Weitere ähnliche Inhalte Ähnlich wie Vi deflection and control of cracking Ähnlich wie Vi deflection and control of cracking (18) Vi deflection and control of cracking 1. 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
2. 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
3. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
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
Deflection and Control of Cracking 92
4. 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
PaBdab nigsñameRbH 93
5. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
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 ⎠ ⎣ ⎠ ⎦
Deflection and Control of Cracking 94
6. 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.
PaBdab nigsñameRbH 95
7. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
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 ¦.
Deflection and Control of Cracking 96
8. 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,
PaBdab nigsñameRbH 97
9. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
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.
Deflection and Control of Cracking 98
10. 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
PaBdab nigsñameRbH 99
11. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
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
Deflection and Control of Cracking 100
12. 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¢aMgxNн 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
PaBdab nigsñameRbH 101
13. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
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
Deflection and Control of Cracking 102
14. 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
PaBdab nigsñameRbH 103
15. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
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
Deflection and Control of Cracking 104
16. 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
PaBdab nigsñameRbH 105
17. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
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
Deflection and Control of Cracking 106
18. 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;.
PaBdab nigsñameRbH 107
19. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
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
Deflection and Control of Cracking 108
20. 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
PaBdab nigsñameRbH 109
21. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
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¦.
Deflection and Control of Cracking 110
22. 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.
PaBdab nigsñameRbH 111
23. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
- 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.
Deflection and Control of Cracking 112
24. 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
PaBdab nigsñameRbH 113
25. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
/
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 .
Deflection and Control of Cracking 114
26. 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;
PaBdab nigsñameRbH 115
27. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
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
Deflection and Control of Cracking 116
28. 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
PaBdab nigsñameRbH 117