Xix introduction to prestressed concrete

T.Chhay NPIC

XVIII. esckþIENnaMBIebtugkugRtaMg
Introduction to Prestressed Concrete

1> ebtugeRbkugRtaMg Prestressed Concrete

k> eKalkarN_énkareFVIeRbkugRtaMg Principles of Prestressing
karGnuvtþeRbkugRtaMgeTAelIGgát;eRKOgbgáúMKWCakarbegáItkugRtaMgGciéRnþy_xagkñúgEdlmanGMeBI
RbqaMgnwgkugRtaMgTajenAkñúgebtugEdl)anbnÞúkxageRkA. karGnuvtþeRbkugRtaMgenHbegáItCaEdnkug
RtaMgEdlGgát;GacTb;Tl;)any:agmansuvtßiPaB. eKGacGnuvtþkMlaMgeRbkugRtaMgmun b¤kñúgeBldMNal
KñaénkarGnuvtþbnÞúkxageRkA. kugRtaMgenAkñúgGgát;eRKOgbgÁúMRtUvEtenAsl; ¬RKb;TIkEnøg nigsMrab;RKb;
sßanPaBénkardak;bnÞúk¦ enAkñúgEdnkMNt;rbs;kugRtaMgEdlsMPar³ GacRTRTg;)anKμanTIkMNt;. CaTU
eTA karGnuvtþkugRtaMg ¬PaKeRcInCakugRtaMgsgát;¦ RtUv)anbegáIteLIgeday high-strength steel
tendon EdlrgkarTaj nig anchor eTAnwgGgát;ebtug. kugRtaMgRtUv)anepÞreTAebtugeday bond

tamépÞrbs; tendon b¤eday anchorage enAxagcugrbs; tendon.
edIm,IgayRsYlkñúgkarBnül; cUrBicarNaFñwmmYyEdleFVIBIebtugsuT§ ehIyFñwmenaHRtUvRTnUv
bnÞúkTMnajxageRkA (external gravity load) dUcbgðajkñúgrUbTI 19>1 a. muxkat;FñwmRtUv)aneRCIs
erIsCamYynwg tensile flexural stress EdlCalkçxNÐeRKaHfñak;sMrab;karKNna dUcenHeKTTYl)an
muxkat;EdlminmanlkçN³esdækic©. mUlehtuKWedaysarebtugxøaMgkñúgkarsgát;CagkarTaj. flexural
tensile strength b¤m:UDuldac; (module of rupture) rbs;ebtug f r esμInwg 0.62 f 'c ¬rUbTI 19>1a¦.

kñúgkarKNnaebtugGarem:Fmμta eKminKit tensile strength rbs;ebtugeT ehIyEdksrés
RtUv)andak;enAkñúgtMbn;Tajrbs;ebtugedIm,ITb;Tl;nwgkugRtaMgTaj b:uEnþebtugTb;Tl;nwgkugRtaMgsgát;
¬rUbTI 19>1 b¦.
kñúgkarKNnaebtugeRbkugRtaMg kugRtaMgsgát;dMbUgRtUv)anGnuvtþeTAkñúgFñwmedIm,IeGaymanGMeBI
Tb;nwgkugRtaMgTajEdlekIteLIgedaysarbnÞúkxageRkA ¬rUbTI 19>1 c¦. RbsinebIkugRtaMgEdldak;
dMbUgenHesμInwgkugRtaMgTajenAsréseRkambMput enaHkugRtaMgTaMgBIrRtUv)anlubecal b:uEnþkugRtaMg
sgát;enAsrésEpñkxagelIbMputnwgmantMélDub. kñúgkrNIenH muxkat;Ggát;TaMgmUlrgkarsgát;. Rb

esckþIENnaMBIebtugeRbkugRtaMg 560

T.Chhay NPIC

sinebIkugRtaMgsgát;EdlGnuvtþdMbUgtUcCagkugRtaMgTajenAsrésEpñkxageRkambMput enaHsrésenA
EpñkxageRkamenHrgkarTaj ÉsrésEpñkxagelIbMputrgkarsgát;.


T.Chhay NPIC

kñúgkarGnuvtþ Ggát;ebtugGacrgeRbkugRtaMgtamviFImYykñúgcMeNamviFIxageRkam³
!> karTajeRkay (Posttensioning): kñúgkareFVI posttensioning, eKTaj steel tendon
eRkayeBlebtugRtUv)ancak; nigkkrwg. kareFVI posttensioning RtUv)aneFVIeLIgtamviFI
saRsþdUcteTA³ dMbUgeKeRbI hydraulic jack Taj steel wire b¤ strand eGaylUt bnÞab;
mkCMnYs jack eday anchorage EdlGacrkSaeGay steel strand enAEtrgkarTaj. CaTU
eTA tendon RtUv)aneFVIeLIgBI wire, strand b¤ bar. eKGacTaj wire nig strand CaRkum)an
EteKTaj bar mþg)anEtmYy. kñúgdMeNIrkareFVI posttensioning, eKdak; steel tendon
eTAkñúgBum<muneBlcak;ebtug ehIy tendon RtUv)ankarBarkars¥itCab;eTAnwgebtugeday
waterproof paper wrapping b¤ metal duct (sheath). tendon Edls¥itCab;eTAnwg

ebutgRtUv)aneKehAfa boded tendon. Unbonded tendon/ RtUv)andak;edayKμan grout
b¤RtUv)anlabeRbg.
@> karTajmun (pretensioning): kñúgkareFVI prettensioning eKTaj steel tendon muneBlcak;
ebtug. eKTb; tendon CabeNþaHGasnñeday abutment ehIyeKkat;va bnÞab;ebtugRtUv)an
cak; nigkkrwg. kMlaMg prestessing RtUv)anepÞreTAebtugeday PaBs¥itenAtamRbEvgrbs;
tendon. CaTUeTA eKeRcIneFVI prettensioning enAkñúgkardæan b¤eragcRkpliteRKOgbgÁúMebtug

eRbkugRtaMgcak;eRsc EdlmankMralrwgmaMCaGciéRnþy_.
#> kareFVIeRbkugRtaMgxageRkA (external prestressing): kñúgkareFVI external pretessing, eK
GnuvtþkMlaMgeRbkugRtaMgeday flat jack EdlRtUv)andak;enAcenøaHcugGgát;ebtug nig
permanent rigid abutments. Ggát;minman prestressing tendon dUcviFITaMgBIrxagelI

EdleKGacehAfakareFVIeRbkugRtaMgxagkñúgeT. External prestressing mingayRsYlkñúg
karGnuvtþeT edaysar shrinkage nig creep enAkñúgebtugEdlnaMeGaymankarkat;bnßy
kugRtaMgsgát;EdlGnuvtþdMbUg.
Profile rbs; tenden GacRtg; ekag b¤ragrgVg;GaRs½yeTAelIkarKNnaGgát;eRKOgbgÁúM.

CaTUeTAeKeRbI straight tendon enAkñúg solid slab nig hollow-cored slab b:uEnþeKeRbI bent tendon
enAkñúgFñwm nigGgát;eRKOgbgÁúMPaKeRcIn. eKeRbI circular tendon enAkñúgeRKOgbgÁúMEdlmanTMrg;mUldUc


T.Chhay NPIC

Ca tank, silo nig pipe. eKGacGnuvtþkMlaMgeRbkugRtaMgEtkñúgmYydMNakkal b¤eRcIndMNakkaledIm,I
karBarebtugkMurGayrgkugRtaMgelIs.
eK)anbegáItnUvRbB½n§eRbkugRtaMgCaeRcIn EdlkñúgcMeNamenaHman Freyssinet, Magnel Blaton,
B.B.R.V., Dywidag, CCL, Morandi, VSL, Western Concrete, Prescon, nig INRYCO. eBlxøH

eKCYbnUvbBaðakñúgkareRCIserIsRbB½n§eRbkugRtaMgsMrab;kargarBiess.
visVkrKYrBicarNanUvktþacMbgbIEdlnaMdl;kareRCIserIsRbB½n§enH³
!> GaMgtg;sIueténkMlaMgeRbkugRtaMgEdlRtUvkar
@> ragFrNImaRtrbs;muxkat; nigKMlatEdlGacmansMrab; tendon
#> tMélénRbB½n§eRbkugRtaMg ¬sMPar³ nigkMlaMgBlkmμ¦
]TahrN_xageRkambgðajBIlkçN³Biessrbs;ebtugeRbkugRtaMg.

]TahrN_ 19>1³ sMrab;FñwmTMrsamBaØEdlbgðajenAkñúgrUbTI 19>2 cUrkMNt;kugRtaMgRtg;muxkat;
kNþalElVgEdlbNþalmkBITMgn;pÞal;xøÜnva nigkrNIénkardak;bnÞúk nigeRbkugRtaMg³
!> bnÞúkGefrBRgayesμI 13.15kN / m
@> bnÞúkGefrBRgayesμI 13.15kN / m nigkMlaMgsgát;tambeNþaycMp©it P = 1132kN
#> bnÞúkGefrBRgayesμI 30.61kN / m nigkMlaMgsgát;tambeNþaycakp©it P = 1132kN Edl
manGMeBIRtg;cMNakp©it e = 10cm
$> bnÞúkGefrBRgayesμI 39.28kN / m nigkMlaMgsgát;tambeNþaycakp©it P = 1132kN Edl
manGMeBIRtg;cMNakp©itGtibrmasMrab;muxkat; e = 15cm
%> bnÞúkGefrGtibrmaenAeBl P = 1132kN EdlmanGMeBIRtg; e = 15cm
eRbI b = 30cm / h = 60cm / f 'c = 31MPa nigkugRtaMgGnuBaØat f 'c = 14.14MPa
dMeNaHRsay³
!> kugRtaMgEdlbNþalmkEtBIbnÞúkefr nigbnÞúkGefr
bnÞúkpÞal;rbs;Fñwm = (0.3 × 0.6)24 = 4.32kN / m
wL2 4.32(7.2 )2
m:Um:g;bnÞúkefr M D.L. = 8 = 8 = 28kN .m
kugRtaMgenARtg;srésEpñkxageRkAbMputEdlbNþalBIbnÞúkefrKW

T.Chhay NPIC

Mc M (h / 2) 6 M
σ= = 3 =
I bh / 12 bh 2
6 × 28
σD = 10 − 3 = ±1.56MPa
0.3 × 0.6 2

m:Um:g;EdlbNþþalBIbnÞúkGefr L1 = 13.15kN / m KW
13.15 × 7.2 2
M L. L. = = 85.2kN .m
8
kugRtaMgEdlbNþalBIbnÞúkGefrKW
6M 6 × 85.2
σ L1 = = 10 − 3 = ±4.73MPa
bh 2
0.3 × 0.6 2

edayeFVIkarbUkbBa©ÚkkugRtaMgEdl)anBIbnÞúkefr nigbnÞúkGefr ¬rUbTI 19>2 a¦ eyIg)an
kugRtaMgxagelI = −1.56 − 4.73 = −6.29MPa ¬rgkarsgát;¦
kugRtaMgxageRkam = +1.56 + 4.73 = 6.29MPa ¬rgkarTaj¦
edaysarkugRtaMgTajFMCagm:UDuldac;rbs;ebtug f r = 0.62 31 = 3.45MPa dUcenHFñwmnwg)ak;.
@> kñúgkrNIEdlkugRtaMgbNþalBIeRbkugRtaMgesμI RbsinebIeKGnuvtþkMlaMgsgát; P = 1132KN
Rtg;TIRbCMuTMgn;rbs;muxkat; enaHmuxkat;tambeNþyFñwmnwgrgkugRtaMgesμI
P 1132
σp = = 10 − 3 = ±6.29MPa
area 0.3 × 0.6
kugRtaMgcugeRkayEdlbNþalBIbnÞúkGefr nigbnÞúkefrbUknwgbnÞúkeRbkugRtaMgenARtg;srés
xagelI nigxageRkambMputKW 12.58MPa nig 0 erogKña ¬rUbTI 19>2 b¦. kñúgkrNIenH kMlaMg
eRbkugRtaMg)anebgáInkugRtaMgsgát;enARtg;srésEpñkxagelIbMputeGaymantMélBIrdg nig)an
kat;bnßykugRtaMgTajenARtg;srésEpñkxageRkameGay esμInwg 0 . kugRtaMgsgát;Gtibrma
12.58MPa mantMéltUcCagkugRtaMgGnuBaØat f 'c = 14.14MPa .

#> sMrab;kugRtaMgEdlbNþalBIeRbkugRtaMgcMNakp©it ¬ e = 100mm ¦
RbsinebIeKGnuvtþkMlaMgeRbkugRtaMg P = 1132 KN enARtg;cMNakp©it e = 100mm BI eRkamTI
RbCMuTMgn;rbs;muxkat; kugRtaMgenARtg;srésEpñkxagelI nigEpñkxageRkambMputRtUv)ankMNt;
dUcxageRkam. m:Um:g;EdlbNþalBIeRbkugRtaMgcMNakp©itKW Pe
P (Pe )c P 6(Pe )
σp =− ± =− ±
A I A bh 2
1132 6(1132 × 0.1) − 3
=− 10 − 3 ± 10
0.3 × 0.6 0.3 × 0.6 2


T.Chhay NPIC

= −6.29 ± 6.29
enAsrésEpñkxageRkam σ p = −12.58MPa nig σ p = 0 enAsrésEpñkxageRkambMput.
BicarNaGMBIkardak;bnÞúkGefr L2 = 30.61kN / m
30.61× 7.2 2
M L. L. = = 198.36kN .m
8
6(198.36) − 3
σ L2 = 10 = ±11.02MPa
0.3 × 0.6 2
kugRtaMgcugeRkayEdlbNþalBIbnÞúkefr/ bnÞúkGefr nigkMlaMgeRbkugRtaMgenAsrésEpñkxagelI
nigEpñkxageRkambMputKW 12.58MPa nig 0 erogKña ¬rUbTI 19>2 c¦. cMNaMfa kugRtaMgcug
eRkaydUcKñanwgkrNIelIkmunenAeBlEdlbnÞúkGefresμInwg 13.15kN / m . edayGnuvtþkMlaMg
eRbkugRtaMgenARtg;cMNalp©it 10cm FñwmenHGacRTbnÞúkGefrEfm eTot ¬17.46kN / m ¦.


T.Chhay NPIC

$> sMrab;kugRtaMgEdlbNþalBIeRbkugRtaMgcakp©itRtg;cMNakp©itGtibrma
snμt;facMNakp©itGtibrmasMrab;muxkat;enHKW e = 15cm .
m:Um:g;Bt;EdlekItedaysarkMlaMgeRbkugRtaMgcakp©itKW Pe = 1132 × 0.15 = 169.8kN .m


T.Chhay NPIC

kugRtaMgEdlbNþalBIkMlaMgeRbkugRtaMgKW
1132 6(169.8) − 3
σp =− 10 − 3 ± 10
0.3 × 0.6 0.3 × 0.6 2

= −6.29 ± 9.41
= −15.7 MPa nig + 3.12

begáInbnÞúkGefrdl; L3 = 39.28kN / m . m:Um:g;Edl)anBIbnÞúkenHKW
39.28 × 7.2 2
M L. L. = = 254.5kN .m
8
kugRtaMgEdlbNþalmkBIbnÞúkGefrKW
6(254.5)
σ L3 = 10 − 3 = ±14.14MPa
0.3 × 0.6 2

kugRtaMgcugeRkayenAsrésEpñkxagelI nigEpñkxageRkambMputEdlbNþalBIbnÞúkefr nigbnÞúk
GefrKW 12.58MPa nig 0 erogKña ¬rUbTI 19>2 d¦. cMNaMfa kugRtaMgcugeRkaydUcKñanwgkrNI
mun²Edr b:uEnþbnÞúkGefr)anekIneLIgdl; 39.28kN / m . kugRtaMgTaj 1.56MPa RtUv)an
begáIteLIgenAsrésEpñkxagelIbMput enAeBlEdleKGnuvtþkMlaMgeRbkugRtaMg. kugRtaMgenH
mantMéltUcCagm:UDuldac;rbs;ebtug f r = 3.45MPa dUcenHvaminekItmansñameRbHenAelIFñwm
eT.
%> eKkMNt;bnÞúkGefrGtibrmaenAeBlkMlaMgeRbkugRtaMgcakp©iteFVIGMeBIenARtg; e = 15cm dUct
eTA. kñúgkrNImun² kugRtaMgsgát;cugeRkayesμInwg 12.58MPa EdltUcCagkugRtaMg GnuBaØat
f 'c = 14.14MPa . dUcenH bnÞúkGefrGacekIneLIgdl; L4 = 43.6kN / m .
43.6 × 7.2 2
M L. L. = = 282.5kN .m
8
6 × 282.5 − 3
σ L4 = 10 = 15.7 MPa
0.3 × 0.6 2
kugRtaMgcugeRkayEdlbNþalBIbnÞúkefr nigbnÞúkGefr L4 ehIynigkMlaMgeRbkugRtaMg KW
− 14.14 MPa nig + 1.56 MPa ¬rUbTI 19>2 e¦. kugRtaMgsgát;KWesμInwgkugRtaMgGnuBaØat

14.14 MPa ehIykugRtaMgTajKWtUcCag modulus of rupture rbs;ebtug 3.45MPa . kñúg

krNIenH eKGacKNnabnÞúkGefrBRgayEdlesμμInwg 43.6kN / m dUcteTA³ bUkkugRtaMgsgát;
GnuBaØatGtibrma 14.14MPa CamYynwgkugRtaMgTajdMbUgenARtg;srésEpñkxagelIbMput


T.Chhay NPIC

1.56 MPa edIm,ITTYl)an 15.7MPa . m:Um:g;EdlnwgbegáItkugRtaMgenAsrésEpñkxagelIbMput
15.7 MPa esμInwg
⎛ bh 2 ⎞
M =σ⎜ ⎟
⎜ 6 ⎟
⎝ ⎠
=
15.7
(0.3)(0.6)2 ⋅103 = 282.6kN.m
6
W L2
M= L
8
eyIgTTYl)an
8 × 282.6
WL = = 43.6kN / m
7.2 2
cMNaMfa³
!> muxkat;ebtugTaMgmUlKWskmμkñúgkarTb;Tl;CamYykMlaMgxageRkA
@> kugRtaMgTajcugRkayenAkñúgmuxkat;tUcCag modulus of rupture rbs;ebtug Edlbgðaj
famuxkat;ebtugminmansñameRbHeRkamGMeBIrbs;bnÞúkGtibrma
#> bnÞúkGnuBaØatenAelIFñwmekIneLIgeRcInKYrsmedaysarkarGnuvtþrbs;kMlaMgeRbkugRtaMg
$> karekIneLIgnUvcMNakp©itrbs;kMlaMgeRbkugRtaMgk¾begáInkMlaMgGnuvtþn_GnuBaØat EdleFVIeGay
kugRtaMgenAelImuxkat;minFMCagkugRtaMgGnuBaØat.

x> karGnuvtþeRbkugRtaMgedayEpñk Partial Prestressing
eKkMNt;Ggát;ebtugeRbkugRtaMgedayEpñk (partially prestressed concrete member) CaGgát;
Edl³
- kugRtaMgxagkñúgEdlmanGMeBITb;EpñkénkugRtaMgEdlekItBIbnÞúkxageRkA
- kugRtaMgTajekItmanenAkñúgebtugeRkamGMeBIbnÞúkeFVIkar (service load)
- EdkBRgwgminEmnCaEdkeRbkugRtaMgRtUv)andak;bEnßmedIm,IbegáInlT§PaBrbs;Ggát; edIm,ITb;
nwgm:Um:g;
eKGacBicarNa partially prestressed concrete tamBIrkrNI³
!> eKeRbIEdkeRbkugRtaMg nigEdkminEmneRbkugRtaMgenAkñúgmuxkat;EtmYy. ExSkabeRbkugRtaMg
begáItkugRtaMgxagkñúgRtUv)anKNnaedIm,ITTYl)an ultimate capacity rbs;muxkat;ebtugEt
mYyEpñkb:ueNÑaH. cMENkÉ capacity EdlenAsl;RtUv)anTTYlBIEdkminEmneRbkugRtaMg


T.Chhay NPIC

Edldak;tamTisdUcKñanwgkabeRbkugRtaMg. EdkEdleRbICaEdkminEmneRbkugRtaMgGacCaRb
ePTEdkFmμta dUcCaEdk carbon steel b¤CaEdk high-tensile-strength. kabeRbkugRtaMgk¾
CaRbePTEdkFmμtadUcEdkminEmneRbkugRtaMgEdr Etvaman ultimate strength esμInwg
1725MPa ¬ 250ksi ¦. kareRCIserIsGaRs½ynwgktþacMbgBIrKW³ PaBdabGnuBaØat nigTMhM

sñameRbHGnuBaØat. dUcKña ACI Code kMNt;nUvpleFobGtibrmaénRbEvgElVgelIkkMBs;én
Ggát;ebtugGarem: sMrab;PaBdab. eKminGnuBaØateGaymanPaBdabFMelIslubCamYynwg
kMBs;rbs;muxkat;ebtugeRbkugRtaMgtUc nigedaysarkareRbIPaKryEdktic. sñameRbHekIt
manenAtMbn;rgkarTajrbs;muxkat;ebtug b¤enARtg;nIv:UEdkedaysareKGnuBaØateGaykug
RtaMgTajekItmaneRkamGMeBI working load. eKGnuBaØatsñameRbHGtibrmaRtwm 0.016in.
(0.41mm) sMrab;Ggát;xagkñúg nig 0.013in. (0.33mm) sMrab;Ggát;xagkñúg.

@> kugRtaMgxagkñúgEdleFVIGMeBIelIGgát;)anEtBI prestrssed steel b:ueNÑaH b:uEnþvaRtUv)anTajCa
mYynwgEdnkMNt;TabCag. kñúgkrNIenHsñameRbHekItmanelOnCagGgát;rgeRbkugRtaMgeBj
eljeRkambnÞúkdUcCaKña.
eKGacBicarNa partially prestresssed concrete kñúgTMrg;kNþalrvagebtugGarem: nigebtug
eRbkugRtaMgeBj (fully prestressed concrete). enAkñúgGgát;ebtugGarem: sñameRbHekItmaneRkamGMeBI
bnÞúkeFVIkar dUcenHeKdak;EdkBRgwgenAkñúgtMbn;Taj. CaTUeTAenAkñúgGgát;ebtugeRbkugRtaMg sñameRbH
minekItmaneRkamGMeBIbnÞúkeFVIkareT. kugRtaMgsgát;EdlbNþalBIkMlaMgeRbkugRtaMgGacesμI b¤elIsBI
kugRtaMgTajEdlbNþalBIbnÞúkxageRkA. dUcenHeKGacBicarNaGgát; partially prestressed concrete
CaGgát;ebtugGarem:EdlkugRtaMgxagkñúgrbs;vamanGMeBITb;nwgEpñkxøH rbs;kugRtaMgEdl)anBIbnÞúkxag
eRkA dUcenHkugRtaMgTajenAkñúgebtugminRtUvFMCagtMélkMNt;eRkambnÞúkeFVIkareT. eKKitvaCaebtug
Garem:enAeBlNaEdlminmankugRtaMgxagkñúgeFVIGMeBIelIGgát;. ebtugeRbkugRtaMgeBjCakMritx<s;bMput
rbs;ebtugeRbkugRtaMgedayEpñk EdlenAkñúgenaHEdkminEmneRbkugRtaMgRtUv)ankat;bnßydl;sUnü.
enAcenøaHGgát;ebtugGarem:EdlmaneRbH nigGgát;ebtugeRbkugRtaMgeBjEdlminmaneRbH eK
manEdnd¾FMsMrab;KNnaebtugeRbkugRtaMgedayEpñk ¬rUbTI 19>3¦. kareRCIserIskMriténkareFVIeRbkug
RtaMgd¾l¥ nwgbegáItnUveRKOgbgÁúMEdlmansuvtßiPaB nigmanlkçN³esdækic©.


T.Chhay NPIC

rUbTI 19>3 bgðajBIExSekagPaBdab-bnÞúkrbs;FñwmebtugGarem:EdlmanbrimaNEdk nigRbePT
EdkxusKña. ExSekag a bgðajBIFñwmebtugGarem: EdlmansñameRbHFmμtaeRkambnÞúktUc Wcr . eKGackM
Nt;m:Um:g;EdleFVIeGayeRbH (cracking moment) dUcxageRkam³
fr I
M cr =
c
Edl m:UDuldac;rbs;ebtug = 0.62 f 'c
fr =

I = m:Um:g;niclPaBén gross concrete section

c = cMgayBIG½kSNWteTAsrésrgkarTajxageRkAbMput

eKGackMNt;ebtugEdleFVIeGayeRbH (cracking load) BI cracking moment enAeBlRbEvg
ElVg nigRbePTénkardak;bnÞúkRtUv)ankMNt;. sMrab;FñwmTMrsamBaØEdlrgbnÞúkcMcMnugenAkNþalElVg
Wcr = (4M cr ) / L .


T.Chhay NPIC

ExSekag e nig f bgðajBIFñwmebtugeRbkugRtaMgeBjEdlmanEdktic nigEdkeRcIn erogKña. Fñwm
ebtugGarem:EdlmanbrimaNEdkeRcIn)ak;edaysarkarEbkebtugmunnwgEdkeTAdl; yield strength b¤
proof stress rbs;va. FñwmmanPaBdabtUc nwgrgkar)ak;edayPaBRsYy (brittle failure). FñwmEdlman

brimaNEdktic)ak;edaysarEdkeFVIkardl; yield nig ultimate strength rbs;va. vabgðajnUv)abdab
nigsñameRbHEdlbNþalBIkarlUtsac;rbs;EdkmuneBlebtugEbkCabnþbnÞab; ehIyFñwm)ak;rlM.
enAcenøaHExSekag a nig e CaEdnd¾FMrbs;FñwmebtugCamYynwgbrimaNERbRbYlrbs;Edk nigrg
nUvbrimaNERbRbYlrbs;kMlaMgeRbkugRtaMg. FñwmEdlrgkMlaMgeRbkugRtaMgtUcenAEk,rExSekag a ehIy
FñwmEdlmaneRbkugRtaMgFMenAEk,rExSekag e . eKeRCIserIsbnSMEdkeRbkugRtaMg nigEdkminEmneRbkug
RtaMgsMrab;karKNnaKWGaRs½yelIkugRtaMgebtugGnuBaØat PaBdab nigTMhMsñameRbHGtibrma.
ExSekag b tMNageGayFñwmEdlnwgeRbHeRkamGMeBIénbnÞúkeFVIkareBjelj. RbsinebIEtEpñk
xøHrbs;bnÞúkGefrekItmanenAelIeRKOgbgÁúMCaerOy² enaH W1 tMNageGaybnÞúkefrsrub nigEpñkxøHrbs;
bnÞúkGefr L1 .
ExSekag c tMNageGayFñwmcab;epþImeRbHeRkamGMeBI working load. kugRtaMgTajGtibrmaenA
kñúgebtug = f r = 0.62 f 'c .
ExSekag d tMNageGayFñwmEdlmankMlaMgeRbkugRtaMgkMNt;. muxkat;eRKaHfñak;rbs;Fñwmnwg
mineRbHeRkambnÞúkeFVIkareBjeljeT b:uEnþvanwgmankugRtaMgTajGtibrma 0 < f r < 0.62 f 'c . ACI
Code GnuBaØatkugRtaMgTajGtibrmaenAkúñgebtugRtwm 0.5 f 'c .

ExSekag e nig e' tMNageGayFwñmebtugeRbkugRtaMgeBjeljEdlminmankugRtaMgTajeRkam
bnÞúkeFVIkar ¬emIlrUbTI 19>4¦.
sar³RbeyaCn_d¾sMxan;bMputrbs;kMlaMgeRbkugRtaMgedayEpñkKWlT§PaBkúñgkarRKb;RKgkMeNag
(camber). edaykat;bnßykMlaMgeRbkugRtaMg camber nwgRtUv)ankat;bnßy ehIysnSMnUvbrimaNEdk

eRbkugRtaMg brimaNkargarkúñgkarTaj nigcMnYn end anchorage.
GaRs½ynwgGaMgtg;sIueténkMlaMgeRbkugRtaMg sñameRbHenAkñúg partially prestressed member
ekIteLIgelOnCagenAkñúg fully prestressed concrete member eRkamGMeBIrbs; service load. enA
eBlEdlsñameRbHekItman m:Um:g;niclPaBRbsiT§PaBrbs;muxkat;eRKaHfñak;RtUv)ankat;bnßy ehIyeK
nwgTTYl)anPaBdabFMCagmun. b:uEnþ kareRbIkMlaMgeRbkugRtaMgedayEpñkKWeKTTYl)anlT§plKYrCaTU
eBjcitþ ehIyvaTTYl)ankareBjniym.


T.Chhay NPIC

K> karcat;cMNat;fñak;Ggát;rgkarBt;ebtugeRbkugRtaMg
Classification of Prestressed Concrete Flexural Members
ACI Code, Section 18.3 )anEckGgát;ebtugeRbkugRtaMgCabIfñak;edayQrelIkugRtaMgTaj
enAelIsrésxageRkAbMput f t enAkñúgtMbn;TajeRkamGMeBIbnÞúkeFVIkardUcxageRkam³

!> fñak; U (uncracked section) Edlman f t ≤ 0.62 f 'c . enAkñúgmuxkat;ebtugEdlKμansñam
eRbHenH eKeRbIlkçN³én gross section edIm,IRtYtBinitüPaBdabeRkamGMeBIbnÞúkeFVIkar.
KμansñameRbHekItmanenAkñúgmuxkat; nigeKminRtUvkar skin reinforcement eT.
@> fñak; T (muxkat;enAkñúg transition zone) Edlman 0.62 f 'c < ft ≤ f 'c . muxkat;
RbePTenHmankugRtaMgTajenAkñúgebtugFMCagm:UDuldac; (modulus of rupture) rbs;ebtug
f r = 0.62 f 'c EdlbegáItnUvkrNIcenøaHmuxkat;eRbH nigmuxkat;Gt;eRbH. enAkñúgkrNIenH

eKeRbIlkçN³én gross section edIm,IRtYtBinitükugRtaMg ehIyeKeRbI bilinear section rbs;
muxkat;eRbHedIm,IKNnaPaBdab. eKmincaM)ac;eRbI skin reinforcement enAkñúgtMbn;TajeT.
#> fñak; C (cracked section) Edlman f t > f 'c . kugRtaMgTajenAkúñgmuxkat;FMCag
modulus of rupture rbs;ebtug 1.6 dg. dUcenH sñameRbHnwgekItmandUckñúgkrNIGgát;eb

tugeRbkugRtaMgedayEpñk. enAkñúgkrNIenH eKeRbIlkçN³énmuxkat;eRbHedIm,IRtYtBinitükug

T.Chhay NPIC

RtaMg sñameRbH nigPaBdab. eKKYreRbIkarpþl;eGayedIm,IRKb;RKgsñameRbH nigeRbI skin
reinforcement dUckarBnül;enAkñúgEpñk 6>7 sMrab;Ggát;ebtugGarem:EdlmankMBs;RbsiT§-

PaB d > 915mm .

2> sMPar³ nigtMrUvkarsMrab;bMerIbMras; Material and Serviceability Requirement

k> ebtug Concrete
lkçN³rbs;ebtugRtUv)anbgðajenAkñúgCMBUk 2. eTaHbICaerOy² Ggát;ebtugGarem:RtUv)anplit
BIebtugEdlmanersIusþg;sgát; 21MPa eTA 35MPa k¾eday k¾Ggát;ebtugeRbkugRtaMgRtUv)anplitBI
sMPar³EdlmanersIusþg;x<s;Cag CaTUeTAsßitenAcenøaH 28MPa eTA 56MPa . eKeRbIebtugersIusþg;x<s;
sMrab;Ggát;ebtugcak;eRsc nigGgát;ebtugeRbHkugRtaMg Edlkarlay karcak; karbgðab; nigkarEfTaMeb
tugsßiteRkamkarRtYtBinitüy:agm:t;ct;.
kugRtaMgGnuBaØatenAkñúgebtugEdleyagtam ACI Code, Section 18.4 mandUcxageRkam³
!> kugRtaMgeRkayeBlepÞreRbkugRtaMg (prestress transfer) nigmuneBl)at;bg;eRbkugRtaMg
(prestress losses):
a. kugRtaMgsgát;GtibrmaesμInwg 0.6 f ci
b. kugRtaMgTajGtibrma ¬elIkElgGVIEdl)anGnuBaØatdUcxageRkam¦ esμInwg 0.25 f ci

c. kugRtaMgTajGtibrmaenARtg;cugénGgát;TMrsamBaØesμInwg 0.5 f ci

Edl f ci CaersIusþg;rbs;ebtugenAeBlepÞr
RbsinebIkugRtaMgTajmantMélFMCagenH eKRtUvdak;EdkenAtMbn;sgát;edIm,ITb;Tl;kMlaMgTaj
srubenAkñúgebtug ¬edayQrelI uncracked gross section¦.
@> kugRtaMgeRkamGMeBIbnÞúkeFVIkareRkayeBlkMhatbg; (loss) TaMgGs; ¬sMrab;fñak; U nigfñak; T ¦
mandUcteTA³ kugRtaMgsgát;Gtibrma 0.45 f 'c EdlbNþalBIkMlaMgeRbkugRtaMgbUknwgbnÞúkefr
nigkugRtaMg 0.05 f 'c EdlbNþalBIkMlaMgeRbkugRtaMgbUknwgbnÞúksrub.
#> kugRtaMgTaMgenHGacmantMélFMCagenH RbsinebIkarBiesaF nigkarviPaKbgðajfakaRbRBwtþeTA
rbs;vaRKb;RKan;.


T.Chhay NPIC

x> EdkeRbkugRtaMg Prestressing Steel
Edk tendon EdleKniymeRbICageKenAkñúgebtugeRbkugRtaMgCa strands ¬b¤kab¦ Edlplit
eLIgCamYynwglYssrésr (wire) CaeRcIn CaTUeTAmancMnYn 7 b¤ 19 . Wire nig bar k¾RtUv)aneKeRbI
R)as;pgEdr. Stand nig wire RtUv)anpliteLIgedayeKarBtam ASTM Standard A421 sMrab;
uncoated stress-relieved wire nig A416 sMrab; uncoated seven-wire stress-relieved strand.

lkçN³rbs;EdkeRbkugRtaMgRtUv)aneGayenAkñúgtarag 19>1.
tarag 19>1
Diameter Area Mass
Type
(mm) (mm2) (kg/m)
Seven-wire strand (grade 250) 6.350 23.2 0.179
7.950 37.4 0.298
9.525 51.6 0.402
11.125 69.7 0.551
12.700 92.9 0.729
15.240 139.4 1.101
Seven-wire strand (grade 270) 9.525 54.8 0.432
11.125 74.2 0.595
12.700 98.7 0.789
15.250 138.7 1.101
Prestressing wire grades (250) 4.877 18.7 0.146
(250) 4.978 19.4 0.149
(240) 6.350 31.6 0.253
(235) 7.010 38.7 0.298
Prestressing bars (smooth) 19.050 283.9 2.232
(grade 145 or 160) 22.225 387.1 3.036
25.400 503.2 3.973
28.575 638.7 5.030
31.750 793.5 6.206
34.925 954.8 7.515
Prestressing bars (deformed) 15.875 180.6 1.458
(grade 150-160) 19.050 271.0 2.218
25.400 548.4 4.480
31.750 806.5 6.535
34.925 1006 8.274

EdkeRbkugRtaMgEdleRbIenAkñúgebtugeRbkugRtaMgRtUvEtmanersIusþg;x<s; CaTUeTAman
ultimate strength f puenAcenøaH 1730MPa eTA 1860MPa . eKcaM)ac;GnuBaØateGayEdkersIusþg;
x<s;mansac;lUtFM nigrkSakugRtaMgenAkñúgebtugeGayRKb;RKan; nigGciéRnþy_bnÞab;BI inelastic
shortening rbs;ebtug.


T.Chhay NPIC

kugRtaMgGnuBaØatenAkñúgEdkeRbkugRtaMgeyagtam ACI Code, Section 18.5 mandUcxageRkam³
!> kugRtaMgGtibrmaEdlbNþalBI tendon jacking force minRtUvFMCagtMélEdltUcCageKkñú
gcMeNam 0.8 f pu nig 0.94 f py . tMélEdltUcCagminRtUvFMCagkugRtaMgEdlENnaMedayGñk
plit tendon b¤ anchorage eT.

@> kugRtaMgGtibrmaenAkñúg pretensioned tendon Pøam²eRkayeBlepÞrminRtUvFMCagtMéltUc
CageKkñúgcMeNam 0.74 f pu nig 0.82 f py .
#> kugRtaMgGtibrmaenAkñúg postensioned tendon eRkayeBl tendon RtUv)an anchor KW
0.70 f pu .

K> EdkBRgwg Reinforcing Steel
CaTUeTA eKeRbIEdkBRgwgminEmneRbkugRtaMgenAkúñgGgát;eRKOgbgÁúMebtugeRbkugRtaMg
CaBiessenAkñúgsMNg;ebtugeRbkugRtaMgcak;eRsc. eKeRbIEdkBRgwgCaEdkkMlaMgkat;TTwg Ca
EdkbEnßmsMrab;kardwkCBa¢Ún nigkarelIkdak;Ggát;cak;eRsc ehIynigeRbIenAkñúgGgát;ebtugeRb
kugRtaMgedayEpñkEdlcUlrYmCamYynwgEdkeRbkugRtaMg. RbePT nigkugRtaMgGnuBaØatrbs;Edk
RtUvENnaMenAkñúgCMBUk 2 nigCMBUk 5 rYcehIy.

3> kMhatbg;eRbkugRtaMg Loss of Prestress

k> Lump-sum losses
kMhatbg;énkMlaMgeRbkugRtaMgCabnþbnÞab;ekItmaneRkayeBlkMlaMgeRbkugRtaMgRtUv)an
epÞrBI jack eTaGgát;ebtug. kMhatbg;eRbkugRtaMgCakarkat;bnßykMlaMgeRbkugRtaMgkñúgmYy
CIviténeRKOgbgÁúM. brimaNkMhatbg;enAkñμúg tendon ERbRbYlcenøaHBI 15% eTA 30% énkugRtaMg
edIm edayGaRs½ynwgktþaCaeRcIn. sMrab;eRKOgbgÁúMebtugGarem:TMgn;FmμtaPaKeRcInEdlsagsg;eday
standard method, tendon stress loss bNþalmkBI elastic shortening, shrinkage, creep nig

relaxation rbs;EdkKWmantMélRbEhlnwg 35ksi(241MPa ) sMrab; pretensioned member nigsMrab;


T.Chhay NPIC

posttensioned member KWRbEhlbwg 25ksi(172MPa) . kMlaMgkkit nig anchorage slip minRtUv)an
rab;bBa©ÚleT.
karENnaMBIrsMrab;kar)a:n;sμankMhatbg;srubenAkñúgGgát;ebtugeRbkugRtaMgRtUv)anbgðajeday
AASTHO nig Posttensioning Institute (PTI). AASTHO ENnaMeGayykkMhatbg;srub

(edayminKitkMlaMgkkit) 45ksi(310 MPa ) sMrab; pretensioned strand nig 33ksi(228MPa ) sMrab;

postentioned strand nig wire enAeBlEdleKeRbIersIusþg;sgát;ebtug f 'c = 35MPa . PTI ENnaM

lump-sum prestress loss sMrab; posttensioned member 35ksi(241MPa ) sMrab;Fñwm nig

30ksi(207 MPa ) sMrab;kMralxNÐ ¬edayminKitkMlaMgkkit¦. eKGaceRbItMélTaMgGs;enH)anluHRta

EteK)aneFVIkar)a:n;RbmaNkMhatbg;eRbkugRtaMgedayRbPBénkMhatbg;nImYy²dac;edayELkBIKña)an
l¥ dUcEdl)anENnaMy:agsegçb.
CaTUeTA RbPBénkMhatbg;eRbkugRtaMgKW
- Elastic shortening rbs;ebtug
- Shrinkage rbs;ebtug
- Creep rbs;ebtug
- Relaxation rbs;Edk tendon
- kMlaMgkkit
- Anchorage set

x> kMhatbg;edaysar Elastic Shortening of Concrete
kar)a:n;RbmaNkMhatbg; elastic shortening rbs;ebtugenAkñúg pretensioned member

RtUv)aneFVIeLIgdUcteTA. BicarNa pretensioned concrete member énmuxkat;efr nigkugRtaMgBRgay
esμItambeNþayG½kSTIRbCMuTMgn;rbs;vaedaysarkMlaMg Fo . eRkayBIkarepÞrkMlaMgeRbkugRtaMgFñwmebtug
nig prestressing tendon rYjxøIedaybrimaNesμIKña edaysarPaBs¥itrvagsMPar³TaMgBIr. dUcenH kMlaMg
eRbkugRtaMgEdlcab;epþIm Fo Føak;mkRtwm Fi ehIykMhatbg;kMlaMgeRbkugRtaMgKW Fo − Fi . dUcKña
strain enAkñúgebtug ε c RtUvEtesμInwgbMErbMrYl strain rbs; tendon Δε s . dUcenH ε c = Δε s b¤

( f c / Ec ) = (Δf s / E s ) ehIykMhatbg;kugRtaMgEdlbNþalBI elastic shortening KW


T.Chhay NPIC

Es nF nF
Δf s = × f c = nf c = i ≈ o (19.1)
Ec Ac Ac
Edl RkLaépÞrbs;muxkat;ebtug
Ac =

n = E s / Ec = pleFobm:UDul (modular ratio)

f c = kugRtaMgrbs;ebtugenARtg;TIRbCMuTMgn;rbs;EdkeRbkugRtaMg

KuNkugRtaMgnwgRkLaépÞrbs;EdkeRbkugRtaMg Asp edIm,ITTYl)ankMlaMgsrub . Elastic loss KW
⎛ nF ⎞
ES = Fo − Fi = Δf s Asp = (nf c )Asp ≈ ⎜ o
⎜ A ⎟ Asp
⎟ (19.2)
⎝ c ⎠
Fi = Fo − (nf c )Asp (19.3)

sMrab;karKNnaGnuvtþn_ kMhatbg;kugRtaMgénkMlaMgeRbkugRtaMg ¬ Δf s kñúgmYyktþaépÞ Asp ¦ mantMél
Rbhak;RbEhlnwg nFo / Ac . RbsinebI kMlaMg Fo manGMeBIRtg;cMNakp©it e enaH elastic loss Edl
bNþalBIvtþmanén Fo nigbnÞúkefrGnuvtþn_enAeBlepÞrKW
ES = −(nf c )Asp ¬EdlbNþalBIeRbkugRtaMg¦ + (nf c )Asp ¬bnÞúkefr¦
⎛ F F e2 ⎞ ⎛M e⎞
ES = Fo − Fi = −⎜ i + i ⎟nAsp + ⎜ D ⎟nAsp (19.4)
⎜ A ⎟
I ⎠ ⎝ I ⎠
⎝
eKGaceRbItMélRbhak;RbEhlén Fi = (0.63 f pu )Asp enAkñúgsmIkarxagelI.
⎡ ⎛ 1 e 2 ⎞⎤
Fo + f c (D.L.)nAsp = Fi ⎢1 + nAsp ⎜ + ⎟⎥ (19.5)
⎜A I ⎟
⎢
⎣ ⎝ ⎠⎥
⎦

Fi =
( )
Fo + nAsp f c (D.L.)
⎛ 1 e2 ⎞
( )
1 + nAsp ⎜ + ⎟
⎜A I ⎟
⎝ ⎠
sMrab; posttensioned member Edl tendon nig individual strand minrgkugRtaMgdMNalKña enaHeK
GackMhatbg;eRbkugRtaMgesμInwgBak;kNþaléntMél ES sMrab; prestensioned member.
dUcKña eKGacyk elastic shortening loss enAkñúgkMralxNÐesμInwgmYyPaKbYnéntMél ES
sMrab; prestensioned member edaysarkarlUtrbs; tendon mYymanT§iBltictYceTAelIkugRtaMgén
tendon d¾éTeTot.

K> kMhatbg;edaysarkarrYmmaD Loss Due to Shrinkage
kMhatbg;eRbkugRtaMgEdlbNþalBIkarrYmmaDKWGaRs½ynwgeBl. eKGac)a:n;RbmaNvadUcxag
eRkam³

T.Chhay NPIC

SH = Δf s (shrinkage) = ε sh E s (19.6)
Edl Es = 2 ⋅105 MPa nig ε sh = shrinkage strain enAkñúgebtug
eKGacsnμt; Strain mFümEdlbNþalBIkarrYmmaDmantMéldUcxageRkam³
- ε sh1 = 0.0003 sMrab; pretensioned member
- ε sh2 = 0.0002 sMrab; posttentioned member
RbsinebIeKGnuvtþ posttensioning kñúgcenøaH 5 eTA 7 éf¶eRkayBIcak;ebtug/ eKGacyk shrin-
kage strain esμInwg 0.8ε sh1 . RbsinebIeKGnuvtþ posttensioning kñúgcenøaH 1 s)aþh_eTA 2 s)aþh_ eK

GaceRbI ε sh = 0.7ε sh1 nigRbsinebIeKGnuvtþ posttensioning eRkayeBlcak;ebtugeRcInCag 2 s)aþh_
enaHeKGacyk ε sh = ε sh2 . eKk¾Gac)a:n;RbmaNkMhatbg;edaykarrYmmaD SH dUcxageRkam³
⎛ 0.06V ⎞
SH = 8.2 × 10 − 6 K sh E s ⎜1 − ⎟(100 − RH )
⎝ S ⎠
Edl V / S = pleFobmaDelIépÞ nig RH = average relative humidity. K sh = 1.0 sMrab; pretensioned
member nigesμInwg 0.8 / 0.73 / 0.64 nig 0.58 sMrab; posttensioned member RbsinebIeKGnuvtþ

posttensioning eRkayeBlcak;ebtug 5 / 10 / 20 nig 30 éf¶ erogKña.

X> kMhatbg;edaysar creep rbs;ebtug
CakMhUcRTg;RTayGaRs½ynwgeBlEdlekIteLIgenAkñúgebtugeRkamGMeBIbnÞúkefr. kMhUc
Creep

RTg;RTayEdlekIteLIgedaysar creep eFVIeGay)at;bg;kMlaMgeRbkugRtaMgBI 5% eTA 7% .
Creep strain ERbRbYlCamYynwgGaMgtg;sIueténkugRtaMgedImenAkñúgebtug relative humidity

nigeBl. eKGackMNt;kMhatbg;kugRtaMgEdlbNþalBI creep dUcxageRkam³
CR = Δf s (creep) = Cc (nf c ) = Cc (ε cr E s ) (19>7)
creepstrain, ε cp
Edl emKuN
Cc = creep =
initial elastic strain, εi
eKGacyktMél Cc dUcxageRkam³

ersIusþg;ebtug f 'c ≤ 28MPa f 'c > 28MPa
Relative humidity 100% 50% 100% 50%
Cc 1− 2 2−4 0.7 − 1.5 1 .5 − 3


T.Chhay NPIC

eKGaceFVI interpolation sMrab;tMélEdlenAcenøaHtMélEdlenAkñúgtaragxagenH. edayKitfa
creep EtBak;kNþalekIteLIgkñúgGMLúg 134 ExdMbUgén 6 ExdMbUgbnÞab;BIkarepÞreRbkugRtaMgeTAebtug

nigeRkamlkçxNÐsMeNImFmμta enaHeKGacsnμt; creep strain sMrab;karKNnaGnuvtþn_dUcxageRkam³
!> sMrab; pretensioned members, ε cr = 7 ⋅10−5 × kugRtaMgenAkñúgebtug
@> sMrab; postensioned members, ε cr = 5.2 ⋅10−5 × kugRtaMgenAkñúgebtug. eKeRbItMélenH
enAeBlEdleKGnuvtþ posttensionning kñúgGMLúg 2 eTA 3 s)aþh_. sMrab;karGnuvtþ posten-
sioning elOnCagenH eKGaceRbItMélkNþal.

eKeRbItMélTaMgenH enAeBlEdlersIusþg;rbs;ebtugenAeBlepÞrKW f 'c ≥ 28MPa . enAeBlEdl
f 'c < 28MPa creep strain KYrekIneLIgkñúgGRta 4 / ersIusþg;Cak;Esþg.

kMhatbg;eRbkugRtaMgsrubEdlbNþalBI creep = ε cr Es (19.8)

g> kMhatbg;edaysar Relaxation rbs;Edk
Relaxation rbs;EdkbNþaleGaymankMhatbg;enAkñúgEdkeRbkugRtaMgGaRs½ynwgeBl Edl
RsedogKñanwg creep enAkñúgebtugEdr. kMhatbg;edaysar relaxation ERbRbYleTAtamRbePTEdk.
CaTUeTA tMélrbs;vaRtUv)anpþl;eGayedayGñkplitEdk. CaFmμta eKsnμt;kMbaatbg;enHesμInwg 3%
énkugRtaMgedImrbs;EdksMrab; posttensioned member nig 2% eTA 3% sMrab; pretensioned
members. RbsinebIeKminmanB½t’manBIkarBiesaFeT eKGacPaKrykMhatbg;sMrab; relaxation enA

1000h dUcxageRkam³

!> enAkñúg low-relaxation strands, enAeBlEdleRbkugRtaMgedImesμInwg 0.7 f pu nig 0.8 f pu /
relaxation (RE) KW 2.5% nig 3.5% erogKña.

@> enAkñúg stress-relieved strand b¤ wire, enAeBlEdleRbkugRtaMgedImesμInwg 0.7 f pu nig
0.8 f pu / relaxation (RE) KW 8% nig 12% erogKña.

c> kMhatbg;edaysarkMlaMgkkit Loss Due to Friction
CamYynwgEdkrg pretensioning kMhatbg;edaysarkMlaMgkkitekItmanenAeBlEdl wires b¤
strand dabtam diaphragm. CaFmμtakMhatbg;enHmantMéltUc ehIyeKGacecalva)an.


T.Chhay NPIC

enAeBlEdl strand dabtam concordant profile enaHkMhatbg;edaysarkMlaMgkkitGacmantMélFM.
enAkrNIEbbenH CaFmμtaeKeRbI]brkrN_Edlvas;bnÞúkCak;EsþgedIm,IkMNt;kMlaMgenAkñúg tendon.
CamYynwgEdkrg posttensioning, T§iBlénkMlaMgkkitmantMélFMedaysarktþacMbgBIrKW
kMeNagrbs; tendon nigkar)at;bg;PaBRtg;rbs;bMBg; (wobble). RbsinebIeKTb;cugbgáb;mçagrbs;
tendon edaykMlaMg P2 nigeKTajcugTMenrmçageTotrbs; tendon edaykMlaMg P edIm,IeGay tendon
1

enaHrGiltamTisrbs;kMlaMg P1 )anluHRtaEt
μα px
P = P2 e
1 (19.9)
Edl μ = emKuNmMukkitsþaTic nig α px = mMurvag P1 nig P2 . CaTUeTAeKKit wobble effect tamviFI
RsedogKña
Px = Ps e −(μα + Kl x )
Ppj = Ppx e
(
+ Kl px + μ pα px )
Ppx = Ppj e
(
− Kl px + μ pα px ) (19.10)

Edl kMlaMgeRbkugRtaMgenARtg;cMnuc x
Ppj =

Ppx = kMlaMgeRbkugRtaMgenARtg; jacking end

μ p = emKuNkMlaMgkkitedaysarkMeNag
α px = bMErbMrYlmMusrubénragtambeNþayrbs;EdkeRbkugRtaMgBI jacking end eTAdl;cMnuc x
KitCara:düg;
=
RbEvgkMeNag
kaMkMeNag
K = emKuNkMlaMgkkit wobble kñúgmYyÉktþaRbEvgrbs; tendon

CakarsMrYl ACI Code eGaynUvsmIkarxageRkamsMrab;krNI (μ pα px + Kl x ) ≤ 0.30 . lT§plEdl
TTYl)anBIsmIkarCatMélRbEhl
(
Ppx = Ppj 1 + Kl px + μ pα px )−1 (AIC Code, eq. 18.2) (19.11)

emKuNkMlaMgkkit μ nig K GaRs½ynwgRbePTén strand b¤ wire, RbePTbMBg; niglkçxNÐépÞ
b:H. ACI Commentaru, Sectin 18.6 nigenAkñúgtarag 19.2 eGaynUvtMélRbhak;RbEhlrbs; μ nig
K.

kMhatbg;edaysarkMlaMgkkitenAkñúg jack ERbRbYl nigGaRs½ynwgktþaCaeRcIn edayrab;
bBa©ÚlTaMgRbEvgrbs; tendon. eKENnaMeGayeRbI accurate load ceel edIm,Ivas;kMlaMgedaypÞal;.


T.Chhay NPIC

kareRbI pressure gauge pþl;nUvlT§plminsuRkit luHRtaEteKeFVIkarEktMrUveTAtamkMlaMgEdleKsÁal;
enAkúñúg tendon.
kMhatbg;edaysarkMlaMgkkitenAkñúg cnchorage KWGaRs½ynwgRbePT anchorage nigbrimaN
én deviation rbs; tendon Edlqøgkat; anchorage. CaFmμtakMhatbg;enHmantMéltUcEdlGac
ecal)an. karENnaMkñúgkrNIBiessKYrTTYl)anBIplitkr.

tarag 19>2 emKuNkMlaMgkkitsMrab; posttensioned tendon
emKuNkMlaMgkkit wobble K emKuNkMlaMgkMlaMgkkit
RbePT tendon
kñúgmYyÉktþaRbEvg (10 −3 ) edaysarkMeNag μ
Tendon in flexible metal sheathing
(grouted)
Wire tendon
3.33 − 5.0 0.15 − 0.25
Seven-wire strand
1.67 − 6.67 0.15 − 0.25
High-strength bars
0.33 − 2 0.08 − 0.30
Pregreased unbonded tendon

Wire tendon and seven-wire strand
1 − 6.67 0.05 − 0.15
Mastic-coated unbonded tendon

Wire tendon and seven-wire strand
0.33 − 0.67 0.05 − 0.15

q> kMhatbg;edaysar Anchor set
enAeBlkMlaMgenAkñúg tendon RtUv)anepÞrBI jack eTA anchorage unit, clnart;cUlkñúgbnþic
rbs; tendon ekIteLIgedaysarkardak; gripping device b¤ wedge. karrGilenHbNþaleGayman
tendon rYjxøI EdleFVIeGay)at;bg;kMlaMgeRbkugRtaMg. RbEvgrGilERbRbYlBI 2.5mm eTA 6mm ehIy

CaTUeTARtUv)ankMNt;edayplitkr. eKGacKNnakMhtbg; anchor set edayrUbmnþxageRkam³
ΔL
Δf s = ΔεE s = × Es (19.12)
L
Edl Δε = GaMgtg;sIueténkarrGil anchor
E s = 2 ⋅10 5 MPa


T.Chhay NPIC

RbEvgrbs; tendon
L=

edaysarkMhatbg;eRbkugRtaMgCacMras;smamaRteTAnwgRbEvgrbs; tendon ¬b¤RbEhlCaBak;kNþal
énRbEvgrbs; tendon RbsinebIvargkugRtaMgBIcugsgxagkñgeBlEtmYy¦PaKrykMhatbg;enAkñúgkug
ú
RtaMgEdknwgRtUv)ankat;bnßyenAeBlEdlRbEvgrbs; tendon ekIneLIg. RbsinebI tendon lUteday
Δε enAeBlepÞr enaHeKecalkMhatbg;eRbkugRtaMgedaysarkarrGil.

]TahrN_ 19>2³ FñwmTMrsamBaØrg pretensionning RbEvg 11m manmuxkat;ctuekaNEkgCamYynwg
b = 45cm nig h = 80cm . KNnakMhatbg;eGLasÞic nigkMhatbg;EdlGaRs½ynwgeBlTaMgGs;. eK
eGay³ kMlaMgeRbkugRtaMgenAeBlepÞrKW Fi = 1935kN / RkLaépÞrbs;EdkeRbkugRtaMgKW Aps =
1935mm 2 / f 'c = 35MPa / Ec = 34500MPa / E s = 2 ⋅105 MPa / profile rbs; tendon manrag

Ca)a:ra:bUl/ cMNakp©itenAkNþalElVg = 15cm nigcMNakp©itenAcug = 0 .
dMeNaHRsay³
!> kMhatbg;edaysar elastic shortening: kugRtaMgEdl)anBIkMlaMgeRbkugRtaMgenAeBlepÞrKW
Fi 1935 3
= 10 = 1000MPa
A ps 1935
f
rbs;EdkeRbkugRtaMg = Es = 210005 = 5 ⋅10 −3
strain
⋅10
s

edayeRbIsmIkar 19>1
E 2 ⋅10 5
n= s = = 5.8 yk 6
E 34500
c
nFi 6 × 1935 3
Δf s = = 10 = 32.25MPa
Ac 450 × 800
edayKitbMErbMrYlcMNakp©ittambeNþayFñwm
Fi
strain enAmuxkat;xagcug =
1935
= 10 3 = 1.56 ⋅10 − 4
AE 450 × 800 × 34500
c c
Fi e 2
strain enAkNþalElVg = AFE
i
+
IEc
c c

bh 3
450(800 )3
I= = = 1.92 ⋅1010 mm 4
12 12
1935 × 150 2
strain = 1.56 ⋅10 − 4 + 10 3 = 2.22 ⋅10 − 4
1.92 ⋅10 × 34500
10


T.Chhay NPIC

mFüm = 1 (1.56 + 2.22)10 − 4 = 1.89 ⋅10− 4
strain
2
kMhatbg;eRbkugRtaMg = strain × Es = 1.89 ⋅10−4 × 2 ⋅105 = 37.8MPa
PaKrykMhatbg; = 1000 = 3.78%
37.8

@> kMhatbg;edaysar shrinkage:
shrinkage strain = 0.0003
Δf s = ε sh E s = 0.0003 × 200000 = 60MPa

PaKrykMhatbg; =
60
1000
= 6%

#> kMhatbg;edaysar creep rbs;ebtug³ edaysnμt; Cc = 2.0 enaH Δf s = Cc (ε cr Es )
Fi
Elastic strain = = 1.56 ⋅10 − 4
Ac Ec

( )
Δf s = 2 1.56 ⋅10 −4 × 200000 = 62.4MPa

PaKrykMhatbg; =
62.4
1000
= 6.24%

b¤edaytMélRbhak;RbEhl eyIgyk ε cr = 7 ⋅10−5 × kugRtaMgenAkñúgebtug
⎛ 1935 ⎞
ε cr = 7 ⋅10 − 5 ⎜ 103 ⎟ = 3.76 ⋅10 − 4
⎝ 450 × 800 ⎠
Δf s = ε cr E s = 3.76 ⋅10 −4 × 200000 = 75.2MPa

PaKrykMhatbg; =
75.2
1000
= 7.52%

vaCatMélEdlmansuvtßiPaB ehIyeKnwgTTYl)anGRtadUcKñasMrab;karKNnaxagelIRbsinebIeKyk
Cc = 2.41 .

$> kMhatbg;edaysar relaxation rbs;Edk³ sMrab; low-relaxation strand eKsnμt;ykkMhatbg;
esμInwg 2.5%
Δf s = 1000 × 2.5% = 25MPa
%> snμt;kMhatbg;edaysarkarBt; kMlaMgkkitrbs; cable spacer nigbøúkxagcugrbs;RbB½n§
pretensioning KW 2% .

Δf s = 0.02 × 1000 = 20MPa
^> kMhatbg;edaysarkMlaMgkkitenAkñúg tendon KWsUnü.
&> kMhatbg;srubmandUcxageRkam


T.Chhay NPIC

Elastic Shortening 37.8MPa 3.78%
Shrinkage loss 60.0MPa 6.00%
Creep of concrete loss 62.4MPa 6.24%
Relaxation of steel loss 25.0MPa 2.50%
Other loss 20.0MPa 2.00%
Total loss 205.2MPa 20.52%
eRbkugRtaMgRbsiT§PaB = 1000 − 167.4 = 832.6MPa
kMlaMgeRbkugRtaMgRbsiT§PaB F = 832.6 ×1935 ⋅10 −3 = 1611kN
F = (1 − 0.167 )Fi = 0.833Fi
sMrab; F = ηFi
dUcenH η = 0.833

]TahrN_ 19>3³ KNnakMhatbg;TaMgGs;én posttensioned beam EdlmanRbEvg 36m . RkLaépÞ
rbs;muxkat;ebtug ( Ac ) = 49 ⋅104 mm 2 / m:Um:g;niclPaB (I g ) = 6.83 ⋅1010 mm 4 / kMlaMgeRbkugRtaMg
enAeBlepÞr (Fi ) = 4950kN / RkLaépÞEdkeRbkugRtaMg (Aps ) = 4840mm 2 / f 'c = 35MPa /
Ec = 34500MPa / nig E s = 2 ⋅10 5 MPa . Profile rbs; tendon manragCa)a:ra:bUl/ cMNakp©itenA

kNþalElVg = 50cm nigcMNakp©itenAxagcug = 0 .
dMeNaHRsay³
!> kMhatbg;eday elastic shortening:
kugRtaMgEdkenAeBlepÞr = AFi =
4950 3
4840
10 = 1022.7 MPa
ps

kugRtaMgenAkúñgebtugRtg;muxkat;xagcug = 49 ⋅104 103 = 10.1MPa
4950

2
kugRtaMgenAkúñgebtugRtg;muxkat;kNþalElVg = Ai + FiIe − MID e
F
c

TMgn;rbs;Fñwm = 49 ⋅10 −2 × 25 = 12.25kN / m
36 2
M D = 12.25 = 1984.5kN .m
8
4950 × 500 2 3 1982.5 × 500 6
kugRtaMgenAkNþalElVg =
4950
49 ⋅10 4
10 3 +
6.83 ⋅1010
10 −
6.83 ⋅1010
10

= 10.1 + 18.12 − 14.5 = 13.72 MPa
10.1 + 13.72
kugRtaMgmFüm =
2
= 11.9MPa


T.Chhay NPIC

mFüm = 11.9 = 34500 = 3.45 ⋅10− 4
strain
Ec
11.9

kMhatbg;eGLasÞicKW Δf s = ε c Es = 3.45 ⋅10 −4 × 2 ⋅105 = 69MPa edaysnμt;faeKTaj tendon
mþgBIrkñúgeBlEtmYy. KUrTImYynwgmankMhatbg;FMCageK b:uEnþKUrcugeRkaynwgmankMhatbg;esμIsUnü.
dUcenH kMhatbg;eGLasÞicmFüm Δf s = 69 / 2 = 34.5MPa .
PaKrykMhatbg; = 10225.7 = 3.37%
34.

@> kMhatbg;edaysarkarrYmmaDrbs;ebtug
Δf s (shrinkage) = 0.0002 E s = 0.0002 × 200000 = 40MPa

PaKrykMhatbg; =
40
1022.7
= 3.91%

#> kMhatbg;;edaysar creep rbs;ebtug³ snμt; Cc = 1.5
Fi 4950
elastic strain = = 10 3 = 2.93 ⋅10 − 4
Ac Ec 49 ⋅ 10 × 34500
4

( )
Δf s (creep) = Cc (ε cr E s ) = 1.5 2.93 ⋅10 −4 × 200000 = 87.9MPa

PaKrykMhatbg; =
87.9
1022.7
= 8.59%

$> kMhatbg;edaysar relaxation rbs;Edk³ sMrab; low-relaxation strand, kMhatbg;KW 2.5%
Δf s = 0.025 × 1022.7 = 25.6MPa
%> karrGilrbs; anchorage: sMrab;karTajEtBIcugmçag snμt;RbEvgrGil 3.8mm
ΔL 3.8
Δf s = Es = 200000 = 21.1MPa
L 36000
edIm,IGnuBaØateGaymankarrGilrbs; anchorage eKRtUvkMNt;kugRtaMgkñúgkarTaj 1022.7 + 21.1
= 1043.8MPa enAelI pressure gauge edIm,ITTYl net stress 1022.7 MPa enAkñúg tendon.

^> kMhatbg;EdlbNþalBIkMlaMgkit³ smIkar parabolic profile KW
e x = 2 (Lx − x 2 )
4e
L
Edl ex = cMNakp©itenARtg;cMgay x Edlvas;BITMr nig e = cMNakp©itenAkNþalElVg
d (e x ) 4e
= 2 (L − 2 x )
dx L
CaCMerl (slope) rbs; tendon enARKb;cMnucTaMgGs;. enARtg;TMr e = 0 eyIgTTYl)an slop
d (e x ) 4e 4 × 500
= = = 0.056
dx L 36000


T.Chhay NPIC

slope enAkNþalElVgesμIsUnü. dUcenH α px = 0.056 . edayeRbI flexible metallic sheath,
μ p = 0.5 nig K = 0.00333 . enAkNþalElVg x = 18m . RtYtBinitüfaetI (μ pα px + Kl x ) ≤ 0.30
μ pα px + Kl x = 0.5 × 0.056 + 0.00333 × 18 = 0.088 < 0.3
(
Ppx = Ppj 1 + Kl px + μ pα px )
= Px (1 + 0.088) = 1.088 Px
= 1.088 × 1022.7 = 1112.7 MPa ¬kMlaMgenAcug jacking¦
Δf s = 1112.7 − 1022.7 = 90MPa

PaKrykMhatbg; =
90
1022.7
= 8.8%

&> kMhatbg;srub
Elastic Shortening 34.5MPa 3.37%
Shrinkage loss 40.0MPa 3.91%
Creep of concrete loss 87.9MPa 8.59%
Relaxation of steel loss 25.6MPa 2.50%
Friction loss 90.0MPa 8.80%
Total loss 278.0MPa 27.17%
eRbkugRtaMgRbsiT§PaB = 1022.9 − 243.5 = 779.2MPa
kMlaMgeRbkugRtaMgRbsiT§PaB (F ) = (1 − 0.238)Fi = 0.762Fi
F = 0.762 × 4950 = 3772kN
sMrab; F = ηFi
dUcenH η = 0.762

4> viPaKGgát;rgkarBt;begáag Analysis of Flexural Members

k> kugRtaMgEdlbNþalBIlkçxNÐmanbnÞúk niglkçxNÐKμanbnÞúk
Stresses Due to Loaded and Unloaded condition
enAkñúgkarviPaKFñwmebtugeRbkugRtaMg CaTUeTAkardak;bnÞúkBIrmaneRKaHfñak;CageK. TImYyKWekIt
manenAeBlepÞr KWenAeBlFñwmrgkMlaMgeRbkugRtaMg Fi ehIyTMgn;rbs;Fñwm b¤bnÞúkefrGnuvtþn_enAxN³én
karepÞrkMlaMgkugRtaMg. eKminKitlkçxNÐbnÞúkefrbEnßm b¤bnÞúkGefreT. kñúglkçxNÐKμanbnÞúkenH kug
RtaMgenAsrésEpñkxagelIbMput nigEpñkxageRkambMputénmuxkat;eRKaHfñak;minRtUvFMCagkugRtaMgenA
eBlepÞr f ci nig f ti sMrab;kugRtaMgrgkarsgát; nigkugRtaMgrgkarTajrbs;ebtugerogKña.


T.Chhay NPIC

krNITIBIrénkardak;bnÞúkekIteLIgenAeBlEdlFñwmrgkMlaMgeRbkugRtaMgeRkayBIekItmankMhat
bg;TaMgGs; nigrgnUvbnÞúkefr nigGefr. enAkñúglkçxNÐmanbnÞúkenH kugRtaMgenAsrésEpñkxagelIbM
put nigEpñkxageRkambMputénmuxkat;eRKaHfñak;dac;xatminRtUvFMCagkugRtaMgGnuBaØat f c nig f t sMrab;
kugRtaMgrgkarsgát; nigkugRtaMgrgkarTajrbs;ebtugerogKña.
lkçxNÐTaMgenHGacsresrCaTMrg;KNitviTüadUcxageRkam³
!> sMrab;lkçxNÐKμanbnÞúk ¬enAeBlepÞr¦
- enAsrésEpñkxagelIbMput
Fi (Fi e ) yt M D yt
α ti = − + − ≤ f ti (19.14)
A I I
- enAsrésEpñkxageRkambMput
Fi (Fi e ) yb M D yb
α bi = − − − ≥ − f ci (19.15)
A I I
@> sMrab;lkçxNÐmanbnÞúk ¬bnÞúkTaMgGs;RtUv)andak;eRkayBIkMhatbg;eRbkugRtaMg¦
- enAsrésEpñkxagelIbMput
- σ t = − F + (FeI)yt − M D yt − M IL yt ≥ − f c
A I
(19.16)

- enasrésEpñkxageRkambMput
- σ b = − F − (FeI)yb − M D yb − M LI yb ≤ f t
A I
(19.17)

Edl Fi nig F = kMlaMgeRbkugRtaMgenAeBlepÞr nigeRkayBIkMhatbg;
f ti nig f t = kugRtaMgrgkarTajenAkñúgebtugenAeBlepÞr nigeRkayBIkMhatbg;

f ci nig f c = kugRtaMgrgkarsgát;enAkúñgebtugenAeBlepÞr nigeRkayBIkMhatbg;

M D nig M L = m:Um:g;EdlbNþalBIbnÞúkefr nigbnÞúkGefr

yt nig yb = cMgayBIG½kSNWteTAsrésEpñkxagelIbMput nigEpñkxageRkambMput

enAkúñgkarviPaK eKsnμt;fasMPar³manlkçN³eGLasÞicenAkúñgEdneFVIkarénkugRtaMgEdlGnuvtþ.

x> EdnkMNt;sñÚl Kern Limits
RbsinebIeKGnuvtþkMlaMgeRbkugRtaMgenARtg;TIRbCMuTMgn;rbs;muxkat; vanwgekItmankugRtaMg
BRgayesμI. RbsinebIGnuvtþkMlaMgeRbkugRtaMgenARtg;cMNakp©it e BIeRkamTIRbCMuTMgn; EdleFVIy:agNa
eGaykugRtaMgenAsrésEpñkxagelIbMputesμIsUnü enaHeKcat;TukkMlaMgeRbkugRtaMgenHmanGMeBIRtg;cMnug

T.Chhay NPIC

lower kern ¬rUbTI 19>5¦. enAkñúgkrNIenH e RtUv)ansMKal;eday K b ehIIykarBRgaykugRtaMgman
ragRtIekaN EdlmankugRtaMgsgát;GtibrmaenasréseRkameKbMput. kugRtaMgenAsrésxagelIbMputKW
Fi (Fi e ) yt
σt = − + =0
A I
I
e = K b = lower kern = (19.17)
Ayt
dUcKña RbsinebIeKGnuvtþkMlaMgeRbkugRtaMgenARtg;cMNap©it e' BIelITIRbCMuTMng; EdleFVIy:agNaeGaykug
RtaMgenAsrésEpñkxageRkambMputesμIsUnü enaHkMlaMgeRbkugRtaMgRtUv)aneKcat;TukfamanGMeBIRtg;cMnuc
upper lower ¬rUbTI 19>5¦. enAkñúgkrNIenHcMNap©it e' RtUv)ansMKal;eday K t ehIykarBRgaykug

RtaMgmanragRtIekaN EdlmankugRtaMgsgát;GtibrimaenAsrésEpñkxagelIbMput. kugRtaMgenAsrés
EpñkxageRkambMputKW
Fi (Fi e ) yb
σb = − + =0
A I
I
e' = K t = upper kern = (19.18)
Ayb
Kern limits énmuxkat;RtIekaNRtUv)anbgðajenArUbTI 19>5.


T.Chhay NPIC

K> karkMNt;tMéléncMNakp©it Limiting Values of Eccentricity
eKGacsresrsmIkarkugRtaMgTaMgbYn ¬BIsmIkar 19.13 dl; 19.16¦CaGnuKmn_éncMNakp©it e
sMrab;lkçxNÐénkardak;bnÞúkepSg². Ca]TahrN_ eKGacsresrsmIkar 19.13 dUcxageRkam
Fi (Fi e ) yt M D yt
σ ti = − + − ≤ f ti
A I I
(Fi e ) yt ≤ f + Fi + M D yt
ti
I A I
I ⎛ Fi M D yt ⎞
e≤ ⎜ + + f ti ⎟ (19.19)
Fi yt ⎝ A I ⎠
RbsinebIeKeRbI lower kern limit K b = I / Ayt / enaH
M D f ti AK b
e ≤ Kb + + (19.20)
Fi Fi
tMél e CacMNakp©itGtibrmaEdlQrelIsrésEpñkxagelIbMputsMrab;lkçxNÐKμanbnÞúk.
dUcKña BIsmIkar 19.14
I ⎛ Fi M D yb ⎞
e≤ ⎜− + + f ci ⎟ (19.21)
Fi yb ⎝ A I ⎠
M f AK
e ≤ − K t + D + ci t (19.22)
Fi Fi
tMél e CacMNakp©itGtibrmaEdlQrelIsrésEpñkxageRkambMputsMrab;lkçxNÐKμanbnÞúk. eKKNna
tMélGtibrma e BIsmIkarelITaMgBIredayyktMélEdltUcCagmkeRbI.
BIsmIkar 19.15
I ⎛ F M T yt ⎞
e≥ ⎜ + − fc ⎟ (19.23)
Fyt ⎝ A I ⎠
M f AK b
e ≥ Kb + T − c (19.24)
F F
Edl M T = m:Um:g;EdlbNþalBIbnÞúkefr nigbnÞúkGefr = (M D + M L ) . tMélenHCacMNakp©itGb,-
brma EdlQrelIsrésEpñkxagelIbMputsMrab;lkçxNÐRTbnÞúk. BIsmIkar 19.16
I ⎛ F M T yb ⎞
e≥ ⎜− + − ft ⎟ (19.25)
Fyb ⎝ A I ⎠
M f AK t
e ≥ Kt + T − t (19.26)
F F


T.Chhay NPIC

tMélenHCacMNakp©itGb,brmaEdlQrelIsrésEpñkxageRkambMput sMrab;lkçxNÐmanbnÞúk. eKKYr
KNnatMélGb,brma e TaMgBIrenHBIsmIkarTaMgBIrxagelI ehIyeKykcMNakp©itGb,brmaNaEdl
mantMélFMCageKmkeRbI.

X> tMélkMNt;émkMlaMgeRbkugRtaMgenAeBlepÞr
Limiting Values of the Prestessing Force at Transfer
edayKitfa F = ηFi Edl η CapleFobén net prestressing force eRkayBIkMhatbg; nig
sMrab;krNIepSg²énkardak;bnÞúk eKGacsresrsmIkar 19.20, 19.22, 19.24 nig 19.26 eLIgvijdUc
xageRkam³
(e − K b )Fi ≤ M D + f ti AK b (19.27)
(e + K t )Fi ≤ M D + f ci AK t (19.28)

(e − K b )Fi ≥ M D + M L − 1 ( f c AK b ) (19.29)
η η η
(e + K t )Fi ≥ M D +
ML
−
1
( f t AK t ) (19.30)
η η η
CMnYsmIkar 19.27 eTAkñúgsmIkar 19.30 eKTTYl)an
⎛1 ⎞ M f AK t
Fi (K b + K t ) ≥ M D ⎜ − 1⎟ + L − t
⎜η ⎟ η − f ti AK t
⎝ ⎠ η
⎡⎛ 1 ⎞ M L ⎛ f t AK t ⎞ ⎤
b¤ Fi ≥
1
⎢⎜ − 1⎟ M D +
⎜η ⎟
(K b + K t ) ⎣⎝ η ⎝
−⎜⎜ η ⎟ − ( f ti AK b )⎥
⎟ (19.31)
⎠ ⎠ ⎦
tMél Fi CatMélGb,brmaénkMlaMgeRbkugRtaMgenAeBlepÞredaymineGayFMCagkugRtaMgGnuBaØateRkam
lkçxNÐmanbnÞúk nigKμanbnÞúk. CMnYssmIkar 19.29 eTAkñúgsmIkar 19.28 edIm,ITTYl)an
⎡⎛ 1 ⎞ M L ⎛ f c AK b ⎞ ⎤
⎜ η ⎟ + ( f ci AK t )⎥
1
Fi ≤ ⎢⎜1 − ⎟ M D −
⎜ η⎟ +⎜ ⎟ (19.32)
(K b + K t ) ⎣⎝ ⎠ η ⎝ ⎠ ⎦
tMél Fi CatMélGtibrmaénkMlaMgeRbkugRtaMgenAeBlepÞredaymineGayelIskugRtaMgGnuBaØateRkam
lkçxNÐmanbnÞúk nigKμanbnÞúk. edayCMnYssmIkar 19.31 eTAkñúgsmIkar 19.32
⎛ 1⎞ 2M L ⎛ f ⎞ ⎛ f ⎞
⎜1 − ⎟ 2 M D −
⎜ η⎟ + ⎜ fti + c ⎟ AKb + ⎜ f ci + t ⎟ AKt ≥ 0
⎜ ⎟ ⎜ (19.33)
⎝ ⎠ η ⎝ η ⎠ ⎝ η⎟ ⎠
smIkarenHbgðajfa Fi max − Fi min ≥ 0 . eKeRbIsmIkarenHsMrab;bgðajfamuxkat;NamYymanlkçN³
RKb;RKan;.


T.Chhay NPIC

]TahrN_ 19>4³ FñwmTMrsamBaØEdlrgeRbkugRtaMgmunEdlmanRbEvg 14.4m manmuxkat;dUcbgðajenA
kñúgrUbTI 19>6 a. FñwmenHRTnUvbnÞúkefr 13.15kN / m ¬edayminrYmbBa©ÚlTMgn;pÞal;¦ nigrgnUvbnÞúk
Gefr 16kN / m . edaysnμt;faEdkeRbkugRtaMgpSMeLIgeday tendon 20 Edl tendon mYymanGgát;
p©it 11.125mm CamYynwg Es = 2 ⋅105 MPa / Fo = 1200MPa nig ultimate strength
f pu = 1725MPa .

!> kMNt;TItaMgEdnkMNt;xagelI nigEdnkMNt;xageRkamrbs; tendon profile ¬TIRbCMuTMgn;rbs;
EdkeRbkugRtaMg¦ sMrab;muxkat;enAkNþalElVg nigsMrab;muxkat;bIepSgeTotenAcenøaHmux
kat;kNþalElVg nigcugFñwm.


T.Chhay NPIC

@> dak; tendon cMTItaMgedIm,IbMeBjEdnkMNt;TaMgenH edayeGay tendon xøHegIbeLIgcab;BI
cMnucmYyPaKbIénRbEvgElVg. RtYtBinitütMélkMNt;énkMlaMgeRbkugRtaMgenAeBlepÞr.
#> RtYtBinitüeLIgvijnUvkMhatbg; edayKit tendon profile Edl)aneRCIserIs nigbMErbMrYlcM
Nakp©it e .
eRbI fci ¬enAeBlepÞr¦ = 28MPa / f 'c = 35MPa / Ec = 27600MPa nig Eci = 24800MPa .

dMeNaHRsay³
!> kMNt;lkçN³rbs;muxkat;
RkLaépÞ = 600 × 150 + 450 × 150 + 300 × 250 = 23.25 ⋅104 mm2


T.Chhay NPIC

kMNt;TIRbCMuTMgn;rbs;muxkat;edayKitm:Um:g;eFob)atrbs;muxkat;
yb =
1
23.25 ⋅ 10 4
(75 ⋅103 ×125 + 90 ⋅103 × 550 + 67.5 ⋅103 × 925) = 522mm
yt = 1000 − 522 = 478mm
KNna gross moment of inertia I g
⎡ 450(150 )3 ⎤ ⎡150(600)3 ⎤
Ig = ⎢ + (450)(150)(403)2 ⎥ + ⎢ + (150)(600 )(28)2 ⎥
⎢
⎣ 12 ⎥ ⎢
⎦ ⎣ 12 ⎥
⎦
⎡ 300(250)3 ⎤
+⎢ + (300 )(250)(397 )2 ⎥
⎢
⎣ 12 ⎥
⎦
= 2.607 ⋅ 1010 mm 4
I 2.607 ⋅ 1010
Kb = = = 235.6mm
Ayt 23.25 ⋅ 10 4 × 478
I 2.607 ⋅ 1010
Kt = = = 214.8mm
Ayb 23.25 ⋅ 10 4 × 522
@> )a:n;RbmaNkMhatbg;eRbkugRtaMg eday Fo = 1200MPa
a. snμt; elastic loss esμI 4% b¤ 0.04 × 1200 = 48MPa

b. kMhatbg;edaysarkarrYmmaDKW 0.0003Es = 0.0003 × 2 ⋅ 105 = 60MPa

c. kMhat;bg;edaysar creep rbs;ebtug ³ kar)a:n;RbmaNdMbUgd¾l¥bMputénkMhatbg;eday

sar creep KW 1.67 dgén elastic loss
1.67 × 48 ≈ 80 MPa
d. kMhatbg;edaysar relaxation énEdkKW 4% ³ 0.04 ×1200 = 48MPa
kMhatbg;GaRs½ynwgeBlKW 60 + 80 + 48 = 188MPa
PaKrykMhatbg; = 1200 = 15.7%
188

e. kMhatbg;srubKW 188 + 48 = 236MPa
PaKryénkMhatbg;srubKW
236
= 19.7%
1200
f. kugRtaMgkMlaMgeRbkugRtaMg
Fi = 1200 − 48 = 1152MPa ¬enAeBlepÞr¦
F = 1200 − 236 = 964 MPa


T.Chhay NPIC

F = ηFi
η = 1− pleFobkMhatbg;GaRs½ynwgeBl
964
= = 0.837
1152
#> EdnkMNt;éncMNakp©it e Rtg;kNþalElVg³ kMNt;kugRtaMgGnuBaØat nigm:Um:g;. enAeBlepÞr
f 'ci = 28MPa / f ci = 0.6 × 28 = 16.8MPa nig f ti = 0.25 f 'ci = 1.32MPa . enAeBlrgbnÞúk

eFVIkar f 'c = 35MPa / fc = 0.45 f 'c = 15.75MPa nig ft = 0.5 f 'c = 2.96MPa .
bnÞúkpÞal;rbs;Fñwm = 23.25 ⋅10−2 × 25 = 5.81kN / m
5.81(14.4 )2
M D ¬bnÞúkpÞal;¦ = = 150.6kN .m
8
2
Ma ¬bnÞúkbEnßm nigbnÞúkGefr¦ = wa8L
=
(13.15 + 16)14.4 2 = 755.6kN.m
8
m:Um:g;srub (M T ) = M D + M L = 906.2kN .m
Fi = kugRtaMgenAeBlepÞr × RkLaépÞEdkeRbkugRtaMg

RkLaépÞrbs; tendon 20 Edl tendon nImYy²manGgát;p©it 11.125mm KW
20 × 69.7 = 1394mm 2
Fi = 1394 × 1152 ⋅ 10 −3 = 1606kN

F = 1394 × 964 ⋅ 10 −3 = 1344kN
a. BicarNamuxkat;enAkNþalElVg
srésxagelIbMput kñúglkçxNÐminrgbnÞúk
M D f ti AK b
e ≤ Kb + +
Fi Fi

≤ 235.6 + 10 +
( )
150.6 3 1.32 23.25 ⋅ 10 4 (235.6) − 3
10 ≤ 374.4mm
1606 1606
srésxageRkambMput kñúglkçxNÐKμanbnÞúk
M D f ci AK t
e ≤ −Kt + +
Fi Fi

≤ −214.8 + 10 +
( )
150.6 3 16.8 23.25 ⋅ 10 4 214.8 − 3
10 ≤ 401.4mm
1606 1606
yktMél e EdltUcCageKkñúgcMeNamlT§plTaMgBIrxagelICatMélGtibrma.


T.Chhay NPIC

dUcenH tMélGtibrmarbs; e = 374mm
srésxagelIbMput kñúglkçxNÐrgbnÞúk
M T f c AK b
e ≥ Kb + −
F F

≥ 235.6 + 10 −
( )
906.2 3 15.75 23.25 ⋅ 10 4 235.6 − 3
10 ≥ 268mm
1344 1344
srésxageRkambMput kñúglkçxNÐmanbnÞúk
M T f t AK t
e ≥ −Kt + −
F F

≥ −214.8 + 10 −
( )
906.2 3 2.96 23.25 ⋅ 10 4 214.8 − 3
10 ≥ 349.5mm
1344 1344
tMélGb,brmarbs; e CatMéltUcCageKkñúgcMeNamlT§plTaMgBIrxagelI.
dUcenH tMélGb,brmarbs; e = 350mm
b. BicarNamuxkat;EdlenAcMgay 2.4m BIkNþalElVg ¬muxkat;elx @ kñúgrUbTI 19>6 a¦³
w
M D ¬bnÞúkpÞal;¦ = R A (4.8) − D (4.8)2
2
= (5.81)(7.2)(4.8) − (4.8)2 = 133.9kN.m
5.81
2
M a = (29.15)(7.2)(4.8) − (4.8)2 = 671.6kN .m
29.15
2
M T = 133.9 + 671.6 = 805.5kN .m
srésxagelIbMput sMrab;lkçxNÐminrgbnÞúk³
133.9 3 1.32(23.25 ⋅ 10 4 )(235.6) − 3
e ≤ 235.6 + 10 + 10 ≤ 364mm
1606 1606
srésxageRkambMput sMrab;lkçxNÐminrgbnÞúk
133.9 3 16.8(23.25 ⋅ 10 4 )214.8 − 3
e ≤ −214.8 + 10 + 10 ≤ 391mm
1606 1606

tMélGtibrmarbs; e = 364mm
srésxagelIbMput sMrab;lkçxNÐRTbnÞúk
805.5 3 15.75(23.25 ⋅ 10 4 )235.6 − 3
e ≥ 235.6 + 10 − 10 ≥ 193mm
1344 1344
sésrxageRkambMput sMrab;lkçxNÐRTbnÞúk
805.5 3 2.96(23.25 ⋅ 10 4 )214.8 − 3
e ≥ −214.8 + 10 − 10 ≥ 274.5mm
1344 1344


T.Chhay NPIC

tMélGb,brmarbs; e = 274.5mm
c. BicarNamuxkat;Rtg;cMgay 4.8m BIkNþalElVg ¬muxkat;elx # kñúgrUbTI 19>6 a¦³
M D ¬bnÞúkpÞal;¦ = 83.7 kN .m

M a = 419.8kN .m

M T = 503.5kN .m
- srésxagelIbMput sMrab;lkçxNÐKμanbnÞúk e ≤ 333mm ¬Gtibrma¦
- srésxageRkambMput sMrab;lkçxNÐKμanbnÞúk e ≤ 360mm
- srésxagelIbMput sMrab;lkçxNÐrgbnÞúk e ≥ −32mm
- srésxageRkambMput sMrab;lkçxNÐrgbnÞúk e ≥ 50mm ¬Gb,brma¦
d. BicarNamuxkat;Rtg;cMgay 0.9m BIcugFñwm ¬RbEvg anchorage¦³

M D ¬bnÞúkpÞal;¦ = 35.3kN .m / M a = 177.1kN .m nig M T = 212.4kN .m

- srésxagelIbMput sMrab;lkçxNÐKμanbnÞúk e ≤ 303mm ¬Gtibrma¦
- srésxageRkambMput sMrab;lkçxNÐKμanbnÞúk e ≤ 330mm
- srésxagelIbMput sMrab;lkçxNÐrgbnÞúk e ≥ −248mm
- srésxageRkambMput sMrab;lkçxNÐrgbnÞúk e ≥ −167mm ¬Gb,brma¦
$> Tendon profile RtUv)anbgðajenAkñúgrUbTI 19>6 b. cMNakp©itEdl)aneRCIserIsenAkNþalElVgKW
e = 364mm EdlvaRKb;RKan;sMrab;muxkat; B enAcMgay 2.4m BIkNþalElVg. TIRbCMuTMgn;énEdk

eRbkugRtaMgmanlkçN³edkcenøaH A nig B nigbnÞb;mkeTreLIgEdlmanlkçN³CabnÞat;cenøaHBI B
eTA E . cMNakp©itenARtg;muxkat; C nig D RtUv)anKNnaedayeRbIbnÞat;eRTt BE EdlmanCMerl
364 / 4.8 = 75.83mm / m . cMNakp©itRtg; C KW 182mm nigRtg; D KW 68mm . Tendon profile

Edl)aneRCIserIsbMeBjlkçxNÐEdnkMNt;xagelI nigEdnkMNt;xageRkamrbs;cMNakp©itenARKb;
muxkat;TaMgGs;.
karelIk tendon eLIgRtUv)aneFVIdUcxageRkam³
a. dak; tendon TaMg 20 ¬Ggát;p©it 11.125mm ¦

enAmYyPaKbIénkNþalElVgrbs;FñwmedaymanKMlat 50mm BIKñadUcbgðajenAkñúgrUbTI 19>6 a.
edIm,IKNnacMNakp©itCak;EsþgenARtg;muxkat;kNþalElVg Kitm:Um:g;sMrab; tendon eFobnwg)at
rbs;muxkat;³


T.Chhay NPIC

cMgayBI)at = 20 (16 × 125 + 4 × 275) = 155mm
1

e ¬kNþalElVg¦ = yb − 155 = 522 − 155 = 367 mm

EdlvaEk,rnwg 364mm Edl)ansnμt;. RbsinebIebIeKdak; tendon BIrenAcMgay 75mm BI
tendon EdlenAxageRkam enaHcMgayBI)atkøayeTACa
1
(16 × 125 + 2 × 250 + 2 × 325) = 158mm
20
enaHcMNakp©itnwgkøayeTACa 522 − 158 = 364mm EdlesμIweTAnwgcMNakp©itEdl)ansnμt;. Ca
karGnuvtþ eKdak; tendon TaMgGs;edaymanKMlatBIKña 50mm .
b. elIkEt tendon EdlenAkNþalcMnYn 12 eGayegIbeLIg. karBRgay tendon enAmuxkat;xag
cugRtUv)anbgðajenAkñúgrUbTI 19>6 a. RtYtBinitücMNakp©itrbs; tendon edayKitm:Um:g;eFobTI
RbCMuTMgn;rbs;muxkat;ebtugsMrab; tendon 12 enAxagelI nig tendon 8 eTotEdlRtUv)andak;
enAxageRkam³
e=
1
(8 × 364 − 12 × 226) = 10mm
20
tMél e enHtUc nigRKb;RKan;. cMNakp©itCak;EsþgenAcMgay 0.9m BImuxkat;xagcug
e=
0.9
(364 − 10) + 10 = 76mm
4.8
cMNakp©itCak;EsþgenAcMgay 2.4m BImuxkat;xagcugKW
e=
1
(364 − 10) + 10 = 187mm
2
%> tMélkMNt;rbs; Fi ³ tMélrbs; Fi EdleRbIsMrab;karKNnaBIxagelIKW Fi = 1606kN .
RtYtBinitü Fi Gb,brmaedayeRbIsmIkar 19.31:
⎡⎛ 1 ⎞ M L ( f t AK t ) ⎤
− ( f ti AK b )⎥
1
Fi min = ⎢⎜ − 1⎟ M D +
⎜η ⎟ −
(K b + K t ) ⎣⎝ ⎠ η η ⎦

10 −3
⎡⎛ 1
⎢⎜
⎞
− 1⎟150.6 ⋅ 106 + −
( )
755.6 ⋅ 106 2.96 × 23.25 ⋅ 10 4 214.8 ⎤
⎥
= ⎢⎝ 0.837 ⎠ 0.837 0.837 ⎥
(235.6 + 214.8) ⎢
(
⎣− 1.32 × 23.25 ⋅ 10 × 235.6
4
) ⎥
⎦

= 1516.8kN
vamantMéltUcCag Fi EdleRbI. RtYtBinitü Fi GtibrmaedayeRbIsmIkar 19.32:
1 ⎡⎛ 1 ⎞ M L f c AK b ⎤
Fi max = ⎢⎜1 − ⎟ M D −
⎜ η⎟ + + f ci AK t ⎥
(K b + K t ) ⎣⎝ ⎠ η η ⎦


T.Chhay NPIC

10 −3
⎡⎛
⎢⎜1 −
1 ⎞ 6 755.6 ⋅ 10
⎟150.6 ⋅ 10 −
6
+
( )
15.75 23.25 ⋅ 10 4 235.6 ⎤
⎥
= ⎢⎝ 0.837 ⎠ 0.837 0.837 ⎥
(235.6 + 214.8) ⎢
(
⎣+ 16.8 23.25 ⋅ 10 214.8
4
) ⎥
⎦

= 2081.9kN
vamantMélFMCag Fi EdleRbI. dUcenHmuxkat;eRKaHfñak;enAkNþalElVgKWRKb;RKan;.
^> RtYtBinitükMhatbg;eRbkugRtaMg edayeyIgman Fo = 1200MPa nig Aps = 1394mm2
kMlaMg Fo srub = 1200 × 1394 × 10−3 = 1672.8kN
Ec = 27600MPa
E
n= s =
Ec
200000
27600
= 7.25 yk n = 7
MD enAkNþalElVg = 150.6kN .m
Fo + nAps f c (D.L.) ×
2
Fi = 3
⎛1 e 2⎞
(
1 + nAps )⎜ + ⎟
⎜A I ⎟
⎝ ⎠
tMélrbs; fc Edl)anBIkarBRgaybnÞúkefrRtUvKuNnwg 2 / 3 edIm,IbgðajBIbMErbMrYlrag)a:ra:bUl
rbs;kugRtaMgbnÞúkefrtambeNþayFñwm Edlpþl;eGaynUvtMélRbhak;RbEhlrbs; Fi )an
RbesIrCag.
a. kMNt;tMélmFümrbs; e2 EdlTTYlenAkñúgFñwm. ExSekagtMNageGay e2 RtUv)anbgðajenA
kñúgrUbTI 19>6 c³
⎡1 ⎤
1 ⎢ (5776 × 0.9) + (5776 × 3.9) + (126720 × 3.9)⎥
1
e 2
¬mFüm¦ =
7.2 ⎢
3 3
⎥
⎣+ (2.4 × 132496 ) ⎦

= 70414.7mm 2

e = 265mm
RkLaépÞrbs;)a:ra:bUlesμInwgRkLaépÞrbs;ctuekaNEkg.
b. kugRtaMgEdlbNþalBIbnÞúkefrenARtg;nIv:Urbs; tendon KW
150.6 × 265
f c (D.L.) = 10 6 = 1.53MPa
2.607 ⋅ 10 10


T.Chhay NPIC

1672.8 ⋅ 103 + 7(1394) × 1.53 ×
2
dUcenH Fi =
⎛
3
70414.7 ⎞
10 − 3 = 1575.1kN
1 + 7(1394 )⎜
1
+ ⎟
⎝ 23.25 ⋅ 10 4
2.607 ⋅ 1010 ⎠
elastic loss KW 1672.8 − 1575.1 = 97.7kN = 5.8% . tMélenHKWFMCag elastic loss Edl)an
snμt; 4% .
kñúgmYyÉktþaRkLaépÞEdk = 1394 103 = 70MPa
elastic loss
97.7

Fi kñúgmYyÉktþaRkLaépÞ =
1575.1 3
10 = 1130MPa
1394
c. kMhatbg;GaRs½ynwgeBl
kMhat;bg;edaysarkarrYmmaDebtug = 60MPa ¬dUcelIkmun¦
kMhatbg;edaysar creep
Fi 1575.1
elastic strain = = 103 = 2.45 ⋅ 10 − 4
Ac Ec ( )
23.25 ⋅ 10 27600
4

Δf s = Cc (ε cr Es )
yk Cc = 1.5 enaH
( )
Δf s = 1.5 2.45 ⋅ 10 −4 200000 = 73.5MPa

PaKrykMhatbg; =
73.5
1130
= 6.5%

kMhatbg;edaysar relaxation rbs;EdkKW 48MPa ¬dUcelIkmun¦. kMhatbg;GaRs½ynwgeBl
esμInwg 60 + 73.5 + 48 = 181.5MPa ehIyPaKrykMhatbg;KW 181.5 / 1130 = 16% Edlman
tMélEk,rnwgtMélEdl)ansnμt; 15.7% .
F = ηFi = (1 − 0.16)Fi = 0.84 Fi

η = 0.84

5> KNnaGgát;rgkarBt;begáag Design of Flexural Members

k> sBaØaNTUeTA General
EpñkBIedIm)anbBa¢ak;fakugRtaMgenAsrésxagelIbMput nigsrésxageRkambMputénmuxkat;eRKaH
fñak;rbs;Ggát;ebtugeRbkugRtaMgminRtUvFMCagkugRtaMgGnuBaØatsMrab;RKb;krNITaMgGs; b¤dMNak;kalén


T.Chhay NPIC

kardak;bnÞúk. bEnßmBIelIlkçxNÐTaMgenH eKRtUvKNnaGgát;ebtugeRbkugRtaMgCamYynwgemKuNsuvtßi-
PaBRKb;RKan;edIm,IRbqaMgnwgkar)ak;. ACI Code tMrUveGaym:Um:g;Edl)anBIbnÞúkemKuN M u minRtUvFM
CagersIusþg;rgkarBt; φM n énmuxkat;Edl)anKNna.
sMrab;krNI tension-controlled section, FñwmebtugeRbkugRtaMgcab;epþIm)ak;enAeBlEdlkug
RtaMgEdkFMCag yield strength rbs;EdkEdleRbIenAkñúgmuxkat;ebtug. EdkeRbkugRtaMgersIusþg;nwgmin
bgðajcMnuc yield c,as;las;dUcEdkFmμtaEdleRbIenAkñúgebtugGarem:eT. b:uEnþeRkamkarbEnßmbnÞúk
strain enAkñúgEdkekIneLIgedayGRtay:agelOn ehIykar)ak;ekIteLIgenAeBl compressice strain

Gtibrmarbs;ebtugmantMélesμInwg 0.003 ¬rUbTI 19>7¦.

EdnkMNt;sMrab;EdkBRgwgrbs;Ggát;rgkarBt;ebtugeRbkugRtaMgEdlGaRs½yeTAtam ACI Code,
Section 18.8 KWQrelI net tensile strain sMrab; tension-controlled, transition b¤ compression-


T.Chhay NPIC

edayeKarBtam ACI Code, Section 10.3 dUcEdl)anBnül;enAkñúgCMBUk 3. em
controlled section

KuNkat;bnßyersIusþg; φ RtUv)aneGayenAkñúgCMBUkTI 3 edayQrelI ACI Code, Section 9.3.

x> muxkat;ctuekaN Rectangular Sections
eKGackMNt; Nominal moment capacity rbs;muxkat;ctuekaNdUcxageRkam ¬eyagtamrUb
TI 19>7¦³
⎛ a⎞ ⎛ a⎞
M n = C⎜ d − ⎟ = T ⎜ d − ⎟ (19.34)
⎝ 2⎠ ⎝ 2⎠
Edl T = Aps f s nig C = 0.85 f 'c ab . sMrab; C = T
Aps f ps ρ p f ps
a= = d (19.35|
0.85 f 'c b 0.85 f 'c
EdlpleFobEdkeRbkugRtaMgKW ρ p = Aps / bd ehIy Aps nig f ps CaRkLaépÞ nigkugRtaMgTajrbs;
EdkeRbkugRtaMg. yk
⎛ f ps ⎞
ωp = ρp⎜
⎜ ⎟ ≤ 0.32 β1
⎟
⎝ f 'c ⎠
ωp
bnÞab;mk a=
0.85
d (19.36)

tMél ω p CakMlaMgenAkñúg tendon EdlRtUv)anvas;edaypÞal;. edIm,IFananUv tesion-controlled
behavior, ACI Code, Section 18.8.1 kMNt;fa ω p minRtUvFMCag 0.32β1 EdkRtUvKñanwg net tensile

strain ε t = 0.005 . cMNaMfa β1 = 0.85 sMrab; f 'c ≤ 28MPa nwgkat;bnßyeday 0.05 sMrab;ral;

7 MPa sMrab; 28MPa < f 'c < 56MPa ehIyesμInwg 0.65 sMrab; f 'c > 56MPa . eKk¾Gacsresr
⎛ a⎞
M n = Aps f ps ⎜ d − ⎟
⎝ 2⎠
⎛ ρ p f ps ⎞
⎜ 1 .7 f ' ⎟
M n = A ps f ps d ⎜1 − ⎟ (19.37)
⎝ c ⎠
⎛ ωp ⎞
⎜ 1 .7 ⎟
M n = A ps f ps d ⎜1 − ⎟ (19.38)
⎝ ⎠
nig M u = φM n
enAkúñgsmIkarBImun f ps CakugRtaMgenAkñúgEdkeRbkugRtaMgenAeBl)ak;. eKminGackMNt;tMél
Cak;Esþgrbs; f ps edaygayRsYleT. dUcenH ACI Code, Section 18.7.2 GnuBaØateGay)a:n;RbmaN
tMél f ps dUcxageRkam.


T.Chhay NPIC

sMrab; bonded tendons
⎡ γp ⎛ f ⎞⎤
f ps = f pu ⎢1 − ⎜ ρ p × pu ⎟⎥
⎜ (19.39)
⎢ β1
⎣ ⎝ f 'c ⎟ ⎥
⎠⎦
sMrab; unbonded tendon enAkñúgGgát;EdlmanpleFobElVgelIkMBs;tUcCag b¤esμI 35
⎛ f 'c ⎞
f ps = ⎜ f se + 69 + ⎟ ≤ f py (19.40)
⎜ 100 ρ p ⎟
⎝ ⎠
RbsinebI f se ≥ 0.5 f pu nigRbsinebI f ps sMrab; unbonded tendon minFMCag f py b¤ f se + 415MPa .
sMrab; unbonded tendon enAkñúgGgát;EdlmanpleFobElVgelIkMBs;FMCag 35
⎛ f 'c ⎞
f ps = ⎜ f se + 69 + ⎟ (19.41)
⎜ 300 ρ p ⎟
⎝ ⎠
b:uEnþminRtUvFMCag f py b¤ f se + 207MPa Edl
γ p = emKuNsMrab;RbePTrbs; tendon eRbkugRtaMg
= 0.55 sMrab; f py / f pu EdlmintUcCag 0.8



f pu = ersIusþg;TajEdlkMNt;rbs;EdkeRbkugRtaMg

f se = kugRtaMgRbsiT§PaBenAkñúgEdkeRbkugRtaMgeRkayeBlkMhatbg;TaMgGs;

f py = specified yield strength rbs;EdkeRbkugRtaMg

enAkñúgkrNIEdl ω p > 0.32β1 FñwmebtugeRbkugRtaMgCa compression-controlled section.
edIm,IFananUv ductile failure eKkMNt; ω p RtwmtMélGtibrma 0.32β1 . sMrab; ω = 0.32β1 nig
a = 0.377 β1d eyIgTTYl)an
⎛ 0.32 β1 ⎞
M n = Aps f ps d ⎜1 − ⎟
⎝ 1.7 ⎠
( )
= ρ p bd f ps d (1 − 0.188β1 )

= ω p f 'c (1 − 0.188β1 )bd 2
( )
= 0.32β1 − 0.06β12 f 'c bd 2 (19.42)
sMrab; f 'c = 35MPa / β = 0.8 . enaH
1

M n = 0.22 f 'c bd 2 = 1.09bd 2
dUcKña/ sMrab; /
f 'c = 28MPa M n = 0.915bd 2 nigsMrab; /
f 'c = 42 MPa M n = 1.238bd 2


T.Chhay NPIC

K> muxkat;Edlmansøab Flanged Sections
sMrab;muxkat;mansøab (T- or I-section) RbsinebIkMBs;bøúk a sßitenAkñúgsøab eKnwgKitvaCa
muxkat;ctuekaNEkg. RbsinebI a sßitenAkñúgRTnug enaHeKKitRTnugCamuxkat;ctuekaNEkgedayeRbI
TTwgRTnug nigTTwgsøabEdlelIs (b − bw ) RtUv)anKitdUcKñanwgebtugGarem: T-section Edl)anBnül;
enAkñúgCMBUk 3 nig4. eKGacKNna design moment strength rbs; flanged section dUcxageRkam
¬emIlrUbTI 19>7¦.
M n = M n1 ¬ersIusþg;m:Um:g;rbs;RTnug¦ + M n2 ¬ersIusþg;m:Um:g;rbs;søabEdlelIs¦
⎛ a⎞ ⎛ hf ⎞
M n = A pw f ps ⎜ d p − ⎟ + A pf f ps ⎜ d p −
⎜ ⎟ (19.43)
⎝ 2⎠ ⎝ 2 ⎟⎠
A pw f ps
M u = φM n nig
a=
0.85 f 'c bw
Edl Apw = Aps − Apf

[
Apf = 0.85 f 'c (b − bw )h f / f ps ]
kMras;rbs;søab
hf =

cMNaMfaRkLaépÞEdkeRbkugRtaMgsrub Aps EckecjCaBIrEpñk Apw nig Apf EdlbegáIt web
moment capacity nig flange moment capacity. sMrab;muxkat;Edlmansøab snÞsSn_EdkBRgwg

(reinforcement index) ω pw minRtUvFMCag 0.32 β1 sMrab; tension-controlled section Edl
⎛ A pw ⎞⎛ f ps ⎞ ⎛f ⎞
ω pw = ⎜ ⎟⎜
⎜ b d ⎟⎜ f ' ⎟ ⎟ = GRtaEdkRTnugeRbkugRtaMg × ⎜ ps ⎟
⎜ f' ⎟
⎝ w ⎠⎝ c ⎠ ⎝ c ⎠

X> EdkBRgwgrgeRbkugRtaMg Nonprestressed Reinforcement
kñúgkrNIxøH eKdak;EdkminrgeRbkugRtaMg As enAkñúgtMbn;Tajrbs;Ggát;rgkarBt;ebtugeRbkug
RtaMgCamYynwgEdkeRbkugRtaMg Aps edIm,IbegáInersIusþg;m:Um:g; (moment strength) rbs;Fñwm. enAkñúg
krNIenH Edksrub ¬ Aps nig As ¦ RtUv)anBicarNaenAkñúgkarviPaKm:Um:g;. sMrab;muxkat;ctuekaNEdl
manEdkrgeRbkugRtaMg nigEdkminrgeRbkugRtaMg eKKNna design moment strength φM n dUcxag
eRkam³
⎛ a⎞ ⎛ a⎞
M n = Aps f ps ⎜ d p − ⎟ + As f y ⎜ d − ⎟ (19.44)
⎝ 2⎠ ⎝ 2⎠


T.Chhay NPIC

Edl a = Aps0.f85 f+' Abs f y / ehIy d p nig d CacMgayBIsrésrgkarsgát;xageRkAbMputeTATIRbCMuTMgn;
ps

c

rbs;EdkrgeRbkugRtaMg nigEdkminrgeRbkugRtaMg erogKña. sMrab;muxkat;Edlmansøab
⎛ a⎞ ⎛ a⎞ ⎛ hf ⎞
M n = A pw f ps ⎜ d p − ⎟ + As f s ⎜ d − ⎟ + A pf f ps ⎜ d p −
⎜ ⎟ (19.45)
⎝ 2⎠ ⎝ 2⎠ ⎝ 2 ⎟⎠
Aps f ps + As f y
a=
0.85 f 'c bw
sMrab;muxkat;ctuekaNEkgEdlmanEdkrgkarsgát; ehIym:Um:g;RtUv)anKiteFobnwgkMlaMg C
⎛ a⎞ ⎛ a⎞ ⎛a ⎞
M n = Aps f ps ⎜ d p − ⎟ + As f y ⎜ d − ⎟ + A's f y ⎜ − d ' ⎟ (19.46)
⎝ 2⎠ ⎝ 2⎠ ⎝2 ⎠
Aps f ps + As f y − A's f y
Edl a=
0.85 f 'c b
smIkarenHmann½yEtenAeBlEdlEdkrgkarsgát; yield. lkçxNÐsMrab;Edkrgkarsgát; yield KW
⎛ Aps f ps + As f y − A's f y ⎞ f ' d ' ⎛ 600 ⎞
⎜
⎜ ⎟ ≥ 0.85β1 c ⎜
⎟
⎟
⎝ bd ⎠ d ⎜ 600 − f y
⎝
⎟
⎠
RbsinebIeKminCYblkçxNÐenHeT enaHEdkrgkarsgát;min yield eT. enAkñúgkrNIenH eKGacecal A's
¬yk A's = 0 ¦ b¤mü:ageTot eKGackMNt;kugRtaMgenAkñúg A's edaykarviPaKTUeTA dUcEdlBnül;enAkñúg
CMBUk 3.
enAeBleKeRbIEdkrgeRbkugRtaMg nigEdkminrgeRbkugRtaMgenAkñúgmuxkat;dUcKña eKKYrGansmIkar
19.39 dUcxageRkam³
⎡ γp ⎛ f ⎞⎤
f ps = f pu ⎢1 − ⎜ ρ p pu + d (ω − ω ')⎟⎥ (ACI Code, Eq. 18.3) (19.47)
⎢ β1 ⎜ f 'c d p ⎟⎥
⎣ ⎝ ⎠⎦
RbsinebIeKKitEdkrgkarsgát;TaMgGs; enAeBlKNna f ps enaHtY
f pu d
ρp + (ω − ω ') ≥ 0.17
f 'c dp

nig d '≤ 0.15d p
Edl d / d ' nig d p CacMgayBIsrésrgkarsgát;xageRkAbMputeTATIRbCMuTMgn;rbs;EdkTajEdlminrg
eRbkugRtaMg Edkrgkarsgát; nigEdkrgeRbkugRtaMg erogKña.
γ p = emKuNsMrab;RbePTrbs; tendon eRbkugRtaMg


T.Chhay NPIC

sMrab; f py / f pu EdlmintUcCag 0.85
= 0 .4


β1 = 0.85 sMrab; f 'c ≤ 28MPa nwgkat;bnßyeday 0.05 sMrab;ekIneLIg 7 MPa rbs; f 'c Et
β1 ≥ 0.65 .
!> sMrab;muxkat;ctuekaNEkg ACI Cose,Section 18.8 kMNt;pleFobEdkdUcxageRkam
¬ ε t ≥ 0.005 sMrab; tension –controlled section¦³
d
wp + ω ≤ 0.32β1
dp
⎛ f ps ⎞ Aps
Edl ωp = ρp⎜
⎜ ⎟
⎟ nig ρp =
bd
¬EdkrgeRbkugRtaMg¦
⎝ f 'c ⎠
⎛ fy ⎞ As
ω = ρ⎜
⎜ ⎟
f 'c ⎟
nig ρ=
bd
¬EdkminrgeRbkugRtaMg¦
⎝ ⎠
@> RbsinebIeKeRbIEdkFmμta A's enAkñúgtMbn;sgát; enaHlkçxNÐkøayeTACa
d
ωp + (ω − ω ') ≤ 0.32β1
dp

Edl ω ' = ρ ' ( f y / f 'c ) nig ρ ' = A's / bd . EdnkMNt;Edk (reinforcement limitation)
mansar³sMxan;edIm,I Fana plastic failure rbs;FñwmebtugGarem:EdkmanEdktic.
#> sMrab;muxkat;mansøab RkLaépÞEdlcaM)ac;edIm,IbegáItersIusþg;rbs;RTnug Apw RtUv)aneRbI
edIm,IRtYtBinitü reinforcement index.
⎛ f ps ⎞
ω pw ¬RTnug¦ = ρ pw ⎜ ⎜ f ' ⎟ ≤ 0.32 β1
⎟
⎝ c ⎠
Apw
Edl ρ pw =
bw d d
RbsinebIeKeRbIEdkminrgeRbkugRtaMg enaH reinforcement limitation KW
d
ω pw + (ωw − ω 'w ) ≤ 0.32β1
d pw
As ⎛ f y ⎞ A' s ⎛ fy ⎞
Edl ωw = ⎜ ⎟
bw d ⎜ f 'c ⎟
nig ω 'w =
bw d
⎜
⎜ f' ⎟
⎟
⎝ ⎠ ⎝ c⎠
enAeBleKmineRbIEdkrgkarsgát; A's enaH ω 'w = 0 . kñúgkarKNna nigkarviPaKGgát;
ebtugeRbkugRtaMgedayEpñk (partially prestressed concrete member) eKRtUvEtCYb
lkçxNÐEdkxagelI.


T.Chhay NPIC

sMrab;fñak; C énGgát;ebtugeRbkugRtaMgrgkarBt; Edl ft > f 'c ¬muxkat;EdleRbH¦
eKKYreRbI crack control provision EdlBnül;enAkñúgEpñk 6>7 . enAeBleRbIsmIkar 6.18
sMrab;KMlatGtibrma s / ACI Code, Section 18.4 kMNt;dUcxageRkam³
a. sMrab; tendon eRbIKMlat s = 17 mm .

b. sMrab;bnSMénEdkminrgeRbkugRtaMg nig tonden eRbIKMlat s = 20mm .

c. sMrab; tendon eRbI Δf ps CMnYseGay f s Edl Δf ps CaPaBxusKñarvagkugRtaMgEdl

KNnaenAkñúg tendon eRbkugRtaMgeRkambnÞúkeFVIkaredayQrelImuxkat;eRbH nig
decompression stress f dc enAkñúg tendon eRbkugRtaMg EdlRtUv)anKiteGayesμInwgeRb

kugRtaMgRbsiT§PaB (effective prestress) f se . cMNaMfa Δf ps minKYrelIsBI 250MPa .
RbsinebIvaticCag b¤esμInwg 140MPa eKminGacGnuvtþKMlatEdlTamTareT.
eKGacsresrsmIkar 8.18 dUcxageRkam³
⎛ 2 ⎞ ⎡⎛ 3700 ⎞ ⎤
s = ⎜ ⎟ ⎢⎜ ⎟ − 2.5Cc ⎥
⎝ 3 ⎠ ⎢⎜ Δf ps ⎟
⎣⎝ ⎠ ⎥
⎦

6> m:Um:g;eRbH Cracking Moment

sñameRbHekItmanenAkñúgFñwmebtugeRbkugRtaMgenAeBlEdlkugRtaMgTajenAsrésEpñkxageRkA
bMputrbs;muxkat;eRKaHfñak;esμI b¤elIs modulus of rupture rbs;ebtug f r . eKGacsnμt;tMélrbs;
f r sMrab;ebtugTMgn;FmμtaeGayesμInwg 0.62 f 'c . kugRtaMgenAsrésxageRkambMputrbs;FñwmTMr

samBaØEdlbegáItedaykMlaMgeRbkugRtaMg nig cracking moment KW
F (Fe ) yb M cr yb
σb = − − +
A I I
enAeBlEdl σ b = f r = 0.62 f 'c enaH cracking moment KW
I ⎛ F (Fe ) yb ⎞
M cr = ⎜ 0.62 f 'c + + ⎟ (19.48)
yb ⎝ A I ⎠
kugRtaMgTajGtibrmaeRkayBIkMhatbg;KW 0.62 f 'c EdltMNageGay f r . enAkñúgkrNIenH Fñwmeb
tugeRbkugRtaMgenArkSamuxkat;minmansñameRbHeRkamGMeBIbnÞúkefr. edIm,IFanaPaBRKb;RKan;ersIusþg;Rb
qaMgnwgsñameRbH ACI Code, Section 18.8.3 TamTarfa ultimate moment strength rbs;Ggát; φM n
y:agticRtUvesμInwg moment cracking 1.2 dg.


T.Chhay NPIC

]TahrN_ 19>5³ sMrab;Fñwmén]TahrN_ 19>4 cUrRtYtBinitü design strength nigtMrUvkarrbs; ACI
Code EdlRbqaMgnwg cracking moment.
dMeNaHRsay³
!> RtYtBinitüemIlfakMBs;bøúkkugRtaMg a sßitenAkñúgsøabb¤Gt;.
Aps f ps
a= (19.35)
0.85 f 'c b
¬én 20 tendon Edl tendon mYy²manGgát;p©it 11.125mm ¦= 1394mm2
Aps

yk f py / fu = 0.85 / ρ p = 0.4 nig γ p / β1 = 0.4 / 0.8 = 0.5 . sMrab; bonded tension
⎛ γp f pd ⎞
f ps = f pu ⎜1 −
⎜ ρp × ⎟ (19.39)
⎝ β1 f 'c ⎟
⎠

d = 1000 − 158 = 842mm
Aps 1394
ρp = = = 3.68 ⋅ 10 − 3
bd 450 × 842
edayeKeGay f pu = 1725MPa
⎡
(
f ps = 1725⎢1 − 0.5 3.68 ⋅ 10 − 3
⎣
)
1725 ⎤
35 ⎥⎦
= 1568.6MPa

1394(1568.6)
a= = 163.3mm
0.85(35)450
edayvamantMélFMCakMras;søab 150mm . dUcenH muxkat;enHeFVIkarCamuxkat;mansøab.
@> sMrab;muxkat;mansøab
⎛ a⎞ ⎛ hf ⎞
M n = Apw f ps ⎜ d − ⎟ + Apf f ps ⎜ d −
⎜ ⎟
⎟
⎝ 2⎠ ⎝ 2 ⎠
Apf =
1
f ps
[0.85 f 'c (b − bw )h f ]
=
1
[0.85(35)(450 − 150)150] = 853.5mm 2
1568.6
Apw = 1394 − 853.5 = 540.5mm 2
Apw f ps 540.5(1568.6)
a= = = 190mm
0.85 f 'c bw 0.85(35)150
⎛ 190 ⎞ − 6 ⎛ 150 ⎞ − 6
M n = 540.5(1568.6)⎜ 842 − ⎟10 + 853.5(1568.6)⎜ 842 − ⎟10 = 1660.2kN .m
⎝ 2 ⎠ ⎝ 2 ⎠


T.Chhay NPIC

φM n = 0.9(1660.2 ) = 1494.2kN .m
RtYtBinitü reinforcement index sMrab;muxkat;mansøab
A pw 540.5
ρ pw = = = 4.28 ⋅ 10 − 3
bw d 150 × 842
f ps
ω pw = ρ pw
f 'c
≤ 0.32 β1 = 0.32 × 0.8 = 0.256 ¬ φ = 0.9 ¦
#> KNnam:Um:g;emKuNxageRkAEdl)anBIbnÞúkefr nigGefr
bnÞúkefr = TMgn;pÞal; + bnÞúkefrbEnßm
= 5.81 + 13.15 = 18.96kN.m
bnÞúkGefr = 16kN / m
U = 1 .2 D + 1 .6 L
14.4 2
Mu = [1.2 × 18.96 + 1.6 × 16] = 1253.3kN .m
8
m:Um:g;emKuNxageRkAenHmantMéltUcCag ultimate moment capacity rbs;muxkat; 1494.2kN.m
dUcenHmuxkat;enHRKb;RKan;.
$> cracking moment ¬smIkar !(>$*¦ KW
I ⎛ F y ⎞
M cr = ⎜ 0.62 f 'c + + (Fe ) b ⎟
yb ⎝ A I ⎠
BI]TahrN_ 19>4 F = 1344kN / A = 23.25 ⋅104 mm2 / e = 364mm / yb = 522mm /
I = 2.607 ⋅ 1010 mm 4 / f 'c = 35MPa nig 0.62 f 'c = 3.67 MPa

⎜ 3.67 + 1344 ⋅ 10 + (1344 ⋅ 103 × 364)
2.607 ⋅ 1010 ⎛ 3
522 ⎞ −6
M cr = ⎟10
522 ⎜ 23.25 ⋅ 10 4
2.607 ⋅ 1010 ⎟
⎝ ⎠

= 961.2kN.m
RtYtBinitüemIlfa 1.2M cr ≤ φM n
1.2 M cr = 1.2(961.2) = 1153.4kN .m
tMélenHKWtUcCag φM n = 1494.2kN .m . dUcenH FñwmenHRKb;RKan;kñúgkarTb;nwgkareRbH.


T.Chhay NPIC

7> PaBdab Deflection

PaBdabrbs;cMnucmYyenAelIFñwmCabMlas;TIsrubrbs;cMnucenaH eTaHCaeLIgelI b¤cuHeRkamEdl
bNþalBIkarGnuvtþrbs;bnÞúkenAelIFñwmenaH. enAkñúgFñwmebtugeRbkugRtaMgEdlRTedayTMrsamBaØ CaTU
eTAkMlaMgeRbkugRtaMgRtUv)aneKGnuvtþenABIeRkamTIRbCMuTMgn;rbs;muxkat; EdlbegáItCaPaBdabeLIgelI
EdleKeGayeQμaHfa camber. TMgn;pÞal;rbs;Fñwm nigbnÞúkTMnajxageRkAEdlmanGMeBIenAelIFñwmnwgeFVI
eGaymanPaBdabcuHeRkam. Net deflection CaplbUkBiCKNiténPaBdabTaMgBIr.
enAkñúgkarKNnaPaBdab eKcaM)ac;RtUvBicarNaTaMgPaBdabry³eBlxøI (short-term deflection)
b¤PaBdabPøam² (immediate deflection) nigPaBdabry³eBlyUr (long-term deflection). edIm,IFa
naeGayeRKOgbgÁúMGaceFVIkareTA)an PaBdabry³eBlxøIGtibrma nigPaBdabry³eBlEvgGtibrmaenA
RKb;dMNak;kalénkardak;bnÞúkEdlmanlkçN³eRKaHfñak;TaMgGs;minRtUvFMCagtMénkMNt;Edl)ankMNt;
eday ACI Code ¬emIlEpñk 6>3¦.
eKGacKNnaPaBdabrbs;Ggát;ebtugeRbkugRtaMgedaysmIkarPabdabsþg;dar b¤eday conven
tional method EdlmanenAkñúgesovePAviPaKeRKOgbgÁúM. ]TahrN_ PaBdabenAkNþalElVgrbs;FñwmTMr

samBaØEdlrgbnÞúkTMnajBRgayesμI w esμInwg 5wL4 / 384EI . m:UDuleGLasÞicrbs;ebtugKW
Ec = 0.043w1.5 f 'c = 4780 f 'c sMrab;ebtugTMgn;Fmμta.

eKKNnam:Um:g;niclPaBrbs;muxkat;ebtug I edayQrelIlkçN³rbs; gross section sMrab;
FñwmEdlKμansñameRbH. eKGaceRbIkrNIenH)anenAeBlEdlkugRtaMgTajGtibrmaenAelIsrésxageRkA
bMputrbs;ebtugminelIs modulus of rupture rbs;ebtug f r = 0.62 f 'c ¬Fñwmfñak; U ¦. enAeBl
EdlkugRtaMgTajGtibrmaEdlWrelIlkçN³rbs; gross section FMCag 0.62 f 'c eKRtUveRbIm:Um:g;nicl
PaBRbsiT§PaB (effective moment of inertia) I e EdlQrelImuxkat;EdlmansñameRbH b¤Gt;man
sñameRbH dUcEdl)anBnül;enAkñúgCMBUk 6 ¬Fñwmfñak; T nig C ¦. PaBdabkNþalE;lVgsMrab;FñwmTMr
samBaØEdlbNþalBIbnÞúkTMnaj nigkMlaMgeRbkugRtaMgRtUv)anbgðajenAkñúgtarag 19>3.

]TahrN_ 19>6³ sMrab;FñwmenAkñúg]TahrN_ 19>4 cUrKNna camber enAeBlepÞr nigbnÞab;mkKNna
PaBdabPøam²EdlrMBwgTukcugeRkayEdlrgbnÞúkeFVIkar.
dMeNaHRsay³
!> PaBdabenAeBlepÞr


T.Chhay NPIC

a. KNnaPaBdabcuHeRkamEdlbNþalBIbnÞúkefrenAeBlepÞr ¬kñúgkrNIenHbnÞúkefrCabnÞúkpÞal;¦
sMrab;FñwmTMrsamBaØEdlrgbnÞúkBRgayesμI
5wL4
Δ D ¬bnÞúkefr¦ =
384 EI
BI]TahrN_ 19>4/ wD = 5.81kN / m / L = 14.4m / Eci = 24800MPa nig
I = 2.607 ⋅ 1010 mm 4


T.Chhay NPIC

5(5.81)14400 4
ΔD =
384(24800 )2.607 ⋅ 1010
= 5mm ¬cuHeRkam¦
b. KNna camber EdlbNþalBIkMlaMgeRbkugRtaMg
sMrab;FñwmTMrsamBaØEdleKelIk tendon xøHenAcMnucmYyPaKbI CamYynwgcMNakp©itenAcMENk
mYyPaKbIkNþalFñwm e1 = 364mm nigenAcugFñwm e2 = 0 .
23(Fi e1 )L2
Δp =
216 Eci I

=
( )
23 1606 ⋅ 103 × 364 14400 2
= −20mm ¬eLIgelI¦
216(24800 )2.607 ⋅ 1010
Camber cugeRkayenAeBlepÞrKW − 20 + 5 = −15mm ¬eLIgelI¦
c. PaBdabeRkambnÞúkeFVIkar

bnÞúkeFVIkarBRgayesμIsrubKW WT = 5.81 + 13.15 + 16 = 35kN / m
nig Ec = 27600MPa . PaBdabcuHeRkamEdlbNþalBI WT KW
5WT L4 5(35)14400 4
Δw = =
384 E I 384(27600 )2.607 ⋅ 1010
= 27 mm ¬cuHeRkam¦
c

Camber EdlekItBIkMlaMgeRbkugRtaMg F = 1344kN nig Ec = 27600MPa KW
23(1344 ⋅ 103 × 364 )
14400 2
Δp = = −15mm
216(27600 )2.607 ⋅ 1010
¬eLIgelI¦
PaBdabPøam²cugeRkayeRkambnÞúkeFVIkarKW
Δ = Δ w − Δ p = 27 − 15 = 12mm ¬cuHeRkam¦

8> KNnasMrab;kMlaMgkat;TTwg Design for Shear

viFIKNnaedIm,IkMNt;EdkkMlaMgTTwg (shear reinforcement) enAkñúgFñwmebtugeRbkugRtaMgesÞIr
EtdUcKñanwgviFIKNnasMrab;FñwmebtugGarem:. eKsnμt;sñameRbHedaysarkMlaMgkat;TTwg (shear crack)
ekIteLIgtammMu 45o edayvas;BIG½kSrbs;Fñwm. CaTUeTA TMrg;énsñameRbHEdlTak;TgnwgkMlaMgkat;TTwg
cMnYnBIrRbePT. mYyRbePTKWbNþalBIT§iBlEdlrYmpSMénkarBt; nigkMlaMgkat;TTwg³ sñameRbHcab;
epþImeLIgedaysñameRbHedaykarBt; ehIybnÞab;mksñameRbHenHcab;epþImgak ehIyraltamTiseRTt
EdlbNþalBIT§iBlénkMlaMgTajGgát;RTUg. RbePTTIBIrCa web-shear cracking ekItmanenAkñúgFñwm
EdlkugRtaMgTajem b¤kugRtaMgTajGgát;RTUg (principal tensile stress) enAkñúgRTnugd¾tUcrbs;vaman

T.Chhay NPIC

tMélFMebIeRbobeFobnwgkugRtaMgBt;. eKRtUveRbIEdkkgedIm,IFana principal tensile stress enAkñúgkrNI
TaMgBIr. eyIgnwgeRbIlkçNvinicä½ykñúgkarKNnarbs; ACI sMrab;kMlaMgkat;TTwg.

k> viFIcMbg Basic Approach
ACI design approachQrelItMrUvkarersIusþg;cugeRkay (ultimate strength requirement)
edayeRbIemKuNbnÞúkEdl)anbgðajenAkñúgCMBUk 3. enAeBlEdlkMlaMgkat;TTwgemKuN Vu FMCagBak;
kNþalén nominal shear strength ( φVc / 2 ) eKmindak;EdkkgeT. . kMlaMgkat;TTwgKNnaEdlRtUv
kar Vu enARtg;muxkat;nImYy²minRtUvFMCag nominal design strength φVn rbs;muxkat;EdlQrelI
nominal shear capacity Edl)anBIkarbUkpSMénebtug nigEdkRTnug.
Vu ≤ φVn ≤ φ (Vc + Vs ) (19.49)
Edl rbs;ebtug
Vc = nominal shear strength

Vs = nominal shear capacity rbs;Edk

φ = emKuNkat;bnßyersIusþg; = 0.75
enAeBlEdlkMlaMgkat;emKuN Vu tUcCag φVu / 2 eKRtUvkardak;EdkkMlaMgkat;Gb,brma.

x> ersIusþg;kMlaMgkat;Edlpþl;edayebtug
Shear Strength Provided by Concrete
ACI Code, Section 11.4 bgðajsmIkary:agsamBaØEdl)anBIkarBiesaFedIm,I)a:n;RbmaN
nominal shear capacity rbs;Ggát;ebtugeRbkugRtaMg EdlenAkñúgGgát;enaH tendon mankMlaMgeRb

kugRtaMg f se y:agtic 40% ersIusþg;Taj f pu ³
⎛ V d⎞
Vc = ⎜ 0.05 f 'c + 4.8 u ⎟bw d
⎜ (ACI Code, Eq. 11.9) (19.50)
⎝ Mu ⎟
⎠
Edl nig M u = kMlaMgkat;emKuN nigm:Um:g;emKuNenARtg;muxkat;EdlBicarNa
Vu

bw = TTwgrbs;RTnug
V d
d ¬kñúgtY u ¦ = cMgayBIsrésrgkarsgát;eTATIRbCMuTMgn;rbs;EdkeRbkugRtaMg
Mu
d ¬enAkñúgsmIkar Vci b¤ Vcw ¦ = tMélEdlFMCageKkñúgcMeNam d xagelI nig 0.8h (ACI Code,
Section 11.4.2)
kareRbIsmIkar 19.50 RtUv)ankMNt;tamlkçxNÐxageRkam³


T.Chhay NPIC

!> Vu d / M u ≤ 1.0 ¬edIm,IKittMéltUcén Vu nig M u ¦
@> Vc ≥ (0.17 f 'c )bwd ¬Vc Gb,brma¦
#> Vc ≤ (0.42 f 'c )bwd ¬Vc Gtibrma¦
karERbRbYlén shear capacity rbs;ebtugsMrab;FñwmebtugeRbkugRtaMgEdlRTedayTMrsamBaØrgnUvbnÞúk
BRgayesμIRtUv)anbgðajenAkúñgrUbTI 19>8. cMNaMfa eKcaM)ac;dak;EdkkMlaMgkat;GtibrmaEk,rTMr nig
Ek,rcMnucmYyPaKbYnénElVgEdl φVs manxiteTArktMélGtibrma. pÞúymkvij FñwmebtugGarem:RtUvkar
EdkkMlaMgkat;TTwg ¬b¤KMlatGb,brma¦ EtEk,rTMrEdl φVs mantMélGtibrma.

eBlxøH tMél Vc EdlKNnaedaysmIkar 19.50 GacmantMéltUc dUcenH ACI Code, Section
11.4.2 eGaynUvviFImü:ageTotsMrab;KNna Vc EdlKitBicarNaersIusþg;bEnßmrbs;ebtugenAkñúgmuxkat;.


T.Chhay NPIC

enAkñúgviFIenH Vc CatMélEdltUcCageKkñúgcMeNamersIusþg;kMlaMgkat;rbs;ebtug Vci nig Vcw ¬rUbTI
19>8¦.
ersIusþg;kMlaMgkat; Vci RtUv)anQrelIkarsnμt;fa flexural-shear cracking ekIteLIgEk,rEpñk
q¶aybMputrbs; flexural cracking EdlenAcMgayRbEhl d / 2 BIcMnucrgbnÞúkénkarfycuHm:Um:g;. ACI
Code kMNt;fa Vci RtUv)anKNnadUcxageRkam³

Vci = (0.05 f 'c )bw d + ⎜Vd + i cr ⎟
⎛ VM ⎞
⎜ ⎟ (19.51)
⎝ M
max ⎠
b:uEnþvaminRtUvtUcCag (0.14 f 'c )bwd Edl
Vd = kMlaMgkat;TTwgEdlbNþalBIbnÞúkefrKμanemKuN

Vi = kMlaMgkat;TTwgemKuNEdlbNþalBIbnÞúkEdlGnuvtþBIxageRkA EdlekItmandMNalKña

CamYynwg M max
M max = m:Um:g;emKuNGtibrmabNþalBIbnÞúkEdlGnuvtþxageRkA

M cr = m:Um:g;eRbH (cracking moment)

eKGackMNt; cracking moment BIsmIkarxageRkam³
M cr = (0.5 f 'c + f pe − f d )
I
(ACI Code, Eq. 11.11) (19.52)
yt
Edl I = m:Um:g;niclPaBEdlTb;Tl;nwgbnÞúkemKuNxageRkA
yt = cMgayBIG½kSTIRbCMuTMgn;én gross section EdlminKitsrésEdkeTAsrésrgkarTajeRkA

bMput
f pe = ersIusþg;rgkarsgát;enAsrésxageRkAbMputrbs;muxkat;ebtugedaysarkMlaMgeRbkug

RtaMgeRkaykMhatbg;
f d = kugRtaMgEdlbNþalBIbnÞúkefrKμanemKuNenAsrésxageRkAbMput EdlkugRtaMgTaj

bNþalBIbnÞúkxageRkA
Web-shear strength Vcw KWQrelIsñameRbHedaykMlaMgkat;TTwgenAkñúgFñwmEdlminEmneRbH

edaysarkarBt;eT. sñameRbHEbbenHekItmanenAEk,rTMrEdlmanRTnugtUc. ACI Code, Section
11.4.2 bBa¢ak;fa Vcw RtUv)anKNnatamsmIkarxageRkam³
Vcw = (0.29 f 'c + 0.3 f pc )bw d + V p (19.53)

Edl V p = bgÁMúkM;laMgeRbkugRtaMgbBaÄrenARtg;muxkat;EdlBicarNa


T.Chhay NPIC

kugRtaMgsgát;enAkñúgebtug ¬eRkaykarGnuBaØatsMrab;kMhatbg;eRbkugRtaMg¦ enARtg;TI
f pc =

RbCMuTMgn;rbs;muxkat;EdlTb;Tl;nwgbnÞúkEdlGnuvtþ b¤enARtg;TIRbsBVrvagRTnug nig
søabenAeBlEdlTIRbCMuTMgn;sßitenAkñúgsøab
müa:gvijeTot eKGackMNt; Vcw CakMlaMgkat;EdlbegáItkugRtaMgTajem (principle tensile
stress) 0.33 f 'c enARtg;G½kSTIRbCMuTMgn;rbs;Ggát; b¤Rtg;RbsBVénsøab nigRTnugenAeBlEdlTIRbCMuTM-

gn;sßitenAkñúgsøab. eKGacsresrsmIkarG½kSdUcxageRkam³
2
⎛ f pc ⎞ f
f t = 0.33 f 'c = 2
vcw +⎜
⎜ 2 ⎟ ⎟ − pc
⎝ ⎠ 2
⎛ f pc ⎞
b¤ Vcw = f t ⎜ 1 +
⎜ ft ⎟
⎟b d
w (19.54)
⎝ ⎠
Edl ft = 0.33 f 'c . enAeBlGnuvtþsmIkar 19.51 nig 19.53 b¤ 19.54 tMélrbs; d RtUv)anKitCacM-
gaycenøaHsrésrgkarsgát; nigTIRbCMuTMgn;rbs; tendon eRbkugRtaMg b:uEnþvaminRtUvtUcCag 0.8h .
muxkat;eRKaHfñak;sMrab;kMlaMgkat;GtibrmaRtUv)anykRtg; h / 2 BIépÞrbs;TMr. eKRtUveRbIEdk
kMlaMgkat;dUcKñaRtg;muxkat;cenøaHTMr nigmuxkat;Rtg; h / 2 .

K> EdkkMlaMgkat; Shear Reinforcement
eKRtUvKNnatMél Vs edIm,IkMNt;RkLaépÞcaM)ac;rbs;EdkkMlaMgkat;
Vu = φ (Vc + Vs ) (19.49)

Vs = (Vu − φVc )
1
(19.55)
φ
sMrab;EdkkgbBaÄr
Av f y d
Vs = (19.56)
s
Vs Av f y d
nig Av = s
f yd
b¤ s=
Vs
(19.57)

Edl Av = RkLaépÞrbs;EdkkgbBaÄr nig s = KMlatrbs;Edkkg. smIkarsMrab;EdkkgeRTt
dUcKñanwgsmIkarEdl)anbgðajenAkñúgCMBUk 8.


T.Chhay NPIC

X> EdnkMNt; Limitation
!> KMlatGtibrma s rbs;EdkkgminRtUvFMCag 3h / 4 b¤ 60cm . RbsinebI V FMCag
max s

enaHKMlatGtibrmaxagelIRtUv)ankat;bnßymkRtwmBak;kNþal (ACI Code,
(4 f 'c )bwd
Section 11.5.4).

@> kMlaMgkat;Gtibrma Vs minRtUvFMCag (8 f 'c )bwd . RbsinebImindUcenaHeT eKRtUvbegáInTMhM
rbs;muxkat; (ACI Code, Section 11.5.6).
#> EdkkMlaMgkat;Gb,brma Av EdlTamTareday ACI Code KW
0.35bw s ⎛b s⎞
Av min = ≤ 0.062 f 'c ⎜ w ⎟ (19.58)
fy ⎜ fy ⎟
⎝ ⎠
enAeBlEdlkMlaMgeRbkugRtaMgRbsiT§PaB f pe ≥ 0.4 f pu EdkkMlaMgkat;Gb,brma Av Kw
Aps f pu s d
Av = (19.59)
80 f y d bw

eKmincaM)ac;ykkMBs;RbsiT§PaB d < 0.8h eT. CaTUeTA smIkar 19.59 pþl;eGaynUvEdk
kMlaMgkat;Gb,brmaFMCagkarpþl;eGayedaysmIkar 19.58.
]TahrN_ 19>7³ sMrab;FñwmenAkñúg]TahrN_ 19>4 cUrkMNt; nominal shear strength nigEdkkMlaMg
kat;caM)ac;. RtYtBinitümuxkat;Rtg; h / 2 nig 3m BIcugrbs;Fñwm. eRbI f y = 400MPa sMrab;EdkkMlaMg
kat; nigbnÞúkGefr = 19.4kN / m .
dMeNaHRsay³
!> sMrab;muxkat;enARtg; h / 2
h 1000
2
=
2
= 500mm BIxagcug
@> bnÞúkBRgayesμIemKuNenAelIFñwm
Wu = 1.2(5.81 + 13.15) + 1.6 × 19.4 = 53.8kN / m
h
enAcMgay
Vu
2
= 53.8(7.2 − 0.5) = 360.5kN

kMNt; M u enARtg;muxkat; h / 2
0.52
M u = 53.8(7.2)0.5 − 53.8 = 187kN .m
2
tMélrbs; d Rtg;muxkat; h / 2 BIxagcug ¬rUbTI 19>6 b¦ KW


T.Chhay NPIC

d = (1000 − 158) −
(4.8 − 0.5) 364 = 516mm
4.8
Vu d 360.5 × 0.516
= = 0.995 ≤ 1.0
Mu 187
dUckarTamTarrbs; ACI Code
⎛ V d⎞
Vc = ⎜ 0.05 f 'c + 4.8 u ⎟bw d
⎜
⎝ Mu ⎟
⎠
( )
= 0.05 35 + 4.8 × 0.995 150 × 516 × 10 −3 = 392.6kN
tMélGb,brmarbs; Vc = 0.17 f 'c bwd = 0.17 35 (150)516 ⋅10−3 = 77.8kN
tMélGtibrmarbs; Vc = 0.42 f 'c bwd = 0.42 35 (150)516 ⋅10−3 = 192.3kN
tMélGtibrmarbs; Vc = 192.3kN manlkçN³lub.
#> viFIepSgeTotEdlbgðajeday ACI Code KWfaeKGacyktMél Vc CatMéltUcCageKkñúgcMeNam Vci
nig Vcw
a. edayQrelI flexural-shear cracking strength

Vci = (0.05 f 'c )bw d + ⎜Vd + i cr ⎟
⎛ VM ⎞
⎜ M ⎟
⎝ max ⎠
KNnatYnImYy²dac;edayELkBIKña
(0.05 f 'c )bwd = 0.05 35 (150)516 ⋅10−3 = 22.9kN
Vd = kMlaMgkat;efrKμanemKuN = (5.81 + 13.15)(7.2 − 0.5) = 129.4kN

M max = m:Um:g;emKuNGtibrma ¬elIkElgTMgn;rbs;Fñwm¦

bnÞúkemKuN = 1.2 × 13.15 + 1.6 × 19.4 = 46.8kN / m
⎡ 0.52 ⎤
M max = 46.8⎢7.2 × 0.5 − ⎥ = 162.6kN .m
⎢
⎣ 2 ⎥⎦
Vi = 46.8(7.2 − 0.5) = 313.6kN

M cr =
I
yt
(
0.5 f 'c + f pe + f d )
/
I = 2.607 ⋅ 1010 mm 4 yb = 522mm

f pe = kugRtaMgrgkarsgát;EdlbNþalBIkMlaMgeRbkugRtaMg
F Feyb
= +
A I
1344 ⋅ 103 1344 ⋅ 103 × 38 × 522
= + = 6.8MPa
23.25 ⋅ 10 4 2.607 ⋅ 1010


T.Chhay NPIC

fd = kugRtaMgbnÞúkefr = M D yb
I
⎡ 0.52 ⎤
MD = 18.96⎢7.2 × 0.5 − ⎥ = 65.9kN .m
⎢
⎣ 2 ⎥⎦
65.9(522 ) ⋅ 10 6
fd = = 1.32 MPa
2.607 ⋅ 1010

M cr =
2.607 ⋅ 1010
522
( )
0.5 35 + 6.8 − 1.32 ⋅ 10 − 6 = 412.4kN .m

dUcenH Vci = 22.9 + 129.4 + 313.6
412.4
162.6
= 947.7kN

Vci minRtUvtUcCag (0.14 f 'c )bw d = 0.14 35 (150)516 ⋅ 10 −3 = 64.1kN

b. ersIusþg;kMlaMgkat;EdlQrelI web-shear cracking KW
Vcw = (0.29 f 'c + 0.3 f pc )bw d + V p
1344 ⋅ 103
f pc = = 5.78MPa
23.25 ⋅ 10 4
d = 516mm b¤ 0.8h = 800mm
yk d = 800mm
1
V p = 1344 × = 101.8kN
13.2
Edl 1 / 13.2 = slop rbs; tendon profile = 364 / 4800
0.29 35 = 1.72MPa
dUcenH Vcw = (1.72 + 0.3 × 5.78) × 150 × 800 ⋅10−3 + 101.8 = 516.3kN
c. edaysar Vcw < Vci dUcenH Vcw = 516.3kN Ca nominal shear strength enARtg;muxkat; h / 2

BIcugrbs;Fñwm. kñúgkrNICaeRcIn Vcw lubRtg; h / 2 BITMr.
$> EdkRTnug (web reinforcement)
Vu = 360.5kN φVcw = 0.75 × 516.3 = 387.2kN
edaysar Vu < φVcw / Vs = 0 dUcenHeRbIEdkkgGb,brma. eRbIEdkkg DB10 .
Av = 2 × 10 2 π / 4 = 157mm 2
KMlatGtibrmaCatMélEdltUcCageKkñúgcMeNam
3
s1 = h = 750mm s2 = 600mm
4
KNna s3 BIsmIkarEdkRTnugGb,brma


T.Chhay NPIC

Aps f pu s d
Av min = × × ×
80 fy d bw
1394 1725 s3 516
157 = × ×
80 400 516 150
s3 = 581mm yk s3 = 500mm
0.35bw s ⎛b s⎞
müa:gvijeTot Av min =
fy
≤ 0.062 f 'c ⎜ w ⎟
⎜ fy ⎟
0.062 f 'c = 0.367
⎝ ⎠
Av f y 157 × 400
s4 = = = 1140mm
0.367bw 0.367 × 150
dUcenHeyIgeRbI DB10 @ 500 .
%> sMrab;muxkat;enAcMgay 3m BIxagcug viFIsaRsþkñúgkarKNnaKWRsedogKñasMrab;muxkat;enARtg; h / 2 .
edayeRbIviFIEdlsMrYlrbs; ACI
Vu = 53.8(7.2 − 3) = 226kN
⎡ 32 ⎤
M u = 53.8⎢7.2 × 3 − ⎥ = 920kN .m
⎢
⎣ 2⎥⎦
d = (1000 − 158) −
(4.8 − 3) 364 = 705mm
4.8
Vu d 226 × 0.705
= = 0.173 < 1.0
Mu 920
⎛
⎜
V d⎞
( )
Vc = ⎜ 0.05 f 'c + 4.8 u ⎟bw d = 0.05 35 + 4.8 × 0.173 150 × 705 × 10 − 3 = 119.1kN
Mu ⎟
⎝ ⎠
Vc min = 77.8kN nig Vc max = 192.3kN
dUcenH Vc = 119.1kN ¬lub¦
^> eRbIsmIkar ACI Code edIm,IKNna Vci nig Vcw . dMbUgKNna Vci ¬EdllubsMrab;muxkat;enH¦
(0.05 f 'c )bwd = 0.05 35 (150)705 ⋅10−3 = 31.3kN
Vd = (5.81 + 13.15)(7.2 − 3) = 79.6kN
⎡ 32 ⎤
M max = 46.8⎢7.2 × 3 − ⎥ = 800.3kN .m
⎢
⎣ 2⎥⎦
Vi = 46.8(7.2 − 3) = 196.6kN
1344 ⋅ 103 1344 ⋅ 103 × 227.5 × 522
f pe = + = 11.9 MPa
23.25 ⋅ 10 4 2.607 ⋅ 1010


T.Chhay NPIC

⎡ 32 ⎤
M D = 18.96 ⎢7.2 × 3 − ⎥ = 324.2kN .m
⎢
⎣ 2⎥⎦
324.2(522 ) ⋅ 10 6
fd = = 6.5MPa
2.607 ⋅ 1010

M cr =
2.607 ⋅ 1010
522
( )
0.5 35 + 11.9 − 6.5 ⋅ 10 − 6 = 417.4kN .m

dUcenH Vci = 31.3 + 79.6 + 196.6
417.4
800.3
= 213.4kN

( )
Vci min = 0.14 f 'c bw d = 0.14 35 (150)705 ⋅ 10 −3 = 87.6kN

dUcnH Vci = 213.4kN

bnÞab;mkKNna Vcw
f pc = 5.78MPa V p = 101.8kN ¬dUcelIkmun¦
d = 705mm b¤ 0.8h = 800mm
yk d = 800mm
dUcenH Vcw = (1.72 + 0.3 × 5.78) × 150 × 800 ⋅10−3 + 101.8 = 516.3kN
tMélrbs; Vcw minmaneRKaHfñak;eT. enARtg;mYyPaKbYnénRbEvgElVg tMélkMlaMgkat;eRKaHfñak;KW
Vci ¬rUbTI 19>8¦.

&> KNnaEdkRTnug
Vu = 226kN φVci = 0.75 × 213.4 = 160kN
Vu = φ (Vc + Vs )

Vs =
1
(226 − 160) = 88kN
0.75
Edkkg DB10 / Av = 157mm2 . RtYtBinitüKMlatGtibrma smax = 500mm ¬dUcelIkmun¦
muxkat;EdlEdlRtUvkar Av = Vsd = 88400 × × 500 = 156
f
s ⋅ 103
705
y

Av EdleRbIKW 157mm2 > 156mm2 . dUcenH yk DB10 @ 400 .


T.Chhay NPIC

9> KNnaCMhandMbUgénGgát;ebtugeRbmugRtaMgrgkarBt;
Preliminary Design of Prestressed Concrete Flexural Members
k> rUbrag nigTMhM Shapes and Dimensions
kar)a:n;sμanTMhMrbs;muxkat;d¾RtwmRtUveFVIeGaykarKNnacMNayeBltic nigminsμúKsμaj. dUc
enH karKNnaCMhandMbUgmansar³sMxan;Nas; edIm,IFanafaTMhMmuxkat;manlkçN³smrmümunnwgcab;
epþImKNnalMGit.
enAkñúgdMNak;kalKNnaCMhandMbUg CaFmμtaeKmanTinñn½yxøHEdlGacCYykñúgkareRCIserIsTMhM
d¾smRsb. ]TahrN_ m:Um:g;Bt;Edl)anBIbnÞúkGnuvtþn_xageRkA kugRtaMgGnuBaØat nigTinñn½ysMrab;kMNt;
kMhatbg;RtUv)andwg b¤RtUv)ankMNt;.
rUbragrbs;muxkat;Ggát;ebtugeRbkugRtaMgGacCactuekaNEkg GkSr T GkSr I b¤RbGb;. kMBs;
srubrbs;muxkat; h RtUv)ankMNt;edaykarBicarNakMBs;rbs;lMh b¤k¾minRtUv)ankMNt;. kareRCIs
erIsTMhMrbs;muxkat;sMrab;karKNnaCMhandMbUgmandUcxageRkam ¬rUbTI 19>9¦³

!> kMBs;srubénmuxkat;KW h = 1 / 20 eTA 1 / 30 énRbEvgElVg. sMrab;karrgbnÞúkF¶n; h = L / 20
nigsMrab;karrgbnÞúkRsal h = L / 30 b¤ h = 43.6 M D + M L Edl h KitCa mm nig
M KitCa kN .m .

@> kMras;rbs;søabxagelIKW h f = h / 8 eTA h / 6 .
#> TTwgrbs;søabxagelIKW b ≥ 2h / 5 .
$> kMras;rbs;RTnugKW bw ≥ 100mm . CaTUeTA eKyk bw = h / 30 + 100 .


T.Chhay NPIC

%> kareRCIserIs bw nig t KWERbRbYleTAtamkarBRgay tendon eRbkugRtaMg edayeFVIy:agNa
rkSakMras;ebtugkarBarEdk.
^> RkLaépÞEdlsmrmürbs;muxkat;ebtugEdlRtUvkarKW
M + ML
Ac (m 2 ) = D
1450h
¬Edl M D + M L KitCa kN .m nig h KitCa m ¦
sMrab;karKNnakñúgkarGnuvtþ nigkarKNnaEdlmanlkçN³esdækic© ]sSahkmμpliteRKOg
sMNg;BIebtug)anbegáItnUvrUbrag nigTMhMbTdæanCaeRcIn EdlGñkKNnaGaceRCIserIsnUvGgát;NaEdl
RtUvkar. taragénmuxkat;sþg;darmanenAkñúg PCI Design Handbook. AASTHO k¾)anENnaMnUv
girder sþg;daredIm,IeRbIR)as;enAkñgsMNg;s<an ¬tarag 19>4¦.
ú
tarag 19>4 AASTHO Girders, ebtugTMgn;Fmμta
A I yb Zb Zt TMgn;
RbePT
(in.2 ) (in.4 ) (in.) (in.3 ) (in.3 ) (lb / ft )
Type II 369 50979 15.83 3220 2527 384
Type III 560 125390 20.27 6186 5070 593
Type IV 789 260741 24.73 10544 8908 822

x> kMlaMgeRbkugRtaMg nigRkLaépÞEdk Shapes and Dimensions
enAeBlEdleKeRCIserIsrUbrag kMBs; nigTMhMepSgeTotrbs;muxkat;rYcehIy eKGac)a:n;sμantM-
élRbhak;RbEhlrbs;kMlaMgeRbkugRtaMg nigRkLaépÞrbs;EdkeRbkugRtaMg Aps . BIm:Um:g; couple
xagkñúg m:Um:g;srub M T EdlekItBIbnÞúkefr nigbnÞúkGefreFVIkaresμInwgkMlaMgTaj T KuNnwgédXñas;
jd .


T.Chhay NPIC

M T = T ( jd ) = C ( jd )
MT
M T = Aps f se ( jd ) Aps =
f se ( jd )
Edl Aps CaRkLaépÞEdkeRbkugRtaMg nig f se CakugRtaMgeRbkugRtaMgRbsiT§PaBeRkaykMhatbg;. tM-
élrbs;édXñas; jd ERbRbYlBI 0.4h eTA 0.8h EdlkñúgkarGnuvtþeKykvaenAkñúgcenøaH 0.6h eTA 0.7h .
eKGaceRbItMélmFüm 0.65 )an. dUcenH
MT
Aps = (19.60)
(0.65h ) f se
ehIykMlaMgeRbkugRtaMgKW
MT
F = T = Aps f se = (19.61)
0.65h
kMlaMgeRbkugRtaMgenAeBlepÞrKW Fi = F / η Edl η CaemKuNénkMhatbg;GaRs½ynwgeBl.
kMlaMgsgát; C enAelImuxkat;esμInwgkMlaMgTaj T ³
C = T = Aps f se

KitCakugRtaMg A = Aps f se = fc1
C
Ac
c

Edl fc1 CakugRtaMgBRgayesμIEdlsnμt;enAelImuxkat;
sMrab;karKNnaCMhandMbUg eKsnμt;srésxageRkArbs;karEbgEckkugRtaMgragRtIekaNesμInwg
kugRtaMgsgát;GnuBaØatGtibrma f ca . dUcenHkugRtaMgmFümKW 0.5 fca = fc1 . kugRtaMgsgát;GnuBaØat
enAkñúgebtugKW fca = 0.45 f 'c . dUcenH eKGacsnμt;RkLaépÞebtugEdlRtUvkar Ac BIkMlaMgTaj T dUc
xageRkam³
T Aps f se Aps f se Aps f se
Ac = = = = (19.62)
f c1 f c1 0.5 f ca 0.225 f 'c
T MT MT MT
Ac = = = = (19.63)
0.5 f ca (0.65h )(0.5 f ca ) 0.33 f ca 0.15 f 'c
karviPaKenHKWQrelIkarKNnasMrab;bnÞúkeFVIkar minEmnsMrab;bnÞúkemKuNeT. cMNakp©it e RtUv)anvas;
BITIRbCMuTMgn;rbs;muxkat;eTATIRbCMuTMgn;rbs;EdkeRbkugRtaMg ehIyeKGacKNnaCatMélRbhak;RbEhl
dUcxageRkam³
MD
e = Kb + (19.64)
Fi
Edl Kb Ca lower kern limit ehIy M D Cam:Um:g;Edl)anBIbnÞúkefrKμanemKuN.


T.Chhay NPIC

10> kugRtaMgbøúkxagcug End-Block Stresses

k> Ggát;rgkugRtaMgTajmun Pretensioned Members
dUcKñanwgEdkenAkñúgGgát;ebtugGarem:EdlRtUvkarRbEvgbgáb;Cak;lak; enaHkMlaMgeRbkugRtaMg
enAkñúgGgát;ebtugrgkugRtaMgmunRtUv)anepÞreTAebtugedaykarbgáb; b¤edaykarf<k;cug b¤edaybnSMTaMgBIr.
enAkñúgGgát;rgkugRtaMgmun eKehARbEvgrbs;EdkeRbkugRtaMgEdlkMlaMgeRbkugRtaMgRtUv)anepÞreTAeb-
tugfa RbEvgepÞr (transfer length) lt . eRkayeBlepÞr kugRtaMgenAkñúg tendon Rtg;cugbMputrbs;Ggát;
esμInwgsUnü b:uEnþkugRtaMgenARtg;cMgay lt BIcugGgát;esμInwgkMlaMgeRbkugRtaMgRbsiT§PaB f pe . transfer
length lt GaRs½ynwgTMhM nigRbePTrbs; tendon, lkçxNÐépÞb:H ersIusþg;ebtug f 'c kugRtaMg nigviFIén

karepÞrkMlaMg. kar)a:n;RbmaN lt EdlGacTTYlyk)anKWesμInwg 50 dgénGgát;p©itrbs; tendon b:uEnþ
sMrab; single wires eKyk lt esμInwg 100 dgénGgát;p©itrbs; wire.
edIm,IeGaykMlaMgTajenAkñúgEdkeRbkugRtaMgbegáIt ultimate flexural strength eBjelj va
RtUvkarnUvRbEvgcMNg (bond length). eKalbMNgKWedIm,IkarBarkarrGilTUeTAmunnwgkar)ak;rbs;Fñwm
eRkam design strength eBjelj. RbEvgbgáb; ld (development length) esμInwgplbUkrvag bond
length nig transfer length. edayEp¥kelIkarBiesaF ACI Code, Section 12.9.1 eGaynUvsmIkar

xageRkamsMrab;KNnaRbEvgbgáb;én three- or seven-wire pretensioning strand:
⎛ 2 ⎞
ld = ⎜ f ps − f se ⎟d e (19.65)
⎝ 3 ⎠
Edl kugRtaMgenAkñúgEdkeRbkugRtaMgeRkam nominal strength
f ps =

f se = kugRtaMgRbsiT§PaBenAkñúgEdkeRbkugRtaMgeRkaykMhatbg;

d b = nominal diameter rbs; wire b¤ strand

enAkñúg pretensioned member kugRtaMgTajx<s;manenAtMbn;xagcug EdleKRtUvkardak;EdkBiess.
EdkenHmanTMrg;CaEdkkgbBaÄrehIyRtUv)anBRgayesμIkñúgKMlat h / 5 Edlvas;BIxagcugrbs;Fñwm. Ca
TUeTA eKdak;EdkkgTImYyenAcMgay 25mm eTA 75mm BIxagcugFñwm. vaCaKarGnuvtþFmμtakñúgkar
bEnßm nominal reinforcement sMrab;cMgay d Edlvas;BIcugrbs;Fñwm. RkLaépÞrbs;EdkbBaÄr Av
EdlRtUv)aneRbIenAtMbn;xagcugGacRtUv)anKNnaedaytMélRbhak;RbEhlBIsmIkarxageRkam³
Fi h
Av = 0.021 (19.66)
f sa lt
Edl f sa = kugRtaMgGnuBaØatenAkñúgEdkkg ¬CaTUeTA 140MPa ¦ nig ld = 50 dgénGgát; tendon


T.Chhay NPIC

]TarhN_ 19>8³ kMNt;EdkkgcaM)ac;EdlRtUvkarenAtMbn;xagcugrbs;FñwmEdleGayenAkñúg]TahrN_
19>4.
dMeNaHRsay³
/
Fi = 1606kN h = 1000mm / /
f sa = 140MPa lt = 50 × 11.125 = 556mm
1606 ⋅ 103 × 1000
dUcenH Av = 0.021
140 × 556
= 433mm 2
h 1000
= = 200mm
5 5
eRbIEdkkgbiTCit DB10 cMnYnbYnkg EdlEdkkgTImYysßitenAcMgay 200mm BITMr
Av ¬Edldak;¦ = 157 × 4 = 628mm 2

x> Ggát;rgkugRtaMgTajeRkay Posttensioned Members

enAkñúg posttensioned concrete member kMlaMgeRbkugRtaMgRtUv)anepÞrBI tendon ¬sMrab;
bonded nig unbonded tendon¦ eTAebtugRtg;cugrbs;Ggát;eday special anchorage device. enAkñúg

anchorage zone Rtg;cugrbs;Ggát; kugRtaMgsgát;mantMélx<s;Nas; ehIy transverse tensile stress


T.Chhay NPIC

ekItman dUcbgðajenAkñúgrUbTI 19>10. enAkñúgkarGnuvtþ eKeXIjfaRbEvgén anchorage zone minFM
CagkMBs;rbs;cugGgát;eT RbsinebImindUcenaHeT sßanPaBénkugRtaMgenAkñúgtMbn;enaHnwgmanlkçN³sμúK
sμaj.
karBRgaykugRtaMgEdl)anBI tendon mYyenAkñúg anchorage zone RtUv)anbgðajenAkñúgrUbTI
19>11. enARtg;cMgay h BImuxkat;xagcug eKsnμt;kugRtaMgBRgayesμIelImuxkat;TaMgmUl. edayKit
ExSkMlaMg (trajectories) CaFatumYy²EdlmanGMeBICa curved strut enaHExSkMlaMgmanGMeBItamTisedk
eTArkExSG½kSrbs;FñwmenAkñúgtMbn; A edaybegáItCakugRtaMgsgát;. enAkñúgtMbn; B ExSkMlaMg)anbEgVr
Tis ehIy strut manGMeBIecjeRkAEdlbegáItCakugRtaMgTaj. enAkñúgtMbn; C strut esÞIrEtRtg;eday
begáItkarBRgaykugRtaMgesμI.

EdkBRgwgEdlRtUvkarsMrab; anchorage zone xagcugrbs; posttentioned member CaTUeTApSM
eLIgedayRkLaénEdkbBaÄr nigEdkedkEdlmanKMlatCit²KñaeBjRbEvgénbøúkxagcugedIm,ITb;Tl;nwg
kugRtaMgpÞúH (bursting stress) nigkugRtaMgTaj. CakarGnuvtþTUeTAeKmineGayKMlatEdkFMCag 75mm
enAkñúgTisnImYy²eT ehIyminRtUvdak;EdkenAcMgayFMCag 50mm BIépÞxagkñúgrbs; bearing plate eT.


Xix introduction to prestressed concrete

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Xix introduction to prestressed concrete