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11. displacement method of analysis slope deflection equations

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  • 1. Department of Civil Engineering NPIC !!> karviPaKtamviFIbMlas;TI³ smIkar slope-deflection Displacement method of analysis: Slope-deflection equations emeronenHnwgerobrab;y:agsegçbnUveKalKMnitsMrab;viPaKeRKOgbgÁúMedayeRbIviFIbMlas;TI. eRkayeBlbgðajBIeKalKMnit eyIgnwgbegáItsmIkarTUeTAén slop deflection ehIybnÞab;mkeyIg nigeRbIvaedIm,IviPaKFñwm nigeRKagminkMNt;edaysþaTic. !!>!> viFIsaRsþTUeTAkñúgkarviPaKtamviFIbMlas;TI (Displacement method of analysis: General procedures) rcnasm<½n§TaMgGs;RtUvEtbMeBjtMrUvkarsmIkarlMnwg TMnak;TMngrvagbnÞúk nigbMlas;TI nigRtUv bMeBjtMrUvkarlkçxNÐRtUvKñaénbMlas;TIedIm,IFananUvsuvtßiPaBrbs;va. enAkñúgkfaxNÐ 10.1 )anerobrab; fa eKmanviFIBIrepSgKñaedIm,IbMeBjtMrUvkarTaMgenHenAeBlviPaKeRKOgbgÁúMminkMNt;edaysþaTic. viFI kmøaMg ¬emeronTI !0¦ Ep¥kelIkarkMNt;kmøaMgEdlCaGBaØatelIs bnÞab;mkbMeBjtMrUvkarlkçxNÐRtUvKña rbs;rcnasm<½n§. eKeFVIrebobenHedaysresrsmIkarbMlas;TIeGayTak;TgnwgkmøaMgedayeRbITMnak;TMng rvagkmøaMg nigbMlas;TI. dMeNaHRsayénsmIkarers‘ultg;CakmøaMgRbtikmμEdlCaGBaØatelIs bnÞab; mkeKeRbIsmIkarlMnwgedIm,IkMNt;kmøaMgRbtikmμEdlenAsl;EdlmanGMeBIelIeRKOgbgÁúM. viFIbMlas;TICaviFIEdlmanlkçN³pÞúyeTAnwgviFIkmøaMg. dMbUgvaRtUvbMeBjtMrUvkarsmIkarlMnwg sMrab;rcnasm<½n§. edIm,IeFVIEbbenH eKRtUvsresrbMlas;TIEdlCaGBaØateGayTak;TgnwgbnÞúkedayeRbI TMnak;TMngrvagkmøaMg nigbMlas;TI. eRkayeBlTTYl)anbMlas;TIrYcehIy eKGackMNt;kmøaMgEdlCa GBaØatBIsmIkarlkçxNÐRtUvKñaedayeRbITMnak;TMngrvagkmøaMg nigbMlas;TI. RKb;viFIbMlas;TITaMgGs; GnuvtþtamviFIsaRsþTUeTAenH. Degrees of freedom enAeBleRKOgbgÁúMrgkmøaMg cMNucCak;lak;enAelIeRKOgbgÁúMEdleKeGay eQμaHfa node nwgrgbMlas;TI. bMlas;TITaMgenHRtUv)aneKKitCa degrees of freedom sMrab;eRKOgbgÁúM ehIyenAkñúgviFIbMlas;TIeKcaM)ac;RtUvkMNt; degrees of freedom TaMgenH edaysarvanwgkøayCaGBaØat enAeBleKGnuvtþviFIbMlas;TI. cMnYnénGBaØatTaMgenHCadWeRkénPaBminkMNt;edaysarsIueNm:aTicrbs; eRKOgbgÁúM. edIm,IkMNt;PaBminkMNt;edaysIueNma:Tic eKGacKitvaeRKOgbgÁúMpÁúMedayes‘rIénGgát;EdltP¢ab; Kñaedaysar node EdlCaTUeTAvasßitenARtg;tMN TMr enAxagcugrbs;Ggát; b¤enARtg;kEnøgbMlas;bþÚrmux kat;Pøam²rbs;Ggát;. enAkñúglMhr (three dimensions), node nImYy²GacmanbMlas;TIlIenEG‘rcMnYn karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -378
  • 2. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa BIr nigbMlas;TImMucMnYnmYYy. elIsBIenH eKGacTb;bMlas;TIrbs; node edayTMr b¤edaykarsnμt;Edl Ep¥kelIkareFVIkarrbs;eRKOgbgÁúM. Ca]TahrN_ RbsinebIeRKOgbgÁúMCaFñwm ehIyeKBicarNaEtkMhUcRTg; RTayEdlbNþalBIm:Um:g;Bt; enaHvaminmanbMlas;TIlIenEG‘rtambeNþayG½kSrbs;FñwmeT edaysar bMlas;TI enHbNþalBIkMhUcRTg;RTayedaysarkmøaMgtamG½kS. edIm,IbBa¢ak;BIeKalKMnitenH eyIgnwg BicarNa]TahrN_xøH edaycab;epþImCamYynwgFñwmenAkñúgrUbTI 11-1a. enATIenH RKb;bnÞúk P Edl GnuvtþeTAelI FñwmnwgeFVIeGay node A vilEtb:ueNÑaH ¬edayecalkMhUcRTg;RTaytamG½kS¦ cMENkÉ node B RtUv)an Tb;mineGaycl½tEtmþg. dUcenH FñwmmanEt degree of freedom EtmYyKW θ A ehIy vaCaeRKOgbgÁúMmin kMNt;edaysIueNm:aTicdWeRkTImYy. FñwmenAkñúgrUbTI 11-1b man node enARtg; A / B nig C dUcenHva man degree of freedom cMnYnbYn EdlkMNt;edaymMurgVil θ A / θ B / θ C nigbMlas; TItamTisQr ΔC . vaCaeRKOgbgÁúMminkMNt;edaysIueNma:TicdWeRkTIbYn. LÚvBicarNaeRKagenAkñúgrUbTI 11-1c. mþgeTot RbsinebIeyIgecalkMhUcRTg;RTaytamG½kSrbs;Ggát; bnÞúk P EdlGnuvtþeTAelI eRKagGaceFVIeGay node B nig C vil ehIy node TaMgenHGacpøas;TItamTisedkedaybrimaNesμIKña. dUcenH eRKagman degree of freedom cMnYnbIKW θ B / θC nig Δ B ehIyvaCaeRKOgbgÁúMminkMNt;eday sIueNm:aTicdWeRkTIbI. Cakarsegçb karkMNt;PaBminkMNt;edaysIueNma:Tic b¤cMnYnrbs; degree of freedom sMrab; eRKOgbgÁúMCaCMhancaM)ac;bMputEdleKRtUveFVImuneK enAeBlGnuvtþviFIbMlas;TI. vakMNt;nUvcMnYnGBaØat EdlQrelIkarsnμt;EdlKitcMeBaHkMhUcRTg;RTayrbs;eRKOgbgÁúM. elIsBIenH enAeBlEdleKsÁal; bMlas;TIrbs; node enaHeKnwgGackMNt;bMlas;TIrbs;eRKOgbgÁúM ehIyeKGacTTYl)ankMlaMgkñúgrbs; Ggát;. Displacement method of analysis: Slope-deflection equations T.Chhay -379
  • 3. Department of Civil Engineering NPIC !!>@> smIkar slope-deflection (Slope-Deflection equation) viFIkmøaMgenAkñúgemeronTI!0 RtUvkarsresrsmIkarEdlTak;TgeTAnwgkmøaMg b¤m:Um:g;EdlCaGBaØat. EteKminGaceRbIvasMrab;eRKOgbgÁúMminkMNt;EdlmandWeRkx<s;)aneT edaysareKRtUvkarbegáItsmIkar lkçxNÐRtUvKñaeRcIn ehIyelIsBIenH smIkarnImYy²Tak;TgnwgGBaØatTaMgGs;EdleFVIeGayeKBi)akedaH RsayRbB½n§smIkar elIkElgEteKeRbIkMuBüÚT½r. edaykareRbobeFob viFI slope-deflection minsμúK sμajdUcviFIkmøaMgeT. dUcEdleyIg)aneXIj vaTamTarkargarticCag TaMgkarsresrsmIkar nigkar edaHRsaysmIkaredIm,IrkbMlas;TI nigkmøaMgkñúg. ehIyelIsBIenH eKGacsresrkmμviFIkMuBüÚT½rCamYy nwgviFIenHy:aggayRsYl ehIyeKGaceRbIvaedIm,IviPaKeRKOgbgÁúMminkMNt;eRcInEbb. viFI slope-deflection RtUv)anbegáIteLIgdMbUgeday ehnrIc Em:ndWLa (Heinrich Manderla) nig GUfU m: (Otto Morh) edIm,IsikSakugRtaMgbnÞab;bnSM (secondary stresses) enAkñúg trusses. eRkay mk enAqñaM1915 CI eG EmnI (G.A. Maney) )aneFVIeGaybec©keTsenHkan;EtRbesIeLIg ehIyeRbIva kñúgkarviPaKFñwm nigeRKagminkMNt;. General Case eKeGayeQμaHviFIenHfa viFI slope-deflection edaysarGBaØatrbs;vaCamMurgVil nig PaBdabEdlekIteLIgedaysarGMeBIrbs;bnÞúkmkelIrcnasm<½n§. edIm,IbegáItTMrg;TUeTAénsmIkar slope- deflection eyIgnwgBicarNaElVgKMrU AB énFñwmCab;dUcbgðajenAkñúgrUbTI 11-2 EdlElVgenH rgGMeBIén bnÞúkNamYy ehIyvaman EI efr. eyIgcg;eFVI eGaym:Um:g;Bt;rbs;Fñwm M AB nig M BA man TMnak;TMngnwg degree of freedom rbs;vaTaMgbI eBalKW θ A / θ B nig Δ EdlGaceFVIeGayman sMrutenAcenøaHTMr. enA kñúgkarbegáItrUbmnþ eyIg Kitfam:Um:g; nigmMurgVilviC¢manenAeBlvaeFVIGMeBIeTA elIFñwmtamTisRTnicnaLikavil ¬rUbTI 11-2¦. elIsBIenH bMlas;TIlIenEG‘r Δ mantMélviC¢mandUcbgðajenAkñúgrUb edaysarbMlas;TIeFVIeGaymMψ rbs; u ElVgvilRsbTisRTnicnaLikavil. eKGacTTYlsmIkar slope-deflection edayeRbIeKalkarN_tMrYtpledayBicarNaBicarNa dac;edayELkBIKñarvagm:Um:g;EdlekItmanenARtg;TMrEdlbNþalBIbMlas;TInImYy² ¬ θ A / θ B nig Δ ¦ nigbnÞúk. karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -380
  • 4. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa bMlas;TICamMuenARtg;cMNuc A/ θ ³ eKman node A énGgát;EdlbgðajenAkñúgrUbTI 11-3a manmMu A rgVil θ A ÉcMENk node B RtUv)anbgáb;. edIm,IkMNt;m:Um:g; M AB EdlcaM)ac;edIm,IeFIVeGaymanbMlas;TI enH eyIgnwgeRbIviFIFñwmnimitþ. sMrab;krNIenH FñwmnimitþRtUv)anbgðajenAkñúgrUbTI 11-3b. cMNaMfa kmøaMgkat;enARtg;cMNuc A' eFVIGMeBIeTAelIFñwmedaymanTisedAcuHeRkam edaysar θ A RsbRTnicnaLika. PaBdabrbs;FñwmBitenAkñúgrUbTI 11-3a RtUvesμIsUnüenARtg; A nig B dUcenHplbUkm:Um:g;enARtg; A' nig B' rbs;Fñwmnimitþk¾RtUvEtesμIsUnüEdr dUcenHeK)an ⎡ 1 ⎛ M AB ⎞ ⎤ L ⎡ 1 ⎛ M BA ⎞ ⎤ 2 L + ∑ M A' = 0 ⎢ 2 ⎜ EI ⎟ L ⎥ 3 − ⎢ 2 ⎜ EI ⎟ L ⎥ 3 = 0 ⎣ ⎝ ⎠ ⎦ ⎣ ⎝ ⎠ ⎦ ⎡ 1 ⎛ M BA ⎞ ⎤ L ⎡ 1 ⎛ M AB ⎞ ⎤ 2 L + ∑ M B' = 0 ⎢ 2 ⎜ EI ⎟ L ⎥ 3 − ⎢ 2 ⎜ EI ⎟ L ⎥ 3 + θ A L = 0 ⎣ ⎝ ⎠ ⎦ ⎣ ⎝ ⎠ ⎦ BIsmIkarTaMgBIxagelI eyIgTTYl)anTMnak;TMngbnÞúk nigbMlas;TIdUcxageRkam 4 EI M AB = θA (11-1) L 2 EI M BA = θA (11-2) L bMlas;TICamMuenARtg;cMNuc B/ θB ³ tam rebobdUcKña RbsinebIcug B rbs;FñwmvileTA rkTItaMgcug eRkay)an θ B ÉcMENkcug A RtUv)anbgáb;Cab; ¬rUbTI 11-4¦ eyIgGac sresrTMnak;TMngrvagm:Um:g; M BA eTAnwgmMu rgVil θ B nigrvagm:Um:g;Rbtikmμ M AB enARtg;CBa¢aMg. lT§plKW 4 EI M BA = θB (11-3) L 2 EI M AB = θB (11-4) L Displacement method of analysis: Slope-deflection equations T.Chhay -381
  • 5. Department of Civil Engineering NPIC bMlas;TIeFob/ Δ ³ RbsinebI node B rbs;Ggát;pøas;TIeFobeTAnwgcMNuc A dUcenHGgát;vilRsbTis RTnicnaLika ¬bMlas;TIviC¢man¦ ehIycugTaMgBIGt;vl nigmanm:Um:g; nigkmøaMgkat;EdlmanTMhMdUcKñaEt i TisedApÞúyKñaekItmanenAkúñgGgát; ¬rUbTI 11-5a¦. dUcelIkmun eKGacP¢ab;TMnak;TMngrvagm:Um:g; M eTA nwgbMlas;TI Δ edayeRbIviFIFñwmnimitþ. enAkñúgkrNIenH Fñwmnimitþ ¬rUbTI 11-5b¦ mancugTaMgBIresrI edaysarFñwm ¬Ggát;¦ BitmanTMrbgáb;. b:uEnþ edaysarbMlas;TIrbs;FñwmBitenARtg;cMNuc B m:Um:g;enA Rtg;cMNuc B' énFñwmnimitþRtUvEtman Δ dUcbgðaj . edayeFVIplbUkm:Um:g;Rtg;cMNuc B' eyIg)an * ⎡ 1 ⎛ M ⎞ ⎤ 2L ⎡ 1 ⎛ M ⎞ ⎤ L + ∑ M B' = 0 ⎢ 2 ⎜ EI ⎟ L ⎥ 3 − ⎢ 2 ⎜ EI ⎟ L ⎥ 3 − Δ = 0 ⎣ ⎝ ⎠ ⎦ ⎣ ⎝ ⎠ ⎦ − 6 EI M AB = M BA = M = 2 Δ (11-5) L tamkarkMNt;sBaØa m:Um:g;enHGviC¢manedaysarvaeFVIGMeBIRcasRTnicnaLikaedIm,IeGayGgát;manlMnwg. m:Um:g;bgáb;cug (Fixed-End Moment)³ enAkñúgkrNIelIkmun eyIg)anKitBITMnak;TMngrvagbMlas; TInigm:Um:g;caM)ac; M AB nig M BA EdleFVIGMeBIenARtg; node A nig B erogKña. b:uEnþ CaTUeTAbMlas;TI lIenEG‘r nigbMlas;TImMuén node ekItBIbnÞúkEdlmanGMeBIenAelIElVgrbs;Ggát; minEmnekItBIm:Um:g;Edl manGMeBIenARtg; node eT. edIm,IbegáItsmIkar slope-deflection, eyIgRtUvbMElgbnÞúkEdlmanGMeBIenA elIElVgeGayeTACam:Um:g;smmUlEdlmanGMeBIenARtg; node ehIybnÞab;mkeRbITMnak;TMngrvagbnÞúk nig bMlas;TIedIm,IedaHRsay. eKGaceFVIvaeTA)anedayrkm:Um:g;RbtikmμTb;Tl;nwgbnÞúkEdlekItmanenARtg; node. Ca]TahrN_ eKmanGgát;TMrbgáb;dUcbgðajenAkñúgrUbTI 11-6a EdlvargbnÞúkcMcMNuc P enARtg; kNþalElVg. FñwmnimitþsMrab;krNIenHRtUv)anbgðajenAkñúgrUbTI 11-6b. edaysareyIgRtUvkareGay mMurgVilenARtg;cugnImYy²esμIsUnü ⎡ 1 ⎛ PL ⎞ ⎤ ⎡ 1 ⎛ M ⎞ ⎤ + ↑ ∑ Fy = 0 ⎢ 2 ⎜ 4 EI ⎟ L ⎥ − 2⎢ 2 ⎜ EI ⎟ L ⎥ = 0 ⎣ ⎝ ⎠ ⎦ ⎣ ⎝ ⎠ ⎦ * düaRkamm:Um:g;EdlbgðajenAlíFñwmnimitþRtUv)ankMNt;edayviFItMrYtplsMrab;FñwmTImrsamBaØ ¬dUckarBnül;enAkñúgkfaxNÐ4-5¦. karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -382
  • 6. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa PL M = 8 m:Um:g;enHRtUv)aneKehAfam:Um:g;bgáb;cug (FEM). cMNaMfa GaRs½ynwgkarkMNt;sBaØa vamantMélGviC¢- manenARtg; node A ¬RcasRTnicnaLika¦ ehIyvamantMélviC¢manenARtg; node B ¬RsbRTnic naLika¦. edIm,IgayRsYlkñúgkaredaHRsaycMeNaT eKRtUvKNnam:Um:g;bgáb;cugsMrab;kardak;bnÞúkdéT eTot ehIyvaRtUv)anerobcMCatarag. edayeFVIplbUk FEM Edl)anKNnasMrab;cMeNaTCak;lak; ¬rUb TI 11-7¦ eyIg)an M AB = (FEM ) AB M BA = (FEM )BA (11-6) smIkar Slope-Deflection³ RbsinebIeKbUkm:Um:g;cugEdlbNþalBIbMlas;TInImYy² ¬smIkar 11-1 rhUtdl; 11-5¦ CamYynwgm:Um:g;cugEdlbNþalBIbnÞúk ¬smIkar 11-6¦ enaHeyIgGacsresrm:Um:g;er- s‘ultg;enAxagcugdUcxageRkam ⎛ I ⎞⎡ ⎛ Δ ⎞⎤ M AB = 2 E ⎜ ⎟ ⎢2θ A + θ B − 3⎜ ⎟⎥ + (FEM ) AB ⎝ L ⎠⎣ ⎝ L ⎠⎦ (11-7) ⎛ I ⎞⎡ ⎛ Δ ⎞⎤ M BA = 2 E ⎜ ⎟ ⎢2θ B + θ A − 3⎜ ⎟⎥ + (FEM )BA ⎝ L ⎠⎣ ⎝ L ⎠⎦ edaysarsmIkarTaMgBIrenHRsedogKña eKGacsresrsmIkareTalEdlGacbgðajsmIkarTaMgBIr enH)an. edayykcugmçagrbs;ElVgCacugCit (near edd N) ehIycugmçageTotCacugq¶ay (far end F) ehIyedayykkMrajrbs;Ggát;Ca k = I / L ehIymMurbs;Ggát;Caψ = Δ / L eyIgGacsresr M N = 2 EK (2θ N + θ F − 3ψ ) + (FEM )N sMrab;ElVgxagkñúg b¤ElVgxagEdlmancugq¶ayCacugbgáb; (11-8) Displacement method of analysis: Slope-deflection equations T.Chhay -383
  • 7. Department of Civil Engineering NPIC karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -384
  • 8. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa Edl MN = m:Um:g;Bt;enAkñúgcugCitrbs;ElVg m:Um:g;enHviC¢man ¬RsbTisRTnicnaLika¦ enAeBlEdl vamanGMeBIelIElVg. E / K = m:UDuleGLasÞicrbs;sMPar³ nigkMrajrbs;ElVg k = I / L . θ N / θ F = mMurgVil b¤bMlas;TImMurbs;cugCit nigcugq¶ayrbs;ElVgenARtg;TMr mMuenHmanxñatra:düg; ehIyviC¢manenAeBlvavilRsbTisRTnicnaLika. ψ = mMurgVilrbs;Ggát;bNþalBIbMlas;TIlIenEG‘r Edl ψ = Δ / L mMuenHmanxñatra:düg; ehIy viC¢manenAeBlvavilRsbTisRTnicnaLika. (FEM )N = m:Um:g;bgáb;cugenARtg;TMréncugCit m:Um:g;enHviC¢man ¬RsbTisRTnicnaLika¦ enA eBlvamanGMeBIelIElVg. m:Um:g;bgáb;cugsMrab;lkçxNÐbnÞúkepSg²RtUv)anerobcMenA kñúgtaragxagelI. BIkarbkRsay smIkar11-8RtUv)ankMNt;edaysmIkarlkçxNÐRtUvKña nigTMnak;TMngrvagbnÞúk nig bMlas;TIEdlBicarNaEteTAelIT§iBlm:Um:g;Bt; edaymin)anKitBIkMhUcRTg;RTayedaysarkmøaMgkat; nigkMhUcRTg;RTayedaysarkmøaMgtamG½kS dUcenHeKKitvaCasmIkar slope-deflection TUeTA. enAeBl eRbIvasMrab;edaHRsaycMeNaT eKRtUvGnuvtþsmIkarenHBIrdgsMrab;ElVgGgát; AB nImYy² eBalKWGnuvtþ BI A eTA B nigBI B eTA A sMrab;ElVg AB enAkñúgrUbTI 11-2. cugElVgRTedayTMrsnøak;³ eBlxøH cugElVgrbs;Fñwm b¤eRKagRtUv)anRTedayTMrsnøak; b¤TMrkl;enAcug q¶ay ¬rUbTI 11-8a¦. enAeBlekItmankrNIEbbenH m:Um:g;enARtg;TMrkl; b¤TMrsnøak;RtUvEtesμIsUnü enaHeKmincaM)ac;kMNt;bMlas;TImMu θ B enARtg;TMrenaHeT eKGacEkERbsmIkar slope-deflection TUeTA dUcenHeKRtUvGnuvtþvaEtmþgKt;eTAelIElVg minEmnBIrdg. edIm,IeFVIEbbenH eyIgnwgGnuvtþsmIkar 11-8 b¤smIkar 11-7 eTAelIcugnImYy²rbs;FñwmenAkñúgrUbTI 11-8. CalT§pl eKTTYl)ansmIkarBIrdUcxag eRkam M N = 2 Ek (2θ N + θ F − 3ψ ) + (FEM )N (11-9) 0 = 2 Ek (2θ F + θ N − 3ψ ) + 0 enATIenH (FEM )F esμIsUnüedaysarcugq¶ayCaTMrsnøak; ¬rUbTI 11-8b¦. elIsBIenH eKGacTTYl)an (FEM )N ¬Ca]TahrN_¦ edayeRbItaragmU:m:g;bgáb;cugxagelI. edayKuNsmIkarTImYyxagelInwgBIr ehIydkvaCamYynwgsmIkarTIBIr edIm,IsMrYl θ F enaHeyIg)an M N = 3Ek (θ N −ψ ) + (FEM )N sMrab;Etcugq¶ayrbs;ElVgRtUv)anRTedayTMrsnøak; b¤TMrkl; (11-10) Displacement method of analysis: Slope-deflection equations T.Chhay -385
  • 9. Department of Civil Engineering NPIC edaysarm:Um:g;enAcugq¶ayesμIsUnü eKGnuvtþsmIkarenHEtmþgb:ueNÑaHsMrab;cugElVg. vasMrYlkarviPaK edaysarsmIkarTUeTA ¬smIkar 11-8¦ TamTarkarGnuvtþBIrdgsMrab;ElVgenH edaysarvaBak;B½n§nwg bMlas;TImMu θ B ¬b¤θ F ¦ enAcugTMr. edIm,IsegçbkarGnuvtþsmIkar slope-deflection, cUrBicarNaFñwmCab;EdlbgðajenAkñúgrUbTI 11- 9 Edlman degree of freedom cMnYnbYn. enATIenH eKGacGnuvtþsmIkar 11-8 cMnYnBIrdgeTAelIElVg nImYy² eBalKWBI A eTA B / BI B eTA A / BI B eTA C / BI C eTA B / BI C eTA D / nigBI D eTA C . smIkarTaMg enHnwgBak;B½n§nwgmMurgVilθ A / θ B / θC nig θ D EdlCaGBaØat. b:uEnþ edaysarm:Um:g;cugenARtg; A nig D sUnü enaHeKmincaM)ac;kMNt; θ A nig θ D eT. eKnwgTTYl)ankaredaHRsayxøICagRbsinebIeKGnuvtþ smIkar 11-10 BI B eTA A nigBI C eTA D ehIybnÞab;mkGnuvtþsmIkar 11-8 BI B eTA C nig BI C eTA B . smIkarTaMgbYnenHnwgBak;B½n§EtnwgmMurgVil θ B nig θ C EdlCaGBaØatEtb:ueNÑaH. !!>#> viPaKFñwm (Analysis of beams) dMeNIrkarkñúgkarviPaK (Procedure for analysis) Degree of Freedom bg;elx b¤GkSrRKb;TMr nigtMN (node) edIm,IkMNt;GtþsBaØaNénElVgrbs;Fñwm b¤eRKagEdlenAcenøaH node. edayKUrrUbragdabrbs;rcnasm<½n§ eKmanlT§PaBkMNt;cMnYn degree of freedom. enATIenH node nImYy²GacmanbMlas;TImMu nigbMlas;TIlIenEG‘r. eKRtUvrkrkSaPaBRtUvKñaenARtg; node edIm,IeFVI karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -386
  • 10. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa eGayGgát;EdltP¢ab;edaytMNbgáb;enARtg; node rgbMlas;TIdUcKña. RbsinebIbMlas;TITaMgenaHCa GBaØat eKKYrsnμt;eGayvaeFVIGMeBItamTisviC¢man edayeFVIeGayGgát; b¤tMNvilRsbTisRTnicnaLika ¬rUbTI 11-2¦. smIkar Slope-Deflection smIkar slope-deflection P¢ab;TMnak;TMngrvagm:Um:g;EdlCaGBaØatEdlGnuvtþenARtg; node CamYynwg bMlas;TIrbs; nodes sMrab;ElVgTaMgGs;rbs;eRKOgbgÁúM. RbsinebIeKmanbnÞúkenAelIElVg eKRtUvKNna FEM edayeRbItaragm:Um:g;cugEdleGayenAkñúgkfaxNÐ 11.2. ehIyRbsinebI node manbMlas;TI lIenEG‘r Δ eKRtUvKNna ψ = Δ / L sMrab;ElVgEk,r. GnuvtþsmIkar 11-8 eTAelIcugrbs;ElVgnImYy² edayeRbIsmIkar slope-deflection BIrdgsMrab;ElVgnImYy². b:uEnþ RbsinebIcugElVgénFñwm b¤eRKagCab; CaTMrsnøak; GnuvtþsmIkar 11-10 EteTAelIcugEdlTb; ehIyeRbIsmIkar slope-deflection Etmþg b:ueNÑaHsMrab;ElVg. smIkarlMnwg sresrsmIkarlMnwgsMrab; degree of freedom EdlCaGBaØatnImYy². eKRtUvsresrsmIkarTaMgenH edayeRbIm:Um:g;Bt;EdlCaGBaØatdUcEdlkMNt;enAkñúgsmIkar slope-deflection. sMrab;Fñwm nigeRKag sresrsmIkarlMnwgénm:Um:g;enARtg;TMrnImYy² nigsMrab;eRKagsresrsmIkarlMnwgénm:Um:g;Bt;Rtg;tMN. RbsinebIeRKageyal b¤manPaBdabtamTisedk eKRtUvP¢ab;TMnak;TMngrvagkmøaMgkat;enAkñúgssreTA nwgm:Um:g;enAxagcugrbs;ssr ¬nwgmanerobrab;enAkñúgkfaxNÐ 11-5¦. CMnYssmIkar slope-deflection eTAkñúgsmIkarlMnwg ehIyedaHRsayrkbMlas;TIrbs;tMNEdl CaGBaØat. bnÞab;mkeKRtUvCMnYslT§plTaMgenHeTAkñúgsmIkar slope-deflection edIm,IkMNt;m:Um:g;Bt; enAxagcugrbs;Ggát;nImYy². RbsinebIlT§plmantémøGviC¢man mann½yfavavilRcasRTnicnaLika cMENkÉm:Um:g; nigbMlas;TIviC¢manvilRsbRTnicnaLika. ]TarhN_ 11-1³ sg;düaRkamkmøaMgkat; nigdüaRkamm:Um:gsMrab;FñwmEdlbgðajenAkñúgrUbTI 11- ; 10a. EI Camantémøefr. dMeNaHRsay³ smIkar slope-deflection enAkñúg]TahrN_enHeKRtUvBicarNaElVgBIr. edaysarKμanElVgNamYymanTMrsnøak; b¤TMrkl; dUcenHeK RtUvGnuvtþsmIkar 11-8. edayeRbIrUbmnþsMrab; FEM EdlerobcMenAkñúgtaragsMrab;bnÞúkRtIekaN Displacement method of analysis: Slope-deflection equations T.Chhay -387
  • 11. Department of Civil Engineering NPIC wL2 6(6 )2 (FEM )BC =− =− = −7.2kN .m 30 30 2 6(6 )2 (FEM )CB = wL = = 10.8kN .m 20 20 cMNaMfa (FEM )BC GviC¢man edaysarvaeFVIGMeBIRcasRTnicnaLikaenARtg;cMNuc B . ehIy (FEM )AB = (FEM )BA = 0 edaysarminmanbnÞúkenAelIElVg AB . edIm,IsÁal;GBaØat ExSekageGLasÞicsMrab;FñwmRtUv)anbgðajenAkñúgrUbTI 11-10b. dUckar bgðaj eKmanm:Um:g;Bt;EdlCaGBaØatcMnYnbYn. manEtmMurgVil θ B enARtg;cMNuc B b:ueNÑaHEdlCaGBaØat. edaysar A nig C CaTMrbgáb; enaHθ A = θC = 0 . ehIy edaysarTMrminRsut ¬vaminmanbMlas;TIeLIg elI b¤cuHeRkam¦ ψ AB = ψ BC = 0 . sMrab;ElVg AB ¬edayKitfa A CacugCit nig B Cacugq¶ay¦ eyIg)an ⎛I⎞ M N = 2 E ⎜ ⎟(2θ N + θ F − 3ψ ) + (FEM )N ⎝L⎠ ⎛I⎞ M AB = 2 E ⎜ ⎟[2(0 ) + θ B − 3(0)] + 0 = EI θB (1) ⎝8⎠ 4 LÚvKitfa B CacugCit nig A Cacugq¶ay eyIg)an ⎛I⎞ M BA = 2 E ⎜ ⎟[2θ B + 0 − 3(0)] + 0 = EI θB (2) ⎝8⎠ 2 tamrebobdUcKña sMrab;ElVg BC eyIg)an karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -388
  • 12. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ⎛I⎞ M BC = 2 E ⎜ ⎟[2θ B + 0 − 3(0)] − 7.2 = 2 EI θ B − 7.2 (3) ⎝6⎠ 3 ⎛I⎞ M CB = 2 E ⎜ ⎟[2(0 ) + θ B − 3(0 )] + 10.8 = EI θ B + 10.8 (4) ⎝6⎠ 3 smIkarlMnwg smIkarTaMgbYnxagelImanGBaØatcMnYnR)aM. smIkarEdlcaM)ac;TIR)aM)anBIlkçxNÐlMnwgrbs;m:Um:g;enARtg; TMr B . düaRkamGgÁesrIrbs;kMNat;Ggát;enARtg;cMNuc B RtUv)anbgðajenAkñúgrUbTI 11-10c. enATI enH M BA nig M BC RtUv)ansnμt;eGayeFVIGMeBIkñúgTisedAviC¢manedIm,IeGayRsbeTAnwgsmIkar slope- deflection . dUcenH * + ∑MB = 0 M BA + M BC = 0 (5) edIm,IedaHRsay CMnYssmIkar (2) nig (3) eTAkñúgsmIkar (5) enaHeyIg)an 6.17 θB = EI edayCMnYstémøenHeTAkñúgsmIkar (1) dl;smIkar (4) eyIg)an M AB = 1.54kN .m M BA = 3.09kN .m M BC = −3.09kN .m M CB = 12.86kN .m témøGviC¢mansMrab; M BC bgðajfam:Um:g;enHeFVIGMeBIRcasRTnicnaLikaenAelIFñwm minEmnRsbRTnicna- LikadUcbgðajenAkñúgrUbTI 11-10b. edayeRbIlT§plenH eKkMNt;kMlaMgkat;enARtg;cugElVgBIsmIkarlMnwg ¬rUbTI 11-10d¦. düaRkamGgÁesrIénFñwmTaMgmUl nigdüaRkamkMlaMgkat; nigdüaRkamm:Um:g;RtUv)anbgðajenAkñúgrUbTI 11- 10e. ]TarhN_ 11-2³ sg;düaRkamkmøaMgkat; nigdüaRkamm:Um:gsMrab;FñwmEdlbgðajenAkñúgrUbTI 11- ; 11a. EI Camantémøefr. dMeNaHRsay³ smIkar slope-deflection enAkñúg]TahrN_enHeKRtUvBicarNaElVgBIr. eKGnuvtþsmIkar 11-8 eTAelIElVg AB . eyIgGaceRbI * RsbRTnicnaLikaenAelIFñwm b:uEnþRcasRTnicnaLikaenAelITMr. Displacement method of analysis: Slope-deflection equations T.Chhay -389
  • 13. Department of Civil Engineering NPIC smIkar 11-10 sMrab;ElVg BC edaysar C CaTMrkl;. edayeRbIrUbmnþsMrab; FEM EdlerobcMenAkñúg tarag eyIg)an 2 (FEM ) AB = − wL =− 1 (40)(6)2 = −120kN .m 12 12 2 (FEM )BA = wL = 1 (40)(6)2 = 120kN .m 12 12 (FEM )BC = − 3PL = − 3(60)(2) = −22.5kN .m 16 16 cMNaMfa (FEM )AB nig (FEM )BC GviC¢man edaysarvaeFVIGMeBIRcasRTnicnaLikaenARtg;cMNuc A nig B erogKña. ehIyedaysarTMrKμansMrut ψ AB = ψ BC = 0 . edayGnuvtþsmIkar 11-8 sMrab;ElVg AB nigedaydwgfa θ A = 0 eyIg)an ⎛I⎞ M N = 2 E ⎜ ⎟(2θ N + θ F − 3ψ ) + (FEM )N ⎝L⎠ ⎛I⎞ M AB = 2 E ⎜ ⎟[2(0) + θ B − 3(0)] − 120 ⎝L⎠ M AB = 0.333EIθ B − 120 (1) ⎛I⎞ M BA = 2 E ⎜ ⎟[2θ B + 0 − 3(0)] + 120 ⎝L⎠ karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -390
  • 14. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa M BA = 0.667 EIθ B + 120 (2) edayGnuvtþsmIkar 11-12 CamYynwg B CacugCit nig C Cacugq¶ay eyIg)an ⎛I⎞ M N = 3E ⎜ ⎟(θ N −ψ ) + (FEM )N ⎝L⎠ ⎛I⎞ M BC = 3E ⎜ ⎟(θ B − 0) − 22.5 ⎝2⎠ M BC = 1.5 EIθ B − 22.5 (3) cgcaMfa eKminGnuvtþsmIkar 11-10 BI B ¬cugCit¦ eTA B ¬cugq¶ay¦ eT. smIkarlMnwg smIkarTaMgbIxagelImanGBaØatcMnYnbYn. smIkarEdlcaM)ac;TIR)aM)anBIlkçxNÐlMnwgrbs;m:Um:g;enARtg; TMr B . düaRkamGgÁesrIRtUv)anbgðajenAkñúgrUbTI 11-11b. eyIg)an + ∑MB = 0 M BA + M BC = 0 (4) edIm,IedaHRsay CMnYssmIkar (2) nig (3) eTAkñúgsmIkar (4) enaHeyIg)an 45 θB = − EI edaysar θ B GviC¢man ¬RcasRTnicnaLika¦ ExSekageGLasÞicsMrab;FñwmRtUv)ansg;y:agRtwmRtUvenAkñúg rUbTI 11-11a. edayCMnYs θ B eTAkñúgsmIkar (1) –(3) eyIg)an M AB = −135kN .m M BA = 90kN .m M BC = −90kN .m edayeRbIlT§plénm:Um:g;TaMgenH eyIgGackMNt;kmøaMgkat;enARtg;cugrbs;ElVgFñwmenAkñúgrUbTI 11- 11c. düaRkamkmøaMgkat; nigdüaRkamm:Um:g;RtUv)anbgðajenAkñúgrUbTI 11-11d. ]TarhN_ 11-3³ kMNt;m:Um:g;enARtg;cMNuc A nig B sRmab;FñwmEdlbgðajenAkñúgrUbTI 11-12a. TMrenARtg; A mansMrut 80mm . yk E = 200GPa nig I = 5(106 )mm 4 . dMeNaHRsay³ smIkar slope-deflection enAkúñgcMeNaTenH eyIgBicarNaEtElVg AB mYyb:ueNÑaH edaysarm:Um:g; M AB EdlbNþalBIFñwmly EdlGacKNnaBIsþaTic. edaysarminmanbnÞúkenAelIElVg AB enaH FEM esμIsUnü. dUcbgðajenAkñúg rUbTI 11-12b bMlas;TIcuHeRkamrbs;cMNuc B eFVIeGayElVg AB vilRsbRTnicnaLika. dUcenH Displacement method of analysis: Slope-deflection equations T.Chhay -391
  • 15. Department of Civil Engineering NPIC 0.08m ψ AB = ψ BA = = 0.02rad 4 kMrajrbs; AB KW k= ( ) ( I 5 10 6 mm 4 10 −12 m 4 / mm 4 = ) ( ) = 1.25 10 − 6 m 3 L 4m edayGnuvtþsmIkar slope-deflection ¬smIkar 11-8¦ eTAelIElVg AB CamYynwgθ A =0 eyIg)an ⎛I⎞ M N = 2 E ⎜ ⎟(2θ N + θ F − 3ψ ) + (FEM )N ⎝L⎠ ( ( ) )[ ( ) ] M AB = 2 200 109 N / m 2 1.25 10 −6 m 3 [2(0 ) + θ B − 3(0.02 )] + 0 (1) M BA = 2(200(10 )N / m )[ .25(10 )m ][2θ 9 1 2 −6 3 B + 0 − 3(0.02 )] + 0 (2) smIkarlMnwg düaRkamGgÁesrIrbs;FñwmenARtg;TMr B RtUv)anbgðajenAkñúgrUbTI 11-12c. eyIg)an + ∑MB = 0 M BA − 8000 N (3m ) = 0 edIm,IedaHRsay CMnYssmIkar (2) eTAkñúgsmIkarenH eyIg)an 1(10 6 ) B − 30(103 ) = 24(103 ) θ θ B = 0.054rad ] dUcenHBIsmIkar (1) nig (2) M AB = −3.00kN .m M BA = 24.0kN .m ]TarhN_ 11-4³ kMNt;m:Um:g;enARtg;TMrrbs;FñwmEdlbgðajenAkñúgrUbTI 11-13a. TMrenARtg; A mansMrut 30mm . yk E = 200GPa nig I = 600(106 )mm 4 . karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -392
  • 16. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ smIkar slope-deflection enAkúñgcMeNaTenH eKmanElVgcMnYnbIEdlRtUvBicarNa. eKeRbIsmIkar 11-8 edaysarTMr A nig D CaTMr bgáb;. ehIymanEtElVg AB b:ueNÑaHEdlman FEM . 2 (FEM ) AB = − wL 1 (20)(7.2)2 = −86.4kN .m = 12 12 wL2 1 (FEM )BA = = (20 )(7.2)2 = 86.4kN .m 12 12 dUcbgðajenAkñúgrUbTI 11-13b, sMrut ¬bMlas;TI¦ rbs;TMr C eFVIeGayψ BC viC¢man edaysarGgát; BC vilRsbRTnicnaLika ehIyψ CD GviC¢man edaysarGgát; CD vilRcasRTnicnaLika. dUcenH 0.03m 0.03m ψ BC = = 0.005rad ψ CD = − = −0.00667rad 6m 4m edaybgðajxñatrbs;kMrajCaEm:Rt eyIg)an 600(106 )(10 −12 ) k AB = = 83.33( − 6 )m 3 10 7.2 600( 6 )(10 −12 ) = 100(10 − 6 )m 3 10 k BC = 6 k CD = ( )( 600 10 10 −12 6 ( ) ) = 133.33 10 −6 m 3 4.5 cMNaMfa θ = θ = 0 edaysar A nig D CaTMrbgáb; ehIyedayGnuvtþsmIkar slope-deflection A D ¬smIkar 11-8¦ BIrdgenAelIElVgnImYy² eyIg)an sMrab;ElVg AB M = 2[200(10 )[83.33( AB 10 )]2(0 ) + θ − 3(0 )] − 86.4 6 −6 B M AB = 33333.3θ B − 86.4 (1) [ ( )[ ( )] M BA = 2 200 10 6 83.33 10 − 6 2θ B + 0 − 3(0 ) + 86.4 ] Displacement method of analysis: Slope-deflection equations T.Chhay -393
  • 17. Department of Civil Engineering NPIC M BA = 66666.7θ B + 86.4 (2) sMrab;ElVg BC [ ( )[ ( )] ] M BC = 2 200 10 6 100 10 − 6 2θ B + θ C − 3(0.005) + 0 M BC = 80000θ B + 40000θ C − 600 (3) [ ( )[ ( )] ] M CB = 2 200 10 6 100 10 − 6 2θ C + θ B − 3(0.005) + 0 M CB = 80000θ C + 40000θ B − 600 (4) sMrab;ElVg CD [ ( )[ ( )] ] M CD = 2 200 10 − 6 133 10 − 6 2θ C + 0 − 3(− 0.00667 ) + 0 M CD = 106666.7θ C + 0 + 1066.7 (5) [ ( )[ ( )] ] M DC = 2 200 10 6 133.33 10 − 6 2(0 ) + θ C − 3(− 0.00667 ) + 0 M DC = 53333.3θ C + 1066.7 (6) smIkarlMnwg smIkarTaMgR)aMmYymanGBaØatR)aMbI. edaysresrsmIkarlMnwgrbs;m:Um:g;sMrab;TMrenARtg;cMNuc B nig cMNuc C ¬rUbTI 10-13c¦ eyIg)an + ∑MB = 0 M BA + M BC = 0 (7) + ∑MC = 0 M CB + M CD = 0 (8) edIm,IedaHRsay CMnYssmIkar (2) nig (3) eTAkñúgsmIkar (7) ehIyCMnYssmIkar (4) nig (5) eTAkñúg smIkar (8) enaHeyIg)an θ C + 3.667θ B = 0.01284 − θ C − 0.214θ B = 0.00250 dUcenH θ B = 0.00444rad θ C = −0.00345rad témøviC¢mansMrab; θC bgðajkarvilRcasRTnicnaLikarrbs;bnÞat;b:HenARtg;cMNuc C ¬rUbTI 11-13a¦. edayCMnYstémøTaMgenHeTAkñúgsmIkar (1)-(6) eyIg)an M AB = 61.6kN .m M BA = 383kN .m M BC = −383kN .m M CB = −698kN .m M CD = 698kN .m M DC = 883kN .m karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -394
  • 18. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa !!>$> viPaKeRKagEdlmineyal (Analysis of Frame: No sidesway) eRKagnwgmineyal b¤pøas;TIeTAeqVg b¤eTAsþaM RbsinebIeKTb;va)anl¥. ]TahrN_RtUv)anbgðaj enAkñúgrUbTI 11-14. ehIyeRKagEdlminmankarTb;)anl¥nwgmineyaleT RbsinragFrNImaRt nigkar rgbnÞúkrbs;vamanlkçN³sIuemRTI dUcbgðajenAkñúgrUbTI 11-15. sMrab;krNITaMgBIr tYψ enAkñúgsmIkar slope-deflection esμIsUnü edaysarm:Um:g;Bt;min)aneFVIeGaytMNmanbMlas;TIlIenEG‘r. ]TahrN_xageRkambgðajBIkarGnuvtþénsmIkar slope-deflection edayeRbIdMeNIrkarkñúgkar viPaKEdlmanerobrab;enAkñúgkfaxNÐ !!># sMrab;eRKagRbePTenH. ]TarhN_ 11-5³ kMNt;m:Um:g;enARtg;tMNnImYy²rbs;eRKagEdlbgðajenAkñúgrUbTI 11-16a. EI mantémøefr. dMeNaHRsay³ smIkar slope-deflection enAkúñgcMeNaTenH eKmanGgát;cMnYnbIEdlRtUvBicarNaKW AB / BC nig CD . edayeRKagenHmanTMrbgáb; enARtg;cMNuc A nig D eyIgnwgGnuvtþsmIkar 11-8 sMrab;karedaHRsayenH. Displacement method of analysis: Slope-deflection equations T.Chhay -395
  • 19. Department of Civil Engineering NPIC BItaragm:Um:g;bgáb;cug/ FEM sMrab; BC KW 5wL2 5(24 )(8)2 (FEM )BC =− =− = −80kN .m 96 96 2 5(24 )(8)2 (FEM )CB = 5wL = = 80kN .m 96 96 cMNaMfa θ A = θ D = 0 ehIyψ AB = ψ BC = ψ CD = 0 edaysareRKagenHnwgmineyal. edayGnuvtþsmIkar 11-8 eyIg)an M N = 2 Ek (2θ N + θ F − 3ψ ) + (FEM )N ⎛ I ⎞ M AB = 2 E ⎜ ⎟[2(0) + θ B − 3(0)] + 0 ⎝ 12 ⎠ M AB = 0.1667 EIθ B (1) ⎛ I ⎞ M BA = 2 E ⎜ ⎟[2θ B + 0 − 3(0 )] + 0 ⎝ 12 ⎠ M BA = 0.333EIθ B (3) ⎛I⎞ M BC = 2 E ⎜ ⎟[2θ B + θ C − 3(0)] − 80 ⎝8⎠ M BC = 0.5EIθ B + 0.25EIθ C − 80 (3) ⎛I⎞ M CB = 2 E ⎜ ⎟[2θ C + θ B − 3(0)] + 80 ⎝8⎠ M CB = 0.8EIθ C + 0.25EIθ B + 80 (4) karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -396
  • 20. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ⎛ I ⎞ M CD = 2 E ⎜ ⎟[2θ C + 0 − 3(0)] + 0 ⎝ 12 ⎠ M CD = 0.333EIθ C (5) ⎛ I ⎞ M DC = 2 E ⎜ ⎟[2(0) + θ C − 3(0)] + 0 ⎝ 12 ⎠ M DC = 0.1667 EIθ C (6) smIkarlMnwg smIkarTaMgR)aMmYymanGBaØatR)aMbI. smIkarlMnwgBIreTot)anBIsmIkarlMnwgrbs;m:Um:g;sMrab;TMrenARtg; cMNuc B nig cMNuc C ¬rUbTI 11-16c¦. eyIg)an M BA + M BC = 0 (7) M CB + M CD = 0 (8) edIm,IedaHRsaysmIkarTaMgR)aMbIenH CMnYssmIkar (2) nig (3) eTAkñúgsmIkar (7) nigCMnYssmIkar (4) nig (5) eTAkñúgsmIkar (8). eyIg)an 0.833EIθ B + 0.25EIθ C = 80 0.833EIθ C + 0.25EIθ B = −80 edayedaHRsayRbB½n§smIkarenHeyIg)an 137.1 θ B = −θ C = EI EdleqøIytbeTAnwgrebobEdleRKagdabdUcbgðajenAkñúgrUbTI 11-16a. edayCMnYsvaeTAkñúgsmIkar (1)-(6) eyIg)an M AB = 22.9kN .m M BA = 45.7 kN .m M BC = −45.7kN .m M BC = 45.7 kN .m M CD = −45.7kN .m M DC = −22.9kN .m edayeRbIlT§plTaMgenH eyIgGackMNt;kmøaMgRbtikmμenARtg;cugrbs;Ggát;nImYy²BIsmIkarlMnwg ehIy eyIgGacsg;düaRkamm:Um:g;sRmab;eRKagdUcbgðajenAkñúgrUbTI 11-16c. ]TarhN_ 11-6³ kMNt;m:Um:g;enARtg;tMNnImYy²rbs;eRKagEdlbgðajenAkñúgrUbTI 11-17a. m:Um:g;niclPaBsMrab;Ggát;nImYy²RtUv)anbgðajenAkñúgrUb. yk E = 200GPa . Displacement method of analysis: Slope-deflection equations T.Chhay -397
  • 21. Department of Civil Engineering NPIC dMeNaHRsay³ smIkar slope-deflection enAkúñgcMeNaTenH eKmanGgát;cMnYnbYnEdlRtUvBicarNa. eyIgnwgGnuvtþsmIkar 11-8 cMeBaHGgát; AB nig BC ehIyGnuvtþsmIkar 11-10 cMeBaHGgát; CD nig CE BIeRBaH D nig E CaTMrsnøak;. edaykMNt;kMrajrbs;Ggát; eyIg)an 160( 6 )(10 −12 ) 80( 6 )( −12 ) = 35.56( 10 )m = 17.78(10 − 6 )m 3 10 −6 3 10 10 k AB = kCD = 4.8 4.5 320(10 )(10 ) 260(106 )(10 −12 ) = 66.67( − 6 )m 3 = 72.23( − 6 )m 3 6 −12 k BC = 10 kCE = 10 4.8 3.6 FEM EdlbNþalBIbnÞúkKW 30(4.8) (FEM )BC =− PL =− = −18kN .m 8 8 PL 30(4.8) (FEM )CB = = = 18kN .m 8 8 wL2 50(3.6 )2 (FEM )CE =− =− = −81kN .m 8 8 edayGnuvtþsmIkar 11-8 nigsmIkar 11-10 eTAelIeRKag nigcMNaMfaθ A = 0 /ψ AB = ψ BC = ψ CD = ψ CE = 0 edaysarminmankareyal dUcenHeyIg)an M N = 2 Ek (2θ N + θ F − 3ψ ) + (FEM )N [ ( )] ( ) M AB = 2 200 10 6 (35.56 ) 10 − 6 [2(0 ) + θ B − 3(0 )] + 0 M AB = 14222.2θ B (1) [ ( )] ( ) M BA = 2 200 10 6 (35.56 ) 10 − 6 [2θ B + 0 − 3(0 )] + 0 M BA = 28444.4θ B (2) karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -398
  • 22. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa [ ( )] ( ) M BC = 2 200 10 6 (66.67 ) 10 −6 [2θ B + θ C − 3(0 )] − 18 M BC = 53333.3θ B + 26666.7θ C − 18 (3) [ ( )] ( ) M CB = 2 200 10 6 (66.67 ) 10 − 6 [2θ C + θ B − 3(0)] + 18 M BC = 26666.7θ B + 53333.3θ C + 18 (4) M N = 3Ek (θ N −ψ ) + (FEM )N [ ( )] ( ) M CD = 3 200 10 6 (17.78) 10 −6 [θ C − 0] + 0 M CD = 10666.7θ C (5) [ ( )] ( ) M CE = 3 200 106 (72.22 ) 10 − 6 [θ C − 0] + 81 M CE = 43333.3θ C − 81 (6) smIkarlMnwg smIkarTaMgR)aMmYymanGBaØatR)aMbI. eKGacsresrsmIkarlMnwgrbs;m:Um:g;cMnYnBIrsMrab;TMrenARtg;cMNuc B nigcMNuc C ¬rUbTI 11-17c¦. eyIg)an M BA + M BC = 0 (7) M CB + M CD + M CE = 0 (8) edIm,IedaHRsay CMnYssmIkar (2) nig(3) eTAkñúgsmIkar (7) ehIyCMnYssmIkar (4)-(6) eTAkñúgsmIkar (8) enaHeyIg)an 81777.7θ Β + 26666.7θ C = 18 26666.7θ B + 107333.3θ C = 63 edaHRsayRbB½n§smIkarenHeyIg)an θ B = 3.124(10 −5 )rad θ C = 5.792(10 −4 )rad edaysartémøTaMgenHviC¢man dUcenHvaeFVIeGayeRKagxUcRTg;RTaydUcbgðajenAkñúgrUbTI 11-17a. edayCMnYstémøTaMgenHeTAkñúgsmIkar (1)-(6) nigedaHRsay eyIg)an M AB = 0.444kN .m M BA = 0.888kN .m M BC = −0.888kN .m M CB = 49.7kN .m Displacement method of analysis: Slope-deflection equations T.Chhay -399
  • 23. Department of Civil Engineering NPIC M CD = 6.48kN .m M CE = −55.9kN .m !!>%> viPaKeRKagEdleyal (Analysis of frames: Sidesway) eRKagnwgeyal b¤cl½teTAxagenAeBlEdlva b¤bnÞúkEdlmanGMeBIelIvamanlkçN³minsIuemRTI. edIm,IbgðajBIT§iBlenH BicarNaeRKagEdlbgðajenAkñúgrUbTI 11-18. enATIenH bnÞúk P eFVIeGayman m:Um:g; M BC nig M CB enARtg;cMNuc B nig C erogKña ehIym:Um:g;TaMgBIrenHminesμIKñaeT. m:Um:g; M BC eFVI eGaycMNuc B cl½teTAsþaM Ém:Um:g; M CB eFVIeGaycMNuc C cl½teTAeqVg. edaysar M BC FMCag M CB enaHlT§plénbMlas;TIsrub Δ rbs;tMN B nig C KWeTAsþaM dUcbgðajenAkñúgrUb . enAeBlGnuvtþsmIkar * slope-deflection eTAelIssrnImYy²rbs;eRKag eyIgRtUvBicarNamMurgVilrbs;ssr ψ ¬edaysar ψ = Δ / L ¦ CaGBaØatenAkñúgsmIkar. CalT§pl eKRtUvkar bBa¢ÚlsmIkarlMnwgbEnßmsMrab;karedaHRsay. enAkñúgkfa- xNÐelIkmun eyIgeXIjfabMlas;TImMu θ EdlCaGBaØatmanTMnak; TMngenAkñúgsmIkar lMnwgénm:Umg;. tamrebobdUcKña enAeBl : ekItmanbMlas;TIlIenEG‘r Δ ¬b¤mMurgVilrbs;ElVg ψ ¦ eyIgRtUv sresrsmIkarlMnwgénkMlaMgedIm,ITTYl)andMeNaHRsayeBj elj. b:uEnþ GBaØatenAkñúgsmIkarTaMgenHRtUvEtBak;B½n§nwg m:Um:g;Bt;Edlman GMeBIenARtg;cugssr edaysarsmIkar slope- deflection Bak;B½n§eTAnwgm:Um:g;TaMgenH. bec©keTssMrab;edaH RsaycMeNaTéneRKagEdleyalRtUv)anbgðajenAkñúg]TahrN_. ]TarhN_ 11-7³ kMNt;m:Um:g;enARtg;tMNnImYy²rbs;eRKagEdlbgðajenAkñúgrUbTI 11-19a. EI mantémøefr. dMeNaHRsay³ smIkar slope-deflection edaysarTMr A nig D CaTMrbgáb; eKGacGnuvtþsmIkar 11-8 sRmab;Ggát;TaMgbIrbs;eRKag. enATIenH eRKagmanPaBeyal edaysarkardak;bnÞúk nigragFrNImaRtrbs;eRKagmanlkçN³minsIuemRTI. enATI * rMlwkfa kMhUcRTg;RTayrbs;Ggát;TaMgbIKWbNþalBIm:Um:g;Bt; edayecalkmøaMgkat;nigkmøaMgtamG½kS. karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -400
  • 24. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa enH eKdak;bnÞúkedaypÞal;eTAelItMN B dUcenHminman FEM eFVIGMeBIeTAelItMNeT. dUcbgðajenAkñúgrUb TI 11-19a eKsnμt;eGaytMN B nigtMN C pøas;TIedaybrimaN Δ esμIKña. Cavi)ak ψ AB = Δ / 4 ehIy ψ DC = Δ / 6 . tYTaMgBIrenHviC¢man edaysarGgát; AB nig CD vilRsbTisRTnicnaLika. edayP¢ab; TMnak;TMngψ AB eTAnwgψ DC eyIg)anψ AB = (6 / 4)ψ DC . edayGnuvtþsmIkar 11-8 eTAelIeRKag eyIg)an ⎛ I ⎞⎡ ⎛6 ⎞⎤ M AB = 2 E ⎜ ⎟ ⎢2(0) + θ B − 3⎜ ψ DC ⎟⎥ + 0 = EI (0.5θ B − 2.25ψ DC ) (1) ⎝ 4 ⎠⎣ ⎝4 ⎠⎦ ⎛ I ⎞⎡ ⎛6 ⎞⎤ M BA = 2 E ⎜ ⎟ ⎢2θ B + 0 − 3⎜ ψ DC ⎟⎥ + 0 = EI (1.0θ C − 2.25ψ DC ) (2) ⎝ 4 ⎠⎣ ⎝4 ⎠⎦ ⎛I⎞ M BC = 2 E ⎜ ⎟[2θ B + θ C − 3(0)] + 0 = EI (0.8θ B + 0.4θ C ) (3) ⎝5⎠ ⎛I⎞ M CB = 2 E ⎜ ⎟[2θ C + θ B − 3(0)] + 0 = EI (0.8θ B + 0.4θ C ) (4) ⎝5⎠ ⎛I⎞ M CD = 2 E ⎜ ⎟[2θ C + 0 − 3ψ DC ] + 0 = EI (0.667θ C − 1.0ψ DC ) (5) ⎝6⎠ ⎛I⎞ M DC = 2 E ⎜ ⎟[2(0) + θ C − 3ψ DC ] + 0 = EI (0.333θ C − 1.0ψ DC ) (6) ⎝6⎠ smIkarlMnwg smIkarTaMgR)aMmYymanGBaØatR)aMbYn. eKGacsresrsmIkarlMnwgrbs;m:Um:g;cMnYnBIrsMrab;TMrenARtg; cMNuc B nigcMNuc C ¬rUbTI 11-17c¦. eyIg)an Displacement method of analysis: Slope-deflection equations T.Chhay -401
  • 25. Department of Civil Engineering NPIC M BA + M BC = 0 (7) M CB + M CD = 0 (8) edaysarvaekItmanbMlas;TItamTisedk Δ / eyIgnwgBicarNarplbUkkmøaMgenAelIeRKagTaMgmUltam Tis x dUcenHeyIgTTYl)an + → ∑ Fx = 0 200 − V A − VD = 0 eKGacP¢ab;TMnak;TMngkmøaMgRbtikmμtamTisedk b¤kmøaMgkat;enAkñúgssrVA nigVD eTAnwgm:Um:g;Bt; edayBicarNadüaRkamGgÁesrIénssrdac;edayELkBIKña ¬rUbTI 11-19c¦. eyIgman M AB + M BA ∑MB = 0 VA = − 4 M + M CD ∑ MC = 0 VD = − DC 6 M AB + M BA M DC + M CD dUcenH 200 + 4 + 6 =0 (9) edIm,IedaHRsay CMnYssmIkar (2) nig(3) eTAkñúgsmIkar (7) ehIyCMnYssmIkar (4) nig(5) eTAkñúg smIkar (8) ehIyCMnYssmIkar (1), (2), (5) nig (6) eTAkñúgsmIkar (9) enaHeyIgTTYl)an 1.8θ Β + 0.4θ C − 2.25ψ DC = 0 0.4θ B + 1.467θ C −ψ DC = 0 800 1.5θ B + 0.667θ C − 5.833ψ DC = EI edaHRsayRbB½n§smIkarxagelI eyIgTTYl)an EIθ B = 243.78 EIθ C = 75.66 EIψ DC = 208.48 cugeRkay edayeRbIlT§plTaMgenH nigedaHRsaysmIkar (1)-(6) eyIg)an M AB = −347 kN .m M BA = −225kN .m M BC = 225kN .m M CB = 158kN .m M CD = −158kN .m M DC = −183kN .m ]TarhN_ 11-8³ kMNt;m:Um:g;enARtg;tMNnImYy²rbs;eRKagEdlbgðajenAkñúgrUbTI 11-20a. TMr enARtg; A nig D CaTMrbgáb; ehIytMN C RtUv)ansnμt;faCatMNsnøak;. EI mantémøefrsMrab;RKb; Ggát;TaMgGs;. karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -402
  • 26. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ smIkar slope-deflection eKGacGnuvtþsmIkar 11-8 eTAelIGgát; AB edaysarvaRtUv)antP¢ab;edaytMNbgáb;enAcugTaMgsgçag. eKGacGnuvtþsmIkar11-10 BI B eTA C nigBI D eTA C edaysarsnøak;enARtg;cMNuc C RTm:Um:g;sUnü. dUc bgðajedaydüaRkamPaBdab ¬rUbTI 11-20b¦ eKmanbM;las;TIlIenEG‘r Δ nigbMlas;TImMuθ B enARtg; cMNuc B EdlCaGBaØat*. edaysar Δ Ggát; AB nig CD vilRsbRTnicnaLika ψ = ψ AB = ψ DC = Δ / 4 . edaydwgfa θ A = θ D = 0 nigvaminman FEM sMrab;Ggát; enaHeyIg)an * bMlas;TImMu θ nig θ enARtg;tMN C ¬snøak;¦ minRtUv)anrab;bBa©ÚlenAkñúgkarviPaKeT edaysareKeRbIsmIkar 11-10. CB CD Displacement method of analysis: Slope-deflection equations T.Chhay -403
  • 27. Department of Civil Engineering NPIC ⎛I⎞ M N = 2 E ⎜ ⎟(2θ N + θ F − 3ψ ) + (FEM )N ⎝L⎠ ⎛I⎞ M AB = 2 E ⎜ ⎟[2(0 ) + θ B − 3ψ ] + 0 (1) ⎝4⎠ ⎛I⎞ M BA = 2 E ⎜ ⎟(2θ B + 0 − 3ψ ) + 0 (2) ⎝4⎠ ⎛I⎞ M N = 3E ⎜ ⎟(θ N − ψ ) + (FEM )N ⎝L⎠ ⎛I⎞ M BC = 3E ⎜ ⎟(θ B − 0) + 0 (3) ⎝3⎠ ⎛I⎞ M DC = 3E ⎜ ⎟(0 − ψ ) + 0 (4) ⎝4⎠ smIkarlMnwg smIkarlMnwgrbs;m:Um:g;enARtg;tMN B ¬rUbTI 11-20c¦. eyIg)an M BA + M BC = 0 (5) RbsinebIeKbUkkmøaMgsRmab;eRKagTaMgmUltamTisedAedk eyIg)an + → ∑ Fx = 0 10 − V A − VD = 0 (6) dUcbgðajenAkñúgdüaRkamGgÁesrIrbs;ssrnImYy² ¬rUbTI 11-20d¦ eyIg)an M AB + M BA ∑MB = 0 VA = − 4 M DC ∑ MC = 0 VD = − 4 dUcenHBIsmIkar (6) M AB + M BA M DC 10 + + =0 (7) 4 4 edayCMnYssmIkar slope-deflection eTAkñúgsmIkar (5) nig (7) ehIyedaysMrYlvaeyIgTTYl)an 3 θB = ψ 4 EI ⎛ 3 15 ⎞ 10 + ⎜ θB − ψ ⎟ = 0 4 ⎝2 4 ⎠ dUcenH θB = 240 21EI ψ= 320 21EI edayCMnYstémøTaMgenHeTAkñúgsmIkar (1)-(4) eyIg)an M AB = −17.1kN .m M BA = −11.4kN .m M BC = 11.4kN .m M DC = −11.4kN .m edayeRbIlT§plTaMgenH eKGackmøaMgRbtikmμcugrbs;Ggát;nImYy²BIsmIkarlMnwg ¬rUbTI 11-20e¦. düaRkamm:Um:g;sMrab;eRKagRtUv)anbgðajenAkñúgrUbTI 11-20f. karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -404
  • 28. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ]TarhN_ 11-9³ cUrBnül;BIrebobkMNt;m:Um:g;enARtg;tMNnImYy²éneRKagBIrCan;EdlbgðajenAkñúg rUbTI 11-21a. EI mantémøefr. dMeNaHRsay³ smIkar slope-deflection edaysarTMrenARtg; A nig F CaTMrbgáb; eKGacGnuvtþsmIkar 11-8 sMrab;Ggát;TaMgR)aMmYyrbs;eRKag. eKminKNna FEM eT edaysarbnÞúkTaMgGs;manGMeBIenARtg;tMN. enATIenH bnÞúkeFVIeGaytMN B nig tMN E pøas;TIedaybrimaN Δ1 ehIyeFVIeGaytMN C nigtMN D pøas;TIedaybrimaN Δ1 + Δ 2 . Ca lT§pl Ggát; AB nigGgát; FE rgkarviledaybrimaNψ 1 = Δ1 / 5 ehIy BC nig DE rgkarvileday brimaN ψ 2 = Δ 2 / 5 . edayGnuvtþsmikar 11-8 eTAelIeRKageyIg)an ⎛I⎞ M AB = 2 E ⎜ ⎟[2(0) + θ B − 3ψ 1 ] + 0 (1) ⎝5⎠ ⎛I⎞ M BA = 2 E ⎜ ⎟[2θ B + 0 − 3ψ 1 ] + 0 (2) ⎝5⎠ ⎛I⎞ M BC = 2 E ⎜ ⎟[2θ B + θ C − 3ψ 2 ] + 0 (3) ⎝5⎠ Displacement method of analysis: Slope-deflection equations T.Chhay -405
  • 29. Department of Civil Engineering NPIC ⎛I⎞ M CB = 2 E ⎜ ⎟[2θ C + θ B − 3ψ 2 ] + 0 (4) ⎝5⎠ ⎛I⎞ M CD = 2 E ⎜ ⎟[2θ C + θ D − 3(0)] + 0 (5) ⎝7⎠ ⎛I⎞ M DC = 2 E ⎜ ⎟[2θ D + θ C − 3(0 )] + 0 (6) ⎝7⎠ ⎛I⎞ M BE = 2 E ⎜ ⎟[2θ B + θ E − 3(0)] + 0 (7) ⎝7⎠ ⎛I⎞ M EB = 2 E ⎜ ⎟[2θ E + θ B − 3(0 )] + 0 (8) ⎝7⎠ ⎛I⎞ M ED = 2 E ⎜ ⎟[2θ E + θ D − 3ψ 2 ] + 0 (9) ⎝5⎠ ⎛I⎞ M DE = 2 E ⎜ ⎟[2θ D + θ E − 3ψ 2 ] + 0 (10) ⎝5⎠ ⎛I⎞ M FE = 2 E ⎜ ⎟[2(0 ) + θ E − 3ψ 1 ] + 0 (11) ⎝5⎠ ⎛I⎞ M EF = 2 E ⎜ ⎟[2θ E + 0 − 3ψ 1 ] + 0 (12) ⎝5⎠ smIkarTaMg 12enHmanGBaØat 18. smIkarlMnwg smIkarlMnwgrbs;m:Um:g;enARtg;tMN B / C / D nig E ¬rUbTI 11-21c¦. eyIg)an M BA + M BE + M BC = 0 (13) M CB + M CD = 0 (14) M DC + M DE = 0 (15) M EF + M EB + M ED = 0 (16) dUcenAkñúg]TahrN_elIkmun kmøaMgkat;enARtg;Kl;rbs;ssrsMrab;RKb;Can;RtUvEteFVIeGaybnÞúktamTis edkmantulüPaB ¬rUbTI 11-21c¦. eyIg)an + → ∑ Fx = 0 40 − VBC − VED = 0 M + M CB M ED + M DE 40 + BC + =0 (17) 5 5 + → ∑ Fx = 0 40 + 80 − V AB − VFE = 0 M + M BA M EF + M FE 120 + AB + =0 (18) 5 5 dMeNaHRsayTamTarnUvkarCMnYssmIkr (1) - (12) eTAkñúgsmIkar (13) - (18) EdlTTYl)anR)aMmYy smIkarEdlmanR)aMmYyGBaØatKW ψ 1 / ψ 2 / θ B / θC / θ D nigθ E . eKGacedaHRsayRbB½n§smIkarenH. eKRtUvCMnYslT§plEdlTTYl)aneTAkñúgsmIkar (1) –(12) EdleFVIeGayeyIgTTYl)anm:Um:g;enARtg;tMN. karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -406
  • 30. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ]TarhN_ 11-10³ kMNt;m:Um:g;enARtg;tMNnImYy²éneRKagEdlbgðajenAkñúgrUbTI 11-22a. EI mantémøefrsMrab;Ggát;nImYy². dMeNaHRsay³ smIkar slope-deflection eKGacGnuvtþsmIkar 11-8 eTAelIGgát;TaMgbI. FEM KW 2 30(3.6 )2 (FEM )BC = − wL =− = −32.4kN .m 12 12 wL2 30(3.6 )2 (FEM )CB = = = 32.4kN .m 12 12 Ggát;eRTt AB eFVIeGayeRKageTreTAsþaMdUcbgðajkñúgrUbTI 11-22a. CalT§pl tMN B nig C rgnUv bMlas;TImMu nigbMlas;TIlIenEG‘r. bMlas;TIlIenEG‘rRtUv)anbgðajenAkñúgrUbTI 11-22b Edl B cl½teTA B' )anTMhM Δ1 ehIy C cl½teTA C ' )an Δ 3 . bMlas;TITaMgenHeFVIeGayGgát;vilRsbRTnicnaLikaψ 1 / ψ 3 niigRcasRTnicnaLika − ψ 2 dUcbgðaj . dUcenH * * rlWkfa kMhUcRTg;RTayEdlbNþalBIkmøaMgtamG½kSRtUv)anecal ehIybMlas;TI BB' nig CC ' RtUv)anKitCaragbnÞat; eday sarCak;Esþgψ 1 nigψ 3 mantémøtUc. Displacement method of analysis: Slope-deflection equations T.Chhay -407
  • 31. Department of Civil Engineering NPIC Δ1 Δ1 Δ3 ψ1 = ψ2 = − ψ3 = 3 3.6 6 dUcbgðajenAkñúgrUbTI 11-22c bMlas;TITaMgbImanTMnak;TMngKña. ]TahrN_ Δ 2 = 0.5Δ1 ehIy Δ 3 = 0.866Δ1 . dUcenH BIsmIkarxagelI eyIg)an ψ 2 = −0.417ψ 1 ψ 3 = 0.433ψ 1 edaseRbIlT§plTaMgenH smIkar slope-deflection sMrab;eRKagKW ⎛I⎞ M AB = 2 E ⎜ ⎟[2(0) + θ B − 3ψ 1 ] + 0 (1) ⎝3⎠ ⎛I⎞ M BA = 2 E ⎜ ⎟[2θ B + 0 − 3ψ 1 ] + 0 (2) ⎝3⎠ ⎛ I ⎞ M BC = 2 E ⎜ ⎟[2θ B + θ C − 3(− 0.417ψ 1 )] − 32.4 (3) ⎝ 3.6 ⎠ ⎛ I ⎞ M CB = 2 E ⎜ ⎟[2θ C + θ B − 3(− 0.417ψ 1 )] + 32.4 (4) ⎝ 3.6 ⎠ ⎛I⎞ M CD = 2 E ⎜ ⎟[2θ C + 0 − 3(0.433ψ 1 )] + 0 (5) ⎝6⎠ ⎛I⎞ M DC = 2 E ⎜ ⎟[2(0) + θ C − 3(0.433ψ 1 )] + 0 (6) ⎝6⎠ smIkarTaMgR)aMmYymanGBaØatR)aMbYn. smIkarlMnwg smIkarlMnwgrbs;m:Um:g;enARtg;tMN B nig C . eyIg)an M BA + M BC = 0 (7) M CD + M CB = 0 (8) eKGacTTYl)ansmIkarcaM)ac;TIbIedayeFVIplbUkm:Um:g;enARtg;cMNuc O ¬rUbTI 11-22d¦. smIkarenH lubbM)at;kmøaMgEkg N A nig N D dUcenH + ∑ MO = 0 ⎛ M + M BA ⎞ ⎛M + M CD ⎞ M AB + M DC − ⎜ AB ⎟(10.2) − ⎜ DC ⎟(12.24) − 108(1.8) = 0 ⎝ 3 ⎠ ⎝ 6 ⎠ − 2.4M AB − 3.4M BA − 2.04 M CD − 1.04M CD − 194.4 = 0 (9) edayCMnYssmIkar (2) nig (3) eTAkñúgsmIkar (7) smIkar (4) nigsmIkar (5) eTAkñúgsmIkar (8) nig smIkar (1), (2), (5) nig (6) eTAkñúgsmIkar (9) eyIg)an 9.72 0.733θ B + 0.167θ C − 0.392ψ 1 = EI 9.72 0.167θ B + 0.533θ C + 0.0784ψ 1 = − EI 58.32 − 1.840θ B − 0.512θ C + 3.880ψ 1 = EI karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -408
  • 32. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa edaHRsayRbB½n§smIkarTaMgenH eyIg)an EIθ B = 35.51 EIθ C = −33.33 EIψ 1 = 27.47 edayCMnYstémøTaMgenHeTAkñúgsmIkar (1) – (6), eyIg)an M AB = −31.3kN .m M BC = 7.60kN .m M CD = −34.2kN .m M BA = −7.60kN .m M CB = 34.2kN .m M DC = −23.0kN .m Displacement method of analysis: Slope-deflection equations T.Chhay -409
  • 33. Department of Civil Engineering NPIC cMeNaT !!>!> kMNt;m:Um:g;enARtg;TMr A nig C bnÞab;mksg; !!>%> kMNt;m:Um:g;enARtg;TMr B bnÞab;mksg;düa- düaRkamm:Um:g;. snμt;fatMN B CatMNkl;. EI Rkamm:Um:g;sMrab;Fñwm. snμt;TMrenARtg; A nig C mantémøefr. CaTMrbgáb;. EI mantémø efr. !!>@> kMNt;m:Um:g;enARtg;TMr A nig B bnÞab;mk !!>^> kMNt;m:Um:g;Bt;enARtg;TMr A / B nig C sg;düaRkamm:Um:g;. EI mantémøefr. bnÞab;mksg;düaRkamm:Um:g;sMrab;Fñwm. snμt;TMr enARtg; A CaTMrsnøak; ehIy B nig C CaTMrkl;. EI mantémøefr. !!>#> kMNt;m:Um:g;enARtg;TMr A nig B bnÞab;mk sg;düaRkamm:Um:g;sMrab;Fñwm. EI mantémøefr. !!>&> kMNt;RbtikmμTMrenARtg; A / B nig C bnÞab;mksg;düaRkamm:Um:g;sMrab;Fñwm. snμt;TMr enARtg; A CaTMrsnøak;. !!>$> kMNt;m:Um:g;enARtg;TMr B nig C bnÞab;mk sg;düaRkamm:Um:g;sMrab;Fñwm. snμt; A / B nig C CaTMrkl; ehIy D CaTMrsnøak;. EI mantémø efr. !!>*> kMNt;m:Um:g;enARtg; B / C nig D bnÞab; mksg;düaRkamm:Um:g;sMrab; ABDE . snμt; A CaTMrsnøak; nig D CaTMrkl; ehIy C CaTMrbgáb;. EI mantémøefr. cMeNaT T.Chhay -410
  • 34. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa !!>(> kMNt;m:Um:g;enARtg; B bnÞab;mksg;düa- Rkamm:Um:g;sMrab;Ggát;nImYy²rbs;eRKag. snμt; A nig C CaTMrsnøak; ehIy B CatMNbgáb;. EI mantémøefr. !!>!@> kMNt;m:Um:g;enARtg; B nig C . snμt; B nig C CaTMrkl; ehIy A nig D CatMNsnøak;. EI mantémøefr. !!>!0> kMNt;m:Um:g;enARtg; B nig D bnÞab;mksg; !!>!#> kMNt;kmøaMgRbtikmμtamTisedk nigtam düaRkamm:Um:g;. snμt; A nig C CaTMrsnøak; ehIy TisQrenARtg; A nig C . snμt; A nig C CaTMr B nig D CatMN nigTMrbgáb;. EI mantémøefr. snøak; ehIy B CatMNbgáb;. yk E = 200GPa . !!>!!> kMNt;m:Um:g;enARtg; B bnÞab;mksg;düa- Rkamm:Um:g;sMrab;Ggát;nImYy²rbs;eRKag. snμt; A nig C CaTMrbgáb; ehIy B CatMNbgáb;. EI !!>!$> kMNt;m:Um:g;Bt;enARtg; A nig B bnÞab;mk mantémøefr. sg;düaRkamm:Um:g;. snμt; B nig C CaTMrkl;. Problems T.Chhay -411
  • 35. Department of Civil Engineering NPIC EI mantémøefr. !!>!*> kMNt;m:Um:g;enARtg;tMNnImYy² nigTMrrbs; eRKag. tMN nigTMrCatMN nigTMrbgáb;. EI man témøefr. !!>!%> kMNt;m:Um:g;Bt;enARtg; A / B nig C bnÞab; mksg;düaRkamm:Um:g;. snμt; A CaTMrbgáb;. EI mantémøefr. !!>!(> eRKagenHeFVIBIbMBg;TIbEdlmantMNbgáb;. RbsinebIvaRTbnÞúkdUcbgðaj cUrkMNt;m:Um:g;Edl ekItmanenARtg;tMN nigTMrnImYy². EI man !!>!^> kMNt;m:Um:g;Bt;enARtg;cugénGgát;rbs; témøefr. eRKag. TMrenARtg; A nig C TMrbgáb; ehIy B CatMNbgáb;. EI mantémøefr. !!>@0> kMNt;m:Um:g;Bt;enARtg;tMNnImYy² nigenA Rtg;TMrbgáb; bnÞab;mksg;düaRkamGgÁesrI. EI mantémøefr. !!>!&> FñwmCab;RTbnÞúkcMcMNucbI. kMNt;m:Um:g;Bt; GtibrmaenAkñúgFñwm nigbnÞab;mksg;düaRkam m:Um:g;. EI mantémøefr. !!>@!> kMNt;m:Um:g;Bt;enARtg;tMN nigTMrnImYy². eKmantMNbgáb;enARtg; B nig C ehIyeKmanTMr bgáb;enARtg; A nig D . EI mantémøefr. cMeNaT T.Chhay -412
  • 36. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa EdlmanTMrsamBaØ. snμt;TMrRtg; A nig E CaTMr snøak; ehIytMNTaMgGs;CatMNbgáb;. EI man témøefr. !!>@@> kMNt;m:Um:g;Bt;enARtg; A / B / C nig D bnÞab;mksg;düaRkamm:Um:g;. Ggát;nImYy²RtUv )antP¢ab;edaytMNbgáb; nigmanTMrbgáb;. EI mantémøefr. !!>@%> edaHRsaycMeNaT !!>@$ edaysnμt;faTMr A nig E CaTMrbgáb;. !!>@^> kMNt;m:Um:g;Bt;enARtg;tMNnImYy²rbs; eRKag. TMr A nig D CaTMrsnøak;. EI mantémø efr. !!>@#> épÞxagrbs;eRKagrgbnÞúkGIuRdUsþaTicdUc bgðaj. kMNt;m:Um:g;Bt;enARtg;tMN nigTMr nImYy². EI mantémøefr. !!>@&> kMNt;m:Um:g;Bt;enARtg;tMNnImYy²rbs; eRKag nigenARtg;TMr A nig D EdlCaTMrbgáb;. EI mantémøefr. !!>@$> kMNt;m:Um:g;Bt;enARtg;tMNnImYy²rbs; eRKag. bnÞúkelIdMbUlRtUv)anepÞreTAédrENg Problems T.Chhay -413
  • 37. Department of Civil Engineering NPIC KMerag !!>!> dMbUlRtUv)anRTedayrnUtEdlQr elIr:tBIr. eKKitfarnUtnImYy²RtUv)anRTedayTMr samBaØ ehIyeKKitfar:txagmuxRtUv)anRTeday ssrEdlCaTMrsnøak;enARtg; A nigCaTMrkl;enA Rtg; B nig C . snμt;fadMbUlRtUv)anplitBIebtug EdlmankRmas; 75mm ehIyrnUtnImYy²man TMgn; 2.5kN . eyagtamkUddMbUlenHrgnUvbnÞúk RBwlEdlmanTMgn; 1.2kN / m2 . rnUtmanRbEvg 8m . sg;düaRkamkmøaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;r:t. snμt;fassrmanlkçN³rwg. cMeNaT T.Chhay -414

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