1. T.Chhay
kugRtaMgpÁÜb
Combined Stress
1> esckþIepþIm Introduction
enAeBlkugRtaMgRsb nigCaRbePTEtmYymanGMeBIRtg;cMnucmYy eKGaceFVIplbUkBiCKNiténkugRtaMg
TaMgenaH)an. ]TahrN_ eKGaceFVIplbUkrvagkugRtaMgEkgTaj nigkugRtaMgEkgsgát;EdlekItBIbnÞúkm:Um:g;Bt;
nigbnÞúktamG½kS)an enAeBlEdlBYkvamanTisRsbKña. enAeBlkugRtaMgminmanTisdUcKña b¤CaRbePTepSgKña
¬kugRtaMgEkg nigkugRtaMgkat;¦ eyIgRtUvbUkvatamlkçN³viucT½r.
enAeBlEdlEpñkrbs;eRKOgma:sIun nigeRKOgbgÁúMRtUv)anKNnaRKb;RKan;edayBicarNaEtkugRtaMgdac;
edayELk enaHvanwgmansßanPaBEdlkugRtaMgpÁÜbGacbegáItCalkçxNÐeRKaHfñak;EdleKKYrEteFVIkarGegát.
vaCakrNIBiess enAeBlEdleKsikSaBIkugRtaMgenAeBl)ak;Cak;Esþg.
2> kugRtaMgpÁÜbénkugRtaMgBt; nigkugRtaMgtamG½kS Combined Axial and Bending Stresses
eRKOgbgÁúMCaeRcInrgnUvbnÞúktamTisTTwg nigbnÞúktamG½kSkñúgeBldMNalKña EdlbegáIt)anCabnSMén
kugRtaMgBt; nigkugRtaMgtamG½kS. rUbTI 17>1 a bgðajBIFñwmTMrsamBaØEdlrgbnÞúkBRgayesμITTwgG½kS w rYm
CamYynwgbnÞúktamG½kS P . bnÞúkTTwgG½kSbegáItm:Um:g;Bt; ehIykarBRgaykugRtaMgelImuxkat;FñwmmanragRtI-
kugRtaMgpÁÜb 1

2. NPIC
ekaN dUcbgðajkñúgrUbTI 17>1 c. eyIgnwgBicarNamuxkat;Rtg;kNþalElVg ¬m:Um:g;Gtibrma¦ ehIyeyIg
kMNt;vaCabøg; X-X. kugRtaMgBt;ERbRbYlBItMélGtibrmaenAsrésxageRkAeTAtMélsUnüenAG½kSNWt. cMNaM
fa edaysarFñwmenAkñúgrUbTI 19>1 a CaFñwmTMrsamBaØ nigrgbnÞúkTTwgG½kSEdlmanTisedAcuHeRkam enaHmux
kat;nwgrgkugRtaMgTajenAEpñkxageRkamG½kSNWt nigrgkugRtaMgsgát;enAEpñkxagelIrbs;G½kSNWt. eKGac
TTYl)anGaMgtg;sIuetrbs;kugRtaMgBt;GtibrmaedayeRbI flexural formula
Mc
sb =
I
edaysarEtFñwmrgbnÞúktamG½kSkñúgeBlCamYyKña ¬bnÞúkenHGacsgát; b¤Taj¦ enaHkugRtaMgtamG½kS
nwgekItmanenAelImuxkat;rbs;FñwmedayBRgayesμI dUcbgðajkñúgrUbTI 19>1 b. eKGacTTYl)anGaMgtg;sIuet
rbs;kugRtaMgtamG½kSBIsmIkarxageRkam
P
st =
A
cMNaMfam:Um:g;Bt;begáIt)anCakugRtaMgTaj nigkugRtaMgsgát;EdlEkgeTAnwgbøg;rbs;muxkat; ¬bøg;X-X¦.
dUcKña bnÞúkTajtamG½kSbegáIt)anCakugRtaMgTajEkgeTAnwgbøg;. edaysarvaCaRbePTkugRtaMgdUcKñaeKGac
eFVIkarbUktamlkçN³BiCKNit dUcbgðajkñúgrUbTI 19>1 d. dMeNIkarbUktamlkçN³BiCKNiténkugRtaMg
RbePTdUcKñaedIm,ITTYl)ankugRtaMgpÁÜbRtUv)anehAfa superposition. eKGnuBaØateGayeFVI superposition
eTA)anEtkñúgkrNIkugRtaMgGtibrmasßitenAkñúgEdneGLasÞic ehIyPaBdabmantMéltUc.
kugRtaMgpÁÜbénkugRtaMgBt; nigkugRtaMgtamG½kSKW
P Mc
s=± ± (17-1)
A I
CaTUeTA eKeRbIsBaØabUksMrab;kugRtaMgTaj nigsBaØadksMrab;kugRtaMgsgát;.
cMNaMfa viFIenHminsUvsuRkiteT sMrab; slender member b¤sMrab;Ggát;Edlman EI tUc EdlbnÞúktam
TisTTwgbegáItPaBdabFM. eyagtamrUbTI 17>2 bnÞúk P EdlKuNnwgPaBdab Δ nwgbegáIt)anCam:Um:g;Bt;
bEnßm ¬eBlxøHRtUv)aneKehAfam:Um:g; " PΔ" ¦. sMrab;bnÞúksgát;tamG½kS P Edlbgðaj m:Um:g;bEnßmRtUv)an
bUkeTAelIm:Um:g;Edl)anBIbnÞúkTTwgG½kS P1 nig P2 . RbsinebI P CabnÞúkTaj enaHeKeFVIpldkrvagm:Um:g; PΔ
nigm:Um:g;Edl)anBIbnÞúkTTwgG½kS. CaTUeTA eKminKitm:Um:g; PΔ eT enAeBlEdlPaBdabmantMéltUc.
Combined Stresses 2

3. T.Chhay
]TahrN_ 17>1³ eKeRbIEdk W 360 × 0.89 CaFñwmTMrsamBaØEdlmanRbEvg 3m . FñwmenHrgbnÞúkBRgayesμI
87.5kN / m edayKitTaMgbnÞúkpÞal;rbs;va nigrgkMlaMgTajtamG½kS 445kN . kMNt;kugRtaMgTaj nigkug-
RtaMgsgát;pÁÜbGtibrma.
dMeNaHRsay³
Edk W 360 × 0.89 manlkçN³muxkat;dUcxageRkam³
A = 11.55 ⋅ 10 −3 m 2
I x = 266 ⋅ 10 −6 m 4
S x = 1.5 ⋅ 10 −3 m3
edIm,IbgðajBIviFI superposition eKRtUvedaHRsayvaedaybMEbkCaBIrEpñk. FñwmEdlrgbnÞúkRtUv)anbgðajenA
kñúgrUbTI 17>3 (a). vabgðajedayrgEtbnÞúktamG½kS kúñgrUbTI 17>3 (b) nigenAkñúgrUbTI 17>3 (c) bgðajBI
FñwmEdlrgEtbnÞúkTTwgG½kS.
sMrab;bnÞúktamG½kS kugRtaMgEkgeBjRbEvgrbs;FñwmKW
445 ⋅ 103
= +38.5MPa ¬rgkugRtaMgTaj¦ ehIyRtUv)anbgðajenAkñúgrUbTI 17>4 (a).
P
st = =
A 11.55 ⋅ 103
kugRtaMgpÁÜb 3

4. NPIC
kugRtaMgEkgEdlbNþalBIbnÞúkTTwgG½kSGaRs½ynwgGaMgtg;sIuetrbs;m:Um:g;Bt;Gtibrma EdlkúñgkrNI
vaekIteLIgenAkNþalElVg.
m:Um:g;Bt;GtibrmaKW
wL2 87.5(3)2
M = = = 98.4kN .m
8 8
BI flexural formula kugRtaMgGtibrmaenAsrésxageRkAbMputEdlekIteLIgedaysarm:Um:g;enHKW
Mc M 98.4 ⋅ 10 6
sb = = = = 65.6 MPa
I S 1.5 ⋅ 106
kugRtaMgenHEkgnwgmuxkat;rbs;FñwmenAkNþalElVg ehIyvafycuHtamlkçN³smamaRt ¬lkçN³
bnÞat;¦ eTAG½kSNWt dUcbgðajenAkñúgrUbTI 17>4 (b).
edIm,ITTYl)ankugRtaMgpÁÜb ¬kugRtaMgers‘ultg;¦ kugRtaMgBt; nigkugRtaMgTajtamG½kSRtUv)anbUk
bBa©ÚltamlkçN³BiCKNit. eKGacsresrsmIkarenHtamEbbKNitviTüadUcxageRkam
P Mc
s=+ ±
A I
BIsmIkarenH eyIg)ankugRtaMgenAsréseRkambMputKW
s = +38.5 + 65.6 = +104.1MPa ¬kugRtaMgTaj¦
kugRtaMgenAsréselIbMputKW
s = +38.5 − 65.6 = −27.1MPa ¬kugRtaMgsgát;¦
dUcEdl)aneXIjenAkñúgrUbTI 17>4 (c) kugRtaMgTajpÁÜbKW + 104.1MPa enARtg;srésxageRkam
bMputénmuxkat;kNþalElVg nigkugRtaMgsgát;pÁÜbKW − 27.1MPa enARtg;srésxagelIbMputénmuxkat;kNþal
ElVg. edaysarm:Um:g;Bt;ERbRbYleTAtamRbEvgrbs;Fñwm dUcenHkugRtaMgBt; nigkugRtaMgpÁÜbnwgmantMélxusKña
Rtg;muxkat;epSgKña. cMNaMfa ExSénkugRtaMgsUnüEdlsßitenAelIG½kSNWtkñúgkrNIEdlmanEtkugRtaMgBt;rMkil
eLIgelI.
]TahrN_ 17>2³ EdkTIbEdlman nominal diameter
205mm RtUv)anbgðajenAkñúgrUbTI 17>5. EdkTIbrg
nUvbnÞúkedk nigbnÞúkbBaÄrdUcbgðajenAkñúgrUb. bnÞúk
bBaÄrrYmbBa©ÚlTaMgTMgn;pÞal;rbs;TIb. kMNt;kugRtaMg
Taj nigkugRtaMgsgát;pÁÜbGtibrmaenARtg;épÞxagelI
rbs;ebtug.
Combined Stresses 4

5. T.Chhay
dMeNaHRsay³
EdkTIbEdlman nominal diameter 205mm manlkçN³muxkat;dUcxageRkam³
A = 5.42 ⋅ 10 −3 m 2
S = 275 ⋅ 10 −6 m3
I = 30.2 ⋅ 10 −6 m 4
muxkat;EdlRtUveFVIkarGegátKWenAkñgbøg;énépÞxagelIrbs;ebtug ¬bøg; A − B ¦.
ú
kugRtaMgenAelIbøg; A − B EdlekItBIbnÞúktamG½kSKW
33.5 ⋅ 103
= −6.18MPa ¬rgkarsgát;¦ vaRtUv)anbgðajenAkñúgrUbTI 17>6 (a).
P
sc = − = −
A 5.42 ⋅ 103
kugRtaMgBt;enAkñúgbøg; A − B EdlekItBIbnÞúkedk H GaRs½ynwgGaMgtg;sIuetrbs;m:Um:g;Bt;.
m:Um:g;Bt;GtibrmaKW
M = Hb = 4.5 × 8.5 = 38.25kN .m
edayeRbI flexural formula kugRtaMgGtibrmaenAsrésxageRkAbMputEdlekItedaysarm:Um:g;enHKW
Mc M 38.25 ⋅ 106
sb = = = = ±139.1MPa RtUv)anbgðajenAkñúgrUbTI 17>6 (b).
I S 275 ⋅ 103
kugRtaMgpÁÜb 5

6. NPIC
kugRtaMgpÁÜbRtUv)anbUktamlkçN³BiCKNiténkugRtaMgTaMgBIrRbePT ehIyRtUv)anbgðajenAkñúgrUbTI
17>6 (c):
P Mc
s=− ±
A I
dUcenH kugRtaMgpÁÜbenARtg;cMnuc A KW
s = −6.18 + 139.1 = +132.92 MPa ¬rgkarTaj¦
kugRtaMgpÁÜbenARtg;cMnuc B KW
s = −6.18 − 139.1 = −145.28MPa ¬rgkarsgát;¦
3> Ggát;rgbnÞúkcakp©it Eccentrically Loaded Members
eKkMNt;Ggát;rgkarsgát;xøICaGgát;Edl)ak;edaykarEbk (crushing) b¤ yielding EdlminEmneday
sarkarekag (buckling) enAeBlEdlvargkarekIneLIgnUvbnÞúksgát;tamG½kS. enAeBlEdlGgát;enHrgbnÞúk
EdlmanGMeBIcMTIRbCMuTMgn;rbs;muxkat; enaHeKsnμt;fakugRtaMgsgát;EdlnwgekIteLIgnwgBRgayesμIeBjmuxkat;
rbs;Ggát;. EtRbsinebIbnÞúkenaHmanGMeBImincMTIRbCMuTMgn; EtmanTisRsbnwgG½kSbeNþayrbs;Ggát; dUcbgðaj
enAkñúgrUbTI 17>7 (a) enaHkugRtaMgEdlekItmanenAelImuxkat;nwgminBRgayesμIeT. ehIykugRtaMgenARKb;cM-
nucTaMgGs;nwgmankugRtaMgpÁÜbRsedogKñanwgkugRtaMgpÁÜbEdlekItmanenAkñúgGgát;EdlrgbnÞúkTTwgG½kS nig
bnÞúktamG½kS.
Combined Stresses 6

7. T.Chhay
cMNakp©it e énbnÞúkCacMgayEdlvas;BITIRbCMuTMgn;rbs;muxkat; O eTAExSskmμrbs;bnÞúk P . kñúg
krNIenH kMlaMg P manGMeBIelIG½kSmYyEdlkat;tamTIRbCMuTMgn;rbs;muxkat;. ehIycMNaMfa edaysarssr
enHxøIdUcenHPaBdabtamTisTTwgRtUv)ansnμt;famantMéltUc nigGacecal)anedayeRbobeFobCamYynwgcM-
Nakp©it e . dUcenHeKGaceRbIviFI superposition )an.
karviPaKGgát;rgbnÞúkcakp©itTak;TgnwgTMnak;TMngbnÞúk-m:Um:g; (load-moment relationship). enA
kñúgrUbTI 17>7 (a) eyIgeXIjfaGgát;rgkarsgát;xøIEdlbnÞúkbBaÄr P manGMeBIcakp©it e enAelIG½kSmYyrbs;
G½kSkat;tamTIRbCMuTMgn;. enAkñúgrUbTI 17>7 (b) kMlaMg P BIrEdlmantMéldUcKña EtmanTisedApÞúyKñamanGM
eBIRtg;TIRbCMuTMgn;rbs;muxkat;. karbEnßmkMlaMgTaMgBIrenHminmanbBaðaGVIeT eRBaHplbUknBVnþénkMlaMgRsbBIr
EdlmanGaMgtg;sIuetesμIKña nigTisedApÞúyKñaesμIsUnü.
bnÞab;mk eyIgCMnYsbnÞúkcakp©itCamYynwgbnÞúkEdlmanGMeBIcMp©itEdlmanTisedAeLIgelIedaym:Um:g;
couple EdlmanGaMgtg;sIuet Pe . dUcEdl)anbgðajenAkñúgrUbTI 17>7 (c) nig (d) kMlaMgsgát; P enARtg;TI
RbCMuTMgn; begáItkugRtaMgsgát;tamG½kSedaypÞal;
P
sc = −
A
ehIym:Um:g; couple Pe EdlmanGMeBIeFobnwgG½kS Y − Y begáIt)ankugRtaMgBt;
Mc Pec
sb = ± =±
I I
kugRtaMgpÁÜbGacsresr)andUcxageRkam
P Pec
s=− ± (17.2)
A I
düaRkaménkugRtaMgpÁÜbRtUv)anbgðajenAkñúgrUbTI 17>7 (e). eKsnμt;fakugRtaMgBt;GtibrmaFMCag
kugRtaMgtamG½kS dUcenHtMbn;énkugRtaMgTajenAxagsþaM nigtMbn;énkugRtaMgsgát;enAxageqVg . RbsinebI
kugRtaMgBt;GtibrmamantMéltUcCagkugRtaMgtamG½kS enaHmuxkat;TaMgmUlnwgrgkugRtaMgsgát; ehIyminman
kugRtaMgTajeT.
]TahrN_ 17>3³ eKeRbIGgát;eQICaGgát;rgkarsgát;xøIdUcbgðajenAkúñgrUbTI 17>7 (a). muxkat;rbs;ssr
man b = 25cm nig d = 40cm . ssrrgnUvbnÞúkcakp©it P = 178kN EdlmanGMeBIRtg; 125mm BIG½kS
Y − Y . edayeyagtamrUbTI 17>7 (a) cUrKNnakugRtaMgpÁÜbenAelIsrésxageRkAénRCug MM nig NN .
dMeNaHRsay³ kugRtaMgsgát;tamG½kSEdlbNþalBIbnÞúk P KW
P 178 ⋅ 103
sc = − =− = −1.78MPa
A 250(400 )
kugRtaMgpÁÜb 7

8. NPIC
m:Um:g;Bt;EdlekItedaybnÞúkcakp©uitKW
M = Pe = 178 × 0.125 = 22.25kN .m
kugRtaMgBt;EdlekIteLIgedaysarm:Um:g;enHKW
Mc Pec
sb = ± =±
I I
Edl c = d / 2 = 200mm nig I eFobnwgG½kS Y − Y RtUv)anKNnadUcxageRkam
bd 3 250(400 )3
Iy = = = 13.33 ⋅ 108 mm 4
12 12
22.25 ⋅ 10 6 × 200
dUcenH sb = ±
13.33 ⋅ 108
= ±3.34 MPa
kugRtaMgpÁÜbKWCaplbUkBiCKNiténkugRtaMgTaMgBIrRbePT
P Pec
s=− ±
A I
kugRtaMgpÁÜbenARCug MM KW
s = −1.78 − 3.34 = −5.12 MPa ¬rgkarsgát;¦
kugRtaMgpÁÜbenARCug NN KW
s = −1.78 + 3.34 = +1.56 MPa ¬rgkarTaj¦
]TahrN_ 17>4³ enAkñúgrUbTI 17>8 bgðajBI press frame. enAeBleKeFVIikarsgát; kMlaMgGtibrma
P = 135kN manGMeBIenAcenøaH jaw xagelI nig jaw xageRkam. KNnakugRtaMgTajGtibrmaenARtg;cMnuc o
nigcMnuc i enAelIbøg; A − A .
Combined Stresses 8

9. T.Chhay
dMeNaHRsay³ kMlaMgcakp©it P )anbegáItkMlaMgRbtikmμ P nigm:Um:g;Rbtikmμ Pe enARtg;bøg; A − A dUcEdl
bgðajenAkñúgdüaRkamGgÁesrI ¬rUbTI 17>9¦.
dMbUgKNnalkçN³muxkat;edayeyagtamrUbTI 17>8 (b). RkLaépÞmuxkat;KW
A = a1 + a2 = (300 × 75) + (225 × 75) = 39375mm 2
eyIgKitTIRbCMuTMgn;rbs;muxkat;edayeFobnwg reference axis. cMNaMfa x CacMgayBITIRbCMuTMgn;
rbs;RkLaépÞ a nimYy²eTA reference axis.
∑ ax a1 x1 + a2 x2 16875(75 / 2) + 2250(225)
ci = = = = 145mm
∑a a1 + a2 39375
ehIy co = 375 − 145 = 230mm
KNnam:Um:g;niclPaBeFobG½kSTIRbCMuTMgn;
I = ∑(I o + ad 2 )
⎡ 225(75)3 ⎛ 75 ⎞ ⎤ ⎡ 75(300 )3
2 ⎤
=⎢ + 16875⎜145 − ⎟ ⎥ + ⎢ + 22500(230 − 150 )2 ⎥ = 515.75 ⋅ 10 6 mm 4
⎢
⎣ 12 ⎝ 2 ⎠ ⎥ ⎢ 12
⎦ ⎣ ⎥
⎦
eKGackMNt;kugRtaMgEdlekItenAelIbøg; A − A edayeRbIsmIkar 17.2. cMNaMfa kMlaMgRbtikmμ
Taj P begáItkugRtaMgTajesμInwg P / A EdlvaBRgayesμIenAelImuxkat;TaMgmUl. tY P / A mansBaØaviC¢man
edaysarvaCakMlaMgTaj. m:Um:g;Bt; Pe nwgbegáItkugRtaMgTajenAelIépÞxagkñúg nigkugRtaMgsgát;enAelIépÞ
xageRkA.
P Pec
s=+ ±
A I
BIdüaRkamGgÁesrI cMNalp©it e = ci + 600 = 745mm
dUcenH kugRtaMgenAépÞxagkñúg ¬cMnuc i ¦KW
135 ⋅ 103 135 ⋅ 103 × 745 × 145
si = +
39375
+
515.75 ⋅ 10 6
= +31.71MPa ¬rgkarTaj¦
nigkugRtaMgenAépÞxageRkA ¬cMnuc o ¦ KW
kugRtaMgpÁÜb 9

10. NPIC
135 ⋅ 103 135 ⋅ 103 × 745 × 230
so = +
39375
−
515.75 ⋅ 10 6
= −41.42 MPa ¬rgkarsgát;¦
4> cMNakp©itGtibrmaedIm,I)ankugRtaMgTajsUnü
Maximum Eccentricity for Zero Tensile Stress
enAkñúgrUbTI 17>10 kugRtaMgpÁÜbekItmanEtenAelIRCug AB nig CD edaysarkarBt;eFobnwgbøg;
Y − Y . kugRtaMgpÁÜbGtibrmaKW
P Pec
s=− ±
A I
cMNaMfa enARCug AB EtgEtrgkugRtaMgsgát; edaysarbnÞúkcMNakp©itenACitRCug AB CagRCug CD .
enARCug CD GacrgkugRtaMgsgát; kugRtaMgTaj b¤sUnüGaRs½ynwgcMNakp©itrbs;bnÞúk. RbsinebIkugRtaMgBt;
mantMélFMCagkugRtaMgtamG½kS RCugxageRkA CD nwgrgkugRtaMgTajdUcbgðajenAkñúgrUbTI 17>10 (e).
müa:gvijeTot RbsinebIkugRtaMgBt;mantMéltUcCagkugRtaMgtamG½kS RCugxageRkA CD nwgrgkugRtaMgsgát;
Combined Stresses 10

11. T.Chhay
dUcbgðajenAkñúgrUbTI 17>10 (f). RbsinebIkugRtaMgBt;mantMG½kS enaHkugRtaMgenARCugxageRkA CD nwgesμI
sUnü dUcbgðajenAkñúgrUbTI 17>10 (g).
edaysarsMPar³xøH ¬dUcCaebtug¦ manersIusþg;TajtUc eKcaM)ac;RtUvdwgBIcMNakp©itGtibrmaedIm,IkMu
eGayekItmankugRtaMgTaj. vaRtUvKñanwglkçxNÐkugRtaMgTajsUnü s = 0 .
Pec P
=
I A
RbsinebIenAkñúgrUbTI 17>10/ AB = b / AC = d / c = d / 2 / I = bd 3 / 12 ehIyRkLaépÞmuxkat;
= bd smIkarxagelIkøayCa
Pe(d / 2 ) P
(bd 3
/ 12 ) =
bd
dUcenHeyIg)an e=
d
6
(17.3)
vaCacMNakp©itGtibrmaEdlminGacekItmankugRtaMgTaj.
RbsinebIcMNakp©it e < d / 6 enaHkugRtaMgBt;nwgmantMélFMCagkugRtaMgtamG½kS ehIykugRtaMgTaj
nwgekItmanenAelIRCugxageRkA CD dUcbgðajenAkñúgrUbTI 17>10 (e). RbsinebIcMNakp©it e ≤ d / 6 enaH
manEtkugRtaMgsgát;eTEdlekItmanenAeBjmuxkat;rbs;Ggát; dUcbgðajenAkñúgrUbTI 17>10 (f) nig (g).
dUcKñasMrab;bnÞúkEdlmanGMeBIenAEpñkmçageTotrbs;bøg; Y − Y . dUcenH RbsinebIbnÞúkcakp©itmanGMeBI
Rtg;tMbn;mYyPaKbIEpñkkNþalrbs;muxkat; ¬ d / 6 + d / 6 énRbEvg d ¦ enavanwgminmanrgkugRtaMgTajeT.
eKeRbIeKalkarN_dUcKñasMrab;bnÞúkcakp©itEdlsßitenAelIG½kS Y − Y . cMNakp©itGtibrmaedIm,I
TTYl)ankugRtaMgTajesμIsUnüKW b / 6 .
5> bnÞúkcMNakp©itminsßitenAelIG½kSTIRbCMuTMgn;
Eccentric Load Not on Centroidal Axis
kugRtaMgpÁÜb 11

12. NPIC
rUbTI 17>11 bgðajBIbnÞúkcMNakp©itEdlminsßitenAelIG½kSNamYyEdlkat;tamTIRbCMuTMgn;. kug-
RtaMgpÁÜbenARKb;cMnucTaMgGs;CaplbUkBiCKNitén
!> kugRtaMgtamG½kSEdlekItBIbnÞúk P EdlmanGMeBIRtg;TIRbCMuTMgn; O
@> kugRtaMgBt;EdlekItBIm:Um:g; Pe1 eFobnwgG½kSBt; Y − Y
#> kugRtaMgBt;EdlekItBIm:Um:g; Pe2 eFobnwgG½kSBt; X − X
eKGacsresrvakñúgTMrg;xageRkam³
P Pe1c1 Pe2 c2
s=− ± ± (17.4)
A Iy Ix
Edl c1 nig c2 CacMgayBIG½kSTIRbCMuTMgn;eTAsrésxageRkArbs;muxkat;EdleKRtUvkarrkkugRtaMgpÁÜb. EdnkM
Nt;énTItaMgrbs;bnÞúkcMNakp©it P edIm,IkMueGaymankugRtaMgTajKWmanlkçN³RsedogKñanwgGVIEdl)anerob
rab;enAkñúgcMnucTI 4 xagelI. cMNakp©itGtibrmatamG½kS½ Y − Y KW b / 6 nigtamG½kS X − X KW d / 6 .
edaysarEtbnÞúkcMNakp©itminsßitenAelIG½kSNamYy ehIyedIm,IkMueGayekItmankugRtaMgTaj enaHbnÞúkcM-
Nakp©itRtUvmanGMeBIenARtg;tMbn;EdlbegáIteLIgedaykarP©ab;cMnucEdnkMNt;EdlmineFVIeGaymankugRtaMg
TajedayGMeBIrbs;bnÞúkcakp©ittamG½kSTaMgBIrxagelI. dUcbgðajenAkñúgrUbTI 17>12 RbsinebIbnÞúkcakp©it
manGMeBIenAelIGgát;rgkarsgát;xøImuxkat;ctuekaNEkgRtg;RkLaépÞqUt enaHkugRtaMgTajnwgminekItmanenAelI
muxkat;enHeT. RkLaépÞqUtenHeKeGayeQμaHfa kern rbs;muxkat; ehIyrUbragrbs;vaGaRs½yeTAnwgrUbrag
rbs;muxkat;.
Combined Stresses 12

13. T.Chhay
]TahrN_ 17>5³ Ggát;xøIrgbnÞúksgát; 450kN . vamancMNakp©itDubdUcbgðajenAkñúgrUbTI 17>13. Ggát;
enHmanmuxkat; 35cm × 30cm . KNnakugRtaMgpÁÜbenARCugTaMgbYn A, B, C nig D . kMNt;TItaMgExSén
kugRtaMgsUnü.
dMeNaHRsay³
kMNt;m:Um:g;niclPaBeFobG½kS X − X nigG½kS Y − Y
db 3 350(300 )3
Ix = = = 78.75 ⋅ 10 7 mm 4
12 12
bd 3 300(350 )3
Iy = = = 107.19 ⋅ 107 mm 4
12 12
kMNt;kugRtaMgnImYy²dac;edayELkBIKña
P 450000
= = 4.29 MPa
A 350(300)
Pe1c1 450000(75)175
= = 5.51MPa
Iy 107.19 ⋅ 107
Pe2c2 45000(65)150
= = 5.57 MPa
Ix 78.75 ⋅ 107
KNnakugRtaMgpÁÜbenARtg;RCugTaMgbYnrbs;muxkat;edayeRbIsmIkar 17.4
P Pe1c1 Pe2 c2
s=− ± ±
A Iy Ix
Rtg;cMnuc A
kugRtaMgpÁÜb 13

14. NPIC
s = −4.29 − 5.51 + 5.57 = −4.23MPa ¬rgkarsgát;¦
Rtg;cMnuc B
s = −4.29 − 5.51 − 5.57 = −15.37 ¬rgkarsgát;¦
Rtg;cMnuc C
s = −4.29 + 5.51 − 5.57 = −4.35MPa ¬rgkarsgát;¦
Rtg;cMnuc D
s = −4.29 + 5.51 + 5.57 = +6.79 MPa ¬rgkarTaj¦
KNna DE BIRtIekaNdUc
DE 350 − DE
=
6.79 4.23
eK)an DE = 215.7mm . dUcKñaeK)an DF = 182.9mm
6> kugRtaMgpÁÜbénkugRtaMgEkg nigkugRtaMgkat; Combined Normal and Shear Stresses
enAkñúg homogeneous elastic beam EdlkugRtaMgsmamaRteTAnwg strain enaHkugRtaMgBIrRbePTnwg
ekItmantMNalKña. kugRtaMgTaMgBIrenaHKW kugRtaMgBt;EdlpSMeLIgedaykugRtaMgTaj nigkugRtaMgsgát;Edl
EkgeTAnwgépÞEdlvamanGMeBI nigkugRtaMgkat;EdlRsbeTAnwgépÞEdlvamanGMeBI. kugRtaMgBt;RtUv)aneKehAfa
kugRtaMgEkg (normal stresses) nigkugRtaMgkat;RtUv)aneKehAfakugRtaMgRsb (tangential stresses). eKGac
KNnakugRtaMgTaMgenHedaysmIkarxageRkam³
BI flexural formula
Mc
sb =
I
BI general shear formula
Combined Stresses 14

15. T.Chhay
VQ
ss =
Ib
BicarNaFatuGnnþtUcrbs;FñwmTMrsamBaØEdlbgðajenAkñúgrUbTI 17>15. eKsikSaFatuenHRtg;cMnucEdl
kMlaMgkat; nigm:Um:g;Bt;minesμIsUnü ehIyminsßitenAsrésxageRkAbMput b¤G½kSNWt. dUcenHFatuenHrgTaMgkug
RtaMgkat; nigkugRtaMgBt;. kugRtaMgTaMgenHRtUv)anbgðajedaymanGMeBIeTAelIépÞrbs;FatuenAkñúgrUbTI 17>15
(b). kugRtaMgRtUv)ansMEdgeday s x nigkugRtaMgkat;RtUv)ansMEdgeday s xy . kugRtaMgpÁÜbCakugRtaMgEkg
Gtibrma nigGb,brma nigkugRtaMgkat;EdlekItmanenAelIbøg;eRTteFobnwgG½kSrbs;Fñwm.
enAelIFatuEdlrgkugRtaMgdUcEdlbgðajenAkñúgrUbTI 17>16 man sx CakugRtaMgEkgEdlmanGMeBI
eTAelIbøg;EdlEkgeTAnwgG½kS X − X . kugRtaMgenHGacCakugRtaMgTaj b¤kugRtaMgsgát; ehIyvaGacekItBI
m:Um:g;Bt; b¤BIbnÞúkedaypÞal;. kugRtaMgkat;enARtg;cMnucdUcKñaEdlmanGMeBIenAelIRKb;épÞrbs;FatuRtUv)ansMEdg
eday sxy . kugRtaMgkat;mantMéldUcKñaehIysßitenAelIbøg;EdlEkgKña. kugRtaMgkat;)anBIm:Um:g;rmYl b¤kMlaMg
kat;.
kugRtaMgpÁÜb 15

16. NPIC
edaybUkkugRtaMgEdlbgðajenAkñúgrUbTI 17>16 eKGackMNt;kugRtaMgenAelIbøg;eRTt. eyIgnwgkMNt;
Tisrbs;bøg;enHedaymMupÁúMrvagG½kS X − X nigG½kSEdlEkgnwgbøg;eRTt. RbsinebIeyIgkat;FatuctuekaNenH
tambøg;eRTt eyIgnwgTTYl)anFatuRtIekaNdUcbgðajenAkñúgrUbTI 17>17 EdlbgðajkñúgTMrg;düaRkamGgÁesrI.
BIrUbTI 17>17 enAelIbøg;eRTtmankMlaMgBIrKWklaMgEkg nigkMlaMgkat;EdlCaGBaØtiRtUvrk. edIm,I
M
gayRsYlkñúgedaHRsaykugRtaMgTaMgGs;RtUv)anbMElgeTACakMlaMgvij ¬cMNaMfa kMlaMgesμInwgplKuNrvag
kugRtaMgnwgRkLaépÞ¦. edaysarFatuEdlbgðajsßitenAkñúgbøg;dUcenHeyIgykkMras;rbs;vaesμImYyÉkta. enA
kñúgrUbenH sn CakugRtaMgEkg ¬Taj b¤sgát;¦ enAelIbøg;eRTt ehIy s's CakugRtaMgRsb ¬kugRtaMgkat;¦ enAelI
bøg;eRTt. eFVIplbUkkMlaMgEkgenAelIbøg;eRTtCamYynwgTMhMrbs;FatuEdlmanRbEvg h / w nig d dUcbgðaj
sn (d )(1) = s x (h ) cos θ + s y (w)(1) sin θ − s xy (h )(1) sin θ − s xy (w)(1) cos θ
edayEckGgÁTaMgBIrnwg (d )(1) eK)an
⎛h⎞ ⎛ w⎞ ⎛h⎞ ⎛ w⎞
sn = s x ⎜ ⎟ cos θ + s y ⎜ ⎟ sin θ − s xy ⎜ ⎟ sin θ − s xy ⎜ ⎟ cos θ
⎝d ⎠ ⎝d ⎠ ⎝d ⎠ ⎝d ⎠
eday sin θ = w / d nig cosθ = h / d
sn = s x cos 2 θ + s y sin 2 θ − 2s xy sin θ cos θ (17.5)
vaCakugRtaMgEkgenAelIRKb;bøg;eRTtTaMgGs;EdlmMupÁúMrvagG½kSEkgeTAnwgbøg;eRTtCamYynwgG½kS X − X esμInwg
θ.
bnÞab;mk eFVIplbUkBiCKNiténkMlaMgRsbeTAnwgbøg;eRTt
Combined Stresses 16

17. T.Chhay
s 's (d )(1) = s xy (h )(1) cos θ + s x (h )(1) sin θ − s y (w)(1) cos θ − s xy (w)(1) sin θ
edayEckGgÁTaMgBIrnwg (d )(1) eK)an
⎛h⎞ ⎛h⎞ ⎛ w⎞ ⎛ w⎞
s 's = s xy ⎜ ⎟ cos θ + s x ⎜ ⎟ sin θ − s y ⎜ ⎟ cos θ − s xy ⎜ ⎟ sin θ
⎝d ⎠ ⎝d ⎠ ⎝d⎠ ⎝d⎠
( () )
s 's = s x − s y sin θ cos θ + s xy cos 2 θ − sin 2 θ (17.6)
vaCakugRtaMgkat;enAelIRKb;bøg;eRTtTaMgGs;EdlmMupÁúMrvagG½kSEkgeTAnwgbøg;eRTtCamYynwgG½kS X − X esμInwg
θ.
RbsinebIbøg;eRTtenHviledaymMu θ eKnwgTTYl)anTItaMgEdleFVIeGaykugRtaMgEkgEdlmanGMeBIelIva
mantMélGtibrma b¤Gb,brma. TItaMgrbs;bøg;sMrab;kugRtaMgEkgGtibrmaKWEkgnwgbøg;sMrab;kugRtaMgEkgGb,-
brma. enAelIbøg;enHminmankugRtaMgkat;eT. eKehAbøg;EdlkugRtaMgEkgkøayCaGtibrma b¤Gb,brmaCabøg;em
(principal plane) ehIyeKehAkugRtaMgEkgEdlmanGMeBIelIbøg;enaHCakugRtaMgem (principal stress).
edIm,IkMNt;tMél θ EdlkMNt;TItaMgbøg;em eyIgeGay s' = 0 enaHeyIg)an
p s
0 = (s − s )sin θ cos θ + s (cos θ − sin θ )
x y p p xy
2
p
2
p
2 s xy
tan 2θ p = − (17.7)
sx − s y
sx − s y
cos 2θ p = ±
(s x − s y )2 + 4s xy
2
2 s xy
sin 2θ p = ±
(s x − s y )2 + 4s xy
2
CMnYstMél cos 2θ p nig sin 2θ p eTAkñúgsmIkar 17.5 edaymansBaØapÞúyKña enaHeyIg)ankugRtaMgemGti brma
nigkugRtaMgemGb,brmadUcxageRkam³
s1, 2
⎛ sx + s y
=⎜
⎞
⎟±
(s x + s y )2 + s 2 (17.8)
⎜ 2 ⎟ xy
⎝ ⎠ 4
edIm,ITTUl)ankugRtaMgkat;elIbøg;eRTtmantMélGtibrmaluHRtaEtG½kSEkgrbs;bøg;eRTtenaHvil)anmMu θ s
ehIyedrIevénsmIkar 17.6 eFobnwgmMu θ RtUvesμInwgsUnü enaHeyIg)an
( )
0 = s x − s y cos 2θ − 2s xy sin 2θ
2 s xy
tan 2θ s = (17.9)
sx − s y
2 s xy
cos 2θ s = ±
(s x − s y )2 + 4s xy
2
sx − s y
sin 2θ s = ±
(s x − s y )2 + 4s xy
2
kugRtaMgpÁÜb 17

18. NPIC
CMnYs cos 2θ s nig sin 2θ s eTAkñúgsmIkar 17.6 eyIgTTYl)an
s 's (max) = ±
(s x − s y )2 + s 2 (17.10)
xy
4
pÞúyKñanwgkugRtaMgem enAelIbøg;eRTtEdlmankugRtaMgkat;Gtibrmak¾enAmankugRtaMgEkgEdr.
kugRtaMgEkgEdlmanGMeBIelIbøg;énkugRtaMgkat;GtibrmaKW
sx + s y
sn = (17.11)
2
smIkar 17.5 rhUtdl; 17.10 mann½ysMrab;TisedAénkugRtaMgdUcbgðajenAkñúgrUbTI 17>16. RbsinebI
kugRtaMgmanTisedApÞúyBIenHsBaØaGacpøas;bþÚr. sMrab;kugRtaMgEkg kugRtaMgTajmansBaØaviC¢man kugRtaMg
sgát;mansBaØadk. sMrab;kugRtaMgkat; sBaØaviC¢maneRbIsMrab;TisedAdUcbgðajkñúgrUb. mMu θ mansBaØaviC¢man
enAeBlEdlvavilBIG½kS X − X RcasRTnicnaLika.
]TahrN_TI 17>6³ FatumYyEdlrgnUvbnÞúkBIrTismankugRtaMgdUcbgðajenAkñúgrUbTI 17>18. (a) kMNt;
kugRtaMgEkg nigkugRtaMgkat;enAelIbøg;EdlG½kS X − X vilRcasRTnicnaLika)anmMu 60o . (b) KNna
kugRtaMgem nigTisrbs;bøg;em. (c) KNnakugRtaMgkat;Gtibrma nigTItaMgrbs;bøg;EdlvaeFVIGMeBI.
dMeNaHRsay³
(a) edIm,IKNnakugRtaMgEkg nigkuugRtaMgkat;eyIgGaceRbIsmIkar 17.5 nig 17.6
tamrUbeyIgman sx = −69MPa / s y = 90MPa / s xy = −48MPa nig θ = 60o
BIsmIkar 17.5 eyIg)ankugRtaMgEkg
sn = s x cos 2 θ + s y sin 2 θ − 2s xy sin θ cos θ
= −69(0.25) + 90(0.75) − 2(− 48)(0.866 )(0.5) = 91.8MPa ¬kugRtaMgTaj¦
Combined Stresses 18

19. T.Chhay
BIsmIkar 17.6 eyIg)ankugRtaMgkat;
( ) (
s 's = s x − s y sin θ cos θ + s xy cos 2 θ − sin 2 θ )
= (− 69 − 90 )(0.433) + (− 48)(− 0.5) = 44.8MPa
cMNaMfasmIkar 17.5 nigsmIkar 17.6 KWQrelITisedAkugRtaMgkat;dUcbgðajenAkñúgrUbTI 17>17.
dUcenHsBaØadksMrab;kugRtaMgkat;bgðajfakugRtaMgEdl)anKNnaenAelIbøg;eRTtmanTisedAdUc
bgðajenAkñúgrUbTI 17>19.
(b) KNnakugRtaMgemBIsmIkar 17.8
s1, 2 =
sx + s y
±
(s x − s y )2 + (− 48)2
2 4
= +10.5 ± 92.9
eyIg)an s1 = 103.4 MPa ¬kugRtaMgTaj¦
s2 = −82.4 MPa ¬kugRtaMgsgát;¦
eKGacKNnamMurbs;bøg;eRTtedayeRbIsmIkar 17.7
2 s xy 2(− 48)
tan 2θ p = − =− = −0.604
sx − s y − 69 − 90
edIm,IgayRsYlkñúgkaryl;BIrebobkMNt; θ p rUbTI 19>20 bgðajBIdüaRkamrbs;GnuKmn_tg;sg;.
kugRtaMgpÁÜb 19

20. NPIC
RbsinebI tan 2θ p GviC¢man enaH 2θ p RtUvsßitenAkñúgkaRdg;TI 2 nigkaRdg;TI 4 .
BIkñúgrUbTI 17>20 eyIg)an
tan 2θ = tan ( o + 2θ )
180
TMnak;TMngmMuenAkñúgkaRdg;TI1 nigkaRdg;TI 2 eyIg)an
tan 2θ = − tan (180 o − 2θ )
TMnak;TMngmMuenAkñúgkaRdg;TI1 nigkaRdg;TI 4 eyIg)an
tan 2θ = − tan (360 o − 2θ )
sMrab; tan 2θ p = −0.604 RbsinebI 2θ p enAkñúgkaRdg;TI 1 ¬RbsinebItMéltg;sg;viC¢man¦ vanwg
esμInwg 31.13o . b:uEnþvaenAkñúgkaRdg;TI 2 dUcenH
2θ p = 180o − 31.13o = 148.47 o
nigenAkñúgkaRdg;TI 4 eyIg)an
2θ p = 360o − 31.13o = 328.47 o
dUcenHbøg;emRtUv)ankMNt;edaybøg;BIrEdlmMu θ p RtUv)anvas;edayRcasRTnicnaLikaBIG½kS
X−X
148.47 o
θp = = 74.24o
2
328.47 o
θp = = 164.24o
2
edIm,IkMNt;bøg;EdlmankugRtaMgemGtibrma eyIgCMnYs θ p = 74.24o eTAkñúgsmIkar 17.5
sn = s x cos 2 θ p + s y sin 2 θ p − 2s xy sin θ p cos θ p
= −69(0.0738) + 90(0.9362 ) − 2(− 48)(0.2614 ) = 103.4 MPa
dUcenHkugRtaMgemTaMgBIr
RtUv)anbgðajenAkñúgrUb
TI 17>21. cMNaMfa
enAelIbøg;emmanEtkug
RtaMgEkgbueNÑaH.
Combined Stresses 20

21. T.Chhay
(c) eKGackMNt;kugRtaMgkat;GtibrmaBIsmIkar 17.10
s's = ±
(s x − s y )2 + s 2
xy
4
=±
(− 69 − 90)2 + (− 48)2 = ±92.9 MPa
4
eKGacTTYl)anmMupÁúMrbs;G½kSEdlEkgnwgbøg;EdlmankugRtaMgkat;GtibrmaedayeRbIsmIkar 17.9
sx − s y − 69 − 90
tan 2θ s = = = 1.656
2 s xy 2(− 48)
edaytg;sg;viC¢mandUcenH 2θ s sßitenAkñúgkaRdg;TI 1 nigkaRdg;TI 3 nigmantMél 58.87o nig
238.87 o erogKña
87 o
dUcenH θ s = 58.2 = 29.44o
o
nig θs =
233.87
2
= 119.44o
cMNaMfa kugRtaMgkat;pÁÜbGtibrmaekItmanenAelIbøg;EdlpÁúMCamYynwgbøg;em)anmMu 45o .
edIm,ITTYl)anTisedArbs;kugRtaMgEkg CMnYs θ s = 29.44o eTAkñúgsmIkar 17.6 eyIg)an
s 's = (s x − s y )sin θ s cos θ s + s xy (cos 2 θ s − sin 2 θ s )
= (− 69 − 90 )(0.4915)(0.8709 ) + (− 48)(0.8709 2 − 0.4915 2 ) = −92.9 MPa
TisedArbs;kugRtaMgEkgRtUv)anbgðajenAkñúgrUbTI 17>22.
eKGackMNt;kugRtaMgEkgEdlmanGMeBIenAelIbøg;énkugRtaMgkat;GtibrmaBIsmIkar 17.11
s x + s y (− 69 + 90)
sn =
2
=
2
= 10.5MPa ¬kugRtaMgTaj¦
]TahrN_ 17>7³ built-up steel beam manElVgxøIRTedayTMrsamBaØRtUv)anbgðajenAkñúgrUbTI 17>23.
enARtg;cMnuc A TIRbsBVrvagRTnug nigsøab (a) KNnakugRtaMgem (b) TItaMgkugRtaMgem (c) KNnakugRtaMg
kugRtaMgpÁÜb 21

22. NPIC
kat;Gtibrma. ecalTMgn;pÞal;rbs;Fñwm nigkugRtaMgRbmUlpþúM. kardak;bnÞúkRtUv)ansnμt;faCakardak;bnÞúk
cMcMnuc.
dMeNaHRsay³ (a) munnwgkMNt;kugRtaMgem eyIgRtUvkMNt;kugRtaMgEkg nigkugRtaMgkat; ¬ s / s nig s ¦
x y xy
m:Um:g;Bt;enAkNþalElVgKW
PL 270 × 0.6
M = = = 40.5kN .m
4 4
kugRtaMgkat;enAkNþalElVg ¬sßitenAEk,rkNþalElVgxageqVg¦ KW
P 270
V= = = 135kN
2 2
m:Um:g;niclPaBrbs;muxkat;eFobG½kSxøaMgedayecal I o rbs;søab
I = ∑ I o 2 + ∑ a1d 2
13 × 2703
= + 2 × 125 × 20 × 1452 = 126.45 ⋅ 106 mm 4
12
kugRtaMgBt;GtibrmaenAsrésxageRkAbMputKW
Mc 40.5 ⋅ 10 6 × 155
sb = ± =± = ±49.6 MPa
I 126.45 ⋅ 10 6
kugRtaMgBt;enARtg;cMnuc A KW
135
sx = 49.6 = 43.2MPa
155
m:Um:g;sþaTicrbs;muxkat; Q KW
Q = 125(20 )(135) = 337500mm3
kugRtaMgkat;enARtg;cMnuc A KW
VQ 135 ⋅ 103 × 337500
s xy = = = 27.7 MPa
Ib 126.45 ⋅ 106 × 13
Combined Stresses 22

23. T.Chhay
kugRtaMgenARtg;cMnuc A RtUv)anbgðajenAkñúgrUbTI 17>24 (a) EdlkugRtaMgTaMgenaHmanGMeBIelIFatuGnnþtUc.
eKGacKNnakugRtaMgemBIsmIkar 17.8
s1, 2 =
sx + s y
±
(s x − s y )2 + s 2
xy
2 4
=
43.2
±
(43.2)2 + 27.7 2
2 4
= 21.6 ± 35.1
Edl s1 = 56.7 MPa ¬kugRtaMgTaj¦
s2 = −13.5MPa ¬kugRtaMgsgát;¦
cMNaMfa kugRtaMgemGtibrma s1 FMCagkugRtaMgBt;GtibrmaenAsrésxageRkAbMput.
(b) TItaMgbøg;em
2 s xy 2(27.7 )
tan 2θ p = − =− = −1.23
sx − s y 43.2
dUcenH 2θ p sßitenAkñúgkaRdg;TI 2 nigkaRdg;TI 4 . RbsinebIvasßitenAkaRdg;TI 1/ 2θ p = 52o . eyIg
eXIjfa vasßitenAkñúgkaRdg;TI 2
2θ p = 180o − 52o = 128o
θ p = 64o
nigvasßitenAkñúgkaRdg;TI 4
2θ p = 360o − 52o = 308o
θ p = 154o
edIm,IkMNt;TItaMgbøg;EdlkugRtaMgemGtibrmamanGMeBIelI CMnYs θ p = 64o eTAkñúgsmIkar 17.5
sn = s x cos 2 θ p + s y sin 2 θ p − 2s xy sin θ p cos θ p
= 43.2(0.192 ) − 2(27.7 )(0.394 ) = −13.5MPa
vaCakugRtaMgemGb,brma. dUcenHkugRtaMgemGtibrma 56.7MPa sßitenAelIbøg;EdlmanG½kSEkgpÁúMmMuCamYy
nwgG½kS X − X )an 154o RcasRTnicnaLika. FatuEdlrgkugRtaMgemRtUv)anbgðajenAkñúgrUbTI 17>24 (b).
(c) kugRtaMgkat;Gtibrma
s's = ±
(s x − s y )2 + s 2
xy
4
=±
(43.2)2 + 27.7 2 = ±35.1MPa
4
TItaMgrbs;bøg;EdlkugRtaMgkat;GtibrmamanGMeBIelI
kugRtaMgpÁÜb 23

24. NPIC
sx − s y 43.2
tan 2θ s = = = 0.78
2 s xy 2(27.7 )
dUcenH 2θ s = 38o nig 2θ s = 218o
eyIg)an θ s = 19o θ s = 109o
kMNt;TisedArbs;kugRtaMgkat;GtibrmaedayCMnYsmMu θ s = 19o eTAkñúgsmIkar 17.6
s 's = (s x − s y )sin θ s cos θ s + s xy (cos 2 θ s − sin 2 θ s )
= 43.2(0.308) + 27.7(0.894 − 0.106 ) = 13.3 + 21.8 = 35.1MPa
tMélviC¢manmann½yfakugRtaMgkat;sßitenAelIbøg;EdlmanG½kSEkgpÁúMmMu 19o CamYynwgG½kS X − X manTisedA
dUcbgðajenAkñúgrUbTI 17>24 (c).
kugRtaMgEkgEdlmanGMeBIenAelIbøg;kugRtaMgkat;GtibrmaKW
s x + s y 43.2
sn =
2
=
2
= 21.6MPa ¬kugRtaMgTaj¦
Combined Stresses 24

25. T.Chhay
7> rgVg;m: Mohr’s Circle
rgVg;m:CaviFIRkahVikkñúgkarrkkugRtaMgEkg nigkugRtaMgkat;EdlmanGMeBIelIbøg;eRTt. viFIenHrkeXIj
edayvisVkrCnCatiGaLWm:g; Otto Mohr enAqñaM 1882.
eyagtamrUbTI 17>25 (a) Ggát;Edlmanmuxkat; A rgnUvbnÞúkTajtamG½kS P . xageRkamCaCMhan
énkarGnuvtþrgVg;m:³
!> sg;RbB½n§kUGredaenEdlmankugRtaMgEkgsßitenAelIG½kSGab;sIus nigkugRtaMgkat;sßitenAlIG½kS
Gredaen ¬rUbTI 17>25 (c)¦.
kugRtaMgpÁÜb 25

26. NPIC
@> edaysarGgát;enHrgkugRtaMgTaj eKKUrrgVg;sßitenAxagsþaMG½kSGredaen. p©itrbs;rgVg;sßitenAelIcM
nuc M ehIyrgVg;enHkat;cMnuc O ¬kugRtaMg sx ¦ nigcMnuc Q ¬kugRtaMg s y ¦. dUcenHGgát;p©it
OQ = P / A .
#> MO tMNageGayG½kS X − X nig MQ tMNageGayG½kS Y − Y . bgVilkaM MO RcasRTnic
naLikaeGay)anmMu 2θ kat;rgVg;Rtg;cMnuc D . mMu θ RtUv)ankMNt;dUckñúgrUbTI 17>25 (b).
$> tMélGab;sIusRtg;cMnuc D CakugRtaMgTajEkg ¬ sn ¦ EdlekItmanenAelIbøg;eRTtEdlBicarNa.
tMélelIG½kSGredaenRtg;cMnuc D CakugRtaMgkat; ¬ s's ¦EdlmanenAelIbøg;eRTt. elIsBIenH
cMNafa tMélGtibrmaenAelIG½kSGredaenrbs;rgVg;CakugRtaMgkat;Gtibrma.
kugRtaMgviC¢manmanTisedAeLIgelI nigeTAxagsþaM. Gredaenrbs;cMnuc D bgðajfakugRtaMgkat;man
tMélGviC¢man dUcenHvamanTisedAdUcbgðajenAkñúgrUbTI 17>26.
eyagtamrUbTI 17>26 eyIgeXIjfaTisedArbs;kugRtaMgkat;GviC¢maneFVIeGayFatuenHvilRcasRTnic
naLika.
]TahrN_ 17>8³ dMuEdkmYymanRbEvg 250mm nigmanmuxkat; 100mm × 100mm rgnUvkMlaMgsgát; 140kN .
edayeRbIrgVg;m: (a) kMNt;kugRtaMgEkgenAelIbøg;eRTtEdlmanmMupÁúMrvagG½kSEkgnwgbøg;eRTt nigG½kS X − X
esμInwg 60o RcasRTnicnaLikadUcbgðajenAkñúgrUbTI 17>27 nig(b) kMNt;tMélkugRtaMgkat;Gtibrma nigTI
taMgrbs;bøg;EdlvaeFVIGMeBIelI.
dMeNaHRsay³
(a) edIm,IKUrrgVg;m: eyIgRtUvkMNt;kugRtaMgsgát;GtibrmaEdlmanGMeBIelIbøg;EdlEkgnwgG½kSbeNþay
rbs;Ggát;
P 140000
sy = − =− = −14 MPa
A 100 × 100
Combined Stresses 26

27. T.Chhay
kaMrbs;rgVg;m:KW P / 2 A = 7MPa
eRkayBIKUrrgVg;m:CamYynwgtMélxagelIeyIg)an
sn = CO = 7 + 7 cos 60o = 10.5MPa ¬kugRtaMgsgát;edaysarvasßitenAxageqVgcMnuc O ¦
s 's = DC = 7 sin 60o = 6.06 MPa ¬viC¢manedaysarvasßitenAxagelIG½kS X − X ¦
kugRtaMgTaMgBIrenHRtUv)anbgðajenAkñúgrUbTI 17>27 (c). cMNaMfa TisedArbs;kugRtaMgkat;viC¢man
eFVIeGayGgÁFatuvilRsbRTnicnaLika.
(b) kugRtaMgkat;GtibrmaEtgEtekItmanenAelIbøg;eRTt 45o . ehIytMélrbs;vaRtUvnwg
ME = 7 MPa . ehIykugRtaMgsgát;enAelIbøg;kugRtaMgkat;GtibrmaCatMélGab;sIusrbs;cMnuc E
EdlRtUvKñanwg MO = 7MPa EdlCakugRtaMgsgát;.
kugRtaMgpÁÜb 27

28. NPIC
8> rgVg;m:sMrab;sßanPaBTUeTArbs;kugRtaMg Mohr’s Circle – The General State of Stress
eKGaceRbIrgVg;m:sMrab;kMNt;GaMtg;sIuet nigTisedArbs;kugRtaMgem k¾dUcCakugRtaMgkat;Gtibrma. xag
eRkamCaCMhankñúgkarsg;rgVg;m:sMrab;sßanPaBTUeTArbs;kugRtaMg³
!> KUrGgÁFatuedaybgðajRKb;kugRtaMgTaMgGs;EdlmanGMeBIelIva dUcbgðajenAkñúgrUbTI 17>28 (a) .
@> KUrRbB½n§kUGredaensMrab;rgVg;.
#> eFVItamkarkMNt;rbs;sBaØa³ TajKWbUk/ sgát;KWdk/ kMlaMgkat;eFVIeGayGgÁFatuvilRsbRTnicnaLikaKWbUk
ehIykMlaMgkat;eFVIeGayGgÁFatuvilRcasRTnicnaLikaKWdk.
$> Rkit sx nig s y enAelIG½kSkugRtaMgEkg ¬edk¦
%> Rkit ± sxy EdlrYmKñaCamYynwgkugRtaMgEkgsMrab;épÞQr nigépÞedk. kMNt;CacMnuc X nig Y .
^> KUrbnÞat; XY nigrgVg;m:.
&> kMNt;kugRtaMgem nigkugRtaMgkat;Gtibrma.
*> kMNt; 2θ p1 nig ¬b£¦ 2θ p2 edayyk MX CaeKal.
(> kMNt; 2θ s .
!0> KUrGgÁFatuEdlvilBIr EdlmYysMrab;kugRtaMgem nigmYyeTotsMrab;kugRtaMkat;Gtibrma.
]TahrN_ 17>9³ GgÁFaturbs;Ggát;mYyrgnUvkugRtaMgdUcbgðajenAkñúgrUbTI 17>29. edayeRbIrgVg;m: (a)
kMNt;GaMgtg;sIuetrbs;kugRtaMgem nigkugRtaMgkat;Gtibrma (b) kMNt;TItaMgrbs;bøg;em nigbøg;rbs;kugRtaMg
kat;Gtibrma (c) bgðajkugRtaMgenAelIGgÁFatuEdlrgkarvil.
Combined Stresses 28

29. T.Chhay
dMeNaHRsay³
(a) edayeyagtamkarkMNt;TisedArbs;sBaØaeyIg)an s x = +70MPa / s y = +90MPa / s xy
enAelIépÞQr = +40MPa nig s xy enAelIépÞedk = −40MPa .
BIrgVg;m:kñúgrUbTI 17>30 eyIg)an
s y − sx 90 − 70
GM = MH = = = 10MPa
2 2
kugRtaMgpÁÜb 29

30. NPIC
MY = (MH )2 + (HY )2 = 10 2 + 40 2 = 41.23MPa
s1 = OM + MQ = (70 + 10 ) + 41.23 = 121.23MPa ¬kugRtaMgTaj¦
s 2 = OM − MP = (70 + 10 ) − 41.23 = 38.77 MPa ¬kugRtaMgsgát;¦
kugRtaMgkat;GtibrmaRtUvKñanwgkaMrbs;rgVg;m:
s ' x = MY = ±41.23MPa
kugRtaMgEkgenAelIbøg;kugRtaMgkatGtibrma
s n = OM = 70 + 10 = 80MPa
(b) mMurgVilrbs;bøg;em
XG 70
2θ p 2 = ∠XMG = tan −1 = tan −1 = 81.87 o
GM 10
θ p 2 = 40.94 o ¬RcasRTnicnaLika¦
2θ p1 = 180 o − 81.87 o = 98.13o
θ p1 = 49.06 o ¬RsbRTnicnaLika¦
mMurgVilrbs;bøg;kugRtaMgkat;Gtibrma
2θ s = 2θ p1 − 90 o
b¤ θ s = θ p1 − 45o = 49.06o − 45o = 4.06o
(c) lT§plBI (a) nig (b) RtUv)anbgðajenAkñúgrUbTI 17>31.
Combined Stresses 30

31. T.Chhay
]TahrN_TI 19>10³ Ggát;Edkrbs;eRKOgma:sIunmYymankMras; 150mm RtUv)anbgðajenAkñúgrUbTI 17>32.
kMNt;kugRtaMgTajGtibrmaekItmanenAelImuxkat; A − A edayminKitkugRtaMgRbmUlpþúM (stress
concentration).
dMeNaHRsay³ lkçN³muxkat;rbs;muxkat; A − A man³
A = (75)(150 ) = 11250mm 2
S=
(150)(70)2 = 140625mm 3
6
75
e= = 37.5mm
2
kugRtaMgTajtamG½kSEdlekItBIbnÞúk P enAelIbøg; A − A KW
P 250000
st = + =+ = +22.2MPa
A 11250
kugRtaMgEdlekItBIm:Um:g; couple Pe EdlmanGMeBIelImuxkat; A − A KW
sb = ±
Mc
=±
Pe
=±
Pe
=±
(250000)(37.5) = ±66.7 MPa
I I /c S 140625
kugRtaMgTajGtibrmaCaplbUkBiCKNiténkugRtaMgTaMgBIr
s = 22.2 + 66.7 = 88.9 MPa
kugRtaMgenHekItmanenARCugxagelIbMputrbs;bøg;énmuxkat; A − A .
kugRtaMgpÁÜb 31

32. NPIC
]TahrN_ 17>11³ KNnakugRtaMgTajGtibrmaenAelIbøg; A − A nig B − B énGgát;Edl)anbgðajenAkñúg
rUbTI 17>34 (a). Ggát;enHmanmuxkat; nigTMhMdUcbgðajenAkñúgrUbTI 17>34 (b).
Combined Stresses 32

33. T.Chhay
dMeNaHRsay³
RkLaépÞ nigm:UDulmuxkat;rbs;Ggát;RtUv)anKNnadUcxageRkam³
A = (50 )(150 ) − (38)(138) = 2256mm 2
S=
(50)(150)2 − (38)(138)2 = 66890mm 3
6 6
kugRtaMgTajGtibrmaenAelIbøg; A − A
P M 31200 (13500000)
s=− + =− + = +188MPa
A S 2256 66890
kugRtaMgTajGtibrmaenAelIbøg; B − B
P M 13500000
s= + = = 202MPa
A S 66890
kugRtaMgpÁÜb 33