More Related Content Similar to Vii. camber, deflection, and crack control Similar to Vii. camber, deflection, and crack control (8) More from Chhay Teng (20) Vii. camber, deflection, and crack control1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
18. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa
Camber, Deflection and Crack Control 424
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. T.Chhay viTüasßanCatiBhubec©keTskm<uúCa
Camber, Deflection and Crack Control 452
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. 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. 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. 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. 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