Vi deflection and control of cracking

Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa

VI. PaBdab nigsñameRbH
1> sñameRbHenAkñúgeRKOgbgÁúMebtug Deflection of Structural Concrete Members

Ggát;ebtugrgkarBt;RtUv)anKNnasMrab;suvtßiPaB nigkareRbIR)as; serviceability . Ggát;Edl)an
KNnaedayeKarBtamsmIkar nigkarkMNt;EdlEcgedaybTdæan ACI Code nwgmansuvtßPaB. dUcenH dUc
karBnül;kñúgemeronmun TMhMGgát;nImYy²Edl)ankMNt; d¾dUcbrimaNEdkcaM)ac;edIm,IrkSanUvlT§PaBm:Um:g;kñúg
esμIb¤FMCagm:Um:g;xageRkA. enAeBlEdl TMhMmuxkat;cugeRkayRtUv)anKNna FñwmRtUv)anepÞógpÞat;sMrab;lkç
xNÐbMerIbMras; serviceability dUcCa sñameRbH nigPaBdab. edIm,IkarBarnUvsñameRbH nigPaBdabelIslb; eK
caM)ac;manPaBrwgRkajrbs;Ggát;RKb;RKan;.
kareRbIR)as; ACI Code provision Edl)anBicarNanUvTMnak;TMng nonlinear relationship rvag
stress nig strain enAkñúgebtug )anpþl;CalT§plnUvmuxkat;tUcCagmuxkat;EdlKNnadayRTwsþIeGLasÞic. bT

dæan ACI Code ,Section 9.4 TTYlsÁal;nUvkareRbIR)as;EdkEdlman yield strength rhUtdl; 560MPa nig
kareRbIR)as;nUvebtugersIusþg;x<s;. kareRbInUvebtug nigEdkersIusþg;x<s;pþl;CalT§plnUvmuxkat;tUcCag ehIy
karbnßynUvPaBrwgRkajTb;nwgkarBt;rbs;Ggát; )aneGayPaBdabekInFMCag.
PaBdabEdlGacGnuBaØatiRtUv)ankMNt;edayktþaepSg²dUcCa RbePTGKar rUbragrbs;GKar vtþman
rbs;Bidan nigCBa¢aMg karxUcxatrMBwgTukEdlbNþalBIPaBdabelIslb; nigRbePT nigTMhMrbs;bnÞúkcl½t.
bTdæan ACI Code , Section 9.5 kMNt;bBa¢ak;kMras;Gb,brmasMrab;Ggát;rgkarBt;mYyTis one-way
flexural members nigkMralxNÐmYyTis dUcbgðajkñúgtaragTI 1. tMélTaMgGs;kñúgtaragKwsMrab;Ggát;Edl

minRT b¤EdlminP¢ab;eTAnwgCBa¢aMgxNÐ b¤sMng;epSgeTotEdlTMngeFVIeGayxUcxatedayPaBdabFM.
taragTI1³ kMras;Gb,brmarbs;Fñwm nigkMralxNÐmYyTis ¬ L RbEvgElVg¦
Yield
Ggát; Strength
TMrsamBaØ cugmçagCab; cugsgçagCab; Cantilever
f y (MPa)

kMralxNÐtan;mYyTis 275 L / 25 L / 30 L / 35 L / 12.5

350 L / 22 L / 27 L / 31 L / 11

400 L / 20 L / 24 L / 28 L / 10
Fñwm b¤kMralxNÐ 275 L / 20 L / 23 L / 26 L / 10
ribbed mYyTis 350 L / 18 L / 20.5 L / 23.5 L/9

400 L / 16 L / 18.5 L / 21 L/8

Deflection and Control of Cracking 90

T.Chhay NPIC

kMras;Gb,brmaEdlbgðajenAkñúgtaragTI1 RtUv)aneRbIsMrab;Ggát;EdlplitBIebtugTMgn;Fmμta
normal-weight concrete, W = 2320kg / m 3 nigsMrab;EdkEdlman yield strength dUcbgðajkñúgtarag.

tMélRtUv)anEktMrUvsMrab;krNIebtugTMgn;Rsal b¤ yield strength rbs;EdkxusBI 400MPa dUcxageRkam³
- sMrab;rbtugTMgn;RsalEdlmanTMgn;maDsßitkñúgcenøaH 1400kg / m3 − 1900kg / m3 tMélenAkñúgtarag
sMrab; f y = 400MPa RtUvKuNnwg (1.65 − 0.0003125Wc ) b:uEnþminRtUvtUcCag 1.09 . Edl Wc Ca
m:as;maDebtug.
- sMrab; yield strength rbs;EdkxusBI 400MPa tMélenAkñúgtarag sMrab; f y = 400MPa RtUvKuNnwg
(0.4 + f y / 689.5) .

2> PaBdabxN³ Instantaneous Deflection

PaBdabrbs;Ggát;ekIteLIgCacMbgedaysarbnÞúkefrbUknwgbnÞúkGefrmYyEpñk b¤TaMgGs;. PaBdab
EdlekIteLIgPøam²bnÞab;BIGnuvtþbnÞúkeKeGayeQμaHfa PaBdabPøam² immediate deflection b¤PaBdabxN³
instantaneous deflection . eRkambnÞúkEdlGnuvtþsßitesßr PaBdabnwgekIneLIgKYreGaykt;sMKal;CamYynwg

eBl. eKmanviFIepSg²sMrab;KNnaPaBdabenAkñúgeRKOgbgÁúMsþaTickMNt; nigsþaTicminkMNt;. karKNnaPaB
dab instantaneous deflection KWQrelI elastic behavior énGgát;rgkarBt;. PaBdabeGLasÞic Δ CaGnu-
Kmn_eTAnwg bnÞúk W ElVg L m:Um:g;niclPaB I nigm:UDuleGLasÞicrbs;sMPar³ E ³
⎛ WL ⎞ ⎛ WL3 ⎞ ⎛ 2⎞
Δ= f⎜ ⎟ = α⎜ ⎟ = K ⎜ ML ⎟ (6-1)
⎝ EI ⎠ ⎜ EI ⎟ ⎜ EI ⎟
⎝ ⎠ ⎝ ⎠
Edl W = bnÞúksrubenAelIElVg ehIy α nig K CaemKuNEdlGaRs½yeTAnwgdWeRkénPaBbgáb;rbs;TMr
bMErbMrYlm:Um:g;niclPaBtambeNþayElVg nigkarBRgaybnÞúk. ]TahrN_ PaBdabGtibrmaénbnÞúkBRgayesμI
enAelIFñwmTMrsamBaØKW³
5WL3 5wL4
Δ= = (6-2)
384 EI 384 EI
Edl W = bnÞúksrubenAelIElVg = wL ¬bnÞúkBRgayesμIelImYyxñatRbEvg × ElVg¦. PaBdabrbs;Fñwm
CamYynwglkçxNÐbnÞúk niglkçxNÐTMrepSg² edayCab;GnuKmn_eTAnwgbnÞúk ElVg nig EI RtUv)aneGayenA kñúg
]bsm<n§½ C nigenAkñúgesovePAviPaKeRKOgbgÁúM.
edaysareKsÁal; W nig L enaHeKRKan;EtKNnam:UDuleGLasÞic E nigm:Um:g;niclPaB I énGgát;eb
tug b¤PaBrwgRkajTb;karBt;rbs;Ggát; EI .
2>1> m:UDuleGLasÞic Modulus of Elasticity
bTdæan ACI Code , Section 8.5 kMNt;bBa¢ak;fa eKGacykm:UDuleGLasÞicrbs;ebtug Ec
PaBdab nigsñameRbH 91

Ec = 0.043Wc1.5 f 'c sMrab;tMél Wc enAcenøaH 1400kg / m3 eTA 2500kg / m3 (6-3)

Ec = 4780 f 'c sMrab;ebtugTMgn;Fmμta normal-weight concrete Wc = 2320kg / m3
CaTUeTAm:UDuleGLasÞicRtUv)ankMNt;edayebtugsMNakKMrYsIuLaMgrgbnÞúkkñúgry³eBlxøI short-term
loading of concrete cylinder. enAkñúgGgát;BitR)akd creep EdlbNþalmkBIbnÞúkEdlGnuvtþmkelICab;

lab; ¬y:agehacNas;k¾manbnÞúkefr¦ manT§iBleTAelIm:UDulenAEpñkrgkarsgát;énGgát;. sMrab;Epñkrgkar
Taj m:UDulenAkñúgEpñkTajRtUv)ansnμt;dUcKñanwgm:UDulenAkñúgEpñksgát; enAeBlEdltMél stress mantMéltUc.
enAeBl stress FM m:UDuleGLasÞicfycuHCaxøaMg. elIsBIenH m:UDuleGLasÞicERbRbYltambeNþayElVgEdl
bNþalmkBIbMErbMrYlénm:Um:g;nigkMlaMgkat;TTwg.
2>2> pleFobm:UDuleGLasÞic Modular Ratio
pleFobm:UDuleGLasÞic n = Es / Ec EdlRtUv)aneRbIenAkñúgkarbMElgRkLaépÞ dUcEdl)anBnül;enA
kñúgemeronTI2 EpñkTI10. vaRtUv)anKitCacMnYnKt; EtminGactUcCag 6 . ]TahrN_ enAeBlEdl
f 'c = 17.5MPa enaH n = 10
f 'c = 20MPa enaH n = 9
f 'c = 30 MPa enaH n = 8
f 'c = 35MPa enaH n = 7
sMrab;ebtugTMgn;Fmμta normal-weight concrete n GacykesμI 42 / f 'c .
2>3> m:Um:g;eRbH Cracking moment
dMeNIrkarénkar)ak;rbs;FñwmTMrsamBaØrgbnÞúkRtUv)anBnül;enAkñúgemeronTI3 EpñkTI3. enAeBlbnÞúk
GnuvtþtUc ¬ekItmanm:Um:g;tUc¦ nig stress enAsésrrgkarTajeRkAbMputnwgmantMéltUcCagm:UDuldac;
modulus of rupture rbs;ebtug f r = 0.623 f 'c . RbsinebIeKbegáInbnÞúkrhUtdl;kugRtaMgTaj tensile

stress xiteTA CitkugRtaMgdac;mFüm f r enaHsñameRbHnwgcab;epþImekIteLIg. RbsinebIkugRtaMgTajFMCag f r

muxkat;nwg eRbH ehIykrNImuxkat;eRbHnwgcab;epþmekIteLIg. enHmann½yfaeKmanbIkrNIRtUvBicarNa³
I
- enAeBlEdlkugRtaMgTaj ft tUcCag f r muxkat;Gt;eRbHTaMgmUlRtUv)anBicarNaedIm,IKNnalkçN³
rbs;muxkat;. enAkñgkrNIenH m:Um:g;niclPaB I g RtUv)aneRbI³ I g = bh3 / 12 Edl bh = muxkat;ebtugTaMg
mUl.
- enAeBlEdlkugRtaMgTaj ft esμInwg fr = 0.623 f 'c sñameRbHnwgcab;epþImekItman ehIym:Um:g;Edl
begáItnUvkugRtaMgenHRtUv)aneKeGayeQμaHfa m:Um:g;eRbH cracking moment . edayeRbIrUbmnþkugRtaMgBt;
f r = M cr
I
c
b¤ M cr = f r Icg (6-4)
g


T.Chhay NPIC

Edl fr = 0.623 f 'c / I g = m:Um:g;niclPaBmuxkat;l¥eBj nig c = cMgayBIGkS½NWteTAsésrrgkar
TajEpñkxageRkAbMput. ]TahrN_ sMrab;muxkat;ctuekaNEkg I g = bh3 / 12 nig c = h / 2
- enAeBlEdlm:Um:g;xageRkAFMCagm:Um:g;eRbH cracking moment M cr enaHkrNImuxkat;eRbHnwgcab;
epþImekIteLIg ehIyebtugenAkñúgtMbn;rgkarTajRtUv)anecal. muxkat;eRbHbMElg transformed cracked
section RtUv)aneRbIedIm,IKNnam:Um:g;niclPaBeRbH cracking moment of inertia I cr edayeRbIRkLaépÞ

ebtugenAkñgtMbn;sgát; nigRkLaépÞEdkbMElg transformed steel area nAs .
]TahrN_6>1³ muxkat;ebtugctuekaNEkgRtUv)anBRgwgedayEdk 3DB28 enAkñúgmYyCYr ehIymanTTwg
300mm nigkMBs;srub 650mm cMENk d = 585mm ¬rUb 6>1¦. cUrKNnam:UDuldac; modulus of rupture

f r / m:Um:g;niclPaBmuxkat;l¥eBj I g nigm:Um:g;eRbH M cr edayeRbI f 'c = 28MPa nig f y = 400MPa .

dMeNaHRsay³
1> m:UDuldac; modulus of rupture fr = 0.623 f 'c = 3.28MPa
2> m:Um:g;niclPaBénmuxkat;l¥eBj I g = bh3 / 12 = 300(650)3 / 12 = 6.87 ⋅109 mm4
3> m:Um:g;eRbH M cr = f r I g / c = 3.28 × 6.87 ⋅109 / 325 = 69.3kN.m

2>4> m:Um:g;niclPaB Moment of inertia
edayKuNm:Umg;niclPaBeTAnwg m:UDuleGLasÞiceyIgTTYl)anPaBrwgRkajTb;nwgkarBt;rbs;Ggát;.
eRkambnÞúktUc m:Um:g;GtibrmaEdlekItmannwgmantMéltUc ehIy tensile stress enAsésrrgkarTajxageRkA
bMputnwgmantMéltUcCagm:UDuldac;rbs;ebtug. enAkñúgkrNIenH muxkat;eRbHbMElgeBj gross transformed
cracked section nwgmanRbsiT§PaBkñúgkarpþl;nUvPaBrwgmaM. enAxN³bnÞúkkargar b¤bnÞúkFM sñameRbHenAtMbn;

TajEdl)anmkBIkarBt;nwgekIteLIg. enARtg;muxkat;eRbH TItaMgGkS½NWtsßitenAx<s; b:uEnþenARtg;muxkat;
kNþalcenøaHsñameRbHenAelIbeNþayFñwm TItaMgGkS½NWtsßitenATab ¬Ek,rEdkrgkarTaj¦. manEtTItaMgTaMg
BIrenHeT Edlmuxkat;eRbHbMElgmanRbsiT§PaBkñúgkarkMNt;PaBrwbRkajrbs;Ggát;. dUcenH m:Um:g;niclPaB

RbsiT§PaBERbRbYly:agxøaMgtambeNþayElVg. enATItaMgm:Um:g;Bt;Gtibrma ebtugeRbH enaHEpñk enAtMbn;rgkar
TajminRtUv)anKitkñúgkarKNnam:Um:g;niclPaB. enARtg;cMnucrbt; kugRtaMgtUc ehIymuxkat;TaMgmUlGacnwg
mineRbH. sMrab;sßanPaBenH nigenAkñúgkrNIsMrab;FñwmEdlmanFñwmERbRbYl dMeNaHRsayd¾RtwmRtUvman
lkçN³sμúKsμaj.
rUb 6>2 a bgðajBIExSekagbnÞúk-PaBdabrbs;FñwmebtugEdlRtUv)aneFVIBiesaFn_rhUtdl;)ak;. Fñwm
RtUv)anRTedayTMrsamBaØ manRbEvg 17 ft nigrgnUvbnÞúkcMcMnucBIrEdlmancMgayBIKña 5 ft edaysIuemRTInwgGkS½.
FñwmRtUv)anrgnUvkardak;bnÞúkBIrCMu³ TImYy ¬ExSekag cy1 ¦ ExSekagbnÞúk-PaBdab CabnÞat;Rtg;eLIgelIdl;
bnÞúk P = 1.7 K enAeBlEdlFñwmcab;epþImekItmanenAkñúgFñwm. bnÞat; a bgðajnUvTMnak;TMngbnÞúk-PaBdab
edayeRbIm:Um:g;niclPaBsMrab;muxkat;bMElgGt;eRbH. eKGacemIleXIjfaPaBdabBitR)akdrbs;FñwmeRkam
bnÞúkEdltUcCagtUcCagbnÞúkeRbH ¬edayQrelImuxkat;Gt;eRbHsac;mYy homogeneous cracked section¦
mantMélEk,reTAnwgPaBdabEdl)anmkBIkarKNna ¬bnÞat; a ¦. ExSekag cy1 bgðajnUvExSekagPaBdabBit
R)akd enAeBlEdlbnÞúkekIneLIgrhUtdl;Bak;kNþalbnÞúkcugeRkay ultimate load. CMral slop rbs;ExS
ekag enARKb;nIv:UbnÞúkTaMgGs; mantMéltUcCagCMralrbs;bnÞat; a edaysarEtsñameRbHekItman ehIyEpñk
EdleRbHénmuxkat;ebtug)ankat;bnßynUvPaBrwgRkajrbs;Fñwm. bnÞab;mkbnÞúkRtUv)andkecj PaBdabEdl
enAesssl; residual deflection RtUv)anGegáteXIjmanenAkNþalElVgFñwm. enAeBlmansñameRbH
karsnμt;BIkarRbRBwtþeTArbs;muxkat;Gt;eRbHeRkambnÞúktUcminRtUv)anykmkGnuvtþ.
sMrab;karGnuvtþn_bnÞúkCMuTIBIr PaBdab ¬ExSekag c ¦ )anekIneLIgkñúgkMritFMCagbnÞat; a edaysarEt
ersIusþg;rbs;sésrebtugTajRtUv)an)at;bg;. enAeBlbnÞúkekIneLIg TMnak;TMngbnÞúk-PaBdabRtUv)anbgðaj
edayExSekag cy2 . RbsinebIbnÞúkenACMuTImYyekIneLIgdl;bnÞúkcugeRkay enaHExSekag cy1 nwgcab;ykKnøg
ExSekag cy2 enARtg;bnÞúkRbEhl 0.6Pu . ExSekag c bgðajBIdMeNIrkarBitR)akdrbs;FñwmsMrab;karbEnßm
bnÞúk b¤kardkbnÞúk.
bnÞat; b bgðajBITMnak;TMngbnÞúk-PaBdabedayQrelImuxkat;bMElgeRbH. eKeXIjfa PaBdabEdl
KNnaedayQrelIeKalkarN_hñwg xusKñaBIPaBdabBitR)akd. rUb 6>2 b bgðajBIbMErbMrYlPaBrwgRkaj
EI rbs;FñwmCamYynwgkarekIneLIgénm:Um:g;. bTdæan ACI Code, section 9.5 ENnaMnUvsmIkaredIm,IkMNt;

m:Um:g;niclPaBRbsiT§PaBEdleRbIenAkñúgkarKNnaPaBdabenAkñúgm:Um:g;rgkarBt;. m:Um:g;niclPaBRbsiT§PaB
EdleGayeday ACI Code (Eq. 9.8) edayQrelIsmIkarEdlesñIredayelak Branson nwgRtUv)anKNna
dUcxageRkam³
⎛ M cr ⎞
3 ⎡ ⎛ M ⎞3 ⎤
⎜ M ⎟ I g + ⎢1 − ⎜ M ⎟ ⎥ I cr ≤ I g
Ie = ⎜ cr (6-5)
⎟ ⎢ ⎝ a ⎟ ⎥
⎜
⎝ a ⎠ ⎣ ⎠ ⎦


T.Chhay NPIC

Edl Ie = m:Um:g;niclPaBRbsiT§PaB
⎛ fr I g ⎞
M cr = m:Um:g;eRbH/ ⎜
⎜ Y ⎟ ⎟ (6-6)
⎝ t ⎠
fr = m:UDuldac;rbs;ebtug fr = 0.623 f 'c (6-7)

M a = m:Um:g;KμanemKuNGtibrmarbs;Ggát;enAxN³EdlPaBdabkMBugRtUv)anKNna

I g = m:Um:g;niclPaBrbs;muxkat;ebtugl¥eBjeFobGkS½TIRbCMuTMgn; edayecalEdl

I cr = m:Um:g;niclPaBénmuxkat;bMElgeRbH

Yt = cMgayBIGkS½TIRbCMuTMgn;edayecalEdkeTAépÞrgkarTaj.


karkMNt;xageRkamRtUv)ankMNt;edaybTdæan³
1> sMrab;FñwmCab; m:Um:g;niclPaBRbsiT§PaBGacRtUv)anyktMélmFüménm:Um:g;niclPaBénmuxkat;
Edlmanm:Um:g;viC¢man nigGviC¢manFMCageK.
2> sMrab;ebtugTMgn;Rsal m:UDuldac; fr edIm,IeRbIenAkñúgsmIkar (6-6) esμInwg
⎛ f ⎞
f r = 0.623⎜ ct ⎟
⎝ 0.556 ⎠
Edl 0.f556 ≤ f 'c
ct (6-8a)

Edl fct CaersIusþg;TajedaykarbMEbk splitting tensile strength. enAeBlEdleKGt;sÁal;
f ct enaH f r GacRtUv)anKitdUcxageRkam³
f r = 0.465 f 'c (6-8b)

sMrab;ebtugxSac;TMgn;Rsal sand-lightweight concrete
f r = 0.532 f 'c (6-8c)

3> sMrab;Ggát;RBIs Ie GacyktMélEdlTTYl)anBIsmIkar (6-5) enAkNþalElVgsMrab;FñwmTMrsamBaØ
nigFñwmCab; nigenATMrsMrab;Fñwm cantilever (ACI Code, section 9.5.2) .
cMNaMfa Ie Edl)anmkBIkarKNnatamsmIkar (6-5) pþl;nUvtMélEdlsßitenAcenøaHm:Um:g;nicl
PaBl¥eBj gross moment of inertia I g nig m:Um:g;niclPaBeRbH cracked moment of inertia
I cr edayGnuKmn_eTAnwgkMritéplpleFob cr . Ggát;ebtugEdlmanbrimaNEdkeRcIn Gacnwg
M
Ma
man Ie xiteTACit Icr b:uEnþsMrab;Ggát;EdlmanRTnug flanged member Gacnwgman Ie xiteTA
Cit I g .
4> sMrab;FñwmCab; tMélRbEhléntMélmFümén Ie sMrab;Ggát;RBIs b¤minRBIs edIm,IeFVIeGaylT§pl
kan;EtRbesIreLIgmandUcxageRkam³
- sMrab;FñwmEdlmancugsgçagCab; Average Ie = 0.7 I m + 0.15( Ie1 + Ie2 ) (6-9)

- sMrab;FñwmEdlmancugmçagCab; Average Ie = 0.85I m + 0.15( Icon ) (6-10)

Edl I m = Ie enAkNþalElVg É Ie1, Ie2 = Ie enAcugFñwmsgçag nig Icon = Ie enARtg;cug
Cab;. Ie GacyktMélmFümén Ie enARtg;muxkat;Edlmanm:Um:g;viC¢man nigGviC¢manFM. eKKYr
eRbI moment envelope kñúgkarKNnatMélviC¢man nigGviC¢manén Ie . enAkñúgkrNIEdlFñwmrg
bnÞúkcMp©itFM manEt Ie kNþalElVgKYrRtUv)aneRbI.
2>5> lkçN³rbs;muxkat; Properties of sections
edIm,IKNnam:Um:g;niclPaBmuxkat;eBj gross section nigmuxkat;eRbH cracked section eKcaM)ac;RtUv
KNnacMgayBIsésrrgkarsgát; compression fiber eTAGkS½NWt neutral axis ¬ x b¤ kd ¦.


T.Chhay NPIC

1> m:Um:g;niclPaBeBj I g ¬ecalmuxkat;EdkTaMgGs;enAkñúgebtug¦
a. sMrab;muxkat;ctuekaNEkgEdlmanTTwg b nigkMBs;srub h enaH I g = bh3 / 12 .

b. sMrab;muxkat; T-section EdlmanTTwgsøab b TTwgRTnug bw nigkMras;RTnug t KNna y

cMgayBIGkS½TIRbCMuTMgn;eTAcMnucx<s;bMputrbs;søab³
⎟ + bw (h − t ) ⎡ (h − t )⎤
⎛ bt 2 ⎞
⎜
⎜ 2 ⎟ ⎢ 2+t ⎥
⎣ ⎦
y=⎝ ⎠ (6-11)
bt + bw (h − t )
bnÞab;mkeTotKNna
⎡ bt 3 t ⎞ ⎤ ⎡ ( y − t )3 ⎤ ⎡ (h − y ) ⎤
2
⎛
Ig = ⎢ + bt ⎜ y − ⎟ ⎥ + ⎢bw ⎥ + bw (6-11a)
⎢ 12
⎣ ⎝ 2⎠ ⎥ ⎢
⎦ ⎣ 3 ⎦ ⎢ ⎥ ⎣ 3 ⎥ ⎦

2> m:Um:g;niclPaBeRbH Icr ³ yk x = cMgayBIGkS½NWtmksésrrgkarsgát;eRkAbMput ¬ x = kd ¦
a. muxkat;ctuekaNEkgCamYyEtnwgEdkrgkarTaj As

i. KNna x BIsmIkarxageRkam
bx 2
− nAs (d − x) = 0 (6-12)
2
ii. KNna Icr = bx3 / 3 + nAs (d − x)2 (6-12a)

b. muxkat;ctuekaNEkgCamYyEdkrgkarTaj As nigEdkrgkarsgát; A's
bx 2
i. KNna x ³ + (n − 1) A's ( x − d ' ) − nAs (d − x) = 0 (6-13)
2
ii. KNna I cr = (bx 3 / 3) + (n − 1) A's ( x − d ' ) 2 + nAs (d − x) 2 (6-13a)

c. T-section CamYyEdkrgkarTaj As

i. KNna x ³
t
bt ( x − ) + bw
(x − t )2 − nA (d − x) = 0 (6-14)
s
2 2
⎡ bt 3 t ⎞ ⎤ ⎡ ( x − t )3 ⎤
2
⎛
ii. KNna Icr =⎢ + bt ⎜ x − ⎟ ⎥ + ⎢bw
⎝ 2⎠ ⎥ ⎢
⎥ + nAs (d − x)
2
(6-14a)
⎢ 12
⎣ ⎦ ⎣ 3 ⎥ ⎦

3> PaBdabry³eBlyUr Long-term Deflection

PaBdabrbs;Ggát;ebtugEdkBRgwgbnþekIneLIgeRkambnÞúkEdlGnuvtþsßitesßr ebIeTaHbICavayWtebI
eRbobeFobCamYyeBlk¾eday. karrYmmaD nig creep CamUlehtueFVIeGayekItmanPaBdabbEnßmEdleKeGay
eQμaHfa PaBdabry³eBlyUr long-term deflection. vargT§iBlCacMbgBI sItuNðPaB sMeNIm Gayurbs;eb
tugenAxN³énkardak;bnÞúk karEfTaMebtug brimaNEdkrgkarsgát; nigdg;sIuetbnÞúkefr. bTdæan ACI Code,


Section 9.5 esñIfa y:agehacNas;tMélEdlTTYledaykarviPaKd¾RtwmRtUvmYy KWPaBdabry³eBlyUrbEnßm
sMrab;Ggát;rgkarBt;ebtugFmμta nigebtugTMgn;RsalKYr)an TTYlBIkarKuNPaBdabPøam²CamYyemKuN
ζ
λΔ = (6-15)
1 + 50 ρ '
Edl λΔ =emKuNsMrab;PaBdabbEnßmEdlbNþalmkBIT§iBlry³eBlyUr
ρ '= A's / bd sMrab;muxkat;enAkNþalElVgrbs;FñwmTMrsamBaØ b¤FñwmCab; b¤sMrab;muxkat;enATMr
rbs;Fñwm cantilever .
ζ = emKuNGaRs½yeBl time-dependent factor sMrab;bnÞúkefrEdlGacnwgykdUcbgðajkñúg
taragTI 6>2.
emKuN λΔ RtUv)aneRbIedIm,IKNnaPaBdabEdlbNþalmkBIbnÞúkefr nigEpñkxøHénbnÞúkGefrEdlnwg
sßitenAzitezrsMrab;ry³eBlmYyRKb;RKan;edIm,IbegáItnUvPaBdabGaRs½ynwgeBlmYyKYreGaykt;sMKal;. em
KuN λΔ CaGnuKmn_eTAnwglkçN³sMPar³ EdlsMEdgeday ζ niglkçN³muxkat; sMEdgeday (1 + 50ρ ' ) .
enAkñúgsmIkar (6-15) T§Blrbs;Edkrgkarsgát;KWTak;TgeTAnwgRkLaépÞebtugCagTak;TgeTAnwgpleFob
rvagEdksgát;elIEdkTaj.
ACI Code Commentary, section 9.5 bgðajnUvExSekagedIm,IKNna ζ sMrab;ry³eBlticCag 60 Ex.

tMélTaMgenHRtUv)anKNnadUcbgðajkñúgtaragTI 6>2.
PaBdabsrubesμInwgplbUkPaBdabPøam² nigPaBdabbEnßmry³eBlyUr. ]TahrN_ PaBdabbEnßm
ry³eBlyUrsrubrbs;FñwmrgkarBt;CamYy ρ '= 0.01 enAry³eBl 5 qñaMesμInwgplKuN λΔ CamYynwgPaBdab
Pøam² Edl λΔ = 2 /(1 + 50 × 0.01) = 1.33 .
tarag 6>2 emKuNGaRs½yeBlsMrab;PaBdabry³eBlyUr
ry³eBl ¬Ex¦ 1 3 6 12 24 36 48 60
ζ 0 .5 1 .0 1 .2 1 .4 1 .7 1 .8 1 .9 2 .0

4> PaBdabGnuBaØati Allowable Deflection

PaBdabKYrminRtUvelIstMéldUcxageRkamGaRs½yeTAtambTdæan ACI Code, Section 9.5 ³
- L /180 sMrab;PaBdabPøam²EdlbNþalmkBIbnÞúkGefrsMrab;dMbUlerobesμIEdlminRTGgát;ÁEdlTMng
eFVIxUcxat.
- L / 360 sMrab;PaBdabPøam²EdlbNþalmkBIbnÞúkGefrsMrab;kMralxNÐEdlminRTGgát;EdlTMngeFVI
xUcxat.


T.Chhay NPIC

- sMrab;EpñkénPaBdabsrubEdlekIteLIgeRkayBIkarP¢ab;GgÁ dUcCa plbUkPaBdabry³eBl
L / 480

yUrEdlbNþalmkBIbnÞúkefrTaMgGs; nigPaBdabPøam²EdlbNþalmkBIbnÞúkGefrbEnßm sMrab;kMral
xNÐ b¤dMbUlEdlRTGgát;EdlTMngnwgeFVIeGayxUcxat.
- L / 240 sMrab;EpñkénPaBdabsrubEdlekIteLIgeRkayBIGgát;RtUv)anP¢ab; sMrab;kMralxNÐ b¤dMbUl
EdlminRTGgát;EdlTMngnwgeFVIeGayxUcxat.
5> PaBdabEdlbNþalmkBIbnSMbnÞúk Deflection Due to Combinations of Load

RbsinebIFñwmrgnUvbnÞúkeRcInRbePT ¬BRgayesμI BRgayminesμI b¤bnÞúkcMcMnuc¦ b¤rgnUvm:Um:g;cug enaH
PaBdabGacRtUv)anKNnasMrab;bnÞúk b¤kMlaMgtamRbePTmYy²EdlGnuvtþmkelIFñwmdac;edayELkBIKña ehIy
PaBdabsrubRtUvKNnaedayviFItMrYtpl superposition. enHmann½yfaPaBdabmYy²RtUveFVIplbUknBVnþCa
mYyKñaedIm,ITTYl)anPaBdabsrub. PaBdabrbs;bnÞúkeRkambnÞúkmYy²RtUv)anbgðajenAkñúgtarag 6>3.
]TahrN_6>2³ KNnaPaBdabxN³kNþalElVgsMrab;FñwmTMrsamBaØdUcbgðajkñúgrUb 6>3 EdlRTnUvbnÞúkefr
BRgayesμI 5.85kN / m nigbnÞúkGefr 8.75kN / m rYmnwgbnÞúkefrcMcMnuc 22.25kN enARtg;kNþalElVg.
eKeGay f 'c = 28MPa / f y = 400MPa / b = 330mm nig d = 530mm nigkMBs;srub 630mm ¬ n = 8 ¦.

dMeNaHRsay³
1> epÞógpÞat;kMBs;Gb,brmaGaRsy½eTAtambTdæan ACI Code tarag 6>1
kMBs;srubGb,brma = 16 = 12200 = 762.5mm
L
16
edaysarkMBs;srubEdleRbIR)as; 630mm < 762.5mm dUcenH eKRtUvkarepÞógpÞat;PaBdab.
2> PaBdabenAkNþalElVgEdlbNþalmkBIbnÞúkBRgayKW
5wL4
Δ1 =
384 Ec I e
PaBdabenAkNþalElVgEdlbNþalmkBIbnÞúkcMcMnucKW

PL3
Δ2 =
48Ec I e
edaysarEteKsÁal; w / P nig L dUcenHeKRtUvKNnam:UDuleGLasÞic
Ec nigm:Um:g;niclPaBRbsiT§PaB I e .

3> m:UDuleGLasÞicrbs;ebtugKW
Ec = 4780 28 = 25293.4 MPa
4> m:Um:g;niclPaBRbsiT§PaB
⎛ M cr ⎞
3 ⎡ ⎛ M ⎞3 ⎤
⎜
Ie = ⎜ ⎟ I g + ⎢1 − ⎜ cr ⎟ ⎥ I cr ≤ I g
⎟
⎝ Ma ⎠ ⎢ ⎜ Ma ⎟ ⎥
⎣ ⎝ ⎠ ⎦

kMNt;tMélTaMgGs;enAGgÁxagsþaM
wL2 PL (8.75 + 5.85)12.22 22.25 × 12.2
Ma = + = + = 339.5kN .m
8 4 8 4
bh3 330(630)3
Ig = = = 6.88 ⋅ 109 mm 4
12 12
M cr =
fr I g
Yt
eday
f r = 0.623 f 'c = 0.623 28 = 3.3MPa
h
Yt = = 315mm
2
3.3 × 6.88 ⋅ 109
M cr = = 72.1kN .m
315
m:Um:g;niclPaBénmuxkat;bMElgeRbH Icr RtUv)anKNnadUcxageRkam³
kMNt;TItaMgGkS½TIRbCMuTMgn;sMrab;muxkat;eRbHedaysmIkarm:Um:g;énmuxkat;bMElgeFobGkS½TIRbCMuTMgn;
eTA 0 yk x = kd = cMgayBIGkS½TIRbCMuTMgn;
bx 2 Es
− nAs (d − x) = 0 n= =8 As = 3041mm 2
2 Ec

165 x 2 + 24328 x − 12893840 = 0 x = 215.38mm
bx3
330(215.38)3
I cr = + nAs (d − x) 2 = + 8 × 3041(530 − 215.38) 2 = 3.5 ⋅ 109 mm 4
3 3
CamYynwgGgÁTaMgGs;Edl)anKNna eyIgTTYl)an
⎛ 72.1 ⎞
3 ⎡ ⎛ 72.1 ⎞3 ⎤
Ie = ⎜ ⎟ 6.88 ⋅ 10 + ⎢1 − ⎜
9
⎟ ⎥3.5 ⋅ 10 = 3.53 ⋅ 10 mm
9 9 4
⎝ 339.5 ⎠ ⎢ ⎝ 339.5 ⎠ ⎥
⎣ ⎦
5> KNnaPaBdabBIbnÞúkepSg²
5wL4
Δ1 ¬bNþalmkBIbnÞúkrayesμI¦ =
384 E I c e
5(8.75 + 5.85) × 122004
Δ1 = = 47.2mm
384 × 25293.4 × 3.53 ⋅ 109


T.Chhay NPIC
3
Δ2 ¬bNþalmkBIbnÞúkcMcMnuc¦ = 48PL I
Ec e
22250 × 12200 3
Δ2 = = 9.43mm
48 × 25293.4 × 3.53 ⋅ 109
PaBdabPøam²srub = Δ1 + Δ2 = 47.2 + 9.43 = 56.63mm
6> eRbobeFobtMélEdl)anmkBIkarKNnaCamYyPaBdabGnuBaØati³
PaBdabPøam²Edl)anmkBIbnÞúkGefrBRgayesμI 8.75kN .m esμInwg 8.75 × 47.2 /(8.75 + 5.85) =
28.29mm . RbsinebIGgát;CaEpñkmYyrbs;kMralxNÐEdlminRT b¤P¢ab;eTAnwgCBa¢aMgxNÐ½ b¤Ggát;epSg²eTot

EdlGacTMngeFVIeGayxUcxatedaysarPaBdabFM enaHPaBdabPøam²GnuBaØatiEdlbNþalmkBIbnÞúkGefresμInwg
L 12200
= = 33.9mm > 28.29mm
360 360
EtRbsinebIGgát;CaEpñkénkMralxNÐdMbUl nigRsedogKñaeTAGVIEdl)anerobrab;xagelI enaHPaBdab
Pøam²GnuBaØatiEdlbNþalmkBIbnÞúkGefrKW 180 = 12200 = 67.8mm > 28.29mm . tMélGnuBaØatiTaMgBIrFM
L
180
CagPaBdabBitR)akd 28.29mm EdlbNþalmkBIbnÞúkGefrBRgayesμI.
]TahrN_6>3³ kMNt;PaBdabry³eBlyUr long-term deflection rbs;FñwmenAkñúg]TahrN_6>2 RbsinebIem
KuNGaRs½yeBl time-dependent factor esμInwg 2.0 .
dMeNaHRsay³
1> bnÞúkEdlGnuvtþn_zitezrbNþaleGaymanPaBdabry³eBlyUrKWekItmkBIbnÞúkGefr EdlrYmmanbnÞúk
efrBRgayesμI 5.85kN / m nigbnÞúkefrcMcMnuc 22.25kN EdlGnuvtþn;enAkNþalElVg.
PaBdabbNþalmkBIbnÞúkBRgayesμI = 5.85 × 47.2 /(8.75 + 5.85) = 18.9mm
PaBdabCaGnuKmn_bnÞat;eTAnwgbnÞúk w ehIytMélepSgeTot ¬ L / Ec / Ie ¦dUcKña
PaBdabbNþalmkBIbnÞúkcMcMnuc = 9.43mm
PaBdabsrubEdlbNþalmkBIbnÞúkzitezr sustained load = 18.9 + 9.43 = 28.33mm
2> sMrab;PaBdabry³yUrbEnßm PaBdabPøam²RtUv)anKuNnwgemKuN λΔ
ζ 2
λΔ = =
1 + 50 ρ ' 1+ 0
kñúgkrNIenH A's = 0 dUcenH λΔ = 2.0
PaBdabry³eBlyUrbEnßm = 2 × 28.33 = 56.66mm
3> PaBdabry³eBlyUrsrubCaplbUkrvagPaBdabPøam²CamYynwgPaBdabry³eBlyUrbEnßm
56.63 + 56.66 = 113.29mm


4> PaBdabbNþalmkBIbnÞúkefrCamYynwgPaBdabry³eBlyUrbEnßmEdlbNþalmkBIkarrYmmaDnig
creep KW 28.33 + 56.66 = 85mm

]TahrN_6>4³ KNnaPaBdabxN³ nigPaBdabenAry³eBl 1qñaM enAcugTMenrrbs;Fñwm cantilever dUcbgðaj
kñúgrUbTI 6>4. FñwmmanRbEvg 6.1m nigRTbnÞúkefrBRgayesμI 5.85kN / m bnÞúkGefrBRgayesμI 5.85kN / m
bnÞúkefrcMcMnuc PD = 13.35kN enAcugTMenr nigbnÞúkGefrcMcMnuc PL = 17.8kN EdlGnuvtþn_enAcMgay 3.05m

T.Chhay NPIC

BITMrbgáb;. eKeGay f 'c = 28MPa / f y = 400MPa / b = 300mm / d = 550mm / nigkMBs;muxkat;srub
635mm ¬sésrEdkrgkarTajKW 6 DB 25 nigEdkrgkarsgát; 2 DB 25 ¦.

dMeNaHRsay³
1> kMBs;Gb,brma L = 6100 = 762.5mm > 635mm dUcenHeKRtUvepÞógpÞat;PaBdab.
8 8
PaBdabGtibrmarbs;Fñwm cantilever KWsßitenAcugTMenr. PaBdabenAcugTMenrKWdUcxageRkam³
PaBdabEdlbNþalmkBIbnÞúkBRgay
wL4
Δ1 =
8 EI
PaBdabEdlbNþalmkBIbnÞúkefrcMcMnucenAcugTMenr
PD L3
Δ2 =
3EI
PaBdabGtibrmaenAcugTMenrEdlbNþalmkBIbnÞúkGefrcMcMnucenA a = 3.05m BIcugbgáb;
P a2 a3
Δ 3 = L (3L − a)
6 EI
b¤ PLEI (1 + 2b )
3
3
a
2> m:UDuleGLasÞicrbs;ebtugTMgn;RsalKW
Ec = 4780 28 = 25293.4 MPa
3> m:Um:g;GtibrmaenAcugbgáb;
wL2 5.85 × 6.12
Ma = + 6.1PD + 3.05PL = + 6.1 × 13.5 + 3.05 × 17.8 = 245.5kN .m
2 2
4> m:Um:g;niclPaBeBj ¬Etebtug¦
bh3 300 × 6353
Ig = = = 6.4 ⋅ 109 mm4
12 12
f r I g 0.623 28 × 6.4 ⋅ 109
5> M cr =
Yt
=
635
= 66.5kN .m

2


6> kMNt;TItaMgGkS½NWt bnÞab;mkkMNt;m:Um:g;niclPaBénmuxkat;bMElgeRbH. Kitm:Um:g;énmuxkat;
eFob GkS½TIRbCMuTMgn;nigdak;eGayvaesμIsUnü. eRbI n = 8 edIm,IKNnamuxkat;bMElgrbs; As nig
eRbI (n − 1) = 7 sMrab;KNnamuxkat;bMElgrbs; A's . yk kd = x
(x )2 + (n − 1) A'
b s ( x − d ' ) − nAs ( d − x) = 0
2

150 x 2 + 30434 x − 13370440 = 0
sMrab;muxkat;enH x = 213.875mm
b 3
I cr = x + (n − 1) A's ( x − d ' ) 2 + nAs (d − x) 2 = 3.8 ⋅ 109 mm 4
3
7> m:Um:g;niclPaBRbsiT§PaBKW
⎛ M cr ⎞
3 ⎡ ⎛ M ⎞3 ⎤
⎜ M ⎟ I g + ⎢1 − ⎜ M ⎟ ⎥ I cr ≤ I g
Ie = ⎜ cr
⎟ ⎢ ⎝ a ⎟ ⎥
⎜
⎝ a ⎠ ⎣ ⎠ ⎦

⎛ 66.5 ⎞
3 ⎡ ⎛ 66.5 ⎞3 ⎤
Ie = ⎜ ⎟ 6.4 ⋅ 109 + ⎢1 − ⎜ ⎟ ⎥3.8 ⋅ 10 = 3.85 ⋅ 10
9 9
⎝ 245.5 ⎠ ⎢ ⎝ 245.5 ⎠ ⎥
⎣ ⎦
8> KNnaPaBdabeRkambnÞúkepSg²EdlmanGMeBIelIFñwm
11.7 × 6100 4
Δ1 ¬bNþalBIbnÞúkBRgayesμI 11.7 kN / m ¦ = = 20.8mm
8 × 25293.4 × 3.85 ⋅ 109
Δ1 ¬bNþalBIbnÞúkefr¦ = 10.4mm
13500 × 61003
Δ 2 ¬bNþalBIbnÞúkefrcMcMnuc¦enAcugTMenr = = 10.5mm
3 × 25293.4 × 3.85 ⋅ 109
Δ 3 ¬bNþalBIbnÞúkGefrcMcMnuc¦enA 3.05m BIcugbgáb;
17800 × 30502 × (3 × 6100 − 3050)
Δ3 = = 4.3mm
6 × 25293.4 × 3.85 ⋅ 109
PaBdabPøam²srub Δ = Δ1 + Δ2 + Δ3 = 20.8 + 10.5 + 4.3 = 35.6mm
9> PaBdabbEnßmry³eBlyUr esμInwgplKuNrvagPaBdabPøam²CamYynwgemKuN λΔ
sMrab;ry³eBl 1qñaM ζ = 1.4
A's 982
ρ'= = = 0.006
bd 300 × 550
1.4
λΔ = = 1.077
1 + 50 × 0.006
PaBdabPøam²srub Δ s EdlbNþalmkBIbnÞúkzitezr sustained load ¬enATIenHmanEtbnÞúkefr
5.85kN / m nig PD = 13.35kN enAcugTMenr¦

Δ s = 10.4 + 10.5 = 20.9mm
PaBdabry³eBlyUrbEnßm = 1.077 × 20.9 = 22.5mm

T.Chhay NPIC

10> PaBdabry³eBlyUrsrubCaplbUkrvagPaBdabPøam²CamYynwgPaBdabry³eBlyUrbEnßm
EdlekItBIkarrYmmaD nig creep .
Total Δ = 35.6 + 22.5 = 58.1mm
]TahrN_6>5³ KNnaPaBdabxN³enAkNþalElVgrbs;FñwmenAkñúgrUb 6>5 EdlFñwmenHmanRbEvg 9.8m .
FñwmenHRtUv)anbnþedayTMrCaeRcInEdlmanRbEvgElVgepSg²Kña. düaRkamm:Um:g;Bt;nigmuxkat;rbs;FñwmenA
kNþalElVg nigTMrk¾RtUv)anbgðaj. FñwmenHRTbnÞúkefrBRgayesμI 61.3kN / m nigbnÞúkGefr 52.5kN / m .
eKeGay f 'c = 20MPa / f y = 400MPa nig n = 9.2 .
m:Um:g;enAkNþalElVg M D = 260kN .m M ( D + L ) = 650kN .m

m:Um:g;enATMrxageqVg A M D = 242.75kN .m M ( D + L) = 569.5kN .m
m:Um:g;enATMrxagsþaM B M D = 293kN .m M ( D + L ) = 735kN .m

dMeNaHRsay³
1> Fñwm AB rgnUvm:Um:g;viC¢manEdleFVIeGaydabcuHeRkamenAkNþalElVg nigm:Um:g;GviC¢manenAcugTaMgBIr
EdleFVIeGaydabeLIgelIenAkNþalElVg. dUcEdl)anBnül;BIxagedIm PaBdabCaGnuKmn_eTAnwg
m:Um:g;niclPaBRbsiT§ Ie . enAkñúgFñwmCab; tMélrbs; Ie EdlRtUv)aneRbICatMélmFümsMrab;tMbn;m:U
m:g;viC¢man nigm:Um:g;GviC¢man. dUcenHmuxkat;bInwgRtUv)anBicarNa muxkat;enAkNþalElVg nigmuxkat;
enATMrTaMgBIr.
2> KNna Ie ³ sMrab;RkLaépÞeBjénmuxkat;TaMgGs; kd = 356mm nig I g = 5.2 ⋅1010 mm4 dUcKña
f r = 0.623 f 'c = 2.8MPa nig Ec = 4780 f 'c = 21376.8MPa sMrab;RKb;muxkat;. tMélrbs;

kd / I cr nig M cr sMrab;muxkat;eRbH/ I e sMrab;EtbnÞúkefr ¬edayeRbI M a sMrab;bnÞúkefr¦ nig I e

sMrab;bnÞúkefr nigbnÞúkGefr ¬edayeRbI M a sMrab;bnÞúkefr nigbnÞúkGefr¦RtUv)anKNna nigbegáItCa
taragdUcxageRkam³
I e ¬ mm 4 ¦ I e ¬ mm4 ¦
muxkat; kd ¬ mm ¦ I cr ¬ mm ¦ M cr ¬ kN .m ¦
4
¬bnÞúkGefr¦ ¬ DL + LL ¦
kNþalElVg 169.8 2.12 ⋅ 1010 226 4.14 ⋅ 1010 2.25 ⋅ 1010
TMr A 281 1.52 ⋅ 1010 409 5.2 ⋅ 1010 2.88 ⋅ 1010
TMr B 320 1.84 ⋅ 1010 409 5.2 ⋅ 1010 2.42 ⋅ 1010
cMNaMfa enAeBlEdlFñwmrgEtbnÞúkefr ehIypleFob M cr / M a FMCag 1.0 enaH Ie esμInwg I g .
3> kMNt;tMélmFümrbs; Ie BIsmIkar (6.9)
I e1 (average) = 0.7(2.25 ⋅ 1010 ) + 0.15(2.88 + 2.42)1010 = 2.37 ⋅ 1010 mm 4

sMrab;bnÞúkefr nigbnÞúkGefr
I e (average for end sections) = 0.5(2.88 + 2.42)1010 = 2.65 ⋅ 1010 mm4

( )
I e 2 (average) = 0.5 2.25 ⋅ 1010 + 2.65 ⋅ 1010 = 2.45 ⋅ 1010 mm 4
sMrab;EtbnÞúkefr
I e (average for end sections) = 5.2 ⋅ 1010 mm 4

( )
I e3 (average) = 0.5 4.14 ⋅ 1010 + 5.2 ⋅ 1010 = 4.67 ⋅ 1010 mm4


T.Chhay NPIC

4> KNnaPaBdabPøam²enAkNþalElVg
5wL4
Δ1 ¬bNþalmkBIbnÞúkBRgay¦ = ¬cuHeRkam¦
384 EI e
M A L2
Δ2 ¬bNþalmkBIm:Um:g;enAcMnuc A / M A ¦ =−
16 EI e
¬eLIgelI¦
2
Δ3 ¬bNþalmkBIm:Um:g;enAcMnuc B / M B ¦ = − 16B L ¬eLIgelI¦
M
EI e
PaBdabsrub Δ = Δ1 − Δ2 − Δ3
PaBdabsMrab;bnÞúkefrBRgayesμI 61.3kN / m edayyk M A ( DL) = 242.75kN .m /
M B ( DL) = 293kN .m nig I e3 = 4.67 ⋅ 1010 mm4 CMnYseTAkñúgsmIkarxagelI eyIgTTYl)an³

Δ = 7.4 − 1.5 − 1.8 = 4.1mm ¬cuHeRkam¦

PaBdabEdlbNþalBIbnSMbnÞúkefr nigbnÞúkGefrRtUv)anKNnaedayykbnÞúkefrbUknwgbnÞúkGefr
= 113.8kN / m / M A ( DL + LL ) = 569.5kN .m / M B ( DL + LL ) = 735kN .m nig

I e 2 = 2.45 ⋅ 1010 mm 4
Δ = 26.1 − 6.5 − 8.4 = 11.2mm ¬cuHeRkam¦
PaBdabPøam²EdlbNþalBIEtbnÞúkGefrKW 11.2 − 4.1 = 7.1mm ¬cuHeRkam¦.
RbsinebIkarkMNt;PaBGnuBaØatiKW L / 480 = 9800 / 480 = 20.4mm enaHmuxkat;KWRKb;RKan;.
cMnucmYycMnYnEdlTak;TgnwglT§pl³
1> RbsinebIeKeRbIEt Ie énmuxkat;kNþalElVg Ie = 2.25 ⋅1010 mm4 enaHPaBdabEdlbNþalmkBI
bnÞúkefr nigbnÞúkGefrRtUv)anKNnaeday KuNtMélEdlTTYl)ankñúgCMhanTI4 CamYynwgpl
eFobéntMél Ie TaMgBIr³
2.45 ⋅ 1010
Δ ¬bnÞúkefr + bnÞúkGefr¦ = 11.2 × = 12.2mm
2.25 ⋅ 1010
PaBxusKñamantMéltUc RbEhl 8% .
2> RbsinebI Ie1 mFüm RtUv)aneRbI Ie1 = 2.37 ⋅1010 mm4 enaH
2.37 ⋅ 1010
Δ ¬bnÞúkefr + bnÞúkGefr¦ = 11.2 × = 11.8mm . PaBxusKñamantMéltUc RbEhl
2.25 ⋅ 1010
5% .

3> edIm,IPaBgayRsYleKGacRbI Ie énmuxkat;kNþalElVg elIkElgEteKRtUvkartMélCak;lak;.


6> PaBdabenAkñúgGgát;rgkarBt; Cracks in Flexural Members

karsikSaBIkarkekItsñameRbH kareFVIkarrbs;sñameRbHenAeBlbnÞúkekIneLIg nigkarRtYtBinitüsñameRbH
mansar³sMxan;Nas;sMrab;karKNnaeRKOgbgÁúMebtugEdkBRgwgd¾Cak;lak;mYy. enAkñúgGgát;rgkarBt; sñam
eRbH ekIteLIgeRkambnÞúkeFVIkar working load nigedaysarebtugexSayTb;nwgkarTaj enaHEdkRtUv)andak;
enAtMbn;TajEdleRbH edIm,ITb;nwgkarTajEdlbgáeLIgedaykMlaMgxageRkA.
sñameRbHedaykarBt;ekIteLIgenAeBlEdlkugRtaMgenAsésrrgkarTajxageRkAeKbMputmantMélelIsBI
m:UDuldac;rbs;ebtug modulus of rupture of concrete . CamYynwgkareRbIR)as;EdkBRgwgEdlmanersIusþg;
x<s; high-strength reinforcing bars sñameRbHGacnwgekItmaneRcInenAelIGgát;ebtugGarem:. CamYynwgkar
eRbIR)as;EdkrgkarTajx<s; high-tensile steel manRbeyaCn_CaeRcIn b:uEnþeKminGacecosputBIsñameRbHEdl
eKmincg;)anenaHeT. sñameRbHEdlmanTMhMFM² )anGnuBaØatieGaymanERcHsIuEdk b¤eFVIeGaymankarRCabTwk
EdleFVIeGay)at;bg;esaPNÐPaBrbs;eRKOgbgÁúM.
sñameRbHekItmanenAelIebtug enAeBlEdlrn§tUc²EdlminGackMNt;)anekItmanenAkñúgFñwmebtugEdlCa
lT§plénkugRtaMgTajxagkñúg. kugRtaMgkñúgTaMgenHGacnwgbNþalmkBIkrNImYy b¤eRcInCagmYyénkrNIxag
eRkam³
- kMlaMgxageRkA dUcCa kMlaMgTajtamGkS½edaypÞal; kMlaMgkat;TTwg m:Um:g;Bt; b¤m:Um:g;rmYl
- karrYmmaD
- creep
- karrIkmaDxagkñúgEdlCalT§plBIkarpøas;bþÚrlkçN³rbs;smasFatupSMebtug
CaTUeTA sñameRbHRtUv)anEbgEckCaBIrRbePTFM²KW³ sñameRbHbnÞab;bnSM secondary cracks nigsñam
eRbHcMbg main cracks.
6>1> sñameRbHbnÞab;bnSM Secondary Cracks
sñameRbHbnÞab;bnSM CasñameRbHd¾tUc²ekItmanenAdMNak;kaldMbUgénkareRbH EdlekIteLIgedaykar
rIkmaDxagkñúg nigkarRbTajKñaénsmasFatuebtug nigedaykugRtaMgEdl)anBIkarBt;tUcbNþalmkBITMgn;
pÞal;rbs;Ggát; nigbnÞúkefrd¾éTeTot. sñameRbHmanbIRbePTKW³
- sñameRbHedaykarrYmmaD shrinkage cracks CasñameRbHsMxan; BIeRBaHvaCHT§iBlelITMrg;ragrbs;kar
eRbHEdlekIteLIgedaybnÞúkenAkñúgGgát;rgkarBt;. enAeBlEdlBYkvaekIteLIg BYkvabegáItnUvKnøgd¾
exSayenAkñúgebtug. enAeBlEdlbnÞúkRtUv)anGnuvtþn_ sñameRbHcab;epþImelcecjrUbragenAelImux
kat;EdlexSayCageK dUcCatambeNþayEdkBRgwg. cMnYn sñameRbHEdlekIteLIgRtUv)ankMNt;eday


T.Chhay NPIC

brimaNénkarrYmmaDrbs;ebtug nigvtþmanénkarbgábTMr. eKBi)akkñúgkarRKb;RKgsñameRbHedaykar
;
rYmmaDenHNas;.
- sñameRbHbnÞab;bnSMedaykarBt; secondary flexural cracks CaTUeTAvamanKMlatFM² ehIysñameRbH
mYy²minmanT§BlkñúgkarbegáItsñameRbHd¾éTeToteT. vaRtUv)anrMBwgnwgekItmaneRkambnÞúktUc dUc
CabnÞúkefr. enAeBlbnÞúkRtUv)anGnuvtþn_CabnþbnÞab;eTA elIFñwmsamBaØ kugRtaMgTajekItmanenA
sésrxageRkam ehIyenAeBlEdlvaFMCagkugRtaMgTajEdl ekItBIkarBt;rbs;ebtug sñameRbHcab;
epþImekIteLIg. vacab;epþImrIkFMCabnþbnÞab; niglatsn§wgeTAkan;GkS½NWt. vaCakarBi)ankñúgkarTsSn_
TaynUvmuxkat;EdlsñameRbHbnÞab;bnSMcab;epþImekItman BIeRBaH ebtugminEmnCasMPar³
homogeneous material nig isotropic material.

elak Salinger nigelak Billing)an):an;RbmaNkugRtaMgEdkmunkareRbHRtwmEtBIRbEhl 42MPa
eTA 49MPa . TMhMeRbHdMbUgRtUv)anrMBwgRbEhl 0.025mm enAsésrTajxageRkAbMputrbs;ebtug.
enAeBlsñameRbHRtUv)anbegáIt kugRtaMgTajrbs;ebtugenAmuxkat;eRbHfycuHdl; 0 ehIysésrEdk
TTYlnUvkMlaMgTajTaMgGs;. enAkñúgxN³enH vaekItmankarsNþkrvagsésrEdk nigebtugedaysar
Etebtug nigEdkmansac;lUtxusKña niglatsn§wgeTAdl;muxkat;Edlebtug nigEdkman strain esμIKña.
rUbTI6>6 bgðajBIKMrUénkarEbgEckkugRtaMgrvagsñameRbHenAkñúgGgát;eRkamkMlaMgTajtamGkS½.

- sñameRbHbnÞab;bnSMERcH corrosion secondary cracks ekIteLIgenAeBlsMeNImmansarFatuKImIdUcCa
sUdüÚmkørY sodium chloride Nacl kabUnDIG‘uksIut carbon dioxide CO2 nig]sμn½G‘uksIuEsnRCabcUl

eTAkñúgépÞebtug ehIybegáItCaERcHsIuEdk. bnSMG‘uksIutEdlbegáItedaykarxUcxatrbs;EdkRKbdNþb;
nUvbrimaNeRcInCagEdk nigCMrujsMBaFemkanicEdlbnþlatsn§wgsñameRbHkan;EtFM. sñameRbHRbePT
enHmaneRKaHfñak;EdlGaceFVIeGayeRKOgbgÁúM)ak;. kar)ak;dMbUlenA Muskegan enArdæ Michigan kñúg
qñaM 1955 bNþalmkBIERcHsIuEdk RtUv)anraykarN_eday Shermer. karrIkraldalnUvsñameRbH
nig karpÞúHebtug spalling of concrete enAelIs<an San Mateo-Hayward kñúgGMLúgeBl 7qñaM enArdæ
California RtUv)anraykarN_eday Stratfull. sñameRbHedayERcHGacRtUv)anbBaÄb;edayeRbInUv

viFIsaRsþsagsg;RtwmRtUv nigebtugEdlmanKuNPaBx<s;.
6>2> sñameRbHcMbg Main Cracks
sñameRbHcMbg main cracks ekIteLIgnUvdMNak;kalbnÞab;BIsñameRbHbnÞab;bnSM secondary cracks.
vaRtUv)anbegáIteLIgedayPaBxusKñaén strain enAkñúgEdk nigebtug enAmuxkat;eRbH. karRbRBwtþeTArbs;sñam
eRbHcMbgpøas;bþÚrenABIrdMNak;kalepSgKña. enAeBlEdkrgkugRtaMgTajtUc cMnYnsñameRbHekIneLIg b:uEnþTMhM
sñamenAmanTMhMtUc EtenAeBlEdlkugRtaMgTajekIneLIg va)aneTAdl;dMNak;kallMnwg. enAeBlEdlkug
RtaMgekIneLIgkan;Etx<s; dMNak;kalTIBIrénkareRbHekIteLIg ehIyTMhMeRbHrIkFM b:uEnþBuMmankarekIneLIgcMnYn
sñameRbHKYreGaykt;sMKal;eT. CaTUeTAsñameRbHmYy b¤BIrcab;epþImBRgIkTMhMFMCagsñameRbHdéT EdlbegáIt
CasñameRbHeRKaHfñak; ¬rUbTI 6>7¦.


T.Chhay NPIC

sñameRbHenAkñúgFñwm nigeRKOgbgÁúMrgkarTaj Edl)ansikSaedayGñkGegát karTsSn_TayBITMhMsñam
eRbH nigkarRKb;RKgsñameRbHCacMENkénbBaðaEdlRtUvsikSa.
TaMgenHRtUv)anBiPakSaenATIenH CamYynigtMrUvkarrbs;bTdæan ACI Code.
- ebIeyagtamRTwsþIsñameRbHEdlmaneRsc TMhMsñameRbH crack width nigKMlatsñameRbH crack
spacing GaRs½yeRcInktþa EdlrYmmanPaKryEdk karBRgayEdkenAkñúgmuxkat;ebtug kugRtaMgBt;

rbs;Edk enAbnÞúkeFVIkar service load kMras;ebtugkarBarEdk niglkçN³rbs;sarFatupSMebtug.
smIkarepSg²sMrab;TsSn_TayTMhM nigKMlatsñameRbHenAkñúgGgát;ebtugGarem:RtUv)anbgðajenAkñúg
snñisiT§sþIGMBIkarkekItkaretags¥it nigkarkekItsñameRbHenAkñúgebtugBRgWgedayEdk Symposium
on Bond and Crack Formation in Reinforced Concrete enA Stockholm RbeTs Sweden kñúgqñaM

1957. elak Chi nigelak Kirstein )anbgðajsmIkarsMrab;TMhMeRbH nigKMlatsñameRbHCaGnu-
Kmn_eTAnwgRkLaépÞRbsiT§PaBrbs;ebtugEdlenACMuvijEdk³ RkLaépÞebtug mUlEdlmanGgát;p©itesμI
Ggát;EdkbYndgRtUv)aneRbIedIm,IKNnaTMhMeRbH. smIkardéTeTotRtUv)anbgðajenATsvtSr_eRkay
²eTot.
elak Gergely nigelak Lutz bgðajnUvrUbmnþxageRkamsMrab;kMNt;TMhMeRbH³
W = 11βf s 3 Ad c × 10 −6 (6-16)

Edl β / A nig f s RtUv)ankMNt;BIxagedIm nig dc = kMras;karBarebtugEdlRtUv)anvas;BIsésrTaj
EpñkxageRkAbMputeTAGkS½EdkCitCageK. tMél β GacRtUv)anykRbEhl 1.2 sMrab;Fñwm nig 1.35 sMrab;kM
ralxNÐ. cMNaMfa f s KitCa MPa nig W KitCa mm .
pleFobrvagTMhMsñameRbHGtibrmaelITMhMsñameRbHmFümRtUv)anrkeXIjsßitenAcenøaH 1.5 nig 2.0
Edl)anraykarN_edayGñkGegátCaeRcIn. tMélmFümeKGaceRbI 1.75 .
- TMhMeRbHEdlGacGnueRKaH tolerable crack width ³ eKminGaceCosputBIkarkekItsñameRbHenAelI
Ggát;ebtugGarem:)aneT. eTaHbICamankarsikSa nigkarsagsg;y:agykcitþTukdak;y:agNak¾eday
k¾vaenAEtkekItnUvsñameRbHb:unsésrsk;Edr. CaTUeTAsñameRbHRtUv)anvas;enAelIépÞrbs;ebtug b:uEnþ
Cak;EsþgsñameRbHmantaMgBInIv:UEdkemø:H EdlCakEnøgekItmanERcHsIuEdk. TMhMsñameRbHGnuBaØati
RtUv)anCHT§iBledaytMrUvkaresaPNÐPaB. sñameRbHEdlGacemIleXIjedayEPñkTeTrmanTMhMRb
Ehl 0.15mm vaGaRs½yeTAnwg texture rbs;épÞebtug. tMélepSg²sMrab;sñameRbHGnuBaØatienAnIv:U
EdkRtUv)anesñIeLIgedayGñkGegátCaeRcIn EdlvasßitenAcenøaH 0.25 nig 0.40mm sMrab;Ggát;xagkñúg
nigsßitenAcenøaH 0.15 nig 0.25mm sMrab;Ggát;xageRkA. TMhMsñameRbH 0.40mm sMrab;Ggát;xagkñúg
nig 0.32mm sMrab;Ggát;xageRkA eRkamlkçxNÐs¶Üt GacRtUv)anGnueRKaH.

- sñameRbH crack control eRcInekItmanenAelIGgát;ebtugGarem:EdleRbIEdkersIusþg;x<s; high-strength
steel . sñameRbHkan;EtFMekItmaneRkambnÞúkeFVIkar edaysarEtkugRtaMgGnuBaØatix<s;. karRKb;RKg

sñameRbHGaRs½yeTAnwgTMhMsñameRbHGnuBaØati . vaCakarRbesIrEdlmansñameRbHtUc²EteRcIn Ca
CagmansñameRbHFM²Ettic. sñameRbHbnÞab;bnSM secondary crack RtUv)ankat;bnßyedaykarRKb;
RKgnUvbrimaNkMe)areRbgTaMgGs; pleFobTWk-sIum:g;t¾ PaBRCabTWkrbs;fμbMEbk nigebtug kMritén
karEfTaMebtug niglkçxNÐbgák;cug end-restraint condition.
ktþaEdlTak;TgnwgkarRKb;RKgsñameRbHcMbg KWkugRtaMgEdk PaBetags¥itrbs;ebtug kar
BRgayEdk Ggát;p©itEdk PaKryEdk kMras;karBarebtug niglkçN³rbs;sarFatupSMebtug. edIm,ICYy
kat;bnßyTMhMsñameRbH eKRtUveFVIeGayktþaEdl)anerobrab;xagelIy:agehacmYymanlkçN³kan;Etl¥.
7> tMrUvkarrbs;bTdæan ACI Code ACI Code Requirement

edIm,IRKb;RKgsñameRbHenAkñúgGgát;ebtugGarem: bTdæan ACI Code, chapter 10 kMNt;nUvtMrUvkarxag
eRkam³
1> manEtEdk deformed bar RtUv)anGnuBaØatieGayeRbICaEdkem main reinforcement
2> EdkrgkarTajKYrEtRtUv)anBRgayenAtMbn;TajGtibrma (ACI Code, section 10.6.3)
3> enAeBlEdlsøabrbs;muxkat;sßiteRkamkarTaj EpñkénEdkemKYrRtUv)anBRgayenAkñúgcMeNamtMél
EdltUcCageKkñúgcMeNamTTwgsøabRbsiT§PaB effective flange width nigmYyPaKdb;énElVg one-
tenth of the span (ACI Code, section 10.6.6).

4> ersIusþg;KNna design yield strength rbs;EdkminKYreGayelIsBI 560MPa (ACI Code, section
9.4).

5> KMlatGtibrma s BIEdkEdlmancMgayeTAépÞebtugrgkarTajCitCageK sMrab;FñwmebtugGarem: nigkM
ralxNÐmYyTisRtUv)ankMNt;
s = 105000 / f s − 2.5Cc (6-17)
b:uEnþminRtUvFMCag 300(280 / f s )
Edl f s = kugRtaMgKNnaenAkñúgEdkenAbnÞúkeFVIkarRtUv)ankMNt;edaypleFobrvagm:Um:g;Kμanem
KuNnigplKuNrvagRkLaépÞEdknigRbEvgédXñas;kñúg f s = M /( As jd ) . ¬mü:agvijeTot
eKGaceRbI f s = 2 f y / 3 edaysarEtkareRbItMélRbEhlrbs; jd = 0.87d ¦.
Cc = kMras;ebtugkarBarEdk clear cover BIépÞrgkarTajCitbMputeTAépÞEdkrgkarTaj.

s = KMlatKitBIGkS½énEdkrgkarTajEdlCitCageKeTAépÞebtugrgkarTaj.


T.Chhay NPIC

karkMNt;enHRtUv)aneRbIsMrab;EtFñwmebtugGarem: nigkMralxNÐmYyTisEdlsßitenAkñúglkçxNÐ
brisßanFmμta. karkMNt;KMlatminTak;TgnwgTMhMEdkeT EdlvanaMeGaymankareRbIR)as;TMhMEdktUc
CagedIm,IbMeBjlkçxNÐKMlat. sMrab;krNIFñwmebtugGarem:EdleRbIEdk f y = 420MPa nig
Cc = 50mm . enaHKMlatGtibrma s RtUv)anKNnadUcxageRkam³

snμt; f s = 0.66 × 420 = 280MPa
s = (105000 / 280 ) − 2.5 × 50 = 250mm
EdldUcKñaeTAnwg 300(280 / 280) = 300mm
6> enAkñúgbTdæanBImun karRKb;RKgsñameRbHRtUv)anQrelIemKuN Z EdlkMNt;dUcxageRkam³
Z = f s 3 Ad c ≤ 31kN / mm sMrab;Ggát;enAxagkñúg (6-18)

Z ≤ 26kN / mm sMrab;Ggát;enAxageRkA (6-19)

Edl f s = kugRtaMgBt;enAbnÞúkeFVIkar nigGacykesμI 2 f y / 3 / A nig dc CatMbn;TajRbsiT§PaB
rbs;ebtug nigkMras;ebtugkarBarEdk erogKña. smIkarenHQrelIsmIkar (6-16) edaysnμt;TMhMsñameRbH
RtUv)ankMNt;Rtwm 0.40mm sMrab;Ggát;enAxagkñúg nig 0.32mm sMrab;Ggát;enAxageRkA. edIm,ITTYl)an
Z tUc eKRtUvbnßykMras;karBarebtug. kMras;ebtugminRtUvelIsBI 50mm .

7> Edkxag skin reinforcement Ask sMrab;FñwmkMBs;eRCA EdlmankMBs;RbsiT§PaB d ≥ 900mm sésr
EdkesþIgKYrRtUbv)anbEnßmenAEk,répÞbBaÄrkñúgtMbn;TajedIm,IRKb;RKgsñameRbHenAelIRTnugBIelI
Edkem. bTdæan ACI Code, section 10.6.7 )anbriyayBIEdkbEnßmEdlCaEdkxag skin
reinforcement Ask . EdkxagKYrRtUv)anBRgayesμIelIépÞTaMgsgçagsMrab;cMgay d / 2 Ek,rEdkrgkar

TajbMput.
KMlat Ssk rvagEdkbeNþayénEdkxagminKYrFMCagtMélEdltUcCageKkñúgcMeNamtMél d / 6 /
300mm nig 1000 Ab /(d − 750) Edl Ab CaRkLaépÞénmuxkat;EdkmYy² KitCa mm 2 ¬rUbTI6>8¦.
⎧ d /6
⎪
S sk = min ⎨ 300
⎪1000 A /( d − 750)
⎩ b

Edkxag k¾GacRtUv)anKitbBa©ÚlkñúgkarKNnaersIusþg; (strength) edayQrelIkarKNnakugRtaMg
stress nigbMErbMrYlrageFob (strain). elIsBIenH EdkxagsrubenAépÞTaMgsgçagénGgát; minKYrelIsBIBak;

kNþalmuxkat;EdkrgkarTajsrub As eT.
eyagtamrUbTI 6>8 RbsinebI b = 400mm / d = 950mm nigeRCIserIsEdk DB10 ¬ Ab =
78mm 2 ¦CaEdkxag enaHKMlatrbs;Edk DB10 EdlCaEdkbeNþayKYryktMéltUcCageKkñúgcMeNam


/
300mm 950 / 6 = 158mm nig (1000 × 78) /(950 − 750) = 390mm . dUcenH eyIgeRCIserIsykKMlat
150mm . EdkxagKYrRtUv)anBRgayedayEdk DB10 kñúgKMlat 150mm elIkMBs; d / 2 sMrab;xagmYy² ¬rUb

TI 6>8¦. RkLaépÞsrubEdkxagsMrab;épÞTaMgBIr = 6 × 78 = 468mm2 EdltUcCag 0.5 As = 0.5 × 4021
= 2010.5mm 2 .

taragxageRkampþl;nUvKMlatGtibrmaenAelIépÞmYy² Ssk RkLaépÞEdkxagGb,brmamYy² Ab .
kMBs;Fñwm d / mm 800 900 1000 1100 1200 1300 1400 1500
KMlatGtibrma Ssk / mm 150 150 150 175 200 200 225 250
Ab ¬Gb,brma¦/ mm 2 7 .5 22.5 37.5 61.25 90 110 146.25 187.5
]TahrN_6>6³ muxkat;FñwmTMrsamBaØRtUv)anbgðajenAkñúgrUb 6>9
a. epÞógpÞat;muxkat;EdktamtMrUvkarrbs; ACI Code

b. kMNt;TMhMeRbHEdlnwgGacekItman

c. epÞógpÞat;emKuN Z tamsmIkar (6-18)

eKeGay f 'c = 28MPa / f y = 400MPa nigEdkkg DB10 .
dMeNaHRsay³
1> rUb 6>9 muxkat; a ³
a. sMrab; 3DB 28 / As = 1847mm 2 / kMras;ebtugkarBarEdk Cc = 60 − 28 / 2 = 46mm / snμt;

f s = 0.66 f y = 0.66 × 400 = 264MPa / KMlatGtibrma

s = 105000 / 264 − 2.5 × 46 = 282.7 mm EdltUcCag 300(280 / 264) = 318.2mm .


T.Chhay NPIC

KMlatEdlpþl;eGay = 0.5(300 − 60 − 60) = 90mm edayKitBIGkS½EdkeTAGkS½Edk EdltUcCag
282.7 mm .

b. sMrab;muxkat;enH dc = 60mm . RkLaépÞebtugrgkarTajRbsiT§PaBsMrab;EdkmYyKW
A = 300(2 × 60 ) / 3 = 12000mm 2
KNnaTMhMsñameRbHtamsmIkar (6-16)

W = 11β f s 3 Ad c × 10 −6

edayGgát;enHCaFñwmenaH β = 1.2 nig f s = 240MPa

W = 11× 1.2 × 2403 12000 × 60 ⋅10 −6 = 0.28mm
EdltUcCag 0.40mm nig 0.32mm sMrab;Ggát;xagkñúg nigsMrab;Ggát;xageRkA.
c. tamsmIkar (6-18)
Z = f s 3 Ad c = 2403 12000 × 60 ⋅10 −3 = 21.5kN / mm

EdltUcCag 31kN / mm nig 26kN / mm sMrab;Ggát;xagkñúg nigsMrab;Ggát;xageRkA.

2> rUb 6>9 muxkat; b ³
a. karKNnaKMlatEdkmanlkçN³RsedogKñacMnuc a xagelI.

b. sMrab;muxkat;enH d c = 60mm ehIyEdkRtUv)andak;CaBIrRsTab;. TIRbCMuTMgn;rbs;EdkKW 90mm

BIsésrxageRkam. RkLaépÞebtugrgkarTajRbsiT§PaBsMrab;EdkmYyKW
A = 300(2 × 90 ) / 6 = 9000mm 2
W = 11× 1.2 × 2403 9000 × 60 ⋅10 −6 = 0.26mm RKb;RKan;
c. tamsmIkar (6-18)
Z = f s 3 Ad c = 2403 9000 × 60 ⋅10 −3 = 19.5kN / mm RKb;RKan;

karBiPakSa
eyIgGaceXIjfaKMlat s enAkñúgsmIkar (6-17) CaGnuKmn_énkugRtaMgkñúgEdkTaj b¤CaGnuKmn_eday
minpÞal;énbMErbMrYlrageFobenAkñúgEdkTaj f s = Es × ε s ehIy Es sMrab;EdkesμI 2.1⋅105 MPa . dUcKñaKM
latk¾GaRs½ynwgkMras;ebtugkarBarEdk Cc . karekIneLIgnUvkMras;ebtugkarBar vanwgkat;bnßyKMlat Edlva
minTak;TgeTAnwgTMhMEdkEdleRbIenAkñúgmuxkat;eT.
enAkñúg]TahrN_enH TMhMsñameRbHEdlrMBwgTukRtUv)anKNnaedaysmIkar (6-16) edIm,Ipþl;eGay
nisiSt nigvisVkrnUvtMrUvkarRKb;RKgsñameRbH nigTMhMsñameRbH. CaTUeTATMhMsñameRbHRtUv)anvas;enAelIFñwm
EdleFVIkarBiesaFn_enAkñúgmnÞIrBiesaFn_ b¤k¾enAkñúgeRKOgbgÁúMCak;EsþgEdlsßiteRkambnÞúk EdleFVIeGayman
sñameRbHeRKaHfñak;ekIteLIgenAkñúgFñwm nigkñúgkMralxNÐ ehIyEdlRtUvkarkarBiesaFn_. RbsinebITMhMsñam
eRbHEdl)anvas;mun nigeRkaykardak;bnÞúkFMCag yield strain rbs;Edk enaHEdkemnwgmanlkçN³)øasic nig
KμanRbsiT§PaB. eKGacvas;TMhMeRbHCak;Esþg)anedaybnÞat;sMrab;vas;sñameRbHEdleKGacrkTij)an. eRkABI
kugRtaMgEdk nigkMras;ebtugkarBarEdk W enAGaRs½ynwgktþaTIbIKW A EdlCaRkLaépÞebtugrgkarTajBT§½CMu
vijEdkrgkarTajmYy.
]TahrN_6>7³ KNnaFñwmTMrsamBaØEdlmanRbEvg 7.3m RTnUvbnÞúkefrBRgayesμI 21.9kN / m nigbnÞúk
GefrBRgayesμI 17.2kN / m . eRCIserIsmuxkat;EdkRKb;RKan; nigepÞógpÞat;KMlatEdkedIm,IbMeBjlkçxNÐ
ACI Code. eKeGay b = 400mm / f 'c = 28MPa / f y = 400MPa PaKryEdk 0.8% nigkMras;ebtugkar

BarEdk Cc = 50mm .

dMeNaHRsay³
1> sMrab;PaKryEdk ρ = 0.8% / Ru = 2.7MPa ¬ φ = 0.9 ¦. m:Um:g;emKuNxageRkAKW
M u = wL2 / 8 nig w = 1.2 × 21.9 + 1.6 × 17.2 = 53.8kN / m


T.Chhay NPIC

M u = 53.8 × 7.32 / 8 = 348.624kN .m

M u = Ru bd 2 d = 568mm As = 0.008 × 400 × 568 = 1817.6mm 2
eRbI 3DB28 ¬ As = 1847.3mm2 ¦mYyCYr. kMBs;srubrbs;muxkat;Fñwm h = 650mm .
CaTUeTA
d = 650 − 50 − 28 / 2 = 586mm ¬rUbTI6>10¦
2> KNnabnÞúkeFVIkar service load nigm:Um:g;eFVIkar service moment
w = 21.9 + 17.2 = 39.1kN / m
M u = 39.1× 7.32 / 8 = 260.5kN .m
3> KNnakMBs;GkS½NWt kd nigRbEvgédXñas; jd tamsmIkar (6-12)
b(kd ) 2 / 2 − nAs (d − kd ) = 0 n = 8 As = 1847.3mm 2 d = 586mm

kd = 174.4mm jd = d − kd / 3 = 527.9mm j = 527.9 / 586 = 0.9
cMNaMfa eKGacyktMélRbEhlrbs; j = 0.87 kñúgkrNIEdl kd minRtUv)anKNna.
4> KNnakugRtaMg f s
M = As ⋅ f s ⋅ jd

f s = 260.5 ⋅106 /(1847.3 × 527.9) = 267 MPa
5> KNnaKMlat s tamrUbmnþ (6-17)
s = 105000 / 267 − 2.5 × 50 = 268.3mm
EdltUcCag 300(280 / 267) = 314.6mm
KMlatEdleGay = 0.5(400 − 64 − 64) = 136mm < 230.8mm


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