1. T.Chhay
kugRtaMg nigbMErbMrYlrageFob
Stresses and strain
1> kugRtaMgrgkMlaMgtamGkS½ (Axial stresses or normal stress)
kMlaMgtamGkS½ CakMlaMgEdlmanGMeBIelIGgÁFatuedayExSskmμrbs;kMlaMgsßitenAelIGkS½beNþay
ehIyEkgCamYynwgmuxkat;rbs;GgÁFatuenaH. kMlaMgtamGkS½EckecjCaBI KWkMlaMgTaj nigkMlaMgsgát;.
kMlaMgTaj CakMlaMgEdleFVIeGayGgÁFatumYylUtEvgCagmun. ¬rUbTI1¦
kMlaMgsgát; CakMlaMgEdleFVIeGayGgÁFatumYyrYjxøICagmun.
edIm,IeGayGgÁFatumYyrgnUvkMlaMgtamGkS½manlMnwg luHRtaNaEtva mankMlaMgxag kñúgmanGaMgtg;-
sIuetFMCagb¤ esμIkMlaMgxageRkA EtRtUvmanTisedApÞúynwgTisedAénkMlaMgxageRkA.
enAeBlEdlGgÁFatumYyrgnUvkMlaMgtamGkS½ vaekItmannUvkugRtaMgEdlbNþalmkBIkMlaMgxageRkA
enaH. kugRtaMgKWCa pleFobkMlaMgxageRkA elIRkLaépÞmuxkat;.
B A C
P P
(a) Metal Bar BC
B A
P P1
(b) Free-body BA
rUbTI1 Prismatic metal bar in tension
P
s=
A
Edl s- kugRtaMg (Pa )
P - kMlaMgxageRkA ( N )
A - RkLaépÞmuxkat; (m )
2
kugRtaMg nigbMErbMrYlrageFob 66

2. T.Chhay
]TahrN_³ bnÞHEdkmYymankMras; 5mm nigmanTTwg 4cm RtUv)anrgnUvkMlaMgTaj 20 KN . bnÞHEdkenH
manrn§BIrEdlmanGgát;p©it 7.5mm dUcbgðajkñúgrUb. kMNt;kugRtaMgrgkMlaMgTajRtg;muxkat; A − A nigRtg;
muxkat; B − B .
A B
20KN 20KN
4cm
A B
dMeNaHRsay³
kMNt;kugRtaMgrgkMlaMgTajRtg;muxkat; A − A
P
s=
A
A = 40 × 5 = 200mm 2
P = 20000 N
20000
⇒s= = 100 mm 2 = 100MPa
N
200
kMNt;kugRtaMgrgkMlaMgTajRtg;muxkat; B − B
P
s=
A
A = 40 × 5 − 2 × 7.5 × 5 = 125mm 2
P = 20000 N
20000
⇒s= = 160 mm 2 = 160MPa
N
125
]TahrN_³ ssreQImYymanmuxkat; 200mm × 200mm 500KN
rgnUvkMlaMgsgát; 500KN dUcbgðajkñúgrUb.
ssreQIenHsßitenABIelIeCIgtagebtugEdlmanTMhM 1.2m ×1.2m .
kMNt;kugRtaMgrgkMlaMgsgát;rbs;ssrBIelIeCIgtag.
kMNt;kugRtaMgrgkMlaMgsgát;rbs;)ateCIgtag. 0.2m
¬edayecalbnÞúkpÞal;rbs;ssr nigeCIgtag¦
dMeNaHRsay³ 1.2m
kMNt;kugRtaMgrgkMlaMgsgát;rbs;ssrBIelIeCIgtag 1.2m
kugRtaMg nigbMErbMrYlrageFob 67

3. T.Chhay
P
s=
A
A = 200 × 200 = 40000mm 2
P = 500000 N
500000
⇒s= = 12.5 mm2 = 12.5MPa
N
40000
kMNt;kugRtaMgrgkMlaMgsgát;rbs;)ateCIgtag
P
s=
A
A = 1200 × 1200 = 1440000mm 2
P = 500000 N
500000
⇒s= = 0.35 mm2 = 0.35MPa
N
1440000
2> kugRtaMgkMlaMgkat; (Shear stresses)
kugRtaMgkMlaMgkat; CakugRtaMgEdlekItmantamTisedARsbCamYymuxkat; EdlkMlaMgmanGMeBIelIva.
eKGacehAvamü:ageTotfa kugRtaMgkMlaMgb:H (tangential stress). rUbTI2
kugRtaMgkMlaMgb:HsMEdgedayrUbmnþxageRkam³
P
ss =
A
Edl ss- kugRtaMgkMlaMgkat; (Pa )
P - kMlaMgkat;xageRkA ( N )
A - RkLaépÞmuxkat;EdlekItmankugRtaMgkMlaMgkat; (m ) 2
a b
P
c d
rUbTI2 Shear stress
]TahrN_³ bnÞHEdkBIrRtUv)antP¢ab;Kña edayeRbIb‘ULúgBIrEdlmanGgát;p©it 12mm ¬emIlrUb¦. bnÞHEdkenH
rgnUvkMlaMgTaj 18KN . kMNt;kugRtaMgkMlaMgkat;kñúgb‘ULúg.
18KN
sheared surface
18KN
(a) Joint (b) sheared bolt
kugRtaMg nigbMErbMrYlrageFob 68

4. T.Chhay
dMeNaHRsay³
b‘ULúgmYymanmuxkat;
πd 2
A= = 113mm 2
4
b‘ULúgmYyrgnUvkMlaMgkat;
18
P= = 9 KN
2
b‘ULúgrgnUvkugRtaMgkt;
P 9000
ss = = = 79.65 N mm 2 = 79.65MPa
A 113
]TahrN_³ bnÞHEdkmYymankMras; t RtUv)aneKykeTAecaHrn§eGaymanGgát;p©it d edayeRbIkMlaMg P .
kMNt;kugRtaMgkMlaMgkat;rbs;EdkenAeBlecaHrn§.
P
d
P
d
t t
shear stress
dMeNaHRsay³
kMNt;kugRtaMgkMlaMgkat;
tamrUbmnþ³ s = PA
s
muxkat;EdlrgkMlaMgKWRkLaépÞxagrbs;rn§
dUcenH A = πdt
P
⇒ ss =
πdt
3> bMErbMrYlrageFobtambeNþay nigbMErbMrYlrag (Tensile and compressive strain and
deformation)
r)armYyenAeBlrgkMlaMgtamGkS½beNþay eBlenaHRbEvgedIm L ebs;vakøayeTACa L' )ann½yfa
r)arenaHmanbMErbMrYlrag δ = L'− L .
bMErbMrYlrageFobKWCapleFobbMErbMrYlragelIRbEvgedIm. bMErbMrYlrageFobRtUv)anKitCa cMnYnBit.
kugRtaMg nigbMErbMrYlrageFob 69

5. T.Chhay
δ
ε= P P
L
δ/2 L δ/2
L'
4> pleFobBr½sug (Poisson’s ratio)
tamkarBiesaFn_ eKsegÁteXIjfa GgÁFatumYymanbMErbMrYlrageFobtambeNþay nigbMErbMrYlrag
eFobtamTTwgsmamaRtKña. tMélénpleFobrvagbMErbMrYlrageFobtamTTwg nigbMErbMrYlrageFobtam
beNþay mantMélefrsMrab;sMPar³eGLasÞic. tMélénpleFobenHeKeGayeQμaHfa pleFobBr½sug.
μ=
bMErbMrYlrageFobTTwg
bMErbMrYlrageFobbeNþ ay
Final transverse Initial transverse
dimension dimension
P P
Initial axial dimension
Final axial dimension
5> bMErbMrYlrageFobb:H (Shear strain)
enAeBlNa EdlkMlaMgkat;GnuvtþeTAelIGgÁFatumYy enaHvanwgekItmanbMErbMrYlragb:H EdleRTteTA
tamTisrbs;kMlaMg. δ bMErbMrYlragb:H enaH ε CabMErbMrYlrageFobesμInwg³
s s
δs
εs =
L
P δs P
L L εs
P P
6> düaRkamkugRtaMg-bMErbMrYlrageFob (Stress-strain diagram)
düaRkamkugRtaMg-bMErbMrYlrageFobrbs;GgÁFatubgðajeGayeyIgeXIjnUvB½t’mancaM)ac;mYycMnYn edIm,I
yl;BIsßanPaBersIusþg;rbs;GgÁFatukñúgkarrgkMlaMgTaj. düaRkamkugRtaMg-bMErbMrYlrageFobKMrU bgðajkñúgrUb
xageRkam RtUv)anykmkbgðajedIm,IbriyayBItMbn;énkarrgkugRtaMgrbs;GgÁFatu.
kugRtaMg nigbMErbMrYlrageFob 70

6. T.Chhay
ExSekag OABCDEF kñúgrUbxagelI bgðajBIExSekagkugRtaMg-bMErbMrYlrageFobenAeBlEdlsMNak
KMrUGgÁFatu RtUv)aneGayTTYlnUvbnÞúkTaj ¬mYyTis¦ ekIneLIgbnþicmþg²CalMdab;. ExSekagenHEckecjCa
eRcIntMbn;dUcxageRkam³
- OAB³ bgðajBIExSekagbnÞat;Rtg; bMErbMrYlrageFobekIneLIgCasmamaRtCamYynwgkugRtaMgEdlGnu
eLameTAtamc,ab;h‘uUk (Hook’s law).
- A³ bgðajBIEdnsmamaRt (limit of proportionality) EdlkugRtaMgekIneLIgmanlkçN³CaExS
bnÞat;Rtg;.
- B³ bgðajBIEdneGLasÞic (elastic limit) EdlCacMnuckugRtaMgGtibrma enAeBlEdleKElgbnÞúk
mkvij RbEvgsMNakKMrUrt;eTArksPaBedImvij.
- C’C³ bgðajBIcMnucrMhUrEpñkxagelI nigEpñkxageRkam (upper/lower yield point) CakugRtaMgEdl
ekItmanEtkarekIneLIgbMErbMrYlrageFob edayKμankarekIneLIgnUvkugRtaMgeToteT. tMélrbs;kug
RtaMgEdlRtUvKñacMnucrMhUrEpñkxagelI C’ GaRs½yeTAnwgragénmuxkat;énsMNakKMrU nigGaRs½yeTA
nwgRbePTGgÁFatuEdlykmkeRbIR)as;kñúgkarBiesaFn_enH. cMENkÉtMélrbs;kugRtaMgEdlRtUvKμanwg
cMnucrMhUrEpñkxageRkam C CakugRtaMgKNna b¤kugRtaMgrMhUr (yield stress) f . cMnucrMhUrEpñkxag
y
eRkamRtUv)anykmkKit RbsinebIGRtaénkardak;bnÞúkyWt EtpÞúyeTAvij cMnucrMhUrEpñkxagelI
RtUv)anykmkKit RbsinebIGRtaénkardak;bnÞúkmansPaBelOn. sMrab;tMbn; OC’ GgÁFatuman
lkçN³yWt (elastic) ehIyCMral E Cam:UDulyauMg (Young’s modulus) b¤ m:UDuleGLasÞic. CaTUeTA
E mantMélesμI 2 ×10 N / mm . bMErbMrYlrageFobenAcMnuckugRtaMgrMhUr mantMélesμI 0.0012 .
5 2
kugRtaMg nigbMErbMrYlrageFob 71

7. T.Chhay
- CD ³ bgðajBIrMhUr)øasÞic (plastic yielding) vaCacMnucEdlbMErbMrYlrageFobekIneLIg bnÞab;BIcMnucrMhUr
edayKμankarekIneLIgnUvkugRtaMg. cMnuc D énEdnrMhUr)øasÞic kugRtaMgEtgmantMélERbRbYl b:uEnþ
bMErbMrYlrageFobmantMél 0.014 . bMErbMrYlrageFobkñúgEdn CD CaTUeTAmantMélesμI 10 dgénbMEr
bMrYlrageFobenAcMnucrMhUr.
- DE³ bgðajBIbMErbMrYlrageFobrbwug (strain hardening) CatMbn;EdlekItmankarekIneLIgbEnßmnUv
kugRtaMg CamYynigkarekIneLIgnUvbMErbMrYlrageFob. bMErbMrYlrageFobekIneLIgy:agrh½sCamYynig
kugRtaMgrhUtdl;bnÞúkGtibrma. tMbn;enHminRtUv)anykmkKitkñúgkarKNnaeT.
- E³ bgðajBIkugRtaMgGtibrma EdlCacMnuckugRtaMgRtUvKñanwgbnÞúkGtibrma. CMralxagedImrbs;ExS
ekagenH mantMélRbEhl 4% énm:UDulyuaMg. CaTUeTAenAeBlEdlbMErbMrYlrageFobxiteTACittMél
y:agtic 0.2 eBlenaHkugRtaMgxiteTACittMélGtibrma.
- EF³ CatMbn;EdlkugRtaMgFøak;cuHCamYynwgkarekIneLIgy:agelOnrbs;bMErbMrYlrageFob rhUtdl;sM
NakKMrUdac;.
- F³bgðajBIkugRtaMgdac; (breaking stress) EdltMélrbs;kugRtaMgRtUvKñanwgbnÞúkdac;.
enARtg;tMbn;eGLasÞic eyIgeXIjfabMErbMrYlrageFob ekIneLIgsmamaRteTAnwgkugRtaMg. tamc,ab;
h‘Uk³
E=
ε
s
sMrab;karGnuvtþn¾kMlaMgmYyTis
Edl E - m:UDuleGLasÞic MPa vakMNt;lkçN³emkanicrbs;rUbFatu kalNaGgÁFatumYyman E
kan;EtFM enaHersIusþg;kñúgkarTaj nigkarBt;rbs;vakna;EtFM.
s - kugRtaMgEdlekIteLIgedaysarkMlaMgtamGkS½ MPa
ε - bMErbMrYlrageFob
edayCMnYs s = P nig ε = δ eTAkñúgsmIkarxagelIeyIg)an
A L
P
PL
E= A=
δ Aδ
L
dUcenH bMErbMrYlragkMNt;eday
PL
δ=
AE
kñúgkrNI EdlGgÁFaturgkMlaMgbITisdUcrUbxageRkamenaH
m:UDuleGLasÞicRtUv)ankMNt;dUcxageRkam³
kugRtaMg nigbMErbMrYlrageFob 72

8. T.Chhay
sz sx sy
E= − − Z
εz εy εx
ε ε
⇔ Eε z = s z − z s x − z s y Pz
εx εy
ε ε
eday μ= z = z
εx εy
X
Py
1 Y
⇒E= ( s z − μ ( s x + s y ))
εz
1
⇒ εz = ( s z − μ ( s x + s y ))
Px Px
E
εz CabMErbMrYlrageFobtamGkS½ Z Py
dUcKñasMrab;bMErbMrYlrageFobtamGkS½ Y nig tamGkS½ X
1
εy = ( s y − μ ( s x + s z ))
E
1
ε x = ( s x − μ ( s y + s z )) Pz
E
G Cam:UDulénPaBrwgrbs;rUbFatuTb;nwgkMlaMgkat;mantMélesμI
ss
G=
εs
RbsinebIeKsÁal; m:UDuleGLasÞicrbs;rUbFatu nigpleFobBr½sugeKGackMNt;m:UDulénPaBrwgtamTMnak;
TMngxageRkam³
E
G=
2(1 + μ )
manxñatdUc E .
G
]TahrN_³ dMumYymanTMgn; 6700 N RtUv)anBüÜredayExSlYsEdkEdlmanRbEvg 7.5m . kugRtaMgrgkMlaMg
TajenAkñúgExSlYsEdkminRtUvmantMélelIsBI s = 138MPa nigbMErbMrYlragsrubminelIsBI 4.5mm .
( all )
kMNt;Ggát;rbs;ExSlYsEdkRKb;RKan;edIm,IBüÜrdMuenH ebIExSlYsEdkenHman E = 207000MPa .
dMeNaHRsay³
muxkat;RtUvkarsMrab;tMélkugRtaMgGnuBaØati
P 6700
A= = = 48.6 × 10 −6 m 2 = 48.6mm 2
s 138 × 10 6
muxkat;RtUvkarsMrab;tMélbMErbMrYlragGnuBaØati
PL 6700 × 7.5
A= = = 0.0539 × 10 −3 m 2 = 53.9mm 2
δE 0.0045 × 207000 × 10 6
muxkat;EdlmantMélFMRtUv)aneRCIserIs
4A 53.9
⇒d = = = 8.28mm
π 0.7854
kugRtaMg nigbMErbMrYlrageFob 73

9. T.Chhay
7> kugRtaMgekItedaysarkMedA (Thermal effects)
CaFmμta RKb;rUbFatuTaMgGs;EtgEtERbRbYlTMhMenAeBlNaEdl vargnUvkMedABIxageRkAERbRbYl. bMEr
bMrYlrag δ rbs;va ERbRbYlsmamaRteTAnwgbMErbMrlénkMedACamYynwgemKuNrIkedaykMedA α mYysMrab;
Y
rUbFatumYy.
δ = αL(ΔT )
Edl - bMErbMrYlragtamRbEvg (m)
δ
L - RbEvgedIm (m )
α - emKuNrIkedaykMedA Linear coefficient of thermal expansion (m o)
m. c
ΔT - bMErbMrYlkMedA ( c)
o
enAeBlEdl GgÁFatuenaHRtUv)ancab;Cab;cugsgxag ehIyTTYlnUvbMErbMrYlkMedA enaHvaekItmannUv
kugRtaMg. enAeBlmanbMErbMrYlkMedA RbEvgrbs;vaRtUvERbRbYl edIm,IrkSanUvRbEvgedImrbs;va enaHvaRtUvrgnUv
kMlaMgtamGkS½.
PL L
δ= =s
AE E
L
⇒ s = αL(ΔT )
E
⇒ s = E.α (ΔT )
Edl - kugRtaMgekIteLIgedaysarkMedAenAeBlcugsgxagrbs;vaRtUv)ancab;Cab; (MPa)
s
E - m:UDuleGLasÞic (MPa)
α - emKuNrIkedaykMedA Linear coefficient of thermal expansion (m o)
m. c
ΔT - bMErbMrYlkMedA ( c)
o
]TahrN_³ pøÚvedIrrbs;]bkrN¾sÞÚcRtUv)andak;dMeLIgedayeGaymancenøaHBIKña 3.2mm enAsItuNðPaB 15 c . o
RbEvgrbs;EdkKW 18m . EdkEdlykmkeRbImanemKuNrIkedaykMedA 11.7 × 10 m m. c nigmanm:UDul
−6
o
eGLasÞic 207000 MPa .
k> kMNt;sItuNðPaB EdleFVIeGaycugEdkb:HKña.
x> kMNt;cnøaH rbs;EdkenAeBlsItuNðPaBcuHdl; − 10 c .
o
K> kMNt;kugRtaMgrgkarsgát;EdlekItedaysarkMedA enAeBlsItuNðaPaBeLIgdl; 45 c . o
dMeNaHRsay³
k> kMNt;sItuNðPaB EdleFVIeGaycugEdkb:HKña
edIm,IeGaycugsgxagrbs;Edkb:HKña luHRtaEtcenøaH 3.2mm rbs;vaesμI 0mm enAeBlEdlsItuNðPaBekIneLIg
kugRtaMg nigbMErbMrYlrageFob 74

10. T.Chhay
δ 3.2 × 10 −3
ΔT = = = 15.2 o c
α .L 0.0000117 × 18
eday ΔT = T f − Ti
⇒ T f = ΔT + Ti
⇒ T f = 15.2 + 15 = 30.2 o c
x> kMNt;cnøaH rbs;EdkenAeBlsItuNðPaBcuHdl; − 10 c o
bMErbMrYlsItuNðPaB ΔT = 15 − (−10) = 25 c o
⇒ δ = α .L.(ΔT )
⇒ δ = 0.0000117 × 18 × 25 = 5.27 × 10 −3 m = 5.27mm
enAeBlEdlkMedAcuHRtCak; EdknwgrYjxøICagmun dUcenHcenøaHEdkXøatq¶ayCagmun
Δ = 3.2 + 5.27 = 8.47 mm
K> kMNt;kugRtaMgrgkarsgát;EdlekItedaysarkMedA enAeBlsItuNðaPaBeLIgdl; 45 c o
eyIgeXIjfa enAsItuNðPaB 30.2 c cenøaHrbs;va)anb:HKña
o
dUcenH enAeBlEdlsItuNðPaBekIneLIgFMCag 30.2 c eBlenaHEdknwgrgnUvkugRtaMgsgát;
o
enAeBlEdlsItuNðPaBekIneLIgdl; 45 c enaHbMErbMrYlsItuNðPaBEdleFVIeGayEdkrgkugRtaMgKW
o
ΔT = 45 − 30.2 = 14.8 o c
⇒ s c = E.α .(ΔT )
⇒ s c = 207000 × 0.0000117 × 14.8 = 35.8MPa
8> eRKOgbgÁúMpÁúMeLIgedaysMPar³BIrb¤eRcIn (Members composed of two or more material)
enAeBlxøHeRKOgbgÁúMRtUv)anpÁúMeLIgedaysMPar³BIrRbePTepSgKña EtCaTUeTArUbFatuEtgmanm:UDul
eGLasÞicepSg²Kña. enAeBlEdlsMPar³enaHTTYlnUvbnÞúk eBlenaHkugRtaMgEdlekItmankñúgsMPar³RtUvEbg
EcktamsmamaRténm:UDuleGLasÞicrbs;va. P
edaysarEtvamanbMErbMrYlragdUcKña enaHkugRtaMgEdlekItman
kñúgsMPar³ A Edlmanm:UDuleLasÞicFM RtUvmantMélFMCagkugRtaMgEdl
ekItmankñúgsMPar³ B Edlmanm:UDuleGLasÞuictUcCag. Wood
δA = δB ⇒ ε A = εB
eday ε = E
s
sA s
⇒ = B
E A EB Steel P
kugRtaMg nigbMErbMrYlrageFob 75

11. T.Chhay
EA
⇒ sA = sB
EB
eday n - CapleFobm:UDuleGLasÞic
EA
n=
EB
⇒ s A = ns B
mü:ageToteday sMPar³TaMgBIr A nig B TTYlnUvbnÞúk P enaH
P = PA + PB
Eteday P = s.A
⇒ P = s A . AA + s B . AB
⇒ P = AA (ns B ) + AB .s B
⇒ P = s B (nAA + AB )
tYr nA RtUv)aneKeGayeQμaHfa RkLaépÞsmmUl. RkLaépÞ A RtUv)anbMElgBIm:UDuleGLasÞic
A A
E eTAm:UDuleGLasÞic E EdlmanRkLaépÞsmmUl nA . enaHeKnwg)ansMPar³TaMgBIrmanm:UDuleGLasÞic
A B A
dUcKña.
P
⇒ sB =
(nAA + AB )
AB .P
⇒ PB =
(nAA + AB )
nig ⇒ PA = P −
AB .P
=
nPAA
(nAA + AB ) (nAA + AB )
]TahrN_³ eKmanEdk steel rod mYyEdlmanGgát;p©it 50mm manRbEvg 254mm RtUv)andak;enAkñúgEdk
TIb brass tube mYyEdl manGgát;p©itxagkñúg 50mm nigGgát;p©itxageRkA 75mm dUcbgðajkñúgrUb.
eRKOgbgÁúMrgnUvkMlaMgTaj EdlmankMlaMg 300kN . sMPar³TaMgBIrRtUv)anpSaP¢ab;edaybnÞHrwg edIm,IeGaysM
Par³TaMgBIreFVIkarCalkçN³EtmYy ehIysMPar³TaMgBIrlUtkñúgRbEvgdUcKña. kMNt;
k> kugRtaMgEdlekItmanenAkñúg steel nig brass Tg;Edg
x> kMlaMgEdlTb;edaysMPar³TaMgBIr Rigid plate
Brass tube
Rigid plate
K> bMErbMrYlragCaRbBn§½
300kN 300kN
E st = 200000 MPa
Ebr = 97000 MPa Steel rod
dMeNaHRsay³
kugRtaMg nigbMErbMrYlrageFob 76

12. T.Chhay
RkLaépÞmuxkat;rbs;Edk steel
πd 2
Ast = = 1963.5mm 2
4
RkLaépÞmuxkat;rbs;Edk brass
π (D 2 − d 2 )
Abr = = 2454.4mm 2
4
pleFobm:UDuleGLasÞic
E st 200000
n= = = 2.06
E br 97000
k> kugRtaMgEdlekItmanenAkñúg steel nig brass
eyIgman P = s (nA + A )
br st br
P 300000
⇒ sbr = = = 46.2 MPa
nAst + Abr 2.06 × 1963.5 + 2454.4
kugRtaMgkñúgEdk steel
s st = nsbr = 2.06 × 46.2 = 95.2 MPa
x> kMlaMgEdlTb;edaysMPar³TaMgBIr
kMlaMgkñúgEdlekItmankñúg brass
Pbr = sbr Abr = 46.2 × 2454.4 = 113400 N = 113.4kN
kMlaMgkñúgEdlekItmankñúg steel
Pst = s st Ast = 95.2 × 1963.5 = 186900 N = 186.9kN
K> bMErbMrYlragCaRbBn§½
tamc,ab;h‘Uk eyIg)anbMErbMrYlrageFob
s st s 95.2
ε= = br = = 0.000476
E st E br 200000
bMErbMrYlragCaRbBn§½
δ = ε .L = 0.000476 × 254 = 0.121mm
9> kugRtaMgelIbøg;eRTt (Stresses of inclined planes)
enAeBlr)armYyrgnUvkMlaMgTaj b¤kMlaMgsgát;tamGkS½ kugRtaMgrgkMlaMgTaj b¤sgát;ekIteLIgelI
bøg;énmuxkat;EkgeTAnwgGkS½beNþayrbs;r)ar. elIsBIenH kugRtaMgrgkMlaMgTaj b¤sgát;EdlmanGaMgtg;
sIuettUcCag EdlCakugRtaMgkMlaMgkat; ekItmanelIbøg;eRTt.
kugRtaMg nigbMErbMrYlrageFob 77

13. T.Chhay
RbsineKkat;r)armYyEdlrgkMlaMgTaj P tambøg; CD Edlbøg;enHeRTtmMu θ CamYybøg;edk. Binitü
GgÁesrIEpñkxageRkam eyIgeXIjfakMlaMg P bMEbk)anbgÁúMkMlaMgBIrKW kMlaMgEkgnwgbøg;énmuxkat; nigkMlaMg
RsbCamYynwgmuxkat;.
P
longitudinal axis
of member
Pcosθ P
Tensile stress
θ θ θ
D Psinθ D D
A B Shear stress
C C C
P P P
(a) Axial loaded (b) Forces on (c) Stresses on
member inclined plane inclined plane
kMlaMgEdlEkgCamYynwgmuxkat; CD: P cosθ
kMlaMgEdlEkgCamYynwgmuxkat; CD: P sin θ
RbsnebIr)arenHmanmuxkat;Ekg A
enaH muxkat;eRTt CD: cosθ
A
bgÁMúkMlaMgRsbEdlmanGMeBIelImuxkat;eRTt begáIt)ankugRtaMgkMlaMgkat;
P sin θ P P
s' s = = sin θ cos θ = sin 2θ
A A 2A
cos θ
tMélénkugRtaMgkMlaMgkat;mantMélGtibrma enAeBlEdl sin 2θ = 1 enaHeyIg)an θ = 45 o
P
⇒ s ' s (max) =
2A
bgÁMúkMlaMgEkgEdlmanGMeBIelImuxkat;eRTt begáIt)ankugRtaMgkMlaMgTaj
P cos θ P
sn = = cos 2 θ
A A
cos θ
kugRtaMg nigbMErbMrYlrageFob 78

14. T.Chhay
tMélénkugRtaMgTajmantMélGtibrma enAeBlEdl cos 2θ = 1 enaHeyIg)an θ = 0 . o
eyIgeXIjfakugRtaMgkat;GtibrmaekItelIbøg;eRTt 45 mantMéltUcCagkugRtaMgTajekItelIbøg;edk (0 ) 2
o o
dg.
]TahrN_³ eKmanGgát;mYymanGgát;p©it 150mm EdleFVIBIsMPar³EdlmanlT§PaBFn;nwgkugRtaMgsgát;
82.7 MPa niglT§PaBFn;nwgkugRtaMgTaj 34.5MPa . kMNt;kMlaMgGtibrmaEdlGnuvtþn¾mkelIGgát;enH.
dMeNaHRsay³
kugRtaMgsgát;GtibrmaEdlmanenAkñúgGgát; ekItmanenAeBl θ = 0 o
P
⇒ s n (max) =
A
150 2
⇒ P = s n (max) . A = 82.7 × π = 1461kN
4
kugRtaMgkat;GtibrmaEdlmanenAkñúgGgát; ekItmanenAeBl θ = 45 o
P
⇒ s ' s (max) =
2A
150 2
⇒ P = 2 s ' s (max) . A = 2 × 34.5 × π × = 1219kN
4
dUcenH kMlaMgGtibrmaEdlGnuvtþn¾mkelIGgát;mantMél 1219kN CatMélEdlmantMéltUcCageKkñúgcMeNam
tMélTaMgBIr.
10> kugRtaMgkMlaMgkat;elIbøg;Ekg (Shear stresses on mutually perpendicular planes)
shear force that result ss2
in shear stress s s1 C D
ss1
H
ss1
A ss2 B
Element
ABCD W
(a) Member stressesd
in shear (b) Stressed element
enARKb;cMnucTaMgGs;rbs;Ggát;EdlrgkugRtaMgkMlaMgkat;enAelIbøg; enaHvaekItmanCamYyKñaenAkug
RtaMgkMlaMgkat;EdlEkgnwgbøg;enaH.
kugRtaMg nigbMErbMrYlrageFob 79

15. T.Chhay
RbsinebIeKmankMlaMgkat;enAelIbøg; EdlbegáIt)ankugRtaMgkat; s enAEpñkxagsþaMénmuxkat;. Binitü
s1
FatuGnnþtUc ABCD EdleyIgsnμt;favamankMras; 1Ékta (1 unit) kMBs; H nig TTwg W .
kMlaMgkat;EdlekItBIkugRtaMg s mkelIFatuGnnþtUc ¬bøg; BD¦
s1
VBD = s s1 .H .1 ↑
kMlaMgkat;EdlmanGMeBImkelIFatuGnnþtUc ¬bøg; AC¦
∑ Y = 0 ⇒ V AC = s s1 .H .1 ↓
edaykMlaMg V nig V manGaMgtg;sIuetesμIKña EtTisedApÞúyKña enaHekItmannUv kMlaMgbgVil
BD AC
C1 = s s1 .H .1.W
eBlenaH FatuGnnþtUc ABCD minmanlMnwg. edIm,IeGayvamanlMnwgeTA)an luHRtaEtvamanm:Um:g;KUr
mYyeTot EdlmanGaMgtg;sIuetesμIKña EtTisedApÞúyKña.
RbsinebI eKmankugRtaMgkat; s ekItmanenAbøg;Epñkxagel I¬bøg; CD¦ enaH eK)ankMlaMgkat;
s2
VCD = s s 2 .W .1 →
edIm,ImanlMnwgtamTisedA X enaHeK)an
∑ X = 0 ⇒ V AB = s s 2 .W .1 ←
⇒ C 2 = s s 2 .W .1.H
∑ C = 0 ⇒ s s1 = s s 2
dUcenH eyIgeXIjfa kugRtaMkMlaMgkat;enAelIbøg;minGacekIteLIgedayÉkÉg)aneT vaRtUvbegáItkug
RtaMgkMlaMgkat;EdlmanGaMgtg;sIuetesμIKñaelIbøg;EkgedIm,IrkSalMnwg.
11> kMlaMgTaj nigkMlaMgsgát;ekItBIkMlaMgkat; (Tension and compression caused by shear)
eKmanFatuGnnþtUcmYy EdlmankMras; 1Ékta (1 unit) kMBs; H nig TTwg W rgnUgkugRtaMgkMlaMg
kat; s . s
Longitunal axis ss .W.1.sinθ
of member
θ
R R
ss .W.1 ss .W.1
θ ss .H.1 ss .H.1 W
ss .H.1.cosθ H ss .W.1.cosθ D s' .D.1
s
ss .H.1.sinθ
H
ss .H.1 θ
R s s .W.1 R
W sn.D.1
(a) Stressed member (b) Stressed on inclined plane
kugRtaMg nigbMErbMrYlrageFob 80

16. T.Chhay
kMlaMgkat;EdlmanGMeBIenAelIbøg;
V = s s .H .1
kMlaMgkat;EdlmanGMeBIenAelIbøg;Ekg
V = s s .W .1
RbsinebIeKkat;FatuGnnþtUcenaH BIRCugEkgmçageTARCugEkgmçag tambøg; R − R . mMu θ CamMuCMral
rbs;bøg; R − R . RbsinebI D CaRbEvgGgát;RTUg ¬bøg; R − R ¦ enaHkMlaMgEdlmanGMeBIelIGgát;RTUgenaHKW
man³
- kMlaMgkat; s' .D.1 s
- kMlaMgTaj s .D.1 n
Edl s' CakugRtaMgkat;EdlmanGMeBIelIGgát;RTYg nig s CakugRtaMgTajEdlmanGMeBIEkgeTAnwg
s n
Ggát;RTUg.
plbUkBiCKNitkMlaMgEdlEkgeTAnwgépÞGgát;RTUg³
s n .D.1 = s s .H .1. cos θ + s s .W .1. sin θ
H W
⇒ s n = s s ( ) cos θ + s s ( ).1. sin θ
D D
eday sin θ =
H
D
nig
cos θ =
W
D
eyIg)an³
s n = s s . sin θ . cos θ + s s . cos θ . sin θ
⇒ s n = s s sin 2θ
edIm,IeGaykugRtaMgTajmantMélGtibrmaluHRtaEt
sin 2θ = 1 ⇒ 2θ = 90 o ⇒ θ = 45 o
plbUkBiCKNitkMlaMgEdlRsbeTAnwgépÞGgát;RTUg³
s ' s .D.1 = s s .W .1. cos θ − s s .H .1. sin θ
W H
⇒ s' s = s s ( ) cos θ − s s ( ) sin θ
D D
eday sin θ =
H
D
nig cos θ =
W
D
eyIg)an³
s ' s = s s . cos θ . cos θ − s s . sin θ . sin θ
⇒ s' s = s s . cos 2θ
enAeBlEdl θ = 45 enaHkugRtaMgkMlaMgkat;EdlekItmanenAelIbøg;Ggát;RTugmantMélesμIsUnü.
o
kugRtaMg nigbMErbMrYlrageFob 81

17. T.Chhay
eyIgeXIjfa enAeBlGgÁFatumYyrgnUvGMeBIkMlaMgkat;TTwgsuT§ enaHkugRtaMgTaj b¤sgát;)anekItman
enAelIbøg;Ggát;RTUgEdlCalT§plénkugRtaMgkMlaMgkat;. kugRtaMgTaj b¤sgát;GtibrmaekItmanenAelIbøg;
eRTtEdlmanmuM θ = 45 ehIymanGaMgtg;sIuetesμIeTAnwgkugRtaMgkat;.
o
]TahrN_³ PaKtUcEdlykecjBIeRKOgbgÁúMCBa¢aMgesþIgragsIuLaMgrgnUvkMlaMgrmYl rgnUvkugRtaMgkMlaMgkat;
suT§dUcbgðajkñúgrUb. KNnaGaMgtg;suIeténkugRtaMgEkg nigkugRtaMgkat;EdlekItmanenAelIbøg;Ggát;RTUg Edl
manmMu θ = 30 nig θ = 45 .
o o
Longitunal axis
of member
θ
70MPa Diagonal
plane
70MPa 70MPa
70MPa
dMeNaHRsay³
KNnaGaMgtg;suIeténkugRtaMgEkg nigkugRtaMgkat;EdlekItmanenAelIbøg;Ggát;RTUg
sMrab;mMu θ = 30 o
kugRtaMgEkgEdlekItmanenAelIbøg;Ggát;RTUg
s n = s s sin 2θ = 70 × sin 60 o = 60.6 MPa
kugRtaMgkat;EdlekItmanenAelIbøg;Ggát;RTUg
s ' s = s s . cos 2θ = 70 × cos 60 o = 35.0 MPa
sMrab;mMu θ = 45 o
kugRtaMgEkgEdlekItmanenAelIbøg;Ggát;RTUg
s n = s s sin 2θ = 70 × sin 90 o = 70MPa
kugRtaMgkat;EdlekItmanenAelIbøg;Ggát;RTUg
s ' s = s s . cos 2θ = 70 × cos 90 o = 0 MPa
kugRtaMg nigbMErbMrYlrageFob 82

18. T.Chhay
lMhat;³
1> cUrkMNt;kugRtaMgEdlekItmanenAelIGgát;EdlrgnUvkMlaMgTajcMGkS½ 5tf . ebIGgát;enaHmanmuxkat;³
k> 5cm × 5cm
x> R100(mm)
K> W 840 × 6.19
2> cUrkMNt;kugRtaMgenAelIkMNat;r)arTaMgBIrxageRkamEdlmanGgát;p©iterogKña 3cm nig 5cm
3> cUrkMNt;nUvmuxkat;EdkEdlmanlkçN³snSMsMéc edayeKeGayvaERbRbYlral; 2mm edIm,IrgnUv
kMlaMgTaj 16tf RbsinebIeKmineGayEdkenaHrgnUvkugRtaMgTajelIsBI 140MPa .
4> kMNt;TMgn;F¶n;bMputrbs;kUnTMgn; W EdlGacBüÜr)anedayr)arBIrdUcbgðajkñúgrUb. kugRtaMgenAkñúgr)ar
AB nig BC minRtUveGaymantMélelIsBI 100 MPa nig 150 MPa erogKña. muxkat;rbs;r)ar
AB = 400mm nig muxkat;rbs;r)ar BC = 200mm . cM³ W = 33.445 N
2 2
5> sMrab; Truss dUcbgðajkñúgrUb cUrKNnakugRtaMgenAkñúgGgát; DF / CE nig BD . muxkat;rbs;Ggát;
nImYy² 1200mm . cM³ s = 187.5MPa(C ) s = 112.5MPa(T ) s = 80.1MPa(C )
2
DF CE BD
kugRtaMg nigbMErbMrYlrageFob 83

19. T.Chhay
6> sMrab; Truss dUcbgðajkñúgrUb cUrkMNt;muxkat;rbs;Ggát; BE / BF nig CF edaykugRtaMgrgkar
TajmineGayelIs 100MPa nig kugRtaMgrgkarsgát;mineGayelIs 80MPa . karkat;bnßykug
RtaMgrgkarsgát;edIm,IeCosvagkar)ak;edaykarPøat;. cM³ A = 625mm (T ) A = 656mm (C )
BE
2
CF
2
ABF = 427.2mm 2 (T )
7> r)ar AB Edlmanm:as; 150kg RtUv)andak;eGayBüÜrTMgn; 2kN dUcbgðajkñúgrUb. r)arRtUv)anRT
edayTMr Pin Rtg; B nigExSkab CD EdlmanGgát;p©it 10mm . kMNt;kugRtaMgenAkñúgExSkab.
cM³ s = 87.08MPa(T )
CD
8> kMNt;TMgn;sIuLaMgF¶n;bMputEdlGacdak;enATItaMgdUcbgðajkñúgrUb edaymineGaykugRtaMgkñúgExSkab
BC elIsBI 50 MPa . ExSkabenHmanmuxkat; 100mm . edayecalTMgn;rbs;r)a AB . cM³
2
W = 6000 N
kugRtaMg nigbMErbMrYlrageFob 84

20. T.Chhay
9> FugmYymanTMgn; 75kN RtUv)anBüÜredayEdkEdlmanGgát;p©it 19mm cMnYnbIedIm. KNnakugRtaMg
TajEdlekItmankñúgEdk edaysnμt;fa EdkrgnUvkMlaMgesμI²Kña.
10> r)arEdkEdlmanmuxkat; 25mm ×100mm nigmankugRtaMgTajGnuBaØati 140mm .
cUrkMNt;nUvkMlaMg TajcMGkS½GnuBaØatiEdlGacGnuvtþmkelIr)arEdkenaH.
11> KNnamuxkat;tMrUvkarsMrab;ssreQIxøIragkaerEdlrgnUvbnÞúksgát; 250kN . kugRtaMgsgát;GnuBaØati
8MPa .
12> KNnaGgát;p©itEdkcaM)ac;sMrab;BüÜrnUvranhal. EdknimYy² RtUv)anBüÜrBIdMbUl RTnUvépÞkMral 7m
2
EdlmanbnÞúksrub 10.5kPa . kugRtaMgTajGnuBaØatirbs;EdkenaH 140MPa .
13> GaluymIjÚ:mmYymanGgát;p©it 25mm manRbEvg 3m rgnUvkMlaMgTajtamGkS½ 67kN .
GaluymI-j:Úmmanm:UDuleGLasÞic E = 70×10 MPa . cUrkMNt; kugRtaMg bMErbMrYlrageFob
3
nigbMErbMrYlrag srub.
14> sésEdkEdlmanGgát;p©it 5mm RbEvg 18m RtUv)aneRbIkñúgkarplitebtugeRbkugRtaMg. KNnakM-
laMgTajcaM)ac; P EdleFVIeGaysésEdkenaHlUt)an 45mm nigKNnakugRtaMgEdlekItmanenAkúñg
sésEdk. m:UDuleGLasÞicrbs;Edk E = 207 ×10 MPa .
3
15> EdksésmanGgát;p©it 19mm nigmanRbEvg 30m RtUv)andak;BüÜr. kMNt;bMErbMrYlragsrub Edl
ekItBIbnÞúkpÞal;xøÜn nigbnÞúkxageRkA 36kN GnuvtþenAcugTMenr. TMgn;maDrbs;Edk 77kN / m .
3
m:UDuleGLasÞicrbs;Edk E = 207 ×10 MPa .
3
16> KNnabMErbMrYlragsrubénr)arEdkEdlmanRbEvg 635mm RbsinebIkugRtaMgTaj 103MPa ekIt
manenAkñúgr)ar. m:UDuleGLasÞicrbs;Edk E = 207 ×10 MPa .
3
17> KNnakMlaMgTajmanGMeBIelIr)arEdkmUlEdlmanGgát;p©it 25mm RbsinebIvaekItmanbMErbMrYlrag
eFob 0.0007 .
18> tMNrRCYsmYyEdlP¢ab;edayb‘ULúg 2RKab;Ggát;p©it 25mm dUcbgðajkñúgrUbxageRkam. edaysnμt;
b‘ULúgTaMgBIrrgnUvkMlaMgesμIKña. KNnakugRtaMgkat;TTwgekItmanenAelIb‘ULúg enAeBlEdltMNrenHrg
nUvbnÞúkTajesμI 125kN .
kugRtaMg nigbMErbMrYlrageFob 85

21. T.Chhay
19> KNnakMlaMgcaM)ac;edIm,IecaHrn§nUvbnÞHEdkEdlmankMras; 5mm . rn§EdlRtUvecaHmanGgát;p©it
27 mm . bnÞHEdkenaHmanersIusþg;nwgkarkat;GnuBaØati 400 MPa .
20> Edk ASTM A36 EdlmanGgát;p©it 50mm rgnUvbnÞúkTaj 270kN . ersIusþg;TajrMhUr 234MPa .
KNnabMErbMrYlrageFobtambeNþay nigtamTTwg. RbsinebIEdkenaHmanRbEvg 1m cUrKNnabMErbM
rYlragsrub.
21> kMNt;bMErbMrYlragRKb;TissMrab;EdkrgbnÞúkdUcbgðajkñúgrUbxageRkam. ersIusþg;EdkrMhUr 230MPa .
22> r)arEdkmYymanRbEvg 3m RtUv)andak;enAcenøaHCBa¢aMgBIr nigbnÞab;mkr)arEdkenHRtUv)andutrhUt
sItuNðPaBekIneLIgdl; 27C . RbsinebICBa¢aMgenaHrMkilmçag 0.38mm . etIr)arEdkenaHrgkugRtaMg
o
sgát;b:unμan?
23> r)arTg;EdgmYymanRbEvg 1m RtUv)andak;dUcbgðajkñúgrUbxageRkam. kMNt;kugRtaMgekItmanenAkñúg
r)arEdlekItBIsItuNðPaBekIndl; 60C .o
24> KNnapleFobBr½sugrbs;EdkEfb Edlmanm:UDuleGLasÞic E = 83000MPa nigm:UDulénPaBrwg
G = 33000 MPa .
25> r)arEdkbIRTkUnTMgn;dUcbgðajkúñgrUbxageRkam. KNnakMlaMgEdlRTedayr)armYy².
kugRtaMg nigbMErbMrYlrageFob 86

22. T.Chhay
26> r)arEdkmYymanmuxkat; 300mm nigmanRbEvg 150m RtUv)andak;BüÜrbBaÄrdUcbgðajkñúgrUb.
2
vaRtUv)andak;BüÜrbnÞúk 20kN . RbsinebIm:as;maDrbs;Edk 7850kg / m nig E = 200,000MPa .
3
kMNt;RbEvglUtsrubrbs;r)arEdkenaH. cM³ y = 54.33mm
27> kMralebtugmYyesμIsac;mYymanm:as; M RtUvP¢ab;edayr)arBIrdUcbgðajkñúgrUb. kMNt;pleFobRkLa
épÞrbs;r)ar edIm,IeGaynIv:Urbs;kMralebtugdUcKña. cM³ Aa = 8.57
As
28> r)ar AB sßitkñúgsPaBedkehIyP¢ab;eTAnwgkabbBaÄrBIrmundak;bnÞúkdUcbgðajkñúgrUb. kMNt;bMlas;
TItamTisbBaÄreRkaydak;bnÞúk P = 50kN . cM³ y = 2.286mm / y = 1.5mm / y = 1.814mm
a s P
kugRtaMg nigbMErbMrYlrageFob 87

23. T.Chhay
29> r)armYymanGgát;p©it 25mm rgnUvkMlaMgTajcMGkS½ 80kN . KNna kugRtaMgcMGkS½ nigkugRtaMgkat;
TTwg EdlekIteLIgenAelIépÞeRTtEdlmanmMu 30 nigKNnakugRtaMgcMGkS½ nigkugRtaMgkat;TTwgGtib
o
rma.
30> kMlaMgcMGkS½manGMeBIelIEdkEdlmanmuxkat; 50mm × 75mm EdleFVIeGayekItmankugRtaMgkat;
TTwg 138MPa elIbøg;eRTtEdlmanmMu 50 . KNnabnÞúkcMGkS½EdlmanGMeBIelIr)ar nigKNnakug
o
RtaMgcMGkS½elIbøg;eRTt.
kugRtaMg nigbMErbMrYlrageFob 88