. / in. dUceXIjkñúgrUbTI 7>20¦ Net incremental creep strain Edlnwgpþl;nUv incremental rotation φn KW sMrab;srésxagelI Δε CR , net = (Δε CR , n − Δε tps , n ) t t (7.19a) sMrab;srésxageRkam ( Δε CRb, net = Δε CRb, n − Δε psb, n ) (7.19b) kMeNInénmMuvil (incremental rotation) KW Δε CR , net − Δε CRb, net t Δφ n = (7.19c) h PaBekag PaBdab nigkarRKb;RKgsñameRbH 445
  40. 40. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa ehIymMuvilsrub (total rotation) køayCa φT = φ n −1 + Δφn (7.20) bMErbMrYlénbMErbMrYlrageFob nigmMuvil (rotation) BI time-step n − 1eTA time-step n RtUv)anbgðajenA kñúgrUbTI 7>17. kareRCIserIscenøaHeBl (time interval) GaRs½ynwgPaBsuRkitEdleKcg;)anBIkarKNna camber. sMrab; time step nImYy² kMeNInbMErbMrYlrageFobEdlbNþalmkBI creep nigkarrYjmaD nig karxatbg;kMlaMgeRbkugRtaMgedaysar relaxation RtUv)anKNnadUcbgðajenAkñúg]TahrN_ 7>7 edIm,I TTYl)ankMeNInkMeNag (curvature increment) Δφ . bnÞab;mk eKnwgTTYl)antMélkugRtaMg bMErbMrYl rageFob nigkMeNagfμIenAcugbBa©b;éncenøaHeBl EdlbEnßm curvature increment Δφn eTAelIkMeNag srub φn −1 enARtg;cMnuccab;epþIméncenøaHeBlEdleKcg;)an dUceGayenAkñúgsmIkar 7.18. Cak;Esþg incremental time-step procedure manlkçN³Evg. eKGacTTYlPaBekagsrub (↑) b¤PaBdab (↓) EdlbNþalBIkMlaMgeRbkugRtaMgBIsmIkar 7.20 δ T = φT kl 2 (7.21) Edl k CaGnuKmn_énElVg nigragFrNImaRtrbs;muxkat; nigragFrNImaRtrbs;EdkeRbkugRtaMg. GñkGegÁtCaeRcIn)anesñInUvTMrg;epSg²sMrab;kar)a:n;RbmaNPaBdabbEnßmGaRs½yniwgeBl Δδ BITMnak;TMngrvagm:Um:g; nigkMeNag φ Edl)anEkERbsMrab; creep. TaMg Tadros nig Dilger ENnaMeGay eFVIplbUk modified curvature tambeNþayElVgrbs;Fñwm xN³Edl Naaman KitPaBdabry³eBl EvgedayeRbIkMeNagkNþalElVg nigkMeNagRtg;TMrRtg;cenøaHeBl t . Ca]TahrN_ smIkarrbs; Naaman sMrab; parabolic tendon KW l2 l2 Δδ (t ) = φ1 (t ) + [φ 2 (t ) − φ1 (t )] 8 48 Edl kMeNagkNþalElVgenAxN³ t φ1 (t ) = φ 2 (t ) = kMeNagelITMrenAxN³ t EdlkñúgenaH φ (t ) = E Mt )I ce ( c Edl Ece (t ) = m:UDulEdlEksMrYltameBl (time adjusted modulus) Ec (t1 ) E ce (t ) = 1 + KC c (t ) EdlkñúgenH Ec (t1 ) = m:UDulrbs;ebtugenAeBlcab;epþIméncenøaHeBl Cc (t ) = emKuN creep enAcugbBa©b;éncenøaHeBl Camber, Deflection and Crack Control 446
  41. 41. Department of Civil Engineering NPIC K> viFIRbhak;RbEhledaycenøaHeBl Approximate Time-Steps Method CaviFIEdlEp¥kelITMrg;y:agsmBaØEdlbUkbBa©ÚlKñanUvPaB- Approximate time-steps method dabTaMgGs;EdlbNþalBIemKuNGaRs½ynwgeBlepSg². RbsinebI Cu CaemKuN creep ry³eBlEvg eKGackMNt;kMeNageRkamGMeBIkMlaMgeRbkugRtaMgRbsiT§PaB Pe tamsmIkarxageRkam ⎛ P + Pe ⎞ e x + (Pi − Pe ) x − ⎜ i Pi e x e φe = ⎟ Cu (7.22) Ec I c Ec I c ⎝ 2 ⎠ Ec I c PaBdabcugeRkayeRkamGMeBI Pe KW ⎛ δi + δe ⎞ δ et = −δ i + (δ i − δ e ) − ⎜ ⎟Cu (7.23a) ⎝ 2 ⎠ ⎛δ +δ ⎞ b¤ δ et = −δ e − ⎜ i e ⎟Cu (7.23b) ⎝ 2 ⎠ edaybEnßmPaBdabedaysarbnÞúkpÞal; δ D nig superimposed dead load δ SD EdlrgT§iBleday- sar creep pþl;nUvkMeNInPaBdabcugeRkayGaRs½ynwgeBlEdlbNþalBIkMlaMgeRbkugRtaMg nigbnÞúk Gcié®nþy_ (sustained load) dUcxageRkam ⎛ δ + δe ⎞ Δδ = −δ e − ⎜ i ⎟Cu + (δ D + δ SD )(1 + Cc ) (7.24a) ⎝ 2 ⎠ ehIy net deflection srubcugeRkayEdlrYmbBa©ÚlTaMgPaBdabedaysarbnÞúkGefrKW ⎛ δi + δe ⎞ δ T = −δ e − ⎜ (7.24b) ⎟Cu + (δ D + δ SD )(1 + Cu ) + δ L ⎝ 2 ⎠ eKGackMNt;PaBdabkMritmFüm (intermediate deflection) edayCMnYs Ct eGay Cu enAkñúgsmIkar 7.24a nig b. Edl t 0.60 Ct = Cu (7.25) 10 + t 0.60 EdlkñúgenaH t 0.60 / (10 + t 0.60 ) CapleFob creep α Brason et al. )anesñInUvsmIkarxageRkamsMrab;TaykarekIneLIgénPaBdabGaRs½ynwgeBl Δδ énsmIkar 7.24 a dUcxageRkam³ ⎡ Δδ = − ⎢η + (1 + η ) k C ⎤δ + k C δ + K k C δ r t ⎥ i ( Pi ) r t i (D ) a r t i (SD ) (7.26) ⎣ 2 ⎦ Edl η = Pe / Pi Ct = emKuN creep enAxN³ t K a = emKuNEdlRtUvnwgGayurbs;ebtugeRkamGMeBIrbs; superimposed load PaBekag PaBdab nigkarRKb;RKgsñameRbH 447
  42. 42. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa = 1.25t −0.118sMrab; moist-cured concrete = 1.13t −0.095 sMrab; steam-cured concrete t = GayuenAeBlrgbnÞúk KitCaéf¶ k r = 1 / (1 + As / A ps ) Edl As / A ps << 1.0 = 1 sMrab;RKb;karGnuvtþTaMgGs; sMrab;kMeNInPaBdab (deflection increment) cugeRkay eKeRbI Cu CMnYseGay Ct enAkñúg smIkar 7.26. sMrab;FñwmminEmnsmas (noncomposite beams) PaBdabsrub δ T ,t køayCa ⎡ ΔP ⎤ δ T , t = −δ pi ⎢1 − + λ (k t Ct )⎥ + δ D [1 + k t Ct ] + δ SD [1 + K a k r Ct ] + δ L (7.27) ⎣ P o ⎦ Edl δp =PaBdabEdlbNþalBIkMlaMgeRbkugRtaMg ΔP = kMhateRbkugRtaMgsrubEdlminrYmbBa©ÚlkMhateRbkugRtaMgeGLasÞicedIm (initial elastic loss) λ = 1 − ΔP / 2 P0 EdlkñúgenaH kMlaMgeRbkugRtaMgenAeBlepÞreRkay elastic loss P0 = = Pi tUcCag elastic loss. sMrab;Fñwmsmas PaBdabsrubKW ⎡ ΔP ⎤ δ T = −δ pi ⎢1 − + K a k r Cu λ ⎥ + δ D [1 + K a k t Cu ] ⎣ P0 ⎦ ⎡ ΔP − ΔPc ⎤ + k r Cu (λ − αλ ')⎥ Ie + δ pi ⎢1 − I comp. ⎣ P0 ⎦ Ic ⎡ I ⎤ + (1 + α )k r Cu δ D + δ D ⎢1 + αk r Cu c ⎥ + δ df + δ L (7.28) I comp ⎢ ⎣ I comp ⎥ ⎦ Edl λ ' = 1 − (ΔPc / 2 P0 ) P0 = kMhatbg;eRbkugRtaMgenAxN³EdleKcak; composite topping slab edayminKitbBa©Úl initial elastic loss δ df =PaBdabedaysar differential shrinkage nig differential creep rvagmuxkat;cak;Rsab; nig composite topping slab = Fycs l 2 / 8 Ecc I comp sMrab;FñwmTMrsamBaØ ¬sMrab;FñwmCab; eRbIemKuNsmrmüenAPaKEbg¦ ycs = cMgayBITIRbCMuTMgn;rbs;muxkat;smaseTATMRbCMuTMgn;rbs; topping slab Camber, Deflection and Crack Control 448
  43. 43. Department of Civil Engineering NPIC kMlaMgEdl)anBI differential shrinkage nig differential creep F= Ecc = m:UDulénmuxkat;smas α = creep strain enAxN³ t EdlEckeday ultimate creep strain = t 0.60 / ( + t 0.60 ) . 10 Cakarsegçb visVkrRtUvvinicä½ykñúgkarkMNt;tMélm:UDulrbs;ebtug Ec eRkamGMeBIénkardak;bnÞúk epSg²eGay)ansuRkit edIm,ITTYl)antMélemKuN creep smrmü. X> karKNnaPaBdabedaykMuBüÚT½r Computer Methods for Deflection Evaluation eKGacKNnaPaBdabedayeRbIkmμviFIepSg²CaeRcIn. kMuBüÚT½rCYyvisVkry:ageRcInsMrab; time- step method. b:uEnþ eKRtUvcaMfaPaBdabeRkamGMeBIkardak;bnÞúkry³eBlxøI nigry³eBlEvgRtUv)anRKb; RKgedaylkçxNÐEdlGacekItmanCaeRcInEdlsßitenAkñúgvIFIénkarkMNt;PaBdabEtmYy. lkçxNÐTaMg enHTak;TgnwglkçN³énsarFatupSMrbs;ebtugEdlCHT§iBldl;PaBdab CaBiessPaBdabry³eBlEvg. dUcenH elIkElgkrNIs<anElVgEdlEvg dUcCa cable-stayed bridges dMeNIrkar nigviFIénkarKNnaPaB dabKYrmankMritERbRbYl ± 40% . karbBa©ÚllkçN³sMPar³eTAkñúgkmμviFIkMuBüÚT½rRtUveFVIeLIgedayRby½tñ RbEygbMputedayEp¥kelIlT§plBiesaFn_RbsinebIElVgrbs;eRKOgbgÁMúEvg. g> PaBdabrbs;Fñwmsmas Deflection of Composite Beams karKNnaPaBdabrbs;FñwmeRbkugRtaMgsmasmanlkçN³RsedogKñanwgkarKNnaPaBdabsMrab; noncomposite section Edr. viFIsaRsþKNnanwgkøayCasμúKsμajCagRbsinebIeKeRbI incremental time-steps method. CMhanbEnßméndMNak;kalsagsg;CaeRcInrbs;Ggát;cak;Rsab; nigsMrab; situ-cast top slab TamTarkarBicarNaénkarERbRbYlm:Um:g;niclPaBBImuxkat;cak;Rsab;eTAmuxkat;smasenA Rtg;dMNak;kalsmrmü. elIsBIenH PaBxusKñaénlkçN³rbs; shrinkage nigkMeNIncenøaHeBl (time- step increments) EdlbNþalBIPaBxusKñaéntMélrbs; shrinkage énmuxkat;cak;Rsab; nigkarbEnßm concrete topping )anbegáInPaBBi)akdl;dMeNIrkarKNna. CasMNagl¥ kareRbIkmμviFIkMuBüÚT½rsMrYlkar KNnaPaBdab nig camber rbs;Ggát;smas)any:ageRcIn. PaBekag PaBdab nigkarRKb;RKgsñameRbH 449
  44. 44. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa 8> PaBdabGnuBaØat Permissible Limits of Calculated Deflection ACI CodeTamTareGayPaBdabEdl)anKNnaRtUvbMeBjtMrUvkar serviceability énPaBdabGnuBaØatGtibrmasMrab;lkçxNÐrcnasm<½n§epSg²Edlmanerobrab;enAkñúgtarag 7>2. cMNaMfa T§iBlry³eBlEvgbgáeGayPaBdab nig camber ekIneLIgeTAtameBl ehIyeFVIeGayebtug nigEdk rgkugRtaMgelIs (overstress). PaBdabGnuBaØatrbs; AASHTO EdlbgðajenAkñúgtarag 7>3 manlkçN³suRkitCageday- sar karb:HTgÁícCalkçN³DINamic (dynamic impact) énbnÞúkcl½tenAelIElVgs<an. Camber, Deflection and Crack Control 450
  45. 45. Department of Civil Engineering NPIC xageRkamCa dMeNIrkarCaCMhan² (step-by-step procedure) sMrab;KNnaPaBdab³ !> kMNt;lkçN³rbs;ebtug edayrYmbBa©ÚlTaMgm:UDuleGLasÞicrbs;ebtug Ec / creep rbs;ebtug @> eRCIserIskMeNInry³eBl (time increment) EdlRtUveRbIenAkñúgkarKNnaPaBdab #> KNnakugRtaMgsrésebtugedaysarbnÞúkTaMgGs;TaMgenAEpñkxagelIbMput nigTaMgenAEpñk xageRkambMput $> KNnabMErbMrYlrageFobdMbUg (initial strains) ε ci enAsrésxagelI nigsrésxageRkam nig mMuvil (rotation) EdlRtUvKña k¾dUcCabMErbMrYl nigmMuvilbnþbnÞab;. eRbIsmIkar ε cbi − ε ci t φi = h ε −ε φe = cbe cte h ε −ε t φ = c cb h εu φu = c %> kMNt;karERbRbYlbMErbMrYlrageFobsrubenAkñúgEdkeRbkugRtaMgedaysar creep, shrinkage nig relaxation EdlGnuvtþCakMlaMg F enARtg;TIRbCMuTMgn;rbs;EdkeRbkugRtaMg. bnÞab;mk KNnakugRtaMgsrésebtugenAnIv:U cgs EdlbNþalBIkMlaMg F . ^> bEnßmlT§plénCMhan % eTAkñúglT§plénCMhan 3. &> GnuvtþdMeNIrkarKNnasMrab;RKb;cenøaHeBl nigbEnßmT§iBlén superimposed dead load. *> bEnßmPaBdabedaysarbnÞúkGefredIm,ITTYl)anPaBdabsrub δT . (> epÞógpÞat;faetI δT Edl)anKNnasßitenAkñúgEdnkMNt;GnuBaØatb¤Gt;. RbsinebImindUecñaHeT eFVIkarpøas;bþÚrmuxkat;. rUbTI 7>18 bgðajBI flowchart sMrab;karKNnaPaBdabeday approximate time-step method. PaBekag PaBdab nigkarRKb;RKgsñameRbH 451
  46. 46. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa Camber, Deflection and Crack Control 452
  47. 47. Department of Civil Engineering NPIC 9> karKNnaPaBdab nigPaBekagry³eBlEvgedayviFIemKuN PCI Long-Term Camber and Deflection Calculation by the PCI Multipliers Method ]TahrN_ 7>6³ edayeKeGay cUrKNnaPaBdab nigPaBekagrbs; boded double f pi = 189,000 psi T-beam enAkñúg]TahrN_ 7>3 eday PCI multiplers method nigepÞógpÞat;fatMélPaBdabbMeBjEdn kMNt;GnuBaØatrbs; ACI. RbsinebIFñwmRtUv)anrg post-tensioned snμt;fa f pi = 189,000 psi eRkay eBl anchorage losses nigeRkayeBllubbM)at; frictional losses edaykarTajBIcugsgçagrbs;cug Fñwm nigbnÞab;mkeKRtUvTajeLIgvijedIm,IFana net prestressing f pi = 189,000 psi munnwgdMeLIg. dUc Kña snμt;faGgát;EdlminEmnCaeRKOgbgÁúMrgbnÞúkEdlP¢ab;eTAnwgeRKOgbgÁúMrgbnÞúkminrgkarxUcxateday sarPaBdab ehIybnÞúkGefrmanlkçN³ transient. yk Ec = 4.03 ⋅106 psi sMrab;bnÞúkTaMgGs;enA kñúgkaredaHRsayenH. dMeNaHRsay³ I g = 86,072in.4 WD = 1,019 plf = 84.9lb / in. PaBekag PaBdab nigkarRKb;RKgsñameRbH 453
  48. 48. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa 5Wl 4 5 × 84.9(60 × 12)4 δD = = = 0.99in. ↓ (14mm ) 384 Eci I g 384 × 3.49 ⋅10 6 × 86,072 WSD = 100 plf = 8.3lb / in. 5 × 8.3(60 × 12 )4 δ SD = = 0.08in. ↓ (2.0mm ) 384 × 4.03 ⋅10 6 × 86,072 WL = 1,100 plf = 91.7lb / in. muxkat;Gt;mansñameRbH ¬emIl]TahrN_ 7>3¦ I e = I g = 86,072in.4 ( f t max < f r = 530 psi ) 5 × 91.7(60 × 12 )4 δL = = 0.93in. ↓ (24mm ) 384 × 4.03 ⋅ 10 6 × 86,072 RbsinebImuxkat;maneRbH eKeRbItMélRbsiT§PaBrbs; I e CMnYseGay I g . kareRbI PCI multi- plier method sMrab;KNnaPaBdabenAeBldMNak;kaldMeLIg (30éf¶) nigenAeBlmanPaBdabcugeRkay edaysar service-load ¬5qñaM¦ taragxageRkamnwgbgðajBItMélrbs;PaBdab nig camber ry³eBlEvg EdlTTYledayeRbIemKuN PCI enAkñúgtarag 7>1. RbsinebImuxkat;lkøayCamuxkat;smaseRkay eBldMeLIg eKeRbI I comp kñúgkarKNna δ L nig δ SD RbsinebIFñwmRtUv)anTl;kñúgGMLúgeBlcak; con- crete topping. ehIyRbsinebIeKeRbIEdkFmμta As enAkñúgFñwmeRbkugRtaMg eKRtUveRbIemKuNEdlkat; bnßy (reduced multiplier). emKuN C1 RtUv)ankat;bnßyedayemKuN C2 Edl C1 + As / A ps C2 = 1 + As / A ps Camber, Deflection and Crack Control 454
  49. 49. Department of Civil Engineering NPIC edaysarEdkFmμtaRKb;RKgkarrIkralFMénsñameRbHedaysarkarBt;begáageRkamGMeBIbnÞúkry³eBl Evg dUcenHPaBrwgRkajrbs;vaRtUv)anbegáIn. Ca]TahrN_ snμt;faeKeRbIEdk 3#5 enAkñúgFñwmeRbkug RtaMg As 3 × 0.31 = = 0.43 Aps 2.142 eyIgTTYl)an C2 = 2.01 Ca]TahrN_énkarEksMrYltMélEdlmanenAkñúgtarag 7>1 tMélrbs; camber edImnwgkøayCa 3.80in. ↑ CMnYseGay 4.63in. ↑ EdlbgðajenAkñúgtarag edayeKeRbIemKuN 2.01 CMnYseGayemKuN 2.45 . eK GaceFVIkarEksMrYlEdlmanlkçN³RsedogKñaeTAelIPaBdabTaMgGs;edayeRbIemKuNEksMrYlEdlRtUvKña. BItarag 7>4/ camber eRkayeBltMeLIg nigeRkayeBlrg superimposed dead load enAGayu 30éf¶ = 1.49in. ↑ (38mm ) . ehIy net camber cugeRkayeRkayGayu 5qñaM = 0.79in. ↑ (20mm ) / PaBdabedaysarbnÞúkGefr = 0.93in. ↓ (24mm) ehIyPaBdabGnuBaØat = l / 240 = (60 × 12) / 240 = 30in.(76mm ) > 0.79in. . enAkñúgkrNIenH RbsinebIeKsnμt;fabnÞúkGefrmanlkçN³ transient enaH vanwgRKb;RKan;. 10> karKNnaPaBdab nigPaBekagry³eBlEvgedayviFIkMeNIncenøaHeBl Long-Term Camber and Deflection Calculation by the Incremental Time-Steps Method ]TahrN_ 7>7³ edaHRsay]TahrN_ 7>6 tam incremental time-steps method edaysnμt;fa f pi = 189,000 psi ehIyeKsegÁteXIjfakMlaMgeRbkugRtaMgmankarekIneLIgenAeBlrgeRbkugRtaMg ¬7éf¶ bnÞab;BIcak;ebtug¦/ 30éf¶bnÞab;BIepÞr ¬kartMeLIg nigkardak; superimposed dead load rYceRsc¦/ 90 éf¶ nig 5qñaM. snμt;fa ultimate creep coefficient Cu = 2.35 sMrab;ebtug nig f py = 230,000 psi sMrab;EdkrgeRbkugRtaMgEdleRbIenAkñúgFñwm. sg;düaRkamTMnak;TMngrvagcamber CamYynwgeBl nigPaB dab CamYynwgeBledayeRbI Ec = 4.03 ⋅ 106 sMrab;RKb; incremental steps TaMgGs;kñúgkaredaHRsay enH edayelIkElgenAeBlepÞr Edl f 'ci = 3,750 psi . snμt;faFñwmenHCaFñwm post-tensioned. yk E ps = 27.5 ⋅ 10 6 psi . dMeNaHRsay³ kugRtaMg/ bMErbMrYlrageFob nigPaBdabxN³ Eci = 57,000 3,750 = 3.49 ⋅ 10 6 psi PaBekag PaBdab nigkarRKb;RKgsñameRbH 455
  50. 50. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa BI]TahrN_ 7>3 nigrUbTI 7>9/ kugRtaMg nigbMErbMrYlragdMbUgsMrab;FñwmenAeBlepÞrEdlbNþalBIkMlaMg eRbkugRtaMg Pi nig Pi + WD mandUcxageRkam kMlaMgeRbkugRtaMg P i kNþalElVg³ f t = +501 psi (3.1MPa ) f b = −3,524 psi (24.3MPa ) 501 εc = t = 144 ⋅ 10 − 6 in. / in. 3.49 ⋅ 10 6 ε cb = −1,010 ⋅ 106 psi TMr³ f t = +92 psi (0.7 MPa ) f b = −2,242 psi(15.5MPa ) ε c = +26 ⋅ 10 −6 in. / in. t ε cb = −642 ⋅ 10 −6 in. / in. cMNaMfa eKRtUveFVIkarKNnam:UDuleGLasÞic Ec sMrab;karpøas;bþÚreBlenAeBlEdlkMeNIncenøaHeBl nImYy²cb;. Cabnþ eyIgman − 1,010 − 144 φci kNþalElVg = × 10 − 6 = −33.94 ⋅ 10 − 6 rad / in. 34 − 642 − 26 φei TMr = × 10 − 6 = −19.65 ⋅ 10 − 6 rad / in. 34 BIrUbTI 7>6 ⎛ l2 ⎞ 2 ⎜ ⎟ + (φe − φc ) l δ i ↑= φc ⎜ ⎟ ⎝8⎠ 24 δ i ↑= −33.94 ⋅10 −6 (60 ×12)2 + (− 19.65 + 33.94)×10 − 6 × (60 ×12)2 8 24 = (60 × 12) 2 × 10 − 6 (− 33.94 × 2 − 19.65) 24 = −1.89in. ↑ (48mm ) cMNaMfa tMélenHdUcKñanwgGVIEdlTTYl)anedaysmIkarm:Um:g;enAkñúg]TahrN_ 7>3 ⎛ 1019 ⎞ 5× ⎜ ⎟(60 × 12 ) 4 4 δD TMgn;pÞal; =+ 5wl = ⎝ 12 ⎠ 384 Ec I g 384 × 3.49 ⋅10 6 × 86,072 = +0.99in. ↓ (25mm ) net camber enAeBlepÞr = −1.89 ↑ +0.99 ↓= −0.90in. ↑ (23mm) Camber, Deflection and Crack Control 456
  51. 51. Department of Civil Engineering NPIC emKuNGaRs½ynwgeBl (a) creep BIsmIkar 3.10 ε CR = Ct ( f cs ) = C1ε cs Ec Edl kugRtaMgebtugenARtg;nIv:U cgs f cs = ε cs = bMErbMrYlrageFobenARtg;nIv:U cgs ε CR = unit creep stain kñúgmYyÉktþakugRtaMgeRkam ultimate creep = Cu / Ec = 2.35 / 4.03 ⋅106 = 0.583 ⋅10 −6 in. / in. kñúgmYyÉktþakugRtaMg cMNaMfa eKRtUvKNna creep strain enARtg;TMRbCMuTMgn;rbs;edIm,IKNnakMhatbg;edaysar creep enAkñúgeRbkugRtaMg. BIsmIkar 3.9b, emKuN creep enAeBlNak¾eday EdlKitCaéf¶KW t 0.60 Ct = Cu 10 + t 0.60 Ca]TahrN_ enAGayu 30éf¶eRkayeBlepÞr ⎛ t 0.60 ⎞ ⎛ 0.60 ⎞ ε 'CR , s = ε 'CR ⎜ ⎟ = 0.583 ⋅10 − 6 ⎜ 30 ⎟ ⎜ 0.60 ⎟ ⎜ 10 + 30 0.60 ⎟ ⎝ 10 + t ⎠ ⎝ ⎠ kñúgmYyÉktþakugRtaMg = 0.254 ⋅10 −6 in. / in. Creep strain enAcenøaHeBlepSgeTotRtUv)anKNnakñúgTMrg;dUcKña. (b) karrYmmaDrbs;ebtug BIsmIkar 3.15a sMrab; moist-cured concrete t ε SH , s = ε SH t + 35 Edl ε SH = 800 ⋅10−6 in. / in. sMrab; moist-cured concrete. 30éf¶eRkayeBlepÞr/ ry³eBlrYmmaD t = 30 éf¶ RbsinebIGgát;CaFñwm post-tensioned ehIy t = 30 + 7 = 37 éf¶ RbsinebIvaCa pretensioned. dUcenH 30 ε SH ,30 = × 800 ⋅10 − 6 = 369 ⋅10 − 6 in. / in. 30 + 35 tamrebobdUcKña eKGacKNna ε SH sMrab;RKb;CMhanepSgdéTeTotEdlerobrab;enAkñúgtarag 7>5. PaBekag PaBdab nigkarRKb;RKgsñameRbH 457

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