6.stresses and strain23

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6.stresses and strain23

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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