This document discusses the effective length factor (K) used for calculating the effective length of slender columns. It provides three methods for determining K based on the restraint conditions at the column ends: 1. Using alignment charts and restraint factors (ψA and ψB) for the column and bracing members. 2. Equations relating K to ψmin for partially restrained columns. 3. A simplified equation for K if the column is hinged at one end. Examples are given to calculate K using the alignment chart method for different bracing conditions. The effective length is important for evaluating the strength and stability of slender columns.

1. T.Chhay NPIC
XII. ssrEvg
1> esckþIepþIm
sMrab;karKNnassrxøIEdl)anBnül;enAkñúgBIremeronxagedIm )ansnμt;fa karPøat; buckling karrYjxøI
eGLasÞic elastic shortening nigm:Um:g;TIBIr secondary moment EdlbNþalmkBIPaBdabtamTTwg lateral
deflection manT§iBlCaGb,brmaeTAelIersIusþg;cugeRkay ultimate strength rbs;ssr dUcenHktþaTaMgenH
minRtUv)anrab;bBa©ÚleTAkñúgdMeNIrkarénkarKNnaeT. b:uEnþ enAeBlEdlssrEvg ktþaTaMgGs;enHRtUvEtyk
mkBicarNa. RbEvgbEnßmnwgbNþaleGaymankarkat;bnßyersIusþg;rbs;ssr edayERbRbYlCamYynwgkMBs;
RbsiT§PaB nigTTwgrbs;muxkat; pleFobrlas; slenderness ratio niglkçxNÐcugssr.
ssrEdlman slenderness ratio FMnwgkat;bnßylT§PaBRTRTg;rbs;ssry:agxøaMg Et slenderness
ratio tUcmann½yfassrxøI ehIykarkat;bnßyersIusþg;GacnwgminKYreGaycab;GarmμN_. pleFobrlas;
slenderness ratio KWCapleFobrvagkMBs;ssr l CamYynwgkaMniclPaB radius of gyration r Edl
r = I / A kñúgenaH I Cam:Um:g;niclPaBénmuxkat; moment of inertia of the section nig A CaRkLa
2
épÞmuxkat;.
sMrab;muxkat;ctuekaNEdlmanTTwg b nigkMBs; h ¬rUbTI 1¦ I = bh / 12 nig A = bh dUcenH
x
3
r = 0.288h ¬b¤ edaytMélRbEhl r = 0.3h ¦. dUcKña I = b h / 12 nig r = 0.288b ¬b¤ r = 0.3b ¦.
x x y
3
y y
sMrab;ssrmUlCamYynwgGgát;p©it D enaH I = I = πD / 64 nig A = πD / 4 dUcenH r = r = 0.25D .
x y
4 2
x y
CaTUeTA ssrGacRtUv)anBicarNa dUcteTA³
1> EvgCamYynwg slenderness ratio FM RtUvkarCnÞl; b¤ shear wall.
2> EvgCamYynwg slenderness ratio lμmEdlbgáeGaymankarkat;bnßyersIusþg;ssr enaHCnÞl;Gac
nwgminRtUvkar Etkarkat;bnßyersIusþg;RtUvEtBicarNa.
3> xøIEdl slenderness ratio tUcEdlbNþaleGaymankarkat;bnßyersuIsþg;sþÜcesþIg. karkat;Gac
RtUv)anecal dUcerobrab;BIemeronmun.
2> RbEvgssrRbsiT§PaB Effective Column Length ( Klu )
pleFobrlas; slenderness ratio l / r GacRtUv)anKNnay:agsuRkitenAeBlEdlRbEvgRbsiT§PaB
rbs;ssr ¬ Kl ¦ RtUv)aneRbI. RbEvgRbsiT§PaBenHGnuKmn_eTAnwgBIrktþaFM²³
u
1> RbEvgKμanTMr unsupported length l sMEdgnUvkMBs;minKitTMrrbs;ssrrvagBIrkMralxNÐ.va
u
RtUv)anvas;Ca clear distance rvagkMralxNÐ Fñwm b¤GgÁeRKOgbgÁúMEdlpþl;nUvTMrxagdl;ssr.
ssrEvg 255

2. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
enAkñúgRbBnæ½kMralxNÐ flat slab CamYynwg column capital kMBs; unsupported height rbs;
ssrRtUv)anvas;BIépÞxagelIrbs;kMralxNÐxageRkameTA)atrbs; column capital. RbsinebI
ssrRtUv)anRTCamYyFñwmEdlmankMBs;x<s;tamTismYyCagtamTismYyeTot enaH l KYrEt u
KNnatamTisTaMgBIr ¬tamTis x nig y ¦énmuxkat;ssr. RtUvEdlFMCagRtUv)anBicarNakñúg
karKNna.
2> emKuNRbEvgRbsiT§PaB K bgðajnUvpleFobéncMgayrvagcMnucénm:Um:g;sUnüenAkñúgssr nigkM
Bs;KñanTMrrbs;ssrkñúgTisedAmYy. ]TahrN_ RbsinebIRbEvgKμanTMr unsupported height rbs;
ssrTMrsnøak; hinged enAcugsgçag ¬Edlkareyalxag sidesway RtUv)anTb;¦ KW l nigcMnucm:U
u
m:g;sUnüenAcug nigKl;ssr EdlenAcugTMrRtIekaN hing enaHemKuN K = l / l KWesμInwg 1. Rb
u u
sinebIssr manTMrbgáb; fixed enAcugsgçag ehIykareyalxag sidesway RtUv)anTb; cMnucrbt;
¬cMnucm:Um:g;sUnü¦ sßitenA l / 4 BIcugTMr. dUcenH K = 0.5l / l = 0.5 ¬rUbTI2¦ edIm,IKNna
u u u
tMéld¾RtwmRtUvrbs; K krNI cMbgBIrRtUv)anBicarNa.
- enAeBleRKagbgÁúMEdlpÁúMeLIgeday Fñwm nigssrRtUv)anBRgwgedayCBa¢aMg shear wall
CnÞl;rwg rigid bracing b¤TMrxagEdl)anmkBIeRKagbgÁúMenACab;nwgva. cugrbs;ssrnwg
sßitenATItaMgdEdl EdlkarrMkilxagrbs;tMNRtUv)ankarBar. CaTUeTAsMrab;eRKagBRgwg
tMélrbs; K KWtUcCagb¤esμInwg 1. ACI code, section 10.12 esñIeGayeRbI K = 1 .
- enAeBleRKagbgÁúMminRtUv)anBRgwg vanwgGaRs½yeTAnwgPaBrwgRkaj stiffness rbs;Fñwm
nigssr edIm,ITb;nwgPaBdabxag. edaysarkarrMkilrbs;tMNrminRtUv)ankarBar eRKag
egakeTAtamTisrbs;bnÞúkxag. tMélrbs; K sMrab;ssr nigeRKagRtUv)aneGayenAkñúg
rUbTI2 edayBicarNakrNITaMgBIr KWenAeBlkareyalxag sidesway RtUv)ankarBar nig
minRtUv)ankarBar.
Slender Column 256

3. T.Chhay NPIC
3> emKuNRbEvgRbsiT§PaB Effective Length Factor ( K )
RbEvgRbsiT§PaBrbs;ssrGacRtUv)anKNnaedayeRbIdüaRkam alignment chart kñúgrUbTI3. edIm,Irk
emKuNRbEvgRbsiT§PaB K dMbUgeKcaM)ac;RtUvKNnarkemKuNTb; restraint factor ψ nig ψ enAxagcugnig
A B
Kl;ssrerogKña Edl
EI / l rbs;ssr
ψ =∑ c
(-1)
∑ EI / l rbs;Fwm
Edl l = RbEvgKitBIGkS½eTAGkS½éntMNrrbs;eRKag
c
l = RbEvgElVgKitBIGkS½eTAGkS½éntMNrrbs;eRKag
¬TaMgBIrsßitenAkñúgbøg;Bt;¦. emKuN ψ enAxagcugKYrEtrYmbBa©ÚlTaMgssr nigFñwmEdlCYbKñaenARtg;
tMNr. sMrab;TMrsnøak; hinged end ψ KWGnnþ nigGacsnμt;esμI 10 . sMrab;TMrbgáb; fixed end ψ KWsUnü nig
Gacsnμt;esμI 1. tMélsnμt;TaMgenHGaceRbI)anedaysarEtenAkñúgeRKagbgÁúMebtugGaem:Kμansnøak;Kaμ nkkit
l¥tex©aH b¤TMrbgát;l¥tex©aHenaHEdr.
dMeNIrkarrk K KWKNna ψ sMrab;cugssr nigψ sMrab;Kl;ssr. dak; ψ nig ψ eTAkñúgdüa
A B A B
Rkam alignment chart énrUbTI3 rYcP¢ab;cMnucTaMgBIredaykat;ExSkNþal EdlbgðajBItMél K . düaRkam
BIrEdlmanlkçN³RsedogKñaRtUv)anbgðaj mYysMrab;eRKagBRgwg Edlkareyalxag sidesway RtUv)an
karBar nigmYyeTotsMrab;eRKagFmμta Edlkareyalxag sidesway minRtUv)ankarBar. karbegáItdüaRkam
enHKWQrelIkarsnμt;fa³
- eRKagbgÁúMpÁúMeLIgedayeRKagctuekaNsIuemRTI
- m:Um:g;Bt;Fñwm)anEckmkssredayTak;TgnwgPaBrwgRkajrbs;va
- ssrTaMgGs;TTYlnUvbnÞúkFMenAeBlCamYyKña
edIm,ICMnYsnUvkareRbIdüaRkam alignment chart EdlbgðajkñúgrUbTI3 ACI Code Commentary )an
esñInUvsmIkarsMrYldUcxageRkamsMrab;KNnaemKuNRbEvgRbsiT§PaB K .
1> sMrab;GgÁrgkarsgát;EdlmankarBRgwg tMélrbs; K GacRtUv)anyktMéltUcCageKkñúgcMeNam
smIkarTaMgBIrxageRkam
K = 0.7 + 0.05(ψ A + ψ B ) (-2)
K = 0.85 + 0.05ψ min (-3)
Edl ψ nig ψ CatMélrbs; ψ enAcugsgçagrbs;ssr nig ψ CatMéltUcbMputéntMélTaMgBIr.
A B min
2> sMrab;GgÁrgkarsgát;EdlKμankarBRgwgEtRtUv)anTb;enAcugsgçag tMélrbs; K Gacsnμt;dUc
xageRkam
ssrEvg 257

4. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
sMrab; ψ < 2 / K = 2020ψ 1 +ψ
m
− m
m (-4)
sMrab; ψ ≥ 2 / K = 0.9 1 +ψ
m m (-5)
Edl ψ CatMélmFümrbs; ψ enAcugsgçagGgÁrgkarsgát;.
m
3> sMrab;GgÁrgkarsgát;KμankarBRgwgmanTMrsnøak; hinged enAcugmçag enaH K GacRtUv)ansnμt;dUc
xageRkam
K = 2 + 0.3ψ (-6)
Edl ψ CatMélenAcugEdlmankarTb;.
Slender Column 258

5. T.Chhay NPIC
]TahrN_ 1³ edayeRbInUvsmIkarxagedIm cUrkMNt;emKuNRbEvgRbsiT§PaB K sMrab;Ggát;rgkarsgát;enAkñúg
eRKagCamYynwglkçxNÐxageRkam³
1> eRKagRtUv)anBRgwgedIm,ITb;nwgkareyalxag sidesway ehIy ψ A = 2 .0 nig ψ B = 3 .0 enAcug
xagelI nigxageRkamrbs;Ggát;.
ssrEvg 259

6. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
2> eRKagminRtUv)anBRgwgedIm,ITb;nwgkareyalxag sidesway eT ehIy ψ A = 2 .0 nig ψ B = 3 .0 .
¬Ggát;RtUv)anbgáb;enAcugsgçag¦.
3> eRKagminRtUv)anBRgwgedIm,ITb;nwgkareyalxag sidesway eT ehIy ψ A = 0 .0 ¬TMrsnøak;¦ nig
ψ = 3 .0 .
B
dMeNaHRsay³
1> BIsmIkar (-2) nig (-3)
K1 = 0.7 + 0.05(2 + 3) = 0.95 < 1.0
K 2 = 0.85 + 0.05(2) = 0.95 < 1.0
eRCIserIsyknUvtMéltUcCageKkñúgcMeNam K nig K . kñúgkrNIenH K = 0.95 .
1 2
2> tMélmFümrbs; ψ = (2 + 3) / 2 = 2.5 . eday ψ > 2 eRbIsmIkar (-5)
m m
K = 0.9 1 + 2.5 = 1.684
3> BIsmIkar (-6)
K = 2 + 0.3(3) = 2.9
4> PaBrwgRkajrbs;Ggát; Member Stiffness ( EI )
PaBrwgRkajrbs;Ggát;eRKagesμInwgplKuNrvagm:UDuleGLasÞic E CamYynwgm:Um:g;niclPaBénmuxkat;
I . tMélén E nig I sMrab;ebtugGaem:GacRtUv)anKNnadUcxageRkam³
1> m:UDuleGLasÞicrbs;ebtugRtUv)anBnül;kñúgemeronTI2. bTdæan ACI Code eGaysmIkarxag
eRkam
E = 0.043w
c
1.5
f' c b¤ E = 4780 f '
c c
sMrab;ebtugTMgn;Fmμta. cMENkÉm:UDuleGLasÞicrbs;EdkKW E = 2.1⋅10 MPa .
s
5
2> sMrab;Ggát;ebtugGaem: m:Um:g;niclPaB I ERbRbYltambeNþayrbs;Ggát; GaRs½yeTAnwgkMriteRbH
nigPaBryEdkEdl)aneRbIR)as;.
edIm,IkMNt;nUvemKuN ψ EI RtUvEt)ankMNt;sMrab;Fñwm nigssr. dUcenH EI GacRtUv)ankM
Nt;dUcxageRkam ¬ACI Code, section 10.11.1¦³
sMrab;Fñwm I = 0.35 I g
sMrab;ssr I = 0.70 I g
sMrab;CBa¢aMg¬KμaneRbH¦ I = 0.70I g
sMrab;CBa¢aMg¬maneRbH¦ I = 0.35I g
Slender Column 260

7. T.Chhay NPIC
sMrab;kMralxNн (flat plate nig flat slab) I = 0.25 I
g
Edl I Cam:Um:g;niclPaBsMrab;muxkat;ebtugeBjeFobGkS½kat;tamTIRbCMuTMgn; edayecal
g
Edk.
3> RkLaépÞmuxkat; A = A ¬RkLaépÞmuxkat;eBj gross-sectional area¦
g
4> m:Um:g;niclPaBKYrEtRtUv)anEckeday (1 + β ) enAeBlEdlbnÞúkxagefr sustained lateral load
d
manGMeBIelIeRKagbgÁúM b¤sMrab;epÞógpÞat;esßrPaB stability check Edl
maximum factored sustained axial load
βd =
total factored axial load
5> EdnkMNt;sMrab;pleFobrlas; Limitation of The Slenderness Ratio ( Klu / r )
5>1> eRKagGt;eyal Nonsway Frames
bTdæan ACI Code, section 10.12 ENnaMnUvEdnkMNt;xagRkamrvagssrxøI nigssrEvgenAkñúgeRKag
BRgwg ¬Gt;eyaK nonsway¦³
1> T§iBlrbs;PaBrlas; slenderness GacRtUv)anecal ehIyssrGacRtUv)anKNnaedayKitCassrxøI
enAeBlEdl³
Klu 12 M 1
≤ 34 − (-7)
r M2
Edl M nig M Cam:Um:g;emKuNenAcugssr ehIy M
1 2 2 > M1 .
ssrEvg 261

8. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
2> pleFob M RtUvcat;Tukfa viC¢manRbsinebIGgát;RtUv)anenAkñúgkMeNageTal single curvature
M 1
2
nigGviC¢mansMrab;kMeNagDub double curvature dUcbgðajkñúgrUbTI4.
3> GgÁ (34 − 12M / M ) KYrminRtUvFMCag 40.
1 2
4> RbsinebIm:Um:g;ssremKuN factored column moment esμIsUnü b¤ e = M / P < e tMélrbs; M
u u min 2
KYrEtRtUv)anKNnaedayeRbIcMNakp©itGb,brma³
emin = (15.24 + 0.03h) (-8)
M 2 = Pu (15.24 + 0.03h) (-9)
Edl M Cam:Um:g;Gb,brma. m:Um:g; M KYrEtRtUv)anBicarNaedayeFobnwgGkS½nImYy²rbs;ssrdac;
2 2
edayELkBIKña. tMél K GacRtUv)ansnμt;esμInwg 1.0 sMrab;eRKagBRgwg braced frame elIkElgEt
vaRtUv)anKNnaedayQrelIkarviPaK EI .
5>2> eRKageyal Sway Frames
enAkñúgGgát;rgkarsgát;minBRgwgTb;nwgkareyalxag sidesway T§iBlrbs;pleFobrlas;
slenderness ratio GacecalenABlEdl
Klu
< 22 (ACI Code, section 10.13) (-10)
r
5>3> pleFobrlas;FM High slenderness ratio
enAeBlEdlGgát;rgkarmYydac;edayELkenAkñúgeRKagmanpleFobrlas; slenderness ratio
Kl / r > 100 viFIm:Um:g; magnifier (moment magnifier method) rbs; ACI Code minGacRtUv)aneRbI
u
ehIykarviPaK rigorous dWeRkTIBIr rigorous second-order RtUv)aneRbICMnYsvij. Et muxkat;GacRtUv)an
tMeLIgedIm,Ikat;bnßypleFob Kl / r . tMé;l 100 bgðajBIkarBiesaFn_Cak;EsþgcMNat;fñak;x<s; (ACI
u
Code, section 10.10.5) .
6> viFIKNnabEnßmm:Umg; Moment-Magnifier Design Method
6>1> esckþIepþIm Introduction
CMhandMbUgkñúgkarKNnam:Um:g;enAkñúgssrEvgKWkMNt;faetIeRKagEdlKNna CaeRKakBRgwg b¤min
BRgwgTb;nwg sidesway . RbsinebImanGgÁBRgwgxag dUcCa shear walls nig shear trusses b¤ssrmanPaBrwg
RkajTTwg lateral stiffness efr enaHPaBdabTTwg lateral deflection mantMéltUc ehIyT§iBlrbs;vaeTAelI
ersIusþg;ssrk¾tUcEdr. eKGacsnμt;faeRKagbgÁúMenAkñúgmYyCan;²RtUv)anBRgwgRbsinebI
Q = ∑ u o ≤ 0.05
PΔ
(-11)
Vus lc
Slender Column 262

9. T.Chhay NPIC
Edl ∑ Pu nig V CabnÞúkbBaÄrsrub nigkMlaMgkat; erogKña ehIy Δ PaBdabeFobdWeRkTImYy first-
us o
order relative deflection rvagkMBUl nig)atrbs;Can;EdlbNþalmkBI V . RbEvg l CaRbEvgrbs;Ggát;rg
us c
karsgát;enAkñúgeRKagbgÁúM edayvas;BIGkS½eTAGkS½rbs;tMNrenAkñúgeRKag.
CaTUeTA Ggát;rgkarsgát;GacrgnUgPaBdabTTwg lateral deflection EdlbNþalmkBIm:Um:g;TIBIr
secondary moment. RbsinebIm:Um:g;TIBIr M ' RtUv)anbEnßmeTAelIm:Um:g;EdlGnuvtþelIssr M enaHm:Um:g;cug
a
eRkayKW M = M + M ' . viFIRbEhl approximate method sMrab;kMNt;m:Um:g;cugeRkay M KWCakarKuNm:U
a
m:g; M edayemKuNEdleKehAfa emKuNbEnßmm:Um:g; (magnifying moment factor) ehIyemKuNenHRtUvEt
a
FMCagb¤esμInwg 1.0 . b¤ M = δM nig δ ≥ 1.0 . m:Um:g; M RtUv)anTTYlBIkarviPaKeRKageGLasÞiceday
max a a
eRbIbnÞúkemKuN ehIyvaCam:Um:g;GtibrmaEdlmanGMeBIenAcugssr b¤enAkñúgssr RbsinebIbnÞúkxagmanvtþman.
RbsinebIT§iBl P − Δ RtUv)anykmkBicarNa vanwgcaM)ac;RtUvEteRbIkarviPaKdWeRkTIBIr edIm,IKitBI
TMnak;TMng nonlinear relationship rvagbnÞúk PaBdabTTwg nigm:Um:g;. eKGaceRbIkmμviFIkMuBüÚTr½edIm,IedaHRsay
va. bTdæan ACI Code GnuBaØatieGayeRbIkarviPaKssrdWeRkTImYy b¤dWeRkTIBIr. karviPaKssrdWeRkTIBIr
RtUv)antMrUveGayeRbIenAeBlEdl Klu / r > 100 . viFIKNnam:Um:g;bEnßmrbs; ACI Code CaviFIsMrYlsMrab;
KNnaemKuNbnÞúkbEnßmTaMgeRKagBRgwg nigeRKagminBRgwg.
6>2> m:Um:g;bEnßmenAkñúgeRKagGt;eyal Magnified Moments in Nonsway Frames
T§iBlrbs;pleFobrlas; slenderness ratio Klu / r enAkñúgGgát;rgkarsgát;éneRKagBRgwgGac
RtUv)anecalRbsinebI Klu / r ≤ 34 − 12M1 / M 2 dUcbgðajenAkñúgEpñk 5>1 . RbsinebI
Klu / r > 34 − 12 M1 / M 2 enaHT§iBlPaBrlas;RtUv)anBicarNa. dMeNIrkarkMNt;emKuNbEnßm δ ns enAkñúg
eRKagmineyalGacRtUv)ansegçbdUcxageRkam (ACI Code, section 10.12)³
1> kNt;faeRKagCaeRKagBRgwgTb;nwg sidesway b¤Gt; rYckMNt;RbEvgKμanTMr lu nigemKuNRbEvg
RbsiT§PaB K ¬ K RtUv)ansnμt;eGayesμI 1.0 ¦
2> KNnaPaBrwgRkajrbs;Ggát; EI edayeRbIsmIkar
0.2 Ec I g + Es I se
EI = (-12)
1 + βd
b¤smIkarEdlsMrYlCag
0.4 Ec I g
EI = (-13)
1 + βd
EI = 0.25Ec I g ¬sMrab; β d = 0.6 ¦ (-14)
Edl Ec = 4780 f 'c
Es = 2.1 ⋅105 MPa
ssrEvg 263

10. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
Ig = m:Um:g;niclPaBénmuxkat;ebtugtamGkS½NamYyEdleyIgBicarNaedayecal As
I se = m:Um:g;nicalPaBénmuxkat;EdkeFobGkS½TIRbCMuTMgn;rbs;muxkat;ebtug
maximum factored axial sustained load 1.2 D (sustained)
βd = =
maximum factored axial load 1.2 D + 1.6 L
cMNaMfa³ β d xagelICapleFobEdlFøab;KNnam:Um:gbEnßmenAkñúgssrEdlbNþalmkBIbnÞúk
;
sustained .
smIkar (-13) nig (-14) manlkçN³suRkitticCagsmIkar (-12) . elIsBIenH smIkar (-14)
TTYl)anedaysnμt; β d = 0.6 CMnYskñúgsmIkar (-13) .
3> kMNt;bnÞúk Euler buckling/ Pc ³
π 2 EI
Pc = (-15)
(Klu )2
eRbItMélrbs; EI / K nig lu dUcKNnaBICMhan 1> nigCMhan 2>.
4> KNnatMélénemKuN Cm edIm,IeRbIenAkñúgsmIkarénemKuNm:Um:g;bEnßm moment-magnifier
factor. sMrab;Ggát;BRgwgedayKμanbnÞúkxag transverse load
0.4M1
Cm = 0.6 + ≥ 0.4 (-16)
M2
Edl M1 / M 2 viC¢manRbsinebIssrRtUv)anBt;kñúgkMeNageTal. sMrab;Ggát;CamYybnÞúkxagenA
cenøaHTMr Cm KYrRtUv)anykesμInwg 1.0 .
5> KNnaemKuNm:Um:g;bEnßm δ ns
Cm
δ ns = ≥ 1.0 (-17)
1 − ( Pu / 0.75Pc )
Edl Pu CabnÞúkemKuN nig Pc nig Cm RtUv)anKNnaBIxagelI.
6> KNnaGgát;rgkarsgát;edayeRbIbnÞúkemKuNtamGkS½ Pu BIkarviPaKeRKagd¾RtwmRtUv nigm:Um:g;
bEnßm magnified moment M c EdlKNnadYcxageRkam³
M c = δ ns M 2 (-18)
Edl M 2 Cam:Um:g;emKuNEdlFMCagEdlekItBIbnÞúk EdllT§plmineyal. sMrab;eRKagBRgwg
Tb;nwg sidesway emKuNeyalKW δ s = 0 . enAkñúgeRKagGt;eyal nonsway frame PaBdab
TTwgRtUv)anrMBwgeGaytUcCagb¤esμInwg H /1500 Edl H CakMBs;srubrbs;eRKag.
6>3> m:Um:g;bEnßmenAkñúgeRKageyal Magnified Moments in sway Frames
Slender Column 264

11. T.Chhay NPIC
T§iBlrbs;PaBrlas;GacRtUv)anecalenAkúñgeRKageyal sway frame ¬KμanBRgwg unbraced¦
enAeBlEdl Klu / r < 22 . karKNnaemKuNbEnßm magnification factored δ s sMrab;eRKageyal ¬Kμan
BRgwg¦ RtUv)ansegçbdUcxageRkam (ACI Code, Section 10.13)³
1> kMNt;faeRKagCaeRKagKμanBRgwgTb;nwg sidesway b¤Gt; rYckMNt;RbEvgKμanTMr lu nigemKuN
RbEvgRbsiT§PaB K EdlGacTTYlBIsmIkar (-4) (-5) nig (-6) b¤düaRkamrUbTI3.
2-4> KNna EI / Pc nig Cm dUceGaykñúgsmIkar (-12) dl; (-16). cMNaMfa βd ¬edIm,IKNna EI ¦KW
CapleFobrvagkMlaMgkat;TTwgefremKuNGtibrma maximum factored sustained shear
tamCan; nigkMlaMgkat;TTwgemKuNsrubenAkñúgCan;enaH.
5> KNnaemKuNm:Um:g;bEnßm moment-magnifier factor/ δ s ³
1
δs = ≥ 1.0 (-19)
1 − (∑ Pu / 0.75∑ Pc )
Edl δ s ≤ 2.5 nig ∑ Pu CaplbUkbnÞúkbBaÄrTaMgGs;enAkñúgmYyCan; nig ∑ Pc CaplbUk
bnÞúksMrab;ssrEdlTb;nwgkareyal sway enAkñúgmYyCan;. dUcKña³
M 2s
δsM s = ≥ Ms (-20)
1 − (∑ Pu / 0.75∑ Pc )
Edl M s Cam:Um:g;emKuNxagcugbNþalmkBIbnÞúkEdlbegáItkareyalEdlTTYlyk)an.
6> KNnam:Um:g;cugbEnßm M1 nig M 2 enAxagcugGgát;rgkarsgát;EtÉg dUcxageRkam³
M1 = M1ns + δ s M1s (-21)
M 2 = M 2ns + δ s M 2 s (-22)
Edl M1ns nig M 2ns Cam:Um:g;EdlTTYlBIlkçxNÐGt;eyal b:uEnþ M1s nig M 2s Cam:Um:g;Edl
TTYl)anBIlkçxNÐeyal. RbsinebI M 2 > M1 BIkarviPaKeRKag enaHkarKNnam:Um:g;bEnßmKW³
M c = M 2ns + δ s M 2 s (-23)
m:Um:g;cug M1 nig M 2 enAkñúgsmIkar (-21) (-22) nig (-23) manm:Um:g;Gt;eyal bUknigm:Um:g;
eyalbEnßm CamYynwglkçxNÐEdl
lu 35
< (-24)
r Pu / f 'c Ag
enAkñúgkrNIenH Ggát;rgkarsgát;KYrEtRtUv)anKNnasMrab;bnÞúkemKuNtamGkS½ Pu nig M c .
b:uEnþkñúgkrNIEdl
lu 35
> (-25)
r Pu / f 'c Ag
Ggát;rgkarsgát;KYrEtRtUv)anKNnasMrab; Pu nigm:Um:g;Gt;eyalbEnßm δ ns M 2 bUkCamYy
nwgm:Um:g;eyalbEnßm δ s M 2 CamYynwgm:Um:g;KNna M c = δ ns M 2ns + δ s M 2s . krNIenHGac
ssrEvg 265

12. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
ekItmansMrab;ssrEvg slender column CamYynwgbnÞúktamGkS½FM enAeBlEdlm:Um:g;Gtibrma
ekItmanenAcenøaHcugssr nigminenAxagcug.
bTdæan ACI Code, section 10.13.4 GnuBaØatinUvviFIepSgeTotsMrab;karKNna δ s M s én
smIkar (-20) edayeRbIsnÞsSn_esßrPaB stability index Q EdleGaykñúgsmIkar (-11) Edl
δ s ≤ 1.5 ³
Ms
δsM s = ≥ Ms (-26)
1− Q
]TahrN_ 2³ muxkat;ssrdUcbgðajkñúgrUbTI5 RTbnÞúktamGkS½ P D nigm:Um:g; M D = 157kN .m
= 605kN
EdlbNþalmkBIbnÞúkefr nigbnÞúktamGk½S PL = 490kN nigm:Um:g; M L = 126kN .m EdlbNþalmkBIbnÞúk
cl½t. ssrCaEpñkrbs;eRKagBRgwg nigmankMeNageTaltamGkS½em. RbEvgKμanTMrrbs;ssrKW lc = 5.8m
ehIym:Um:g;enAcugTaMgsgçagrbs;ssrmantMélwsμIKña. epÞógpÞat;muxkat;ssredayeRbI f 'c = 28MPa nig
f y = 400MPa .
dMeNaHRsay³
1> KNnabnÞúkcugeRkay ultimate load³
Pu = 1.2 PD + 1.6 PL = 1.2 × 605 + 1.6 × 490 = 1510kN
M u = 1.2M D + 1.6M L = 1.2 × 157 + 1.6 × 126 = 390kN .m
M 390
e= u = = 258.3mm
Pu 1510
2> RtYtBinitüemIlfaetIssrEvgb¤xøI. edaysareRKagRtUv)anBRgwg snμt; K = 1.0
r = 0.3h = 0.3 × 550 = 165mm nig lu = 5.8m
Klu 5800
= = 35.15
r 165
sMrab;ssrBRgwg RbsinebI Klu / r ≤ 34 − 12M1 / M 2 T§iBlénPaBrlas;GacRtUv)anecal. eday
sarssrekagedaykMeNageTal enaH M1 / M 2 viC¢man. dUcenH
Slender Column 266

13. T.Chhay NPIC
M1
34 − 12 = 34 − 12 = 22
M2
edaysar Klu / r = 35.15 > 22 enaHT§iBlénPaBrlas;RtUv)anBicarNa.
3> KNna EI BIsmIkar (-12)³
A. KNna E c
Ec = 4780 f 'c = 4780 28 = 25293.4MPa
Es = 2.1 ⋅105 MPa
B. m:Um:g;niclPaBKW
350(550) 4π 282
3
Ig = = 4852.6 ⋅ 106 mm 4 As = A's = = 2463mm 2
12 4
⎛ 550 − 120 ⎞
2
I se = 2 × 2463⎜ ⎟ = 227.7 ⋅10 mm
6 4
⎝ 2 ⎠
pleFobm:Um:g;GefrKW
1.2 × 605
βd = = 0.48
1510
C. PaBrwgRkajKW
0.2 Ec I g + Es I se
EI =
1 + βd
0.2 × 25293.4 × 4852.6 ⋅106 + 2.1 ⋅105 × 227.7 ⋅106
= = 48.9 ⋅1012 N .mm 2
1 + 0.48
4> KNna P c
π EI
2
π 2 48.9 ⋅1012
Pc = = = 14346.72kN
(Klu )2 (5800) 2
5> KNna C BIsmIkar (-16)³
m
M1
Cm = 0.6 + 0.4 ≥ 0.4
M2
= 0.6 + 0.4(1) = 1.0
6> KNnaemKuNm:Um:g;bEnßmBIsmIkar (-17)³
Cm 1
δ ns = = = 1.16
1 − ( Pu / 0.75Pc ) 1 − [1510 /(0.75 × 14346.72]
7> KNnam:Um:g;KNna design moment nigbnÞúkKNna design load edaysnμt; φ = 0.65
1510
Pn = = 2323kN
0.65
390
Mn = = 600kN .m
0.65
KNna M c = 1.16 × 600 = 696kN .m
ssrEvg 267

14. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
KNnacMNakp©it e=
696
2323
= 300mm
8> kMNt;ersIusþg; nominal load strength énmuxkat;edayeRbI e = 300mm edayeRbIsmIkar (-4) emeron
eRKOgbgÁúMrgkarsgát; nigkarBt;³
Pn = 8.33a + 926.58 − 2.46 f s (I)
h 550
e' = e + d − = 300 + 490 − = 515mm
2 2
1 ⎡ ⎛ a⎞ ⎤
Pn = ⎢8.33a⎜ 490 − 2 ⎟ + 926.58(490 − 60 )⎥
515 ⎣ ⎝ ⎠ ⎦
−3 2
= 7.93a − 8.1 ⋅ 10 a + 773.65 (II)
BIsmIkar (I) nig (II) eyIgTTYl)an a = 267mm / f = 338MPa nig P = 2319.2kN .
s n
edaysarEtersIusþg;bnÞúk load strength P = 2319.2kN nigbnÞúktMrUvkar required load
n
P = 2323kN mantMélRbhak;RbEhlKña enaHmuxkat;RtUv)ancat;TukfaRKb;RKan;. RbsinebI
n
muxkat;minRKb;RKan; RtUvtMeLIgmuxkat;Edk.
9> epÞógpÞat;tMélsnμt; φ
a = 267 mm c = 314.12mm d t = 490mm
⎛ dt − c ⎞
εt = ⎜ ⎟0.003 = 0.00168 < 0.002
⎝ c ⎠
dUcenH φ = 0.65
]TahrN_ 3³ epÞógpÞat;PaBRKb;RKan;rbs;ssrkñúg]TahrN_TI2 RbsinebIRbEvgKμanTMr unsupported length
l = 3m . kMNt;bnÞúk nominal load GtibrmaenAelIssr.
u
dMeNaHRsay³
1> bnÞúkEdlGnuvtþKW P = 2323kN nig M = 600kN .m
n n
2> epÞógpÞat;PaBEvgxøIrbs;ssr³ l = 3m / r = 0.3 × 550 = 165mm nig K = 1.0 ¬eRKagRtUv)anBRgwg
u
Tb;nwgkareyalxag sidesway ¦.
Klu 3000
= = 18.2
r 165
epÞógpÞat; Kl u / r = 34 − 12M 1b / M 2b
34 − 12(1) = 22
eday Klu
r
= 18.2 < 22
enaH T§iBlénPaBrlas;Gacecal)an.
Slender Column 268

15. T.Chhay NPIC
3> kMnt;lT§PaBRTbnÞúk nominal load ebs;ssrxøI dUcBnül;enAkñúg]TahrN_TI4 én emeroneRKOgbgÁúM
rgkarsgát; nigkarBt;;. eyIgTTYl)an Pn = 2574.9kN ¬sMrab; e = 258.3mm ¦
EdlFMCagbnÞúkcaM)ac; Pn = 2323kN .
]TahrN_ 4³ epÞógpÞat;PaBRKb;RKan;rbs;ssrkñúg]TahrN_TI2 RbsinebIeRKagminRtUv)anBRgwgTb;nwgeyal
xag sidesway emKuNbgáb;cug end-restraint factor KW ψ A = 0.8 nig ψ B = 2 ehIyRbEvgKμanTMr
unsupported length KW lu = 4850mm .
dMeNaHRsay³
1> kMNt;tMél K BIdüaRkam alignment chart rUbTI3 sMrab;eRKagminBRgwg. P¢ab;tMél ψ A = 0.8
nig ψ B = 2 kat;ExS K Rtg; K = 1.4 .
Klu 1.4 × 4850
= = 41.15
r 165
2> sMrab;eRKagKμanBRgwg RbsinebI Klu / r ≤ 22 ssrGacRtUv)anKNnadUcssrxøI. edaysarEttM
él Klu / r = 41.15 > 22 eKRtUvEtKitBIT§iBlénPaBrlas;.
3> KNnaemKuNm:Um:g;bEnßm δ ns eKeGay Cm = 1 / K = 1.4 / EI = 48.9 ⋅1012 N .m2
¬BI]TahrN_TI2¦ nig
π 2 EI π 2 × 48.9 ⋅ 1012
Pc = = = 10468.1kN
(Klu )2 (1.4 × 4850)2
Cm 1
δ ns = = = 1.24
⎛ Pu ⎞ ⎛ 1510 ⎞
1− ⎜ 1− ⎜
⎜ 0.75 × P ⎟
⎟ ⎟
⎝ 0.75 × 10468.1 ⎠
⎝ c⎠
4> BI]TahrN_TI2 Pu = 1510kN nig M u = 390kN .m b¤ Pn = 2323kN nig M n = 600kN .m
m:Um:g;KNna M c = 1.24 × 600 = 744kN.m dUcenH
δ ns M n 744
e= = = 320.3mm
Pn 2323
5> epÞógpÞat;PaBRKb;RKan;rbs;ssrxøIsMrab; Pn = 2323kN / M c = 744kN .m nig e = 320.3mm .
viFIsaRsþkñúgkaredaHRsayRtUv)anBnül;kñúg]TahrN_TI4 én emeroneRKOgbgÁúMrgkarsgát; nigkar
Bt;.
6> BI]TahrN_TI4 én emeroneRKOgbgÁúMrgkarsgát; nigkarBt; eyIg)an
Pn = 8.33a + 926.58 − 2.46 f s
h 550
e' = e + d − = 320.3 + 490 − = 535.3mm
2 2
1 ⎡ ⎛ a⎞ ⎤
Pn = ⎢8.33a⎜ 490 − 2 ⎟ + 926.58(490 − 60)⎥
535.3 ⎣ ⎝ ⎠ ⎦
ssrEvg 269

16. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
= 7.625a − 0.00778a 2 + 744.31
eyIgTTYl)an a = 259.86mm
dUcenH c = 305.7mm nig Pn = 2200kN . lT§PaBRTbnÞúkrbs;ssr Pn = 2200kN tUc
CagbnÞúkEdlRtUvRT Pn = 2323kN . dUcenHmuxkat;minRKb;RKan;.
7> begáInmuxkat;EdkBI 4DB28 eTA 4DB30 ehIyeFVIkarKNnaepÞógpÞat;eLIgvij enaHeyIg
TTYl)an Pn = 2335kN / ε t < 0.002 nig φ = 0.65 .
]TahrN_ 5³ KNnassrkaer:xagkñúgsMrab;Can;TImYyénGKarkariyal½y8Can;. kMBs; clear height énCan;
TImYyKW 4.9m nigkMBs;sMrab;Can;d¾éTeTotKW 3.4m . GKarenHman 24RbGb; ¬rUbTI6¦ ehIyssrminRtUv)an
BRgwgTb;nwgkareyalxag sidesway. bnÞúkEdlGnuvtþmkelIssrxagkñúgCan;TImYy bNþalmkBITMnajEpndI
nigxül;dUcxageRkam³
bnÞúkefrtamGkS½ = 1690kN
bnÞúkGefertamGkS½ = 623kN
bnÞúkxül;tamGkS½ = 0kN
m:Um:g;bnÞúkefr = 43.4kN.m ¬xagelI¦ 73.2kN.m ¬xageRkam¦
m:Um:g;bnÞúkGefr = 27.1kN.m ¬xagelI¦ 48.8kN.m ¬xageRkam¦
m:Um:g;bnÞúkxül; = 67.8kN.m ¬xagelI¦ 67.8kN.m ¬xageRkam¦
EI / l sMrab;Fñwm = 40 ⋅ 106 kN.mm
eRbI f 'c = 35MPa / f y = 400MPa nigtMrUvkarrbs;bTdæan ACI Code. snμt;fa bnÞúkEdlmanGMeBI
elIssrxageRkAesμI 2 / 3 énssrxagkñúg ehIybnÞúkEdlmanGMeBIelIssrRtg;RCugesμI 1 / 3 énssrxagkñúg
ehIy β d = 0.55 .
dMeNaHRsay³
1> KNnabnÞúkemKuNedayeRbIkarpSMbnÞúk. sMrab;bnÞúkTMnajEpndI³
Slender Column 270

17. T.Chhay NPIC
Pu = 1.2 D + 1.6 L = 1.2 × 1690 + 1.6 × 623 = 3024.8kN
M u = M 2ns = 1.2M D + 1.6M L = 1.2 × 73.2 + 1.6 × 48.8 = 165.92kN .m
sMrab;bnÞúkTMnajEpndI nigbnÞúkxül;
Pu = (1.2 D + 0.5L + 1.6W ) = 1.2 × 1690 + 0.5 × 623 + 0 = 2339.5kN
M uns = M 2 ns = 1.2M D + 1.6M L = 1.2 × 73.2 + 1.6 × 48.8 = 165.92kN .m
M us = M 2 s = 1.6M w = 1.6 × 67.8 = 108.48kN .m
bnSMbnÞúkepSgeTotEdlminsMxan;
Pu = 0.9 D + 1.6W = 0.9 × 1690 + 1.6 × 0 = 1521kN
M 2 = 0.9 M D = 1.2 × 73.2 = 87.84kN .m
M 2 s = 1.6M w = 1.6 × 67.8 = 108.48kN .m
M M 165.92
e = u = 2ns = = 54.85mm
Pu Pu 3024.8
emin = (15.24 + 0.03h) = 15.24 + 0.03 × 460 = 29.04mm < 54.85mm
2> eRCIserIsmuxkat;dMbUgrbs;ssredayEp¥kelIbnSMbnÞúkTMnajEpndIedayeRbItaragb¤düaRkam.
eRCIserIsmuxkat;ssr 460 × 460 CamYynwgEdk DB32 cMnYn4edIm ¬rUbTI7¦.
3> epÞógpÞat; Klu / r
460 4
Ig = = 37.3 ⋅ 108 mm 4 Ec = 28278.9MPa
12
sMrab;ssr I = 0.7 I g
sMrab;ssrEdlmankMBs; 4.9m
EI 0.7 × 37.3 ⋅ 108 × 28278.9
= = 15.1 ⋅ 109 N .mm
lc 4900
sMrab;ssrEdlmankMBs; 3.4m
EI 0.7 × 37.3 ⋅ 108 × 28278.9
= = 21.7 ⋅ 109 N .mm
lc 3400
ssrEvg 271

18. Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa
sMrab;Fñwm EI g / lb = 40 ⋅109 N .mm / I = 0.35I g nig
EI / lb = 0.35 × 40 ⋅ 109 = 14 ⋅ 109 N .mm
ψ (top ) = ψ (bottom) = ∑ (EI / lc ) = (15.1 + 21.7) ⋅ 109 = 1.3
∑ (EI / lb ) 2 × 14 ⋅ 109
BItarag alignment chart K = 1.4 sMrab;eRKagKμanBRgwg nig K = 0.8 sMrab;eRKag BRgwg.
Klu 1.4 × 4900
= = 49.7
r 0.3 × 460
EdlFMCag 22 nigtUcCag 100 . dUcenH eKRtUvBicarNaBI slenderness ratio.
4> epÞógpÞat; lu / r = 4900 /(0.3 × 460) = 35.5
35 35
= = 54.8 (-24)
Pu / f 'c Ag 3024800 /(35 × 460 × 460)
edaysarEt lu / r < 54.8 m:Um:g; nonsway moment mincaM)ac;bEnßm.
5> KNna Pc
Ec = 28278.9MPa Es = 2.1 ⋅ 105 MPa
2
460 4 4π 32 2 ⎛ 340 ⎞
Ig = = 37.3 ⋅ 108 mm 4 I se = ⎜ ⎟ = 93 ⋅ 10 mm
6 4
12 4 ⎝ 2 ⎠
β d = 0.55
0.2 Ec I g + Es I se
EI =
1 + βd
0.2 × 28278.9 × 37.3 ⋅ 108 + 2.1 ⋅ 105 × 93 ⋅ 106
EI = = 26.2 ⋅ 1012 N .mm 2
1 + 0.55
edIm,IKNna δ s / β d = 0 enaH EI = 1.55 × 26.2 ⋅1012 = 40.63N .mm2
π 2 EI π 2 × 26.2 ⋅ 1012
Pc =
( Kl ) 2
=
(0.8 × 4900) 2
= 16827.86kN ¬BRgwg¦
u
π EI
2
π 2 × 40.63 ⋅ 1012
Pc =
( Klu ) 2
=
(1.4 × 4900) 2
= 8521.15kN ¬KμanBRgwg¦
sMrab;mYyCan;enAkñúgGKar eKmanssrxagkñúg 14 ssrxageRkA 18 nigssrkac;RCug 4 .
2 1
∑ Pu = 14(2339.5) + 18( × 2339.5) + 4( × 2339.5) = 63946.3kN
3 3
2
∑ Pc = 14(8521.15) + 22( × 8521.15) = 244273kN
3
1
δs = = 1.54
63946.3
1− ( )
0.75 × 244273
EdlFMCag 1 nigtUcCag 2.5 smIkar (-19)
M c = M 2ns + δ s M 2 s = 165.92 + 1.54 × 108.48 = 333kN .m
Slender Column 272

19. T.Chhay NPIC
6> bnÞúkKNnaKW Pu = 2339.5kN nig M c = 333kN .m
333
e= = 142.34mm
2339.5
emin = (15.24 + 0.03h) = 15.24 + 0.03 × 460 = 29.04mm < e
tamkarviPaK sMrab; e = 142.34mm nig A = 1608.5mm2 ¬ φ = 0.65 ¦ lT§PaBRTbnÞúkrbs;
ssrmuxkat; 460 × 460 KW φPn = 2348.1kN nig φM n = 334.2kN .m dUcenHmuxkat;KWRKb;
RKan;. ¬dMeNaHRsaymanlkçN³RsedogKñaeTAnwg]TahrN_TI4 kñúg emeroneRKOgbgÁúMrgkar
sgát; nigkarBt;. tMél a = 242.86mm / c = 303.57mm / f s = 190.6MPa /
f 's = 400MPa / φPb = 1676.8kN nig eb = 218mm ¦.
400 − 303.57
ε t = 0.003 = 0.00095 < 0.002 / φ = 0.65
303.57
ssrEvg 273