Iii flexural analysis of reinforced concrete

Department of Civil Engineering viTüasßanCatibec©keTskm<úCa

III. viPaKFñwmebtugGarem:rgkarBt;begáag
Flexural Analysis of Reinforced Concrete Beam

1> karsnμt; Assumption

ebtugGarem:CasMPar³minEmnsac;mYy BIeRBaHvaekIteLIgedaysMPar³BIrRbePTKW ebtug nigEdk. dUc
enHeRKOgbgÁúMebtugGarem:EdlkMNt;edayersIusþg;cugeRkay RtUvkMNt;tamkarsnμt;xageRkam³
- bMErbMrYlrageFobrbs;ebtug nigEdkRtUvEtmantMéldUcKña mann½yfaPaBs¥itrvagebtug nigEdkman
tMélRKb;RKan;.
- bMErbMrYlrageFobrbs;ebtugRtUvEt smamaRteTAnwgcMgayBIGkS½NWt
- m:UDuleGLasÞicrbs;Edk RtUvEtyk E = 2 ×10 MPa . kugRtaMgrbs;EdkkñúgtMbn;eGLasÞicRtUvEt
s
5

mantMélesμIplKuNrvag bMErbMrYlrageFobCamYynwgm:UDuleGLasÞic.
- muxkat;Rtg;enAEtRtg;eRkayeBlrgkarBt;
- ersIusþg;Tajrbs;ebtugRtUv)anecal BIeRBaH ersIusþg;TajebtugmantMéltUcCagersIusþg;sgát;dl;
eTA 10 dg ehIysñameRbHrbs;ebtugRtUvsnμt;faKμanT§iBl nigmü:ageTotmuneBleRbH muxkat;ebtugTaMgmUl
manRbsiT§PaBkñúgkarTb;nwgm:Um:g;xageRkA.
- sac;lUteFobGtibrmarbs;ebtugkMritRtwm 0.003
- rUbragénkarBRgaykugRtaMgsgát;rbs;ebtug snμt;manragctuekaNEkg
2> RbePTénkar)ak;edaykarBt; nigEdnkMNt;sac;lUteFob
k> kar)ak;edaykarBt;
eRKOgbgÁúMrgkarBt; Gac)ak;edaybIkrNIGaRs½yeTAnwgPaKryEdkEdl)andak;enAkñúgmuxkat;ebtug³
- EdkGaceTAdl;cMnucyarmunebtugeFVIkardl;ersIusþg;Gtibrma. kñúgkrNIenH kar)ak;bNþalmkBI

sac;lUteFobrbs;EdkmantMélFMCagb¤esμI 0.005 . muxkat;manbrimaNEdktic ehIyRtUv)aneKeGayeQμaHfa
muxkat;rgkarTaj tension-controlled section .
- EdkGaceTAdl;cMnucyarGMLúgeBlEdlebtugeFVIkardl;ersIusþg;GtibrmaEdr. muxkat;RtUv)aneK

eGayeQμaHfa muxkat; balanced section .
- ebtugGacEbkmuneBlEdlEdkeFVIkardl;cMnucyar bNþalmkBIPaKryEdkeRcInenAkñúgmuxkat;. kñúg

krNIenHebtug)aneFVIkardl;ersIusþg;Gtibrma ehIymansac;lUteFobGtibrma 0.003 Edr b:uEnþkugRtaMgrbs;

Flexural Analysis of Reinforced Concrete Beam 18

T.Chhay NPIC

EdkmantMélticCagersIusþg;KNna Edl f < f . sac;lUteFobrbs;EdkmantMéltUcCagb¤esμI 0.002 .
s y

muxkat; enHRtUv)aneKeGayeQμaHfa muxkat;rgkarsgát; compression-controlled section.
eK)ansnμt;faebtugEbkedaysarkMlaMgsgát; enAeBlEdlsac;lUteFobrbs;ebtugmantMél
0.003 Etkñúgkar BiesaFn_tMélenHERbRbYlBI 0.0025 → 0.004 .

kñúgkarKNnaFñwm eKeRCIserIsykmuxkat;rgkarTaj edayeGayEdkeFVIkardl;ersIusþg;KNna muneb
tugEbk. sñameRbHrbs;ebtugrIkFMeLIg² EdlCasBaØaRbkasGasnñmuneBlebtugEbk ehIyrcnasm<n§½)ak;
Ebk.
kñúgkarKNnaFñwm edayeRCIserIsykmuxkat;rgkarsgát; nigbalanced section ebtugEbkPøam²
rcnasm<n§½ )ak;EbkmYyrMBicedayKμankarRbkasGasnñ. kareRCIserIsmuxkat;EbbenHRtUv)aneCosvag.
x> EdnkMNt;sac;lUteFobsMrab;muxkarrgkarTaj nigrgkarsgát;
karKNnapþl;eGaycMeBaHkarKNnaebtugGarem:sMrab;muxkat;rgkarTaj b¤sgát;. muxkat;TaMgBIr
RtUv)ankMNt;edaysac;lUteFobsuT§ net tension strain (NTS). elIsBIenHlkçxNÐBIreTot)anekItKW
lkçxNÐbMErbMrYlrageFobtulüPaB balanced strain condition niglkçxNÐkñúgtMbn; transition region
condition. lkçxNÐTaMgbYnenHRtUv)ankMNt;dUcxageRkam³

- muxkat;rgkarsgát; Camuxkat;Edlsac;lUteFobsuT§ (NTS) sMrab;EdkrgkarTajEpñkxageRkAbMput
mantMél tUcCagbMErbMrYlrageFobrgkarsgát; enAeBlEdlbMErbMrYlrageFobrbs;ebtugrgkarsgát;mantMél
esμI 0.003 . krNIenHekIteLIgCaTUeTAcMeBaH ssrEdlrgbnÞúktamGkS½ nigm:Um:g;.
- muxkat;rgkarTaj Camuxkat;Edlsac;lUteFob (NTS) sMrab;EdkrgkarTajEpñkxageRkAbMputman
tMélFMCag b¤esμI 0.005 kñúgkrNIEdlebtugmanbMErbMrYlrageFobdl;EdnkMNt; 0.003 .
- muxkat;Edlsac;lUteFoblUteFob (NTS) sMrab;EdkrgkarTajEpñkxageRkAbMputmantMélsßitenA
cenøaHmuxkat;rgkarsgát; nigmuxkat;rgkarTaj KWenAcenøaH 0.002 → 0.005 Camuxkat; transition region
- lkçxNÐbMErbMrYlrageFobtulüPaB ekItmanenAkñúgmuxkat; enAeBlEdlbMErbMrYlrageFobEdktMbn;
TajmantMélesμI ε = E kñúgxN³Edlebtugrgkarsgát;manbMErbMrYlrageFobmantMélesμI 0.003 .
f
s
y

s

Section condition Concrete strain Steel strain Note ( f y = 400MPa)
Compression-controlled 0.003 ε t ≤ f y Es ε t ≤ 0.002
Tension-controlled 0.003 ε t ≥ 0.005 ε t ≥ 0.005
Transition region 0.003 f y E s ≤ ε t ≤ 0.005 0.002 ≤ ε t ≤ 0.005
Balanced strain 0.003 ε s = f y Es ε s = 0.002
Transition region 0.003 0.004 ≤ ε t < 0.005 0.004 ≤ ε t < 0.005

viPaKFñwmebtugGarem:rgkarBt;begáag 19


>
>


T.Chhay NPIC

3> emKuNbnÞúk
bnÞúkEdlmanGMeBIelIeRKOgbgÁúMRtUv)anKuNCamYynwgemKuNbnÞúk edIm,IkarBarkar)ak;Pøam² nigpþl;
nUvkarKNnamYyEdlmanlkçN³esdækic©. emKuNbnÞúkGaRs½ynwgRbePTbnÞúk nigkarbnSMbnÞúk. emKuNbnÞúk
sMrab;bnÞúkGefr KW 1.6 ÉemKuNbnÞúksMrab;bnÞúkefr KW 1.2 . dUcenHkarbnSMbnÞúksMrab;bnÞúkGefr nigbnÞúkefrKW
U = 1 .2 D + 1 .6 L
Edl U - bnÞúkKNnacugeRkay
L - bnÞúkGefr

D - bnÞúkefr

4> emKuNkat;bnßyersIusþg;
emKuNkat;bnßyersIusþg; φ mantMéltUcCag 1 . emKuNkat;bnßyersIusþg;GaRs½ynwgRbePTén
eRKOgbgÁúM³
- sMrab;muxkat;rgkarTaj φ = 0.90
- sMrab;muxkat;rgkarsgát;
k> CamYyEdkkgvNÐ φ = 0.70
x> CamYyEdkkgdac;² φ = 0.65
- sMrab;ebtugsuT§ φ = 0.55
- sMrab;kMlaMgkat; nigkMlaMgrmYl φ = 0.75
- sMrab;RTnab;enAelIebtug φ = 0.65
- sMrab;KMrU strut and tie φ = 0.75

5> karEbgEckkugRtaMgsgát;smmUl
kugRtaMgEbgEckkñúgebtugrgkarsgát;enAxN³eBl)ak;RtUv)ansnμt;famanragctuekaNEkg
ctuekaNBñay ExSekag)a:ra:bUl b¤ragNamYyepSgeTotGaRs½yedaykaryl;RBmKñaenAeBleFVIBiesaFn_.
enAeBlEdlFñwmerobnwg)ak; srésEdk)aneFVIkardl;cMnucyarmun RbsinebImuxkat;enaHmanbrimaN
Edktic under-reinforced section ehIykñúgkrNIenHsrésEdkeFVIkardl;kgRtaMgKNna. RbsinebImuxkat;
u
manEdkeRcIn ebtugEbkmun ehIybMErbMrYlrageFobRtUv)ansnμt;faesμI 0.003 .
kMlaMgsgát; C ekItmanenAkñúgtMbn;sgát; ehIykMlaMgTaj T ekItmanenAtMbn;TajEdlsßitenAelInIv:U
Edk. eKsÁal;TItaMgénkMlaMg T BIeRBaHvamanGMeBIRtYtsIuKñanwgGkS½TIRbCMuTMgn;rbs;Edk. ÉTItaMgrbs;kMlaMg

C eKGacsÁal;)an luHNaEteKsÁal;maDéntMbn;sgát; ehIyeBlenaHeKGackMNt;)annUvTItaMgTIRbCMuTMgn; )an.
RbsinebIeKsÁal;TItaMgrbs;kMlaMgTaMgBIr enaHeKGackMNt;nUvRbEvgédXñas; EdlCacMgayBIkMlaMgTaj
mkkMlaMgsgát;.

RbsinebIebtugEbk enAbMErbMrYlrag ε ' = 0.003 ehIyRbsinebIEdkeFVIkardl;cMnucyar f = f enaH
c s y

muxkat;Camuxkat; balanced section. kMlaMgsgát; C RtUv)ansMEdgedaymaDénbøúkkugRtaMg
EdlmanragminÉksNæan ekItmanelIépÞctuekaNqñÚt bc .
maDénkugRtaMgsgát;snμt;esμI C = bc(α f ' )
1 c

Edl α f ' CakugRtaMgmFüménbøúkkugRtaMgminÉksNæan
1 c

TItaMgrbs;kMlaMgsgát; KWmancMgay z BIsrésEpñkxagelIbMputEdlGaccat;TukCaEpñkéncMgay c ¬cM
gayBIsrésEpñkxagelImkGkS½NWt¦.
z = α 2c
α1 = 0.72 sMrab;ebtugEdlmanersIusþg; f 'c ≤ 28MPa .
α RtUv)ankat;bnßyeday 0.04 ral; 7 MPa sMrab;ebtugEdlmanersIusþg; f ' c > 28MPa .
1

α1 = 0.425 sMrab;ebtugEdlmanersIusþg; f ' ≤ 27.6MPa .
c

α RtUv)ankat;bnßyeday 0.025 ral; 7 MPa sMrab;ebtugEdlmanersIusþg; f ' c > 28MPa .
1

edIm,IsMrYldl;karKNnakMlaMgkñúgénmuxkat; ACI code )anyknUvkugRtaMgEbgEckkñúgmuxkat;rag
ctuekaNEkg EdlmantMél 0.85 f ' BRgayesμIelItMbn;sgát;smmUl EdlxNÐ½edaybnÞat;RsbnwgGkS½NWt
c

EdlmanRbEvg a = β c . 1

β = 0.85 sMrab;ebtugEdlmanersIusþg; f ' ≤ 28MPa .
1 c

f ' −28
β = 0.85 − 0.05(
1
c
) sMrab;ebtugEdlmanersIusþg; 28MPa < f ' ≤ 56MPa .
c
7
β = 0.65 sMrab;ebtugEdlmanersIusþg; f ' > 56MPa .
1 c


T.Chhay NPIC

sMrab;muxkat;ragctuekaNEkg RkLaépÞtMbn;sgát;mantMélesμI ba ehIytMélkugRtaMgBRgayesIμKW
0.85 f ' Edlpþl;nUvmaDkugRtaMgsrubesμInwg 0.85 f ' ab ehIyRtUvKñanwgkMlaMgsgát; C . sMrab;muxkat;epSg
c c

BIragctuekaNEkg kMlaMgsrubesμInwgplKuNRkLaépÞtMbn;sgát;CamYynwg 0.85 f ' . c

6> srésEdkrgkMlaMgTajénmuxkat;ctuekaNEkgrgkarBt;
PaKryEdkenAkñúgmuxkat;ebtugkñúglkçxNÐ balanced RtUv)aneKeGayeQμaHfa balanced steel ratio

ρ EdlCapleFobrvagmuxkat;Edk A nigmuxkat;RbsiT§PaB bd
b s

As
ρb =
bd
Edl - TTwgmuxkat;eRKOgbgÁúMtMbn;sgát;
b

d - cMgayBIsésrEpñkxageRkAbMputmkTIRbCMuTMgn;EdkrgkMlaMgTaj ¬kMBs;RbsiT§PaB¦

smIkarlMnwgBIr EdlCaeKalkarN_kñúgkarviPaK nigKNnaeRKOgbgÁúMehIymantMélRKb;muxkat; nigRKb;
RbePTbnÞúkKW³
- kMlaMgsgát;RtUvmantMélesμIkMlaMgTaj C = T
- ersIusþg;m:Um:g;Bt;xagkñúg M esμIeTAnwgplKuNrvagkMlaMgsgát; b¤kMlaMgTajCamYynwgédXñas;
n

M = C (d − z ) = T (d − z ) nig M = φM Edl φ emKuNkat;bnßyersIusþg;
n u u


kareRbIR)as;nUvsmIkarTaMgenHRtUv)anBnül;sMrab;muxkat;ragctuekaNEkgCamYyEdktMbn;Taj. mux
kat;GacCa muxkat; balanced section muxkat;Edktic muxkat;EdkeRcIn GaRs½yedaykareRbIR)as;nUvPaKry
Edk.
k> muxkat; balanced section

CMh‘anTI1³ BIdüaRkamsac;lUteFob eyIg)an
cb 0.003
=
d − cb fy
Es
c 0.003
⇒ b =
d fy
0.003 +
Es
edayCMnYs E s = 200000MPa
600
⇒ cb = ( )d
600 + f y

CMh‘anTI2³ BIsmIkarlMnwg eyIg)an
C = T ⇒ 0.85 f 'c ab = As f y
As f y
⇒a=
0.85 f 'c b
Edl a - CaRbEvgbøúkrgkarsgát; mantMélesμInwg β c
1 b

edaysarvaCamuxkat; balanced section dUcenHPaKryEdkRtUv)aneRbIKW
As
ρb =
bd
⇒ As = ρ bbd
CMnYs A eTAkñúgsmIkarxagelI
s

⇒ 0.85 f 'c ab = ρ bbdf y


T.Chhay NPIC
0.85 f 'c a 0.85 f 'c
⇒ ρb = = ( β1cb )
f yd f yd

CMnYstMél c b =(
600
600 + f y
)d eTAkñúgsmIkarxagelI eyIg)an
f 'c 600
ρ b = 0.85β1 ( )
f y 600 + f y

CMh‘anTI3³ BIsmIkarlMnwgénm:Um:g;xagkñúg eyIg)an
M n = C (d − z ) = T (d − z )

sMrab;muxkat;ragctuekaNEkg cMgay z = a
2
a a
⇒ M n = C (d − ) = T (d − )
2 2
sMrab;muxkat; balanced section b¤muxkat;EdlmanbrimaNEdktic
T = As f y

dUcenH M = A f (d − a )
n
2
s y

m:Um:g;kñúgxagelIEdl)anKNna RtUvkat;bnßyedayemKuN φ
As f y
⇒ φM n = φAs f y (d − )
1.7 f 'c b
smIkarenH sresredayCab;GBaØti ρ
ρbdf y ρf y
⇒ φM n = φf y ρbd (d − ) = φf y ρbd 2 (1 − )
1.7 f 'c b 1.7 f 'c
eyIgGacsresrsmIkarxagelIenHCa
φM n = Ru bd 2

Edl R = φf ρ (1 − 1.ρff ' )
u
7
y
y

c

pleFobrvagRbEvgbøúkkugRtaMgsgát;smmUl a nig kMBs;RbsiT§PaBénmuxkat; d
a ρf y
=
d 0.85 f 'c

x> PaKryEdkGtibrma
PaKryEdkGtibrma ρ EdlGaceRbIenAkñúgmuxkat;ebtugEdlmanEtEdkrgkMlaMgTaj QrelIeKal
max

karN_sac;lUteFobsuT§enAkñúgEdkrgkMlaMgTaj PaKryEdk balanced nigersIusþg;rbs;Edk.
TMnak;TMngrvagPaKryEdkenAkúñgmuxkat; ρ nigsac;lUteFobsuT§ ε t


fy fy
0.003 + 0.003 +
ρ
=
Es
ρ b 0.003 + ε t
b¤ ε t =(
ρ
Es
) − 0.003

ρb
sMrab; f y = 414MPa nigsnμt; f y / Es = 0.00207

- sMrab;muxkat;rgkMlaMgTaj ⇒ ε ≥ 0.005 snμt; ε = 0.005 ¬b¤ dc ≤ 0.375 ¦
t t
t

d - cMgayBIsésEpñkxageRkAbMput eTAGkS½EdkTajCYrTI1
t

ρ 0.00507
=
ρb 0.008
kñúgkrNIEdl ρ = ρ max

⇒ ρ max = 0.63375ρ b
PaKryEdkenHeFVIeGayFñwmmanlkçN³yWtRKb;RKan;munnwg)ak;
Casegçb³
sMrab; f = 276MPa ⇒ ρ = 0.5474ρ
y max b

f y = 345MPa ⇒ ρ max = 0.5905ρ b

f y = 517 MPa ⇒ ρ max = 0.6983ρ b

sMrab;muxkat;rgkMlaMgTaj φ = 0.9
- sMrab;muxkat;enAkñúgtMbn; transition region snμt; ε t = 0.004 ¬minRtUvtUcCag 0.004 ¦
b¤ 0.6 > d > 0.375
c

ρ 0.00507
=
ρb 0.007
kñúgkrNIEdl ρ = ρ max t

⇒ ρ max t = 0.724 ρ b


T.Chhay NPIC

sMrab;muxkat;enAkñúgtMbn; transition region φ < 0.9
250
⇒ φ = 0.65 + (ε t − 0.002)( )
3
]TahrN_1³ sMrab;muxkat;dUcbgðajkñúgrUb
k> kMNt;muxkat;Edk balanced section
x> muxkat;EdkGtibrmaEdlGnuBaØatieday ACI Code sMrab;muxkat;rgkMlaMgTaj nig sMrab;muxkat;enAkñúg
tMbn; transition region
K> TItaMgGkS½NWt nigRbEvgbøúkkugRtaMgsgát;sMrab;muxkat;rgkMlaMgTaj
smμtikmμ³ f ' = 28MPa nig f = 400MPa
c y

dMeNaHRsay³
k> kMNt;muxkat;Edk balanced section
f 'c 600
ρ b = 0.85β1 ( )
f y 600 + f y

eday f ' = 28MPa
c f y = 400MPa nig β
1 = 0.85
28 600
⇒ ρ b = 0.852 ( ) = 0.030345
400 600 + 400
muxkat;EdkEdldak;kñúgmuxkat;ebtugedIm,I)anlkçxNÐ balanced KW
Asb = ρ bbd = 0.030345 × 40 × 65 = 78.897cm 2
x> muxkat;EdkGtibrmasMrab;muxkat;rgkMlaMgTaj
fy
0.003 +
Es
ρ max = ( ) ρb
0.003 + ε t
sMrab; ε t = 0.005
0.005
⇒ ρ max = ρ b = 0.625ρb = 0.625 × 0.030345 = 0.019
0.008


⇒ As max = ρ b maxbd = 0.019 × 40 × 65 = 49.4cm 2 sMrab; φ = 0.9
muxkat;EdkGtibrmasMrab;muxkat;kñúgtMbn; transition region
fy
0.003 +
Es
ρ max = ( ) ρb
0.003 + ε t
sMrab; ε t = 0.004
0.005
⇒ ρ max = ρ b = 0.714 ρb = 0.714 × 0.030345 = 0.0217
0.007
⇒ As max = ρ b maxbd = 0.0217 × 40 × 65 = 56.42cm 2sMrab; φ = 0.817
K> TItaMgGkS½NWt nigRbEvgbøúkkugRtaMgsgát;sMrab;muxkat;rgkMlaMgTaj
As max f y 49.4 × 400
amax = = = 20.76cm
0.85 f 'c b 0.85 × 28 × 40
cMgayBIsésrEpñkxagelImkGkS½NWtKW
a 20.76
c= = = 24.42cm
β1 0.85
]TahrN_2³ kMNt;ersIusþg;m:Um:g;KNna nigTItaMgGkS½NWténmuxkat;ctuekaNEkgdUcbgðajkñúgrUbxageRkam.
RbsinebIeKeRbIEdk 3DB30 ersIusþg;ebtug f ' = 20MPa nig f = 400MPa c y

dMeNaHRsay³
muxkat;Edk 3DB30 ⇒ A = 21.195cm s
2

PaKryEdkeRbIR)as;kñúgebtug ρ = bd = 30.× 55 = 0.0128
A 21 195 s

PaKryEdk balanced kñúgebtug ρ = 0.85β ff ' ( 600 + f ) = 0.021675
600
b 1
c

y y

× 400
RbEvgbøúkkugRtaMgsgát; a = 0.85 ff ' b = 021.19520 × 30 = 16.62cm
A s

.85 ×
y

c

TItaMgGkS½NWt c = β = 16..85 = 19.55cm
a
0
62
1

fy
0.003 +
sac;lUtEdksuT§ εt = (
ρ
Es
) − 0.003 = 0.0055 > 0.005

ρb
⇒ muxkat;rgkMlaMgTaj ⇒ φ = 0.9
ersIusþg;m:Um:g;xagkñúgKNna
a 16.62
φM n = φAs f y (d − ) = 0.9 × 21.195 × 400 × (55 − ) × 10 −3 = 356.25kN .m
2 2
]TahrN_3³ kMNt;ersIusþg;m:Um:g;KNna nigTItaMgGkS½NWténmuxkat;ctuekaNEkgdUcbgðajkñúgrUbxagelI.

T.Chhay NPIC

EteKeRbIEdk 3DB32 vij ersIusþg;ebtug f ' = 20MPa nig f = 400MPa c y

dMeNaHRsay³
muxkat;Edk 3DB32 ⇒ A = 24.1152cm s
2

PaKryEdkeRbIR)as;kñúgebtug ρ = bd = 24.1152 = 0.0146
A
30 × 55
s

PaKryEdk balanced kñúgebtug ρ = 0.85β ff ' ( 600 + f ) = 0.021675
600
b 1
c

y y

RbEvgbøúkkugRtaMgsgát; a = 0.85 ff ' b = 2485 × 20××400 = 18.91cm
A
0.
.1152
s y

30
c

TItaMgGkS½NWt c = β = 18..85 = 22.25cm
a
0
91
1

fy
0.003 +
sac;lUteFobEdksuT§ εt = (
ρ
Es
) − 0.003 = 0.0044 < 0.005

ρb
⇒ muxkat;enAkñúgtMbn; transition region ⇒ φ = 0.65 + (ε − 0.002)( 250 ) = 0.85
3
t

ersIusþg;m:Um:g;KNna φM = φA f (d − a ) = 0.85 × 24.1152 × 400 × (55 − 16262 ) ×10
n
2
s y
. −3
= 373.43kN .m

sMrab;muxkat;rgkMlaMgTaj ε = 0.005 t

0.005
⇒ ρ max = ρ b = 0.625ρb = 0.625 × 0.021675 = 0.01355
0.008
As max = ρ max bd = 0.01355 × 30 × 55 = 22.3575cm 2 < 24.1153cm 2

RbEvgbøúkkugRtaMgsgát; a = 0.85 ff ' b = 2285 × 20××400 = 17.535cm
A
0.
.3575
30
s y

c

a 17.535
⇒ φM n = φAs f y (d − ) = 0.9 × 22.3575 × 400 × (55 − ) × 10 −3 = 372.11kN .m
2 2
eyIgeXIjfa tMélénersIusþg;mantMélesÞIresμIKña EdleKGacTTYlyk)an.
K> PaKryEdkGb,brma
RbsinebIm:Um:g;Gnuvtþn_mkelIFñwmmantMéltUc ehIyTMhMénmuxkat;FMCagGVIEdlRtUvkarsMrab;Tb;Tl;nwg
m:Um:g; enaHkarKNnanwgbgðajeGayeXIjmuxkat;EdktUc b¤k¾Kμan. RbsinebImindak;sésrEdk Fñwmrgm:Um:g;
nwgkar)ak;Pøam². ACI Code kMNt;nUvmuxkat;EdkGb,brma A s min

b d nig ≥
f' 1.4
A =
s min
c
w b d w
4f y f y

sMrab;krNIFñwmragGkSr T EdlsøabrgkMlaMgTaj enaHmuxkat;EdkRtUvyktMéltUcCageKevagsmIkar
xagelI nigxageRkam

f 'c
As min = bw d
2 fy

Edl bw = b sMrab;muxkat;ragctuekaNEkg
bw CaTTwgsøab
7> muxkat;lμm
muxkat;EdlmanlkçN³lμm RbsinebIersIusþg;m:Um:g;kñúgénmuxkat;FMCag b¤esμIm:Um:g;xageRkA
φM ≥ M . viFIsaRsþGacsegçbdUcxageRkam³
n u

- KNnam:Um:g;xageRkAEdlGnuvtþn_mkelIeRKOgbgÁúM M u

M u = 1.2M D + 1.6M L
- KNna φM sMrab;muxkat;EdlsésrEdkrgkMlaMgTaj
n

+ RtYtBinitüfa ρ < ρ < ρ min max

+ kMNt; a = nigRtYtBinitü ε sMrab; φ
A f s y
t
0.85 f ' b c

kMNt; φM = φA f (d − a )
+ n
2
s y

- RbsinebI φM ≥ M enaHmuxkat;manlkçN³lμm
n u

]TahrN_4³
eKmanFñwmTMrbgáb;mYyEdlmanRbEvg 2.5m .
FñwmenHmanmuxkat;ragctuekaNEkgdUcbgðaj
kñúgrUb. FñwmRTbnÞúkefr EdlrYmmanbnÞúkpÞal;xøÜn
rbs;vasrub 22kN / m nigbnÞúkGefr 13kN / m .
edayeRbI f ' = 28MPa nig f = 400MPa
c y

cUrepÞógpÞat;fa FñwmenHmansuvtßiPaBRKb;RKan;kñúg

T.Chhay NPIC

karRTbnÞúkxagelI
dMeNaHRsay³
bnÞúkKNna
Wu = 1.2 D + 1.6 L = 1.2 × 22 + 1.6 × 13 = 47.2kN / m
m:Um:g;KNna
L2 2.52
M u = Wu = 47.2 = 147.5kN .m
2 2
muxkat;Edk
As = 11.3982cm 2
PaKryEdkenAkñúgmuxkat;ebtug
As 11.3982
ρ= = = 0.012256
bd 20 × 46.5
PaKryEdk balance
f 'c 600
ρ b = 0.85β1 ( ) = 0.030345
f y 600 + f y

RbEvgbøúkkugRtaMgsgát;
As f y 11.3982 × 400
a= = = 9.578cm
0.85 f 'c b 0.85 × 28 × 20
TItaMgGkS½NWt
a 9.578
c= = = 11.268cm
β1 0.85
sac;lUteFobEdksuT§
fy
0.003 +
εt = (
ρ
Es
) − 0.003 = 0.00938 > 0.005 ⇒ muxkat;rgkMlaMgTaj φ = 0.9
ρb
a 9.578
φM n = φAs f y (d − ) = 0.9 × 11.3982 × 400 × (46.5 − ) × 10− 3 = 171.155kN .m
2 2
muxkat;manlkçN³RKb;RKan;
φM n > M u
]TahrN_5³ eKmanFñwmmuxkat;mYymanRbEvg
6m . FñwmenHmanmuxkat;dUcbgðajkñúgrUb.

edayeRbI f ' = 20MPa nig f = 400MPa
c y

kMNt;bnÞúkGefrrayesμIGnuBaØati. FñwmenHmin

manbnÞúkefrGVIeRkABITMgn;xøÜnvaeT.
dMeNaHRsay³
TMgn;pÞal;rbs;Fñwm
WD = 30 × 52.5 × 10 −4 × 24 = 3.78kN / m
muxkat;Edk
As = 14.71875cm 2
As f y 14.71875 × 400
a= = = 17.32cm
0.85 f 'c b 0.85 × 20 × 20
PaKryEdkeRbIR)as;enAkñúgmuxkat;ebtug
As 14.71875
ρ= = = 0.009345
bd 30 × 52.5
PaKryEdk balanced kñúgebtug
f 'c 600
ρ b = 0.85β1 ( ) = 0.021675
f y 600 + f y

sac;lUteFobEdksuT§
fy
0.003 +
εt = (
ρ
Es
) − 0.003 = 0.0086 > 0.005 ⇒ muxkat;rgkMlaMgTaj φ = 0.9
ρb
a 17.32
φM n = φAs f y (d − ) = 0.9 ×14.71875 × 400 × (52.5 − ) × 10 −3 = 232.3kN .m
2 2
edayeGay M = φM u n

mü:ageTot M = 1.2Mu D + 1.6M L
3.78 × 6 2 W
232.3 = 1.2( ) + 1.6( L × 6 2 ) = 20.412 + 7.2WL
8 8
232.3 − 20.412
WL = = 29.43kN / m
7.2
]TahrN_6³ RtYtBinitümuxkat;dUcbgðajkñúgrUbxageRkam edIm,ITb;Tl;nwg
m:Um:g;KNna 41kN.m . edayeRbI f ' = 20MPa nig f = 235MPa .
c y

dMeNaHRsay³
muxkat;Edk 7.5cm

As = 3.3912cm 2


T.Chhay NPIC

PaKryEdkeRbIR)as;enAkñúgmuxkat;ebtug
As 3.3912
ρ= = = 0.00377
bd 20 × 45
PaKryEdkGb,brmaeRbIR)as;enAkñúgmuxkat;ebtug
f 'c 1.4
ρ min = max( , ) = max(0.004756,0.00596) = 0.00596
4 fy fy

⇒ ρ < ρ min
⇒ As min = 0.00596 × 20 × 52.5 = 6.258
dUcenHeKRtUveRbIEdk 3DB18 ⇒ A = 7.63cm s
2
> 6.258cm 2

As f y 7.63 × 235
a= = = 5.274cm
0.85 f 'c b 0.85 × 20 × 20
a 5.274
φM n = φAs f y (d − ) = 0.9 × 7.63 × 235 × (45 − ) ×10 − 3 = 68.4kN .m
2 2
⇒ φM n > M u
dUcenH Edk 3DB18 RKb;RKan;edIm,ITb;Tl;nwgm:Um:g;KNnaxageRkA .
8> bNþúMénEdk
enAeBlEdlkarKNnamuxkat;EdkRtUvkarsMrab;ebtugmanbrimaNeRcIn ]TahrN_ enAeBlEdl ρ max

RtUv)aneRbI eBlenaHeKBi)akkñúgkarBRgayEdkeTAkñúgmuxkat;ebtug.
ACI Code )anGnuBaØatieGayEdk beNþayGacdak;CabNþúMEdl

manTMrg;dUcbgðaykñúgrUb.bNþúMénEdkcab;BIbYn GaceFVIeTA)anedayman
EdkkgBT§½ CMuvij. kareFVIbNþúMEdkkgenHk¾GacRbRBwtþeTA)ansMrab;ssr.
bNþúMénEdk RtUv)ancat;TukCaEdkmYyedImsMrab; kMNt;KMlatEdk nigkMras;karBarebtug. Ggát;p©it
énEdkeTal RtUv)anbMEbkBIRkLaépÞsmmUlrbs;bNþúMEdk.
segçb³ karkMNt;EdkrgkMlaMgTajsMrab;muxkat;ctuekaNEkg
1> kMNt;PaKryEdkeRbIR)as;enAkñúgebtug ρ = bd
A s

2> kMNt;PaKryEdk balanced ρ = 0.85β ff ' ( 600 + f ) nigPaKryEdkGtibrma
b
600
1
c

y y

fy
0.003 +
ρ max = (
0.008
Es
) ρb sMrab;muxkat;rgkMlaMgTaj. dUcKña kMNt;PaKryEdkGb,brma

f 'c 1.4
ρ min = max( , )
4 fy fy

3> RbsinebI ρ < ρ < ρ kMNt; a = 0.85 ff ' b / c / ε nig φ = 0.9 . RbsinebI ρ < ρ
min max
A s y
t min
c

PaKryEdkEdleRbIR)as;kñúgebtugminRKb;RKan; eTaHCay:agNaPaKryEdkEdleRbIR)as;kñúgebtug
RtUvEt ρ ≥ ρ . RbsinebI ρ ≥ ρ enaH φ < 0.9 .
min max

4> kMNt;ersIusþg;m:Um:g;xagkñúgKNna φM = φA f (d − a )
n
2
s y

9> muxkat;ctuekaNEkgCamYyEdkrgkMlaMgsgát;
enAkñúgmuxkat;ebtug muxkat;EdkEdlTb;nwgm:Um:g;Bt; RtUv)ankMNt;ecjBIbnÞúkxageRkAEdlmanGMeBI
elIeRKOgbgÁúM edayeFVIy:agNaeGayersIusþg;m:Um:g;xagkñúgFMCag b¤esμInwgm:Um:g;xageRkA. b:uEnþenAeBlEdlmux
kat;ebtug ¬TTwg nigkMBs;RbsiT§PaB¦ mantMéltUcenaH ρ RtUv)aneRbI. RbsinebIm:Um:g;xageRkAFMCag
max

ersIusþg;m:Um:g;xagkñúg enaHbrimaNEdksgát; nigEdkTajRtUv)anbEnßm.
Edksgát;RtUv)aneRbI enAeBlEdlmuxkat;ebtugRtUv)ankMNt;edaymUlehtusßabtükmμ. pl
RbeyaCn_rbs;Edksgát;KW kat;bnßyPaBdabry³eBlyUr nigedIm,IgayRsYldak;Edkkg.
muxkat;EdkDubmanBIrkrNIEdleKRtUvBicarNa GaRs½yeTAnwgkareFVIrbs;Edkdl;cMnucyar b¤Gt;.
k> enAeBlEdksgát;eFVIkardl;cMnucyar


T.Chhay NPIC

m:Um:g;xagkñúgGacRtUv)anEckecjCaBIr dUcbgðajkñúgrUb M Cam:Um:g;EdlekItBIkMlaMgsgát;rbs;ebtug
u1

nigkMlaMgTajsmmUlrbs;Edk A sMrab;muxkat;eKal. M Cam:Um:g;bEnßmEdlekItBIkMlaMgsgát;enAkñúg
s1 u2

Edksgát; A' nigkMlaMgTajenAkñúgEdkrgkMlaMgTajbEnßm A .
s s2

m:Um:g; M Cam:Um:g;Edl)anBImuxkat;sMrab;EdkrgkarTajeKal
u1

T1 = Cc ⇒ As1 f y = 0.85 f 'c ab
As1 f y
⇒a=
0.85 f 'c b
a
M u1 = φAs1 f y (d − )
2
fy
0.003 +
karkMNt; M RtUveGay ρ < bd nigtUcCag b¤esμI ρ = ( 0.008E ) ρ sMrab;eGaymuxkat;
A
u1 1
s1
max
s
b

rgkarTajeKal.
BicarNaelIm:Um:g; M edaysnμt;fa muxkat;Edkrgkarsgát; A' eFVIkardl;cMnucyar
u2 s

M u 2 = φAs 2 f y (d − d ' )

M u 2 = φA' s f y (d − d ' )

d' - CacMgayBIsésEpñkxageRkAbMputeTAGkS½Edkrgkarsgát;
kñúgkrNIenH A = A' begáItnUvkMlaMgesμIKñaTisedApÞúyKña
s2 s

m:Um:g;srub esμInwgplbUkénm:Um:g; M nig M u1 u2

a
φM n = M u1 + M u 2 = φ[ As1 f y (d − ) + A' s f y (d − d ' )]
2
muxkat;EdksrubEdleRbIsMrab;karTajCaplbUkénbrimaNEdk A nig A s1 s2

dUcenH A = A + A = A + A'
s s1 s2 s1 s

⇒ As1 = As − A' s
( A − A's ) f y
⇒a= s
0.85 f 'c b

dUcenHeK)an φM a
= φ[( As − A' s ) f y (d − ) + A' s f y (d − d ' )]
n
2
fy
0 . 003 +
nigeyIgman ρ 1 = ( ρ − ρ ' ) ≤ ρ max = ρ b ( (1)
0 . 008
Es
)

sMrab; f = 414MPa enaH ( ρ − ρ ' ) ≤ 0.63375ρ / φ = 0.9 nig ε = 0.005 kar)ak;rbs;FñwmbNþal
y b t

mkBIEdksrubrgkarTajeFVIkardl;cMnucyar ehIykarEbkPøam²rbs;ebtugRtUv)aneCosvag.

RbsinebI ρ 1 = ( ρ − ρ ' ) > ρ max enHmuxkat;sßitenAtMbn; transition region Edl
fy
0.003 +
( ρ − ρ ' ) ≤ ρ max,t = ρ b (
0.007
Es
) kñúgkrNIenH
φ < 0.9 sMrab; M nig φ = 0.9 sMrab; M enaH
u1 u2

eK)an a
φM n = φ[( As − A' s ) f y (d − )] + 0.9 A' s f y (d − d ' )
2
cMNaMfa ( A − A' ) ≤ ρ bd
s s max,t

enAkñúgtMbn;sgát; kMlaMgEdkrgkarsgát;KW C = A' ( f s s y − 0.85 f 'c )

edayKitfaépÞebtugEdlCMnYsedayépÞEdk A' enaH s

T = As f y = Cc + C s = 0.85 f 'c ab + A' s ( f y − 0.85 f 'c )

⇒ As f y − A' s f y + 0.85 f 'c A' s = 0.85 f 'c ab

eday 0.85 f ' ab = A c s1 fy

⇒ As f y − A' s f y + 0.85 f 'c A' s = As1 f y

EckGgÁTaMgBIrnwg bdf y

⇒ ρ − ρ ' (1 − 0.85
f 'c
fy
) = ρ1 Edl ρ 1 =
As1
bd
≤ ρ max

fy
0.003 +
dUcenH ρ − ρ ' (1 − 0.85 ff ' ) ≤ ρ c
max = ρb (
0.008
Es
) (2)
y

PaKryEdkrgkarTajsrubGtibrma ρ EdleRbIenAkñúgmuxkat;ctuekaNEkg enAeBlEdlEdkrgkar
sgát;eFVIkardl;cMnucyar
Maxρ = ( ρ max + ρ ' )
mann½yfa muxkat;EdkrgkarTajsrubeRbIenAkñúgmuxkat;ctuekaN enAeBlEdkrgkarsgát;eFVIkardl;
cMnucyar MaxA = bd ( ρ + ρ ' )
s max

edIm,IeGaydwgfa Edkrgkarsgát;eFVIkardl;cMncyar eyIgRtUvBinitüsac;lUteFob edayeGay
u
fy
ε 's ≥ ε y =
Es


T.Chhay NPIC

tamrUbxagelI eyIg)an
c 0.003 600
= =
d' fy 600 − f y
0.003 −
Es
600
⇒c=( )d '
600 − f y

eyIgman A f = 0.85 f ' ab
s1 y c

b:uEnþ A = A − A' nig ρ = ρ − ρ '
s1 s s 1

dUcenHeyIg)an ( A − A' ) f = 0.85 f ' ab
s s y c

⇒ ( ρ − ρ ' )bdf y = 0.85 f 'c ab
f 'c a
⇒ ( ρ − ρ ' ) = 0.85( )( )
fy d

eday a = β c = β ( 600 − f
1
600
1 )d '
y

dUcenH ( ρ − ρ ' ) = 0.85β ( ff ' )( d ' )( 600 − f
d
1
600 c
)=K
y y

RbsinebI ( ρ − ρ ' ) ≥ K enaHEdkrgkarsgát;eFVIkardl;cMnucyar.
eyIgeXIjfa enAeBlEdlbrimaNEdkrgkarTajeKal A ekIneLIg enaH T nig C k¾mantMélkan;
s1 1 1

EtFMEdr ehIyGkS½NWtnwgFøak;cuH eBlenaHsac;lUteFobrbs;Edkrgkarsgát;k¾ekIneLIg rhUtdl;cMnucyar.



]TahrN_7³ FñwmctuekaNEkg EdlmanTTwg 30cm nigkMBs;RbsiT§PaB d = 60cm . EdkrgkarTajman
6 DB 28 tMerobCaBIrCYr ÉEdkrgkarsgát;man 2DB 22 . kMNt;ersIusþg;m:Um:g;xagkñúgRbsinebIeKeRbI

f ' = 28MPa nig f = 400MPa .
c y

dMeNaHRsay³
muxkat;EdkrgkarTaj A = 36.93cm PaKryEdkrgkarTaj ρ = 30 ×93 = 0.02052
s
2 36.
60


T.Chhay NPIC

muxkat;Edkrgkarsgát; A' = 7.6cm PaKryEdkrgkarsgát; ρ ' = 307×660 = 0.0042
s
2 .

muxkat;EdkrgkarTajeKal A = 29.33cm PaKryEdkrgkarTajeKal ρ = 30 ×.33 = 0.01629
s1
2 29
60
1

f 'c d ' 600 28 6 600
K = 0.85β1 ( )( )( ) = 0.852 = 0.01517
f y d 600 − f y 400 60 600 − 400

eday ( ρ − ρ ' ) ≥ K enaHEdkrgkarsgát;eFVIkardl;cMnucyar
sMrab; f ' = 28MPa nig f = 400MPa ⇒ ρ = 0.030345 ⇒ ρ
c y b max = 0.019

eday ( ρ − ρ ' ) < ρ ⇒ φ = 0.9max

ersIusþg;m:Um:g;xagkñúg
a
φM n = φ[( As − A' s ) f y (d − ) + A' s f y (d − d ' )]
2
( As − A's ) f y
⇒a= ⇒ a = 16.43cm
0.85 f 'c b
16.43
⇒ φM n = 0.9[29.33 × 400 × (60 − ) + 7.6 × 400 × (60 − 6)] × 10−3 = 694.5kN .m
2
viFImü:ageTot epÞógpÞat;faetIEdkrgkarsgát;eFVIkardl;cMnucyarb¤enA
a 16.43
c= = = 19.33cm
0.85 0.85
sac;lUteFobEdkrgkarsgát; ε ' = c −c d ' × 0.003 = 1919.33 6 × 0.003 = 0.00207
.33 −
s

sac;lUteFobrbs;Edk ε = 0.002 y

eday ε ' > ε ⇒ Edkrgkarsgát;eFVIkardl;cMnucyar
s y

dt − c (60 + 6) − 19.33
εt = ( )0.003 = × 0.003 = 0.007 > 0.005
c 19.33
b¤ d
c 19.33
=
60
= 0.322 < 0.375

muxkat;EdkrgkarTajsrub
MaxAs = bd ( ρ max + ρ ' ) = 30 × 60 × (0.019 + 0.0042) = 41.76cm 2 > As RtwmRtUv
x> enAeBlEdksgát;eFVIkarmindl;cMnucyar
dUckarbkRsayxagelI RbsibebI ( ρ − ρ ' ) < 0.85β ( ff ' )( d ' )( 600 − f
d
600
1
c
)=K
y y

enaHEdksgát;eFVIkarmindl;cMnucyareT. enHbgðajfa RbsinebI ( ρ − ρ ' ) < K EdkrgkarTajeFVIkar
dl;cMnucyarmun ebtugmansac;lUteFobGtibrma 0.003 ehIyEdkrgkarsgát;k¾eFVIkarmindl;cMnucyarEdr.
pleFob d ' c kan;EtFM mann½yfakalNaeKdak;Edkrgkarsgát;enACitGkS½NWt enaHsac;lUteFobrbs;Edk
rgkarsgát;kan;EttUc.

RbsinebIEdksgát;eFVIkarmindl;cMnucyar dMeNaHRsayTUeTAGaceFVIeTA)anedayQrelIeKalkarN_
sþaTic.
c − d' c − d'
ε ' s = 0.003( ) f ' s = E s ε ' s = 600( )
c c
edayeGay C c = 0.85 f 'c β1cb
c − d'
C s = A' s ( f ' s −0.85 f 'c ) = A' s [600( )0.85 f 'c ]
c
edaysar T = A f s y = Cc + C s enaH
c − d'
As f y = 0.85 f 'c β1cb + A' s [600( )0.85 f 'c ]
c
⇒ (0.85 f 'c β1b)c 2 + [(600 A' s ) − (0.85 f 'c A' s ) − As f y ]c − 600 A' s d ' = 0

smIkarenHmanTMrg; A c + A c + A = 0
1
2
2 3

eRkayeBlKNna c
KNna f ' = 600( c −c d ' ) KNna a = β c KNna C A' [600( c −c d ' )0.85 f ' ] nigKNna
s 1 s s c

Cc = 0.85 f 'c β1cb
a
φM n = φ[Cc (d − ) + C s (d − d ' )]
2
enAeBlEdksgát;eFVIkarmindl;cMnucyar/ f 's < f y nigEdkTajsrubRtUvkarsMrab;muxkat;ctuekaN
EkgKW³
f 's ρ ' f 's
MaxAs = ρ max bd + A' s = bd ( ρ max + )
fy fy

edayEckGgÁTaMgBIrnwg bd eyIg)anPaKryEdk
MaxAs f'
Maxρ = ≤ ρ max + ρ ' s
bd fy

b¤ ( ρ − ρ ' ff ' ) ≤ ρ
s
max
y

−
kñúgkrNIenH a = A 0f.85 fA'' bf '
s y s s

c

a
φM n = φ[( As f y − A' s f ' s )(d − ) + A' s f ' s (d − d ' )]
2
segçb³ viFIsaRsþviPaKmuxkat;CamYyEdkrgkarsgát;
1> kMNt; ρ / ρ ' / ( ρ − ρ ' ) dUcKñakMNt; ρ / ρ max min

2> kMNt; K = 0.85β ( ff ' )( d ' )( 600 − f )
d
1
600 c

y y


T.Chhay NPIC

3> RbsinebI ( ρ − ρ ' ) ≥ K enaHEdkrgkarsgát;eFVIkardl;cMnucyar f ' = f . RbsinebI s y

( ρ − ρ ' ) < K enaHEdkrgkarsgát;eFVIkarmindl;cMnucyar f ' < f . s y

4> RbsinebIEdkrgkarsgát;eFVIkardl;cMnucyar
k> BinitüemIl ρ ≥ ( ρ − ρ ' ) ≥ ρ b¤ ε ≥ 0.005 / eRbI φ = 0.9
max min t

−
x> kMNt; a = ( A .85Af'' )bf
0
s s y

c

K> kMNt; φM = φ[( A − A' ) f (d − a ) + A' f (d − d ' )]
n s
2
s y s y

X> muxkat;EdkrgkarTajGtibrma A EdlGaceRbIenAkñúgmuxkat;KW s

MaxAs = bd ( ρ max + ρ ' ) ≥ As
5> RbsinebIEdkrgkarsgát;eFVIkarmindl;cMnucyar
k> KNnacMgayGkS½NWt c edayeRbIsmIkar T = C + C s c

x> kMNt; f ' = 600( c −c d ' )
s

K> RtYtBinitü ( ρ − ρ ' ff ' ) ≤ ρ b¤ MaxA EdlGaceRbIenAkñúgmuxkat; RtUvEtFMCagb¤esμI A
s
max s s
y

Edl)aneRbI
f 's
MaxAs = bd ( ρ max + ρ ' ) ≥ As
fy
−
X> kMNt; a = A 0f.85 fA'' bf ' b¤ a = β c
s y s s
1
c

g> kMNt; φM = φ[( A f − A' f ' )(d − a ) + A' f ' (d − d ' )]
n s
2
y s s s s

]TahrN_8³ kMNt;ersIusþg;m:Um:g;kñúgénmuxkat;dUcbgðajkñúgrUb edayeRbI f ' = 35MPa / f = 400MPa . eK c y

eRbIEdkrgkarsgát; 3DB25 Edl A' = 14.72cm nigEdlrgkarTaj 6DB32 Edl A = 42.39MPa .
s
2
s


dMeNaHRsay³
kMNt; ρ = bd = 35 ×.39 = 0.02125 / ρ ' = bd = 35 ×72 = 0.00738 / ( ρ − ρ ' ) = 0.01387
A 42
s

57
A' 14.
57
s

eday f ' = 35MPa ⇒ β = 0.85 − 0.05( f ' 7−28 ) ⇒ β = 0.85 − 0.05( 35 − 28 ) = 0.8
c 1
c
7
1

kMNt; K = 0.85β ( ff ' )( d ')( 600 − f ) = 0.85 × 0.8( 400 )( 6.5 )( 600600400 ) = 0.020355
1
d
600c 35
57 −
y y

eday ( ρ − ρ ' ) < K enaHEdkrgkarsgát;eFVIkarmindl;cMnucyar
f 'c 600
ρ b = 0.85β1 ( ) = 0.0357
f y 600 + f y
0.005
ρ max = 0.0357 = 0.02231
0.008
muxkat;rgkarTaj ⇒ φ = 0.9
( ρ − ρ ' ) < ρ max

kMNt;cMgayGkS½NWt c
C = 0.85 f ' ab eday a = β c = 0.8c ⇒ C = 0.85 × 35 × 0.8c × 350 = 8330c
c c 1 c

C s = A' s ( f ' s −0.85 f 'c )
c − d' c − 65 c − 65
eday f ' s = 600(
c
) ⇒ C s = 1472[600(
c
) − 0.85 × 35] = 883200(
c
) − 43792

T = As f y = 4239 × 400 = 1695600 N
c − 65
⇒ 1695600 = 8330c + 883200( ) − 43792
c

⇒ 8330c 2 − 856192c − 57408000 = 0

⇒ c = 149mm = 14.9cm

⇒ a = 0.8 ×14.9 = 11.92cm
c − d' 14.9 − 6.5
kMNt; f ' s = 600(
c
) ⇒ f ' s = 600
14.9
= 339MPa

kMNt; C = 0.85 f ' ab ⇒ C = 0.85 × 35 ×119.2 × 350 = 1241170 N = 1241.17kN
c c c

kMNt; C = A' ( f ' −0.85 f ' ) ⇒ C = 1472(339 − 0.85 × 35) = 455216 N = 455.216kN
s s s c s

edIm,IkMNt;ersIusþg;m:Um:g;kñúg eKRtUvKitm:Um:g;eFobGkS½EdkTaj A s

a 0.1192
φM n = φ[Cc (d − ) + C s (d − d ' )] = 0.9[1241.17(0.57 − ) + 455.216(0.57 − 0.065)]
2 2
φM n = 863.38kN .m

RtYtBinitü ( ρ − ρ ' ff ' ) ≤ ρ s
max ⇒ (0.02125 − 0.00738
339
400
) = 0.015 < 0.02231
y

kMNt;muxkat;EdkTajGtibrma MaxA s = bd ( ρ max + ρ '
f 's
fy
)


T.Chhay NPIC

MaxAs = 35 × 57(0.02231 + 0.00738
339
400
) = 56.99cm 2 > 42.39cm 2 RtwmRtUv
c
=
14.9
d t 57 + 9 − 6.5
= 0.25 < 0.375RtwmRtUv
d −c
εt = t
c
0.003 = 0.009 > 0.005 muxkat;rgkarTaj
10> viPaKmuxkat;GkSret T nigmuxkat;GIu I
CaFmμtakMralxNÐ nigFñwmRtUv)aneKcak;CamYyKña edIm,IbegáItCaeRKOgbgÁúMEtmYy monolithic
structure. kMralxNÐmankMras;esþIgCagFñwm. eRkamGMeBIénkugRtaMgBt; EpñkénkMralxNÐEdlCaEpñkrbs;Fñwm

rgnUvkugRtaMgsgát; GaRs½yeTAelITItaMgGkS½NWt. EpñkénkMralxNÐEdleFVIkarCamYyFñwmRtUv)aneKeGay
eQμaHfa søab flange EdlbgðajkñúgrUbedayépÞ bt . EpñkénFñwmEdlenAsl; EdlbgðajedayépÞ (h − t )b w

RtUv)aneKeGayeQμaHfa RTnug stem b¤ web.
sMrab;muxkat;GkSr I mansøabBIr KWsøabrbkarsgát; EdlcUlrYmeFVIkar nigsøabrgkarTaj EdlKμanRb
siT§PaB BIeRBaHvaenABIeRkamGkS½NWt ehIyEdlminRtUv)aneKykvamkKit. dUcenH karviPaK nigkarKNna
Fñwmmuxkat; I manlkçN³dUcKñanwgFñwmmuxkat; T .

k> TTwgRbsiT§PaB
sMrab;muxkat;GkSr T EdlsøabmanRbEvgEvg kugRtaMgsgát;manragCa):ar:abUl EdltMélGtibrmasßit
enAelIFñwm ehIytMélGb,brmasßitenAcMgay x BImuxrbs;Fñwm. ehIykugRtaMgk¾ERbRbYlBIsésEpñkxagelI
søab


mksésEpñkxageRkamsøab BIGtibrma mkGb,brma. tMélbMErbMrYlenHGaRs½yeTAnwgTItaMgGkS½NWt.

kugRtaMgsmmUl CakugRtaMgBRgayesμImanGMeBIelITTwgsøabsmmUl b . TTwgRbsiT§PaB b RtUv)an
e e

eKkMNt;edayGnuKmn¾eTAnwg³
- RbelaHElVg s 1

- TTwgRTnug b w

- TMnak;TMngrvagkMras;kMralxNÐ nigkMBs;srubrbs;Fñwm
- lkçxNÐTMrrbs;Fñwm ¬samBaØ b¤Cab;¦
- lkçxNÐbnÞúk ¬BRgayesμI b¤cMcMnuc¦
- pleFobrvagRbEvgFñwmcenøaHm:Um:g;sUnü nigTTwgRTnug nigcMgayrvagRTnug

ACI Code )ankMNt;nUvTTwgRbsiT§PaBedaykMNt;yktMélGb,brmaénsmIkarxageRkam³
-b =
e
L
4
Edl L CaRbEvgFñwm
- b = 16t + b Edl t kMras;kMralxNÐ nig b TTwgRTnug
e w w


T.Chhay NPIC

- b = b Edl b cMgayBIcenøaHGkS½kMralxNÐ
e

muxkat;ragGkSr T b¤muxkat;ragGkSr I GacRtUvviPaKCaragctuekaNEkg b¤ragGkSr T GaRs½yelITI
taMgGkS½NWt.
x> muxkat;GkSret T RtUv)anKitCaragctuekaNEkg
kñúgkrNIenH kMBs;énbøúkkugRtaMgsmmUl a sßitenAkñúgsøab a ≤ t begáIt)anCaépÞkugRtaMgsgát;esμI
nwg b a . muxkat;ebtugBIeRkamGkS½NWtRtUv)aneKsnμt;faKμanRbsiT§iPaB ehIymuxkat;RtUv)aneKKitfaman
e

EdkrgkarTaj Edl)anBnül;BIxagelI edayRKan;EtCMnYs b eday b . e

dUcenH a = 0.85 ff' b
A s y

c e

nig φM = φA f (d − a )
n s y
2
RbsinebI kMBs; a ekIneLIgeday a = t enaH φM n
t
= φAs f y (d − )
2
kñúgkrNIenH t = 0.85 ff' b b¤ A = 0.85 ff ' b t
A s y
s
c e

c e y

sMrab;karviPaKenH A ≤ A nig ε s s max t ≥ 0.005



K> viPaKmuxkat;ragGkSret T
kñúgkrNIenH GkS½NWtsßitenAelIRTnug. EpñkxøHrbs;ebtugenAkñúgRTnugmanRbsiT§PaBkñúgkarTb;Tl;
nwgm:Um:g;xageRkA.
kMlaMgsgát; C = 0.85 f ' [b t + b (a − t )]
c e w

TItaMgrbs; C sßitenAelITIRbCMuTMgn;rbs;épÞragGkSr T enAcMgay z BIsésEpñkxageRkAbMput.


T.Chhay NPIC

karviPaKmuxkat;ragGkSr T manlkçN³RsedogKñanwgkarviPaKmuxkat;ebtugEdlEdkrgkarsgát;
edaycat;TuképÞebtug (b − b )t smmUleTAnwgEdksgát; A' . karviPaKenHEckecjCaBIrEpñkdUcbgðajkñúg
e w s

rUbxageRkam³

- muxkat;eKalragctuekaNEkg b d nigmuxkat;Edk A . kMlaMgsgát; C = 0.85 f ' ab nigkMlaMg
w s1 1 c w

T = A f ehIyRbEvgédXñas; (d − ) .
a
1 s1 y
2
- muxkat;Edlmansøabebtugsgxag 2 × [(b − b )t ] / 2 begáIt)anCakMlaMgsgát;edayKuNCamYy
e w

0.85 f ' nigRbEvgédXñas;esμInwg (d − ) . RbsinebI A Camuxkat;EdkTajEdlbegáItkMlaMgesμInwg
t
c sf
2
kMlaMgsgát;EdlbegáItedayebtugsøabsgxag dUcenH A = 0.85 f ' ft (b − b ) sf
c e w

y

muxkat;Edksrub A EdleRbIkñúgmuxkat;GkSr T KW³ A = A + A
s s s1 sf

b¤ A = A − A
s1 s sf

muxkat;GkSr T sßitkñúgsßanPaBlMnwg dUcenH C = T / C = T nig C = C + C 1 1 2 2 1 2 = T1 + T2 + T

BicarNaelIsmIkar C = T sMrab;muxkat;eKalctuekaNEkg eK)an
1 1

A f = 0.85 f ' ab b¤ ( A − A ) f = 0.85 f ' ab
s1 y c w s sf y c w

A − )
dUcenH a = (0.85 fA' b f
s sf y

c w

cMNaMfa b RtUv)aneRbIedIm,IkMNt; a .
w

ersIusþg;énm:Um:g;kñúgénmuxkat;CaplbUkénm:Um:g;BIr M nig M u1 u2

φM n = M u1 + M u 2


a a
M u1 = φAs1 f y (d − ) = φ ( As − Asf ) f y (d − )
2 2
( As − Asf ) f y
Edl As1 = As − Asf a= nig
0.85 f 'c bw
t
M u 2 = φAsf f y (d − )
2
a t
φM n = φ[( As − Asf ) f y (d − ) + Asf f y (d − )]
2 2
BicarNaelImuxkat;RTnug b d / sac;lUteFobsuT§ ε GackMNt;BI a / c nig d dUcxageRkam³
w t t

RbsinebI c = βa nig d = h − 6.5cm bnÞab;mk ε = 0.003 (c −cd ) sMrab;muxkat;rgkarTajenAkñúg
t t
t

1

RTnug/ ε ≥ 0.005 .
t

karKNnaersIusþg;m:Um:g;kñúgsMrab;muxkat;GkSr T b¤muxkat;GkSr I GacKNnaedayeRbIsmIkarxagelI
EteKcaM)ac;RtUvRtYtBinitülkçxNÐxageRkam³
- PaKryEdkTajsrubeFobRkLaépÞRbsiT§iPaBRTnugRtUvFMCag b¤esμI ρ min

As
ρw = ≥ ρ min
bw d
f 'c 1.4
ρ min = ≥
4 fy fy

- RtYtBinitü sac;lUteFobsuT§FMCag b¤esμI ε ≥ 0.005 sMrab;muxkat;rgkarTaj t

- muxkat;EdkGtibrma MaxA enAkñúgmuxkat;GkSr T RtUvEtFMCag b¤esμI muxkat;EdkEdl)aneRbI A
s s

sMrab;muxkat;rgkarTaj CamYy φ = 0.9
MaxAs = Asf ( flange) + ρ max (bw d )( web)
1
MaxAs = [0.85 f 'c t (b − bw )] + ρ max (bw d )
fy

PaKryEdkeFobnwgRTnug ρ w =
As
bw d
≤ ( ρ max +
Asf
bw d
)

⇒ ρ w − ρ f ≤ ρ max

smIkarTUeTAsMrab;KNna MaxA enAkñúgmuxkat;GkSr T enAeBl a > t GackMNt;tam
s

C = 0.85 f 'c [(be − bw )t + abw ]

sMrab; ε = 0.003 nig ε = 0.005 / d = 0.003.003.005 = 0.375 sMrab;RTnug
c
c 0
t
+0
dUcenH a = β c = 0.375β d
1 1

muxkat;EdkGtibrmaesμInwg C
f y


T.Chhay NPIC

dUcenH MaxA s = 0.85
f 'c
fy
[(be − bw )t + 0.375β1bw d ]

segçb³ viFIsaRsþviPaKmuxkat;GkSret T b¤GkSrGil L páab;
1> kMNt;TTwgRbsiT§PaB b nigkMNt; ρ / ρ e max min

2> kMNt; a = 0.85 ff' b
A s y

c e

3> RbsinebI a < t enaHmuxkat;eFVIkarCaragctuekaNEkg
- kMNt; φM = φA f (d − a )n
2
s y

cMNaMfa³ c = βa nig ε = 0.003 (c −cd ) ≥ 0.005 sMrab;muxkat;rgkarTaj φ = 0.9
t
t

1

- RtYtBinitü ρ w =
As
bw d
≥ ρ min

- MaxA s =
1
fy
[0.85 f 'c t (b − bw )] + ρ max (bw d ) ≥ As

4> RbsinebI a > t enaHmuxkat;eFVIkarCaragGkSret
k> kMNt; A = 0.85 f ' ft (b − b )
sf
c w

y

( As − A' s ) f y
x> kMNt; a = 0.85 f ' b
c

K> RtYtBinitü ρ − ρ ≤ ρ eFobnwgRkLaépÞRTnug
w f max

Edl ρ = bAd nig ρ = bA d
w
s
f
sf

w w

b¤RtYtBinitü MaxA s = 0.85
f 'c
fy
[(be − bw )t + 0.375β1bw d ] ≥ As / sMrab; φ = 0.9
A − )
X> kMNt; a = (0.85 fA' b f s sf y

c w

g> kMNt; φM = φ[( A − A ) f (d − a ) + A f (d − 2 )]
n
2
t
s sf y sf y

]TahrN_9³ FñwmebtugGarem:EdlmanRbEvg 4.5m ehIymanKMlatBImYyeTAmYyRbEvg 2m . FñwmenHRTkM
ralxNÐEdlmankMras; 10cm . kMNt;nUversIusþg;m:Um:g;kñúgrbs;FñwmkNþal. eKeRbI f ' = 20MPa nig c

f = 400MPa .
y



dMeNaHRsay³
kMNt;TTwgRbsiT§iPaB
L 450
be = min{16t + bw ; ; b} = min{16 × 10 + 25; ;200} = 112.5cm
4 4
kMNt;kMBs;bøúkkugRtaMg
a=
A f
s y

0.85 f ' b
/ A = 14.72cm
s
2

c e
14.72 × 400
a= = 3.08cm < t
0.85 × 20 × 112.5
dUcenHeyIgRtUvKNnaCaragctuekaNEkgEdlmanTTwg b = 112.5cm e

PaKryEdkGb,brma ρ = 4 ff ' ≥ 1f.4 ⇒ ρ = 0.0035
min
c
min
y y

PaKryGtibrma ρ max = 0.625 × 0.85β 1
f 'c
fy
(
600
600 + f y
) = 0.01355

PaKryEdkeFobnwgépÞRkLaRTnug ρ w =
As
=
14.72
bw d 25 × 40
= 0.01472 > 0.0035

TItaMgGkS½NWt c = βa = 3..08 = 3.62cm
0 85 1

sac;lUteFobsuT§rbs;Edk ε = 0.003( d c− c ) = 0.003( 403−.62.62 ) = 0.03 > 0.005 ⇒ φ = 0.9
t
3 t

KNna φM = φA f (d − a ) = 0.9 ×1472 × 400(400 − 30.8 ) = 203807232 N .mm = 203.81kN .m
n s
2
y
2
epÞógpÞat;muxkat;Gtibrma MaxA = f [0.85 f ' t (b − b )] + ρ (b d ) ≥ A
1
s c w max w s
y


T.Chhay NPIC

MaxA = 37.22cm 2 > As RtwmRtUv
]TahrN_10³ KNnaersIusþg;m:Um:g;kñúgénmuxkat;GkSr T dUcbgðajkñúgrUb edayeRbI f 'c = 25MPa nig
f = 400 MPa .
y

dMeNaHRsay³
eKeGay b = b = 90cm / b = 25cm / d = 43cm nig A
e e s = 36.93cm 2
×
KNna a = 0.85 ff' b = 036.9325400 = 7.72cm > t
A s y

.85 × × 90
c e

eday a > t sikSaCaragGkSr T
KNna A = 0.85 f ' ft (b − b ) = 24.17cm
sf
c w 2

y

⇒ As1 = As − Asf = 12.76cm 2

epÞógpÞat; ε t

As1 f y 12.76 × 400
a ( web) = = = 9.6cm
0.85 f 'c bw 0.85 × 25 × 25
a( web)
c= = 11.29cm
β1
d t = 52 − 6.5 = 45.8cm
dt − c
ε t = 0.003( ) = 0.00917 > 0.005 ⇒ φ = 0.9
c
RtYtBinitü A s min = ρ min bw d = 0.0035 × 25 × 43 = 3.76cm 2 < 36.93cm 2 RtwmRtUv
KNna φM a t
= φ[( As − Asf ) f y (d − ) + Asf f y (d − )]
n
2 2
96 70
φM n = 0.9[(3693 − 2417)400(430 − ) + 2417 × 400(430 − )
2 2
φM n = 519172920 N .mm = 519.173kN .m

11> TMhMénmuxkat;FñwmGkSr T Éeka
eBlxøH FñwmGkSr T Éeka RtUv)aneRbIedm,IbEnßmépÞrgkarsgát;. muxkat;enHRtUv)aneKeRbIsMrab;Fñwm
EdleKcak;TukCamun.

ACI Code )anENnaMnUvTMhMmuxkat;sMrab;GkSr T ÉekadUcxageRkam³
- kMras;søab t RtUvFMCag b¤esμIBak;kNþalTTwgRTnug b
w

- TTwgsrubrbs;søab b RtUvEttUcCag b¤esμIbYndgTTwgRTnug b w

11> muxkat;GkSr L páab;
Fñwmmuxkat;GkSr L páab;CaFñwmEdlRTkMralxNÐEpñkxageKbMput.
TTwgRbsiT§PaBrbs;muxkat;enHRtUv )ankMNt;nUvtMélGb,brmaénsmIkarxageRkam³
- (b − b ) ≤ 12
e w
L

- (b − b ) ≤ 6t
e w

- (b − b ) ≤ 2
e w
l

Edl L - RbEvgFñwm
l - KMlatFñwm


T.Chhay NPIC


Iii flexural analysis of reinforced concrete

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