1. T.Chhay NPIC
IV. karKNnaFñwmebtugGarem:rgkarkac;begáag
1> kMBs;RbsiT§iPaBsMrab;Fñwm nigkMralxNÐ
kMBs;rbs;Fñwmx<s; b¤TabGaRs½yeTAnwgRbEvgrbs;Fñwm nigbnÞúkxageRkAEdlvaRtUvRT. edIm,IepÞógpÞat;
CamYYyPaBdab xageRkamenHCarUbmnþsMrab;kMNt;kMBs;FñwmGaRs½y nwgRbEvgFñwm.
kMralxNÐ L/20 L/24 L/28 L/10
mYyTis
Fñwm L/16 L/18.5 L/21 L/8
TTwgFñwmRtUv)ankMNt;ecjBIkMBs;rbs;Fñwm edaykMNt;enAcenøaH 1 d → 1 d .
3 2
2> muxkat;ctuekaNEkgCamYyEdkrgkarTaj
xageRkamCarUbmnþsMrab;karKNnamuxkat;rgkarTaj
f 'c 600
ρ b = 0.85β1 ( )
f y 600 + f y
fy
0.003 +
Es
ρ max = ρ b ( )
0.008
sMrab; f y = 400MPa
⇒ ρ max = 0.625ρ b
ehIy β 1 sMrab;ebtugEdlmanersIusþg; f ' ≤ 28MPa .
= 0.85 c
f ' −28
β = 0.85 − 0.05(
1 ) sMrab;ebtugEdlmanersIusþg; 28MPa < f ' ≤ 56MPa .
c
c
7
β = 0.65 sMrab;ebtugEdlmanersIusþg; f ' > 56MPa .
1 c
tMélénsac;lUteFobrbs;Edkkan;EtFM bgðajBIPaBsVitrbs;ebtugGarem:kan;EtFM )ann½yfaenAeBl
muxkat;ebtugkan;EtFMCamYyPaKryEdkticPaBsVitkan;EtFM pÞúymkvijenAeBlmuxkat;ebtugkan;EttUcCamYy
PaKryEdkkan;EtFMenaHPaBsVitrbs;ebtugkan;EttUc. xageRkamCataragbgðajBIPaKryEdksMNUmBrGa
Rs½yersIusþg;ebtug nigersIusþg;Edk.
Flexural Design on Reinforced Concrete Beams 53

2. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
taragTI1³ PaKryEdksMNUmBr ρ s
f 'c ( MPa) f y (MPa) %ρ s
20 235 1.4
400 1.2
28 400 1.4
500 1.2
35 400 1.6
500 1.4
sac;lUteFobEdksuT§ ε edIm,IeGaymuxkat;rgkarTajRtUv)ankMNt;FMCag 0.005 enaH φ = 0.9 .
t
) sac;lUteFobEdksuT§ ε enAcenøaH 0.004 → 0.005 .
250
φ = 0.65 + (ε − 0.002)(
t t
3
smIkarKNnaersIusþg;m:Um:g;kñúgmanTMrg;dUcxageRkam³
φM n = M u = Ru bd 2
Edl R = φρf (1 − 1.ρff ' ) = φR
u
7
y
y
n
c
Edl φ = 0.9 sMrab;muxkat;rgkarTaj
nig φ < 0.9 sMrab;muxkat;enAkñúgtMbn; transition region
dUcenH φM = M = φA f (d − 1.A f f' b )
n u
7
s y
s y
c
dUcKña φM = M = φρf bd (− 1.ρff ' )
n u
7
y
2 y
c
kñúgkrNIEdleyIgsÁal;m:Um:g;KNnaxageRkA nigsÁal;ersIusþg;sMPar³ eyIgenAsl;GBaØtibIeTotEdl
minTan;sÁal; kñúgenaHman TTwgFñwm b kMBs;RbsiT§iPaB d nigPaKryEdk ρ . dMeNaHRsayGaceFVIeTA)anluH
RtaEteyIgRtUveFVIkarsnμt;cMeBaHGBaØtiBIr. CaTUeTA ρ RtUv)ansnμt; ¬edayeRbI ρ ¦ nigdUcKña b k¾RtUv)an max
snμt;.
xageRkamCaviFIsaRsþkñúgkaredaHRsayedaysÁal; M / f ' nig f ³ u c y
- RbsinebI ρ RtUv)ansnμt; ¬edayeGayenAcenøaH ρ nig ρ ¦ enaHeyIgGackMNt; 1
2 max
1
2 b
ρf y
Ru = φρf y (1 − )
1.7 f 'c
Mu
⇒ bd 2 =
Ru
CaTUeTApleFob d ERbRbYlBI 2 → 3 ¬eKniymyk 2 ¦
b
dUcenH eKGackMNt; b nig d
dUcKña eyIg)an A = ρbd s
- RbsinebI b nig d RtUv)aneGay enaHPaKryEdkEdlRtUvkarkMNt;tamrUbmnþxageRkam³
karKNnaFñwmebtugGarem:rgkarkac;begáag 54

3. T.Chhay NPIC
0.85 f 'c 4M u
ρ= (1 − 1 − )
fy 1.7φf 'c bd 2
⇒ As = ρbd
- RbsinebI b RtUv)ansnμt;bEnßmBIelI ρ enaHeKRtUv
KNna R = φρf (1 − 1.ρff ' )
u y
7
y
c
KNna d = Mu
Ru b
RbsinebI d = 2 enaH d =
b
3
2M u
Ru b
⇒ As = ρbd
3> KMlatEdk nigRsTab;karBarEdk
k> KMlatEdk
sésrEdkRtUv)antMerobedaymanKMlaty:agticbMputesμInwgGgát;p©itEdk b:uEnþminRtUvtUcCag 25mm
edIm,IeGayeKGacbgðab;ebtug)any:aggayRsYlkñúgeBlcak;ebtug. KMlatEdksMrab;kartMerobEdkbBaÄrEdl
maneRcInCagmYyRsTab; minRtUvmantMéltUcCag 25mm Edr. RbsinebIKMlatEdkmanTMhMtUcenaHl,ayeb
tugminGacBT§½CMuvijEdk)anl¥eT.
x> RsTab;karBarEdk
RsTab;karBarEdk Casac;ebtugEdlenAcenøaHépÞxageRkA nigépÞrbs;Edk. eKcaM)ac;RtUvkarRsTab;kar
BarEdkeRBaHvamanplRbeyaCn_bYny:ag³
Flexural Design on Reinforced Concrete Beams 55

4. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
- edIm,Ietags¥itsésrEdkeTAnwgebtugEdleFVIeGaysMPar³TaMgBIreFVIkarCamYyKña. T§iBlrbs;PaB
s¥itGaRs½yeTAnwgkMras;RsTab;karBar.
- edIm,IkarBarsésrEdkeTAnwgERcHsIuEdk.
- edIm,IkarBarkar)at;bg;ersIusþg;EdkEdlbNþalmkBIkMedA. kMras;RsTab;karBar 20mm GacTb;Tl;
nwgePøIgeqH)an 1em:ag.
- sMrab;yanþdæan eragcRk cMNtrfynþ RsTab;karBarbEnßmRtUv)aneKdak;BIelIkMralxNÐEfmeTot
edIm,IkarBarkarswkercrwlEdlbNþalmkBIcracrN_.
kMras;RsTab;karBarEdkGaRs½yeTAnwgmCÄdæanEdleRKOgbgÁúMenaHsßitenA. xageRkamCatarag
bgðajBIkMras;RsTab;karBarGb,brma³
mCÄdæan kMras;karBarEdk (mm)
ebtugcak;pÞal;nwgdI 75
ebtugcak;pÞal;nwgdI b¤halxül;
+ Ggát;p©itEdkFMCag 16mm 50
+ Ggát;p©itEdktUcCag 16mm 35
ebtugmincak;pÞal;nwgdI b¤minhalxül;
- kMralxNÐ CBa¢aMg
+ Ggát;p©itEdkFMCag 36mm 35
+ Ggát;p©itEdktUcCag 36mm 20
- Fñwm ssr 35
- kMralekag
+ Ggát;p©itEdkFMCag 20mm 20
+ Ggát;p©itEdktUcCag 20mm 15
K> TTwgGb,brmarbs;muxkat;ebtug
smIkarTUeTAedIm,IkMNt;TTwgGb,brmarbs;muxkat;ebtugGacsresrdUcxageRkam³
b = nD + (n − 1) s + 2(φEdkkg ) + 2(RsTab;karBarEdk )
min
Edl n - cMnYnEdkbeNþay
D - Ggát;p©itEdkEdlFMCageK
s - KMlatEdk
karKNnaFñwmebtugGarem:rgkarkac;begáag 56

5. T.Chhay NPIC
X> kMBs;Gb,brmarbs;muxkat;ebtug
kMBs;rbs;muxkat;ebtugRtUv)ankMNt;edayGaRs½ynigRsTab;Edk.
- EdkmYyRsTab;³ h = d + D + 50mm
1
2
1
- EdkBIrRsTab;³ h = d + D + 60mm
2 2
]TahrN_1³ kMNt;muxkat;Edk nigmuxkat;ebtugedIm,ITb;Tl;nwgm:Um:g;KNna 490kN .m edayeRbIPaKryEdk
Gtibrma ρ sMrab;muxkat;rgkarTaj. smμtikmμ³ f ' = 20MPa nig f = 400MPa .
max c y
dMeNaHRsay³
eday f ' = 20MPa / f = 400MPa / β = 0.85 nig φ = 0.9 sMrab;muxkat;rgkarTaj
c y 1
f 'c 600
ρ b = 0.85β1 ( )
f y 600 + f y
20 600
ρ b = 0.85 2 ( ) = 0.021675
400 600 + 400
fy
0.003 +
Es
ρ max = ρ b ( ) = 0.625 ρ b = 0.01355
0.008
ρ max f y 0.01355 × 400
⇒ Ru (max) = φρ max f y (1 − ) = 0.9 × 0.01355 × 400(1 − ) = 4.1MPa
1.7 f 'c 1.7 × 20
Mu 490 × 10 6
⇒ bd = 2
= = 119.5 ×10 6 mm 3
Ru (max) 4.1
dUcenHsMrab;karsnμt; b / kMNt; d nig A = ρbd s
b = 200mm / d = 773mm / A = 20.95cm
s
2
b = 250mm / d = 691.5mm / A = 23.42cm s
2
b = 300mm / d = 630mm / A = 25.61cm ¬ 6DB 25 ¦
s
2
b = 400mm / d = 546.5mm / A = 29.62cm s
2
* kareRCIserIskMBs;RbsiT§iPaBGaRs½ynwgktþaxageRkam³
- kMBs;bnÞb;³ TTwgrbs;muxkat;tUcpþl;nUvkMBs;FñwmFM
Edlkat;bnßylMhrkMBs;. elIsBIenH FñwmeRCA ceg¥ótkat;bnßyersIusþg;m:Um:g;edaykarxUcRTg;RTayxag
lateral deformation.
- brimaN nigkarBRgaysésrEdk³ Fñwmceg¥ótRtUvkarsésrEdkeRcInCagmYyRsTab; dUcenHvabegáIt
kMBs;Fñwm.
- kMras;CBa¢aMg³ RbsinebIbøúksIum:g;t_RtUv)aneRbI TTwg b RtUv)aneRCIserIsesμInwgkMras;CBa¢aMg. sMrab;
GKarCBa¢aMgxagRkas;CagCBa¢aMgkñúg.
Flexural Design on Reinforced Concrete Beams 57

6. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
* kareRCIserIsmuxkat;EdkGaRs½ynwgktþaxageRkam³
- kartMerobEdkRKb;RKan;enAkñúgmuxkat; CaTUeTA mYyRsTab; b¤BIrRsTab; nigbMeBjtamlkçxNÐ ACI
Code sMrab;KMlatEdkGb,brma.
- épÞmuxkat;EdkeRCIserIsRtUvmantMélEk,rbMputépÞsésrEdktMrUvkar
dUcenHeyIgeRCIserIsyk b = 300mm / d = 630mm / A = 25.61cm ¬ 6 @ DB25 ¦ RtUvtMerobBIrRsTab;
s
2
kMNt;kMBs;Fñwm h = d + D + 60mm = 630 + 25 + 60 = 715mm
dUcenHyk h = 750mm ⇒ d = 665mm
]TahrN_2³ edaHRsay]TahrN_1 edayeRbI ρ RbEhl 1% nig b = 35cm
dMeNaHRsay³
eday f ' = 20MPa / f = 400MPa / ρ = 0.01355 sMrab;muxkat;rgkarTaj
c y max
ρf y 0.01 × 400
Ru = φρf y (1 − ) ⇒ Ru = 0.9 × 0.01 × 400(1 − ) = 3.1765MPa
1.7 f 'c 1.7 × 20
M u 490 × 106
⇒ bd 2 = = = 154257830mm3
Ru 3.1765
sMrab; b = 35cm ⇒ d = 665mm
As = 0.01 × 35 × 66.5 = 23.275cm 2
edayeRCIserIs 4DB28 mYyRsTab; A = 24.62cm s
2
epÞógpÞat;TTwg b = nD + (n − 1)s + 95mm edayyk s = D
min
⇒ b = 7 D + 95mm = 7 × 28 + 95 = 291mm < 35cm epÞógpÞat;
min
kMNt;kMBs;Fñwm h = d + D + 50mm = 665 + 28 + 50 = 729mm yk h = 75cm
min
2 2
edayEdkEdldak;mantMélFMCagEdkKNna dUcenHeyIgGacbnßykMBs;FñwmBI 75cm → 72cm
sMrab;karERbRbYlkMBs;Fñwm A = 23.275( 72 ) = 24.24cm < 24.62cm
s
75 2 2
d = 720 − 64 = 656mm
RtYtBinitüersIusþg;m:Um:g;
ρ=
24.62
35 × 72
= 0.0098 < ρ max ⇒ muxkat;rgkarTaj
As f y 24.62 × 400
a= = = 16.55cm
0.85 f 'c b 0.85 × 20 × 35
a 165.5
φM n = φAs f y (d − ) = 0.9 × 2462 × 400 × (656 − ) = 508.083 × 106 N .mm = 508.083kN .m > 490MPa
2 2
RtYtBinitüsac;lUteFobEdksuT§ ε t
karKNnaFñwmebtugGarem:rgkarkac;begáag 58

7. T.Chhay NPIC
0.005
εt = ( ρ ) − 0.003
ρb
20 600
ρ b = 0.85 2 ( ) = 0.021675
400 600 + 400
ρ 0.0098
= = 0.452
ρb 0.021675
εt =
0.005
0.452
− 0.003 = 0.008 > 0.005 epÞógpÞat;
mü:ageTot c=
a
=
16.55
0.85 0.85
= 19.47
c 19.47
dt
=
65.6
= 0.3 < 0.375 epÞógpÞat;
]TahrN_3³ kMNt;muxkat;EdksMrab;muxkat;eGayxageRkam b = 25cm nig h = 50cm EdlTb;nwgm:Um:g;
KNna M = 200kN .m . smμtikmμ³ f ' = 28MPa nig f = 400MPa .
u c y
dMeNaHRsay³
edaysnμt;Edk DB25 mYyRsTab;
d = 500 − 70 = 430mm = 43cm
eday f 'c = 28MPa / /
f y = 400 MPa β1 = 0.85
f 'c 600
ρ b = 0.85β1 ( )
f y 600 + f y
28 600
ρb = 0.852 ( ) = 0.030345
400 600 + 400
f
0.003 + y
Es
ρ max = ρb ( ) = 0.625ρb = 0.019
0.008
ρ=
0.85 f 'c
fy
(1 − 1 −
4M u
1.7φf 'c bd 2
) edaysnμt; φ = 0.9
0.85 × 28 4 × 200 × 106
ρ=
400
(1 − 1 −
1.7 × 0.9 × 28 × 250 × 4302
) = 0.01356 < ρ max muxkat;rgkarTajBitR)akd
As = ρbd = 0.01356 × 25 × 43 = 14.58cm2
DB 25 ⇒ As 25 = 4.9cm 2
A
⇒n= s =3
As 25
dUcenH A = 3DB25 = 14.7cm
s
2
]TahrN_4³ kMNt;muxkat;EdkcaM)ac;sMrab; b = 35cm RbsinebIvaRbQmnwgm:Um:g;KNna M u = 425kN .m .
smμtikmμ³ f ' = 28MPa nig f = 400MPa .
c y
Flexural Design on Reinforced Concrete Beams 59

8. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
dMeNaHRsay³
sMrab; f ' = 28MPa / f = 400MPa / β
c y 1 = 0.85
ρ = 0.030345 / ρ = 0.019
b max
sMrab;muxkat;rgkarTaj φ = 0.9
edayeRbI ρ = 0.019
max
ρmax f y 0.019 × 400
⇒ Ru (max) = φρmax f y (1 − ) = 0.9 × 0.019 × 400(1 − ) = 5.75MPa
1.7 f 'c 1.7 × 28
Mu 425 × 106
bd 2 = = = 73.913 × 106 mm3
Ru (max) 5.75
sMrab; b = 35cm ⇒ d = 46cm
⇒ As = 0.019 × 35 × 46 = 30.59cm 2
edayeRbIEdk φ 32 4edIm 4DB32 = 32.154cm 2
EdkRtUv)antMerobmYyCYr ⇒ b = nD + (n − 1)s + 95mm
min
edayyk D = s ⇒ b = 7 D + 95mm = 7 × 32 + 95 = 319mm < 350mm RtwmRtUv
min
kMBs;Fñwm h = d + D + 50mm = 460 + 32 + 50 = 526mm yk 53cm
2 2
BiPakSa³
edaysarsésrEdkeRbIR)as; 32.154cm FMCagsésrEdktMrUvkar 30.59cm
2 2
32.154
⇒ρ= = 0.02 > ρ max
35 × 46
sac;lUteFobEdksuT§ εt = (
0.005
) − 0.003 = 0.0046 > 0.004 ⇒
ρ
muxkat;sßitkñúgtMbn; transition region
ρb
250
⇒ φ = 0.65 + (ε t − 0.002)( ) = 0.867 < 0.9
3
As f y 32.154 × 400
a= = = 15.44cm
0.85 f 'c b 0.85 × 28 × 35
a 154.4
φM n = φAs f y (d − ) = 0.867 × 3215.4 × 400(460 − ) = 426.86 × 106 N .mm = 426.86kN .m ≈ 425kN .m
2 2
eyIgeXIjfa sMrab;bMErbMrYlmuxkat;Edk 32.154 − 30.59 = 1.546cm RbEhl 5% enaHersIusþg;m:Um:g;man
2
tMélRbEhlKña edaysar φ fycuH. dUcenHkarKNnamuxkat;edayeGaymuxkat;rgkarTaj φ = 0.9 man
lkçN³esdækic©.
karKNnaFñwmebtugGarem:rgkarkac;begáag 60

9. T.Chhay NPIC
4> muxkat;ctuekaNEkgCamYyEdkrgkarsgát;
RbsinebIm:Um:g;KNnamantMélFMCagersIusþg;m:Um:g;kñúg enaHmuxkat;RtUv)armuxkat;EdkbEnßmsMrab;tMbn;
rgkarTaj nigrgkarsgát;. sésrEdkrgkarsgát;pþl;nUvkMlaMgsgát;bEnßmeTAelIkMlaMgsgát;rbs;ebtug.
k> edaysnμt;EdkrgkarTajmanmYyRsTab;
viFIsaRsþKNnasMrab;muxkat;ctuekaNEkgCamYyEdkrgkarsgát; enAeBleKsÁal; M / f ' / b / d u c
nig d ' mandUcxageRkam³
- KNnaPaKryEdk balanced ρ nigPaKryEdkGtibrma ρ edayeRbIsmIkarxageRkam³
b max
fy
0.003 +
ρ b = 0.85β1
f 'c
(
600
f y 600 + f y
) nig ρ max = ρb (
0.008
Es
)
kMNt;muxkat;EdkGtibrmasMrab;rgkarTaj A s1 = ρ maxbd
- KNna R edayeRbI ρ (φ = 0.9)
u (max) max
ρ max f y
Ru (max) = φρmax f y (1 − )
1.7 f 'c
- KNnaersIusþg;m:Um:g;kñúgEdlekItedaysarmuxkat;EdkrgkarTaj M edayeRbI R u1 u (max)
M u1 = Ru (max)bd 2
+ RbsinebI M u1 enaHRtUvkarEdkrgkarsgát;
< Mu
+ RbsinebI M > M enaHminRtUvkarEdkrgkarsgát;eT. KNnaPaKryEdk ρ tamrUbmnþ
u1 u
) nigKNna A = ρbd .
0.85 f 'c 4M
ρ= (1 − 1 − u
1.7φf ' bd 2 s
fy c
- KNna M = M − M CaersIusþg;m:Um:g;EdlekItBIEdkrgkarsgát;
u2 u u1
- KNna A BI M = φA f (d − d ' ) nigbnÞab;mkKNna A = A + A
s2 u2 s2 y s s1 s2
- KNnakugRtaMgenAkñúgEdkrgkarsgát;dUcxageRkam
c − d'
+ KNna f ' = 600( s )≤ f y
c
+ b¤KNna ε ' BIdüaRkambMErbMrYlrageFob nig f ' = ε ' E . RbsinebI ε ' = ε enaHEdkrgkar
s s s s s y
sgát;yar ehIy f ' = f . s y
+ KNna A' BI M = φA f ' (d − d ' ) .RbsinebI f ' = f enaH A' = A . EtebI f ' < f
s u2 s2 s s y s s2 s y
enaH A' > A ehIy A' = A ( ff ' ) .
s s2 s s2
y
s
- eRCIserIsmuxkat;EdksMrab; A nig A' edIm,IeGaysmlμmnwgTTwg b . CaTUeTA Edk A eRcIntM
s s s
erobCaBIrRsTab; É A' eRcIntMerobCamYyRsTab;.
s
Flexural Design on Reinforced Concrete Beams 61

10. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
- KNna h = d + 65mm sMrab;EdkrgkarTajmYyRsTab; nig h = d + 90mm sMrab;Edkrgkar
TajBIrRsTab;. RtYtBinitüfa [ ρ − ρ ' ( ff ' )] < ρ edayeRbI d fμI b¤RtYtBinitü
s
max
y
As (max) = bd [ ρ max + ρ ' (
f 's
fy
)] ≥ As eday ρ = bd nig ρ '= bd' . karRtYtBinitüenHmincaM)ac;eT
A s A s
RbsinebI ρ RtUv)aneRbIenAkñúgmuxkat;eKal.
max
- RbsinebIcaM)ac; eKRtUvKNnaersIusþg;m:Um:g;énmuxkat;cugeRkay φM ehIyeRbobeFobCamYy M n u
edayeGay φM ≥ M . n u
- RtYtBinitüsac;lUtEdksuT§ ε = ( d c− c )0.003 ≥ 0.005 t
t
x> edaysnμt;EdkrgkarTajmanBIrRsTab;
kúñgkrNIEdkrgkarTajmanBIrRsTab; eKGacsnμt;fa d = h − 90mm nig
d = h − 65mm = d + 25mm . eKmanviFIBIry:agkñúgkarkMNt;muxkat;sésrEdk³
t
- vIFITI1³ edaysnμt; sac;lUteFobEdksuT§enAnIv:UTIRbCMuTMgn;EdkTajesμI 0.005 b¤ ε = 0.005 s
enAnI v:U d . kñúgkrNIenH sac;lUteFobEdksuT§ sMrab;sésrEdkRsTab;eRkambMputmantMélFMCag
0.005 .
d −c
ε =( t )0.003 > 0.005 . dUcenHeyIgedaHRsaydUckrNIxagelI EdkrgkarTajmanmYyRs
t
c
Tab;.
- vIFITI2³ edaysnμt; ε = 0.005 enAnIv:UEdkRsTab;eRkameK d . kñúgkrNIenH sac;lUtEdkeFob
t t
enAnIv:UTIRbCMuTMgn;EdkTajmantMéltUcCag 0.005 ³ ε = ( d c− c )0.003 < 0.005 EdlenAEtGac s
t
TTYlyk)an. dMeNaHRsaytamviFITI2enH segçbdUcxageRkam³
+ KNna d = h − 65mm / c = ( )d nig a = β c
3
t t 1
8
+ KNnakMlaMgsgát;enAkñúgebtug C = 0.85 f ' ab = T = A f 1 c 1 s1 y
kMNt; A . KNna M = φA f (d − a ) . ρ = bd / φ = 0.9
s1
2
u1
A
s1 y 1
s1
+ KNna M = M − M edaysnμt; d ' = 65mm
u2 u u1
+ KNna A BI M = φA f (d − d ' ) eday f ' = f . muxkat;EdkrgkarTajsrub
s2 u2 s2 y s y
As = As1 + As 2
+ KNnakugRtaMgenAkñúgEdkrgkarsgát;dUcxageRkam
c − d'
× KNna f ' = 600(s )≤ f y
c
karKNnaFñwmebtugGarem:rgkarkac;begáag 62

11. T.Chhay NPIC
b¤KNna ε ' BIdüaRkambMErbMrYlrageFob nig f ' = ε ' E . RbsinebI ε ' = ε enaHEdkrg
× s s s s s y
karsgát;yar ehIy f ' = f . s y
× KNna A' BI M = φA f ' (d − d ' ) . RbsinebI f ' = f enaH A' = A . EtebI
s u2 s2 s s y s s2
f ' < f enaH A' > A ehIy A' = A ( ) .
f y
s y s s2 s s2
f' s
]TahrN_5³ eKmanFñwmmYymanmuxkat; b = 25cm nigkMBs; h = 55cm EdlRtUvRTnUvm:Um:g;Bt;KNna
M = 300kN .m . KNnasésrEdktMrUvkar. smμtikmμ f ' = 20MPa nig f = 345MPa .
u c y
dMeNaHRsay³
- kMNt;ersIusþg;m:Um:g;EdlekItBIEdkrgkarTajCaeKalsMrab;lkçxNÐmuxkat;rgkarTaj
sMrab; f ' = 20MPa nig f = 345MPa / β = 0.85
c y 1
f 'c 600 20 600
ρb = 0.85β1 ( ) = 0.852 ( ) = 0.0266
f y 600 + f y 345 600 + 345
fy 345
0.003 + 0.003 +
Es
ρ max = ρb ( ) = 0.0266( 200000 ) = 0.0157
0.008 0.008
ρ max f y 0.0157 × 345
Ru (max) = φρmax f y (1 − ) = 0.9 × 0.0157 × 345(1 − ) = 4.1MPa
1.7 f 'c 1.7 × 20
edaysnμt;EdkrgkarTajmanBIrRsTab;
d = h − 90mm = 550 − 90 = 460mm
⇒ φM n = Ru (max)bd 2 = 4.1 × 250 × 4602 = 216.89 × 106 N .mm = 216.89kN .m < M u = 300 MPa
dUcenHmuxkat;enHRtuvkarEdkrgkarsgát;edIm,ITb;Tl;nwgm:Um:g;EdlenAsl;.
- KNna A / M nig M
s1 u1 u2
As1 = ρ maxbd = 0.0157 × 25 × 46 = 18.06cm 2
M u1 = φM n = 216.89kN .m
⇒ M u 2 = M u − M u1 = 300 − 216.89 = 83.11cm 2
- KNna A nig A' Edl)anBI M edaysnμt; d '= 6cm
s2 s u2
M u 2 = φAs 2 f y (d − d ' )
Mu2 83.11× 106
As 2 = = = 669mm2 = 6.69cm2
φf y (d − d ' ) 0.9 × 345 × (460 − 60)
EdkrgkarTajsrub A = A + A = 18.06 + 6.69 = 24.75cm
s s1 s2
2
Edkrgkarsgát; A' = A = 6.69cm eFVIkardl;yar
s s2
2
- RtYtBinitüEdksgát;eFVIkardl;yar
Flexural Design on Reinforced Concrete Beams 63

12. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
fy 345
εy = = = 0.001725
200000 200000
1806 × 345
eday a=
As1 f y
=
0.85 f 'c b 0.85 × 20 × 250
= 146.6mm
a 146.6
c= = = 172.47mm
β1 0.85
172.47 − 60
ε 's = (
172.47
) × 0.003 = 0.00196 > ε y eFVIkardl;yar
- RtYtBinitü ε t
edayeyIgeRbI ρ nig R mkeRbIsMrab;edaHRsay edayeGaysac;lUteFobEdksuT§enAnIv:UTIRbCMuTMgn;Edk
max u
ε = 0.005 . dUcenHeyIgRtUvkMNt; ε sMrab;EdkenARsTab;eRkameKbMput.
s t
d t = 550 − 60 = 490mm
d −c 490 − 172.47
εt = ( t
c
)0.003 = (
172.47
)0.003 = 0.0055 > 0.005 RtwmRtUv
dUcenH EdktMrUvkarsMrab;karTaj A = 24.75cm eRbI 5DB25
s
2
sMrab;karsgát; A' = 6.69cm eRbI 2DB22
s
2
]TahrN_6³ eKmanFñwmmYymanmuxkat; b = 30cm nigkMBs; h = 50cm EdlRtUvRTnUvm:Um:g;Bt;KNna
M = 400kN .m . KNnasésrEdktMrUvkar. smμtikmμ f ' = 28MPa nig f = 400MPa .
u c y
dMeNaHRsay³ manBIrviFIsaRsþ
vIFITI1³
- kMNt;ersIusþg;m:Um:g;Gtibrmaénmuxkat;EdlmanEtEdkrgkarTajCaeKal
sMrab; f ' = 28MPa / f = 400MPa / β = 0.85
c y 1
ρ = 0.030345 / ρ = 0.019
b max
sMrab;muxkat;rgkarTaj φ = 0.9
karKNnaFñwmebtugGarem:rgkarkac;begáag 64

13. T.Chhay NPIC
ρmax f y 0.019 × 400
⇒ Ru (max) = φρmax f y (1 − ) = 0.9 × 0.019 × 400(1 − ) = 5.75MPa
1.7 f 'c 1.7 × 28
edaysnμt; d = h − 90 = 500 − 90 = 410mm EdkmanBIrRsTab;
⇒ M u1 = Ru bd 2 = 5.75 × 300 × 4102 = 290 × 106 N .mm < 290MPa
dUcenHRtUvkarEdksgát;
- KNna A , M , A nig A
s1 u2 s2 s
As1 = ρ maxbd = 0.019 × 30 × 41 = 23.37cm 2
M u 2 = M u − M u1 = 400 − 290 = 110kN .m
M u 2 = φAs 2 f y (d − d ' ) edaysnμt; d '= 60mm
110 × 106
⇒ As 2 = = 873mm 2 = 8.73cm 2
0.9 × 400 × (410 − 60)
muxkat;EdkTajsrub A = A + A = 23.37 + 8.73 = 32.1cm
s s1 s2
2
eRbIEdk 5DB30
- RtYtBinitüEdkrgkarsgát;eFVIkardl;yarb¤Gt;
f 'c d ' 600
ρ − ρ ' ≥ K = 0.85β1 ( )( )
f y d 600 + f y
28 60 600
K = 0.852 ( )( ) = 0.00444
400 410 600 + 400
A 23.37
ρ − ρ ' = s1 = = 0.019 < K
bd 30 × 41
Edkrgkarsgát;eFVIkarmindl;yareT dUcenH
⇒ f 's < f y
- KNna f ' ³ s
c − d'
f 's = 600( ) ≤ fy
c
kMNt; c BI A s1 = 23.37cm2
As1 f y 2337 × 400
a= = = 130.92mm
0.85 f 'c b 0.85 × 28 × 300
a 130.92
c= = = 154.02mm
β1 0.85
154.02 − 60
f 's = 600( ) = 366.3MPa ≤ f y
154.02
- KNna A' BI M s u2 = φA's f 's (d − d ' )
110 × 106
A's =
0.9 × 366.3 × (410 − 60)
= 953mm 2 = 9.53cm 2 eRbIEdk 2DB25
eyIgeXIjva eKarBtamlkçxNÐ
Flexural Design on Reinforced Concrete Beams 65

14. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
f 's
(ρ − ρ ' ) ≤ ρ max
fy
- RtYtBinitü sac;lUteFobEdksuT§ ε enAnIv:UEdkRsTab;xageRkam ¬eRBaHeyIgsnμt;fa sac;lUteFobEdksuT§
t
ε enAnIv:UTIRbCMuTMgn;Edk¦
s
d t = 500 − 60 = 440mm
d −c 440 − 154.02
εt = ( t
c
)0.003 = (
154.02
)0.003 = 0.0056 > 0.005 RtwmRtUv
vIFITI2³
edayeRbIEdkrgkarTajBIrRsTab; nigsac;lUteFobEdksuT§enARsTab;eRkambMput ε t = 0.005
- KNna d = 500 − 60 = 440mm BIdüaRkamsac;lUteFob
t
c 0.003 0.003
= = = 0.375
dt 0.003 + ε t 0.008
⇒ c = 0.375d t = 165mm
⇒ a = 0.85c = 140.25mm
- kMlaMgsgát;kñúgebtug
C1 = 0.85 f 'c ab = 0.85 × 28 × 140.25 × 300 = 1001385 N = 1001.385kN
¬sMrab;muxkat;manEdkrgkarTajCaeKal¦
eday C = T ⇒ A = T = 1001385 = 2503mm
1 1
f
s1
400
1 2
= 25.03cm 2
y
d = 500 − 90 = 410mm
a 140.25
M u1 = φAs1 f y (d − ) = 0.9 × 2503 × 400 × (410 − ) = 306.25 × 106 N .mm = 306.25kN .m
2 2
A 25.03
ρ1 = s1 = = 0.0203
bd 30 × 41
- eday M u > M u1 dUcenHmuxkat;RtUvkarEdkrgkarsgát;
karKNnaFñwmebtugGarem:rgkarkac;begáag 66

15. T.Chhay NPIC
M u 2 = 400 − 306.25 = 93.75kN .m
M u2 93.75 × 106
⇒ As 2 = = = 744mm 2 = 7.44cm 2
0.9 f y (d − d ' ) 0.9 × 400 × (410 − 60)
muxkat;EdkrgkarTajsrub A = A + A = 25.03 + 7.44 = 32.47cm eRbIEdk 5DB30
s s1 s2
2
- RtYtBinitü Edkrgkarsgát;eFVIkardl;yarb¤Gt;
28 60 600
K = 0.852 ( )( ) = 0.00444
400 410 600 + 400
( ρ − ρ ' ) = ρ1 < K ⇒ Edkrgkarsgát;eFVIkarmindl;yareT dUcenH f 's < f y
c − d' 165 − 60
f 's = 600( ) = 600( ) = 381.82MPa
c 165
KNna A' BI M s u2
93.75 × 106
⇒ A's =
M u2
=
0.9 f 's (d − d ' ) 0.9 × 381.82 × (410 − 60)
= 779mm 2 = 7.79cm 2 eRbIEdk 2DB25
- RtYtBinitüersIusþg;m:Um:g;kñúg
As = 5 DB30 = 35.325cm 2 A's = 2 DB 25 = 9.81cm 2 As1 = As − A's = 25.515cm 2
a
φM n = φ[ As1 f y (d − ) + A's f 's (d − d ' )]
2
140.25
φM n = 0.9[2551.5 × 400 × (410 − ) + 981× 381.82 × (410 − 60)] = 430.18 × 106 N .mm
2
d −c 410 − 165
eyIgeXIjfa εs = (
c
)0.003 = (
165
)0.003 = 0.0045
5> KNnamuxkat;GkSret T
kñúgkarKNnamuxkat;GkSr T edaysÁal;m:Um:g;KNna M / kMras;søab t nigTTwgsøab b . kMras;rbs;
u
RTnug b ERbRbYlBI 20cm → 50cm . GBaØtiBIreTotEdlRtUvkarKNnaKW kMBs;RbsiT§iPaB d nigmuxkat;Edk
w
A . eKmanBIkrNIEdlCYbRbTH³
s
- enAeBleKsÁal; d
+ RtYtBinitüfa muxkat;eFVIkarCaragctuekaNEkg b¤GkSret T edaysnμt; a = t
KNnaersIusþg;m:Um:g;sMrab;søabTaMgmUl
t
φM nf = φ (0.85 f 'c )bt (d − )
2
RbsinebI M > φM enaH a > t KNnaCaragGkSret T. RbsinebI M
u nf u < φM nf enaH a < t KNna
CaragctuekaNEkg.
Flexural Design on Reinforced Concrete Beams 67

16. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
+ RbsinebI a < t enaHKNna ρ = 0.85 f ' (1 −
f
c
1−
4M u
1.7φf 'c bd 2
) / KNna A = ρbd .
s
y
epÞógpÞat; ρ ≥ ρ .
w min
+ RbsinebI a > t enaHKNna A sMrab;Epñksøabsgxag ¬>>>¦ sf
Asf = 0.85 f 'c (b − bw )t / f y
t
M u 2 = φAsf f y (d − )
2
m:Um:g;EdlTb;edayRTnugKW
M u1 = M u − M u 2
KNna ρ edayeRbI M / b nig d
1 u1 w
0.85 f 'c 4 M u1
ρ1 = (1 − 1 − )
fy 1.7φf 'c bwd 2
nigKNna A = ρ b d s1 1 w
muxkat;Edksrub A = A + A s s1 sf
bnÞab;mkRtYtBinitü A ≤ A dUcKña RtUvRtYtBinitü ρ
s s max w =
A
bw d
≥ ρ min
+ RbsinebI a = t enaH A = φ 0.85f f ' bt s
c
y
- enAeBleKminsÁal; d nig A karKNnaRtUveFVItamviFIsaRsþxageRkam
s
+ snμt; a = t nigKNnabrimaNEdksrub A EdlRtUvkarsMrab;Tb;nwgkMlaMgsgát;kñúgsøab sft
TaMgmUl bt
0.85 f 'c bt
Asft =
fy
+ kMNt; d BI A nig a = t tamrUbmnþxageRkam³
sft
t
M u = φAsft f y (d − )
2
RbsinebI eKyktam d KNnarkeXIjenaH A = A nig h = d + 65mm sMrab;
× s sft
EdkmanmYyRsTab; b¤ h = d + 90mm sMrab;EdkmanBIrRsTab;.
× RbsinebI eKyk d fμIFMCag d KNnaenaHmuxkat;eFVIkarCaragctuekaNEkg.
1
ehIy ρ = 0.85 f ' (1 − 1 − 1.7φfM bd ) / KNna A = ρbd .
f
4c
'
u
2 s
y c
karKNnaFñwmebtugGarem:rgkarkac;begáag 68

17. T.Chhay NPIC
RbsinebI eKyk d fμItUcCag d KNnaenaHmuxkat;eFVIkarCaragGkSr T. ehIy
× 21
muxkat;EdkcugeRkay A FMCag A . kñúgkrNIenH eKRtUveFVIdUckrNIxagelI ¬>>>¦ edIm,IkM
s sft
Nt;muxkat; A . s
]TahrN_7³ eKmanFñwmmYymanGkSr T EdlmanRTnug b = 25cm RbEvgsøab b = 100cm kMras;søab w
t = 10cm nigkMBs;RbsiT§iPaB d = 37cm . kMNt;muxkat;EdkcaM)ac; edIm,ITb;nwgm:Um:g; M = 375kN .m . u
smμtikmμ f ' = 20MPa nig f = 400MPa .
c y
dMeNaHRsay³
- RtYtBinitüGkS½NWt edaysnμt; a = t = 10cm
t 100
φM n = φ 0.85 f 'c bt (d − ) = 0.9 × 0.85 × 20 × 1000 × 100(370 − ) = 489.6 × 106 N .mm
2 2
eday φM > M ⇒ muxkat;manlkçN³CactuekaNEkg
n u
- kMNt;muxkat;EdkrgkarTaj
0.85 f 'c 4M u 0.85 × 20 4 × 375 × 106
ρ= (1 − 1 − )= (1 − 1 − ) = 0.00845
fy 1.7φf 'c bd 2 400 1.7 × 0.9 × 20 × 1000 × 3702
As = ρbd = 0.00845 × 100 × 37 = 31.265cm 2
edayeRbI 5DB30 = 35.325cm 2
- RtYtBinitü ρ = bAd = 25.× 37 = 0.0382 > ρ
w
35 325 s
min =
1.4
400
= 0.0035
w
0.85 f 'c
As max = [(be − bw )t + 0.375β1bw d ] = 44.41cm 2 > As
fy
dUcKña a = 0.85 ff ' b = 035.325 ××400 = 8.31cm
A s y
.85 × 20 100
c
37 − 9.78
c=
β1
a
=
8.31
0.85
= 9.78cm nig ε s = 0.003(
9.78
) = 0.00835 > 0.005
]TahrN_8³ RbBn§½kMralxNÐdUcbgðajkñúgrUb EdlpÁúMeLIgedaykMralxNÐEdlmankMras; t = 8cm EdlRT
edayFñwmRbEvg L = 430cm EdlmanKMlatBIKña l = 300cm KitBIGkS½mkGkS½. FñwmmanRTnug b = 35cm w
Flexural Design on Reinforced Concrete Beams 69

18. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
nigkMBs; RbsiT§PaB d = 47cm . kMNt;muxkat;EdkcaM)ac;edIm,ITb;nwgm:Um:g;KNna M u = 575kN .m .
smμtikmμ f ' = 20MPa nig f = 400MPa .
c y
dMeNaHRsay³
- kMNt;RbEvgsøab
⎧16t + bw ⎧16 × 80 + 350 = 1630mm
⎪ L ⎪ 4300
be = min ⎨ = min ⎨ = 1075mm
⎪ 4 ⎪ 4
⎩ l ⎩ 3000mm
dUcenH b = 1075mm
e
- RtYtBinitüTItaMgGkS½NWt edaysnμt; a = t
t
φM n = φ 0.85 f 'c bt (d − )
2
80
φM n = 0.9 × 0.85 × 20 × 1075 × 80(470 − ) = 565.794 N .mm = 565.794kN .m < 575kN .m
2
eday φM < M ⇒ muxkat;manlkçN³CaragGkSr T dUcenH a > t .
n u
- kMNt;muxkat;EdksrubsMrab;Tb;Tl;CamYykMlaMgsgát;kñúgsøab
0.85 f 'c (b − bw )t 0.85 × 20 × (1075 − 350)80
Asf = = = 2456mm 2
fy 400
t 80
⇒ M u 2 = φAsf f y (d − ) = 0.9 × 2456 × 400(470 − ) = 380.188 × 106 N .mm
2 2
0.85 f 'c 4 M u1
⇒ ρ1 = (1 − 1 − )
fy 1.7φf 'c bwd 2
0.85 × 20 4 × 194.812 × 106
ρ1 = (1 − 1 − = 0.0077
400 1.7 × 0.9 × 20 × 350 × 4702
enaH A = ρ b d = 0.0077 × 35 × 47 = 12.67cm
s1 1 w
2
⇒ muxkat;EdkTajsrub A = 12.67 + 24.56 = 37.23cm
s
2
eRbI 6DB30 BIrRsTab;
- kMBs;srubrbs;muxkat; h = 470 + 90 = 560mm
0.85 f 'c
As max = [(be − bw )t + 0.375β1bwd ] = 50.87cm 2 > As
fy
karKNnaFñwmebtugGarem:rgkarkac;begáag 70

19. T.Chhay NPIC
42.39 × 400
- RtYtBinitü ε / t a=
As f y
0.85 f 'c b
=
0.85 × 20 × 107.5
= 9.278cm
a 9.278
c= = = 10.91cm
β1 0.85
dt = 56 − 6 = 50cm
50 − 10.91
⇒ ε t = 0.003( ) = 0.0107 > 0.005
10.91
]TahrN_9³ sMrab;RbBn§½kMralxNÐmYy EdlmanTTwgsøabRbEvg b = 122cm TTwgRTnug b = 40 ehIykM e w
ralxNÐmankMras; t = 10cm . KNnamuxkat;GkSr T edIm,IrgnUvm:Um:g;KNna M = 1100kN .m . smμtikmμ u
f ' = 20MPa nig f = 400MPa .
c y
dMeNaHRsay³
- edaysarminsÁal;kMBs;RbsiT§PaB eyIgeGay a = t
kMNt; muxkat;Edk A sMrab;søabTaMgmUl
sft
0.85 f 'c bt 0.85 × 20 × 122 × 10
Asft = = = 51.85cm 2
fy 400
eGay M t
= φAsft f y (d − )
u
2
d KNna
Mu t 1100 × 106 100
⇒d = + = + = 639.3mm
φAsft f y 2 0.9 × 5185 × 400 2
eyIgeXIjfa RbsinebI d = 63.93cm enaH A = A s sft
- RbsinebI d > 63.93cm / eyIgsnμt;yk d = 67cm enaH a < t muxkat;manlkçN³CaragctuekaNEkg
PaKryEdk RtUv)anKNnatamrUbmnþxageRkam³
0.85 f 'c 4M u
ρ= (1 − 1 − )
fy 1.7φf 'c bw d 2
0.85 × 20 4 × 1100 × 106
⇒ρ= (1 − 1 − = 0.006
400 1.7 × 0.9 × 20 × 1220 × 6702
enaHmuxkat;Edk A = ρbd = 0.006 × 122 × 67 = 49.04cm
s
2
- RbsinebI d < 63.93cm / eyIgsnμt;yk d = 60cm enaH a > t muxkat;manlkçN³CaragGkSret T .
0.85 f 'c t (b − bw ) 0.85 × 20 × 10 × (122 − 40)
Asf = = = 34.85cm 2
fy 400
Flexural Design on Reinforced Concrete Beams 71

20. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
t 100
M u 2 = φAsf f y (d − ) = 0.9 × 3485 × 400(600 − ) = 690 × 106 N .mm
2 2
M u1 = 1100 − 690 = 410kN .m
sMrab;muxkat;EdkrgkMlaMgTajeKal b w /
= 40cm d = 60cm nig M u1 = 410kN .m
M u1 410
Ru = = = 2847.2 kN m 2 = 2.85MPa
bw d 2
0.4 × 0.602
0.85 f 'c 4 Ru 0.85 × 20 4 × 2.85
⇒ ρ1 = (1 − 1 − )= (1 − 1 − ) = 0.0088
fy 1.7φf 'c 400 1.7 × 0.9 × 20
As1 = ρ1bwd = 0.0088 × 40 × 60 = 21.12cm2
⇒ As = As1 + Asf = 34.85 + 21.12 = 55.97cm2
eRbIEdk 7 DB32 = 56.27cm 2
- RtYtBinitü ε t
As1 f y 21.12 × 400
a= = = 12.42cm
0.85 f 'c bw 0.85 × 20 × 40
a 12.42
c= = = 14.6cm
0.85 0.85
dt = 63cm
⎛d −c⎞
⇒ ε t = 0.003⎜ t
⎝ c ⎠
⎟ = 0.0099 > 0.005 muxkat;rgkarTaj
- KNnamuxkat;EdksrubGtibrma
f 'c 20
As max = 0.85 [(be − bw )t + 0.375β1bw d ] = 0.85 [(1220 − 400)100 + 0.375 × 0.85 × 400 × 600)]
fy 400
As max = 6736mm 2 = 67.36cm 2 > 56.27cm 2 RtwmRtUv
karKNnaFñwmebtugGarem:rgkarkac;begáag 72