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6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
6.stresses and strain23
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6.stresses and strain23

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  • 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

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