Más contenido relacionado La actualidad más candente (20) Similar a Xi members in compression and bending (20) Xi members in compression and bending1. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
XI. eRKOgbgÁúMrgkarsgát; nigrgkarBt;
1> esckþIepþIm
eRKOgbgÁúMbBaÄrCaEpñkmYyrbs;eRKagsMNg; EdlrgkMlaMgsgát; nigm:Um:g;. kMlaMgTaMgenH )anBIkMlaMg
xageRkAdUcCa bnÞúkefr bnÞúkGefr nigbnÞúkxül;. kMlaMgRtUv)ankMNt;eday karKNnaedayéd b¤edaykMuBüÚT½r
EdlQrelIeKalkarN¾sþaTic nigviPaKeRKOgbgÁMú (structural analysis). Ca]TahrN¾ sMrab;rUb (1) bgðajfa
eRKagQrelITMr hinged BIr EdlrgbnÞúkemKuNBRgayesμI enAelIGgát; BC. düaRkamm:Um:g;Bt;RtUv)anKUrenA
EpñkxagTaj. kMNat;ssr AB nig CD rgnUvkMlaMgsgát; nigm:Um:g;Bt;. pleFobrvagm:Um:g;Bt; nigkMlaMg
sgát; RtUv)aneGayeQμaHfa cMgaycakp©it e Edl e = M . e CacMgayBITIRbCMuTMgn;)aøsÞic(plastic centroid)
n
P
n
énmuxkat;eTA cMnucénbnÞúkmanGMeBI. TIRbCMuTMgn;)aøsÞic (plastic centroid) RtUv)anTTYledaykarkMNt;TItaMgkM
laMgpÁÜbbegáItedaysésEdk nigebtug edaysnμt;kugRtaMgsgát;sMrab;Edk f nigkugRtaMgsgát;sMrab;ebtug
y
0.85 f ' . sMrab;muxkat;sIuemRTI TIRbCMuTMgn;)aøsÞic (plastic centroid) RtYtsIuKñaCamYyTIRbCMuTMgn;rbs;muxkat;.
c
sMrab;muxkat;minsIuemRTI TIRbCMuTMgn;)aøsÞic (plastic centroid) RtUv)ankMNt;edayeRbIm:Um:g; eFobnwgGkS½
arbitrary axis.
rUbTI1³ eRKagTMr pin BIrCamYynwgdüaRkamm:Um:g;
]TahrN_1³ kMNt;TIRbCMuTMgn;)aøsÞic (plastic centroid)
énmuxkat;dUcbgðajkñúgrUbTI2. smμtikmμ³ f 'c = 28MPa
nig f = 400MPa .
y
dMeNaHRsay³
!> kugRtaMgsgát;sMrab;ebtugRtUv)ankMNt;yk 0.85 f ' rUbTI2³ TIRbCuMTMgn;)øasÞic (P.C)énmuxkat;
c
F = kMlaMgkñúgrbs;ebtug = 0.85 f ' A
c c g
Members in Compression and Bending 196
2. T.Chhay NPIC
= (0.85 × 28) × 350 × 500 = 4165kN
Fc sßitenAelITIRbCMuTMgn;énmuxkat;ebtug ¬enAcMgay 250mm BIGkS½ A − A ¦
@> kMlaMgenAkñúgsésEdk
π × 282
Fs1 = As1 f y = 4 × × 400 = 985.2kN
4
π × 282
Fs 2 = As 2 f y = 2 × × 400 = 492.6kN
4
#> kMNt;m:Um:g;eFob A − A
(4165 × 250) + (985.2 × 65) + (492.6 × 435)
x= = 233.85mm
4165 + 985.2 + 492.6
dUcenH TIRbCMuTMgn;)aøsÞic (plastic centroid) RtUvsßitenAelIcMgay 233.85mm BIGkS½ A − A
$> RbsinebI A = A ¬muxkat;sIuemRTI¦ dUcenH x = 250mm BIGkS½ A − A .
s1 s2
2> karsnμt;sMrab;KNnassr
GaRs½ytam ACI Code EdnkMNt;sMrab;karKNnassrkMNt;dUcxageRkam³
!> kMhUcRTg;RTayeFob strain enAkñúgebtug nigEdk RtUvsmamaRteTAnwgcMgayBIGkS½NWt.
@> RtUvEtbMeBjlkçxNÐ smIkarlMnwgénkMlaMg nigPaBRtUvKñaénkMhUcRTg;RTayeFob strain
compatibility.
#> kMhUcRTg;RTayeFobrbs;ebtugrgkarsgát;EdleRbIR)as;GtibrmaKW 0.003 .
$> ersIusþg;rbs;ebtugrgkarTajGacRtUvecal.
%> kugRtaMgenAkñúgEdkKW f = εE ≤ f .
s s y
^> bøúkkugRtaMgGackMNt;manragctuekaNCamYykugRtaMg 0.85 f ' BRgayBIRbEvg a = β c . Edl
c 1
c CacMgayBIGkS½NWt nig
⎧ 0.85 f 'c ≤ 28MPa
⎪ f ' −28
β1 = ⎨0.85 − 0.05( c
7
) sMrab;ebtugEdlmanersIusþg; 28MPa < f 'c ≤ 56 MPa
⎪ f 'c > 56MPa
⎩ 0.65
3> düaRkamGnþrkmμbnÞúk-m:Um:g; Load-moment interaction diagram
enAeBlEdlbnÞúktamGkS½RtUv)anGnuvtþmkelIssrxøI krNIdUcxageRkamGacekIteLIg edayGaRs½y
eTAnwgTItaMgGnuvtþbnÞúkedayeFobeTAnwg TIRbCMuTMgn;)aøsÞic (plastic centroid). rUb
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 197
3. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
kMlaMgsgát;tamGkS½ P CakMlaMgsgát;tamGkS½EdlmantMélFMGnuvtþenAelITIRbCMuTMgn;)aøsÞic (plastic
o
centroid) e = 0 nig M = 0 . kar)ak;rbs;ssr ekIteLIgedayebtugEbk nigEdkeFVIkardl; yielding.
n
vaRtUv)ansMEdgeday P enAelIExSekag.
o
rUbTI3 a³ düaRkamGnþrkmμbnÞúk-m:Um:g;
!> Maximum nominal axial load P : CakrNIEdlkMlaMgtamGkS½GnuvtþeTAelImuxkat;CamYy cM
n max
gaycakp©it eccentricity Gb,rma. tam ACI Code, P = 0.80P sMrab;ssrEdkkgdac;² tie
n max o
column nig P = 0.85P sMrab;ssrEdlmanEdkkgdUcrWus½r spirally reinforced column . kar)ak;
n max o
ekIteLIgedayebtugEbk nigEdkeFVIkardl; yielding.
@> Compression failure: CakrNIEdlbnÞúktamGkS½FMGnuvtþenAcMgaycakp©ittUc. bnÞúktamGkS½kñúg
krNIenHmantMélERbRbYlBI tMélGtibrma P = P eTAtMélGb,rma P = P (balanced load). s
n n max n b
r)ak;edayebtugEbkenAEpñkrgkarsgát;CamYYynwgbMErbMrYlrageFob strain = 0.003 ÉcMEnkkugRtaMgkñúgEdk
¬EpñkrgkarTaj¦ KWtUcCag yield strength f < f . kñúgkrNIenH P > P nig e < e .
s y n e n
#> Balanced condition P : lkçxNÐenHekItmaneLIgenAeBlEdl bMErbMrYlrageFobrgkarsgát;
b
compression strain enAkñúgebtugmantMélesμI 0.003 ehIybMErbMrYlrageFobrbs;EdkrgkarTajmantMél
Members in Compression and Bending 198
4. T.Chhay NPIC
εy =
fy
Es
. kar)ak;rbs;ebtugekIteLIgtMNalKñanwgEdk yield. m:Um:g;EdlekItedaysarbnÞúkenHRtUv)an
eKehAfa balanced moment M cMgaycakp©itRtUv)aneKehAfa balanced eccentricity e = M .
b
P
b
b
b
$> Tension failure: CakrNIekItmanenAxN³Edl bnÞúktamGkS½tUc nigcMNakp©itFM ehIyEdlmanm:U
m:g;FM. muneBl)ak; kugRtaMgTajekItmanenAelIEpñkd¾FMénmuxkat; bNþaleGayEdkrgkarTaj yield muneBl
rUbTI3 b³ muxkat;ssrEdlbgðajBITItaMgbnÞk P sMrab;lkçxNÐbnÞúkepSg²
ú n
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 199
5. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
ebtugEbk. enAeBl)ak; bMErbMrYlrageFobrbs;EdkrgkarTajmantMélFMCagbMErbMrYlrageFob yield ε y
ehIybMErbMrYlrageFobenAkñúgebtugesIμ 0.003 . krNIenHekItmanBI Balanced condition eTAdl; pure
flexure P < P nig e > e .
n b e
%> Pure flexure: muxkat;kñúgkrNIenHrgm:Um:g;Bt; M Edl P = 0 . kar)ak;dUcKñanwg kar)ak;rbs;
n n
FñwmrgkarBt;. cMNakp©itRtUv)ansnμt;fa Gnñn.
4> karpþl;nUvsuvtßiPaB Safety provisions
!> emKuNbnÞúksMrab; bnÞúk gravity nigbnÞúkxül;
U = 1 .4 D
U = 1 .2 D + 1 .6 L
U = 1.2 D + 1.6 L + 0.8W
U = 1.2 D + 1.0 L + 1.6W
U = 0.9 D + 1.6W
bnSMbnÞúkemKuNEdlmantMélsFMCageKRtUv)anykmkeRbIR)as;sMrab;karKNna.
@> emKuNkat;bnßyersIusþg; φ eRbIsMrab;KNnassrGaRs½yeTAnwgkrNIxageRkam³
k> enAeBl P = φP ≥ 0.1 f ' A eBlenaH φ = 0.65 sMrab;ssrEdkkgdac;² tie column
u n c g
ehIy φ = 0.7 sMrab;muxkat;ssr EdlmanEdkkgdUcrWus½r spirally reinforced column .
krNIenHssrRtUv )anrMBwg fa)ak;edaykarrsgát;.
rUbTI4³ tMélemKuN φ
Members in Compression and Bending 200
6. T.Chhay NPIC
x> muxkat;EdlbMErbMrYlrageFobrgkarTajsuT§ net tensile strain ε sMrab;ersIusþg;Fmμta t
nominal strength enAkñúgEdkrgkarTajeRkAeKbMput KWsißtenAcenøaH 0.005 nig 0.002 (transition region)
φ ERbRbYlCalkçN³bnÞat;cenøaH 0.9 nig 0.67 b¤ ¬ 0.7 ¦.
sMrab;muxkat;EdlmanEdkkgdUcrWus½r spiral section
⎡ 1 5⎤
) b¤ φ = 0.7 + 0.2 ⎢
200
φ = 0.7 + (ε − 0.002)( t − ⎥ (-1)
3 c/d 3 ⎣ t ⎦
sMrab;muxkat;epSgeTot
φ = 0.65 + (ε t − 0.002)(
250
) b¤ φ = 0.65 + 0.25⎡ c /1d
⎢
5⎤
− ⎥ (-2)
3 ⎣ t 3⎦
K> enAeBl P = 0 kñúgkrNIrgkarBt;suT§ φ = 0.9 sMrab; muxkat; tension-control section
u
nig ERbRbYlBI 0.9 nig 0.65 b¤ ¬ 0.7 ¦ enAkñúgtMbn; transion region.
5> Balanced condition – muxkat;ctuekaN
Balanced condition ekItmanenAkñúgmuxkat;ssrenAeBlEdl bnÞúkEdlGnuvtþmkelImuxkat;ssr
Edlman nominal strength begáItbMErbMrYlrageFobesμI 0.003 enAkñúgsésrgkarsgát;rbs;ebtug nigbMEr
bMrYlrageFobesμI ε = E enAkñúgr)arEdkrgkarTajkñúgeBldMNalKña. enHKWCakrNIBiessEdl GkS½NWt
f
y
y
s
GacRtUv)ankMNt;BI strain diagram edaysÁal;tMélFMbMput. enAeBlEdlbnÞúkcakp©itmantMél FMCag P b
enaHeKehAmuxkat;enaHfa compression control. pÞúymkvijeKehAfa tension control .
karviPaK balanced column section GacRtUv)anBnül;dUcxageRkam³
!> yk c CacMgayBIsésrgkarsgát;q¶aybMputmkGkS½NWt. BI strain diagram
b
cb (balanced) 0.003
= (-3)
d fy
0.003 +
Es
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 201
7. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
eday E s = 200000MPa
600d
⇒ Cb =
600 + f y
kMBs;bøúkrgkarsgát;smmUl equivalent compressive block
⎛ 600 ⎞
ab = β1cb = ⎜ ⎟β d (-4)
⎜ 600 + f ⎟ 1
⎝ y ⎠
⎧ 0.85 f 'c ≤ 28MPa
⎪ f ' −28
Edl β1 = ⎨0.85 − 0.05( c
7
) sMrab;ebtugEdlmanersIusþg; 28MPa < f 'c ≤ 56 MPa
⎪ f 'c > 56MPa
⎩ 0.65
@> BIsmIkarlMnwg plbUkkMlaMgtamTisedkesIμsUnü
Pb − Cc − C s + T = 0
Edl Cc = 0.85 f 'c ab nig T = A f
s y (-5)
Cs = A' ( f 's −0.85 f 'c ) enAeBlEdlEdkrgkarsgát;eFVIkardl; yield
f 's = f y
⎛ c − d'⎞
f ' = 600⎜
s ⎟ ≤ f pÞúymkvij
y
⎝ c ⎠
⇒ Pb = 0.85 f 'c ab + A's ( f 's −0.85 f 'c ) − As f y (-6)
#> cMNakp©it e RtUv)anvas;BI plastic centroid nig e' RtUv)anvas;BITIRbCMuTMgn;énEdkrgkarTaj.
b
e' = e + d " ¬sMrab;krNIrnH e' = e + d " ¦ Edl d " CacMgayBITIRbCMuTMgn;)øasÞiceTATIRbCMuTMgn;Edkrgkar
b
Taj. e RtUv)anKNnaedayKitm:Um:g;Rtg; plastic centroid
b
a
Pb eb = Cc (d − − d " ) + C s (d − d '− d " ) + Td " (-7)
2
b¤ a
Pb eb = 0.85 f 'c ab(d − − d " ) + A' ( f 's −0.85 f 'c )(d − d '−d " ) + As f y d "
2
(-8)
cMNakp©it balanced eccentricity
Mb
eb = (-9)
Pb
sMrab;muxkat;minEmnctuekaN eyIgeRbIviFIsaRsþdUcKñakñúgkarviPaK edayKitRkLaépÞBitR)akdrbs;ebtug
rgkarsgát;.
emKuNkat;bnßyersIusþg; φ sMrab; balanced condition CamYy f = 400MPa RtUv)ansnμt;yk 0.65
y
b¤ 0.7 . enHedaysar ε = ε = E = 0.002 .
f
s t
y
s
]TahrN_2³ kMNt;kMlaMgsgát; balanced compressive force P rYckMNt; e nig M sMrab;muxkat;
b b b
bgðajkñúgrUb. eKeGay f ' = 27MPa nig f = 400MPa .
c y
dMeNaHRsay³
Members in Compression and Bending 202
8. T.Chhay NPIC
!> sMrab; balanced condition bMErbMrYlrageFobenAkñúgebtugKW 0.003 ehIy bMErbMrYlrageFobenAkñúgEdk
400
εy = = 0.002
200000
@> TItaMgGkS½NWt
600
cb = d = 0.6 × 500 = 300mm
600 + f y
rUbTI6³ balanced condition
ab = β1cb = 0.85 × 300 = 255mm
@> RtYtBinitü Edkrgkarsgát;/ BI strain diagram
ε 's c − d ' 300 − 50
= =
0.003 c 300
⇒ ε 's = 0.0025 > ε y dUcenHEdkrgkarsgát; yield
⎛ c − d" ⎞
b¤RtYtBinitütam f 's = 600⎜
⎝ c ⎠
⎟ ≤ fy
⎛ 300 − 50 ⎞
f 's = 600⎜ ⎟ = 500MPa > 400 MPa
⎝ 300 ⎠
dUcenH f ' = f = 400MPa
s y
$> KNnakMlaMgmanGMeBImkelImuxkat;
Cc = 0.85 f 'c ab = 0.85 × 27 × 255 × 350 = 2048.3kN
Ts = As f y = 28 2 × π × 400 = 985.2kN
C s = A' s ( f y − 0.85 f 'c ) = 28 2 π (400 − 0.85 × 27) = 928.7 kN
%> KNna P nig e e b
Pb = Cc + Cs − T = 2048.3 + 928.7 − 985.2 = 1991.8kN
a
M b = Pb eb = Cc (d − − d " ) + C s (d − d '− d " ) + Td "
2
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 203
9. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
255
M b = 1991.8eb = 2048.3(500 − − 225) + 928.7(500 − 50 − 225) + 985.2 × 225 = 732.8kN .m
2
732.8
⇒ eb = = 0.368m
1991.8
^> sMrab; balanced condition φ = 0.65
φPb = 0.65 × 1991.8 = 1294.67kN
φM b = 0.65 × 732.8 = 476.32kN .m
6> muxkat;ssreRkamGMeBIbnÞúkcakp©it Column sections under eccentric loading
sMrab;krNIBIr enAeBlEdlmuxkat;ssr)ak;edaykarsgát;b¤karTaj smIkarlMnwgmUldæanBIrGac
RtUv)aneRbIsMrab;viPaKssrEdlrgbnÞúkcakp©it.
rUbTI7³ krNITUeTA muxkat;ctuekaNEkg
!> plbUkkMlaMgtamGkS½edk b¤tamGkS½QRtUvesμIsUnü
@> plbUkm:Um:g;eFobnwgGkS½NamYyRtUvesμIsUnü
eyagtamrUb eKGacsresrsmIkarTaMgBIrxagelI dYcxageRkam
!> P − C − C + T = 0
n c s (-10)
Edl C = 0.85 f ' ab
c c
C = A' ( f ' −0.85 f ' )
s s s ¬RbsinebIEdkrgkarsgát; yield enaH f ' = f ¦
c s y
T=A f s s ¬RbsinebIEdkrgkarTaj yield enaH f = f ¦ s y
@> Kitm:Um:g;Rtg;cMnuc A s
a
Pn e'−Cc (d − ) − Cs (d − d ' ) = 0 (-11)
2
Edl e' = e + d " ¬ d " CacMgayBITIRbCMuTMgn;)øasÞiceTATIRbCMuTMgn;rbs;EdkrgkarTaj¦
b¤ e' = e + d −
h
2
sMrab;muxkat;ssrEdlmanEdksIuemRTI
Members in Compression and Bending 204
10. T.Chhay NPIC
1⎡ a ⎤
Pn = ⎢Cc (d − 2 ) + C s (d − d ' )⎥ (-12)
e' ⎣ ⎦
Kitm:Um:g;Rtg; C c
⎡ a ⎤ a a
Pn ⎢e'−(d − )⎥ − T (d − ) − C s ( − d ' ) = 0 (-13)
⎣ 2 ⎦ 2 2
a a
T (d − ) + Cs ( − d ' )
Pn = 2 2 (-14)
a
(e'+ − d )
2
RbsidnebI A = A' ehIy s s f s = f 's = f y enaH
As f y (d − d ' ) As f y (d − d ' )
Pn = = (-15)
a h a
e'+ − d e− +
2 2 2
h a
Pn (e − + )
As = A's = 2 2 (-16)
f y (d − d ' )
7> ersIusþg;rbs;ssrsMrab;kar)ak;edaykarTaj Stregth of columns for tension failure
enAeBlEdlssrrgbnÞúkcakp©itCamYynwgcMNakp©it e FM enaHeKrMBwgfassrnwg)ak;edaykarTaj.
ssr)ak;edayEdkeFIVkardl; yield ebtugEbkenAeBlEdl strain rbs;EdkFMCag ε (ε = f E ) . Kñúg y y
y
s
krNIenH nominal strength P nwgmantMéltUcCag P b¤k¾ cMNakp©it e = M P FMCag balanced
n b
n
n
eccentricity e . edaysarkñúgkrNIxøHeKmankarBi)akkñúgkarTsSn_TayfavaCamuxkat; tension control
b
b¤ compression control enaHeKGacsnμt;fa tension failure GacekIteLIgenAeBl e > d . Karsnμt;enH
GaceFVIeLIgenAeBleRkay.
smIkarlMnwgTUeTA
P − C − C + T = 0 nig
n c s
P e'−C (d − ) − C (d − d ' ) = 0 GacRtUv)aneRbIR)as;sMrab;KNna nominal strength rbs;ssr.
a
n c s
2
!> sMrab;kar)ak;edaykarTaj EdkrgkarTaj yield f = f . snμt;fakugRtaMgEdkrgkarsgát; s y
f' = f .
s y
@> KNna P = C + C − T n c s
Edl C = 0.85 f ' ab c c
Cs = A's ( f y − 0.85 f 'c )
T = As f y
#> KNna P edayKitm:Um:g;Rtg; A
n s
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 205
11. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
a
Pn e' = Cc (d − ) + C s (d − d ' )
2
Edl e' = e + d " b¤ e' = e + d − h enAeBl A = A'
2
s s
$> BICMhan @ nig # eyIg)an
1⎡ a ⎤
Cc + C s − T = ⎢Cc (d − 2 ) + C s (d − d ' )⎥
e' ⎣ ⎦
vaCasmIkarTIdWeRkTI2 EdlmanGBaØti a . CMnYstMél C / C nig T ehIyedaHRsayrk a .
c s
%> eRkayBICMnYs C / C nig T smIkardWeRkTI2 Gacsresry:agsMrYldUcxageRkam
c s
Aa 2 + Ba + C = 0
Edl A = 0.425 f 'c b
B = 0.85 f 'c b(e'− d ) = 2 A(e'− d )
C = A's ( f 's −0.85 f 'c )(e'− d + d ' ) − As f y e'
− B ± B 2 − 4 AC
⇒a=
2A
RbsinebI f ' −0.85 f ' < 0 RtUvykvaesμI 0 .
s c
^> CMnYs a eTAkñúsmIkarCMhan @edIm,ITTYl P . m:Um:g; M kMNt;tam
n n
M n = Pn e
&> RtYtBinitüemIlfaetIEdkrgkarsgát; yield dUckarsnμt;b¤Gt;. RbsinebI ε ' ≥ ε enaH Edkrgkar s y
sgát; yield . pÞúymkvij f ' = E ε ' . Gnuvtþn_CMhan @ dl;% mþgeTot. ε ' = [(c − d ' ) / c]0.003 /
s s s s
ε =
y
E
f
y
nig c = a / β . 1
s
*> RtYtBinitüfamuxkat;Ca tension control . Tension control enAeBlNa e > e b¤ P < P . b n b
(> Net tensile strain ε enAkñúgmuxkat; CaFmμtaFMCag limit strain sMrab;muxkat; compression-
t
controlled section 0.002 . dUcenHtMélénemKuNkat;bnßyersIusþg; φ ERbRbYlcenøaHBI 0.65 ¬b¤ 0.70 ¦
nig 0.90 . smIkar φ = 0.7 + (ε − 0.002)( 200 ) b¤ φ = 0.7 + 0.2⎡ c /1d − 5 ⎤ sMrab;muxkat; EdlmanEdk
t ⎢ ⎥
3 3 ⎣ t ⎦
kgdUcrWus½r spiral section nig φ = 0.65 + (ε t − 0.002)(
250
) b¤ φ = 0.65 + 0.25⎡ c /1d
⎢
5⎤
− ⎥ sMrab;muxkat;
3 ⎣ t 3⎦
epSgeTot RtUv)aneRbIsMrab;KNnark emKuNkat;bnßyersIusþg; φ .
]TahrN_3³ kMNt; nominal compressive strength P sMrab;muxkat;EdleGaydUckñúgrUbxageRkam Rbsin
n
ebI e = 500mm .
dMeNaHRsay³
Members in Compression and Bending 206
12. T.Chhay NPIC
!> eday e = 500mm > d = 485mm snμt;famuxkat;)ak;kñúglkçxNÐ tension failure condion controls
¬EdlRtUveFVIkarRtYtBinitüenAeBleRkayeTot¦. Strain rbs;EdkrgkarTaj ε GacFMCag ε dUcenHeyIg
s y
ykkugRtaMg f . edaysnμt;faEdkrgkarsgát; yield f ' = f EdlRtUvRtYtBinitüenAeBleRkay.
y s y
rUbTI8³ ]TahrN_TI3 kar)ak;edaykarTaj Tension failure
@> BIsmIkarlMnwg P = C + C − T
n c s
Edl C = 0.85 f ' ab = 0.85 × 27 × 350a = 8.03akN
c c
282 π
C s = A's ( f y − 0.85 f 'c ) = 4 (400 − 0.85 × 27) = 928.68kN
4
T = As f y = 28 2 π 400 = 985.2kN
⇒ Pn = 8.03a + 928.68 − 985.2 = 8.03a − 56.52 ¬1¦
#> Kitm:Um:g;Rtg; A s
1⎡ a ⎤
Pn = ⎢Cc (d − 2 ) + C s (d − d ' )⎥
e' ⎣ ⎦
edayTIRbCMuTMgn;)øasÞic plastic centroid sßitenAelITIRbCMuTMgn;énmuxkat; d "= 210mm .
e' = e + d " = 500 + 210 = 710mm
1 ⎡ ⎤
Pn =
710 ⎣
a
⎢8.03a(485 − 2 ) + 928.68(485 − 65)⎥ = −0.0056a + 5.49a + 549.36
⎦
2
¬2¦
$> pÁúMsmIkar ¬1¦ nig ¬2¦ eyIg)an
0.0056a 2 + 2.54a − 605.88 = 0
⇒ a = 172.74mm
%> Pn = 8.03 × 172.74 − 56.52 = 1330.58kN
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 207
13. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
M n = 1330.58 × 0.5 = 665.29kN .m
^> RtYtBinitüfa Edkrgkarsgát; yield b¤Gt;
172.74
c= = 203.22mm
0.85
203.22 − 65
ε 's =
203.22
0.003 = 0.00204 > ε y =
400
200000
= 0.002 dUcenHEdksgát; yield
RtYtBinitü strain enAkñúgEdkTaj
⎛ 485 − 203.22 ⎞
εs = ⎜ ⎟0.003 = 0.00416 > ε y
⎝ 203.22 ⎠
RbsinebIEdksgát;Gt; yield eRbI f ' = ε ' E rYceFVIkarKNnaeLIgvij.
s s s
&> KNna φ ³ eday ε = 0.00416 muxkat;sßitenAkñúgtMbn; transition region
t
⎛ 250 ⎞
φ = 0.65 + (ε t − 0.002 )⎜ ⎟ = 0.83
⎝ 3 ⎠
φPn = 0.83 × 1330.58 = 1104.38kN
φM n = 0.83 × 665.29 = 552.19kN .m
8> ersIusþg;rbs;ssrsMrab;kar)ak;edaykarsgát; Stregth of columns for compression failure
RbsinebIbnÞúkGnuvtþn_sgát; P FMCagbnÞúk balanced force P b¤cMNakp©it e = M tUcCag e enaH
n b
P
n
b
n
ssrnwgrMBwgfaRtUv)ak;edaykarsgát;. kñúgkrNI compression controls ehIy strain rbs;ebtugnwgmantMél
0.003 Edl strain rbs;EdkmantMéltUcCag ε . PaKeRcInrbs;muxkat;ssrnwgrgkarsgát;. GkS½Nwtxit
y
eTArkEdkrgkarTaj edaybegáInmuxkat;sgát; dUcenHcMgayeTAGkS½NWt c > c . b
rUbTI9³ düaRkam strain enAeBl compression controls
Members in Compression and Bending 208
14. T.Chhay NPIC
edaysareKBi)akkñúgkarTsSn_TaynUvmuxkat;ssrfa tension failure b¤ compression failure
eK)ansnμt;fa enAeBl e < 2d enaHssr)ak;eday compression failure EdlRtUvepÞógpÞat;enAeBleRkay.
3
edIm,IKNna nominal load strength P eKeRbIeKalkarN_sþaTic. karviPaKmuxkat;ssrsMrab;kar)ak;eday
n
karsgát; compression failure eKGaceRbIsmIkar P − C − C + T = 0 nigsmIkar
n c s
P e'−C (d − ) − C (d − d ' ) = 0 nigdMeNaHRsaymYykñúgcMeNamdMeNaHRsayxageRkam.
a
n c s
2
8>1> dMeNaHRsay Trial solution
dMeNaHRsayenHRtUv)ansegçbdUcCMhanxageRkam³
!> KNnacMgayeTAGkS½NwtsMrab;muxkat; balanced section c b
600d t
cb = (-3)
600 + f y
@> kMNt; P edayeRbIlkçxNÐlMnwg
n
Pn = Cc + Cs − T (-10)
#> KNna P edayKitm:Um:g;Rtg;EdkrgkarTaj A
n s
a
Pn e' = Cc (d − ) + Cs (d − d ' ) (-11)
2
Edl e' = e + d "kñúgkrNITUeTA
b¤ e' = e + d − enAeBl A = A'
h
2
s s
Cc = 0.85 f 'c ab
Cs = A's ( f 's −0.85 f 'c )
T = As f s
$> edaysnμt;tMél c > c KNna a = β c . snμt;
b 1 f 's = f y
%> KNna f s
⎛d −c⎞
f s = ε s Es = 600⎜ t ⎟ ≤ fy
⎝ c ⎠
^> CMnYstMélEdlrkeXIjeTAkñúgsmIkarCMhan @ nigCMhan # edIm,Irk P nig P . RbsinebI n1 n2
P ≈ P eRCIsyktMéltUcCageK b¤mFümPaKén P nig P . EtebI P mantMélxusKñaq¶ayBI P eK
n1 n2 n1 n2 n1 n2
RtUvsnμt; c b¤ a fμI ehIyeFVIkarKNnaeLIgvijcab;epþImBICMhan $ rhUtdl; P ≈ P . ¬eKGacTTYlyk)an n1 n2
ebI P nig P xusKña 1% ¦.
n1 n2
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 209
15. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
&> epÞógpÞat;fa Edkrgkarsgát; yield edayKNna ε ' = 0.003[(c − d ') / c] ehIyeRbobeFobCamYy
s
εy =
fy
E
. enAeBlEdl ε ' ≥ ε Edkrgkarsgát; yield RbsinebImindUcenaHeT f ' = ε ' E b¤
s
s y s s s
⎛ c − d' ⎞
f s = 600⎜ ⎟ ≤ fy
⎝ c ⎠
*> epÞógpÞat;fa e < e b¤ P > P sMrab; compression failure.
b n b
(> sMrab;muxkat; compression controlled section CaTUeTA net tensile strain ε enAkñúgmuxkat;tUc
t
Cag 0.002 . dUcenH emKuNkat;bnßyersIusþg; φ = 0.65 ¬b¤ 0.70 sMrab;ssrEdleRbIEdkkgCab;¦.
]TahrN_4³ kMNt; nominal compressive strength P sMrab;muxkat;EdleGay RbsinebI e = 254mm .
n
rUbTI10³ ]TahrN_TI4 Compression controls
dMeNaHRsay³
!> edaysar e = 254mm < 23d = 333.33mm . snμt; compression failure. karsnμt;enHRtUvepÞógpÞat;enA
eBleRkay. KNnacMgayeTAGkS½NwtsMrab; balanced section c : b
600d t 600 × 500
cb = = = 300mm
600 + f y 600 + 400
@> BIsmIkarlMnwg
Pn = Cc + Cs − T (-10)
Edl Cc = 0.85 f 'c ab = 0.85 × 27 × a × 350 = 8.03akN
C s = A' s ( f y − 0.85 f 'c ) = 282 π (400 − 0.85 × 27) = 928.68kN
edaysnμt;Edkrgkarsgát; yield ¬karsnμt;enHRtUvepÞógpÞat;enAeBleRkay¦
T = A f = 28 πf = 2.46 f kN ¬ f < f ¦
s s
2
s s s y
Pn = 8.03a + 928.68 − 2.46 f s
#> Kitm:Um:g;Rtg; A s
Members in Compression and Bending 210
16. T.Chhay NPIC
1⎡ a ⎤
Pn = ⎢Cc (d − 2 ) + Cs (d − d ' )⎥ (-11)
e' ⎣ ⎦
TIRbCMuTMgn;)øasÞicsßitenAelITIRbCMuTMgn;rbs;muxkat; d "= 225mm
e' = e + d " = 254 + 225 = 479mm
1 ⎡ ⎛ a⎞ ⎤
Pn = ⎢8.03a⎜ 500 − 2 ⎟ + 928.8(500 − 50)⎥ = 8.38a − 0.0084a + 872.57
2
479 ⎣ ⎝ ⎠ ⎦
$> edaysnμt; c = 342mm EdlmantMélFMCag c b = 300mm
a = 0.85 × 342 = 290.7 mm
CMnYstMél a eTAkñúgsmIkarkñúgCMhanTIBIrxagelIeyIg)an
Pn1 = 8.38 × 290.7 − 0.0084 × 290.7 2 + 872.57 = 2598.78kN
%> KNna f BIdüaRkam strain enAeBlEdl c = 340mm
s
500 − 342
fs = 600 = 277.19 MPa
342
f 277.19
ε s = εt = s = = 0.00139
Es 200000
^> edayCMnYs a = 290.7mm nig f s = 277.19MPa eTAkñúgsmIkarCMhanTImYyedIm,IKNna P n2
Pn 2 = 8.03 × 290.7 + 928.68 − 2.46 × 277.19 = 2581.11kN
eday P nig P mantMélxusKñamindl; 1% dUcenHeyIgyk P = 2581.11kN
n1 n2 n
M n = Pn e = 2581.11 × 0.254 = 655.6kN .m
&> epÞógpÞat;fa Edkrgkarsgát; yield BIdüaRkam strain
342 − 50
ε 's = 0.003 = 0.00256 > ε y = 0.002
342
dUcenH Edkrgkarsgát; yield dUckarsnμt;.
*> P = 2581.11kN > P = 1991.8kN ehIy e = 254mm < e = 368mm bgðajfavaCamuxkat;
n b b
compression control dUckarsnμt;. cMNaM³ eKGaceFVIkarsakl,gKNnaedIm,IeGay P nig P mantMél n1 n2
kan;EtesμIkña.
(> KNna φ
d t = d = 500mm c = 342mm
500 − 342
εt ¬enAnIv:UedkrgkarTaj¦ = 0.003
342
= 0.00139 < 0.002 enaH φ = 0.65
φPn = 0.65 × 2581.11 = 1677.72kN
φM n = 0.65 × 655.6 = 426.14kN .m
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 211
17. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
8>2> dMeNaHRsayviPaKcMnYn Numerical Analysis Solution
enAeBl compression control karviPaKssrGaceFVIeTA)anedaykat;bnßykarKNnamkRtwmsmIkar
dWeRkTI3 EdlmanTMrg; Aa + Ba + Ca + D = 0 rYcedaHRsayrktMél a edayviFIcMnYn numerical method
3 2
b¤ a GacTTYl)anBIm:asIunKitelx. BIsmIkarlMnwg
Pn = Cc + C s − T = (0.85 f 'c ab) + A's ( f y − 0.85 f 'c ) − As f s (-10)
Kitm:Um:g;Rtg;EdkTaj A s
1⎡ a ⎤ 1⎡ a ⎤
Pn = ⎢Cc (d − 2 ) + Cs (d − d ' )⎥ = e' ⎢0.85 f 'c ab(d − 2 ) + A's ( f y − 0.85 f 'c )(d − d ' )⎥ (-11)
e' ⎣ ⎦ ⎣ ⎦
BIdüaRkam strain
a
(d − )
⎛d −c⎞ β1
εs = ⎜ t ⎟0.003 = 0.003
⎝ c ⎠ a
β1
kugRtaMgenAkñúgEdkTajKW
600
f s = ε s E s = 200000ε s = ( β1 d − a )
a
edayCMnYstMél f eTAkñúgsmIkar (-10) nigedaHRsaysmIkar (-10) nig (-11) eRkayBIsMrYlrYceyIg
s
TTYl)an
⎛ 0.85 f 'c b ⎞ 3
⎜ ⎟a + [0.85 f 'c b(e'− d )]a + [ A's ( f y − 0.85 f 'c )(e'−d + d ' ) + 600 As e' ]a − 600 As e' β1d = 0
2
⎝ c ⎠
enHCasmIkardWeRkTI3 EdlmanTMrg; Aa 3
+ Ba 2 + Ca + D = 0
Edl A = 0.852f ' b c
B = 0.85 f 'c b(e'− d )
C = A's ( f y − 0.85 f 'c )(e'− d + d ' ) + 600 As e'
D = −600 As e' β1d
enAeBlEdleKKNna)antMél A / B / C nig D enaH a GacRtUv)anKNnaedayviFIsakl,g b¤
TTYl)anedaypÞal;Bim:asIunKitelx. dMeNaHRsaysmIkardWeRkTI3 GacTTYl)anedayeRbIviFI Newton-
Raphson . viFIenHmanRbsiT§PaBkñúgkaredaHRsayrkb¤srbs; f ( x ) = 0 . vaTak;Tgnwgbec©keTssamBaØ
ehIyeKqab;TTYl)ancMelIyedayeFVItamCMhanxageRkam³
!> eGay f (a) = Aa + Ba + Ca + D nigKNna A / B / C nig D
3 2
@> KNnaedrIevTImYyén f (a) ³ f ' (a) = 3 Aa + 2Ba + C 2
#> edaysnμt;tMéldMbUg a KNnatMélbnÞab;
o
Members in Compression and Bending 212
18. T.Chhay NPIC
f ( ao )
a1 = ao −
f ' ( ao )
$> edayeRbItMél a KNna a dUcsmIkarxagelI
1 2
f (a1 )
a2 = a1 −
f ' (a1 )
%> Gnuvtþn_nUvviFIenHrhUtdl;)antMélsuRkitmYy a ≅ a . kñúgkrNIviPaKssrenAeBl compression
n n −1
control tMél a EtgEtFMCag a . dUcenH eKcab;epþImCamYy a = a ehIyGnuvtþsmIkarxagelIBIrdgedIm,I
b o b
TTYl)ancMelIy.
]TahrN_5³ eFVI]TahrN_TI4eLIgvijedayeRbIviFI numerical analysis
dMeNaHRsay³
!> KNna A / B / C nig D nigkMNt; f (a)
0.85 f 'c b 0.85 × 27 × 350
A= = = 4016.25
2 2
B = 0.85 f 'c b(e'− d ) = 0.85 × 27 × 350(479 − 500) = −168682.5
C = A' s ( f y − 0.85 f 'c )(e'−d + d ' ) + 600 As e'
C = 282 π (400 − 0.85 × 27)(479 − 500 + 50) + 600 × 282 π × 479
C = 734800328.08
D = −600 As e' β1d = −600 × 282 π × 479 × 0.85 × 500 = −300844190383.4
f (a) = 4016.25a 3 − 168682.5a 2 + 734800328.08a − 300844190383.4
@> KNnaedrIevTI1
f ' (a) = 12048.75a 2 − 337365a + 734800328.08
#> eGay a o = ab = 255mm sMrab;muxkat; balanced section c b = 300mm nig a
b = 255mm
f (255)
a1 = 255 − = 295.39
f ' (255)
$> nigKNna a ³ 2
f (295.39)
a2 = 295.39 − = 292.4mm
f ' (295.39)
tMélrbs; a mantMélRsedognwg a enAkñúg]TahrN_TI3. CMnYstMél a eTAkñúgsmIkar (-10) b¤ (-11)
eKTTYl)an P = 2594.66kN
n
8>3> dMeNaHRsayRbEhl Approximate Solution
smIkar approximate RtUv)anesñIreLIgeday Whitney edIm,IedaHRsayrk nominal compressive
strength sMrab;ssrxøI enAeBl compression control.
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 213
19. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
bhf 'c A's f y
Pn = + (-17)
3he e
+ 1.18 + 0.5
d 2
(d − d ' )
smIkarxagelIenHGaceRbIeTA)ansMrab;EtssrEdlmansésEdksIuemRTItMerobEtmYyRsTab; ehIyRsbeTA
nwgGkS½énkarBt;.
smIkar approximate TI2 RtUv)anesñIeLIgeday Hsu
1.5
Pn − Pb ⎛ M n ⎞
+⎜ ⎟ = 1.0 (-18)
Po − Pb ⎜ M o ⎟
⎝ ⎠
Edl =
Pn nominal axial strength énmuxkat;ssr
=
Pb , M b nominal load nig nominal moment énmuxkat; balanced section
M n = nominal bending moment = Pn ⋅ e
enAeBl e=0 P = 0.85 f ' ( A − A ) + A f
Po = nominal axial load o c g st st y
A = gross area énmuxkat; = bh
g
A = muxkat;EdkbeNþayminEmnkugRtaMgsrub
st
]TahrN_6³ kMNt; nominal compressive strength P sMrab;muxkat;EdleGaydUckñúg]TahrN_TI4
n
edaysmIkar ¬*>#>!¦ nig ¬*>#>@¦edayeRbInUvcMNakp©itdUcKña e = 254mm rYceRbobeFobcMelIy.
dMeNaHRsay³
!> dMeNaHRsaytamsmIkar Whitney
k> lkçN³énmuxkat; b = 350mm / h = 550mm / d = 500mm / d ' = 50mm / A' = 2463mm 2
nig (d − d ' ) = 450mm
x> GnuvtþsmIkar Whitney
350 × 550 × 27 2463 × 400
Pn = + = 2745.15kN
3 × 550 × 254 254
+ 1.18 + 0.5
500 2 450
φPn = 0.65 × 2745.15 = 1784.35kN
K> P EdlKNnaedaysmIkar Whitney CatMélEdlminsnSMsMécenAkñúg]TahrN_enH ehIytMél
n
P = 2745.15kN KWFMCagtMélsuRkit P = 2581.11kN EdlKNnaedaysmIkarsþaTickñúg]TahrN_TI4.
n n
@> dMeNaHRsaytamsmIkar Hsu
k> sMrab; balanced condition P = 1991.8kN nig M = 732.8kN ¬]TahrN_TI2¦
b b
x> P = 0.85 f ' ( A − A ) + A f = 0.85 × 27 × (550 × 350 − 2 × 2463) + 2 × 2463 × 400
o c g st st y
Po = 6275.22kN
Members in Compression and Bending 214
20. T.Chhay NPIC
Pn − 1991 .8
1 .5
⎛ 0254 Pn ⎞
K> +⎜
6275 .22 − 1991 .8 ⎝ 732 .8 ⎠
⎟ =1
edayKuNnwg 1000 ehIyedaHRsayrk P n
0.23346 Pn + 0.00654 Pn1.5 = 1465
Edaykarsakl,gtMél P = 2587.65kN EdlmantMélRbEhl 2581.11kN EdlKNnaedaysþa
n
Tic.
9> ]TahrN_sMrab;düaRkamGnþrkmμ Interaction Diagram Example
enAkñúg]TahrN_TI2 bnÞúk balanced load P , M nig e RtUv)anKNnasMrab;muxkat;dUckñúgrUbTI6
b b b
¬ e = 368mm ¦. dUcKña enAkñúg]TahrN_TI3 nigTI4 load capacity sMrab;muxkat;dUcKñaRtUv)anKNnasMrab;
b
krNIenAeBl e = 500mm ¬tension failure¦ nigenAeBl e = 254mm ¬compression failure¦. tMélTaMg
enHnwgRtUvbgðajenAkñúgtaragTI1.
edIm,IKUrdüaRkamGnþrkmμbnÞúk-m:Um:g; tMélepSg²én φP nig φM RtUv)anKNnasMrab;tMél e epSg²
n n
Edl e ERbRbYlBI e = 0 eTA e = Gtibrma sMrab;krNIm:Um:g;Bt;suT§ pure moment enAeBl P = 0 . düa n
Rkamnþr kmμbnÞúk-m:Um:g;RtUv)anbgðajkñúgrUbTI11. bnÞÞúk φP = 4078.90kN CabnÞúkcMGkS½tamRTwsþI enAeBl
no
e = 0 . Et ACI Code GnuBaØatibnÞúkGb,brmaRtwmEt 0.8φP = 3263.12kN EdlRtUvKñanwg cMNakp©wtGb,
no
brma. cMNaMfa sMrab;kar)ak;edaykarsgát; compression failure e < e nig P > P ehIysMrab;kar)ak;
b n b
taragTI1³taragKNnasegçb rUbTI11³düaRkamGnþrkmμbnÞúk-m:Um:g;
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 215
21. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
edaykarTaj tension failure e > e nig P < P . krNI e = Gtibrma ssrrgnUvm:Um:g;Bt;suT§dUckrNIFñwm.
b n b
10> ssrmuxkat;ctuekaNCamYyEdkxag Rectangular columns with side bars
enAkñúgmuxkat;ssrxøH EdkRtUv)aneKdak;BRgaytamRCugTaMgGs;. EdkxagRtUv)andak;tamkMBs;énmux
kat;edaybEnßmeTAelIEdkTaj nigEdksgát; A nig A' ehIyRtUv)aneKkMNt;eGayeQμaH A ¬rUbTI12¦.
s s ss
kñúgkrNIenH viFIsaRsþkñúgkarKNnaEdl)anBnül;rYcmkehIyGacRtUv)anGnuvtþ edayKitBicarNabEnßmkar
pøas;bþÚr strain tamkMBs;énmuxkat; nigTMnak;TMngkMlaMgenAkñúgEdkxagnImYy²eTAkñúgtMbn;sgát; b¤tMbn;Taj
énmuxkat;. kMlaMgTaMgenHRtUv)anbUkbEnßmeTAelI C C nig T edIm,IkMNt; P smIkarmanragdUcxag
c s n
eRkam³
Pn = Cc + ∑ C s − ∑ T (-10a)
]TahrN_TI7 Bnül;BIkarKNnaenH. cMNaMfa RbsinebIEdkxagsßitenAEk,rGkS½NWt ¬rUb12 b¦ strain
nigkMlaMg enAkñúgEdkmantMéltUcNas;EdleKGacecal)an. cMENkEdkEdlsßitenAEk,r A nig A' mantM s s
élFMKYrsm nigCYybegáInlT§PaBRTRTg;bnÞúkénmuxkat;.
rUbTI12³ EdkxagenAkñúgmuxkat;ctuekaNEkg
]TahrN_7³ kMNt;bnÞúk m:Um:g; M nigcMNakp©it e sMrab;muxkat;bgðajkñúgrUbTI13. edayeRbI
Pb b b
f ' = 28MPa nig f = 400 MPa .
c y
dMeNaHRsay³ muxkat; balanced section RtUv)anKNnadUcKñanwg]TahrN_TI2Edr. eKeGay
π 32
b = h = 550mm / d = 485mm nig d ' = 65mm . A = A' = 5
2
= 4021.24mm (5DB32 ) /
s s
2
4
nigEdkxag 6DB32 ¬ 3DB32 sMrab;mçag¦.
1> KNnacMgayeTAGkS½NWt
Members in Compression and Bending 216
22. T.Chhay NPIC
⎛ 600 ⎞
cb = ⎜ ⎟d = ⎛ 600 ⎞485 = 291mm
⎜ 600 + f ⎟ t ⎜ 600 + 400 ⎟
⎝ ⎠
⎝ y ⎠
ab = 0.85cb = 0.85 × 291 = 247.35mm
2> KNnakMlaMgenAkñúgebtug nigEdk tamry³rUb 13 a . enAtMbn;sgát;
Cc = 0.85 f 'c ab = 0.85 × 28 × 247.35 × 550 = 3237.81kN
C s = A' s ( f ' s −0.85 f 'c )
enAnIv:U − 65mm
⎛ c − d'⎞ ⎛ 291 − 65 ⎞
f ' s1 = 600⎜ ⎟ = 600⎜ ⎟ = 466 MPa > 400MPa
⎝ c ⎠ ⎝ 291 ⎠
dUcenH f 's = 400MPa
Cs1 = 4021.24(400 − 0.85 × 28) = 1512.79kN
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 217
23. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
enAnIv:U − 170mm
⎛ c − d'⎞ ⎛ 291 − 170 ⎞
f 's 2 = 600⎜ ⎟ = 600⎜ ⎟ = 249.48MPa
⎝ c ⎠ ⎝ 291 ⎠
π 32 2
Cs 2 = 2 × (249.48 − 0.85 × 28) = 363kN
4
enAnIv:U − 275mm
⎛ c − d' ⎞ ⎛ 291 − 275 ⎞
f 's 3 = 600⎜ ⎟ = 600⎜ ⎟ = 32.99MPa
⎝ c ⎠ ⎝ 291 ⎠
π 32 2
Cs 3 = 2 × (32.99 − 0.85 × 28) = 14.78kN
4
enAkñúgtMbn;Taj ¬nIv:U − 380mm ¦
Members in Compression and Bending 218
24. T.Chhay NPIC
380 − 291
ε s4 = × 0.003 = 917.53 ⋅ 10 −6
291
f s 4 = 200000 × 917.53 ⋅ 10 −6 = 183.5MPa
π 32 2
T1 = 2 × (183.5) = 295.16kN
4
T2 = 4021.24 × 400 = 1608.5kN
3> KNna Pb = Cc + ∑ C s − ∑ T
Pb = 3237.81 + (1512.79 + 363 + 14.78) − (295.16 + 1608.5) = 3224.72kN
4> KNnam:Um:g;Rtg;TIRbCMuTMgn;)øasÞic
M b = 3237.81× 151.325 + 1512.79 × 210 + 363 × 105 + 295.16 × 105 + 1608.5 × 210
M b = 1214.54kN .m
M 1214.54
eb = b = = 0.377m
Pb 3224.72
5> KNna φ sMrab; balanced section ε t /
= ε y = 0.002 φ = 0.65
φPn = 0.65 × 3224.72 = 2096.07kN
niig φM b = 0.65 ×1214.54 = 749.45kN .m
]TahrN_8³ edaHRsay]TahrN_TI7 eLIgvijenAeBlEdl e = 152mm .
dMeNaHRsay³
1> edaysar enaHvaCalkçxNÐ)ak;edaykarsgát; compression failure
e = 152mm < eb = 326mm
condition. snμt; c = 399.5mm ¬edaykarsakl,g¦ nig a = 399.5 × 0.85 a = 339.58mm
¬ rUbTI13 b ¦.
2> KNnakMlaMgenAkñúgebtug nigEdk
Cc = 0.85 × 28 × 339.58 × 550 = 4445.1kN
dUcKñanwgkrNI balanced f s1 = 400MPa nig C s1 = 1512.79kN
f s 2 = 344.68MPa nig Cs 2 = 516.13kN
f s3 = 186.98MPa nig Cs 3 = 262.48kN
f s 4 = 29.29MPa nig Cs 4 = 8.83kN
f s 5 = 128.41MPa nig T = 516.37 kN
3> KNna Pn = Cc + ∑ C s − ∑ T = 6228.96kN
M n = Pn ⋅ e = 6228.96 × 152 = 946.8kN .m
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 219
25. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
4> KNna Pn edayKitm:Um:g;Rtg; As
1⎡ a ⎤
Pn = ⎢Cc (d − 2 ) + Cs1 (d − d ' ) + C s 2 (d − d '− s ) + C s 3 (d − d '−2s ) + C s 4 (d − d '−3s )⎥
e' ⎣ ⎦
h 550
e' = e + d − = 152 + 485 − = 362mm
2 2
s = 105mm KMlatrvagEdkxag ¬efr sMrab;]TahrN_enH¦
⎡ ⎛ 339.58 ⎞ ⎤
1 ⎢4445.1⎜ 485 − ⎟ + 1512.79(485 − 65) + 516.13(485 − 65 − 105)⎥
Pn = ⎝ 2 ⎠
362 ⎢ ⎥
⎢+ 262.48(485 − 65 − 2 × 105) + 8.83(485 − 65 − 3 × 105)
⎣ ⎥
⎦
Pn = 6230kN
5> KNna φ
d t = d = 485mm c = 399.5mm
εt ¬enAnIv:UEdkTaj¦ = 0.03(dt − c ) / c = 0.03(485 − 399.5) / 399.5 = 0.00064
eday ε t < 0.002 enaH φ = 0.65
φPn = 0.65 × 6228.96 = 4048.8kN
φM n = 0.65 × 946.8 = 615.42kN .m
cMNaM³ RbsinebIEdkxagminRtUv)anKit enaH
Pb = 3142.1kN
Pn ¬enA e = 152mm ¦ = 4592.23 + 1512.79 − 422.48 = 5682.54kN
RbsinebIeKKitEdkxagenaH Pb ekIneLIgRbEhl 2.6% nig Pn ekIneLIgRbEhl 9.6% .
11> lT§PaBRTbnÞúkrbs;ssrmuxkat;mUl Load Capacity of Circular Columns
11>1 lkçxNÐ Balanced Condition
tMélénbnÞúk balanced load Pn nig m:Um:g; balanced moment M n sMrab;muxkat;mUlGacRtUv)ankM
Nt;edayeRbIsmIkarlMnwgdUckrNImuxkat;cuekaNpgEdr. sésrEdkenAkñúgmuxkat;rgVg;EdlRtUv)antMerobeTA
tamcMNayBIGkS½TIRbCMuTMgn;)øasÞicERbRbYl KWGaRs½yeTAnwgcMnYnEdkenAkñúgmuxkat;. bBaðacMbgKWrkkMBs;bøúk
sgát; a nigkugRtaMgenAkñúgsésrEdk. ]TahrN_xageRkamBnül;BIkarviPaKmuxkat;eRkamlkçxNÐ balanced
condition. nitiviFIdUcKñaGacRtUv)aneRbIedIm,IviPaKmuxkat;sMrab; tension control b¤ compression control.
]TahrN_9³ kMNt;bnÞúk balanced load Pn nig m:Um:g; balanced moment M n sMrab;ssrmuxkat;rgVg;Edkkg
vNÐGgát;p©it 400mm CamYynig 8DB28 dUcbgðajkñúgrUbTI14. eKeGay f 'c = 28MPa nig Fy = 400MPa .
dMeNaHRsay³
Members in Compression and Bending 220
26. T.Chhay NPIC
1> edaysarEtEdksIuemRTInwgGkS½ A − A Edlkat;tamTIRbCMuTMgn;rgVg; enaHTIRbCMuTMgn;)øasÞicsßitenAelI
GkS½enaH.
2> kMNt;TItaMgTIRbCMuTMgn;GkS½NWt
fy
d t = 329.34mm εy =
Es
cb 0.003 600
= =
d t 0.003 + ε y 600 + f y
600
cb = 329.34 = 197.6mm
600 + 400
ab = 167.96mm
3> kMNt;lkçN³rbs;cMerokrgVg; circular segment ¬rgVg;TI15¦
RkLaépÞcMerokrgVg; = r 2 (α − sin α cos α ) (-19)
TItaMgTIRbCMuTMgn; x ¬BITIRbCMuTMgn; 0¦
2 (r sin 3 α )
x= (-20)
3 α − sin α cos α
Z =r−x (-21)
⎛ a⎞
r cos α = (r − a ) b¤ cos α = ⎜1 − ⎟
⎝ r⎠
(-22)
⎛ 167.96 ⎞
cos α = ⎜1 − ⎟ = 0.16
⎝ 200 ⎠
/
⇒ α = 80.79o sin α = 0.987 nig α = 1.41rad
RkLaépÞcMerokrgVg; = 2002 (1.41 − 0.16 × 0.987)
= 50083.2mm 2
⎛2⎞ 200 × 0.987 3
x =⎜ ⎟ = 102.39mm
⎝ 3 ⎠ (1.41 − 0.987 × 0.16)
Z = 200 − 102.39 = 97.61mm
4> kMNt;kMlaMgsgát; Cc
Cc = 0.85 f 'c × RkLaépÞcMerokrgVg;
= 0.85 × 28 × 50083.2 = 1192kN
vaeFVIGMeBIenA 102.39mm BITIRbCMuTMgn;ssr
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 221
27. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
Members in Compression and Bending 222
28. T.Chhay NPIC
5> KNna strain, stress nig kMlaMgenAkñúgEdkrgkarTaj nigEdkrgkarsgát;.
kMNt;bMErbMrYlrageFob strain BIdüaRkambMErbMrYlrageFob.
sMrab; T1
ε = ε y = 0.002 f s = f y = 400MPa
π 282
T1 = 2 × 400 = 492.6kN
4
sMrab; T2
55.98
ε s3 = ε y = 8.5 ⋅ 10 − 4
131.74
f s 3 = 8.5 ⋅ 10 −4 × 200000 = 170MPa
π 282
T2 = 2 × 170 = 209.36kN
4
sMrab; Cs1
126.94
ε s1 = × 0.003 = 1.93 ⋅10−3
197.6
f s1 = 1.93 ⋅ 10 −3 × 200000 = 386MPa < 400 MPa
282 π
C s1 = 2 (386 − 0.85 × 28) = 446.05kN
4
sMrab; Cs2
51.18
ε s2 = × 0.003 = 7.77 ⋅10 − 4
197.6
f s 2 = 7.77 ⋅10−4 × 200000 = 155.4 MPa
282 π
cs 2 = 2 (155.4 − 0.85 × 28) = 162.07kN
4
6> kugRtaMgenAkñúgEdkrgkarsgát;RtUv)ankat;bnßy edIm,IKitenAkñúgebtugEdlCMnYsedayEdk.
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 223
29. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
kMlaMg balanced KW Pb = Cc + ∑ Cs − ∑ T
Pb = 1192 + (446.05 + 162.07) − (492.6 + 209.36) = 1098.16kN
sMrab;muxkat; balanced ε t = 0.002 nig φ = 0.65
φPb = 713.8kN
7> Kitm:Um:g;Rtg;TIRbCMuTMgn;)øasÞic ¬GkS½ A − A kat;tamGkS½TIRbCMuTMgn;¦ sMrab;kMlaMgTaMgGs;
M b = Pb eb = [Cc × 102.39 + Cs1 × 129.34 + Cs 2 × 53.58 + T1 × 129.34 + T2 × 53.58)
M b = 263.36kN .m
φM b = 171.18kN .m
263.36
eb = = 239.8mm
1098.16
11>2 ersIusþg;rbs;muxkat;mUlsMrab;kar)ak;edaykarsgát; Strength of circular column
for compression failure
muxkat;ssreRkamkMlaMgcMNakp©it GacRtUv)anviPaKtamCMhandUcmuxkat; balanced Edr. va
TTYl)anedaykarsnμt; C > Cb b¤ a > ab nigKNnakMlaMgenAkñúgebtug nig EdkenATItaMgepSgKñaedIm,I
kMNt; Pn1 = Cc + ∑ Cs − ∑ T . dUcKña M n GacRtUv)anKNnaedayKitm:Um:g;Rtg;TIRbCMuTMgn;)øasÞic
¬TIRbCuMTMgn;rbs; muxkat;¦ ehIykMNt; Pn2 = Men . RbsinebItMél Pn1 nig Pn2 minRbhak;RbEhl
KñaeTenaH snμt; C b¤ a fμI ehIyeFVIkarKNnaeLIgvij ¬emIlcMnucTI8¦. tMélxusKñarvag Pn1 nig Pn2 sßit
enArgVg; 1% . muxkat; Camuxkat; compression controls enAeBl e < eb b¤ Pn > Pb .
sMrab;]TahrN_ RbsinebIvaTamTarkMNt;ersIusþg;rbs;muxkat;ssrenAkñúg]TahrN_TI9 enAeBlEdl
e = 150mm Pn GacRtUv)anKNnaedayCMhandUcnwg]TahrN_TI 9.
1> eday e = 150mm tUcCag eb = 239.8mm lkçxNÐ)ak;edaykarsgát; compression failure
condion ekIteLIg.
2> snμt; c = 225mm ¬edaykarsμan¦ > Cb = 197.6 nig a = 191.25mm
3> KNna x = 89.63mm / Z = 110.37mm RkLaépÞcMerokrgVg; = 59332.97mm2
4> -5> KNnakMlaMg³ Cc = 1412.125kN / Cs1 = 463.29kN Cs 2 = 228.73kN / T1 = 342.66kN /
T2 = 93.84kN
6> KNna Pn1 = Cc + ∑ Cs − ∑ T = 1667.64kN
7> Kitm:Um:g;Rtg;GkS½ssr ¬TIRbCMuTMgn;)øasÞic¦³
M n = 248.1kN .m
Members in Compression and Bending 224
30. T.Chhay NPIC
Pn 2 =
Mn
e
= 1653.97 mm EdlmantMélRbEhl Pn1 ¬tMélxusKñaRbEhl 1% ¦. dUcenH
Pn = 1653.97 kN
cMNaM³ RbsinebIEdkkgrbs;ssrCaEdkkgvNÐdUcrWus½renaH φ = 0.70 .
smIkartMélRbEhl approximate equation sMrab;karKNna Pn sMrab;muxkat;mUl enAeBl
compression controls RtUv)anesñIeLIgedayelak Whitney
Ag f ' c Ast f y
Pn = + (-23)
⎡ 9.6he ⎤ ⎛ 3e ⎞
⎢ + 1.18⎥ ⎜
⎜ D + 1⎟
⎟
⎢ (0.8h + 0.67 Ds ) ⎝ s ⎠
2
⎣ ⎥
⎦
Edl RkLaépÞmuxkat; gross area
Ag =
h = Ggát;p©itmuxkat;
Ds = Ggát;p©itmuxkat;ssrEdlvas;BITIRbCMuTMgn;EdkmçageTATIRbCMuTMgn;EdkmçageTot
Ast = RkLaépÞmuxkat;EdkbBaÄr
e = cMNakp©itEdlvas;BITIRbCMuTMgn;)øasÞic
]TahrN_10³ KNnaersIusþg;kMlaMgsgát; nominal Pn sMrab;muxkat;sMrab;]TahrN_TI9 edayeRbIsmI
kar Whitney RbsinebIcMNakp©it e = 150mm
dMeNaHRsay³
1> e = 150mm tUcCag eb = 239.8mm . tamkarKNnadUceBlmun bgðajfamuxkat;ssrCamuxkat;
compression controls.
2> edayeRbIsmIkar Whitney
h = 400mm
π π
Ag =
4
h2 =
4
400 2 = 125663.7mm 2 /
D s = 400 − 120 = 280mm
π × 28 2
As = 8 × = 4926mm 2
4
125663.7 × 28 4926 × 400
Pn = + = 1785.94kN
⎡ 9.6 × 400 × 150 ⎤ ⎛ 3 × 150 ⎞
⎢ + 1.18⎥ ⎜ + 1⎟
⎢ (0.8 × 400 + 0.67 × 280 ) ⎝ 280 ⎠
2
⎣ ⎥
⎦
3> M n = Pn e = 1785.94 × 0.15 = 267.89kN .m
tMél Pn enAeBlenHFMCagtMél Pn = 1653.97kN EdlKNnaenAeBlmunedaysþaTic.
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 225
31. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
11>3 ersIusþg;rbs;muxkat;mUlsMrab;kar)ak;edaykarTaj Strength of circular column
for tension failure
kar)ak;edaykarTajsMrab;ssrmUlenAeBlbnÞúkRtUv)anGnuvtþn_enARtg;cMNakp©it e > eb b¤
Pn < Pb . enAkñúgkrNIenH muxkat;ssrGacRtUv)anviPaKtamCMhandUckarviPaKmuxkat; balanced nigdUc
kñúg]TahrN_TI8. karviPaKRtUv)aneFVIeLIgedaysnμt; C < Cb b¤ a < ab rYcehIyGnuvtþn_tamCMhanBnül;
kñúgEpñkTI11>1. cMNaMfa edaysarEtsésrEdkRtUv)antMerobedaymancenøaHefrtambrimaRtmuxkat;rgVg;
enaHEdkTaj As Edlpþl;eGayGacmantMéltUc ehIylT§PaBRTbnÞúkk¾køayCamantMéltUc. dUcenH
eK)anENnaMeGayeCosvagkareRbIR)as;muxkat;mUlsMrab;krNIkar)ak;edaykarTaj tension failure.
12> karviPaK nigkarKNnassredayeRbIdüaRkam Analysis and Design of
Column Using Charts
karviPaKmuxkat;ssrEdl)anBnül;BIxagedImKWQrelIeKalkarN_sþaTic. sMrab;karviPaK b¤kar
KNnassrCaCMhandMbUg düaRkamb¤taragBiessGacRtUv)aneRbIedIm,IkMNt; φPn nig φM n sMrab;muxkat;
EdleGay nigkar KNnamuxkat;EdkcaM)ac;sMrab; Pu nig M u EdleGay. düaRkam nigtaragenHRtUv)ane)aH
Bum<pSayeday viTüasßanebtugGaemric American Concrete Institute (ACI) viTüasßanebtugBRgwgedayEdk
Concrete Reinforcing Steel Institute (CRSI) nigsmaKmsIum:g;t_Br½Eln Porland Cement Association
(PCA). karKNnassrcugeRkayRtUvEteFVIeLIgedayQrelIsmIkarsþaTic edaykarKNnaedayéd b¤eday
kmμviFIkMuBüÚTr½. kareRbIdüaRkam ACI RtUv)anbgðajenAkñúg]TahrN_xageRkam. düaRkamRtUv)anbgðajkñúgrUb
TI16 nigrUbTI17. Tinñn½yTaMgenHRtUv)ankMnt;sMrab;mxkat;ssrdUcbgðajenARCugxagelIEpñkxagsþaMéntarag.
u
]TahrN_11³ kMnt;sésrEdkcaM)ac;sMrab;ssr short tied column dUcbgðajenAkñúgrUbTI 18 a
edIm,IRTnUvbnÞúkemKuN 2150kN nigm:Um:g;emKuN 440kN.m . ssrmanTTwg 350mm nigbeNþaysrub
h = 500mm . eRbI f 'c = 28MPa / f y = 400MPa .
dMeNaHRsay³
1> cMNakp©it e = M u = 2150 = 204.65mm
Pu
440
yk d = 500 − 60 = 440mm
γh = 500 − 120 = 380mm enaH γ =
380
= 0.76
500
2> eday e = 204.65 < d snμt;famuxkat;)ak;edaykarsgát; compression-controlled section
CamYynwg φ = 0.65
Members in Compression and Bending 226
32. T.Chhay NPIC
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 227
33. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
Members in Compression and Bending 228
34. T.Chhay NPIC
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 229
35. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
Members in Compression and Bending 230
36. T.Chhay NPIC
2150
Pn = = 3307.7kN
0.65
nig Mn =
440
0.65
= 676.9kN .m
Pn 3307.7 ⋅ 10 3
Kn = = = 0.675
f 'c Ag 28 × 350 × 500
⎛e⎞ ⎛ 204.65 ⎞
Rn = K n ⎜ ⎟ = 0.675⎜ ⎟ = 0.276
⎝h⎠ ⎝ 500 ⎠
3> BItaragkñúgrUbTI16 sMrab; γ = 0.7 / ρ = 0.048 dUcKñasMrab; γ = 0.8 / ρ = 0.043
eday interpolation sMrab; γ = 0.76 / ρ = 0.045
As = 0.045 × 500 × 350 = 7875mm 2
eRbI 10DB32 (As = 8042.48mm 2 )/ R)aMedImenAtamRCugxøI. eRbIEdkkg DB10 @ 350mm
¬rUbTI18 a¦
]TahrN_12³ eRbItaragedIm,IkMNt;bnÞúkersIusþg; φP rbs;ssrxøIdUcbgðajkñúgrUbTI 18 b EdlGnuvtþ
n
enAcMgaycMNakp©it e = 305mm . eRbI f 'c = 35MPa nig f y = 400MPa .
dMeNaHRsay³
A. lkçN³rbs;muxkat;³ H = 600mm / γh = 600 − 120 = 480mm ¬cMgayrvagEdlTaj
32 2 × π
8×
nigEdk sgát;¦. γ = 600 = 0.8 ehIy
480
ρ= 4
600 × 350
= 0.03
B. eday e < d / snμt;vaCamuxkat; compression-controlled section.
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 231
37. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
yk ε t = 0.002 / fs
fy
= 1.0 ehIy φ = 0.65 BItaragkñúgrUbTI 17 eKTTYl)an
Pn
K n = 0.36 =
35 × 600 × 350
dUcenH Pn = 2646kN
C. EpÞógpÞat;karsnμt;sMrab;muxkat; compression controlled³
sMrab; K n = 0.36
e
Rn = K n = 0.183
h
BIdüaRkameyIgTTYl)an ρ = 0.019 < 0.03 / dUcenH Pn > 2646kN ¬edIm,I)an ρ = 0.03 ¦
D. karsakl,gelIkTI2³ yk ε t = 0.0015 / f s = 0.0015 × 200000 = 300MPa
f s 300
= = 0.75 ρ = 0.03 K n = 0.43
f y 400
Pn = 0.43 × 35 × 600 × 350 = 3160.5kN
E. EpÞógpÞat;karsnμt;³ sMrab; K n = 0.43 / Rn = K n h = 0.219
e
BItarag ρ = 0.03 dUcGVIEdleGay
dUcenH Pn = 3160.5kN
φPn = 0.65 × 3160.5 = 2054.3kN nig φM n = 626.6kN .m
tamkarviPaK φPn = 2027kN ¬mantMélRbEhlKñanwgkarKNnaedayeRbItarag¦.
13> karKNnassreRkambnÞúkcakp©it Design of Columns Under Eccentric
Loading
karKNnassrmanlkçN³sμúKsμajCagkarviPaKssr edaysarEtbnÞúkxageRkA Pu nigm:Um:g; M u
Casmμtikmμ ehIyeKRtUvkarkMNt;nUvGBaØtiCaeRcIndUcCa b / h / As / A's CamYynwgkarkMNt;rbs; ACI Code.
vaCakarGnuvtþn_TueTAedaysnμt;dMbUgnUvmuxkat;ssr ehIykMnt;brimaNmuxkat;EdkRtUvkar. RbsinebIGñk
KNnaRtUvkardUrmuxkat;EdkKNna enaHmuxkat;ssrk¾RtUv)anEkERbeTAtamenaHEdr. ]TahrN_xageRkam
bgðajBIkarKNnassr.
13>1 KNnassrsMrab;kar)ak;edaykarsgát; Design of Column for Compression
Failure
sMrab; compression failure eKniymeRbI As = A's sMrab;muxkat;ctuekaN. cMNakp©it e = M u .
Pu
edayQrenAelItMélrbs; e eKman2krNIRtUv)anbegáIteLIg
1> enAeBlEdl e ≤ 100mm krNIcMNakp©itGb,brmaGacekItman EdlGaceKNnaedayeRbIrUbmnþ
Members in Compression and Bending 232
38. T.Chhay NPIC
[ (
Pu = φPn = φK 0.85 f 'c Ag + Ast f y − 0.85 f 'c )] Edl φ = 0.65 nig K = 0.80 sMrab;ssrEdl
man dkkgdac;² nig φ = 0.70 nig K = 0.85 sMrab;ssrEdlmanEdkvNн ¬sUmemIl]TahrN_kñúg
emeronssrrgkMlaMgcMGkS½¦. sMrab;krNIepSgBIenH GñkKNnaGacGnuvtþtamkrNITI2. krNIbnÞúk
enHRbRBwtþeTAsMrab;ssrGKarCan;eRkaménGKareRcInCan; Edlm:Um:g; M u )anmkBIRbBn§½mYyCan; nig
Pu )anmkBIbnÞúkRKb;Can;EdlmanGMeBIenABIelIva.
2> tMbn; compression failure KWtMbn;EdlsßitenAcenøaHGkS½QreTAbnÞat; balanced load dUcbgðaj
kñúgrUbTI 3 nigrUbTI11. kñúgkrNIenH muxkat; bh GacRtUv)ansnμt; ehIybnÞab;mkmuxkat;EdkRtUv)an
KNnasMrab; Pu nig M u EdleGay. CMhanénkarKNnaRtUv)ansegçbdUcxageRkam³
k> snμt;muxkat;kaer b¤ctuekaN bh rYckMNt; d / d ' nig e = M u
P u
A' s f y
x> edaysnμt; As = A's KNna A's BIsmIkar Pn = 3he 'c
bhf
+
e
edayeRbI
+ 1.18 + 0.5
d2 (d − d ' )
TMhMmuxkat;Edl)ansnμt; nig φ = 0.65 sMrab;ssrEdleRbIEdkkgFmμta. yk As = A's rYc
eRCIserIsmuxkat;RKb;RKan;. kMNt;muxkat;BitR)akdEdleRbIsMrab; As nig A's . mü:agvij
eToteKGaceRbIdüaRkam ACI.
+
K> epÞógpÞat;fa 1% ≤ ρ g = As bhA's ≤ 8% . RbsinebI ρ g mantMéltUc kat;bnßymuxkat;
snμt; b:uEnþBRgIkmuxkat;RbsinebIeKcg;)anmuxkat;EdktUc.
X> epÞógpÞat;PaBRKb;RKan;rbs;muxkat;cugeRkayedayKNna φPn BIsmIkarsþaTic
dUcBnül;kñúg]TahrN_xagedIm. φPn ≥ Pu .
g> kMNt;EdkkgcaM)ac;.
rUbmnþRbhak;RbEhl approximate formula y:agsamBaØ sMrab;kMNt;muxkat;ssrdMbUg bh b¤PaK
ryEdksrub total steel retio ρ g KW
Pn = K c bh 2 b¤ Pu = φPn = φK c bh 2 ¬-24¦
Edl K c mantMéldUcbgðajkñúgtaragTI2 nigbgðajkñúgrUbTI19 sMrab;Edk f y = 400MPa nig
As = A' s . xñatrbs; K c KW kN / m 3 .
taragTI2 tMélrbs; K ¬ f c y = 400MPa ¦
Kc
ρ g (% )
f 'c (28MPa) f 'c (35MPa) f 'c (42MPa)
1% 24817 30246 35286
4% 37574 43003 48044
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 233
39. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
8% 54675 60103 65144
3
K c ( kN/m )
69000
64000
59000
54000
49000
44000
39000 f'c=28MPa
34000 f'c=35MPa
29000 f'c=42MPa
24000
1 2 3 4 5 6 7 8
ρ g (%)
rUbTI19 tMél K c nig ρ g
tMél K c RtUv)anbgðajkñúgtaragTI2 CatMélRbhak;RbEhl niggayRsYleRbI edaysar K c ekIn
eLIgmþg 5429 sMrab;karekIneLIgrbs; f 'c mþg 7 . sMrab;muxkat;dUcKña enAeBlEdlcMNakp©it e = M u
Pu
ekIneLIg Pn fycuH dUcenH K c fycuH. dUcenH tMél K c sMEdgbnÞúk Pn enAelIdüaRkamGnþrkmμcenøaH
0.8Pno nig Pb dUcbgðajkñúgrUbTI 3 nigTI 11.
Linear interpolation GacRtUv)aneRbI. ]TahrN_ K c = 46124.5 sMrab; ρ g = 6% nig
f 'c = 28MPa . CMhankñúgkarKNnamuxkat;ssrGacRtUv)ansegçbdUcxageRkam³
1> snμt;muxkat;dMbUgsMrab;muxkat;ssr bh
2> KNna K c = (φbh 2 )
Pu
3> kMNt; ρ g BItaragTI 2 sMrab; f 'c EdleGay
4> kMNt; As = A's = ρ g2bh rYceRCIserIsEdkbBaÄr nigEdkkg.
5> kMNt; φPn énmuxkat;cugeRkaytamsmIkarsþaTic ¬dMeNaHRsayCak;lak;¦. tMélén φPn KYrEt
mantMélFMCagb¤esμI Pu . RbsinebImindUecñaHeT EktMrUv bh b¤ ρ g .
Members in Compression and Bending 234
40. T.Chhay NPIC
mü:agvijeTot RbsinebIeKcg;)anPaKryEdksrubCak;lak; ]TahrN_ ρ g = 6% bnÞab;mkGnuvtþdUc
xageRkam³
1> snμt; ρ g dUcEdlTamTar nigbnÞab;mkKNna e = M u Pu
2> edayQrelI f 'c nig ρ g EdleGay/ kMNt; K c BItaragTI2
3> KNna bh 2 = φPu bnÞab;mkeRCIserIs b nig h . GnuvtþCMhan 4 nig 5 eLIgvij.
Kc
eKKYrEtepÞógpÞat;fa 1% ≤ ρ g ≤ 8% . dUcKña epÞógpÞat;fa c
Edl)anmkBIkarKNnatamsþaTicmantMélFMCag cb = 600 +dft sMrab; compression failure .
600
y
]TahrN_13³ kMNt;muxkat;EdkTaj nigmuxkat;Edksgát;sMrab;ssrEdleRbIEdkkgFmμtamanmuxkat;
400 × 600 edIm,IRTbnÞúk Pu = 3470kN nig M u = 530kN .m . edayeRbI f 'c = 28kN nig f y = 400MPa .
dMeNaHRsay³
1> KNna e = M u = 3470 = 152.74mm . eyIgman h = 600mm yk d = 550mm nig d ' = 50mm
P
530
u
edaysar e < 2 d = 366.67mm snμt;fa compression failure.
3
2> snμt; As = A's . kMNt;tMéldMbUgrbs; A's tamrUbmnþ
bhf 'c A' s f y
Pn = + (-17)
3he e
+ 1.18 + 0.5
d 2 (d − d ' )
sMrab; P 3470
Pn = u =
φ 0.65
= 5338.5kN
A' s = 4271.8mm 2 = As
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 235
41. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
eRbIEdk DB32 ⇒ n = 6 edIm
6 DB32 = 4825.5mm 2 sMrab; As nig A' s ¬rUbTI 20¦
3> ρ g = 2 × 4825.5 = 4% EdlvaRtUvEttUcCag 0.08 nig FMCag 0.01
400 × 600
4> epÞógpÞat;muxkat;edaysmIkarsþaTictamCMhanénkarKNnakñúg]TahrN_TI4 eKTTYl)an
a = 430.18mm / c = 506.09mm / Cc = 4095.32kN
C s = 4825.5(400 − 0.85 × 28) = 1815.35kN
⎛d −c⎞ ⎛ 550 − 506.09 ⎞
f s = 600⎜ ⎟ = 600⎜ ⎟ = 52.06MPa
⎝ c ⎠ ⎝ 506.09 ⎠
T = As f s = 4825.5 × 52.06 = 251.21kN
Pn = Cc + C s − T = 5659.5kN > 5338.5kN
cMNaMfa RbsinebI φPn < Pu cUrdMeLIg As nig A's rYceFVIkarKNnaeLIgvij.
5> epÞógpÞat; Pn edayeRbIsmIkar Pn = e' ⎡Cc ⎛ d − a ⎞ + Cs (d − d ' )⎤ Edl e' = e + d − h
1
⎢ ⎜ ⎟ ⎥
⎣ ⎝ 2⎠ ⎦ 2
eyIgTTYl)an Pn = 5659kN
6> sMrab; muxkat; balanced section
⎛ 600 ⎞
cb = ⎜ ⎟d t = 600 550 = 330mm
⎜ 600 + f y ⎟ 1000
⎝ ⎠
edaysarEt c = 506.09mm > cb = 330mm vaCakrNI compression failure dUckarsnμt;.
7> edayeRbIEdkkgmanGgát;p©it 10mm
KMlatEdkkg
⎧48φ ⎧48 ×10 ⎧480
⎪ ⎪ ⎪
min ⎨16d = min ⎨16 × 32 = min ⎨512 = 400
⎪ b ⎪ 400 ⎪400
⎩ ⎩ ⎩
dUcenHeRbIEdkkg DB10 @ 400 .
]TahrN_14³ eFVI]TahrN_TI13 eLIgvijedayeRbIsmIkar ¬-24¦
dMeNaHRsay³
1> muxkat;ssrEdleGay 400 × 600
2> kMNt; K c BIsmIkar ¬-24¦
3> K c = φbh 2 = 0.65 ×3470× 0.6 2 = 37073 m3
Pu
0 .4
kN
Members in Compression and Bending 236
42. T.Chhay NPIC
4> BItaragTI 2 b¤rUbTI19 sMrab; K c = 37073 mm3 f 'c = 28MPa eday interpolation
kN
4 −1
eyIgTTYl)an ρ g = 1 + (37073 − 24817) 37574 − 24817 = 3.88%
5> KNna As = A's = ρbh / 2 = 0.0388(400)(600)/ 2 = 4656mm 2
eRbIEdk DB32 ⇒ n = 6 edIm
6> 6DB32 = 4825.5mm 2
7> kMNt; φPu edayeFVItamCMhan 4-7 sMrab;]TahrN_TI13. Pn = 5659kN > Pn = 5338.5kN
dUcenHmuxkat;EdkRKb;RKan;
8> RbsinebImuxkat;minRKb;RKan; b¤ φPn < Pn tMeLIgmuxkat; As nig A's rYceFVIkarepÞógpÞat;eLIgvij
edIm,ITTYl)antMélEk,r.
]TahrN_15³ KNnamuxkat;ssrctuekaNEkgedIm,IRTbnÞúk Pu = 3150kN nig M u = 630kN .m
CamYynwgPaKryEdksrub ρ g RbEhl 4% . eRbI f ' = 28MPa / f = 400MPa nig b = 450mm .
c y
dMeNaHRsay³
1> KNna e = M u = 3150 = 0.2m . snμt; compression failure ( φ = 0.65 ) ¬RtUvepÞógpÞat;enA
Pu
630
eBleRkay¦ ehIy As = A's
2> sMrab; ρ = 4% nig f 'c = 28MPa enaH K c = 37574 ¬taragTI2¦
3> KNna bh 2 BIsmIkar (-24): Pu = φK cbh 2 rW 3150 = 0.65(37574)(0.45)h 2 dUcenH
h = 0.535m dUcenHyk h = 550mm .
KNna As = A's = 0.04(450 × 550) = 4950mm 2 . eRCIserIs 5DB36
2
( As = 5089.4mm ) dUcbgðajkñúgrUbTI 21. eRbIEdkkg DB12 @ 450 .
2
4> epÞógpÞat; muxkat;cugeRkayedaykarviPaK RsedogKñanwg]TahrN_TI4 eyIgTTYl)an
a = 327.8mm / c = 385.65mm / Cc = 0.85 f 'c ab = 3510.7kN / f ' s = 400 MPa /
⎛d −c⎞
C s = A' s ( f y − 0.85 f 'c ) = 1914.6kN / f s = 600⎜ ⎟ = 146.79 MPa / nig T = 747 kN
⎝ c ⎠
dUcenH Pn = Cc + Cs − T = 4678.3kN ehIy Pu = φPn = 3041kN < 3150kN
edaysarmuxkat;minRKb;RKan; eyIgRtUvtMeLIgmuxkat;Edk b¤muxkat;ebtug rYceFVIkarepÞógpÞat;eLIg
vij. ¬yk h = 600mm ¦ rUbTI21.
5> sMrab;muxkat; balanced section
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 237
43. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
⎛ 600 ⎞
cb = ⎜ ⎟d t = 318mm < c = 436.32mm
⎝ 600 + 400 ⎠
¬ d = 530mm ¦
dUcenH vaCa compression failure dUckarsnμt;.
13>2 KNnassrsMrab;kar)ak;edaykarTaj Design of Column for tension Failure
Kar)ak;edaykarTaj tension failure ekItmanenAeBl Pn < Pu b¤k¾cMNakp©it e > eb dUckarBnül;
enAkñúgEpñkTI7. kñúgkarKNnassr Pu nig M u RtUv)aneGay ehIyvaTamTarnUvkarkMNt;muxkat;ssr nigmux
kat;Edk. vaGacRtUv)ansnμt; ¬dUckarENnaM¦ favaCa tension control enAeBlNa
⎧530 sMrab; h < 600mm
M u
>⎨
600 sMrab; h ≥ 600mm
. kñúgkrNIenH muxkat;ssrGacRtUv)ansnμt; ehIybnÞab;mk As nig
Pu ⎩
A' s RtUv)ankMNt;. düaRkam ACI GacRtUv)aneRbIedIm,IKNna ρ g sMrab;muxkat;EdleGayCamYynwg As
= A's . cMNaMfa φ ERbRbYlcenøaH 0.65(0.7) nig 0.9 dUckarBnül;kñúgEpñkTI 4.
enAeBl tension controls EdkTaj yields b:uEnþEdksgát;Gac yields nigmin yields. karsnμt;dMbUg
f ' s = f y nig As = A' s . smIkar (-16) ¬kñúgEpñkTI 6¦ GacRtUv)aneRbIedIm,IKNnatMéldMbUgrbs; As nig
A' s .
⎛ h a⎞
Pn ⎜ e − + ⎟
As = A' s = ⎝
2 2⎠
(-16)
f y (d − d ')
edaysar a minRtUv)andwgenAeLIy snμt; a = 0.4d nig Pu = φPn bnÞab;mk
Pu (e − 0.5h + 0.2d )
As = A' s = (-25)
φf y (d − d ' )
muxkat;ssrcugeRkayKYrRtUv)anepÞógpÞat;edaysmIkarsþaTicedIm,Ibgðajfa φP n ≥ Pu . ]TahrN_
TI16 Bnül;BIviFIsaRsþkñúgkarKNnaenH.
Members in Compression and Bending 238
44. T.Chhay NPIC
enAeBlbnÞúk P mantMéltUcNas;ebIeRbobeFobCamYynwgm:Um:g; M TMhMrbs;muxkat;GacRtUv)ankM
u u
Nt;edayeRbIEt M EtmYy)anehIy edaysnμt;fa P = 0 . muxkat;cugeRkayKYrRtUv)anepÞógpÞat;eday
u u
smIkarsþaTic. krNIenHekIteLIgsMrab;eRKagGKarmYyCan; b¤BIrCan; EdlGKarenaHRtUv)aneKeRbIsMrab;eFVICa
saltaMgBiBN’ b¤k¾GKarTaMgLayNaEdlmanlkçN³dUcKñaenaH. sMrab;krNIenH A' GacRtUv)ansnμt;eGay s
mantMéltUcCag A . karKNnay:aglMGitsMrab;saltaMgBiBN’kMBs;mYyCan;Edlmansnøak;BIrRtUv)anBnül;
s
enAkñúgCMBUkTI 16 FñwmCab; nigeRKag.
]TahrN_16³ kMNt;sésrEdkcaM)ac;sMrab;ssrragctuekaNEkg 400 × 560 EdlmanEdkkg
FmμtaRTbnÞúk P = 1140kN nig M = 850kN .m . eRbI f ' = 28MPa nig f = 400MPa
u u c y
dMeNaHRsay³
1> KNna e = M
P
u
=
850
1140
= 0.7456m . yk d = 560 − 60 = 500mm . edaysar
u
Mu
Pu
= 745.6mm > 530mm b¤edaysar e > d snμt;fassrenH)ak;edaykrNI tension failure
enaH φ = 0.9 ¬RtUvepÞógpÞat;enAeBleRkay¦.
2> snμt; A = A' nig f ' = f nigeRbIsmIkar (-25) edIm,IkMnt; A nig A' . eday P = 1140kN /
s s s y s s u
e = 745.6mm / h = 560mm / d = 500mm / nig d ' = 60mm
1140 ⋅ 10 3 (745.6 − 0.5 × 560 + 0.2 × 500 )
As = A' s = = 4070.71mm 2
0.9 × 400(500 − 60 )
eRbI 5DB32 (4021.24mm ) sMrab; A nig A' . ¬rUbTI22¦
2
s s
3> epÞógpÞat; ρ = 2400 × 560) = 0.0359 EdltUcCag 0.08 nigFMCag 0.01 .
(4021.24
g
4> epÞógpÞat;kareRCIserIsmuxkat;edaysmIkarsþaTic EdlkarKNnaRsedogKñanwg]TahrN_TI3
eRKOgbgÁúMrgkarsgát; nigrgkarBt; 239
45. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
a. kMNt;tMélrbs; a edayeRbIsmIkarTUeTA Aa + Ba + C = 0 CamYynwg
2
e' = e + d − = 965.6mm / A = 0.425 f ' b = 4760 / B = 2 A(e'− d ) = 4432512 /
h
c
2
C = A' ( f − 0.85 f ' )(e'− d + d ') − A f e' = −758040793 . eKTTYl)an a = 147.62mm
s y c s y
nig c = a / 0.85 = 173.67mm .
b. epÞógpÞat; f ' ³ f ' = 600⎛ c −c d ' ⎞ = 600⎛ 173.67.6760 ⎞ = 392.71MPa
s ⎜
⎝
s ⎟
⎠
⎜
⎝ 173
−
⎟
⎠
c. KNna a eLIgvij
C = A' s ( f ' s −0.85 f ' c )(e'−d + d ') − As f ' s e' = −773454123.3
eKTTYl)an a = 150.25mm nig c = 176.77mm
d. epÞógpÞat; f ' ³ f ' = 600⎛ c −c d ' ⎞ = 600⎛ 176.77.7760 ⎞ = 396.34MPa
s ⎜
⎝
s ⎟
⎠
⎜
⎝ 176
−
⎟
⎠
KNna C = 0.85 × 28 × 150.25 × 400 = 1430.38kN
c
C s = A' s ( f ' s −0.85 f ' c ) = 4021.24(396.34 − 0.85 × 28) = 1498.07kN
T = As f y = 4021.24 × 400 = 1608.5kN
e. Pn = C c + C s − T = 1319.95kN
5> KNna φ ³ ε = 0.003⎛ d c− c ⎞ = 0.0055 edaysarEt ε = 0.0055 > 0.005 enaH φ = 0.9
⎜ t ⎟ t
t
⎝ ⎠
6> φP = 0.9 × 1319.95 = 1187.95kN > 1140kN muxkat;RKb;RKan;
n
14> karBt;tamBIrTis Biaxial Bending
karviPaK nigkarKNnassreRkamGMeBIbnÞúkcakp©itEdl)anBiPakSaknøgmk CakrNIkarBt;mYyTis.
enHmann½yfa P GnuvtþenAelIGkS½ y ¬rUbTI23¦ begáIt)anbnSMénkMlaMgcMGkS½ P nigm:Um:g;Bt;CMuvijGkS½ x
n n
esμInwg M nx = Pn e y b¤ P GnuvtþenAelIGkS½ x ¬rUbTI24¦ CamYynwgcMNakp©it e begáIt)anbnSMénkMlaMgcM
n x
GkS½ P nigm:Um:g;Bt; M ny = Pn e x .
n
Members in Compression and Bending 240