More Related Content Similar to Iv.flexural design of prestressed concrete elements Similar to Iv.flexural design of prestressed concrete elements (20) More from Chhay Teng (20) Iv.flexural design of prestressed concrete elements1. T.Chhay
IV. karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt;
Flexural Design of Prestressed Concrete Elements
1> esckþIepþIm Introduction
kugRtaMgBt;CalT§plénbnÞúkxageRkA nigm:Um:g;Bt;. kñúgkrNICaeRcIn vaCaGñkkMNt;kñúgkar
eRCIserIsTMhMFrNImaRtrbs;ebtugeRbkugRtaMgedayminKitfavargkarTajCamun (pretensioned) b¤rg
karTajCaeRkay (post-tensioned) eT. dMeNIrkarKNnacab;epþImCamYynwgkareRCIserIsmuxkat; bzm
nigedaykarsakl,g nigkarEktMrUveKnwgTTYl)anmuxkat;cugeRkayCamYynwgTMhMlMGitrbs;muxkat;
ehIynwgTMhM nigKnøgrbs;EdkeRbkugRtaMg. muxkat;RtUvbMeBjnUvkarkMNt;rbs;kugRtaMgBt;EdlRtUvkar
rbs;ebtug nigEdk. bnÞab;BIenH vaRtUv)anviPaK nigbMeBjktþamYycMnYneTotdUcCa lT§PaBrgkarkat;
lT§PaBrgkarrmYl PaBdab nigsñameRbH.
edaysarTinñn½ysMrab;karviPaKxusKñaBITinñn½yEdlcaM)ac;sMrab;karKNna karKNnaTaMgGs;Ca
karviPaK. dMbUgeKsnμt;lkçN³muxkat;FrNImaRtEdlRtUvrgeRbkugRtaMg nigbnÞab;mkeKcab;epþÍmkMNt;
faetImuxkat;GacrgkMlaMgeRbkugRtaMg nigkMlaMgGnuvtþn_xageRkA)anedaysuvtßiPaBb¤k¾Gt;. dUcenHeyIg
RtUvyl;BIeKalkarN_mUldæanénkarviPaK nigkarKNnamuxkat;EdlmanlkçN³sMrYly:agxøaMgEdl)an
ENnaMkñúgemeronenH. dUc)aneXIjBICMBUkTI1 lkçN³emkanicmUldæanrbs;sMPar³ eKalkarN_lMnwgrbs;
m:Um:g; couple xagkñúg nigeKalkarN_eGLasÞicéntMrYtpl (superposition) RtUv)aneRbIenARKb;dMNak;
kalénkardak;bnÞúk.
eKKNnamuxkat;ebtugGarem:rgkugRtaMgBt;EtkñúgsßanPaBkMNt;énkugRtaMgenAeBl)ak;sMrab;
muxkat;EdleRCIserIs RbsinebIvabMeBjnUvtMrUvkard¾éTeTotdUcCa serviceability, lT§PaBkñúgkarkat;/
nigPaBs¥itrvagebtug nigEdk. b:uEnþ kñúgkarKNnaGgát;ebtugeRbkugRtaMg eKcaM)ac;RtUveFVIkarRtYtBinitü
bEnßmeTotenAeBlepÞrkMlaMg nigsßanPaBkMNt;enAeBlrgbnÞúkeFVIkar k¾dUcCasßanPaBkMNt;enA
eBl)ak;. karRtYtBinitüTaMgenHmansar³sMxan;sMrab;Fanafa sñameRbHedaysarbnÞúkeFVIkarGac
ecal)an ehIyeKGacRKb;RKg)annUvT§iBlry³eBlyUrrbs;PaBdab nigPaBekag.
eKeRbIsBaØadkedIm,IsMKal;kugRtaMgsgát; ehIyeKeRbIsBaØabUkedIm,IsMKal;kugRtaMgTajenAkñúg
muxkat;ebtug. ragekag (convex or hogging shape) rbs;Ggát;bgðajm:Um:g;GviC¢man ehIyragpt
(concave or sagging) bgðajmU:m:g;viC¢man dUcbgðajenAkñúgrUbTI 4>1.
Flexural Design of Prestressed Concrete Elements 90
2. NPIC
mindUcKñaniwgkrNIGgát;ebtugGarem: kugRtaMgrbs;ebtugERbRbYleTAtamdMNak;kalepSg²én
kardak;bnÞúkefr nigbnÞúkGefr. xageRkamCakarsegçbénkardak;bnÞúkTaMgenH³
eRkayeBlGnuvtþkMlaMgeRbkugRtaMgedIm Pi kMlaMgenHRtUv)anepÞrBIkabeRbkugRtaMgeTAebtug.
TMgn;pÞal;TaMgGs; WD manGMeBIeTAelIGgát;rYmCamYynwgkMlaMgeRbkugRtaMgedIm RbsinebIGgát;
enaHRTedayTMrsamBaØ ¬vaminmanTMrenAkNþalElVg¦.
bnÞúkefrbEnßmTaMgGs; WSD edayrYmTaMg topping sMrab; composite action RtUv)anGnuvtþ
eTAelIGgát;.
kMhatbg;kMlaMgeRbkugRtaMgry³eBlxøIbMputekItman EdlnaMeGaymankarkat;bnßykMlaMg
eRbkugRtaMg Peo .
Ggát;rgnUvbnÞúkeFVIkareBjeljCamYynwgkMhatbg;ry³eBlyUrEdlbNþalmkBI creep,
shrinkage nig stand relaxation EdlnaMeTAdl; net prestressing force Pe .
bnÞúkelIsEdlmanGMeBIelIGgát;ekItmaneRkamlkçxNÐxøHEdlnaMdl;sßanPaBkMNt;enAeBl)ak;.
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 91
3. T.Chhay
rUbTI 4>2 bgðajBICMhanénkardak;bnÞúk nigkarBRgaykugRtaMgelImuxkat;EdlRtUvnwgkardak;
bnÞúktamCMhannImYy². ehIyrUbTI 4>3 bgðajBIdüaRkambnÞúk-kMhUcRTg;RTay ¬ekag b¤pt¦ sMrab;
kardMNak;kalénkardak;bnÞúktaMgBIeBlTTYlT§iBlénTMgn;pÞal;rhUtdl;eBl)ak;.
2> kareRCIserIslkçN³FrNImaRténmuxkat;
Selection of Geometrical Properties of Section Components
k> eKalkarN_ENnaMTUeTA General Guideline
eRkamlkçxNÐbnÞúkeFVIkar FñwmRtUv)ansnμt;famanlkçN³esμIsac; (homogenous) nigeGLasÞic.
ehIyeKsnμt; ¬edaysarkarrMBwgTuk¦ fakMlaMgsgát;eRbkugRtaMgEdlbBa©ÚneTAebtugesÞIreFVIeGaysrés
rgkarTajrbs;FñwmekItmansñameRbH dUcenHeKcat;Tukmuxkat;FñwmCamuxkat;KμansñameRbH (uncracked
Flexural Design of Prestressed Concrete Elements 92
4. NPIC
section) . karviPaKkugRtaMgrbs;FñwmeRbkugRtaMgeRkamlkçxNÐTaMgenHminxusKñaBIkarviPaKkugRtaMgrbs;
FñwmEdk ¬Edlkan;Etc,as;CagenH KW beam column¦. vaEtgEtmankMlaMgtamG½kSEdlbNþalBI
kMlaMgeRbkugRtaMgeTaHbICaman b¤Kμanm:Um:g;Bt;EdlbNþalBIbnÞúkpÞal; b¤bnÞúkxageRkAd¾éTeTotk¾eday.
dUc)aneXIjenAkñúgCMBUk1 vaCakarRbesIrEdlKnøgrbs;EdkeRbkugRtaMgcakp©itenARtg;muxkat;
eRKaHfñak; dUcCamuxkat;kNþalElVgsMrab;FñwmTMrsamBaØ nigmuxkat;elITMrsMrab;FñwmCab;. RbsinebIeK
eFVIkareRbobeFobrvagmuxkat;ctuekaN muxkat;EdlmansøabminsIuemRTImanRbsiT§PaBCagedaykareRbI
R)as;ebtug nigkarRbmUlpþúMebtugenAkñúgtMbn;sgát;énmuxkat;EdleKRtUvkarCageK.
x> m:UDulmuxkat;Gb,brma Minimum Section Modulus
edIm,IKNna nigeRCIserIsmuxkat; CadMbUgeKRtUvkMNt;m:UDulmuxkat;EdlRtUvkar Sb nig S t .
RbsinebI³
f ci = kugRtaMgsgát;GnuBaØatGtibrmaenAkñúgebtugeRkayeBlepÞrPøam² munnwgmankMhatbg;
= 0.60 f 'ci
kugRtaMgTajGnuBaØatGtibrmaenAkñúgebtugeRkayeBlepÞrPøam² munnwgmankMhatbg;
f ti =
= 3 f 'ci psi (0.25 f 'ci MPa ) ¬eKGacbegáIntMélenHdl; 6 f 'ci psi (0.5 f 'ci MPa )
enARtg;TMrsMrab;Ggát;TMrsmBaئ
f c = kugRtaMgsgát;GnuBaØatGtibrmaenAkñúgebtugeRkayeBlepÞrenAeBlrgbnÞúkeFVIkar
= 0.45 f 'c b¤ 0.60 f 'c enAeBlGnuBaØatedaykUd
f t = kugRtaMgTajGnuBaØatGtibrmaenAkñúgebtugeRkayeBlepÞrenAeBlrgbnÞúkeFVIkar
= 6 f 'ci psi (0.5 f 'ci MPa ) ¬eKGacbegáIntMélenHdl; 12 f 'ci psi ( f 'ci MPa ) enA
kñúgRbB½n§mYyTis RbsinebIeKRtUvkarKNnaPaBdabry³eBlyUr¦.
kugRtaMgsrésxageRkACak;EsþgenAkñúgebtugminGacFMCagkugRtaMgGnuBaØatEdl)anerobrab;xag
elIeLIy.
edayeRbImuxkat;minsIuemRTIGt;eRbH karsegçbénsmIkarkugRtaMgEdl)anBICMBUk 1EpñkTI 3 sM
rab;dMNak;kalénkardak;bnÞúkepSg²mandUcxageRkam³
kugRtaMgenAeBlepÞr Stress at Transfer
Pi ⎛ ect ⎞ M D
ft =− ⎜1 − 2 ⎟ − t ≤ f ti (4.1a)
Ac ⎝ r ⎠ S
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 93
5. T.Chhay
Pi ⎛ ecb ⎞ M D
fb = − ⎜1 + 2 ⎟ + ≤ f ci (4.1b)
Ac ⎝ r ⎠ Sb
Edl Pi CakMlaMgeRbkugRtaMgedIm. eKKYreRbIbgÁúMkMlaMgedkrbs; Pi edIm,ITTYl)antMélkan;EtsuRkitCag.
EtsMrab;karGnuvtþTaMgGs;eKmin)anKitdl;PaBRbesIrenHeT.
kugRtaMgRbsiT§PaBeRkaykMhatbg; Effective Stress after Losses
⎛ ect ⎞ M D
Pe
ft =− ⎜1 − 2 ⎟ − t ≤ f t (4.2a)
⎝
Ac r ⎠ S
P ⎛ ec ⎞ M
f b = − e ⎜1 + 2b ⎟ + D ≤ f c (4.2b)
Ac ⎝ r ⎠ Sb
kugRtaMgénbnÞúkeFVIkarcugeRkay Service-load Final Stresses
Pe ⎛ ect ⎞ M T
ft =− ⎜1 − 2 ⎟ − t ≤ f c (4.3a)
Ac ⎝ r ⎠ S
P ⎛ ecb ⎞ M
f b = − e ⎜1 + 2 ⎟ + T ≤ f t (4.3b)
Ac ⎝ r ⎠ Sb
Edl M T = M D + M SD + M L
Pi = kMlaMgeRbkugRtaMgedIm
Pe = kMlaMgeRbkugRtaMgRbsiT§PaBeRkayeBlxatbg;kMlaMgeRbkugRtaMg
t bgðajfasrésxagelI nig b bgðajfasrésxageRkam
e = cMNakp©itrbs; tendon BITIRbCMuTMgn;rbs;munkat;ebtug cgc (center of gravity of
concrete section) q
r 2 = kaer:énkaMniclPaB
S t / Sb = m:UDulmuxkat;srésxagelI nigxageRkamrbs;muxkat;ebtug
dMNak;kalénkacuHfykMlaMgsgát; (decompression) bgðajkarekIneLIgbMErbMrYlrageFob
rbs;EdkEdlbNþalBIkarekIneLIgrbs;bnÞúk taMgBIdMNak;kalEdlkMlaMgeRbkugRtaMgRbsiT§PaB Pe
eFVIGMeBIEtÉkÉgrhUtdl; dMNak;kalEdlbnÞúkbEnßmeFVIeGaykugRtaMgsgát;rbs;ebtugenARtg;nIv:U cgs
kat;bnßydl;sUnü¬emIlrUb TI 4>3¦. enARtg;dMNak;kalenH bMErbMrYlkugRtaMgebtugEdlbNþalBI
decompression KW
Pe ⎛ e2 ⎞
f decomp = ⎜1 + ⎟ (4.3c)
Ac ⎜ r2 ⎟
⎝ ⎠
Flexural Design of Prestressed Concrete Elements 94
6. NPIC
TMnak;TMngenHQrelIkarsnμt;fabMErbMrYlrageFob (strain) rbs;ebtug nigEdkeRbkugRtaMgEdls¥itCab;
eTAnwgebtugEk,reFVIeGaykarekIneLIgénkugRtaMgEdkesμInwgkarfycuHénkugRtaMgebtug.
1. FñwmEdlmancMNakp©itEdkeRbkugRtaMgERbRbYl
Beam with Variable Tendon Eccentricity
FñwmrgnUvkMlaMgeRbkugRtaMgCamYynwg tendon Edl harped b¤ draped. CaTUeTAcMNakp©itGti-
brmaEtgEtsßitenARtg;muxkat;kNþalElVgsMrab;krNIFñwmTMrsamBaØ. edaysnμt;fakMlaMgeRbkugRtaMg
RbsiT§PaBKW
Pe = γPi
Edl γ CapleFobkMlaMgeRbkugRtaMgEdlenAsl; (residual prestress ratio) kMhatbg;énkMlaMgeRb
kugRtaMgKW
Pi − Pe = (1 − γ )Pi (a)
RbsinebIkugRtaMgsrésxageRkAbMputrbs;ebtugCak;EsþgsmmUleTAnwgkugRtaMgGnuBaØat BIsmIkar 4.1a
nig b eyIgTTYl)anbMErbMrYlkugRtaMgenHeRkayeBlxatbg;kMlaMgeRbkugRtaMgdUcxageRkam³
⎛ M ⎞
Δf t = (1 − γ )⎜ f ti + tD ⎟ (b)
⎝ S ⎠
⎛ M ⎞
Δf b = (1 − γ )⎜ − f ci + D ⎟
⎜ (c)
⎝ Sb ⎟
⎠
BIrUb 4>4 (a) edaysarm:Um:g;bnÞúkefrbEnßm M SD nigm:Um:g;bnÞúkGefr M L manGMeBIeTAelIFñwm kugRtaMg
suT§ (net stress) enAsrésxagelIKW
f nt = f ti − Δf t − f c
b¤ f nt = γf ti − (1 − γ ) tD − f c
M
S
(d)
Net stress enAsrésxageRkamKW
f bn = f t − f ci − Δf b
b¤ f bn = f t − γf ci − (1 − γ ) D
M
Sb
(e)
BIsmIkar (d) nig (e) muxkat;EdlRtUveRCIserIsmanm:UDulmuxkat;dUcxageRkam
St ≥
(1 − γ )M D + M SD + M L (4.4a)
γf ti − f c
ehIy Sc ≥
(1 − γ )M D + M SD + ML (4.4b)
f t − γf ci
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 95
8. NPIC
2. FñwmEdlmancMNakp©itEdkeRbkugRtaMgefr
Beam with Constant Tendon Eccentricity
FñwmEdlmancMNakp©itEdkeRbkugRtaMgefrCaFñwmEdlman tendon Rtg; dUckñúgkrNIFñwmeRbkug
RtaMgTMrsamBaØcak;eRscEdlmantMéllμm. edaysar tendon mancMNakp©itFMenARtg;TMr vaeFVIeGay
mankugRtaMgTajFMenAsrésxagelIedayminmankarkat;bnßyNamYyedaym:Um:g;bnÞúkbEnßm M D +
M SD + M L eT. b¤eKGacniyaymü:ageTotfa sMrab;FñwmEbbenH muxkat;eRKaHfñak;KWsßitenARtg;TMr
ehIykarBRgaykugRtaMgenARtg;TMrRtUv)anbgðajenAkñúgrUbTI 4>4 (b). dUcenH
Δf t = (1 − γ )( f ti ) (a’)
ehIy Δf b = (1 − γ )(− f ci ) (b’)
Net stress enAsrésxagelI sMrab;lkçxNÐbnÞúkeFVIkareRkaykMhatbg;KW
f nt = f ti − Δf t − f c
b¤ f nt = γf ti − f cs (c’)
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 97
9. T.Chhay
Edl fcs CakugRtaMgbnÞúkeFVIkarCak;EsþgenAkñúgebtug. Net stress enAsrésxageRkamsMrab;lkçxNÐ
bnÞúkeFVIkareRkaykMhatbg;KW
f bn = f t − f ci − Δf b
b¤ Δf bn = f t − γf ci (d’)
BIsmIkar (c’) nig (d’) muxkat;EdlRtUveRCIserIsRtUvmanm:UDulmuxkat;dUcxageRkam³
M D + M SD + M L
St ≥ (4.5a)
γf ti − f c
M + M SD + M L
ehIy Sb ≥ D
f t − γf ci
(4.5b)
cMNakp©itEdlRtUvkarenARtg;muxkat;eRKaHfñak; dUcCaRtg;TMrsMrab;muxkat;EdlmanlkçN³RsedogKñanwg
GVIEdlRtUvkaredaysmIkar 4.5a nig b KW
( )S
t
ee = f ti − f ci (4.5c)
P i
RkaPictMNageGaym:UDulmuxkat;rbs; nominal section RtUv)anbgðajenAkñúg rUbTI 4>5. eKGaceRbIva
kñúgkareRCIserIsmuxkat;sakl,gdMbUgkñúgdMeNIrkarKNna.
Flexural Design of Prestressed Concrete Elements 98
10. NPIC
tarag 4>1 eGaynUvtMélm:UDulmuxkat;énmuxkat;ctuekaNEkg PCI sþg;dar. tarag 4>2
eGaynUvxñatxageRkAénmuxkat;GkSr T rbs; PCI sþg;dar nigmuxkat;GkSr I rbs; AASTHO erogKña
k¾dUcCam:UDul muxkat;srésxagelIénmuxkat;TaMgenaHEdlRtUvkarkñúgkareRCIserIsmuxkat;bzmsMrab;kar
viPaKeRkamlkçxNÐbnÞúkeFVIkar. tarag 4>4 (a) pþl;nUvxñatlMGiténragFrNImaRt “as built” én PCI
sþg;dar nigmuxkat; AASTHO ehIytarag 4>4 (b) pþl;nUvlkçN³muxkat;rbs; girder EdleRbIenA
kñúgrdæepSg². lkçN³ bulb section manenAkñúg]bsm<½n§ (appendix) C.
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 99
14. NPIC
3> ]TahrN_sMrab;karKNnaeRkamlkçxNÐbnÞúkeFVIkar
Service-Load Design Examples
k> cMNakp©itrbs;EdkeRbkugRtaMERbRbYl Variable Tendon Eccentricity
]TahrN_ 4>1³ KNnaFñwmeRbkugRtaMgmuxkat;GkSr T Dub sMrab;eFVIcMNtrfynþ. FñwmenHmanRbEvg
60 ft (18.3m ) nwgRtUv)anRTedayTMrsamBaØ. EdkeRbkugRtaMgEdleRbIenAkñúgFñwmenHRtUv)an harped.
eKeRbIkugRtaMgGnuBaØatrbs; ACI 318 Building code. FñwmenHRtUvRTbnÞúkeFVIkarbEnßm 1,100 plf
(16.1kN / m ) nigbnÞúkefrbEnßm 100 plf (1.5kN / m ) nigminman concrete topping eT. snμt;faeKeFVI
FñwmenHedayeRbIebtugTMgn;Fmμta (normal-weight concrete) Edlman f 'c = 5,000 psi (34.5MPa )
ehIykugRtaMgebtugenAeBlepÞr f 'ci esμInwg 75% én f 'c . ehIysnμt;fakMhatbg;GaRs½ynwgeBl
rbs;kMlaMgeRbkugRtaMgedImesμInwg 18% énkMlaMgeRbkugRtaMgedIm ehIy ultimate strength rbs;Edk
eRbkugRtaMg f pu = 270,000 psi (1,862MPa ) sMrab; stress-relieved tendon nig f 't = 12 f 'c psi
( f 'c MPa ) .
dMeNaHRsay³
γ = 100 − 18 = 82%
f 'ci = 0.75 × 5,000 = −3,750 psi (25.9MPa )
eRbI f 't = 12 5,000 = 849 psi(5.9MPa ) CakugRtaMgrgkarTajGtibrma ehIysnμt;TMgn;xøÜn
Rbhak;RbEhlnwg 1,000 plf (14.6kN / m).
kMNt;m:Umg;Edl)anBITMgn;pÞal;
wl 2 1,000(60 )2
MD = = × 12 = 5,400,000in. − lb(610kN .m )
8 8
ehIym:Um:g;Edl)anBIbnÞúkbEnßmKW
M SD + M L =
(1,100 + 100)(60)2 × 12 = 6,480,000in. − lb(732kN .m )
8
muxkat;eRKaHfñak;sßitenAEk,rkNþalElVg CakEnøgEdlm:Um:g;Edl)anBIbnÞúkefr nigbnÞúkefr
bEnßmmantMélGtibram nigedaysar tendon RtUv)an harped dUcenHkñúgkrNIPaKeRcInmuxkat;
eRKaHfñak;RtUv)anykenARtg; 0.40L BITMr Edl L CaElVgFñwm. BIsmIkar 4.4a nig b eyIg)an
St ≥
(1 − γ )M D + M SD + M L
γf ti − f c
≥
(1 − 0.82)5,400,000 + 6,480,000 = 3,104in3 (50,860cm3 )
0.82 × 184 + 2,250
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 103
15. T.Chhay
Sb ≥
(1 − γ )M D + M SD + M L
f t − γf ci
≥
(1 − 0.82)5,400,000 + 6,480,000 = 2,766in3 45,330cm3
( )
849 + (0.82 × 2,250 )
BI eRCIserIs nontopped normal weight concrete double-T 12DT
PCI design handbook
34 168-D1 edaysarvamantMélm:UDulmuxkat;srésxageRkamEk,rtMélEdlRtUvkarCageK.
lkçN³muxkat;rbs;ebtugmandUcxageRkam³
Ac = 978in.2 ct = 8.23in.
I c = 86,072in.4 cb = 25.77in.
I
r 2 = c = 88.0in.2 e c = 22 . 02 in .
Ac
S t = 10,458in.3 ee = 12.77in.
Sb = 3,340in.3 WD = 1,019 plf
V
= 2.39in.
S
KNna strands nigRtYtBinitükugRtaMg
BIrUbTI 4>7 TMgn;xøÜnEdlsnμt;mantMélEk,rTMgn;xøÜnCak;Esþg.
KNnam:Um:g;Edl)anBITMgn;pÞal;Cak;EsþgBIm:Um:g;Edl)anBITMgn;pÞal;snμt;
1,019
MD = × 5,400,000 = 5,502,600in. − lb
1,000
f pi = 0.70 × 270,000 = 189,000 psi
f pe = 0.82 f pi = 0.82 × 189,000 = 154,980 psi
Flexural Design of Prestressed Concrete Elements 104
16. NPIC
(a) viPaKkugRtaMgenAeBlepÞr
BIsmIkar 4.1a
Pi ⎛ ect ⎞ M D
ft =− ⎜1 − 2 ⎟ − t ≤ f ti = 184 psi
Ac ⎝ r ⎠ S
P ⎛ 22.02 × 8.23 ⎞ 5,502,600
bnÞab;mk 184 = − i ⎜1 −
978 ⎝ 88.0
⎟−
⎠ 10,458
Pi = (184 + 526.16)
978
= 655,223lb
1.06
cMnYn tendon EdlRtUvkar =
655,223
189,000 × 0.153
= 22.66 edImtendon EdlmanGgát;p©it 1 / 2in.
sakl,g tendon Ggát;p©it 1 / 2in. cMnYn 16 edIm sMrab;muxkat;sþg;dar
Aps = 16 × 0.153 = 2.448in.2 ( .3cm 2 )
15
Pi = 2.448 × 189,000 = 462,672lb(2,058kN )
Pe = 2.448 × 154,980 = 379,391lb(1,688kN )
(b) viPaKkugRtaMgeRkamGMeBIbnÞúkeFVIkarenAkNþalElVg
Pe = 379,391lb(1,688kN )
100(60 )212
M SD = = 540,000in. − lb(61kN .m )
8
1,100(60 )212
ML = = 5,940,000in.lb(788kN .m )
8
m:Um:g;srub M T = M D + M SD + M L = 5,502,600 + 6,480,000
= 11,982,600in. − lb(1,354kN .m )
BIsmIkar 4.3a
Pe ⎛ ect ⎞ M T
ft =− ⎜1 − 2 ⎟ − t
Ac ⎝ r ⎠ S
379,391 ⎛ 22.02 × 8.23 ⎞ 11,982,600
=− ⎜1 − ⎟−
978 ⎝ 88.0 ⎠ 10,458
= 411 − 1146 = −735 psi < f c = −2250 psi O.K.
(c) viPaKkugRtaMgRtg;muxkat;TMr
ee = 12.77in.(324mm )
f ti = 6 f 'ci = 6 3,750 ≅ 367 psi
f t = 12 f 'c = 12 5,000 = 849 pis
(i) enAeBlepÞr
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 105
17. T.Chhay
462,672 ⎛ 12.77 × 8.23 ⎞
ft =− ⎜1 − ⎟ − 0 = +92 psi (T )
978 ⎝ 88.0 ⎠
462,672 ⎛ 12.77 × 25.77 ⎞
fb = − ⎜1 + ⎟ + 0 = −2,240 psi (C )
978 ⎝ 88.0 ⎠
< f ci = −2,250 psi O.K.
RbsinebI fb > fci / eKRtUveFVIkarpøas;bþÚrcMNakp©it.
(ii) eRkamGMeBIbnÞúkeFVIkar
379,391 ⎛ 12.77 × 8.23 ⎞
ft =− ⎜1 − ⎟ − 0 = +75 psi (T )
978 ⎝ 88.0 ⎠
379,391 ⎛ 12.77 × 25.77 ⎞
fb = − ⎜1 + ⎟ + 0 = −1.840 psi (C )
978 ⎝ 88.0 ⎠
< f ci = −2,250 psi O.K.
TTYlykmuxkat;sMrab;lkçxNÐbnÞúkeFVIkaredayeRbI strand Ggát;p©it 1 / 2in.(12.7mm) cMnYn 16
edImedaymancMNakp©itenAkNþalElVg ec = 22.02in.(560mm) nigcMNakp©itenAcugTMr ee = 12.77in.
(324mm ) .
x> cMNakp©itrbs;EdkeRbkugRtaMERbRbYledayminmankarkMNt;kMBs;
Variable Tendon Eccentricity with No Height Limitation
]TahrN_ 4>2³ KNnamuxkat;GkSr I sMrab;FñwmEdlmanElVg 65 ft (19.8m) Edlmanm:UDulmuxkat;dUc
xageRkam. cUreRbInUvkugRtaMgGnuBaØatdUcKñaEdl)aneGayenAkñúg]TahrN_ 4>1.
S t EdlRtUvkar = 3,570in.3 (58,535cm3 )
Sb EdlRtUvkar = 3,780in.3 (61,940cm3 )
Flexural Design of Prestressed Concrete Elements 106
18. NPIC
dMeNaHRsay³
edaysarm:UDulmuxkat;enAsrésxagelI nigsrésxageRkamesÞIresμIKña eKGaceRCIserIsmux
kat;sIuemRTI)an. bnÞab;mk viPaKmuxkat;enAkñúgrUbTI 4>8 EdleRCIserIsedaykarsakl,g nigEktMrUv.
viPaKkugRtaMgenAeBlepÞr
BIsmIkar 4.4d
ct
f ci = f ti − ( f ti − f ci )
h
= +184 −
21.16
(+ 184 + 2,250) ≅ −1,104 psi(C )(7.6MPa )
40
Pi = Ac f ci = 377 × 1,104 = 416,208lb(1,851kN )
393(65)2
MD = × 12 = 2,490,638in. − lb(281kN .m )
8
BIsmIkar 4.4c cMNakp©itEdlRtUvkarenARtg;muxkat;m:Um:g;GtibrmaenAkNþalElVgKW
(
ec = f ti − f ci )
St M D
Pi
+
Pi
= (184 + 1,104 )
3,572 2,490,638
+
416,208 416,208
= 11.05 + 5.98 = 17.04in.(433mm )
edaysar cb = 18.84in. nigedaysnμt;fakMras;ebtugkarBarEdk 3.75in. sakl,g
ec = 18.84 − 3.75 ≅ 15.0in.(381mm )
RkLaépÞ tendon EdlRtUvkar P
Ap = i =
416,208
f pi 189,000
(
= 2.2in 2 14.2cm 2 )
cMnYn tendon = 02153 = 14.38 edIm
.
.2
sakl,g tendon Ggát;p©it 1 / 2in.(12.7mm) cMnYn 13 edIm/ Ap = 1.99in.2 (12.8cm2 ) / ehIy
kMlaMgeRbkugRtaMgedImCak;Esþg
Pi = 189,000 × 1.99 = 376,110lb(1,673kN )
RtYtBinitükugRtaMgsrésxageRkArbs;ebtug
BIsmIkar 4.1a
Pi ⎛ ect ⎞ M D
ft =− ⎜1 − 2 ⎟ − t
Ac ⎝ r ⎠ S
376,110 ⎛ 15.0 × 21.16 ⎞ 2,490,638
=− ⎜1 − ⎟−
377 ⎝ 187.5 ⎠ 3,340
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 107
19. T.Chhay
= +691.2 − 745.7 = −55 psi (C ) minmankugRtaMgTajenAeBlepÞr (O.K.)
BIsmIkar 4.1b
Pi ⎛ ecb ⎞ M D
fb = − ⎜1 + 2 ⎟ +
Ac ⎝ r ⎠ Sb
376,110 ⎛ 15 × 18.84 ⎞ 2,490,638
=− ⎜1 + ⎟+
377 ⎝ 187.5 ⎠ 3,750
= −2,501.3 + 664.2 = −1,837 psi (C ) < f ci = 2,250 psi O.K.
viPaKkugRtaMgenAeBlrgbnÞúkeFIVkar
BIsmIkar 4.3a
Pe ⎛ ect ⎞ M T
ft =− ⎜1 − 2 ⎟ − t
Ac ⎝ r ⎠ S
Pe = 13 × 0.153 × 154,980 = 308,255lb(1,371kN )
m:Um:g;srub M T = M D + M SD + M L
= 2,490,638 + 7,605,000 = 10,095,638in. − lb(1,141kN .m )
308,255 ⎛ 15.0 × 21.16 ⎞ 10,095,638
ft =− ⎜1 − ⎟−
377 ⎝ 187.5 ⎠ 3,340
= +566.5 − 3,022.6 = −2,456 psi (C ) > f c = −2,250 psi
dUcenH eKRtUvdMeLIgkMBs;rbs;muxkat; b¤eRbIebtugEdlmanersIusþg;FMCag.
edayeRbI f 'c = 6,000 psi
f c = 0.45 × 6,000 = −2,700 psi O.K.
Pe ⎛ ecb ⎞ M T 308,255 ⎛ 15.0 × 18.84 ⎞ 10,095,638
fb = − ⎜1 + 2 ⎟ + =− ⎜1 + ⎟+
Ac ⎝ r ⎠ Sb 377 ⎝ 187.5 ⎠ 3,750
= −2,050 + 2,692.2 = 642 psi (T ) O.K.
RtYtBinitümuxkat;Rtg;TMr
kugRtaMgGnuBaØati f 'ci = 0.75 × 6,000 = 4,500 psi
f ci = 0.60 × 4,500 = 2,700 psi
f ti = 3 f 'ci = 201 psi sMrab;kNþalElVg
f ti = 6 f 'ci = 402 psi sMrab;elITMr
f c = 0.45 f 'c = 2,700 psi
f t1 = 6 f 'c = 465 psi
f t 2 = 12 f 'c = 930 psi
(a) enAeBlepÞr
Flexural Design of Prestressed Concrete Elements 108
20. NPIC
kugRtaMgsgát;srésxageRkArbs;muxkat;elITMr
⎛ ecb ⎞
pi
fb = − ⎜1 + 2 ⎟ + 0
⎝
Ac r ⎠
376,110 ⎛ e × 18.84 ⎞
− 2,700 = − ⎜1 + ⎟
377 ⎝ 187.5 ⎠
dUcenH e = 16.98in.
dUcenHsakl,g ee = 12.49in.
376,110 ⎛ 12.49 × 21.16 ⎞
ft =− ⎜1 − ⎟−0
377 ⎝ 187.5 ⎠
= 409 psi (T ) > f ti = 402 psi
376,110 ⎛ 12.49 × 18.84 ⎞
fb = − ⎜1 + ⎟+0
377 ⎝ 187.5 ⎠
= 2,250 psi < f ci = 2,700 psi
dUcenHeRbIEdkFmμtaenAsrésxagelIRtg;muxkat;elITMredIm,ITTYlykkugRtaMgTajkñúgebtugTaMg
Gs; b¤eRbIebtugEdlmanersIusþg;FMCagsMrab;muxkat;enH b¤k¾kat;bnßycMNakp©it.
(b) enAeBlrgbnÞúkeFVIkar
308,255 ⎛ 12.49 × 21.16 ⎞
ft =− ⎜1 − ⎟ − 0 = 335 psi (T ) < 930 psi O.K.
377 ⎝ 187.5 ⎠
308,255 ⎛ 12.49 × 18.84 ⎞
fb = − ⎜1 + ⎟ + 0 = −1,844 psi (C ) < −2,700 psi O.K.
377 ⎝ 187.5 ⎠
dUcenH eKGacTTYlykFñwmebtugeRbkugRtaMgEdlmanmuxkat;GkSr I kMBs; 40in.(102cm)
eRbIebtugTMgn;FmμtaEdlmanersIusþg; 6,000 psi(41.4MPa ) CamYynwg tendon Ggát;p©it
1 / 2in.(12.7 mm ) EdlmancMNakp©itenAkNþalElVg ec = 15.0in.(381mm ) nigcMNakp©itenARtg;
muxkat;xagcug ee = 12.5in.(318mm)
eKGaceRbIviFImü:ageTotsMrableFVIkaredaHRsay edaybnþeRbI f 'c = 5,000 psi b:uEnþeFVIkarpøas;bþÚrcMnYn
EdkeRbkugRtaMg nigcMNakp©it.
K> cMNakp©itrbs;EdkeRbkugRtaMefr Constant Tendon Eccentricity
]TahrN_ 4>2³ edaHRsay]TahrN_ 4>2 edaysnμt;fakabeRbkugRtaMgmancMNakp©itefr. eRbIebtug
TMgn;FmμtaEdlmanersIusþg; f 'c = 5,000 psi(34.5MPa) ehIykugRtaMgTajGnuBaØatGtibrmarbs;eb
tugKW ft = 12 f 'c = 849 psi .
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 109
21. T.Chhay
dMeNaHRsay³ edaysar tendon mancMNakp©itefr ehIym:Um:g;edaysarbnÞúkefr m:Um:g;edaysarbnÞúk
efrbEnßm nigm:Um:g;edaysarm:Um:g;GefrRtg;muxkat;elITMrrbs;FñwmsamBaØesμIsUnü dUcenHeKRtUvKNna
FñwmenHedayeRbImuxkat;Rtg;TMr. m:UDulmuxkat;EdlRtUvkarenARtg;TMrEdl)anBIsmIkar 4.5a KW
M D + M SD + M L
St ≥
γf ti − f c
M + M SD + M L
Sb ≥ D
f t − γf ci
snμt; WD = 425 plf . bnÞab;mk
425(65)2
MD = × 12 = 2,693,438in. − lb(304kN .m )
8
M SD + M L = 7,605,000in. − lb(859kN .m )
dUcenH m:Um:g;srub M T = 10,298,438in. − lb(1,164kN .m )
ehIyeyIgk¾mankugRtaMgGnuBaØatdUcxageRkam
f ci = −2,250 psi
f 'ci = −3,750 psi
f ti = 6 f 'ci = 367 psi sMrab;muxkat;elITMr
f c = −2,250 psi (15.5MPa )
f t = 849 psi
γ = 0.82
m:UDulmuxkat;EdlRtUvkar
St =
10,298,438
0.82 × 367 + 2,250
)
= 4,035.8in.3 61,947cm3 (
Sb =
10,298,438
849 + 0.82 × 2,250
)
= 3,823.0in.3 62,713cm3 (
sakl,gelIkTI 1³ edaysar S EdlRtUvkar = 4,035.8 psi FMCag S rbs;muxkat;enA
t t
kñúg]TahrN_ 4>2 dUcenHeRCIserIsmuxkat;GkSr I Edlman h = 44in. dUcbgðajenAkñúgrUbTI 4>9.
lkçN³muxkat;rbs;vamandUcxageRkam³
I c = 92,700in.4
r 2 = 228.9in.2
Ac = 405in.2
ct = 23.03in.
Flexural Design of Prestressed Concrete Elements 110
22. NPIC
S t = 4,303in.3
cb = 20.97in.
Sb = 4,420in.3
WD = 422 plf
BIsmIkar 4.5c cMNakp©itEdlRtUvkarRtg;muxkat;elITMrEdlCamuxkat;eRKaHfñak;KW
( )S
t
ee = f ti − f ci
P i
Edl f ci = f ti − t ( f ti − f ci )
c
h
= 367 −
23.03
(367 + 2,250) = −1,002 psi(6.9MPa )
44
nig Pi = Ac f ci = 405 × 1,002 = 405,810lb(1,805kN )
dUcenH ee = (367 + 1,002) 405030 = 13.60in.(346mm)
4,
,810
RkLaépÞEdkeRbkugRtaMgEdlRtUvkarKW
= 2.15in.2 ( .4cm 2 )
P 405,810
Ap = i = 14
f 189,000
pi
dUcenHeyIgsakl,geRbIEdkeRbkugRtaMgEdlmanGgát;p©it 1 / 2in. .
cMnYn tendon EdlRtUvkarKW
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 111
23. T.Chhay
2.15 / 0.153 = 14.05
dUcenHeRbI tendon Ggát;p©it 1 / 2in.(12.7mm) cMnYn 14 edIm. CalT§pl
Pi = 14 × 0.153 × 189,000 = 404,838lb(1,801kN )
(a) viPaKkugRtaMgenAeBlepÞrenARtg;muxkat;xagcug
BIsmIkar 4.1a
pi ⎛ ect ⎞ M D 404,838 ⎛ 13.60 × 23.03 ⎞
ft =− ⎜1 − 2 ⎟ − t = − ⎜1 − ⎟−0
Ac ⎝ r ⎠ S 405 ⎝ 228.9 ⎠
= +368.2 psi (T ) ≅ f ti = 367 O.K.
BIsmIkar 4.2b
Pi ⎛ ecb ⎞ M D 404,838 ⎛ 13.6 × 20.97 ⎞
fb = − ⎜1 + 2 ⎟ + =− ⎜1 + ⎟+0
Ac ⎝ r ⎠ Sb 405 ⎝ 228.9 ⎠
= −2,245 psi (C ) ≅ f ci = −2,250 O.K.
eKk¾GaceRbIvatMélTaMgenHsMrab;muxkat;kNþalElVgpgEdr edaysarcMNakp©it e efr.
(b) viPaKkugRtaMgenAeBlrgbnÞúkeFVIkarcugeRkayenARtg;TMr
Pe = 14 × 0.153 × 154,980 = 331,967lb(1,477kN )
m:Um:g;srub M T = M D + M SD + M L = 0
BIsmIkar 4.3a
Pe⎛ ect ⎞ M T
ft =− ⎜1 − 2 ⎟ − t
Ac⎝ r ⎠ S
331,967 ⎛ 13.60 × 23.03 ⎞
=− ⎜1 − ⎟ − 0 = 302 psi (T ) < f t = 849 psi O.K.
405 ⎝ 228.9 ⎠
BIsmIkar 4.3b
Pe ⎛ ecb ⎞ M T 331,967 ⎛ 13.6 × 20.97 ⎞
fb = − ⎜1 + 2 ⎟ + =− ⎜1 + ⎟+0
Ac ⎝ r ⎠ Sb 405 ⎝ 228.9 ⎠
= −1,841 psi (12.2MPa )(C ) < f c = −2,250 psi O.K.
(c) viPaKkugRtaMgenAeBlrgbnÞúkeFVIkarcugeRkayenAkNþalElVg
m:Um:g;srub M T = M D + M SD + M L = 10,298,438in. − lb
dUcenHkugRtaMgsrésxageRkArbs;ebtugEdlbNþalBI M T KW
= −2,555 psi (C )(17.6MPa )
MT 10,298,438
f1t = t
=−
S 4,030
= +2,330 psi (T )(16.1MPa )
M 10,298,438
f1b = T =
Sb 4,030
dUcenH kugRtaMgsrésxageRkArbs;ebtugcugeRkayKW
Flexural Design of Prestressed Concrete Elements 112
24. NPIC
f t = +302 − 2,555 = −2,253 psi (C ) ≅ f c = −2,250 psi TTYlyk)an
f b = −1,841 + 2,330 = +489 psi (T ) < f t = 849 psi O.K.
dUcenH TTYlykmuxkat;sakl,gEdlmancMNakp©itefr e = 13.6in.(345mm) sMrab; tendon
Ggát;p©it 1 / 2in.(12.7mm) cMnYn 14 srés.
4> kareRCIserIsmuxkat; niglkçN³rbs;Fñwmd¾RtwmRtUv
Proper Selection of Beam Sections and Properties
k> eKalkarN_ENnaMTUeTA General Guidelines
muxkat;ebtugeRbkugRtaMgmindUc steel-rolled section eT eRBaHvaminTan;manlkçN³sþg;dar
eBjeljenAeLIy. kñúgkrNICaeRcIn visVkrKNnaeRKOgbgÁúMRtUvEteRCIserIsRbePTmuxkat;edIm,IeRbI
R)as;enAkñúgKMeragenaH. enAkñúgkarKNnaFñwmTMrsamBaØPaKeRcIn cMgayBI cgc nigExS cgs EdleKsÁal;
CacMNakp©it e smamaRteTAnwgkMlaMgeRbkugRtaMgEdlRtUvkar.
CaTUeTA edaysarEteKKNnaeRcIneRbIm:Um:g;kNþalElVg eRBaHvamantMélFMCageK. cMNakp©it
enAkNþalElVgkan;EtFM kMlaMgeRbkugRtaMgEdlRtUvkarkan;EttUc ehIyvapþl;nUvlkçNesdækic©kan;Et
xøaMgkñúgkarKNna. sMrab;cMNakp©itFM eKRtUvkarRkLaépÞebtugenAsrésxagelIFMEdr. dUcenH muxkat;
GkSr T nigmuxkat;GkSr I EdlmansøabFMCamuxkat;Edlsaksm. CaTUeTA muxkat;xagcugEtgCamux
kat;tan;edIm,IeCosevogcMNakp©itFMenAelIbøg;m:Um:g;sUnü ehIyk¾edIm,IbegáInlT§PaBTb;kMlaMgkat;énmux
kat;elITMr nigkarBar anchorage zone failure.
muxkat;epSgeTotEdleKeRbIPaKeRcInEdrKW muxkat;GkSr T Dub. muxkat;enHbEnßmGtßRbeyaCn_
eTAmuxkat;GkSr T eTaledIm,IPaBgayRsYl nigesßrPaBkñúgkarelIkdak; nigdMeLIg. rUbTI 4>10 bgðaj
BIRbePTmuxkat;EdleKeRcIneRbICaTUeTA. muxkat;d¾éTeTotdUcCakMralRbehagkñúg (hollow-core slab)
muxkat;Gt;sIuemRTI k¾RtUv)aneRbICaTUeTApgEdr. cMNaMfa eKeRbImuxkat;mansøabCMnYseGaymuxkat; ctu-
ekaNtan;EdlmankMBs;dUcKñaedayminman)at;bg;ersIusþg;rgkarBt;eT. b:uEnþ eKeRbImuxkat;ctuekaNCa
girder EdlmanElVgxøI.
eKeRbImuxkat;GkSr I CaRbePTFñwmkMralEdlmankMralxNÐsmascak;BIelIsMrab;eeRKOgbgÁMúcMNt
rfynþEdlmanElVgEvg. CaTUeTA eKeRcIneRbImuxkat;GkSr T EdlmansøabxageRkamF¶n;dUcbgðajenA
kñúgrUbTI 4>10 (d) enAkñúgeRKOgbgÁúMs<an. eKeRbImuxkat; T Duby:agTUlMTUlayenAkñúgRbB½n§kMralxNÐ
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 113
25. T.Chhay
rbs;GKar k¾dUcCageRKOgbgÁúMcMNtrfynþ edaysarRbeyaCn_énskmμPaBsmasrbs;søabFMxagelI
EdlmanTTwgBI 10 ft eTA 15 ft .
kMralRbehagkñúgCacMerokFñwmmYyTisRbehagkñúgEdlGacdMeLIgCakMralxNÐ)any:aggayRsYl.
eKGaceRbIr:tRbehagragRbGb;Car:tFñwmsMrab;ElVgEvg EdleKsÁal;vaCaRbB½n§kMralkMNat;s<an
(segmental bridge deck system). kMNat;r:t (segmental girder) enHmanlT§PaBTb;karrmYlFM
ehIypleFoblT§PaBTb;karBt;elITMgn;xøÜnrbs;vaFMCagRbePTmuxkat;RbB½n§eRbkugRtaMgd¾éTeTot.
x> RkLaépÞTaMgmUl muxkat;bMElg nigvtþmanrbs;bMBg;
Gross Area, the Transformed Section, and the Presence of Ducts
CaTUeTA RkLaépÞrbs;muxkat;TaMgmUlrbs;muxkat;ebtug (gross cross sectional area ) KWRKb;
RKan;sMrab;eRbIenAkñúgkarKNna muxkat;ebtugeRbkugRtMgeRkamlkçxNÐbnÞúkeFVIkar. kñúgxN³EdlGñk
KNnaxøHeBjcitþnwgkarKNna EdlmanlkçN³suRkitCagtamry³kareRbImuxkat;bMElg. PaBsuRkit
Edl)anBIkarKitbBa©ÚlkarcUlrYm énmuxkat;rbs;EdkeTAkñúgPaBrwgRkaj (stiffness) rbs;ebtugmin
Flexural Design of Prestressed Concrete Elements 114
26. NPIC
RtUv)anKitfaCakarcaM)ac;enaHeT. enA kúñgFñwmrgeRbkugRtaMgCaeRkay (post-tensioned beam) Edl
bMBg;RtUv)ankMe)arebtug (grout), gross cross section enAEtRKb;RKan;sMrab;RKab;KNnaTaMgenH. man
EtkñúgkrNIs<anElVgEvg nigFñwmeRbkugRtaMgEdlplitCalkçN³]sShkmμEdlmanRkLaépÞEdkeRbkug
RtaMgFMeT EdleKRtUveRbImuxkat;bMElg b¤muxkat;ebtugsuT§ (net concrete area) EdlminKitbMBg;.
K> Envelope sMrab;kardak;kabeRbkugRtaMg
Envelopes for Tendon Placement
kugRtaMgTajenAsrésxageRkAbMputrbs;ebtugeRkamlkçxNÐbnÞúkeFVIkarminGacFMCagkugRtaMg
GnuBaØatEdleGayeday code dUcCa ACI, PCI, AASTHO b¤ CEB-FIP eT. dUcenH eKcaM)ac;RtUv
begáItnUvtMbn;kMNt;mYyenAkññúgmuxkat;ebtugEdlCa envelope EdleKGacGnuvtþkMlaMgeRbkugRtaMgeday
mineFVIeGaymankugRtaMgTajenAsrésxageRkAbMputrbs;ebtug. BIsmIkar 4.1a eyIgman
Pi ⎛ ect ⎞
ft = 0 = − ⎜1 − 2 ⎟
Ac ⎝ r ⎠
2
eK)an e=
r
ct
dUcenH cMnucsñÚlxageRkam (lower kern point)
r2
Kb =
ct
dUcKña BIsmIkar 4.1b RbsinebI fb = 0 enaHeK)an − e = r 2 / cb EdlsBaØadktMNageGaytMNag
eGaykarvas;eLIgelIBIG½kSNWt ÉcMNakp©itviC¢manCakarvas;cuHeRkam. dUcenH upper kern point KW
r2
Kt =
cb
BIkarkMNt;cMnucsñÚlxagelI nigxageRkammk eyIgeXIjy:agc,as;fa³
(a) RbsinebIkMlaMgeRbkugRtaMgmanGMeBIenAxageRkam lower kern point vanwgekItmankugRtaMgTaj
enAsrésxagelIrbs;muxkat;ebtug.
(b) RbsinebIkMlaMgeRbkugRtaMgmanGMeBIenAxagelI upper kern point vanwgekItmankugRtaMgTaj
enAsrésxageRkamrbs;muxkat;ebtug.
eKGackMNt;cMnucsñÚlxagsþaM nigxageqVgénExSsIuemRTIbBaÄrrbs;muxkat;tamlkçN³dUcKña dUc
enHeKnwgTTYl)anépÞsñÚlsMrab;GnuvtþkMlaMgeRbkugRtaMgeTAelIEdkeRbkugRtaMg. rUbTI 4>11 bgðajBI
sñÚlsMrab;muxkat;ctuekaN.
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 115
27. T.Chhay
X> plRbeyaCn_énkardak;kabeRbkugRtaMgCa curved b¤ harped
Advantages of Curved or Harped Tendons
eTaHbICaeKeRbIEdkeRbkugRtaMgRtg;y:agTUlMTUlayenAkñúgFñwmRbEvglμmEdlcak;eRsck¾eday
k¾CaTUeTAeKeRbIkabeRbkugRtaMgEdlmanTMrg;ekagenAkñúgGgát;rgkarTajCaeRkay (post-tensioned
element) Edlcak;enAnwgkEnøgEdr. eKEck tendon EdlminRtg;CaBIrRbePT³
(a) Draped: manTMrg;ekagdUc)a:ra:bUl RtUv)aneKeRbIenAkñúgFñwmEdlrgbnÞúkxageRkABRgayesμICa
bzm.
(b) Harped: tendon eRTtEdlminCab; ¬tamn½yKNitviTüa¦ enARtg;bøg;rgbnÞúkcMcMnuc RtUv)aneK
eRbIenAkñúgFñwmEdlrgbnÞúkcMcMnucTTwgG½kSCabzm.
rUbTI 4>12/ 4>13 nig 4>14 bgðajBI alignment, m:m:g;Bt; nigkarBRgaykugRtaMgsMrab;Fñwm
EdlrgkMlaMgeRbkugRtaMgedaykabeRbkugRtaMgRtg;/ draped/ nig harped erogKña. düaRkamTaMgenHcg;
bgðajBIplcMeNjEpñkesdækic©rbs; draped nig harped tendon elIEdkeRbkugRtaMgRtg;. enAkñúgrUbTI
4>12 Rtg;muxkat; 1-1 kugRtaMgTajrbs;ebtugEdleKminR)afñacg;)an)anbgðajenAsrésxagelI.
muxkat; 1-1 enAkñúgrUbTI 4>13 nig 4>14 bgðajfakugRtaMgsgát;rayesμIRbsinebI tendon eFVIGMeBIenA
Rtg; cgc énmuxkat;enARtg;TMr. plRbeyaCn_epSgeTotrbs; draped nig harped tendon KWvaGnuBaØat
eGayFñwmeRbkugRtaMgRTbnÞúkF¶n; edaysarT§iBllMnwgrbs;bgÁúMkMlaMgbBaÄrrbs;kabeRbkugRtaMgmin
Rtg;. niyaymü:ageTot kMlaMgeRbkugRtaMgEdlRtUvkar Pp sMrab; parabolic tendon enAkñúgrUbTI 4>13
nig Ph sMrab; harped tendon enAkñúgrUbTI 4>14 mantMéltUcCagkMlaMgEdlRtUvkarenAkñúg straight
Flexural Design of Prestressed Concrete Elements 116
28. NPIC
tendon enAkñúgrUbTI 4>14. dUcenH sMrab;kMritkugRtaMgdUcKña eKRtUvkarcMnYn strand ticCagsMrab;krNI
draped b¤ harped tendon nigeBlxøHeKGaceRbImuxkat;ebtugtUcCagkñúgkarKNnaedayTTYl)annUv
lT§plRbkbedayRbsiT§PaB ¬eRbobeFob]TahrN_ 4>2 nig 4>3 mþgeTot¦.
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 117
31. T.Chhay
g> Limiting-Eccentricity Envelopes
eKcg;)ancMNakp©itKNnarbs; tendon tambeNþayElVgEdleFVIy:agNaminbegáItkugRtaMg
TajenAsrésxageRkAbMputrbs;muxkat;FñwmEdleRKaHfñak;. RbsinebIeKmincg;)ankugRtaMgTajtam
beNþayElVgrbs;FñwmenAkñúgrUbTI 4>15 EdleRbI draped tendon eKRtUvkMNt;cMNakp©itRtg;muxkat;
tambeNþayFñwm. RbsinebI M D Cam:Um:g;TMgn;pÞal; ehIy M T Cam:Um:g;srubEdlekItBIbnÞúkTTwgG½kS
TaMgGs; enaHédXñas;rbs;m:Um:g; couple EdlbegáIteday center-of-pressure line (C-line) nigG½kSTI
RbCMuTMgn;rbs;EdkeRbkugRtaMg (cgs line) EdlekItBI M D nig M T KW amin nig amax erogKña dUc
bgðajkñúgrUbTI 4>15.
Lower cgs Envelop
édXñas;Gb,brmarbs; tendon couple KW
MD
amin = (4.7a)
Pi
smIkarenHkMNt;cMgayGtibrmaenABIxageRkam bottom kern EdlCaTItaMgrbs;ExS cgs dUcenH C-line
minFøak;enABIxageRkamExS bottom kern )aneT GBa©wgehIyvaGackarBarmineGaymankugRtaMgTajenA
srésxagelIbMput)an.
Flexural Design of Prestressed Concrete Elements 120
32. NPIC
dUcenH limiting bottom eccentricity KW
eb = (amin + kb ) (4.7b)
Upper cgs Envelop
édXñas;Gtibrmarbs; tendon couple KW
MT
amax = (4.7c)
Pe
smIkarenHkMNt;cMgayGb,brmaenABIxageRkam top kern EdlCaTItaMgrbs;ExS cgs dUcenH C-line
minsßitenABIxagelIExS top kern )aneT GBa©wgehIyvaGackarBarmineGaymankugRtaMgTajenAsrés
xageRkambMput)an.
dUcenH limiting top eccentricity KW
et = (amax − kt ) (4.7d)
kUdxøHGnuBaØateGayeRbIkugRtaMgTajkMNt;sMrab;enAeBlepÞr nigenAeBlrgbnÞúkeFVIkar. enAkñúgkrNI
EbbenH eKGacGnuBaØateGayExS cgs GacsßitenAxageRkA limiting cgs envelop Edl)anbgðajenA
kñúgsmIkar 4.7a nig c bnþicbnþÜc.
RbsineKbEnßmcMNakp©itbEnßmenAelI cgs-line envelop enaHvanwgeFVIeGaymankugRtaMgTaj
kMNt;enAelIsrésxagelI nigxageRkamrbs;ebtug. kugRtaMgxagelI nigxageRkambEnßmKW
f (t ) =
Pi e'b ct
(4.8a)
Ic
nig P e' c
f (b ) = e t b
Ic
(4.8b)
Edl t nig b tMNageGaysrésxagelI nigxageRkam erogKña. BIsmIkar 4.6 cMNakp©itbEnßmEdl
RtUvbEnßmeTAelIsmIkar 4.7b nig d KW
f (t ) Ac kb
e'b = (4.9a)
Pi
f (b ) Ac kt
nig e't =
Pe
(4.9b)
EnvelopEdlGnuBaØatkugRtaMgkMNt;RtUv)anbgðajenAkñúgrUbTI 4>16. eKKYrcMNaMfa enAeBl
upper envelop enAxageRkAmuxkat; ehIykugRtaMgenAmantMélkMNt;GnuBaØat enaHbgðajfamuxkat;Kμan
lkçN³esdækic©eT. bMErbMrYlcMNakp©it b¤kMlaMgeRbkugRtaMgeFIVeGaykarKNnakan;EtRbesI.
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 121
33. T.Chhay
c> Envelopes EdkeRbkugRtaMg Prestressing Tendon Envelopes
]TahrN_ 4>4³ ]bmafaFñwmenAkñúg]TahrN_ 4>2 Ca post-tensioned bonded beam ehIyEdkeRbkug
RtaMgmanrag)a:ra:bUl. kMNt; limiting envelop sMrab;TItaMgrbs; tendon EdlkMritkugRtaMgsrésrbs;
ebtugminFMCagkugRtaMgGnuBaØat. Kitfamuxkat;Rtg;cMnuckNþalElVg mYyPaKbYnénElVg nigcugFñwmCa
muxkat;EdlRtUvKNna. snμt;fatMélrbs;kMhatbg;eRbkugRtaMgdUcKñaenAkñúg]TahrN_ 4>2 b:uEnþ
Pi = 549,423lb / Pe = 450,526lb / f 'c = 6,000 psi / ec = 13in nig ee = 6in .
dMeNaHRsay³ BI]TahrN+_ 4>2 eyIgGacsegçbm:Um:g;KNnarbs;FñwmGkSr I niglkçN³muxkat;Edl
RtUvkardUcxageRkam³
Pi = 549,423lb(2,431kN )
Pe = 450,526lb(2,004kN )
M D = 2,490,638in. − lb(281kN .m )
M SD + M L = 7,605,000in. − lb(859kN .m )
M T = M D + M SD + M L = 10,095,638in. − lb(1,141kN .m )
(
Ac = 377in.2 2,536cm 2 )
f 'c = 6,000 psi
(
r 2 = 187.5in.2 1,210cm 2 )
ct = 21.16in.(537mm )
cb = 18.84in.(479mm )
Flexural Design of Prestressed Concrete Elements 122
34. NPIC
edaysarEtm:Um:g;Bt;enAkñúg]TahrN_enH)anmkBIbnÞúkBRgayesμI TMrg;rbs;düaRkamm:Um:g;man
ragCa)a:ra:bUl CamYynwgm:Um:g;EdlmantMélsUnüenARtg;cugTMrrbs;FñwmsamBaØ. dUcenH m:Um:g;enARtg;
mYyPaKbYnénRbEvgElVgKW
M D = 0.75 × 2,490,638 = 1,867,979in. − lb(211kN .m )
M T = 0.75 × 10,095,638 = 7,571,729in. − lb(856kN .m )
BIsmIkar 4.6a nig b, kern point limit KW
r 2 187.5
kt = = = 9.95in.(253mm )
cb 18.84
r 2 187.5
kb = = = 8.86in.(225mm )
ct 21.16
Lower envelop
BIsmIkar 4.7a cMgayGtibrmaEdlExS cgs RtUv)andak;BIeRkam bottom kern edIm,IkarBarkug
RtaMgTajenAsrésxagelIbMputRtUv)ankMNt;dUcxageRkam
(i) kNþalElVg
= 4.53in.(115mm )
M D 2,490,638
amin = =
Pi 549,423
eyIgTTYl)an e1 = kb + amin = 8.86 + 4.53 = 13.39in.(340mm)
(ii) mYyPaKbYnénElVg
= 3.40in.(340mm )
1,867,979
amin =
549,423
eyIgTTYl)an e2 = 8.86 + 3.40 = 12.26in.(311mm )
(iii) elITMr
amin = 0
eyIgTTYl)an e3 = 8.86 + 0 = 8.86in.(225mm )
Upper envelop
BIsmIkar 4.7b cMgayGtibrmaEdlExS cgs RtUv)andak;BIeRkam top kern edIm,IkarBarkugRtaMg
TajenAsrésxageRkambMputRtUv)ankMNt;dUcxageRkam
(i) kNþalElVg
= 22.41in.(569mm )
M T 10,095,638
amin = =
Pe 450,526
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 123
35. T.Chhay
eyIgTTYl)an e1 = amax − kt = 22.41 − 9.95 = 12.46in.(316mm)
kMras;ebtugkarBarEdk = 3.0in.
cMNaMfa e1 minGacFMCag cb ebImindUecñaHeT tendon nwgenAxageRkAmuxkat;.
(ii) mYyPaKbYnénElVg
= 16.80in.(427mm )
7,571,729
amin =
450,526
eyIgTTYl)an e2 = 16.80 − 9.95 = 6.85in.(174mm )
(iii) elITMr
amin = 0
eyIgTTYl)an e3 = 0 − 9.95 = 9.95in.(− 253mm) ¬9.95in. sßitenABIelIExS cgs¦
sMrab;kargarGnuvtþn_ snμt;fakugRtaMgsrésTajGtibrmaeRkamlkçxNÐbnÞúkeFVIkarsMrab;kargarbegáIt
cgs envelope minRtUvFMCag f t = 6 f 'c = 465 psi sMrab;srésxagelI nigxageRkam. BIsmIkar 4.9a
cMNakp©itbEnßmEdlRtUvbEnßmeTAelI lower cgs envelope edIm,IGnuBaØateGaymankugRtaMgTajkMNt;
enAsrésxagelIKW
f (t ) Ac kb 465 × 377 × 8.86
e'b = = = 2.83in.(72mm )
Pi 549,423
dUcKña BIsmIkar 4.9b cMNakp©itEdlRtUvbEnßmeTAelI upper cgs envelop edIm,IGnuBaØateGayman
kugRtagTajkMNt;enAsrésxageRkamKW
f (b ) Ac kt 465 × 377 × 9.95
e't = = = 3.87in.(98mm )
Pe 450,526
dUcenH eyIgmantaragsegçbBI cgs envelope cMNkp©itdUcxageRkam³
Flexural Design of Prestressed Concrete Elements 124
36. NPIC
rUbTI 4>17 bgðajBI cgs envelope sMrab;kugRtaMgTajesμIsUnü nigkugRtaMgTajkMNt;enAkñúg
ebtug.
q> karkat;bnßykMlaMgeRbkugRtaMgenAEk,rTMr
Reduction of Prestress Force near Support
dUc)aneXIjBI]TahrN_ 4>3 nigEpñk K nig g xagelI straight tendon enAkñúg pretensioned
member GacbNþaleGaymankugRtaMgTajFMenAsrésxageRkArbs;ebtugenARtg;TMr edaysarGvtþ-
mankugRtaMgm:Um:g;Bt;Edl)anBITMgn;pÞal; nigbnÞúkbEnßm. eKmanviFIFmμtaBIrkñúgkarkat;bnßykugRtaMg
enARtg;muxkat;TMrEdlbNþalmkBIkMlaMgeRbkugRtaMg. viFITaMgBIrenaHKW³
- pøas;bþÚrcMNakp©itrbs;kabxøHedayelIkBYkvaeLIgeTAkan;tMbn;TMrdUcbgðajenAkñúgrUbTI 4>18
(a). viFIenHkat;bnßytMélm:Um:g;.
- eRsabkabxøHedaybMBg;)aøsÞiceTAkan;tMbn;TMr dUcbegðIjenAkñúgrUbTI 4>18(b). viFIenHkat;bnßy
EpñkénkugRtaMgepÞrrbs;kabenAcMgayxøHBImuxkat;TMrénFñwmeRbkugRtaMgTMrsamBaØ.
cMNaMfakabEdlelIkeLIgk¾RtUv)aneRbIenAkñúgFñwmeRbkugRtaMgElVgEvgEdlrgeRbkugRtaMgCa
eRkaypgEdr. eKminRtUvkarEpñkminCab;rbs; tendon edaylkçN³RTwsþI edayelIkvaeLIgelI. kMhat
bg;edaysarkMlaMgkkitbEnßmedaysarExSekagbBa©ÚleTAkñúgkarKNna b¤karviPaKmuxkat;.
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 125
37. T.Chhay
5> End Block at Support Anchorage Zones
k> karEbgEckkugRtaMg Stress Distribution
kugRtaMgsgát;cMcMnucd¾FMenAkñúgG½kSbeNþayekItmanenARtg;muxkat;TMrenAelIkMNat;d¾tUcénépÞ
rbs;cugFñwm ¬TaMgenAkñúg pretensioned beam nig post-tensioned beam¦ EdlbNþalmkBIkMlaMg
eRbkugRtaMgd¾FM. enAkñúg pretensioned beam bnÞúkepÞrcMcMnucrbs;kMlaMgeRbkugRtaMgeTAelIebtugEdl
B½T§CMuvijekIteLIgbnþicmþg²rhUtdl;vakøayeTACamanlkçN³BRgayesμIelIRbEvg lt BIépÞénmuxkat;TMr.
enAkñúg post-tensioned beam karEbgEck nigkarepÞrkMlaMgbnþicmþg²tamrebobenHminGaceFVI
eTA)aneT edaysarkMlaMgmanGMeBIedaypÞal;eTAelIépÞrbs;cugFñwmtamry³ bearing plate nig anchors.
ehIy tendon xøH b¤k¾TaMgGs;enAkñúg post-tensioned beam RtUv)anelIkeLIg b¤ draped eTAkan;srés
xagelItamry³EpñkénRTnugrbs;muxkat;ebtug.
edaysarkarpøas;bþÚrkugRtaMgsgát;tamG½kSBIcMcMnuceTABRgayesμIminsnSwm² vabegáIteGayman
kugRtaMgTajTTwg (transverse tensile stress) FMkñúgTisbBaÄr dUcenHehIy longitudinal bursting
cracks k¾ekItmanenA anchorage zone. enAeBlEdlkugRtaMgFMCagm:UDulkat;rbs;ebtug end block
Flexural Design of Prestressed Concrete Elements 126
38. NPIC
nwgeRbHtambeNþay elIkElgEteKdak;EdkbBaÄrsmRsb. TItaMgrbs; concrete-bursting stress nig
resulting bursting crack k¾dUcCa surface-spalling crack KWGaRs½ynwgTItaMg nigkarEbgEckkMlaMg
cMcMnuctamTisedkEdlGnuvtþedayEdkeRbkugRtaMgeTAelI end bearing plate.
eBlxøHeKcaM)ac;begáInRkLaépÞrbs;muxkat;eTArkTMredayeFVIkarBRgIkRTnugbnþicmþg²eGayesμI
TTwgrbs;søabenARtg;TMr kñúgeKalbMNgedIm,IeFVIkarelIk tendon eLIgelI ¬emIlrUbTI 4>19(a)¦. b:uEnþ
karekIneLIgRkLaépÞmuxkat;EbbenHmin)ancUlrYmkarBar bursting b¤ spalling crack eT ehIyvak¾min
manT§iBlkñúgkarkat;bnßykMlaMgTajtamTTwgenAkñúgebtugEdr. tamBit TaMglT§plénkarBiesaF
nigkarviPaKedayRTwsþIén three-dimension stress problem bgðajfakugRtaMgTajGacekIneLIg.
dUcenH eKRtUvkardak; anchorage reinforcement caM)ac;enAkñúgtMbn;epÞrkMlaMgkñúgTMrg;Edkkg
biTCit (closed ties b¤ stirrup) b¤]bkrN_ anchorage edaydak;B½T§CMuvijEdkeRbkugRtaMgemTaMgGs; nig
EdkBRgwgFmμtatambeNþay. rUbTI 4>20 bgðajBIKnøgkugRtaMgTaj nigKnøgkugRtaMgsgát;.
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 127
39. T.Chhay
x> RbEvgbgáb; nigRbEvgepÞrenAkñúgGgát;rgeRbkugRtaMgCamun
nigkarKNna Anchorage Reinforcement
Development and Transfer Length in Pretensioned Members and
Design of Their Anchorage Reinforcement
edaysarkMlaMgTaj (jacking force) RtUv)anRbElgeTAelIGgát;rgeRbkugRtaMgCamun enaH
kMlaMgeRbkugRtaMgRtUv)anepÞredaylkçN³DINamictamry³épÞb:HrvagEdkeRbkugRtaMg nigebtugeTAeb
tugEdlB½T§CMuvijEdkeRbkugRtaMg. PaBs¥itrvagEdkeRbkugRtaMg nigebtugelIRbEvgkMNt;rbs;Edk
eRbkugRtaMgepÞrkMlaMgeRbkugRtaMgcMcMnucsnSwm²eTAmuxkat;TaMgmUlrbs;ebtugRtg;bøg;EdlecjBI end
block eTAkan;kNþalElVg. RbEvgbgáb;kMNt;TMhMkMlaMgeRbkugRtaMgEdlGacekItmantambeNþayElVg.
RbEvgbgáb;kan;EtEvg kMlaMgeRbkugRtaMgkan;EtFM.
Ca]TahrN_ sMrab; 7-wire strand Ggát;p©it 1 / 2in. EdlmanRbEvgbgáb; 40in.(102cm) begáIt
kugRtaMg 180,000 psi(1,241MPa ) b:uEnþCamYynwgRbEvgbgáb; 70in.(178cm) begáItkugRtaMg 206,000 psi
(1,420MPa ) . BIrUbTI 4>21 vabgðajy:agc,as;faRbEvgbgáb; ld EdlbegáItkugRtaMgeBjeljCabnSM
rvagRbEvgepÞr (transfer length) lt nigRbEvgs¥itedaykarBt; (flexural bond length) l f .
1 ⎛ f pe ⎞
lt = ⎜ ⎟d b (xñat US) (4.10a)
1,000 ⎜ 3 ⎟ ⎝ ⎠
Flexural Design of Prestressed Concrete Elements 128
40. NPIC
⎛ f pe ⎞
lt = ⎜
⎜ 20.7 ⎟d b
⎟ ( xñat SI)
⎝ ⎠
f pe
b¤ lt =
3000
db (4.10b)
nig lf =
1
1,000
(
f ps − f pe d b ) ( xñat US) (4.10c)
lf =
1
6.9
(
f ps − f pe d b ) ( xñat SI)
Edl kugRtaMgenAkñμúgEdkeRbkugRtaMgenAeBl nominal strength
f ps =
f pe = eRbkugRtaMgRbsiT§PaBeRkayeBlxatbg;
d b = nominal diameter rbs;EdkeRbkugRtaMg
edaybBa¢ÚlsmIkar 4.10b nig 4.10c eyIg)an
1 ⎛ ⎞
(xñat US)
2
ld min = ⎜ f ps − f pe ⎟d b (4.10d)
1,000 ⎝ 3 ⎠
1 ⎛ ⎞
ld min =
6.9 ⎝
2
⎜ f ps − f pe ⎟d b
3 ⎠
( xñat SI)
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 129
41. T.Chhay
smIkar 4.10d eGaynUvRbEvgbgáb;caM)ac;Gb,brmasMrab;EdkeRbkugRtaMg. RbsinebIeKeRsab
EdkeRbkugRtaMgxøHeq<aHeTAkan;cugFñwmedIm,Ikat;bnßykugRtaMgs¥itenAEk,rxagcug enaHkugRtaMgepÞrenAkñúg
tMbn;enaHRtUv)ankat;bnßy ehIyeKcaM)ac;RtUveFVIkarEksMrYledaybegáInRbEvgbgáb; ld .
!> KNnaEdktMbn;epÞrenAkñúgFñwmrgeRbkugRtaMgCamun
Design of Transfer Zone Reinforcement in Pretensioned Beams
tamkarBiesaF Mattock et al. )anbegáItsmIkarEdl)anBIkarBiesaFsMrab;rkkMlaMgEdkkg
srub F dUcxageRkam³
Pi h
F = 0.0106 (4.11)
lt
Edl h CakMBs;rbs;FñwmrgeRbkugRtaMgCamun ehIy lt Ca transfer length. RbsinebIeKykkugRtaMg
mFümenAkñúgEdkkgRtwmBak;kNþalkugRtaMgGnuBaØatGtibrma f s rbs;Edk enaH F = 1 / 2( At f s ) .
edayCMnYsvacUleTAkñúgsmIkar 4.11 eyIgTTYl)an³
Ph
At = 0.021 i
f l
¬xñat Us¦ (4.12)
s t
At = 21,000 ¬xñat IS¦
Pi h
f s lt
Edl At CaRkLaépÞsrubrbs;Edkkg ehIy f s ≤ 20,000 psi(138MPa) sMrab;karRKb;RKgsñameRbH.
@> kareRCIserIsEdkenAkñúgFñwmrgeRbkugRtaMgCamun
Reinforcement Selection in Pretensioned Beams
]TahrN_ 4>5³ KNna anchorage reinforcement EdlRtUvkaredIm,IkarBar bursting crack b¤
spalling crack EdlekItmanenAkñúgFñwmén]TahrN_ 4>2.
dMeNaHRsay³ Pi = 376,110lb(1,673kN )
BIsmIkar 4.12 At = 0.021 Pi lh
fs t
BIsmIkar 4.10b RbEvgepÞrKW lt = ( f pe / 3,000)db . dUcenH edaysar f pe = 154,980 psi nig
d b = 1 / 2in. eyIgman
× 0.5 = 25.83in.(66cm )
154,980
lt =
3,000
dUcenH Ph
At = 0.021 i
f s lt
Flexural Design of Prestressed Concrete Elements 130
42. NPIC
edaysar f s ≤ 20,000 psi / eyIgTTYl)an
376,110 × 40
At = 0.021
20,000 × 25.83
(
= 0.61in.2 3.9cm 2 )
sakl,gEdkkgbiTCit #3
2 × 0.11 = 0.22in.2 ¬Ggát;p©it 9.5mm ¦
cMnYnEdkkgGb,brma = 0..22 = 2.78
0 61
eRbIEdkkg #3 cMnYnbIkgedIm,Ih‘MuB½T§EdkembeNþay. cgh‘uMB½T§EdkeRbkugRtaMgCamYy helical
steel wire elIRbEvgepÞr lt edIm,ITTYl)ankarepÞrEdlmanRbsiT§PaBl¥.
K> Post-tensioned Anchorage Zones: Linear Elastic and Strut-and-Tie
Theories
eKGacKit anchorage zone CamaDebtugEdlkMlaMgeRbkugRtaMgcMcMnucenARtg; anchorage
device BRgayCalkçN³smamaRttamTisTTwgeBjépÞTaMgmUlrbs;muxkat;ebtugtambeNþayElVg.
RbEvgrbs;tMbn;enHGnuvtþtameKalkarN_ St. Venant EdlkugRtaMgkøayCaBRgayesμIenAcMgayRbhak;
RbEhlmYyBImux anchorage device esμInwgkMBs; h rbs;muxkat;. RBIsTaMgmUlEdlman RbEvgepÞr h
Ca anchorage zone srub.
dUcenHtMbn;enHpSMeLIgedayBIrEpñk³
- General Zone: karraldalTUeTAéntMbn;enHRsedogKñanwg anchorage zone srub. dUcenH
RbEvglatsn§wgtambeNþayFñwmesμInwgkMBs;muxkat; h enAkñúgkrNIsþg;dar.
- Local Zone: tMbn;enHCaRBIsbEnßménebtugEdlB½T§CMuvij nigenABIxagmux anchorage device
Pøam² nigBI confining reinforcement. emIlépÞqUtenAkñúgrUbTI 4>22 (c) nigTMhMrbs;vaenA
kñúgrUbTI 4>22 (a). rUbenHbgðajBIkarEbgEckkugRtaMgTaj nigkugRtaMgsgát;enAkñúg local
zone nig stress contour rbs;vaEdlTTYl)anBI finite element analysis rbs; Rutgers test.
RbEvgrbs; tMbn;enHCatMélFMCageKkñúgcMeNamTTwgGtibrma b¤RbEvgrbs; anchorage device.
eKeRCIserIs confining reinforcement eBj anchorage zone edIm,IkarBar bursting nig
splitting EdlekItBIkMlaMgsgát;cMcMnucFMEdlbBa¢Úntamry³ anchorage device. elIsBIenH eKRtUvRtYt
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 131
43. T.Chhay
Binitü bearing stress enAelIebtugkñúg local zone EdlbNþalmkBIkMlaMgsgát;d¾FMenH edIm,IFana
favaminFMCag allowable compressive bearing stress rbs;ebtug.
Flexural Design of Prestressed Concrete Elements 132
44. NPIC
!> viFIKNnasMrab; General Zone Design Method for General Zone
eKmanviFIbIEdlGacKNna anchorage zone
- Linear Elastic Stress analysis approach Including Use of Finite Element: viFIenH
Bak;B½n§nwgkarKNnasßanPaBlMGitrbs;kugRtaMgdUcCa linearly elastic. karGnuvtþén finite
element method manPaBlM)akxøHkñúgkarbegáItKMrUrEdlmansñameRbHd¾RtwmRtUvenAkñúgebtug.
Et CamYynwgkarsnμt;d¾smRsbmYyeKGacTTYl)annUvlT§plEdlGacTTYlyk)anmYy.
- Equilibrium-Based Plasticity Approach dUcCa Strut-and Tie Method: viFI strut-and-tie
pþl;nUvKnøgd¾l¥rbs;kMlaMgeRbkugRtaMgEdlmanTMrg;dUcCaeRKOgbgÁúM truss EdlkMlaMgkñúgrbs;va
eKarBeTAtameKalkarN_lMnwgTUeTA. Ultimate load EdlBüakrN_edayviFIenHTTYlykBI
kar)ak;énbgÁúM strut b¤ tie NamYy. viFIenHEtgEtpþl;nUvlT§plEdlmansuvtßiPaBsMrab;kargar
Gnuvtþn_.
- Approach Method: viFIenHGnuvtþsMrab;muxkat;ctuekaNEdlmindac;.
@> viFIviPaK Linear Elastic sMrab;kMNt; Confining Reinforcement
Linear Elastic Analysis Method for Confining Reinforcement Determination
Anchorage zone rgnUvkugRtaMgbIkMritdUcbgðajenAkñúgrUbTI 4>22 (a) nig stress contour
zone:
- High bearing stress BImux anchorage device. eKRtUvkarebtugEdlmankarRtYtBinitüd¾Rtwm
RtUvedIm,IkarBarkar)ak;edaykugRtaMgsgát;énkMNat;rgkarsgÁt;dUcbgðajenAkñúgRkLaépÞqñÚtén
rUbTI 4>22(a) nig 4>22(c).
- Extensive tensile-bursting stress enAkñúg tensile contour areas EdlEkgeTAnwgG½kSrbs;
tendon dUcbgðajenAkñúgrUbTI 4>22(a) nig (b) nig enAkñúgrUbTI 4>23(b).
- kugRtaMgsgát;FMenAkñúgEdnkugRtaMg (stress field) RkLaépÞ D nig E enAkñúgrUbTI 4>22(a).
eKGaceRbI linear elastic stress analysis edIm,ITay)annUvTItaMgrbs;sñameRbH nigpþl;nUv
kar)a:n;sμan Rbhak;RbEhlmYyEdlGacTTYlyk)anBIrMhUrkugRtaMgeRkayeBleRbH. RkLaépÞEdk
TajRtUv)ankMNt;edIm,ITb;Tl;kMlaMgTajsrubEdlTTYl)anBIkarRbmUlpþMúkugRtaMgTajenAkñúgebtug.
eKRtUvbEnßmEdkrgkarsgát;enAkñúgtMbn;sgát; RbsinebIkMlaMgsgát;FM.
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 133
45. T.Chhay
EdlbgðajenAkñúgrUbTI 4>22 pþl;nUvkarKNnasßan
Linearly elastic finite element analysis
PaBrbs;kugRtaMgenAkñúg anchorage zone suRkitCag. b:uEnþ CMhanénkarKNnaRtUvkarefr³evlaeRcIn
Cag nigcMNayeRcInCag. lT§plRtUv)ankMNt;edaysarPaBBi)akkñúgkarbegáItKMrUEdlmansñameRbH
enAkñúgebtugd¾RtwmRtUv. eKGaceRbI nonlinear finite element analysis edIm,ITaynUv post-cracking
response.
Flexural Design of Prestressed Concrete Elements 134
46. NPIC
rUbTI 4>23 bgðajBI linearly elastic end block forces. vabgðajBIkMlaMg end-block
nigkugRtaMgsrésEdlbNþalBIkMlaMgeRbkugRtaMg Pi k¾dUcCatMélm:Um:g;Bt;sMrab;kMBs;eRbH y EdlGac
ekItmannImYy² BIelI)atFñwm CD . tMélm:Um:g;Gtibrma M max kMNt;TItaMgén horizontal bursting
crack. m:Um:g;enHRtUv)anTb;Tl;edaym:Um:g; couple EdlekItBIkMlaMgTaj T én vertical anchorage
zone reinforcement nigkMlaMgsgát; C Edlpþl;eday end-block concrete xN³EdlkMlaMgkat;tam
Tisedk V enARtg; crack spite surface RtUv)anTb;Tl;eday aggregate interlock force.
tamkarGegát Edkkg vertical anchorage zone Edlpþl;kMlaMg T RtUv)anEbgEckelItMbn;Edlman
TTwg h / 2 BIépÞxagcugrbs;Fñwm EdldUcCa X enAkñúgrUbTI 4>23 GacERbRbYlBI h / 5 eTA h / 4 .
BIsmIkarlMnwgrbs;m:Um:g;
M max
T= (4.13)
h−x
ehIyRkLaépÞrbs;EdkbBaÄrEdlRtUvkarsrubkøayCa
T
At = (4.14)
fs
EdlkugRtaMgEdk f s EdlRtUv)aneRbIenAkñúgkarKNnaenHminRtUvFMCag 20,000 psi(138.5MPa ) sMrab;
karRKb;RKgTTwgsñameRbH.
Casegçb nigCMnYseGay linear elastic finite element analysis eKGacTTYldMeNIrkar
Edl)anENnaM eTaHbICaminminsUvsuRkitdUckarKNna anchorage y:aglMGitEdlnwgpþl;eGayenA
kñúg]TahrN_ 4>6 Epñk (a) k¾eday.
#> Strut-and-Tie Method for Confining End-Block Reinforcement
Strut-and-tie concept KWQrelI plasticity approach Edl)a:n;RbmaNkMlaMgenAkñúg
anchorage zone edayes‘rIén strut sgát;Rtg; nig tie TajRtg;EdlP¢ab;KñaRtg;cMnucmYyEdleKehAfa
node edIm,IkøayeTACa truss Éktþa. kMlaMgsgát;RtUv)anTb;Tl;eday plastic compressive strut
ehIykMlaMgTajRtUv)anTb;Tl;edayEdkminEmneRbkugRtaMg b¤edayEdkeRbkugRtaMg. Yield strength
rbs; anchorage confining reinforcement RtUv)aneRbIedIm,IkMNt;RkLaépÞsrubrbs;EdkEdlcaM)ac;
eRbIenAkñúg anchorage block. rUbTI 4>24 bgðajBIkMlaMgeRbkugRtaMgcMp©it nigcakp©it P BImuxcMnucén
karGnuvtþkMlaMgTaMgenHtamry³ anchorage device eTAkan;cugén general zone EdlkugRtaMgkøayCa
rayesμItameKalkarN_ St. Venatn.
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 135
50. NPIC
eRkayBIekItmansñameRbHKYreGaycab;GarmμN_mk KnøgkugRtaMgsgát;enAkñúgebtug)anRbmUlpþúM
KñaeTACaExSRtg;EdlGacKitdUcCa straight compressive strut rgkMlaMgsgát;tamG½kSmYy. Srut TaMg
enHnwgkøayCacMENkrbs; truss ÉktþaEdkkugRtaMgTajemRtUv)anKitCa tension tie enAkñúg truss Éktþa
EdlmanTItaMgrbs; node RtUv)ankMNt;edayTisedArbs; compression strut. rUbTI 4>25 (a) bgðajBI
karbegáIt strut nigrUbTI 4>25(b) bgðajBI truss EdlekItBI strut-and-tie sMrab; multiple anchorage
enAkñúgmuxkat;GkSr T. rUbTI 2>26 segçbBIKMnitén strut nig tie enAkñúg anchorage zone. rUbTI
2>27 bgðajBI standard strut-and-tie truss sMrab;krNIcMp©it nigcakp©iténmuxkat;tan; nigmuxkat;man
søabEdleGayeday ACI 318-99 Code.
eKsnμt; tension tie enAkñúg truss sib,nimitþmancMgay h / 2 BI anchorage device. karsnμt;
enHGacGnuvtþeTA)anCamYynwgTItaMgrbs;kMlaMgTaj T enAkñúgrUbTI 4>23 én elastic stress-analysis
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 139
51. T.Chhay
approach. Epñk (b) én]TahrN_ 4>6 bgðajBIKnøgEdlsnμt;sMrab; anchorage zone enAkñúg I-beam
EdleKBicarNa.
⎛ a⎞
Tburst = 0.25 ∑ Psu ⎜1 − ⎟ (4.15a)
⎝ h⎠
d burst = 0.5(h − 2e ) (4.15b)
Edl ∑ Psu =plbUkénkMlaMg tendon emKuNsrub
a = kMBs;rbs; anchorage device b¤RkumeTalén closely-spaced device
e = cMNakp©itén anchorage device b¤Rkumén closely-spaced device BITIRbCMuTMgn;rbs;mux
kat;Fñwm
h = kMBs;rbs;muxkat;
eKeRbI anchorage device Ca closely-spaced device RbsinebIKMlatBIG½kSeTAG½kSrbs;vamin
FMCag 1.5 dgénTTwgrbs; anchorage device.
4. Allowable Bearing Stresses
GnuBaØatGtibrmaenARtg; anchorage device seating minRtUvFMCagtMél
Bearing stress
RsedogKñaBIrEdlTTYl)anBIsmIkar 4.16a nig 4.16b dUcxageRkam³
f b ≤ 0.7φf 'ci A / Ag (4.16a)
f b ≤ 2.25φf 'ci (4.16b)
Edl kMlaMg tendon emKuNGtibrma Pu EckCamYynwg effective bearing area Ab
fb =
f 'ci = ersIusþg;sgát;rbs;ebtugenAeBlrgkugRtaMg
A = RkLaépÞGtibrmaéncMENkrbs;épÞEdlRTEdlmanragFrNImaRtRsedogKñanwgRkLaépÞrg
bnÞúk ehIyRtYtsIuKña.
Ag = gross area rbs; bearing plate
Ab = effective net area rbs; bearing plate EdlRtUv)anKNnaedaydkRkLaépÞ As BIRkLa
épÞRbehagenAelI bearing plate.
smIkar 4.16a nig 4.16b mann½yEtRbsinebIeKdak; general zone reinforcement nigRbsinebIRbEvg
énkarlatsn§wgrbs;ebtugtambeNþayG½kSrbs; tendon BImux anchorage device esμIBIrdgRbEvgén
local zone y:agtic.
Flexural Design of Prestressed Concrete Elements 140
52. NPIC
X> KNnaEdk End Anchorage sMrab;FñwmeRbkugRtaMgrgkarTajCaeRkay
Design of End Anchorage Reinforcement for Post-tensioned Beams
]TahrN_ 4>6³ KNna end anchorage reinforcement sMrab; post-tensioned beam enAkñúg]TahrN_
4>2 EdleGayTMhM RbePT nigkarBRgayEdk. eRbIebtugTMgn;Fmμta f 'c = 5,000 psi(34.5MPa) .
snμt;facugFñwmCabøúkctuekaNEdllUtcUleTAkñúgElVg 40in.(104cm) BIeRkay anchorage
device bnÞab;mkkat;bnßykMras;RTnug 6in. . edaHRsaybBaðaedayeRbI (a) linear elastic stress
analysis method, (b) plastic strut-and-tie method. KUrKMrU truss Edl)ankMNt;.
dMeNaHRsay³
(a) edaHRsayeday linear elastic stress method³
!> begáItKMrUén tendon edaymancMNakp©it ee = 12.49in.(317mm) BI]TahrN_ 4>2.
cb = 18.84in.
dUcenHcMgayBIsrésxageRkamrbs;Fñwm = cb − ee = 6.35in.(161mm)
sMrab;cMgayTIRbCMuTMgn;rbs; tendon Ggát;p©it 1 / 2in. cMnYn 13 edIm = 6.35in. BIsrésxageRkamFñwm
sakl,gkartMerobCaCYredkdUcxageRkam
CYredkTI 1³ 5 tendon enAcMgay 2.5in
CYredkTI 2³ 5 tendon enAcMgay 7.0in.
CYredkTI 3³ 3 tendon enAcMgay 11.5in.
cMgayénTIRbCMuTMgn;rbs; tendon = 5 × 2.5 + 5 ×13.0 + 3 × 11.5 ≅ 6.35in.
7
O.K.
@> Ultimate forces enAkñúgCYredkén tendon nig bearing capacity rbs;ebtug
kMlaMg Pu1 CYredkTI 1³ 5 × 0.153 × 270,000 = 206,550lb(919kN )
kMlaMg Pu 2 CYredkTI 2³ 5 × 0.153 × 270,000 = 206,550lb(919kN )
kMlaMg Pu3 CYredkTI 3³ 3 × 0.153 × 270,000 = 123,930lb(551kN )
#> Elastic analysis énkMlaMg
EckkMBs;FñwmCacMerokEdlmankMBs; 4in. dUcbgðajenAkñúgrUbTI 4>28(a) nigsnμt;fakugRtaMg
ebtugrbs;cMeroknImYy²esμInwgkugRtaMgenARtg;G½kSrbs;cMerokenaH. bnÞab;mkKNnakMeNInm:Um:g;Edl
bNþalBIkugRtaMgxagkñúg nigkMlaMgeRbkugRtaMgxageRkA Pi eFobnwgbøg;edknImYy²edIm,IkMNt; net
moment enAelImuxkat;. Net moment GtibrmanwgkMNt;TItaMgrbs; potential horizontal bursting
crack nigEdkEdlRtUvdak;edIm,IkarBarsñameRbHEdlnwgekItmanenaH. edayeRbIsBaØabUk (+) sMrab;
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 141
53. T.Chhay
m:Um:g;vilRsbRTnicnaLika. BI]TahrN_ 4>2 kMlaMgeRbkugRtaMgedImmuneBlxatbg;KW Pi = 376,110lb
(1,673kN ) . BIrUbTI 4>28 m:Um:g;xagkñúgebtugenARtg;bøg; 4in. BIsrésxageRkamKW
M c 4 = 2,117 × 4 × 18 × (2in.) = 304,848in. − lb
= 0.3 ⋅ 10 6 in. − lb(34.4kN .m )
nigenARtg;bøg; 8in. BIsrésxageRkamKW
18 + 10
M c8 = 2,117 × 4 × 18 × (6in.) + 1,851 × 4 × × (2in.)
2
= 1,121,856in. − lb = 1.12 ⋅ 10 6 in. − lb(127 kN .m )
m:Um:g;kMlaMgeRbkugRtaMgenARtg;bøg; 8in. BIsrésxageRkamKW
M c8 = 376,110 × (8 − 6.35) = −620,582in. − lb
= −0.62 ⋅ 10 6 in. − lb(70.1kN .m )
Net moment KW = 1.12 ⋅106 − 0.62 ⋅106 = 0.50 ⋅106 in. − lb(56.6kN .m)
tamrebobdUcKña eyIgGacrk net moment sMrab;cMerokd¾éTeTot ehIytMélrbs;vaRtUv)anerobdak;enA
kñúgtarag 4>5. BItaragenH net moment GtibrmaKW + M max = +0.75 ⋅106 in. − lb(84.6kN .m)
enARtg;bøg;edk 6.35in. BIsrésxageRkamrbs;Fñwm (bursting potential crack effect) ehIy
Flexural Design of Prestressed Concrete Elements 142
54. NPIC
− M max = −0.20 ⋅ 106 in. − lb enARtg;bøg; 24in.(64cm) BIxagelIsrésxageRkamrbs;Fñwm (spalling
potential crack effect) .
$> KNna anchorage reinforcement
BIsmIkar 4>11 nigedaysnμt;vaG½kSrbs;kMlaMgTajbBaÄr T KWenARtg;cMgay x ≈ 15in.
eyIgTTYl)an
M max 0.75 ⋅ 106
T= = = 30,000lb(133kN )
h−x 40 − 15
edayGnuBaØatkugRtaMgEdkGtibrma f s = 20,000 psi ¬kUdGnuBaØat 0.60 f y = 36,000 psi ¦
Bursting zone reinforcement KW
At =
Tb 30,000
=
f s 20,000
(
= 1.50in 2 968mm 2 )
dUcenH sakl,gEdkkgbiTCit #3 ³ (As = 2 × 0.11 = 0.22in.2 )
cMnYnEdkkgEdlRtUvkar = 1..50 = 6.82 kg
0 22
eRbIEdkkg #3 cMnYn 6 kg bEnßmBIelIEdkkgsMrab;Tb;nwgkMlaMgkat;.
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 143
55. T.Chhay
Spalling zone force
− 0.2 × 106
Ts = = 8,000lb
40 − 15
dUcenH T
As = s =
8,000
f s 20,000
( )
= 0.40in.2 250mm 2
dUcenH eyIgman
cMnYnEdkkg #3 EdlRtUvkar = 0..40 = 1.82 kg
0 22
eRbIEdkkg #3 cMnYnBIrkgbEnßmeTot.
dUcenH cMnYnEdkkgsrub = 6.82 + 1.82 + 4 = 12.64 kg
eRbIEdkkgbiTCit #3 cMnYn 12 kg. dak;EdkkgbBa©ÚleTAkñúgtMbn;sgát;enAkñúgrUbTI 4>23.
dak;Edkkg #3 KMlatBIKña 3in. edayKitBIG½kSeTAG½kS edayEdkkgTImYycab;epþImCamYynwgKM
lat 3in. BIcugFñwm. ehIy dak;Edk #3 RbEvg 10in. cMnYn 4 edImEdlmanKMlatBIKña 3in. KitBIG½kSeTA
G½kS nigmanKMlat 2in. BIépÞxagcugRtg;TItaMg anchorage edaysarsñameRbHGacekItmantamTis
bBaÄr nigTisedk. RbsinebImantMrUvkarrbs;plitkr eKRtUvbEnßm spiral reinforcement BIxageRkam
anchor.
(b) edaHRsayeday plastic Strut-and-tie method³
!> begáItKMrUén tendon EdlmancMNakp©it ee = 12.49in.(317mm)
BI]TahrN_ 4>2 cb = 18.84in. dUcenHcMgayBIsrésrbs;Fñwm = cb − ee = 6.35in.(161mm)
sMrab;cMgayTIRbCMuTMgn;rbs; strand Ggát;p©it 1 / 2in. cMnYn 13 edImEdlesμInwg 6.35in BIsrés
xageRkamrbs;Fñwm sakl,gkartMerob tendon CaCYredkEdlmancMgayBIsrésxageRkamdUcteTA³
CYredkTI 1³ 5 tendon enARt;g 2.5in.
CYredkTI 2³ 5 tendon enARt;g 7.0in.
CYredkTI 3³ 3 tendon enARt;g 11.5in.
cMgayénTIRbCMuTMgn;rbs; tendon = 5 × 2.5 + 5 ×13.0 + 3 ×11.5 ≅ 6.35in. O.K.
7
@> Ultimate force enAkñúgCYredkrbs; tendon nig bearing capacity rbs;ebtug
kMlaMg Pu1 CYredkTI 1³ 5 × 0.153 × 270,000 = 206,550lb(919kN )
kMlaMg Pu 2 CYredkTI 2³ 5 × 0.153 × 270,000 = 206,550lb(919kN )
kMlaMg Pu3 CYredkTI 3³ 3 × 0.153 × 270,000 = 123,930lb(551kN )
Flexural Design of Prestressed Concrete Elements 144
56. NPIC
kMlaMgsgát; ultimate srub = 206,550 + 206,550 + 123,930 = 537,830lb(2,389kN )
RkLaépÞsrubrbs; rigid bearing plate EdlRT Supreme 13-chucks anchorage device
= 14 × 11 + 6 × 4 = 178in.2 ( cm 2 )
113
Bearing stress Cak;Esþg f b = = 3020 psi(20.8MPa )
537,380
178
BIsmIkar 4.16(a) nig (b), bearing pressure GnuBaØatGtibrmaenAelIebtugKW
f b ≤ 0.7φf 'ci A / Ag
f b ≤ 2.25φf 'ci
snμt;fa ersIusþg;ebtugdMbUgenAeBlEdlrgkugRtaMgKW f 'ci = 0.75 f 'c
= 0.75 × 5,000 = 7,750 psi
RkLaépÞcMp©it A rbs;ebtugEdlman bearing plate ≅ 18 ×14 + 10 × 7 = 322in.2
Bearing stress GnuBaØat f b = 0.70 × 0.90 × 3,750
322
= 3,178 psi > 3,020 psi O.K.
178
Bearing stress BIsmIkar 4.14(b) Gt;lub.
#> KUr strut-and-tie model
RbEvgcMgaysrub a enAkñúgrUbTI 4>25 rvagkMlaMg Pu1 nig Pu3 = 11.5 − 2.5 = 9.0in.
dUcenHcMgay a / 2 BImux anchorage = 9.0 / 2 = 4.5in.
sg; strut-and-tie edaysnμt;vadUcbgðajenAkñúgrUbTI 4>29.
TMhMFrNImaRtsMrab;rkbgÁúMkMlaMgedkBI tie 1 − 2 nig 2 − 3 EdlmantMélkUtg;sg; 26.5 / 15.5
nig 13.0 / 15.5 erogKña. BIsþaTic viPaK truss enAkñúgrUbTI 4>29 edayTTYl)ankMlaMgGgát;dUcxag
eRkam³
= 211,982lb(942kN ) rgkarTaj
26.5
tie 1 − 2 = 123,930 ×
15.5
= 173,235lb(728kN ) rgkarTaj
13
tie 2 − 3 = 206,550 ×
15.5
eRbItMélEdlFMCagkñúgcMeNamtMélTaMgBIredIm,IeRCIserIsEdkkgbiTCitEdlrgkarTaj.
sakl,gEdkkg #3 Edlman tensile strength kñúgmYykg = φf y Av
= 0.90 × 60,000 × 2(0.11) = 11,880lb
cMnYnEdkkgEdlRtUvkar =
211,982
11,880
= 17.8
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 145
57. T.Chhay
sMrab;EdkkgrgkarTaj a − b − c enAkñúgrUbTI 4>29 eRbIkMlaMg Pu = 173,235lb edIm,Idak;EdkkgbBaÄr
#4 BIxagmux anchorage device. cab;epþImEdkkgTImYyenAcMgay 1 1 in. BIxagcug rigid steel plate
2
EdlepÞrkMlaMgBI anchorage device eTAebtug.
cMnYnEdkkg = 0.9 × 60,000 × 2(0.20) = 8.0
173,235
eRbIEdkkg #4 cMnYn 8 kgEdlmancMgayBIKña 1 14 in. BIG½kSeTAG½kS ¬12.7mm @ 32mm ¦ Edl
manEdkkgTImYycab;epþImenAcMgay 1 12 in. BIxagmux anchorage device.
eKRtUvkarEdkkgEt 13 CMnYseGay 17.8 Edl)anBIkarKNna edaysarEpñkrbs;tMbn;RtUv)an
Tb;Tl;edayEdkkg #4 . eRbIEdkkg #3 cMnYn 13 EdlmanKMlatBIKña 2 12 in. BIG½kSeTAG½kS ¬12.7mm
@ 57 mm ¦ bnÞab;BIEdkkg #4 EdlmancMgaysrubTaMgGs; 40in.(104cm ) .
cMNaMfaviFIenHRtUvkar confining tie eRcInCag elastic solution kñúgEpñk (a). rUbTI 4>30
bgðajBI anchorage zone confining reinforcement lMGitEdl)anBI strut-and-tie analysis.
Flexural Design of Prestressed Concrete Elements 146
58. NPIC
6> KNnaFñwmsmasrgkarBt; Flexural Design of Composite Beams
muxkat;smas FmμtaCaeRKOgbgÁúMeRbkugRtaMgcak;Rsab;EdlenABIelIva kMralxNÐRtUv)ancak;enA
kardæan ehIyvaeFVIkarCamYyKña ¬rUbTI 4>31¦. eBlxøH eKTl; prestressed element kñúgGMLúgeBlcak;
nigEfTaM situ-cast top slab. kñúgkrNIEbbenH TMgn;kMralxNÐeFVIGMeBIEtelImuxkat;smas Edlmanm:U
Dulmuxkat;FMCagmuxkat;cak;Rsab;. dUcenH karKNnakugRtaMgebtugRtUv)anykmkKitenAkñúgkarKNna.
karEbgEckkugRtaMgebtugEdlbNþalBIGMeBIsmasRtUv)anbgðajenAkñugrUbTI 4>32.
k> krNIkMralxNÐminmanTl; Unshored Slab Case
BIsmIkar 4.2a nig b smIkarkugRtaMgsrésebtugxageRkAbMputmuncak;kMralxagelIKW
Pe ⎛ ect ⎞ M D + M SD
ft =− ⎜1 − 2 ⎟ − (4.17)
Ac ⎝ r ⎠ St
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 147
59. T.Chhay
P ⎛ ec ⎞ M + M SD
nig f b = − e ⎜1 + 2b ⎟ + D
Ac ⎝ r ⎠ Sb
(4.18)
Edl S t nig Sb Cam:UDulmuxkat;rbs;muxkat;cak;Rsab;Etb:ueNÑaH ehIy M SD Cam:Um:g;dak;BIelIbEnßm
dUcCaebtugkMral.
eRkayeBlkMralcak;enAnwgkEnøgkkrwg ehIyvaGaceFVIkarlkçN³smasmk vaGacmanm:UDul
muxkat; Sct nig Scb FMCagmun CamYynwgkarrMkileLIgelIeTArksrésxagelIrbs;ExS cgc. kugRtaMg
srésebtugcUlrYmCamYynwgsmIkar 4.17 nig 4.18 sMrab;srésxagelI nigxageRkamrbs;Epñkcak;
Rsab;rbs;muxkat;smas ¬nIv:U AA enAkñúgrUbTI 4>32(e)¦ KW
⎛ ect ⎞ M D + M SD M CSD + M L
Pe
ft =− ⎜1 − 2 ⎟ − − (4.19a)
⎝Ac r ⎠ St Sc t
P ⎛ ec ⎞ M + M SD M CSD + M L
nig f b = − e ⎜1 + 2t ⎟ + D
Ac ⎝ r ⎠ Sb
+
S cb
(4.19b)
Edl M CSD CabnÞúkefrsmasdak;BIelIbEnßmeRkayeBldMeLIg dUcCaenAeBleFVIkar. ehIy Sct nig
Scb Cam:UDulmuxkat;rbs;muxkat;smasenAnIv:UénsrésxagelI nigxageRkam erogKña rbs;muxkat;cak;
Rsab;.
kugRtaMgenAnIv:UsrésxagelI nigxageRkamrbs;kMralcak;enAnwgkEnøg ¬nIv:U BB nig AA rbs;mux
kat; 4>32 (e)¦ KW
M CSD + M L
f ts = − t
(4.20a)
S cb
Flexural Design of Prestressed Concrete Elements 148
60. NPIC
+ ML
nig M
f bs = − CSD
Sbcb
(4.20b)
Edl M CSD + M L Cam:Um:g;bEnßmEdlekIneLIgeRkayeBlekItmanskmμPaBsmas ehIy Scb nig Sbcb
t
Cam:UDulmuxkat;rbs;muxkat;smassMrab;srésxagelI nigxageRkam AA nig BB erogKña enAkñúgrUbTI
4>32(e).
x> krNIkMralxNÐTl;eBj Fully Shored Slab Case
kñúgkrNIkMralcak;enAkEnøgRtUv)anRTeBjrhUtdl;ekItmanskmμPaBsmas kugRtaMgsrés
ebtugmuneBlRT nigmuneBlcak;ebtugkMralxagelIEdlkøayBIsmIkar 4.18 nig 4.19KW
⎛ ect ⎞ M D
Pe
ft =− ⎜1 − 2 ⎟ − t (4.21a)
⎝
Ac r ⎠ S
P ⎛ ec ⎞ M
nig f b = − e ⎜1 + 2b ⎟ + D
Ac ⎝ r ⎠ Sb
(4.21b)
eRkayeBlkMralxagelIcak;rYc ehIyskmμPaBsmasekItmanenAeBlebtugkkrwg smIkar 4.19a nig b
sMrab;FñwmEdlRtUv)anRTeRkayeBldMeLIgnwgkøayeTACa
⎛ ect ⎞ M D M SD + M CSD + M L
Pe
ft =− ⎜1 − 2 ⎟ − t − (4.22a)
⎝
Ac r ⎠ S t
Sc
P ⎛ ecb ⎞ M M + M CSD + M L
nig f b = − e ⎜1 + 2 ⎟ + D + SD
Ac ⎝ r ⎠ Sb S cb
(4.22b)
cMNaMfaeKRtUveFVIkarRtYtBinitüsMrab;kugRtaMgkat;tamTisedkEdlekItmanenARtg;épÞb:HrvagebtugEdl
cak;enAnwgkEnøg CamYynwgFñwmcak;Rsab; ¬nwgbgðajenAkñúgCMBUk 5¦.
K> TTwgsøabRbsiT§PaB Effective Flange Width
karKNnamuxkat;Ggát;ebtugeRbkugRtaMgrgkarBt; 149
61. T.Chhay
edIm,IkMNt;skmμPaBsmastamRTwsþIEdlTb;Tl;kugRtaMgBt; eKRtUveFVIkarkMNt;TTwgkMralxNÐ
EdlGaccUlrYmy:agmanRbsiT§PaBenAkñúgekIneLIgPaBrwgRkaj (stiffness) EdlTTYl)anBIskmμPaB
smas.
rUbTI 4>33 nigtarag 4>6 eGaynUvtMrUvkarrbs; ACI nig AASTHO sMrab;kMNt;TTwgsøabxag
elIRbsiT§PaB (effective top slange width) rbs;muxkat;smas. RbsinebIersIusþg;rbs;ebtugEdlcak;
BIxagelIxusBIersIusþg;rbs;muxkat;cak;eRsc eKRtUvEktMrUvTTwg b edayKitBIm:UDuleGLasÞicxusKñarbs;
ebtugTaMgBIr edIm,IFanafabMErbMrYlrageFobrbs;sMPar³TaMgBIrenARtg;épÞb:HdUcKña. TTwgEksMrYlrbs;
kMralxagelIsMrab;KNnam:Um:g;niclPaBsmas I cc KW
bm =
Ect
(b ) = ncb (4.23)
Ec
Edl m:UDuleGLasÞicrbs;ebtugEdlcak;BIxagelI
Ect =
Ec = m:UDuleGLssÞicrbs;ebtugcak;Rsab;
enAeBlEdlkMNt;TTwgEksMrYl bm rYcehIy eKRtUvBicarNaersIusþg;ebtugrbs;muxkat;smasTaMgmUlCa
ersIusþg;EdlFMCag.
7> Summary of Step-by-Step Trial-and Adjustment Procedure
for the Service-Load Design of Prestressed member
!> eGaynUvGaMgtg;sIuetbnÞúkefrEdldak;BIelIbEnßm WSD / GaMgtg;sIuetbnÞúkGefr WL / RbEvg
kMNt; nigkMBs;kMNt;/ ersIusþg;sMPar³ f pu / f 'c / RbePTebtug nigeBlxøHRbePTeRbkug
RtaMg dUcCaTajCamun b¤CaeRkay.
Flexural Design of Prestressed Concrete Elements 150