Vii. camber, deflection, and crack control

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

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


- 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.



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.



(
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



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



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.



]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



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



(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



#> 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 ⎠ ⎦



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



3
⎛M ⎞
I e = I cr + ⎜ cr
⎜M ⎟ ( )
⎟ I g − I cr ≤ I g (7.10a)
⎝ a ⎠
eKGacsresrsmIkar 7.10a kñúgTMrg;



⎛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



( )
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

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.



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;.



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.



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)



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.



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



= −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



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 ⎠


− 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



⎛ 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



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



⎛ 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



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.



#> 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



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.



%> 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.


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 ⎠



= 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.



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



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¦



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



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



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



= 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



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.



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.



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.



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.



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



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



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)



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.


Vii. camber, deflection, and crack control

Vii. camber, deflection, and crack control

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