10.column6
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1. This document discusses short compression members (columns) subjected to axial loads and eccentric loads. It provides equations to calculate the stresses in columns based on the applied load and member properties. 2. For a short column with an axial load, the stress is calculated as the load divided by the cross-sectional area. 3. For a column with an eccentric load applied through the centroidal axis, additional equations are provided to calculate the stresses due to the eccentric moment. 4. For a column with an eccentric load not applied through the centroidal axis, more complex equations are needed to calculate the stresses due to multiple eccentric moments.
![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](https://image.slidesharecdn.com/10-column6-100713081237-phpapp01/85/10-column6-2-320.jpg)
![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](https://image.slidesharecdn.com/10-column6-100713081237-phpapp01/85/10-column6-3-320.jpg)
![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](https://image.slidesharecdn.com/10-column6-100713081237-phpapp01/85/10-column6-4-320.jpg)
10.column6
- 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