Viii shear and diagonal tension

T.Chhay NPIC

VIII. kMlaMgkat;TTwg nigkMlaMgTajGgát;RTUg
Shear and Diagonal Tension

1> esckþIepþIm
enAeBlEdlFñwmTMrsamBaØrgnUvbnÞúk m:Um:g;Bt; nigkMlaMgkat;TTwgnwgekIteLIgelIRbEvgFñwm. edIm,IRT
nUvbnÞúkenHedaysuvtßiPaB FñwmRtUv)aneFVIkarKNnaeLIgedIm,ITb;nwgkMlaMgTaMgBIrRbePTenH. karKNnasMrab;
karBt;RtUv)aneFVIeLIgdMbUgeK edIm,IkMNt;muxkat;Fñwm nigEdkemcaM)ac; dUcEdl)anENnaMBIemeronmun.
bnÞab;mkFñwmRtUv)anKNnaedIm,ITb;nwgkMlaMgkat;TTwg. kñúgkrNIEdlEdkkgminRtUv)andak; enaHFñwm
nwg)ak;edaykMlaMgkat;TTwg. kar)ak;edaykMlaMgkat;TTwg ekIteLIgedaymanPaBdabtUc nig)at;bg;nUv
lkçN³yWt ehIymin)anRbkasGasnñenAmuneBl)ak;eT. sMrab;kar)ak;edaykarBt;begáag ekIteLIgeday
karekIneLIgnUvPaBdabbnþicmþg² nigmansñameRbH dUcenHvamankarpþl;sBaØaRbkasGasnñmuneBl)ak;cug
eRkay. karKNnasMrab;kMlaMgkat;TTwg RtUv)aneFVIeLIgedIm,IFananUvkar)ak;edaykMlaMgkat;TTwgekIteLIg
eRkaykar)ak;edaykarBt;begáag.

2> kugRtaMgkMlaMgkat;enAkñúgFñwmebtugGarem:
rUbmnþTUeTAsMrab;kugRtaMgkMlaMgkat;TTwgenAFñwmrUbFatusac;mYy (homogenous beam) karBRgaykug
RtaMg
ν=
VQ
Ib
¬*>!¦
Edl V - kMlaMgkat;srubenAmuxkat;EdlRtUvsikSa

Q - m:Um:g;sþaTiceFobGkS½NWténmuxkat;EdlxNÐedaybnÞat;EdlRtUvsikSakugRtaMgkMlaMgkat;

kMlaMgkat;TTwg nigkMlaMgTajGgát;RTUg 125

Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
- m:Um:g;niclPaBénmuxkat;eFobGkS½NWt
I

b - TTwgFñwménmuxkat;EdlRtUvsikSakugRtaMgkMlaMgkat;

karBRgaykugRtaMgm:Um:g;Bt; nigkMlaMgkat;TTwgGaRs½yeTAnwgRTwsþIbTeGLasÞicsMrab;muxkat;Fñwm
ctuekaN dUcbgðajkñúgrUbTI8>2.
kugRtaMgm:Um:g;Bt;
Mc
f =
I
Edl kugRtaMgkMlaMgkat;enARKb;cMnucTaMgGs;RtUv)anKNnatamrUbmnþkugRtaMgkMlaMgkat;
VQ
ν=
Ib
kugRtaMgkMlaMgkat;GtibrmasßitenAelIGkSNWt nigmantMélesμI 1.5v ¬kMlaMgkat;TTwgmFüm¦
a

Edl ν = bh . ExSekagkMlaMgkat;TTwgmanrag):ar:abUl.
a
V

sMrab;FñwmebtugsésEdkrgkarTaj (singly reinforced concrete beam) karBRgaykugRtaMg
kMlaMgkat;TTwgenAelIGk½SNWtmanrag):ar:abUl. enAeRkamGk½SNWt kugRtaMgkMlaMgkat;TTwgGtibrmamantMél
efr ehIy)anrkSatMélenHRtwmnIv:UEdkrgkarTaj BIeRBaHvaKμankarpøas;bþÚrkMlaMgTajBIcMnucGk½SNWtehIymü:ag
eTotkMlaMgTajkñúgebtugRtUv)anecal. kugRtaMgkMlaMgkat;TTwgmantMélesμIsUnü enAeRkamnIv:UEdk ¬rUbTI8>
3¦.
sMrab;ebtugsésrEdkEdlmanEdkrgkarsgát; nigmuxkat;GkSret karBRgaykugRtaMgkMlaMg
kat;TTwgRtUv)anbgðajkúñgrUbTI8>3. eyIgsegÁteXIjfa kMlaMgkat;TTwgesÞIrEtTaMgGs; RtUv)anTb;edayRT
nugEdlsøabTb;nwgPaKrytUcbMput. sMrab;karGnuvtþn_esÞIrTaMgGs; eKecalnUvlT§PaBTb;kMlaMgkat;TTwgrbs;
søab.

Shear and Diagonal Tension 126

T.Chhay NPIC

eyagtamrUbTI8>1 edayykFñwmmYykMNat;tUc dx mkviPaK eyIgeXIjfa m:Um:g;Bt;enAcug
sgxagénkMNat; M nig M minmantMélesμIKñaeT. edaysar M < M dUcenHedIm,IrkSalMnwgsMrab;kMNat;
1 2 1 2

dx kMlaMgsgát; C RtUvmantMélFMCag C ¬rUbTI8>4¦. dUcenHkugRtaMgkMlaMgkat;TTwg v ekItmanenAelImux
2 1

kat;edk a − a1 b¤ b − b1 ¬rUbTI8>4 a¦. kugRtaMgkMlaMgEkg (normal stresses) nigkugRtaMgkMlaMgkat;TTwg
(shear stresses) enAelIGgát;tUc enAkMritnIv:U a − a1 b¤ b − b1 RtUv)anbgðajenAkñúg¬rUbTI8>4 b¦. cMNaMfa kug

RtaMgkM;laMgEkg (normal stresses) enAnIv:UénGkS½NWtKW 0 b:uEnþkMlaMgkat;TTwgmantMélGtibrma. kMlaMgkat;
TTwgedkesμInwgkMlaMgkat;TTwgbBaÄr dUcbgðajenAkñúg ¬rUbTI8>4 b¦. enAeBlEdl kugRtaMgkMlaMgEkgmantM
élesμIsUnü b¤tUc enaHkrNIkMlaMgkat;TTwgsuT§GacekItman. kñúgkrNIenH kugRtaMgTajGtibrma f t manGMeBI
tammMu 45o ¬rUbTI8>4 c¦.
kugRtaMgTajsmmUleTAnwgkugRtaMgem dUcbgðajkñúgrUbTI 8>4 d. kugRtaMgemenHRtUv)aneK
ehAfakugRtaMgTajGgát;RTUg. enAeBlkugRtaMgTajGgát;RTUgmantMélesμIersIusþg;Tajrbs;ebtug sñameRbH
Ggát;RTUgekIteLIg. karviPaKy:agsegçbenHBnül;BIKMniténkMlaMgTajGgát;RTUg nigsñameRbHGgát;RTUg. kar
eFVICak;EsþgmanlkçN³sμúKsμajCag ehIyvaTTYlT§iBlBIktþaepSg². sMrab;bnSMénGMeBIénkMlaMgkat;TTwg
nigkMlaMgEkgenAcMnucNamYyenAkñúgFñwm kMlaMgTajGgát;RTUg (principal stresses) Gtibrma nigGb,brma f p
RtUv)aneGayedaysmIkarxageRkam³


2
⎛f ⎞
fp =
f
2
± ⎜ ⎟ + v2
⎝2⎠
¬*>@¦
Edl f =GaMgtg;sIueténkugRtaMgEkgEdlbNþalmkBIkarBt;
v = kugRtaMgkMlaMgkat;

kar)ak;edaykMlaMgkat;TTwgenAkñúgFñwmebtugPaKeRcInTMngCaekIteLIgenAkEnøgEdlkMlaMg
kat;TTwgGtibrma CaTUeTAenAEk,rTMrénGgát;. PsþútagTImYykar)ak;EdlKYreGayP½yxøayKWkarekIteLIgnUv
sñameRbHGgát;RTUg.


T.Chhay NPIC

3> kareFVIkarrbs;FñwmedayKμanEdkkMlaMgkat;TTwg
ebtugexSaykñúgkarTaj ehIyFñwmGac)ak;RbsinebImuxkat;EdkminRtwmRtUvRtUv)anpþl;eGay. kugRtaMg
TajekItmanenAkñúgFñwmbNþalmkBIkMlaMgTajtamGkS½ kMlaMgBt; kMlaMgkat;TTwg kMlaMgrmYl b¤bnSMénbnÞúk
TaMgenH. TItaMgénsñameRbHenAkñúgFñwmebtugGaRs½ynwgTisedAénkugRtaMgem (principal stresses). bnSMénkug
RtaMgkMlaMgEkg normal stress nigkugRtaMgkMlaMgkat;TTwg begáIt)ankMlaMgTajtamGgát;RTUg (diagonal
tension) GtibrmaEdlsßitenARbEhlcMgay d BImuxénTMr.

kareFVIkarrbs;FñwmebtugGarem:edaymanb¤KμanEdkkMlaMgkat;TTwg RtUv)anBiesaFeRkamGMeBIénkarekIn
eLIgénbnÞúkdUc)anerobrab;enAkñúgemeronTI3. enAkñúgkarBiesaFn_Fñwm sñameRbHbBaÄrEdlekItBIkarBt;ekIt
eLIgenAenAelImuxkat;Edlmanm:Um:g;Bt;Gtibrma enAeBlEdlkugRtaMgTajenAkñúgebtugelIsBIm:UDuldac;
(module of rupture) rbs;ebtug b¤ f r = 0.623 f 'c . sñameRbHeRTtenAkñúgRTnugekItmanenAkñúgtMNak;kal

bnÞab;enATItaMgEk,rTMr.
sñameRbHeRTtEdlekItmanenAkñúgFñwmEdlminTan;)aneRbHBIdMbUg CaTUeTARtUv)aneKeGayeQμaHfa
sñameRbHkMlaMgkat;RTnug (web-shear crack). RbsinebIsñameRbHeRTtcab;epþImenABIelIsñameRbHEdlekItBI
karBt;EdlmanRsab; ehIyrIksayenAkñúgFñwm enaHsñameRbHRtUv)aneKeGayeQμaHfa sñameRbHkMlaMgkat;rg
karBt; (flexural-shear crack) rUbTI8>5. sñameRbH web-shear crack ekItmanenAkñúgFñwmEdlmankMlaMgkat;
FM nigm:Um:g;Bt;tUcenAkñúgRTnugesþIg. vaCasñameRbHminFmμtaehIyGacekItmanenAEk,rcMnucrbt;énFñwmCab; b¤
Ek,rTMrénFñwmsamBaØ.
sñameRbH flexural-shear crack CaRbePTsñameRbHFmμtaEdleKGaceXIjmanenAelIFñwm. dMbUgsñam
eRbH flexural crack ekItmanbBaÄrenAelIFñwm bnÞab;mksñameRbHeRTtcab;epþImekItmanBIelIkMBUlénsñameRbH
flexural crack enAeBlEdlkugRtaMgkMlaMgkat;TTwgekItmanenAkñgtMbn;enaH. enAkñugtMbn;EdlmankugRtaMg

kMlaMgkat;TTwgFM FñwmRtUvEtBRgwgedayEdkkg stirrup b¤EdkBt; (bent bar) edIm,IeFVIeGayFñwmmanlkçN³sVit
(ductile) Edlmin)ak;b¤ dac;. edIm,IeCosvagkar)ak;edaykMlaMgkat; munkar)ak;edaykarBt; emKuNsuvtßiPaB

FMRtUv)anpþl;eGayedIm,ITb;nwgkar)ak;edaykMlaMgkat;TTwg. ACI Code kMNt;emKuNkat;bnßyersIusþg;
φ = 0.75 sMrab;kMlaMgkat;TTwg.
ersIusþg;kMlaMgkat;TTwgenAkñúgGgát;ebtugGarem:RtUv)anekIteLIgedaybnSMénkMlaMgemkanicxageRkam
¬rUbTI8>5¦³
- ersIusþg;kMlaMgkat;TTwgénebtugminTan;eRbH Vz
- karepÞrkMlaMgkat;TTwgrvagGnþrépÞ (interface shear transfer) Va EdlbNþalmkBIkarbgçaMKñarvag
fμbMEbktambeNþayépÞd¾KRKatrbs;sñameRbH

- GMeBIrbs;FñÚr (arch action)
- GMeBIEdkf<k; (dowel action) Vd EdlbNþalmkBIersIusþg;énr)arEdkbeNþayeTAkMlaMgkat;TTwg
tamTTwg (transverse shearing force)
bEnßmeTAelIkMlaMgTaMgenH EdkkMlaMgkat;TTwg (shear reinforcement) begáInersIusþg;kMlaMgkat;TTwg
Vs edayGaRs½yeTAelIGgát;p©it nigKMlatEdkkgEdleRbIenAkñúgGgát;ebtug. RbsinebIEdkkMlaMgkat;TTwg

minRtUv)andak;enAkñúgFñwmragctuekaN enaHsmamaRténkMlaMgkat;TTwgEdlTb;edaykMlaMgemkanicepSg²KW³
BI 20% eTA 40% edayVz / BI 35% eTA 50% eday Va / BI 15% eTA 25% eday Vd .
4> T§iBlm:Um:g;eTAelIersIusþg;kMlaMgkat;
sMrab;FñwmTMrsamBaØeRkamGMeBIbnÞúkBRgayesμI muxkat;kNþalElVgrgnUvm:Um:g;Bt;FM nigkMlaMgkat;TTwg
tUc b¤esμIsUnü EdlpÞúyBImuxkat;enAEk,rTMrEdlm:Um:g;Bt;mantMéltUc ÉkMlaMgkat;TTwgmantMélFM ¬rUbTI8>1¦.
kMlaMgkat;TTwg nigm:Um:g;mantMélFMenAEk,rTMrkNþalsMrab;FñwmCab;. enATItaMgEdlkMlaMgkat;TTwgFM nigm:Um:g;
Bt;tUc enaHvanwgmankareRbHedaykarBt;tictYc ehIykugRtaMgmFüm v = V / bd . kugRtaMgkMlaMgTajtam
Ggát;RTUgCakugRtaMgEdleRTtedaymMuRbEhl 45o ¬rUbTI8>4 c ¦. sñameRbHGgát;RTUgGacrMBwgfanwgekItman
enAeBlEdlkugRtaMgkM;laMgTajGgát;RTUgEdlsßitenAEk,rtMbn;GkS½NWtxiteTACit b¤elIsersIusþg;kMlaMgTaj
rbs;ebtug. CaTUeTA ersIusþg;kMlaMgkat;TTwgcugeRkay (ultimate shear strength) ERbRbYlcenøaHBI
0.29 f 'c nig 0.42 f 'c . eRkayBIkarBiesaFn_eTAelIFñwmCaeRcInGMBIkMlaMgkat;TTwg nigkMlaMgTajtam

Ggát;RTUg eK)anrkeXIjfaenAkñúgtMbn;EdlmankMlaMgkat;TTwgFM nigm:Um:g;Bt;tUc enaHsñameRbHkMlaMgTaj
Ggát;RTUgRtUv)anbegáIteLIgenAeBlkMlaMgkat;TTwgmFüm
V = 0.29 f ' b d
cr c w ¬*>#¦
Edl bw CaTTwgRTnugmuxkat;GkSret b¤TTwgmuxkat;ctuekaN
d CakMBs;RbsiT§PaBrbs;Fñwm

enATItaMgEdlkMlaMgkat;TTwg nigm:Um:g;Bt;mantMélFM enaHsñameRbHedaykarBt; (flexural crack)
RtUv)anekIteLIgdMbUg. enAdMNak;kalTImYy sñameRbHxøHBt;kñúgTisedAGgát;RTUgenAeBlEdlkugRtaMgkMlaMg
TajGgát;RTUg EdlsßitenABIcugxagelIénsñameRbHTaMgenaHFMCagkugRtaMgkMlaMgTajrbs;ebtug.
RbsinebIeKeGaym:Um:g;FMmanGMeBIelIFñwm sMrab;muxkat;EdlmanbrimaNEdkRKb;RKan; enaHkMlaMg nominal
shear force enAeBlEdlsñameRbHekItmanRtUv)aneGaydUcxageRkam

Vcr = 0.16 f 'c bw d ¬*>$¦


T.Chhay NPIC

tMélenHKWtUcCagtMélEdleGayedaysmIkar ¬*>#¦ eRcInCaBak;kNþalenAeBlEdlm:Um:g;Bt;man
tMéltUcNas;. enHmann½yfam:Um:g;Bt;FMkat;bnßytMélrbs;kugRtaMgkM;laMgkat;TTwgenAeBlEdlsñameRbH
ekIteLIg. smIkarxageRkamRtUv)anesñIeLIgedIm,ITsSn_TaynUvkugRtaMg nominal shear stress enAeBlEdl
sñamGgát;RTUgRtUv)anrMBwgfaekItman³
⎡ ⎛ V d ⎞⎤
vcr =
V
= ⎢0.16 f 'c + ⎜17.2 ρ w u ⎟⎥ ≤ 0.29 f 'c
⎜ M ⎟
¬*>%¦
b d w ⎣ ⎝ u ⎠⎦
ACI Code, Section 11.3.2 )anyksmIkarenHsMrab;KNnakMlaMg nominal ultimate shear force
EdlTb;edayebtug
Vc = (0.16 f 'c + 17.2 ρ w
Vu d
Mu
)bw d ≤ 0.29 f 'c bw d ¬*>^¦
Edl ρ w = As / bwd / d CakMBs;RbsiT§PaBrbs;muxkat;Fñwm/ bw CaTTwgRTnugsMrab;muxkat;GkSret
b¤TTwgénmuxkat;ctuekaN É Vu nig M u CakMlaMgkat;TTwgcugeRkay (ultimate shearing force) nigm:Um:g;Bt;
cugeRkay (ultimate bending moment) EdlekIteLIgkñúgeBldMNalKñaenAelImuxkat;sikSa.
tMélén Vu d / M u minRtUvFMCag 1.0 sMrab;smIkar ¬*>^¦. RbsinebI M u mantMélFMenAkñúgsmIkar
¬*>^¦ enaHtYrTIBIrnwgmantMéltUcesÞIEtesμIsUnü enaH vc xiteTArk 0.16 f 'c . RbsinebI M u mantMéltUc


enaHtYrTIBIrnwgmantMélFM ehIytMél 0.29 f 'c lub. eRkABIsmIkar ¬*>^¦ ACI Code, Section
11.3.1GnuBaØatieGayKNnaersIusþg;kMlaMgkat;rbs;ebtugdUcxageRkam³

Vc = (0.17 f 'c )bw d ¬*>&¦
1> kñúgkrNIsMrab;kMlaMgsgát;tamGkS½ N u
Vc = (0.16 f 'c + 17.2 ρ w
Vu d
Mm
)bw d ≤ 0.29 f 'c bw d ¬*>*¦
⎛ 4h − d ⎞
M m = M u − Nu ⎜ ⎟
⎝ 8 ⎠
Edl ρw = s
A
bw d
h= kMBs;srubrbs;Fñwm
Vu d / M u GacFMCag 1.0 b:uEnþ Vc minRtUvFMCag

Vc = bw d (0.29 f 'c ) 1 +
Nu
3.45 Ag
¬*>(¦
Edl Ag CaRkLaépÞTaMgGs; (gross section) KitCa mm 2

mü:agvijeTot Vc GacRtUv)anKNnaeday
Vc = bw d (2 + 0.145
Nu
Ag
) f 'c ¬*>!0¦
2> kñúgkrNIsMrab;kMlaMgTajtamGkS½ N u
Vc = bw d (2 + 0.58
Nu
Ag
) f 'c ¬*>!!¦
Edl N u mantMélGviC¢mansMrab;kMlaMgTaj

RbsinebI Vc GviC¢man enaH Vc RtUv)anykesμIsUnü.
5> FñwmmanEdkkMlaMgkat;
EdkEdleRbIedIm,ITb;nwgkMlaMgkat; EdleKeRbIGacmaneRcInRbePTepSg²
a. Edkkg EdleKeRbIedaydak;EkgeTAnwgEdkbeNþay ¬Edkem¦ b¤RtUv)aneKdak;eRTt mMuEdl

eKniymeRbI 45 . EdkkgEdleKniymeRbImanmuxkat; DB10 nig DB12 .
o

b. EdkBt; EdlCaEpñkmYyrbs;EdkbeNþayEdleKBt;eLIg ¬enAkEnøgEdleKbBaÄb;¦ edaymMu

30 nig 60 CaTUeTA 45 .
o o o

c. bnSMrvagEdkkg nigEdkBt;

d. sMNaj;Edk CamYynwgsMNaj;EkgeTAnwgGkS½


T.Chhay NPIC

e. EdkkgvNÐ EdleKeRbIsMrab;ssr

ersIusþg;kMlaMgkat;TTwgrbs;FñwmebtugGarem:RtUv)anbegáIneLIgedaykareRbInUvEdkkMlaMgkat;TTwg.
munnwgekItnUvsñameRbHGgát;RTUg EdkkMlaMgkat;TTwgCYyersIusþg;kMlaMgkat;TTwgtictYcbMput. eRkayeBlEdl
sñameRbHkMlaMgkat;TTwgekIteLIg EdkkMlaMgkat;TTwgbegáInersIusþg;kMlaMgkat;rbs;Fñwm ehIykMlaMgkñgmþg
eTotEdlekIteLIgenAmuxkat;eRbH. enAeBlbrimaNEdkkMlaMgkat;TTwgtUc kar)ak;EdlekIteLIgedaysar
EdkenARTnugeFVIkardl; yield GacnwgekIteLIg b:uEnþRbsinebIbrimaNEdkkMlaMgkat;TTwgFM enaHkar)ak;eday
shear-compression failure GacnwgekIteLIg TaMgenHCaGVIEdleyIgKYreCosvag.

ebtug Edkkg (stirrups) nigEdkdgErk (bent bars) eFVIGMeBIrYmKñaedIm,ITb;nwgkMlaMgkat;TTwg.
edaysarersIusþg;rgkarsgát;x<s; ebtugedIrtYCaGgát;rgkarsgát;Ggát;RTUgénRbBn§½Fñwm cMENkÉEdkkgedIrtYCa
Ggát;rgkarTajbBaÄr. kMlaMgsgát;Ggát;RTUg k¾dUcCabgÁúMkMlaMgbBaÄrrbs;va mantMélesμInwgkMlaMgTajenAkñúg
Edkkg. EdkBt;dgErk (bent-up reinforcement) k¾edIrtYdUcCaGgát;TajenAkñúg truss Edr ¬rUbTI *>^¦.
CaTUeTA karcUlrYmrbs;EdkkMlaMgkat;TTwgeTAkñúgersIusþg;kMlaMgkat;TTwgrbs;FñwmebtugGarem:
GacRtUv)anBiBN’nadUcxageRkam³

- vaTb;Tl;EpñkxøHénkMlaMgkat;TTwg/ V s

- vabegáInGaMgtg;sIueténkMlaMgkat;rvagGnþrépÞ/ Va ¬rUbTI *>%¦/ edayTb;Tl;nwgkarrIkFMénsñam
eRbHeRTt (inclined crack).
- vabegáInnUvkMlaMgf<k; (dowel force)/ Vd ¬rUbTI *>%¦/ enAkñúgEdkbeNþay
- GMeBITb; (confining action) rbs;EdkkgeTAelIebtugGacbegáInerIusþg;rbs;va
- GMeBITb; (confining action) rbs;EdkkgeTAelIebtugbegáInnUvsmtßPaBrgVilénsnøak;)øasÞic
(rotation capacity of plastic hinge) EdlekItmanenAkñúgeRKOgbgÁúM indeterminate structure

eRkambnÞúkcugeRkay nigbegáInRbEvgEdl yield GacekItmanenAelIva.
eday V CaersIusþg;kMlaMgkat;rbs;muxkat;ebtugGarem:enaH
n

V =V +V
n c s ¬*>!@¦
Edl V - ersIusþg;kMlaMgkat;)anBIebtug
c

V - ersIusþg;kMlaMgkat;)anBIEdk
s

RbsinebIV kMlaMgkat;Edl)anBIbnÞúkxageRkAenaH
u

V ≤ φV = φ (V + V )
u n c s ¬*>!#¦
Edl V = 1.2V + 1.6V
u nig φ = 0.75
D L

V RtUv)anKNnaedaykarviPaK truss ¬rUbTI *>&¦. sMrab;sñameRbH 45 niges‘rIénEdkkg b¤Edk
s
o

dgErk. kMlaMgkat;bBaÄr V esμInwgplbUkbgÁúMkMlaMgbBaÄrénkMlaMgTajEdlekItmanenAkñúgEdkeRTt
s

V = nA f sin α
s v yt ¬*>!$¦
Edl A -muxkat;kat;EdkkMlaMgkat;CamYyKMlat s
v

f - ersIusþg;EdkkMlaMgkat;
yt

eday ns = aa + a a 1 1 2


T.Chhay NPIC

d = a1a4 = aa1tg 45o ¬BIRtIekaN aa a ¦ 1 4

d = a1a4 = aa2tgα ¬BIRtIekaN aa a ¦ 1 2

⇒ ns = d (cot 45o + cot α ) = d (1 + cot α )
d
⇒ n = (1 + cot α )
s
eKTTYl)an A f d
s
A f d
Vs = v yt sin α (1 + cot α ) = v yt (sin α + cos α )
s
¬*>!%¦
sMrab;krNIEdkkgbBaÄr α = 90 o

V =
s
A f d
s
v
b¤ s = A Vf d
yt v yt
¬*>!^¦
s

sMrab;krNIEdkkgbBaÄr α = 45 o

V = 1.4
s
A f d
s
b¤ s = 1.4 A Vf d
v yt v yt
¬*>!&¦
s

sMrab;krNIEdkdgErkEtmYy b¤RkumEdkenAmYykEnøg
V = A f sin α
s v ytb¤ A = f V α sin
v
s
¬*>!*¦
yt

sMrab; α = 45o

Av = 1.4
Vs
f yt
¬*>!(¦

6> tMrUvkarrbs; ACI Code sMrab;karKNnakMlaMgkat;TTwg
1> muxkat;eRKaHfñak;sMrab;karKNnaersIusþg;kMlaMgkat;TTwgmFüm
Critical section for nominal shear strength calculation
ACI Code, Section 11.1.3 GnuBaØateGayykmuxkat;eRKaHfñak;sMrab;karKNnaersIusþg;kMlaMgkat;
mFüm enAcMgay d BIépÞmuxénTMr. karENnaMenHQrenAelIPaBCak;EsþgEdlsñameRbHeRTtdMbUgeKTMngCaekIt
eLIgenAelIFñwmRtg;cMgay d BITMrEdleRcInelcecjenAeBleFVIBiesaFn_. muxkat;eRKaHfñak;enHRtUv)anGnuBaØat
enAkñúglkçxNÐEdlRbtikmμTMrbBa¢ÚnkMlaMgsgát;eTAkñúgtMbn;cug/ bnÞúkRtUv)anGnuvtþenAelI b¤enAEk,rkMBUlén
Ggát;ehIyKμanbnÞúkcMcMnucGnuvtþenAcenøaHépÞénTMr nigTItaMgénmuxkat;eRKaHfñak;. bTdæank¾kMNt;Edrfa Edk
kMlaMgkat;TTwgRtUv)andak;enAcenøaHépÞénTMr nigcMgay d .
2> muxkat;EdkGb,brmasMrab;EdkkMlaMgkat;TTwg
vtþmanrbs;EdkkMlaMgkat;TTwgenAkñúgFñwmebtugTb;Tl;nwgkarrIkraldalénsñameRbHeRTt. mü:agvij
eTot PaBsVit (ductility) ekIneLIg ehIyva)anRbkasGasnñmuneBl)ak;. RbsinebIKμanEdkkMlaMgkat;TTwg
enaHFñwmmanlkçN³RsYyehIy)ak;edaymin)anR)ab;mun. dUcenH muxkat;EdkkMlaMgkat;TTwgRtUv)ankMNt;

eday Code. ACI Code, Section 11.5.5 tMrUvEdkkgTaMgGs;eGaymanRkLaépÞEdkkMlaMgkat;TTwgGb,br-
ma Av esμInwg
b s b s
Av = 0.062 f 'c w ≥ 0.35 w
f f
¬*>@0¦
yt yt

Edl bw CaTTwgénRTnug nig s CaKMlatrbs;Edkkg. eKRtUvkarbrimaNEdkkMlaMgkat;TTwgGb,-
brmaenAeBlEdl V > 0.5φV elIkElg
u c

- kMralxNÐ nigeCIgtag
- rnUtebtug
- FñwmEdlmankMBs;tUcCag max{250mm; 2.5 dgkMras;énsøab; ; 1.5 énTTwgRTnug }
RbsinebI 0.062 f 'c = 0.35 enaH f 'c = 31.9MPa . enHman½yfaenAeBlEdl f 'c < 32MPa enaH
tMélGb,brma Av = 0.35bw s / f yt manlkçN³lb; EtenAeBlEdl f 'c ≥ 32MPa enaHtMélGb,brma
Av = 0.062 f 'c bw s / f yt manlkçN³lub. karekIneLIgnUvRkLaépÞEdkkMlaMgkat;TTwgsMrab;

f 'c ≥ 32MPa KWedIm,IkarBarCamunnUvkar)ak;edaykMlaMgkat;TTwg (shear failure )Pøam² enAeBlEdlekIt

mansñameRbHeRTt.
vaCakarGnuvtþn_mYyd¾FmμtakñúgkardMeLIgkMras;kMralxNÐ kMras;eCIgtag b¤kMBs;Fñwmrak;edIm,IbegáIn
lT§PaBTb;Tal;nwgkMlaMgkat;TTwg. EdkkgGacnwgKμanT§BlenAkñúgGgát;rak; edaysartMbn;rgkarsgát;
rbs;vamankMBs;tUcEmnETn nigminmanTMBk;RKb;RKan;EdlRtUvkarsMrab;Edkkg. sMrab;FñwmEdlminrak;
eKminRtUvkarEdkkMlaMgkat;TTwgenAeBlEdl Vu < 0.5φVc .
RkLaépÞEdkkMlaMgkat;TTwgGb,brmaGacnwgRtUv)anTTYledayeRbIEdkkg DB10 dak;enAKMlatGti-
brma S max . RbsinebI f yt = 400MPa ehIyEdkkg DB10 manragGkSr U ¬eCIgBIr¦RtUv)aneRbI enaH
smIkar ¬*>@0¦køayeTACa
Av f yt Av f yt
S max =
(0.062 f ' )b
≤
0.35b
¬*>@!¦
c w w

sMrab; f 'c < 32MPa / S max = 157 × 400 / 0.35bw = 179400 / bw
sMrab; f 'c = 32MPa / S max = 179000 / bw ¬*>@@¦
sMrab; f 'c = 35MPa / S max = 171200 / bw
RbsinebIeKeRbIEdk DB12 manragGkSr U enaH
sMrab; f 'c < 32MPa / S max = 258250 / bw
sMrab; f 'c = 32MPa / S max = 25750 / bw ¬*>@#¦

T.Chhay NPIC

RtUvcgcaMfa S max minRtUvFMCag 600mm b¤ d / 2 eT.
taragTI 1 pþl;nUv S max edayQrelIsmIkar ¬*>@@¦ nig ¬*>@#¦. KMlatcugeRkayKYrEtRtUvrMkil
eTArktMélEdltUc. ]TahrN_ S max = 515mm køayeTACa S max = 500mm .
taragTI1> tMélrbs; S max = Av f yt / 0.35bw = 60cm . enAeBlEdl f yt = 400MPa nig f 'c < 32MPa

bw (cm) 25 30 35 40 45 50 55 60 bw

S max (cm) DB10 60 55 50 40 35 35 30 25 179400 / bw

S max (cm) DB12 60 60 60 60 55 50 45 40 258250 / bw

3> kMlaMgkat;TTwgGtibrmaEdlTb;edayEdkkMlaMgkat;TTwg V s

edIm,IkarBarkar)ak; shear-compression failure EdlebtugGacEbkedaykugRtaMgkMlaMgkat;TTwgFM
nigkugRtaMgkMlaMgsgát;enAkñúgtMbn;eRKaHfñak; enABIelIkMBUlénsñameRbHGgát;RTUg ACI Code, Section
11.5.6.8, tMrUvfa V ≤ 0.67 f ' b d . RbsinebI V > 0.67 f ' b d enaHeKRtUvtMelIgmuxkat;ebtug.
s c w s c w

edayQrenAelIkarkMNt;enH³
RbsinebI f 'c = 20MPa enaH Vs ≤ 3bwd b¤ Vs / bwd ≤ 3MPa
RbsinebI f 'c = 28MPa enaH Vs ≤ 3.5bwd b¤ Vs / bwd ≤ 3.5MPa
RbsinebI f 'c = 35MPa enaH Vs ≤ 4bwd b¤ Vs / bwd ≤ 4MPa
4> KMlatEdkkgGtibrma
edIm,IFanafasñameRbHGgát;RTUgRtUvkat;Edkkgy:agehacmYy enaH ACI Code, Section 11.5.4 tMrUv
fa KMlatrvagEdkkgminKYrelIs d / 2 b¤ 600mm RbsinebI V ≤ 0.33 f ' b d ¬edayQrelIkarsnμt;fa
s c w

sñameRbHGgát;RTUgekItmantammMu 45o niglatsn§wgtamcMgayedkRbEhlcMgay d . enAkñúgtMbn;kMlaMgkat;
TTwgFM Edl Vs > 0.33 f 'c bwd KMlatEdkkgGtibrmacenøaHEdkkgminRtUvFMCag d / 4 . karkMNt;enH
caM)ac;edIm,IFanaeGaysñameRbHGgát;RTUgkat;Edkkgy:agehacbI. enAeBlEdl V > 0.67 f ' b d kar s c w

kMNt;énKMlatGtibrmaminRtUv)anGnuvtþ ehIyTMhMrbs;muxkat;ebtugKYrRtUv)antMeLIg.
karkMNt;TIBIrsMrab;KMlatGtibrmaénEdkkgk¾GacTTYl)anBIlkçxNÐmuxkat;EdkkMlaMgkat;TTwgGb,
brma. Av Gb,brma RtUv)anTTYlenAeBlKMlat s Gb,brma ¬smIkar *>@!¦.
karkMNt;TIbIsMrab;KMlatGtibrmaesμInwg 600mm enAeBlEdl V ≤ 0.33 f ' b d nigesμInwg
s c w

300mm enAeBlEdl 0.33 f 'c bw d < Vs ≤ 0.67 f 'c bw d . tMéltUcCageKénKMlatGtibrmaRtUv)anyk


mkeRbI. tMrUvkarKMlatGtibrmaén ACI Code FanaKMlatCitKñarbs;EdkkgedIm,Icab;EdkrgkarTajbeNþay
enAkñúgFñwm edayehtuenHva)anbegáInlT§PaBbgáb;Edkrbs;va Vd ¬rUbTI *>%¦.
5> ersIusþg; yield rbs;EdkkMlaMgkat;TTwg
ACI Code, Section 11.5.2. tMrUveGayersIusþg; design yield strength rbs;EdkkMlaMgkat;TTwg

minKYrelIs 420MPa . mUlehtuEdlenABIeRkaykarsMercenHKWedIm,IkMNt;TMhMsñameRbHEdlbNþalmkBI
kMlaMgTajGgát;RTUg nigedIm,IFanafaEKmrbs;sñameRbHenArkSaépÞb:HCitKñaedIm,IbegáInkMlaMgbBa¢ÚnénkMlaMg
kat;rvagGnþrépÞ Va ¬rUbTI *>%¦. sMrab;sésrEdkfñaMgGMeBAsMrab;pSar (welded deformed wire fabric)
ersIusþg; design yield strength minKYrelIs 560MPa .
6> TMBk;rbs;Edkkg
ACI Code, Section 11.5.2. tMrUvfaEdkkMlaMgkat;TTwgRtUv)andak;enAEk,rseésrrgkarsgát;eRkA

bMput nigsésrrgkarTajeRkAbMputtamEtGaceFVIeTA)an CamYynwgtMrUvkarrbs; code sMrab;kMra;karBarEdk
edaysarEt enAeBlEdlbnÞúkEdlmanGMeBIenAelIFñwmxiteTACitbnÞúkcugeRkay (ultimate load) sñameRbHkM
laMgTaj edaykarBt; (flexural tension crack) bnøayy:ageRCAcUleTAkñúgFñwm. dUcKña edIm,IeGayEdkkg
TTYl)annUversIusþg; yield eBj vaRtUvkarnUvTMBk;Edll¥. enAeBlEdlbnÞúkEdlmanGMeBIelIFñwmxiteTACitbnÞúk
cugeRkay (ultimate load) kugRtaMgenAkñúgEdkkg)aneTAdl;kugRtaMg yield rbs;va enAcMnucEdlsñameRbH
Ggát;RTUgkat;cMEdkkgenaH. tMrUvkarrbs; ACI Code sMrab;TMBk;Edkkg/ Section 12.13 dUcxageRkam³
- karBt;nImYy²enAkñúgEpñkCab;énEdkkgGkSr U Fmμta b¤BhuEdkkgGkSr U KYrBT§½CMuvijEdk
beNþay (ACI Code, Section 12.13.3) emIlrUbTI *>*a.
- Code GnuBaØateGayeRbInUvTMBk; standard 90o / 135o b¤ 180o CMuvijEdkbeNþysMrab;Edkkg
DB16 . RbsinebIEdkkg DB19 / DB 22 nig DB 25 CamYynwg f yt > 280MPa enaH Code,

Section 12.13.2 tMrUvTMBk; standard bUknwgRbEvgbgáb; 0.17d b f yt / f 'c cenøaHBak;kNþal

kMBs;Fñwm nigEpñkxageRkAénTMBk;. RbsinebIEdkRtUvBt;edaymMu 90o RbEvgBnøÚtminRtUvtUcCag
12d b . sMrab;Edk DB16 b¤EdkkgTMhMtUcCagenH RbEvgBnøÚtKW 6d b (ACI Code, Section 7.1)

emIlrUbTI *>*.
- RbsinebIEdkkgGkSr U DubRtUv)aneRbIedIm,IpÁúMCaEdkkgbiTCit RbEvgRCYs (lap length)
minRtUvtUcCag 1.3ld (ACI Code, Section 12.13.5) emIlrUbTI *>*c .
- sésrEdkEdlpSar (welded wire fabric) RtUv)aneRbIsMrab;EdkkMlaMgkat;TTwgenAkñúg]sSahkmμ
plitTukmun (precast industry) . TMBk;lMGitRtUv)anpþl;eGayenAkñúg ACI Code, Section
12.13.2.3 nigenAkñúgesckþIBnül; (commentary) rbs;va.


T.Chhay NPIC

- EdkkgbiTCitRtUv)anpþl;eGaysMrab;FñwmEdlrgnUgkMlaMgrmYl (ACI Code, Section 7.11).
- FñwmEdlenABT§½CMuvijeRKOgbgÁúMRtUveRbIEdkkgbiTCitedIm,IrkSa structural integrity rbs;Ggát;
(ACI Code, Section 7.13.2.2).

7> EdkkgenAEdlenAEk,rTMr
ACI Code, Section 11.1.3 kMNt;faEdkkgkMlaMgkat;TTwgEdlpþl;eGayenAcenøaHépÞTMr nigmuxkat;

eRKaHfñak; (critical section) EdlsßitenAcMgay d BITMr KYrRtUv)anKNnasMrab;kMlaMgkat;TTwg Vu dUcKña enAnwg
muxkat;eRKaHfñak;. vaCakarGnuvtþn_Fmμtaedaydak;EdkkgTImYyenAcMgay s / 2 BIépÞénTMr Edl s CaKMlat
EdlKNnaedaysmIkar ¬*>!^¦ sMrab; Vu enAmuxkat;eRKaHfñak;.

8> RbEvgRbsiT§PaBrbs;EdkdgErk
manEtbIPaKbYnRtg;cMnuckNþalénEpñkeRTténEdkbeNþayRtUv)anKitfamanRbsiT§PaBsMrab;EdkkM
laMgkat;TTwg. enHmann½yfa KMlatGtibrmarbs;EdkdgErkKW 0.75(d − d ' ) . BIrUbTI *>( RbEvgRbsiT§PaB
rbs;EdkdgErkKW 0.75(d − d ' ) / sin 45o = 0.75(1.414)(d − d ' ) = 1.06(d − d ' ) . KMlatGtibrma s esμInwgcM
gayedkEdl)anBIkarTMlak;cMeNalEkgénRbEvgRbsiT§PaBEdkdgErk. dUcenH
S max = 1.06(d − d ' ) cos 45o b¤ S max = 1.06(d − d ' )0.707 = 0.75(d − d ' )

7> karKNnaEdkkgbBaÄr
eKRtUvkarEdkkg (stirrup) enAeBlEdl Vu > 12 φVc . EdkkgGb,brmaRtUv)aneRbIenAeBlEdl
1 φV < V < φV . kñúgkrNIenHeKeRbIEdkkg DB10 EdlRtUv)andak;nUvKMlatGtibrma. enAeBlEdl
2 c u c

Vu > φVc eKRtUvEtdak;EdkkgCamYyKMlattUcCagKMlatGtibrma ehIyGacRtUv)anKNnaedayeRbIsmIkar

¬*>!^¦³ S = Av f yt d / Vs .
EdkkgEdlRtUv)aneRbICaTUeTAenAkñúgmuxkat;ebtugCaEdkkg DB10 nig DB12 GkSr U eCIgBIr
CamYynwg f yt = 400MPa . RbsinebI DB10 RtUv)aneRbIenaH smIkar¬*>!^¦køayCa³
S Av f yt 157 × 400 62800
d
=
Vs
=
Vs
=
Vs
¬*>@$¦
RbsinebI DB12 RtUv)aneRbIenaH
S Av f yt 226 × 400 90400
d
=
Vs
=
Vs
=
Vs
¬*>@%¦
pleFobKMlatEdkkgelIkMBs;RbsiT§PaB d rbs;Fñwm GaRs½ynwg Vs . tMélén S / d sMrab;tMél
epSgKñaén Vs enAeBlEdl f yt = 400MPa RtUv)aneGayenAkñúgtaragTI2 nigtaragTI3 sMrab;Edk DB10 nig
Edk DB12 erogKña. tMéldUcKñaRtUv)anbgðajCadüaRkamdUcenAkñúgrUbTI 8>10 nigrUbTI 8>11.


T.Chhay NPIC

taragTI2> pleFob S / d sMrab;tMél V ¬ fs yt /
= 400MPa S / d = 62800 / Vs ¦ DB10
Vs (kN ) 125.6 142.7 190.3 237.9 251.2 285.5 330.5 380.6 418.7 475.8 592.5
S /d 0.5 0.44 0.33 0.264 0.25 0.22 0.19 0.165 0.15 0.132 0.106

taragTI3> pleFob S / d sMrab;tMél V ¬ fs yt /
= 400MPa S / d = 90400 / Vs ¦ DB12
Vs (kN ) 180.8 225 265 310 361.6 445 490.0 535 665 775 850
S /d 0.5 0.40 0.34 0.29 0.25 0.20 0.18 0.17 0.14 0.12 0.11


Tamtarag nigdüaRkamxagelIeyIgGacsnñidæan)anfa³
- RbsinebIeKeRbI DB10 / S = d / 2 enAeBlEdl Vs ≤ 125.6kN . enAeBlEdl Vs ekIneLIg
S / d fycuHtamExSekageTArktMél 0.132 enAeBl Vs = 475.8kN . RbsinebIKMlatGb,brma

RtUv)ankMNt;Rtwm 75mm enaH d ≥ 568mm . enAeBlEdl Vs > 251.2kN enaH S ≤ d / 4 .
- RbsinebIeKeRbI DB12 / S = d / 2 enAeBlEdl Vs ≤ 180.8kN . enAeBlEdl Vs ekIneLIg
S / d fycuHtamExSekageTArktMél 0.14 enAeBl Vs = 665kN . RbsinebIKMlatGb,brma

RtUv)ankMNt;Rtwm 75mm enaH d ≥ 535mm . enAeBlEdl Vs > 361.6kN enaH S ≤ d / 4 .
- RbsinebIeKeRbIEdkkgGkSr U f yt = 280MPa enaHeKRtUvKuN S / d edaytMél 7 /10 b¤CaTUeTA
f yt / 400 .

8> segçbviFIsaRsþKNnaEdkkgbBaÄr
CMhankñúgkarKNnaEdkkgbBaÄrsMrab;kMlaMgkat;TTwg edayeyagtam ACI Code GacRtUv)ansegçb
dUcxageRkam³
a. kMNt;kMlaMgkat;KNna V BIbnÞúkEdlGnuvtþn¾mkelIeRKOgbgÁúM. kMlaMgkat;KNnaRKITic
u

EdlRtUvykmksikSasßitenARbEvg d BImuxénTMr.
b. kMNt; φV = φ 0.17 f ' b d b¤ φV = φ (0.16 f ' + 17.2 ρ
Vd
c c w )b d ≤ φ 0.29 f ' b d
c c w
u
w c w
M u

bnÞab;mkKNna 1 φV
2
c

c. k> RbsinebI V < 1 φV muxkat;minRtUvkarEdkkg
2
u c

x> RbsinebI 1 φV < V ≤ φV eRbImuxkat;EdkkgGb,brma
2
c u c

K> RbsinebI V > φV muxkat;Edkkg RtUvKNnadUcxageRkam
u c

d. kMNt;kMlaMgkat;EdlTb;eday Edkkg
Vu − φVc
Vs =
φ
e. kMNt; V = 0.33 f ' b d nig V = 0.67 f ' b d = 2V . RbsinebI V > V
c1 c w c2 c w c1 s c2

tMeLIgmuxkat;.
f. kMNt;KMlatrbs;Edk s = A Vf d 1
v yt

s

g. kMNt;KMlatEdkGtibrmaEdlGnuBaØatieday ACI Code. KMlatEdkGtibrmaCatMéltUcbMput
én s nig s
2 3


T.Chhay NPIC

k> s 2
d
2
= ≤ 60cm RbsinebI V ≤ V = 0.33
s c1 f 'c bw d
d
s2 = ≤ 30cm
4
RbsinebI V < V ≤ V
c1 s c2

x> 3A f
s3 = v yt ≥
bw
16 Av f yt
bw f 'c

k> RbsinebI s < s eRbI s
h. 1 max 1

x> RbsinebI s > s eRbI s 1 max max

i. ACI Code min)ankMNt;nUvKMlatGb,brmaeT. eRkamlkçxNÐFmμta KMlatGb,brma S

RtUv)ansnμt;ykesμInwg 75mm sMrab; d ≤ 50cm nigmanKMlatGb,brmaesμInwg 100mm
sMrab;FñwmeRCA (deep beam) . RbsinebI S mantMéltUcenaH eKGactMeLIgmuxkat;Edkkg
b¤eRbIEdkkgeCIgeRcIn ¬rUbTI 8>8¦.
j. sMrab;muxkat;mUl RkLaépÞEdleRbIsMrab;KNna Vc = plKuNGgát;p©itCamYykMBs;RbsiT§PaB

d / Edl d = 0.8 énGgát;p©it/ ACI Code, Section11.3.3 .

]TahrN_1³ FñwmTMrsamBaØmanmuxkat;ctuekaN b = 30cm / d = 55cm nig h = 60cm ehIyRtUv)anBRgwg
eday 4DB25 . epÞógpÞat;faetImuxkat;enHRKb;RKan;b¤Gt;sMrab;kMlaMgkat;TTwgemKuN (ultimate shear force)
xageRkam. RbsinebIva minRKb;RKan; cUrKNnaEdkkMlaMgkat;TTwgkñúgTMrg;CaEdkkgGkSr U . edayeRbI
f 'c = 28MPa nig f yt = 400MPa .

k> Vu = 50kN x> Vu = 110kN K> Vu = 240kN X> Vu = 345kN g> Vu = 570kN
dMeNaHRsay³
CaTUeTA bw = b = 300mm / d = 550mm nig
φVc = φ (0.17 f 'c )bd = 0.75(0.17 28 )300 × 550 × 10 −3 = 111.3kN
1 φV = 55.65kN
2 c

( ) ( )
Vc1 = 0.33 f 'c bd = 0.33 28 300 × 550 × 10 −3 = 288kN

Vc 2 = (0.67 f 'c )bd = 576kN

k> Vu = 50kN < 1 φVc = 55.565kN
2
/ muxkat;RKb;RKan; edayminRtUvkarEdkkMlaMgkat;TTwg.
x> Vu = 110kN > 1 φVc / b:uEnþvatUcCag φVc = 111.3kN . eday Vs = 0 dUcenH muxkat;RtUvkarEdk
2

kMlaMgkat;TTwgGb,brma. eRbI DB10 CaEdkkgGkSr U enAKMlatGtibrma.
π
Av = 2 × 10 2 = 157mm 2
4
KMlatGtibrmaCatMéltUcCageKkñúgcMeNam
S 2 = d / 2 = 275mm yk 250mm ¬lub¦

S 3 = Av f yt / 0.35bw = 157 × 400 /(0.35 × 300) = 598mm yk 550mm ¬b¤eRbItaragTI1¦
S 4 = 600mm
dUcenHeRbIEdkkg DB10 @ 250mm
K> Vu = 240kN > φVc / RtUvkarEdkkMlaMgkat;TTwg. karKNnaGaceFVIeLIgCaCMhanxageRkam³
KNna Vs = (Vu − φVc ) / φ = (240 − 111.3) / 0.75 = 171.6kN
edaysar Vs < Vc1 enaH S max = d / 2 ≤ 600mm
eRCIserIs DB10 CaEdkkgGkSr U nigKNnaKMlatRtUvkaredayQrelI Vs
Av f yt d 157 × 400 × 550
S1 = = = 201mm yk 200mm
Vs 171600
KNnaKMlatGtibrma³ S 2 = 250mm / S3 = 550mm nig S 4 = 600mm dUcenH S max = 250mm
edaysar S = 200mm < S max = 250mm
dUcenHeRbIEdkkg DB10 @ 200mm
X> Vu = 345kN > φVc /RtUvkarEdkkMlaMgkat;TTwg.
KNna Vs = (Vu − φVc ) / φ = (345 − 111.3) / 0.75 = 311.6kN
edaysar Vs > Vc1 enaH S max = d / 4 ≤ 300mm yk 125mm
edaysar Vc1 < Vs < Vc2 enaHeKGaceRbIEdkkg edaymincaM)ac;tMeLIgmuxkat;ebtug.
eRCIserIs DB10 CaEdkkgGkSr U nigKNnaKMlatRtUvkaredayQrelI Vs
Av f yt d 157 × 400 × 550
S1 = = = 110mm yk 100mm
Vs 311600
KNnaKMlatGtibrma³ S 2 = d / 4 = 137.5mm yk 125mm / S3 = 550mm nig S 4 = 300mm
dUcenH S max = 125mm
edaysar S = 100mm < S max = 125mm
dUcenHeRbIEdkkg DB10 @100mm
g> Vu = 570kN > φVc / RtUvkarEdkkMlaMgkat;TTwg.
KNna Vs = (Vu − φVc ) / φ = (570 − 111.3) / 0.75 = 611.6kN
edaysar Vs > Vc2 enaHmuxkat;minRKb;RKan;. eKRtUvkartMeLIgTMhMrbs;muxkat;mYy b¤k¾TaMgBIr.
cMNaM³ taragTI 2 nigrUbTI 8>10 k¾GacRtUv)aneRbIedIm,IKNnaKMlat S sMrab; K> nig X> )anpgEdr.
1> sMrab; K> Vs = 171.6kN BIrUbTI 8>10 ¬b¤taragTI 2> sMrab;EdkkgGkSr U DB10 ¦ eyIgTTYl)an
S / d = 0.37 dUcenH S1 = 203.5mm EdltUcCag d / 2 = 250mm . cgcaMfa S max EdlQrelI Vs

KW d / 2 minEmn d / 4 eT. dUcKñaBItaragTI 1> eyIgTTYl)an S3 = Av f yt / 0.35bw = 550mm .

T.Chhay NPIC

2> sMrab; X> Vs = 311.6kN / S / d = 0.18 enaH S1 = 100mm / Vs = 311.6kN > 251.2kN enaH
S max = d / 4 RtUv)aneRbI.

]TahrN_2³ FñwmTMrsamBaØEdlmanRbEvg 5.2m nigmanRbEvgcenøaHssr (clear span) 4.9m edayRTnUv
bnÞúkBRgayesμIefr 65kN / m nigbnÞúkBRgayesμIGefr 55kN / m . TMhMrbs;Fñwm nigsésrEdkRtUv)anbgðaj
enAkñúgrUbTI 8>12. epÞógpÞat;muxkat;sMrab;kMlaMgkat;TTwg nigKNnaEdkkMlaMgkat;TTwgcaM)ac;. eKeGay
f 'c = 20MPa nig f y = 400MPa .

dMeNaHRsay³
eKeGay bw = 350mm / d = 580mm
1> KNnakMlaMgkat;TTwgemKuN (ultimate shear) BIbnÞúkxageRkA³
bnÞúkBRgayesμIemKuN = 1.2 × 65 + 1.6 × 55 = 166kN / m
166 × 4.9
Vu ¬enABImuxépÞTMr¦ = = 406.7 kN
2
KNna Vu ¬enAcMgayBImuxépÞénTMr¦ = 406.7 − 0.58 ×166 = 310.42kN
2> KNna φVc :
φVc = φ (0.17 f 'c )bw d = 0.75(0.17 20 )350 × 580 × 10 −3 = 115.75kN
1 φV = 57.87 kN
2 c


KNna Vc1 = 0.33 f 'c bwd = 0.33 20 × 350 × 580 ×10−3 = 299.6kN
KNna Vc2 = 2Vc1 = 599.2kN
3> eday Vu > φVc dUcenHmuxkat;RtUvkarEdkkMlaMgkat;TTwg. cMgay x' EdlenARtg;cMgayenHmuxkat;
ebtugminRtUvkarEdkkMlaMgkat;TTwg ¬enA 12 φVc ¦KW³
⎛ 406.7 − 57.87 ⎞ 4.9
x' = ⎜ ⎟ = 2.10m
⎝ 406.7 ⎠ 2
4> KNna Vs = (Vu − φVc ) / φ = (310.42 − 115.75) / 0.75 = 259.56kN . edaysar Vs < Vc1 enaH
S max = d / 2 ≤ 600mm RtUv)anBicarNa ¬b¤eyagtamrUbTI 8>10 b¤taragTI2³ Vs > 251.2kN ¦.

5> KNnaEdkkg³ eRCIserIsEdkkgGkSr U DB10 / Av = 157mm 2 . KNna S1 edayQrelI
Vs = 259.56kN / S1 = Av f yt d / Vs = 140mm yk 125mm ¬b¤yk S / d = 0.24 BItaragTI2

b¤BIrUbTI 8>10¦
6> KNnaKMlatGtibrma³ S 2 = d / 2 = 580 / 2 = 290mm yk 250mm /
S 3 = Av f yt / 0.35bw = 500mm ¬b¤eRbItaragTI1¦/ S 4 = 600mm . dUcenH S max = 250mm .

7> edaysar S1 = 125mm < S max = 250mm eRbI DB10 @125mm
8> KNna Vs sMrab;KMlatGtibrma 250mm
As f yt d 157 × 400 × 580 − 3
Vs = = 10 = 145.7 kN
s 250
φVs = 109.3kN
φVc + φVs = 115.75 + 109.3 = 225kN
cMgay x1 EdlenARtg;cMgayenHmuxkat;GaceRbIKMlat s = 250mm
⎛ 406.7 − 225 ⎞ 4.9
x1 = ⎜ ⎟ = 1.09m
⎝ 406.7 ⎠ 2
edaysar x1 mantMéltUc eRbI s = 125mm sMrab;cMgayFMCag b¤esμI 1.09m . cMNaMfa RbsinebI x1 Evg
KMlatenAcenøaH 150mm eTA 250mm GacRtUv)anbEnßm.
9> EdkkgRtUv)anBRgaydUcxageRkam³
dak;EdkkgTI1enAcMgay S / 2 BImuxépÞénTMr
EdkkgTImYyenA S / 2 = 125 / 2 = 62.5mm yk 50mm
R)aMbYnEdkkgmanKMlat S = 125mm = 1125mm
srub 1175mm > 1090mm
bYnEdkkgmanKMlat S = 250mm = 1000mm
srub 2175mm < 2450mm

T.Chhay NPIC

cMnYnEdkkgsrubsMrab;FñwmKW 2(1 + 9 + 4) = 28 . karBRgayEdkkgRtUv)anbgðajenAkñúgrUbTI 8>13
ÉkMlaMggkat;TTwgEdl)anKNnaRtUv)anbgðajenAkñúgrUbTI 8>12.
10> dak;Edkkg DB12 cMnYnBIredIm enABIelImuxkat;FñmedIm,IedIrtYrCaEdkkgBüÜr.
w

9> kMlaMgkat;TTwgEdlbNþalBIbnÞúkGefr
enAkñúg]TahrN_TI2 TaMgbnÞúkefr nigbnÞúkGefrRtUv)ansnμt;faBRgayesμIeBjtambeNþayFñwm Edl
begáIt)ankMlaMgkat;TTwgsUnüenAkNþalElVg. CaFmμta bnÞúkefrBitCaBRgayeBjelIbeNþayFñwm EtbnÞúk
GefrGacGnuvtþeBj b¤k¾GnuvtþEtEpñkxøHrbs;Fñwm EdltMrUveGaymankMlaMgkat;TTwgGtibrmaekIteLIgenA
kNþalElVg b¤muxkat;kMNt;NamYy. rUbTI 8>14 a bgðajBIFñwmTMrsamBaØCamYynwgbnÞúkBRgayesμIGnuvtþeBj
beNþayFñwm. kMlaMgkat;TTwgERbRbYlCaragbnÞat;tambeNþayFñwm CamYynwgkMlaMgkat;TTwgGtibrmaenATMr
A.

kñúgkrNIEdlbnÞúkGefr W2 = 1.6WL kMlaMgkat;TTwgGtibrmamanGMeBIenARtg;TMr A enAeBlEdl
W2 GnuvtþeBjElVgFñwm ¬rUbTI 8>14 a ¦. kMlaMgkat;TTwgGtibrmaekItmanenAkNþakElVgRbsinebIbnÞúkGefr

RtUv)andak;EtBak;kNþalFñwm BC ¬rUbTI 8>14 b ¦ EdlbegáIt)an Vu enAkNþalElVgesμInwg W2 L / 8 . dUc
enH kMlaMgkat;TTwgKNnaRtUv)anbegáIteLIgedaykarbEnßmkMlaMgkat;TTwgGtibrmaEdlbNþalmkBIbnÞúk
Gefr ¬EdlRtUv)andak;enAelIRbEvgepSg²énElVg¦ eTAelIkMlaMgkat;TTwgGefr ¬rUbTI 8>14 c ¦. vaCakar
Gnuvtþn_FmμtaedayKitkMlaMgkat;TTwgGtibrmaRtg;TMr A esμInwg Wu L / 2 = (1.2WD + 1.6WL ) L / 2 / b:uEnþ Vu
enAkNþalElVgesμI W2 L / 8 = (1.6WL ) L / 8 CamYybnÞat;Rtg;ERbRbYltambeNþay AC nig CB dUcbgðajenA
kñúg rUbTI 8>14 d. karKNnasMrab;kMlaMgkat;TTwgenAkñúkrNIenHnwgGnutþdUcKñanwgkarBnül;kñúg]TahrN_2.
RbsinebI karerobrab;xagelIenHGnuvtþeTAFñwmkñúg]TahrN_2 enaH Vu ¬enATMr A ¦ = 406.7kN nig Vu ¬enA
kNþalElVg¦ = (1.6 × 55)4.9 / 8 = 53.9kN .


T.Chhay NPIC

]TahrN_3³ Fñwm cantilever RbEvg 3m manmuxkat;ctuekaNEkg nigRTnUvbnÞúkemKuNBRgayesμI nigcMcMnuc
¬bnÞúkpÞal;xøÜnRtuv)anrab;bBa©ÚlrYc¦ dUcbgðajenAkñgrUb 8>15. edayeRbI f 'c = 28MPa nig f y = 400MPa

KNnaEdkkMlaMgkat;TTwgcaM)ac;sMrab;dak;kñúgFñwmTaMgmUl edayeyagtam ACI Code.

dMeNaHRsay³
1> KNnakMlaMgkat;TTwgtambeNþayFñwmEdlbNþalmkBIbnÞúkxageRkA
Vu ¬enATMr¦ = 80 × 3 + 89 + 36 = 365kN

Vud ¬enAcMgay d ¦ = 365 − 80
510
= 351.4kN
3000
Vu ¬enAcMgay 1.2m xageqVg¦ = 365 − 80 × 1.2 = 269kN

Vu ¬enAcMgay 1.2m xagsþaM¦ = 269 − 89 = 180kN

Vu ¬enAcugTMenr¦ = 36kN

düaRkamkMlaMgkat;TTwgRtUv)anbgðajenAkñúgrUbTI 8>15.
2> KNna φVc ³
φVc = φ (0.17 f 'c bd ) = 0.75(0.17 28 )300 × 510 × 10 −3 = 103.2kN
1 φV = 51.6kN
2 c

edaysar Vud > φVc muxkat;ebtugRtUvkarEdkkMlaMgkat;TTwg. KNna
Vc1 = 0.33 f 'c bd = (0.33 28 )300 × 510 × 10 −3 = 267.2kN


Vc 2 = 2Vc1 = 534.4kN
cMgay x EdlenARtg;enHmuxkat;minRtUvkarEdkkMlaMgkat;TTwg ¬enA 12 φVc = 51.6kN ¦ EdlRtVv)an
vas;BITMr A .
⎛ 180 − 51.6 ⎞
x = 1.2 + ⎜ ⎟1.8 = 2.8m
⎝ 180 − 36 ⎠
¬ 200mm BIcugTMenr¦. dUcKña x1 sMrab; φVc KW 2.16m BI A ¬ 840mm BIcugTMenr¦
3> Epñk AC ³ kMlaMgkat;TTwgKNna Vu = Vud = 351.4kN . KNna Vs = (Vu − φVc ) / φ
= (351.4 − 103.2) / 0.75 = 330.9kN . edaysarEt Vc1 < Vs < Vc 2 enaH S max ≤ d / 4 RtUv)an

BicarNa ¬b¤epÞógpÞat;edayeRbIrUbTI 8>10¦.
4> KNnaEdkkg³ eRCIserIsEdkkgGkSr U DB10 / Av = 157mm 2 . KNna S1 ¬QrelI Vs ¦
Av f yt d 157 × 400 × 510
S1 = = = 100mm
Vs 330900
eRbI 100mm ¬b¤TTYl s / d = 0.19 BIrUb 8>10¦.
5> KNnaKMlatGtibrma³ S 2 = d / 4 = 510 / 4 = 127.5mm yk 125mm
Av f yt 157 × 400
S3 = = = 550mm ¬BItaragTI1 sMrab; b = 300mm ¦
0.35b w 0.35 × 300

S 4 = 300mm
dUcenH S max = 125mm
6> eday S = 100mm < S max = 125mm dUcenHeRbIEdkkg DB10 @100mm
7> enAcMnuc C / kMlaMgkat;TTwgKNna Vu = 269kN > φVc enaH Vs = (269 − 103.2) / 0.75 = 221kN .
S1 = Av f yt d / Vs = 145mm

Vs = 221kN < Vc1 = 267.2kN S 2 = d / 2 = 255mm ¬b¤ 250mm ¦
enaH S1 = 145mm b¤ 125mm
S1 = 145mm < S 2

8> edaysarKMlat 125mm nig 100mm mantMélEk,rKña eRbIEdkkg DB10 @100mm sMrab;Epñk AC .
9> Epñk BC
A. Vu = 180kN > φVc

Vs = (180 − 103.2) / 0.75 = 102.4kN < Vc1 = 267.2kN
B. S1 = Av f yt d / Vs = 157 × 400 × 510 / 102400 = 313mm

C. S 2 = d / 2 = 510 / 2 = 255mm¬b¤tUcCag S3 = 550mm nig S 4 = 600mm ¦.
yk S max = 250mm . eRbIEdkkg DB10 @ 250mm sMrab;Epñk BC .


T.Chhay NPIC

10> karBRgayEdkkgedayvas;ecjBITMr A ³ dak;EdkkgTImYyenA S = 50mm
2

12 × 100 = 1200mm

7 × 250 = 1750mm
srub 3000mm
karBRgayEdkkgRtUv)anbgðajenAkñúgrUbTI 8>16. EdkkgsrubmancMnYn 20 .

10> kugRtaMgkMlaMgkat;TTwgenAkñúgGgát;EdlmankMBs;ERbRbYl
edaysarEtkugRtaMgkMlaMgkat;TTwg v CaGnuKmn_énkMBs;RbsiT§PaB d dUcenHkugRtaMgkat;TTwgERbRbYl
tambeNþayFñwmebtugBRgwgedayEdkCamYynwgkBs;ERbRbYl. enAkñúgFñwmEbbenH ¬rUbTI 8>17¦ eKBicarNa
elIGgÁGnnþtUc dx . kMlaMgsgát; C enAelImuxkat;NamYyesμInwgm:Um:g;EckeGayédXñas; b¤ C = M / y .
edrIevTImYyén C KW³
ydM − Mdy
dC =
y2
RbsinebI C1 FMCag C2 enaH C1 − C2 = dC = vbdx
ydM − Mdy dM M
vbdx = 2
=
− 2 dy
y y y
1 ⎛ dM ⎞ M ⎛ dy ⎞
v= ⎜ ⎟− ⎜ ⎟
yb ⎝ dx ⎠ by 2 ⎝ dx ⎠

edaysar y = jd / dM / dx esμInwgkMlaMgkat;TTwg V nig d ( jd ) / dx CaCMral (slope)/
M ⎡d ⎤
( jd )⎥ nig v = ¬*>@^¦
V V M
v= − 2 ⎢ dx
± tan α
bjd b( jd ) ⎣ ⎦ bjd b( jd ) 2

Edl V nig M CakMlaMgkat;TTwg nigm:Um:g;xageRkA ehIy α CamMuCMralénépÞmYyrbs;FñwmeTAnwgépÞmYy
eTotrbs;Fñwm. sBaØabUkRtUv)aneRbIenAeBlEdlkMBs;fycuHÉm:Um:g;ekIneLIg b:uEnþsBaØadkRtUv)aneRbIenAeBl
kMBs;ekIneLIgehIym:Um:g;k¾ekIneLIg. rUbmnþenHRtUv)aneRbIenAeBlEdlmMuCMraltUc EdlmMu α ≤ 30o .



TMrg;samBaØénsmIkar ¬*>@^¦GacRtUvbegáIteLIgedaysMrYl j enaHeyIgTTYl)an
v=
V M
± 2 tan α
bd bd
¬*>@&¦
sMrab;viFIKNnaersIusþg; smIkarxageRkamGacRtUv)aneRbI
vu =
Vu
±
M
φbd φbd 2
tan α ¬*>@*¦
sMrab;kMlaMgkat;TTwg
φVn = Vu ±
Mu
d
tan α ¬*>@(¦


T.Chhay NPIC

rUbTI 8>18 bgðajBIFñwm cantilever CamYynwgbnÞúkcMcMnucenAcugTMenr. m:Um:g;nigkMBs; d ekInelIgkñúg
TisedAeTArkTMr. kñúgkrNIenH sBaØadkRtUv)aneRbIenAsmIkar ¬*>@&¦ ¬*>@*¦ nig¬*>@(¦. dUcKña sBaØadk
RtUv)aneRbIsMrab;muxkat; t enAkñúgFñwmTMrsamBaØdUcbgðaj ehIysBaØabUkRtUv)aneRbIsMrab;muxkat; Z Edlm:U
m:g;ekIneLIgenAeBlEdlkMBs;fycuH.
enAkñúgkrNICaeRcIn karERbRbYlkMBs;rbs;FñwmekItmanenAeLIEpñkrbs;FñwmEdlenAEk,rTMr ¬rUbTI
8>18¦.
karBesaFn_eTAelIFñwmCamYykMBs;ERbRbYlbgðajfa FñwmEdlmankMBs;FMenATMrCaTUeTA)ak;edaysar
kMlaMgkat;TTwgsgát;. ÉFñwmEdlmankMBs;tUcenATMrCaTUeTA)ak;edaysarPaBKμanesßrPaB EdlbNþalmkBI
karraldalénsñameRbHemenAkñúgFñwmeLIgelI ehIybnÞab;mksñameRbHenaHraldaltamTisedkenAelImuxkat;
FñwmEpñkxagelI. karBiesaFn_k¾)anbgðajEdrfa sMrab;FñwmEdlmankMBs;ERbRbYl ¬rUbTI 8>18¦ CamYynwgmMu
eRTt α RbEhl 10o nigrgnUvkMlaMgkat;TTwg nigkMlaMgBt; ersIusþg;kMlaMgkat;TTwgrbs;ebtug VCV Gac
RtUv)anKNnaeday
VCV = Vc (1 + tan α ) ¬*>#0¦
Edl VCV = ersIusþg;kMlaMgkat;TTwgrbs;FñwmCamYynwgkMBs;ERbRbYl
Vc = ⎢0.16 f 'c + ⎜17.2 ρ w u ⎟⎥bw d ≤ (0.29 f 'c )bw d
⎡ ⎛ V d ⎞⎤
⎜ ACI Code Eq.11.6
⎣ ⎝ M ⎟
u ⎠⎦
mMuEdlbegáIteLIgedayTisrbs;Edk. vaRtUv)anKitfaviC¢mansMrab;FñwmEdlmankMBs;tUc
α=
enATMr nigGviC¢mansMrab;FñwmEdlmankMBs;FMenATMr ¬rUbTI 8>18¦
d s = kMBs;RbsiT§PaBrbs;FñwmenATMr

ACI Code Eq.11.3 CasmIkarRtUv)ansMrYl nigGaceRbIedIm,IKNna Vc ³

Vc = (0.17 f 'c )bw d ¬*>#!¦
]TahrN_4³ KNnaFñwm cantilever dUcbgðajenAkñúgrUbTI 8>19 eRkamGMeBIbnÞúkemKuN. FñwmenHkMBs;srub
enAcugTMenr 300mm ehIyekIneLIgeTArkTMr. edayeRbIPaKryEdk ρ = 1.5% / f 'c = 28MPa /
f y = 400MPa nig b = 250mm .

dMeNaHRsay³
1> M u ¬TMr¦ = 36.5 × 2.52 / 2 + 62 × 2.5 = 269kN.m
2> sMrab; ρ = 1.5% / Ru = 4.72MPa
M 269 ⋅ 10 6
d= = = 477.5mm
Ru b 4.72 × 250


As = 0.015 × 250 × 477.5 = 1790mm 2 ¬eRbIEdk 3DB28 ¦ yk d = 490mm / h = 550mm .

3> KNnasMrab;kMlaMgkat;TTwg³ kMlaMgkat;TTwgGtibrmaenATMrKW 62 + 36.5 × 2.5 = 153.25kN . Eday
sarmuxkat;FñwmERbRbYl m:Um:g;RtUv)anBicarNakñúgkarKNnakMlaMgkat;TTwg. edaysarkMBs;FñwmekIn
eLIgCamYym:Um:g;ekIneLIg sBaØadkRtUv)aneRbIenAkñúgsmIkar ¬*>@*¦
Vu Mu
vu = − (tan α )
φbd φbd 2
edIm,Irk tan α / yk d enAcugTMenresμI 490mm nig d enAKl;TMresμI 240mm
490 − 240
tan α = = 0.1
2500
269 ⋅10 6
vu¬TMr¦ =
153250
−
0.75 × 250 × 490 0.75 × 250 × 490 2
0.1 = 1.07 MPa

4> kugRtaMgkMlaMgTTwgenAcugTMenrKW Vu / φbd ¬ M u = 0 ¦
62000
vu = = 1.38MPa
0.75 × 250 × 240
5> enAcMgay d = 490mm BImuxépÞénTMr kMBs;RbsiT§PaBKW 441mm ¬BIrUbFrNImaRt¦
Vu = 153.25 − 36.5 × 0.49 = 135.4kN
2.012
Mu ¬enAcMgay 490mm BITMr¦=
62 × 2.01 + 36.5 ×
2
= 198.4kN.m

135.4 ⋅10 3 198.4 ⋅10 6 × 0.1
vu = − = 1.09MPa
0.75 × 250 × 441 0.75 × 250 × 4412


T.Chhay NPIC

6> enAkNþalElVg ¬1.25m BITMr¦
d = 365mm
Vu = 153.25 − 36.5 × 1.25 = 107.6kN
1.25 2
M u = 62 × 1.25 + 36.5 = 106kN .m
2
107.6 ⋅10 3 106 ⋅10 6 × 0.1
vu = − = 1.15MPa
0.75 × 250 × 365 0.75 × 250 × 365 2
dUcKña enAcMgay 1.9m BITMr ¬ 0.6m BIcugTMenr¦
d = 300mm Vu = 83.9kN M u = 43.8kN vu = 1.23kN
enAcMgay 2.2m BITMr ¬ 0.3m BIcugTMenr¦
d = 270mm Vu = 73kN M u = 20.2kN vu = 1.29kN
tMélTaMgGs;enHRtUv)anbgðajenAkñúgrUbTI 8>20
7> kugRtaMgkMlaMgkat;TTwgedayebtugKW
0.17 28 = 0.9MPa
kugRtaMgkMlaMgkat;TTwgGb,brmaEdlRtUvTb;edayEdkkMlaMgkat;TTwg
vus = 1.38 − 0.9 = 0.48MPa
¬ vu nig vus RtUv)anekIneLIgedaypleFob 1/ φ kñúgsmIkar 8>28¦
8> eRCIserIsEdkkg DB10 EdlmanEdkBIr
Av = 2 × 78.5 = 157mm 2
Av f yt 157 × 400
S ¬caM)ac;¦
= =
v s bw 0.48 × 250
= 523mm

¬sMrab; ¦
S max
d
2
eTA
= 245mm 120mm enAcugTMr
Av f yt 157 × 400
¬sMrab;Gb,brma ¦
S max Av = =
0.35bw 0.35 × 250
= 718mm

9> epÞógpÞat;KMlatGtibrma (d / 2) : vus ≤ 0.33 f 'c
0.33 f 'c = 0.33 28 = 1.74MPa > 0.48MPa

10> karBRgayEdkkg ¬cMgayBIcugTMenr¦
EdkkgcMnYnmYymancMgay 50mm = 50mm
EdkkgcMnYndb;mancMgay120mm = 1200mm
EdkkgcMnYnbImancMgay175mm = 525mm
EdkkgcMnYnbI;mancMgay 200mm = 600mm

srub = 2375mm
dUcenHenAsl;cMgay 125mm BIépÞémmuxTMr.


T.Chhay NPIC

11> Ggát;rgkarBt;eRCA
Ggát;rgkarBt;KYrRtUv)anKNnaCaFñwmeRCARbsinebIpleFobén clear span ln ¬Edlvas;BIépÞmuxTl;Kña
rbs;TMr rUbTI 8>21¦ elIkMBs;srub h mantMéltUcCag 4 (ACI Code, Section 11.8) . Ggát;KYrrgnUvbnÞúk
enAelIépÞEdlQmnwgépÞénTMr EdlGaceFVIeGay strut rgkarsgát;GacbegáIteLIgenAcenøaHbnÞúk nigTMr ¬rUbTI
8>22¦. RbsinebIbnÞúkGnuvtþenA)at b¤épÞxagrbs;FñwmeRCA smIkarKNnakMlaMgkat;TTwgsMrab;FñwmFmμta
Edl)aneGayBIxagmuxKYrRtUv)aneRbI. Ca]TahrN_ FñwmeRCAKWCaFñwmElVgxøIEdlRTbnþúkF¶n;/ CBa¢aMgbBaÄreRkam
bnÞúkTMnajEpndI (gravity load), shear wall, nigkMralxNÐ½rgnUvbnÞúkedk.
niymn½yrbs;Ggát;rgkarBt;eRCAk¾RtUv)anbgðajenAkñúg ACI Code, Section 10.7.1. vabgðajfa
Ggát;rgkarBt;EdlmanpleFob ln / h < 4 nigtMbn;rgbnÞúkcMcMnucsßitenAcMgayBIrdgénkMBs;rbs;Ggát;BIépÞén
TMrRtUv)ancat;TukCaGgát;rgkarBt;eRCA. FñwmEbbenHKYrRtUv)anKNnaedayKitnUvkarBRgay nonlinear énkug
RtaMg nigkarPøat;xag (lateral buckling) rUbTI 8>22 a.

rUbTI 8>22 a bgðajBIkarBRgaykugRtaMgeGLasÞic enARtg;muxkat;kNþalElVgénFñwmeRCA nigrUbTI
8>22 b bgðajBIExSekagkugRtaMgemenAkñúgFñwmeRCAEdlrgbnÞúkenAépÞxagelI (top-load deep beam). ExS
Cab;bgðajBI karBRgaykugRtaMgTaj ÉExSdac;²bgðajBIkarBRgaykugRtaMgsgát;. eRkambnÞúkF¶n; sñameRbH
bBaÄreRTtekItmanenAkñgebtugkúñgTisedAEkgnwgkugRtaMgTajem ehIyesÞIrEtRsbeTAnwgExSKnøgdac; ¬rUbTI
8>22 c ¦. dUcenH eKRtUvkarTaMgEdkedk nigEdkbBaÄredIm,ITb;nwgkugRtaMgem. elIsBIenH EdkrgkarBt;
edaykarTaj (tensile flexural reinforcement) RtUv)andak;enARbEhlmYyPaKR)aMenA)atrbs;FñwmtamKnøg
kugRtaMgTaj ¬rUbTI 8>22b ¦. CaTUeTA karviPaKFñwmeRCAmanlkçN³sμúKsμaj nigGacGnuvtþedayeRbIKMrU truss
b¤edIm,ITTYl)anlT§plkan;EtsuRkiteKeRbIviFI finite element b¤viFIRsedogKña. edIm,IgayRsYlkñúgkarKNna
kMlaMgkat;TTwgénFñwmeRCA eKGacGnuvtþtamCMhanEdl)anerobrab;xageRkam³



1> muxkat;eRKaHfñak; (critical section) ³ RbsinebImuxkat;eRKaHfñak;sMrab;KNnakMlaMgkat;TTwg
enAkñúgFñwmeRCAEdlRTbnÞúkbBaÄrGnuvtþenAépÞxagelIrbs;Fñwm sßitenAcMgay X BIépÞénTMr enaHcM
gay X GacRtUv)ankMNt;dUcxageRkam ¬rUbTI 8>23¦³

T.Chhay NPIC

a. sMrab;FñwmeRCAEdlRTbnÞúkBRgayesμI X = 0.15ln / Edl ln = clear span .
b. sMrab;bnÞúkcMcMnuc X 1 = 0.5a1 ¬TMrxageqVg¦ b¤ X 2 = 0.5a2 ¬TMrxagsþaM¦ rUbTI 8>23/
Edl a1 nig a2 esμInwg shear span Ek,rTMrnImYy². Shear span CacMgayBIbnÞúkcMcMnuc
eTAépÞénTMr.
enAkñúgRKb;krNITaMgGs; cMgay X / X 1 nig X 2 dac;xatminRtUvFMCagkMBs;RbsiT§PaB d .

2> ersIusþg;kMlaMgkat;TTwgGtibrma φVn ³ ersIusþg;kMlaMgkat;TTwgGtibrma φVn sMrab;Ggát;rgkar
Bt;eRCAminKYrmantMélFMCagtMélxageRkam ¬ φ = 0.75 ¦³
sMrab; ldn < 2 / φVn = φ 0.67 f 'c bwd ¬*>#@ a ¦
sMrab; 2 ≤ ldn ≤ 5 / φVn = φ 0.055⎛10 + ldn ⎞ f 'c bwd
⎜ ⎟ ¬*>#@ a ¦
⎝ ⎠
b¤yk φVn = φ 0.83 f 'c bwd ¬*>## ¦
krNITaMgBIr manEcgenAkñúg ACI Code, Section 11.8.3. RbsinebI Vu > φVn enaHeKRtUvtMeLIg
muxkat;ebtug.
3> a. ersIusþg;kMlaMgkat;TTwgrbs;ebtug Vc ³ ersIusþg;kMlaMgkat;Fmμta (nominal shear strength)
Vc rbs;ebtugGacRtUv)anKNnadUcxageRkam³

Vc = 0.17 f 'c bw d ¬*>#$ ¦


Vc enHRsedogKñanwgersIusþg;kMlaMgkat;TTwgebtugsMrab;FñwmFmμta dUcenAkñúgEpñkxagmuxénem
eronenH.
b. mü:agvijeTot eKGaceRbIsmIkarmYyepSgeTotEdlmanTak;Tgnwgm:Um:g;emKuN
nigkMlaMgkat;TTwgemKuNenAmuxkat;eRKaHfñak;
⎛ 2.5M u ⎞ ⎡ ⎛ V d ⎞⎤
Vc = ⎜ 3.5 −
⎜ V d ⎟
⎟ ⎢0.16 f 'c + ⎜17.2 ρ w u ⎟⎥bw d
⎜ M ⎟
¬*>#% ¦
⎝ u ⎠⎣ ⎝ u ⎠⎦
b:uEnþ Vc minKYrelIsBI 0.5 f 'c bwd
tMél (3.5 − 2.5M u / Vu d ) minKYrFMCag 2.5 nigminKYrtUcCag 1. tMélén M u nig Vu
RtUv)anykenARtg;muxkat;KNnaeRKaHfñak;. ersIusþg;kMlaMgkat;FMénsmIkar ¬*>#$ ¦ RtUv)aneRbICamYyKMnitfa
sñameRbHEdltUcesÞIrEtemIlmineXIjGacekItmanenAkñúgFñwmeRCA nigGacGueRKaH)an. sñameRbHGaccab;epþIm
ekItmanenARbEhlmYyPaKbIénbnÞúkemKuN.
4> EdkkMlaMgkat;TTwg³ enAeBlkMlaMgkat;TTwgemKuN Vu > φVc eKRtUvdak;EdkkMlaMgkat;TTwg
Edlcat;Tukfa Vu = φ (Vc + Vs ) b¤ Vs = (Vu − φVc ) / φ . CMhanénkarKNnamandUcxageRkam³
a. kMNt; Vs ³ kMlaMgTb;edayEdkkMlaMgkat;TTwg Vs RtUv)ankMNt;BIsmIkarxageRkam³
⎡ A ⎛ 1 + l n / d ⎞ Avh ⎛ 11 − l n / d ⎞⎤
Vs = ⎢ v ⎜ ⎟+ ⎜ ⎟⎥ f y d ¬*>#^ ¦
S⎣ v ⎝
12 ⎠ S h ⎝ 12 ⎠⎦
Edl Av = RkLaépÞsrubénEdkkMlaMgkat;TTwgbBaÄrEdlmanKMlat S v ehIyEkgeTAnwgEdk
emrgkarTajedaykarBt;énépÞxagTaMgBIrrbs;Fñwm
Avh = RkLaépÞsrubénEdkkMlaMgkat;TTwgedkEdlmanKMlat S h RsbnwgEdkemrgkar

TajedaykarBt;énépÞxagTaMgBIrrbs;Fñwm
b. KMlatEdkkMlaMgkat;TTwgKW³

KMlatQrGtibrma S v ≤ d ≤ 300mm
5
KMlatedkGtibrma S h ≤ d ≤ 300mm
5
c. EdkkMlaMgkat;TTwgGb,brma³ RkLaépÞEdkkMlaMgkat;TTwgbBaÄrKW Av = 0.0025bw S v .

RkLaépÞEdkkMlaMgkat;TTwgedkKW Avh = 0.0015bw S h .
d. EdkkMlaMgkat;TTwgRtUvkarenARtg;muxkat;eRKaHfñak;KYrRtUv)anlatsn§wgeBjRbEvg nigkMBs;

rbs;FñwmeRCA.
e. sMrab;FñwmeRCACab; EdkkMlaMgkat;TTwgdUcKñaKYrRtUv)aneRbIenARKb;ElVg RbsinebIElVgTaMgenaH

manRbEvgesμIKñaCamYybnÞúkRsedogKña.

T.Chhay NPIC

5> EdkrgkarBt;énFñwmeRCA³ dMeNIrRbRBwtþeTAénkarBt;rbs;FñwmeRCAKWmanlkçN³sμúKsμaj nigTam
TarkarviPaKkugRtaMg nigbMErbMrYlrageFobtamlkçN³ nonlinear tamkMBs;rbs;Fñwm. sMrab;kar
KNnadMbUg viFId¾sMrYlxageRkamGacRtUv)aneRbI³
φM n = φAs f y ( y )
Edl y = édXñas; = (d − a / 2) . edaysartMél (d − a / 2) mankarBI)akkñgkarKNna/ éd
Xñas; y GacRtUv)anKNnaRbEhlykesμInwg 0.6h sMrab; ln / h = 1 nigmantMél
esμInwg 0.8h sMrab; ln / h = 2 . viFanéRtzan (Linear interpolation) GacRtUv)an
eRbIedIm,IKNna y enAeBl ln / h ERbRbYlcenøaH 1.0 nig 2.0 . dUcenH³
M
As = u
φyf
¬*>#&¦
y

tMélén As minGactUcCagEdkrgkarBt;Gb,brmaEdlRtUvkarsMrab;FñwmFmμtaEdlnwgeGayenA
eBlbnÞab; edaysnμt; d = 0.9h .
As ¬Gb,brma¦ = ¬*>#*¦
0.25 f 'c 1.38
bw d ≥ bw d
f y f y

GgÁTIBIrlub enAeBlEdl f 'c < 30MPa . cMNaMfa f y nig f 'c KitCa MPa .
EdkrgkarTajedaykarBt; (flexural tension reinforcement) KYrdak;enA h / 4 eTA h / 5 én
Fñwm nigKYrmanKMlatRKb;RKan;tambeNþay)attMbn;Taj. EdkrgkarTajKYrEtf<k;eTAkñúgTMr
eGay)anl¥.
sMrab;karviPaK nigkarKNnaEdlmanlkçN³suRkwt nigsMrab;FñwmeRCACab; viFI nonlinear Edl
manlkçN³hμt;ct;GacRtUv)aneRbIedIm,IbrimaNd¾RtwmRtUv nigkarBRgayénEdkrgkarTaj.
]TahrN_5³ FñwmeRCATMrsamBaØmanElVgRbEvg 4.2m man clear span RbEvg ln = 3.6m kMBs;srub
h = 2.5m nigTTwg b = 0.4mm . FñwmeRCARTedaybnÞúkeFVIkarefBRgayesμI 600kN / m ¬rYmbBa©ÚlbnÞúkpÞal;

xøÜn¦ nigbnÞúkGefr 320kN / m enAelIEpñkxagelIénFñwm. KNnasésrEdkrgkarBt; nigEdkkMlaMgkat;TTwg
sMrab;FñwmenH edayeRbI f 'c = 28MPa nig f y = 400MPa ¬rUbTI8>24¦.
dMeNaHRsay³
1> KNnasMrab;EdkTb;nwgm:Um:g;
Wu = 1.2WD + 1.6WL = 1.2 × 600 + 1.6 × 320 = 1232kN / m
Wu L2 1232 × 4.2 2
Mu = = = 2716.56kN .m
8 8
l n 3 .6
= = 1.44
h 2 .5


kMNt;édXñas;/ y . sMrab; ln / h = 1 / y = 0.6d nigsMrab; ln / h = 2 / y = 0.8d dUcenHsMrab;
l n / h = 1.44 / y = 0.688d ¬eday interpolation ¦ = 0.688 × 0.9 × 2.5 = 1.55m ¬edaysnμt;

d = 0 .9 h ¦
Mu 2716.56 ⋅10 6
As = = = 4868mm 2
φyf y 0.9 × 1550 × 400
edaysar f 'c < 30MPa
As ¬Gb,brma¦ =
1.38 1.38
bw d = 400 × 2250 = 3105mm 2
f y 400

dUcenH As = 4868mm 2 lb;. eRbI 10DB25 ¬ 4909mm 2 ¦EdlmanR)aMedImenAépÞmYy² edayBRgay
kñúgkMBs; h / 5 = 500mm EdlCatMbn;TajénFñwm. KMlatrbs;Edk = 500 / 5 = 100mm . EdkTaMgenH
KYrbgáb;cUleTAkñúgTMreGay)anl¥.
2> KNnasMrab;kMlaMgkat;TTwg
k> KNna Vu nig M u enAcMgay x = 0.15ln = d BImuxépÞénTMr
0.15l n = 0.15 × 3.6 = 0.54m < 2.25m

KNna Vu = 1232 ×
3.6
2
− 1232 × 0.54 = 1552.3kN

0.54 2
M u = 1232 × 3.6 × 0.54 − 1232 × = 1017.9kN .m
2
Mu 1017.9
= = 0.29
Vu d 1552.3 × 2.25
x> KNna Vc ³
Mu
3 .5 − 2 .5 = 3.5 − 2.5(0.29 ) = 2.775 > 2.5
Vu d
dUcenH eRbI 2.5 . enAkñúgkrNIenH KNna M u / Vu d edIm,IeRbIkñgkarKNna Vc ³
2.5 = 3.5 − 2.5M u /(Vu d )
Mu
Vu d
= 0 .4 b¤
Vu
Mud
= 2 .5

A 4909
ρw = s = = 0.00545
bw d 400 × 2250

[ ]
Vc = 2.5 0.16 28 + (17.2 × 0.00545 × 2.5) 400 × 2250 × 10 −3 = 2432.2kN
Vc ≤ 0.5 f 'c bw d = 0.5 28 × 400 × 2250 = 2381.2kN

dUcenH Vc = 2381.2kN lb;. φVc = 1785.9kN


T.Chhay NPIC

K> KNna Vs = (Vu − φVc ) / φ edaysar φVc = 1785.9kN > 1552.3kN enaH Vs = 0
dUcenHeKRtUvkarEdkkMlaMgkat;TTwgGb,brma.
X> KNnaEdkkMlaMgkat;TTwg³
edaysnμt;eRbIEdk DB12 sMrab;dak;enAépÞsgçagTaMgtamTisedAedk nigTisedAQrenaH
π
Av = Avh = 2 × 12 2 = 226mm 2
4
KMlatGtibrmaGnuBaØatrbs;EdkQr nigEdkedk
S v = S vh =
d 2250
5
=
5
= 450mm > 300mm yk Sv = S nh = 300mm
EdkQrGb,brma
Av ¬Gb,brma¦= 0.0025 × 400 × 300 = 300mm 2 > 226mm 2

bBa©úHKMlatEdkkgbBaÄr Sv = 400 × 0.0025 = 226mm
226

EdkedkGb,brma
Avh ¬Gb,brma¦= 0.0015 × 400 × 300 = 180mm 2 < 226mm 2

dUcenH eRbI DB12 @ 200 sMrab;TisbBaÄr
nig DB12 @ 300 sMrab;Tisedk
3> RbsinebIeyIgeRbI Vc = 0.17 f 'c bwd enaH Vc = 0.17 28 × 400 × 2250 ×10 −3 = 809.6kN nig
φVc = 607.2kN < 1552.3kN . dUcenH eKRtUvkarEdkkMlaMgkat;TTwg.
1552.3 − 607.2
Vs = = 1260.1kN
0.75
edaysnμt;eRbIEdk DB12 sMrab;dak;enAépÞsgçagTaMgtamTisedAedk nigTisedAQrenaH
π
Av = Avh = 2 × 12 2 = 226mm 2
4
edaysnμt;faKMlatrbs;EdkTaMgBIrTisedAesμInwg Sv = S h = S nig ln / d = 3.6 / 2.25 = 1.6
⎡A ⎛ 1 + l n / d ⎞ Avh ⎛ 11 − l n / d ⎞⎤
Vs = ⎢ v ⎜ ⎟+ ⎜ ⎟⎥ f y d
⎣ Sv ⎝ 12 ⎠ S h ⎝ 12 ⎠⎦
⎡ 226 ⎛ 1 + 1.6 ⎞ 226 ⎛ 11 − 1.6 ⎞⎤
1260100 = ⎢ ⎜ ⎟+ ⎜ ⎟⎥ 400 × 2250
⎣ S ⎝ 12 ⎠ S ⎝ 12 ⎠⎦
S = 161mm yk S = 150mm EdltUcCag Sv ¬Gtibrma¦ nig S h ¬Gtibrma¦. eRbI S = 150mm sM
rab;TaMgKMlatedk nigKMlatQr.
Av ¬Gb,brma¦= 0.0025 × 400 × 150 = 150mm 2 < 226mm 2

Avh ¬Gb,brma¦= 0.0015 × 400 × 150 = 90mm 2 < 226mm 2


dUcenH eRbI DB12 @150mm enAelIépÞTaMgBIrTaMgTisedAedk nigTisedAQr. sMNaj;EdkpSarGac
RtUv)aneRbIedIm,ICMnYskarBRgayEdkEdlCadMeNaHRsaymYyEdlmanlkçN³snSMsMécCag. kar
BRgayEdkenAkñúgmuxkat;ebtugRtUv)anbgðajenAkñúgrUbTI 8>24.

]TahrN_6³ FñwmeRCA strut and tie
FñwmeRCATMrsamBaØEdlman clear span = 3.6m kMBs;srub = 1.8m nigTTwg = 450mm . FñwmenHRT
ssrkaerEdlmanRCug = 450mm enAkNþalElVgEdlRTnUvbnÞúkefr = 1335kN nigbnÞúkGefr = 1070kN .
KNnaFñwmenHedayeRbIviFI strut and tie. eKeGay f 'c = 28MPa nig f y = 400MPa ¬rUbTI8>25¦.


T.Chhay NPIC

dMeNaHRsay³
1> KNnabnÞúkemKuN ¬rUbTI 8>25¦
TMgn;Fñwm = 4.5 ×1.8 × 0.45 × 25 = 91.1kN
edaysarTMgn;pÞal;rbs;FñwmmantMéltUcebIeFobCamYybnÞúkcMcMnucenAkNþalElVg dUcenHbEnßmvaeTA
kñúgbnÞúkGefrcMcMnucEdlmanGMeBIrenAkNþalElVg
Pu = 1.2 D + 1.6 L = 1.2(1335 + 91.1) + 1.6 × 1070 = 3423.32kN

R A = RB = 1711.66kN
2> epÞógpÞat;faetIFñwmenHeRCAtamkarEcgrbs; ACI Code, Section 11.8: clear span ln = 3.6m nig
h = 1.8m ehIy l n / d = 2 < 4 dUcenH FñwmenHCaFñwmeRCA.

3> KNnaersIusþg;kMlaMgkat;TTwgGtibrmaénmuxkat;Fñwm³
yk Vu enARtg; A = R A = 1771.66kN nigsnμt;yk d = 0.9h = 0.9 ×1.8 = 1.62m
Vn = 0.83 f 'c bw d = 0.83 28 × 450 × 1620 × 10 −3 = 2743.1kN

φVn = 0.75 × 2743.1 = 2057.3kN > Vu ¬OK¦
4> eRCIserIsKMrU truss
eRCIserIsKMrU truss RtIekaN. snμt;facMnuc node eFVIGMeBIsßitenAGkS½TMr nigenAcMgay 150mm BI
EKm)at b¤EKmkMBUlFñwm ¬rUbTI8>26¦. KMrU strut and tie pÁúMeLIgedayGgát; tie AB mYynigGgát;
strut BIr AD nig DB . dUcKña RbtikmμenARtg;cMnuc A nigcMnuc B nigbnÞúk Pu Rtg;cMnuc D tMNagCa

strut bBaÄr.

RbEvg strut Ggát;RTUg AD = 1.52 + 2.0252 = 2.52m
yk θ CamMurvagGgát; strut nigGgát; tie enaH
tan θ =
1.5
2.025
= 0.7407 enaH θ = 36.5o > 26o ¬OK¦
5> KNnakMlaMgenAkñúgGgát; truss ³
kMlaMgsgát;enAkñúgGgát; strut AD = FAD = FBD = 1711.66 21..52 = 2875.6kN
5
kMlaMgTajenAkñgGgát; tie AB = FAB = 2875.6 22..025 = 2311kN
52
6> KNnaersIusþg;RbsiT§PaB f ce . snμt;EdkTb; (confining reinforcement) RtUv)andak;edIm,IkarBar
kMlaMgbMEbk (splitting force). Ggát; strut AD nig DB tMNageGayGgát;rgkarsgát;ragdb
(bottle-shape compression member) dUcenH β s = 0.75 .

f ce = 0.85β s f 'c = 0.85 × 0.75 × 28 = 17.85MPa


Ggát; strut bBaÄrenAcMnuc A / B nig D manmuxkat;esμI (uniform section) dUcenH β s = 1.0
f ce = 0.85β s f 'c = 0.85 × 1× 28 = 23.8MPa
tMbn;cMnuc (nodal zone) D mankMlaMg C − C − C dUcenH β s = 1.0 . ersIusþg;RbsiT§PaBenA nodal
zone D KW

f ce = 0.85β s f 'c = 0.85 × 1× 28 = 23.8MPa
edaysarEtGgát; strut AD nig DB P¢ab;eTAcMnucepSgeTot enaH f ce = 17.85MPa lb;elI nodal
zone TaMgGs;.

7> KNna nodal zone
k> KNna nodal zone enAcMnuc A ³ snμt;fakMlaMgén nodal zone mankugRtaMgdUcKñaKW 17.85MPa
ehIyépÞEkgeTAnwgkMlaMgEdlRtUvKña
φFn ≥ Fu b¤ φf ce Acs ≥ Fu
Edl φ = 0.75 sMrab; strut, tie nig node.
RbEvgénépÞedk ab ¬rUbTI8>27 a¦ esμInwg
Fu /(φf ce b) = 1711.66 ⋅10 3 /(0.75 × 17.85 × 450) = 284mm

BIFrNImaRt RbEvg ac = 284
2311
1711.66
= 383mm

dUcKña RbEvg bc = 284
2875.6
1711.66
= 477mm


T.Chhay NPIC

TIRbCMuTMgn;rbs; nodal zone sßitenA 383 / 2 = 191.5mm BI)aténFñwm ¬eyIg)ansnμt; 150mm ¦
x> KNna nodal zone enAcMnuc D ¬rUbTI8>27 b¦³
RbEvgénépÞedk de = 3423.32 ⋅103 /(0.75 ×17.85 × 450) = 568mm
RbEvgén df = ef = 568 3423.32 = 477mm
2875.6

RbEvgrbs; fg = 477 2 − ( 568 ) 2 = 383mm
2
dUcenH TIRbCMuTMgn;én nodal zone sßitenA 383 / 3 = 128mm BIépÞxagelIrbs;Fñwm¬eyIg)ansnμt;
150mm ¦

8> KNnaEdkQr nigEdkedk³
k> EdkQr³ mMurvagEdkQr nigGgát; strut KW 53.5o BIrUbTI 8>27 a. eRbIEdk DB16 EdlmanKMlat
π
300mm / As = 2 × 16 2 = 402mm 2 ¬eCIgBIr¦/ sin 53.5o = 0.804
4
( Asi / bs S ) sin γ i = 402 /(450 × 300) × 0.804 = 0.0024

x> Edkedk³ mMurvagEdkedk nigGgát; strut KW 36.5o BIrUbTI 8>27 a. eRbIEdk DB16 EdlmanKMlat
π
300mm / As = 2 × 16 2 = 402mm 2 ¬eCIgBIr¦/ sin 36.5o = 0.595
4
( Asi / bs S ) sin γ i = 402 /(450 × 300) × 0.595 = 0.0018
K> ( Asi / bs S ) sin γ i ¬srub¦ = 0.0024 + 0.0018 = 0.0042 > 0.003 ¬OK¦
9> KNnaGgát; tie edk AB ³
k> KNna As ³
Fu = φAs f y As = 2311 ⋅10 3 / (0.75 × 400) = 7703mm 2

eRbI 12DB30 ¬ As = 8482mm 2 ¦ dak;CabIbICYrdUcbgðajkñúgrUbTI 8>27 c.
x> KNnaRbEvgf<k; (anchorage length)³ RbEvgf<k;RtUv)anvas;BIcMnuccab;BI nodal zone
¬rUbTI8>28¦.
tan 36.5o = 190 / x

x = 257 mm
RbEvgTMBk;Edlman = 257 + 142 + 225 − 35 ¬kMras;ebtugkarBarEdk¦ = 589mm
RbEvgTMBk;caM)ac;sMrab;Edk DB30 KW 47.5 × 30 = 1425mm > 589mm
dUcenHeRbITMBk; 90o cgP¢ab;CamYyEdkssr
( )
l dh = 0.02ψ e λf y d b / f 'c


ψ e = λ = 1.0 d b = 30

l dh = (0.24 × 400 )30 / 28 = 544mm < 589mm


T.Chhay NPIC

]TahrN_7³ eKmanFñwmTMrsamBaØmYyEdlmanRbEvg 6m ¬ clear span ¦ rgnUvbnÞúkBRgayefr
47.5kN / m nigbnÞúkBRgayGefr 25kN / m . FñwmenHmanmuxkat; b = 35cm nig d = 55cm .

FñwmenHBRgwgedayEdk 4DB25 BRgaymYyCYr. cUrkMNt;nUvmuxkat;caM)ac;sMrab;kMlaMgkat;TTwg. smμtikmμ³
f ' = 28MPa nig f = 280MPa .
c yt


dMeNaHRsay³
k> bnÞúkemKuN ¬bnÞúkKNna¦
1.2 D + 1.6 L = 1.2 × 47.5 + 1.6 × 25 = 97 kN / m
x> kMlaMgkat;TTwgKNnaenARtg;muxxagssr
6
Vu = 97 × = 291kN
2
K> kMlaMgkat;TTwgKNnaenAcMgay d BIépÞxagssr
Vud = 291 − (0.55 × 97) = 237.65kN
X> ersIusþg;kMlaMgkat;TTwgEdlTb;edayebtug
Vc = 0.17 f 'c bd = 0.17 28 × 350 × 550 = 173.2kN

φVc = 130kN
1
φVc = 65kN
2
g> kMlaMgkat;TTwgEdlEdkRtUvTb;
Vu − φVc 237.65 − (0.75 × 173.2)
Vs = = = 143.5kN
φ 0.75
c> cMgayBImuxépÞssrmk 1
2
φVc = 65kN
291 − 65
x' = × 3 = 2.33m
291
q> KNnaEdkkg
1> eRCIserIsEdk RB10 ¬EdkkgmaneCIgBIr¦
⇒ Av = 2 × 78.5mm 2 = 157 mm 2
A f d 157 × 280 × 550
⇒ s1 = v yt = = 168.5mm < 600mmm
Vs 143.5 × 103
dUcenHykKMlat 160mm RtYtBinitüKMlatGtibrma
d 550
s2 = = = 275mm
2 2
3A f 3 × 157 × 280
s3 = v yt = = 376.8mm
b 350
s1 < s2 < s3

2> RtYtBinitüKMlatGtibrmaesμI d
4
Vc1 = 0.33 f 'c bd = 0.33 × 28 × 350 × 550 = 336.1kN

Vc 2 = 2Vc1 = 2 × 336.1kN = 672.2kN


T.Chhay NPIC

eday V < V dUcenHKMlatEdkGtibrmakMritRtwm s = d = 275mm
s c1
2
max

C> kMlaMgkat;TTwgEdlTb;edayEdksMrab;KMlat s = 2 = 275mm
d
max

Av f yt d 157 × 280 × 550
Vs ( for smax = 275mm) = = = 87.9kN
smax 275

φVs = 0.75 × 87.9kN = 65.94kN
cMgayBImuxTMr eTAdl;EdkkgEdlmanKMlat s max = 275mm
291 − (130 + 65.94)
x1 = × 3 = 0.98m
291
dUcenH sMrab; 0.98m BImuxénTMr eRbIEdkkg RB10 KMlat 160mm nigsMrab;EpñkenAsl; eRbIEdkkg
Gb,brma ¬KMlatGtibrma¦
Q> karBRgayEdkkg
Edkkg1 manKMlat 2 = 80mm
s

Edkkg6 manKMlat 160mm = 960mm
srub 1040mm = 1.04m > 0.98m
Edkkg6 manKMlat 270mm = 1620mm
srub 2660mm = 2.66m < 3m
nigEdkkgcugeRkay (3 − 2.66) = 0.34m
srubEdkkgTaMgGs;EdlRtUveRbIsMrab;RbEvgFñwm 6m man 28kg.


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