1. T.Chhay
ssr
Columns
1> esckþIepþIm Introduction
eRKOgbgÁúMénrcnasm<n§½ nigeRKOgm:asIunEdlrgnUvkMlaMgsgát;tamGkS½RtUv)aneKeGayeQμaHfassr
RbsinebIkMBs;rbs;vaFMeRcInCagTMhMmuxkat;TTwg. ssrTaMgLayNaEdlrgbnÞúkRtYtsIuKñaCamYyGkS½beNþay
rbs;ssr RtUv)aneKeGayeQμaHfassrcMp©it axial loaded column EtkrNIenHkMrmanNas; dUcenHssr
cMp©it CassrEdlrgnUv bnÞúktamGkS½EdlmanGMeBIelItMbn;sñÚlrbs;ssrEdlmanragCactuekaNesμI. ssr
TaMgLayNaEdlrgbnÞúktamGkS½EdlmanGMeBIeRkAtMbn;sñÚlssrrbs;ssr RtUv)aneKeGayeQμaHfassr
cakp©it eccentricity loaded column. kñúgkarsikSaBIssr eKEckssrCaBIrRbePTKW³ ssrxøI nigssrEvg.
2> ssrxøIrgbnÞúkcMpM©it Axially loaded short compression member
kalNassrxøIrgbnÞúkcMp©it enaHssrnwgEbk.
kugRtaMgrbs;ssrxøI s = PA
Edl P - bnÞúkcMcMnuc
A - muxkat;rgsMBaF
s - kugRtaMgrgkMlaMgsgát;
3> ssrxøIrgbnÞúkcakpM©itmYyTis Eccentricity load on centroidal axis
RbsinebI P CabnÞúkEdlmanGMeBIenAelIGkS½mYyénTIRbCMuTMgn; enaHcMNakp©it e CacMgayBITItaMgbnÞúk
P mkTItaMgTIRbCMuTMgn;enaH.
P centroidal axis P P
e e
eccentric loads additional loads
O
kugRtaMgrgkarsgát; )anBIbnÞúkcMcMnuc P
X centroid
Y
P
P
sc = − M
c c
N
A M N b
d
kugRtaMgrgkarsgát; )anBIkMlaMgbgVil Pe
d
s=-P
c A
Pec
sb = −
I s=+Pec
b I
kugRtaMgrgkarTaj )anBIkMlaMgbgVil Pe s=- Pec
b I
s=-P +Pec
A I
Pec
sb = + s=-P -Pec
A I
I
dUcenHkugRtaMgsrubEdlekItBIkMlaMgcakp©it KWCaplbUkBiCKNiténkugRtaMg
EdlekItBIkMlaMgcMp©it nigkugRtaMgEdlekItBIkMlaMgbgVil
P Pec
s=− ±
A I
ssr 93

2. T.Chhay
cMNakp©itGtibrma edIm,IeGaykugRtaMgsrubrgkarTajesμIsUnü
P Pec
0=− +
A I
eday c=
d
2
/
I = IY =
bd 3
12
¬bnÞúkxageRkA eFVIGMeBIcakp©iteFobGkS½ Y ¦
ed12 1
⇒ =
2bd 3 bd
⇒e=
d
6
sMrab;krNI bnÞúkxageRkAeFVIGMeBIcakp©iteFobGkS½ Y
sMrab;krNI bnÞúkxageRkAeFVIGMeBIcakp©iteFobGkS½ X cMNakp©itGtibrmaEdleFVIeGaykugRtaMgrgkarTajesμI
sUnü KW e = b
6
kñúgkrNI e > d b¤ e > b enaHsésEpñkxageRkAmçagrbs;muxkat;rgkarTaj nigmçageTotrgkarsgát;.
2 2
]TahrN_³ ssrxøIeFVIGMBIeQIdUcbgðajkñúgrUbxagelI rgnUvkMlaMgsgát;cakp©it P = 40kN EdlGnuvtþenAcMgay
2cm BIGkS½ Y . ssrenHmanmuxkat; b = 10cm nig d = 20cm . kMNt;kugRtaMgsrubEdlmanGMeBIenAsésEpñk
xageRkAbMputrbs;muxkat;.
dMeNaHRsay³
kugRtaMgrgkMlaMgsgát; )anBIbnÞúk P
P
sc = −
A
Edl P = 40kN
nig A = b.d = 10 × 20 = 200cm 2
40kN
⇒ sc = − = −2MPa
200cm 2
kMlaMgbgVil
M = Pe
Edl e = 2cm = 0.02m
⇒ M = 40 × 0.02 = 0.8kN .m
kugRtaMgEdlekItBIkMlaMgbgVil M
Mc
sb =
I
eday d 20
c=
2
=
2
= 10cm = 0.1m
nig I = IY =
bd 3 10 × 203
12
=
12
= 6666.67cm 4 = 6.67 ×10 −5 m 4
0.8 × 0.1
⇒ sb = = ±1.12 MPa
6.67 × 10 −5
kugRtaMgsrubsMrab;sésEpñkxageRkA muxkat; M
ssr 94

3. T.Chhay
stc = −2 − 1.12 = −3.12MPa
kugRtaMgsrubsMrab;sésEpñkxageRkA muxkat; N
stt = −2 + 1.12 = −0.8MPa
4> ssrxøIrgbnÞúkcakpM©itBIrTis Eccentricity load not on centroidal axis
ssrxøIrgbnÞúkp©itBIrTis mancMNakp©itBIrKW e cMNakp©iteFobGkS½ Y nig e cMNakp©iteFobGkS½ X .
1 2
dUcenHkugRtaMgsrubEdlekItBIkMlaMgcakp©itBIrTis KWCaplbUkBiCKNitén³
- kugRtaMgEdlekItBIkMlaMg P EdleFVIGMeBIcMTIRbCMuTMgn;énmuxkat; P
e1 centroidal axis
- kugRtaMgEdlekItBIkMlaMgbgVil Pe EdleFVIGMeBIBt;eFobGkS½ Y
1
X
e2
Y
O
centroid
- kugRtaMgEdlekItBIkMlaMgbgVil Pe EdleFVIGMeBIBt;eFobGkS½ X
2
A
C
B c2
D
c2
P Pe1c1 Pe2c2 c1 c1 b
s=− ± ± d
A IY IX
edIm,IeGay muxkat;rgkugRtaMgTajesμIsUnüluHRtaEt e = d eFobGkS½ Y nig e = b eFobGkS½ X .
6
1
6
2
EtedaysarvargbnÞúkcakp©itBIrTisenaH kugRtaMgTajesμIsUnüenAeBlNaEdl kern d
6
bnÞúkcakp©itmanGMeBIenAelIcMnuc EdlP¢ab;BIcMnuc d → b .
6 6
b
6 X
RkLaépÞEdlpÁúM)anmanragFrNImaRtCactuekaNesμI eKeGayeQμaHfa sñÚl Y
(kern). dUcenH enAeBlNaEdlbnÞúkmanGMeBIeRkAépÞenH enaHmuxkat;enaH
minmanrgkugRtaMgTajeT rgEtkugRtaMgsgát;suT§ RKan;EtmantMélxusKña. d
b
]TahrN_³
ssrxøIrgbnÞúksgát; P = 50kN cakp©itBIrTisdUcbgðajkñúgrUbxagelI Edlman e = 6cm nig e = 2cm .
1 2
ssrmanmuxkat; 15× 25cm . kMNt;kugRtaMgsrubenARCugTaMgbYnrbs;va.
dMeNaHRsay³
kugRtaMgsrub sMrab;ssrxøIrgGMeBIbnÞúkcakp©itBIrTiskMNt;tam
P Pe1c1 Pe2 c2
s=− ± ±
A IY IX
m:Um:g;niclPaBeFobGkS½ X
db3 0.25 × 0.153
IX = = = 7.03 ×10−5 m 4
12 12
m:Um:g;niclPaBeFobGkS½ Y
bd 3 0.15 × 0.253
IY = = = 19.5 × 10 −5 m 4
12 12
kugRtaMrgkMlaMgsgát;
P 50
= = 1.33MPa
A 0.25 × 0.15
ssr 95

4. T.Chhay
kugRtaMrgkMlaMgbgVileFobGkS½ Y
Pe1c1 50 × 0.06 × 0.125
= = 1.92 MPa
IY 19.5 ×10 −5
kugRtaMrgkMlaMgbgVileFobGkS½ X
Pe2 c2 50 × 0.02 × 0.075
= = 1.07 MPa
IX 7.03 × 10 −5
kugRtaMgsrubRtg;cMnuc A
s = −1.33 − 1.92 + 1.07 = −2.18MPa rgkarsgát;
kugRtaMgsrubRtg;cMnuc B
s = −1.33 − 1.92 − 1.07 = −4.32 MPa rgkarsgát;
kugRtaMgsrubRtg;cMnuc C
s = −1.33 + 1.92 − 1.07 = −0.48MPa rgkarsgát;
kugRtaMgsrubRtg;cMnuc D
s = −1.33 + 1.92 + 1.07 = +1.66MPa rgkarTaj
5> ssrEvgrgbnÞúkcMpM©it Axially loaded slender compression member
enAeBlEdlssrEvgrgbnÞúkcMp©itkan;EtekIneLIg² enaHssrEvgenaHminEbkeT Etva)ak;eTAvijeday
sarEtRbEvgrbs;vaEvg )ann½yfabnÞúkEdlssrEvgGacRT)anGaRs½yeTAnwgRbEvg muxkat; nigTMr. GñkR)aCJ
lIGUNat GWeL (Leonhard Euler 1707-1783) )anrkeXIjnUvbnÞúkRKITicsMrab;ssrEvgEdlmanTMrsamBaØ
sgxagEdlcab;epþImekag (buckling).
π 2 EI
Pe =
L2
Edl - bnÞúkRKITic b¤bnÞúkcMp©itEdleFVIeGaymanPaBekag (kN )
Pe
E - m:UDuleGLasÞicrbs;sMPar³ ( MPa)
I - m:Um:g;niclPaBénmuxkat;EdlmantMéltUcCageK (m ) 4
L - RbEvgrbs;ssr (m)
kugRtaMgRKItic EdleFVIeGayssrekItmanPaBekag
Pe π 2E
se = =
A (L / r)2
Edl r=
I
A
kaMniclPaB (m)
L
r
- pleFobPaBrlas;rbs;ssr (slenderness ratio)
]TahrN_³
ssr 96

5. T.Chhay
ssrEvgEdlmanTMrsamBaØsgxagrgbnÞúksgát;cMp©it. ssrenHeFVIBIsMPar³Edlmanm:UDuleGLasÞic
E = 200000MPa nigmanersIusþg; s = 235MPa . muxkat;rbs;ssrenHmanragCargVg;EdlmanGgát;p©it
y
d = 15cm . kMNt;bnÞúkGtibrmaEdlssrenHGacRT)anedayKñaPaB ekagkñúgkrNI³
- ssrmanRbEvg 2m
- ssrmanRbEvg 4m
dMeNaHRsay³
edayssrmanmuxkat;ragrgVg;EdlmanGgát;p©it d = 15cm
πd 2
⇒ A= = 0.0177m 2
4
d
⇒r= = 0.0375m
4
- sMrab;ssrmanRbEvg 2m
π 2E 3.14 2 × 200000
⇒ se = 2
= = 693.25MPa > 235MPa
(L / r) ( 2 / 0.0375) 2
bnÞúkGtibrmaEdlssrGacRT)anKW
P = s y . A = 235 × 0.0177 = 4.16MN = 4160kN
- sMrab;ssrmanRbEvg 4m
π 2E 3.14 2 × 200000
⇒ se = = = 173.31MPa < 235MPa
(L / r)2 (4 / 0.0375) 2
bnÞúkGtibrmaEdlssrGacRT)anKW
P = se . A = 173.31× 0.0177 = 3.07 MN = 3070kN
6> RbEvgRbsiT§PaB Effective length
P P
P P P
Effective length
L=actual Effective length (KL)=0.7 Effective length Effective length
column length (KL)=1 (KL)=0.5 (KL)=2
P P
P P
(a) Pinned (b) Pinned/fixed (c) Fixed (c) Fixed/free
ssr 97

6. T.Chhay
xagelICarUbbgðajBIRbEvgRbsiT§PaBrbs;ssr. RbEvgRbsiT§PaB CaRbEvgenAcenøaHcMnucbegáag
(contraflexure) ¬cMnucEdlmanm:Um:g;Bt;esμIsUnü¦ énrUbragdabrbs;ssr. lkçxNÐénTMrcugssrCHT§iBl
y:agxøaMgeTAelI RbEvgeFVIkarrbs;ssr. RbEvgRbsiT§PaBrbs;ssrERbRbYleTAtamRbePTénTMr.
K - CaemKuNRbEvgRbsiT§PaBrbs;ssr
dUcenHbnÞúkRKITicEdlssrGacRT)an tamrUbmnþ Euler sMrab;RKb;lkçxNÐTMrKW
π 2 EI
Pe =
KL2
dUcKña kugRtaMgrbs;ssrEvgKW
Pe π 2E
se = =
A ( KL / r ) 2
ssr 98