1. This document discusses the flexural analysis of reinforced concrete beams. It includes assumptions made for the analysis, procedures for determining the moment capacity, and calculations for strain conditions in different sections.
2. Methods are described for determining the moment capacity based on the reinforcement ratio and limiting the flexural strain to 0.003. Equations are provided to calculate the strain in the concrete and steel based on the section type (e.g. tension controlled, compression controlled).
3. Procedures for calculating the service load moment capacity using factors for dead and live loads are outlined. Equations are given for calculating the service load bending moment.
6. 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
viPaKFñwmebtugGarem:rgkarBt;begáag 23
8. 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
viPaKFñwmebtugGarem:rgkarBt;begáag 25
12. 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
viPaKFñwmebtugGarem:rgkarBt;begáag 29
18. 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.
viPaKFñwmebtugGarem:rgkarBt;begáag 35
20. 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.
viPaKFñwmebtugGarem:rgkarBt;begáag 37
24. 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
viPaKFñwmebtugGarem:rgkarBt;begáag 41
28. 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
viPaKFñwmebtugGarem:rgkarBt;begáag 45
30. 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
viPaKFñwmebtugGarem:rgkarBt;begáag 47
32. 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
viPaKFñwmebtugGarem:rgkarBt;begáag 49
34. 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.
viPaKFñwmebtugGarem:rgkarBt;begáag 51