Iv.flexural design of prestressed concrete elements

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

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

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


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;TMrsmBaØ¦
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


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 ⎟
⎝ ⎠


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


T.Chhay

cMNakp©itEdlRtUvkarrbs;EdkeRbkugRtaMgenARtg;muxkat;eRKaHfñak; dUcCamuxkat;kNþalElVg KW
( )S
t
MD
ec = f ti − f ci + (4.4c)
P i Pi
Edl f ci CakugRtaMgrbs;ebtugenAeBlepÞrRtg;nIv:UénTIRbCMuTMgn; cgc rbs;muxkat;ebtug ehIy
Pi = f ci Ac
dUcenH
f ci = f ti −
ct
( f ti − f ci ) (4.4d)
h


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’)


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.


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.

T.Chhay


NPIC


T.Chhay


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


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


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


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 )


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


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

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 .


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.


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


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


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


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


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.


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


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¦.


T.Chhay


NPIC


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.

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.


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 )


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


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


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;.


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


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;.


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 ⎟ ⎝ ⎠


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)


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


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


T.Chhay

Binitü bearing stress enAelIebtugkñúg local zone EdlbNþalmkBIkMlaMgsgát;d¾FMenH edIm,IFana
favaminFMCag allowable compressive bearing stress rbs;ebtug.


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.


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.


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.


T.Chhay


NPIC


T.Chhay


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


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.


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 )
#> 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;


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


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;.

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.
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 Pu 2 CYredkTI 2³ 5 × 0.153 × 270,000 = 206,550lb(919kN )


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


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.


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


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


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


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.

Iv.flexural design of prestressed concrete elements

Iv.flexural design of prestressed concrete elements

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