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12. displacement method of analysis moment distribution

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12. displacement method of analysis moment distribution

  1. 1. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa !@> karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; Displacement method of analysis: Moment distribution viFIEbgEckm:Um:g;CakarviPaKtamviFIbMlas;TIEdlgayRsYlkñúgkarGnuvtþenAeBlEdleKkMNt; efreGLasÞic (elastic constant). CMBUkenHnwgbgðajCadMbUgnUvniymn½y nigeKalKMnitsMxan;² sRmab; viFIEbgEckm:Um:g; ehIybnÞab;mkGnuvtþviFIenHedIm,IedaHRsaycMeNaTEdlBak;B½n§nwgFñwm nig eRKagmin kMNt;edaysþaTic. karGnuvtþcMeBaHeRKageRcInCab;RtUv)anbgðajenAkñúgCMBUkcugeRkay. !@>!> eKalkarN_TUeTA nigniymn½y (General principles and definitions) karsikSaviPaKeRKOgFñwm nigeRKagEdleRbIviFIEbgEckm:Um:g;EdlbegáIteLIgedayhaDIRkUs (Hardy Cross) enAkñúgqñaM1930. enAeBlenaH viFIenHTTYlkarcab;GarmμN_Pøam²enAeBlEdleKpSBV pSayvaCaelIkdMbUg ehIyeKTTYlsÁal;favaCaviFIEdlmankareCOnelOneTAmuxKYreGaykt;sMKal;kñúg karviPaKeRKOgbgÁúMenAkñúgstvtSTI20. viFIEbgEckm:Um:g;CaviFIRbhak;RbEhlEdleKGacTTYl)ankMritsuRkiteTAtamtMrUvkarEdleKcg; )an. viFIenHcab;epþImedaysnμt;tMNnImYy²rbs;eRKagCatMNbgáb;. bnÞab;mkedaykarRsay nigkar Tb;tMNnImYy²bnþbnÞab;Kña enaHm:Um:g;Bt;enARtg;tMNRtUv)anEbgEck nigRtUv)aneFIVeGaymantMélesμIKña rhUtdl;tMNenaHvildl; b¤esÞIrdl;TItaMgcugeRkay. eyIgnwgeXIjfaviFIenHmanlkçN³sarcuHsareLIg niggayRsYlkñúgkarGnuvtþ. b:uEnþ munnwgBnül;BIbec©keTsénviFIEbkEckm:Um:g; eyIgRtUvdwgBIniymn½y nigeKalKMnitxøH. karkMNt;sBaØa³ eyIgnwgbegáItkarkMNt;sBaØadUcKñanwgkarkMNt;sBaØasMrab;smIkar slope-deflection Edlm:Um:g;viC¢manCam:Um:g;EdleFVVIeGayGgát;vilRsbRTnicnaLika ehIym:Um:g;GviC¢manCam:Um:g;EdleFVI eGayGgát;vilRcasRTnicnaLika ¬rUbTI 12-1¦. m:Um:g;bgáb;cug (FEMs)³ m:Um:g;enARtg;CBa¢aMg b¤enARtg;tMNbgáb;énGgát;rgbnÞúkRtUv)aneKehAfam:Um:g; bgáb;cug (fixed-end moment). eKGackMNt;m:Um:g;TaMgenHBItaragenAkñúgemeronTI11 m:Um:g;bgáb;cug Displacement method of analysis: Moment distribution T.Chhay -415
  2. 2. Department of Civil Engineering NPIC GaRs½yeTAnwgRbePTénkardak;bnÞúk. Ca]TahrN_ FñwmEdlrgbnÞúkdUcbgðajenAkñúgrUbTI 12-2 man m:Um:g;bgáb;cug FEM = PL / 8 = 800(10) / 8 = 1000 N .m . eKRtUvcMNaMGMBIGMeBIrbs;m:Um:g;TaMgenHmkelI Fñwm nigkarkMNt;sBaØarbs;eyIg dUcenHeyIgeXIjfa M AB = −1000 N .m ehIy M BA = +1000 N . emKuNkMrajsMrab;Ggát; (member stiffness factor)³ eKmanFñwmdUcbgðajenAkñúgrUbTI 12-3 Edl cugmçagRtUv)anRTedayTMrsnøak; nigmçageTotRTedayTMrbgáb;. GMeBIrbs;m:Um:g; M eFVIeGaycug A vil )anmMurgVil θ A . enAkñúgemeronTI11 eyIg)anP¢ab;TMnak;TMngrvag M nig θ A edayeRbIviFIFñwmnimitþ Edl M = (4 EI / L )θ A enAkñúgsmIkar 11-1. tYenAkñúgrg;Rkck 4 EI K= (12-1) L cugbgáb;q¶ay RtUv)aneKeGayeQμaHfaemKuNkMrajenARtg; A . emKuNkMrajCabrimaNénm:Um:g; M EdlcaM)ac;edIm,I eFVIeGaycug A rbs;Fñwmvil)an θ A = 1rad . emKuNkMrajsMrab;tMN (joint stiffness factor)³ RbsinebIGgát;CaeRcInRtUv)antP¢ab;edaytMNbgáb; ehIycugq¶ayrbs;vaCaTMrbgán; enaHedayeKalkarN_tMrYtpl emKuNkMralsrubenARtg;tMNCaplbUkén emKuNkMrajsMrab;Ggát;enARtg;tMN eBalKW KT = ∑ K . Ca]TahrN_ cUrBicarNatMNrbs;eRKagRtg; A enAkñúgrUbTI 12-4a. eKkkMNt;témøCaelxrbs;emKuNkMrajsMrab;Ggát;nImYy²BIsmIkar 12-1 ehIybgðajvaenAelIrUb. edayeRbItémøTaMgenH emKuNkMrajsrubéntMN A KW KT = ∑ K = 4000 + 5000 + 1000 = 10000 . TMhMenHCabrimaNénm:Um:g;EdlRtUvkaredIm,IbgViltMNeday)anmMu 1rad . emKuNEbgEck (Distribution factor DF)³ RbsinebIeKGnuvtþm:Um:g; M eTAelItMNbgáb; enaHGgát; karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -416
  3. 3. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa EdltP¢ab;eTAnwgtMNenaHnwgpþl;nUvcMENkénm:Um:g;Tb;caM)ac;edIm,IbMeBjlMnwgrbs;m:Um:g;enARtg;tMN. cMENkénm:Um:g;Tb;srubEdlpþl;eGayedayGgát;RtUv)aneKeGayeQμaHfaemKuNEbgEck. edIm,ITTYl )antémøTaMgenH BinitüemIltMNEdlP¢ab;eday n Ggát;. RbsinebIm:Um:g; M eFVIeGaytMNvil)an θ enaH Ggát; i nImYy²nwgviledaybrimaNdUcKñaenH. RbsinebIemKuNkMrajrbs;Ggát;TI i Ca K i enaHm:Um:g;Edl cUlrYmedayGgát;enaHKW M i = K iθ . edaysarsmIkarlMnwgRtUvkar M = M1 + M n = K1θ + K nθ = θ ∑ K i enaHemKuNEbgEcksRmab;Ggát;TI i KW Mi K iθ DFi = = M θ ∑ Ki edaysRmYltY θ eyIgeXIjfaemKuNEbgEcksRmab;Ggát;esμInwgplEckénemKuNkRmajrbs;Ggát;elI emKuNkRmajsrubsMrab;tMN EdlCaTUeTAeKsresr K DF = (12-2) ∑K Ca]TahrN_ emKuNEbgEcksMrab;Ggát; AB / AC nig AD enARtg;tMN A ¬enAkñúgrUbTI 12-4a¦ KW DFAB = 4000 / 10000 = 0.4 DFAC = 5000 / 10000 = 0.5 DFAD = 1000 / 10000 = 0.1 CalT§pl RbsinebI M = 2000 N .m eFVIGMeBIenARtg;cMNuc A ¬rUbTI 12-4b¦ m:Um:g;lMnwgEdlGnuvtþ edayGgát;mkelItMN ¬enAkñúgrUbTI 12-4c¦ KW M AB = 0.4(2000 ) = 800 N .m M AC = 0.5(2000) = 1000 N .m M AD = 0.1(2000 ) = 200 N .m emKuNkRmajeFobrbs;Ggát; (member relative-stiffness factor)³ eBlxøH Fñwm b¤eRKagCab;RtUv )aneFVIBIsmÖar³EtmYy dUcenHGgát;rbs;vaTaMgGs;RtUvEtmanm:UDuleGLasÞic E dUcKña. kñúgkrNIEbbenH Displacement method of analysis: Moment distribution T.Chhay -417
  4. 4. Department of Civil Engineering NPIC eKGacminKitemKuN 4E EdlmanenAkñúgsmIkar 12-1 BIPaKykénsmIkar 12-2 enAeBlkMNt;emKuN EbgEcksMrab;tMN. dUcenH eKmanlkçN³gayRsYledayRKan;EtkMNt;emKuNkRmajeFobrbs;Ggát; I KR = (12-3) L cugbgáb;q¶ay ehIyeRbIvasRmab;KNnaemKuNEbgEck. emKuNdMENl (carry-over factor)³ BicarNaFñwmenAkñúgrUbTI 12-3 mþgeTot. enAkñúgemeronTI 11 eyIgeXIjfa M AB = (4EI / L)θ A ¬smIkar 11-1¦ ehIy M BA = (2EI / L)θ A ¬smIkar 11-2¦. edayedaHRsayrk θ A ehIydak;eGaysmIkarTaMgenHesμIKñaeyIg)an M BA = M AB / 2 . eyIgGac niyayfa m:Um:g; M enARtg;TMrsnøak;eFVIeGaymanm:Um:g; M ' = M / 2 enARtg;CBa¢aMg. emKuNdMENlCa cMENkén M EdlRtUv)anepÞrBIsnøak;eTACBa¢aMg. dUcenH enAkñúgkrNIrbs;FñwmEdlmancugq¶ayCaTMrbgáb; emKuNdMENlKW + 1 / 2 . sBaØabUkmann½yfam:Um:g;TaMgBIreFVIkarkñúgTisedAdUcKña. !@>@> karEbgEckm:Um:g;sRmab;Fñwm (Moment distribution for beams) karEbkEckm:Um:g;KWQrelIeKalkarN_Rsay nigTb;tMNrbs;rcnasm<½n§bnþbnÞab;KñaedIm,IGnuBaØat eGaymankarEbgEckm:Um:g; nigeFVIeGaym:Um:g;mantulüPaBenARtg;tMN. meFüa)ayd¾l¥kñúgkarBnül;viFI KWkarBnül;edayeRbI]TahrN_. eKmanFñwmmYyEdlmanm:UDuleGLasÞic E efr nigmanTMhM nigbnÞúkdUcbgðajenAkñúgrUbTI 12-6a. munnwgeyIgcab;epþIm dMbUgeyIgRtUvKNnaemKuNEbgEckenARtg;cugTaMgBIrrbs;ElVgnImYy². edayeRbIsmIkar 12-1 ¬ K = 4EI / L ¦ emKuNkRmajenAEpñksgxagrbs; B KW 4 E (120)( 6 ) = 4 E (40)( 6 )mm 4 / m 10 K BA = 10 3 4 E (240 )(10 6 ) K BC = = 4 E (60)(106 )mm 4 / m 4 dUcenH edayeRbIsmIkar 12-2 DF = K / ∑ K sRmab;cugEdltP¢ab;eTAniwgtMN B eyIg)an 4 E (40) DFBA = = 0.4 4 E (40) + 4 E (60 ) 4 E (60) DFBC = = 0.6 4 E (40 ) + 4 E (60) enARtg;CBa¢aMg ¬tMN A nigtMN C ¦ emKuNEbgEckGaRs½ynwgemKuNkRmajrbs;Ggát; nigemKuN kRmajrbs;CBa¢aMg. edaysarenAkñúgRTwsþI eKRtUvkarm:Um:g;EdlmanTMhMGnnþedIm,IbgVilCBa¢aMgeGay)an karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -418
  5. 5. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa mYyra:düg; ¬emKuNkRmajCBa¢aMgKWGnnþ¦. dUcenHsRmab;tMN A nig C eyIgman 4 E (40) DFAB = =0 ∞ + 4 E (40) 4 E(60) DFCB = =0 ∞ + 4 E (60 ) cMNaMfa eKGacTTYl)anlT§plxagelIRbsinebIeKeRbIemKuNkRmajeFob K R = I / L ¬smIkar 12-3¦ sRmab;karKNna. elIsBIenH plbUkénemKuNEbgEckRtg;tMNEtgEtesμInwgmYy elIkElgEtRtg;TMr bgáb;. eRkayeBlKNnaemKuNEbgEck eyIgnwgkMNt;m:Um:g;bgáb;cug. manEtElVg BC b:ueNÑaHEdlrg bnÞúk ehIyedayeRbItaragenAkñúgemeronTI 11 eyIg)an wL2 6000(4 )2 (FEM )BC =− =− = −8000 N .m 12 12 2 6000(4 )2 (FEM )CB = wL = = 8000 N .m 12 12 eyIgcab;epþImedaysnμt;fatMN B RtUv)anTb; b¤bgáb;. m:Um:g;bgáb;cugenARtg;cMNuc B RTElVg BC kñúgTI Displacement method of analysis: Moment distribution T.Chhay -419
  6. 6. Department of Civil Engineering NPIC taMgbgáb; b¤TItaMgTb;dUcbgðajenAkñúgrUbTI 12-5b. CakarBit enHvaminEmnCasßanPaBlMnwgCak;EsþgenA Rtg;tMN B edaysarm:Um:g;enAelIEpñknImYy²rbs;tMNenHRtUvEtesμIKña nigmanTisedApÞúyKña. edIm,IEk sMrYlva eyIgRtUvGnuvtþm:Um:g; 8000 N.m EdlmanTisedApÞúyKñaeTAelItMN nigGnuBaØateGaytMNvileday esrI ¬rUbTI 12-5c¦. CalT§pl Epñkénm:Um:g;enHRtUv)anEbgEckeTAElVg BC nig BA edayeKarBtam emKuNEbgEck ¬b¤emKuNkRmaj¦ rbs;ElVgenARtg;tMN. m:Um:g;enAkñúg BA KW 0.4(8000) = 3200 N.m ehIym:Um:g;enAkñúg BC KW 0.6(8000) = 4800 N.m . cugeRkay edaysarkarvilEdlekItmanenARtg; B m:Um:g;enHRtUv)anepÞr edaysarm:Um:g;EdlekItmanenARtg;cugq¶ayrbs;ElVg. edayeRbIemKuNdMENl + 1 / 2 enaHlT§plRtUv)anbgðajenAkñúgrUbTI 12-5d. ]TahrN_enHbgðajfaCMhandMbUgEdlcaM)ac;enAeBlEbgEckm:Um:g;enARtg;tMNKW³ kMNt;m:Um:g; GtulüPaBEdlmanGMeBIenARtg;tMNEdlbgáb;dMbUg/ RsayPaBbgáb;rbs;tMNenaH/ rYcGnuvtþm:Um:g; GtulüPaBEdlmanTMhMesμIKña EtTisedApÞúyKñaedIm,IEktRmUvlMnwg/ EbgEckm:Um:g;kñúgcMeNamElVgEdlt P¢ab;/ ehIyepÞrm:Um:g;kñúgElVgrbs;vaeTAcugmçageTot. CaTUeTA CMhanTaMgenHRtUv)anbgðajenAkñúgtarag ¬rUbTI 12-5e¦. enATIenH Dist, CO bgðajfam:Um:g;RtUv)anEbgEck nigepÞr. enAkñúgkrNIBiessenH eK RtUvkareFVIkarEbgEckm:Um:g;EtmYyCMub:ueNÑaH edaysarCBa¢aMgenARtg; A nig C RsUbykm:Um:g; ehIymin mantMNq¶ayÉNaeTotEdleKRtUveFVIeGaymantulüPaB b¤RsaykarTb;edIm,IbMeBjlkçxNÐlMnwgrbs; tMNeT. eRkayeBlEbgEcktamrebobenH eKeFVIplbUkm:Um:g;Rtg;tMNnImYy²EdleFVIeGayeKTTYl)an lT§plcugeRkayEdlbgðajenAkñúgbnÞat;xageRkamrbs;taragénrUbTI 12-5e. cMNaMfa LÚvtMN B sßitkñúgsßanPaBlMnwg. edaysar M BC GviC¢man enaHm:Um:g;enHmanGMeBIelIElVg BC RcasRTnicnaLika dUcbgðajenAelIdüaRkamGgÁesrIrbs;ElVgFñwmenAkñúgrUbTI 12-5f. edaysareyIgsÁal;m:Um:g;cug eyIg GacKNnakmøaMgkat;enAxagcugrbs;Ggát;BIsmIkarlMnwgedayGnuvtþeTAelIElVgnImYy². LÚvBicarNaFñwmdUcKña EtTMr C CaTMrkl; ¬rUbTI 12-6a¦. enAkñúgkrNIenHmanEtGgát;mYyenA Rtg;tMN C dUcenHemKuNEbgEcksRmab;Ggát; CB enARtg;tMN C KW 4 E (60) DFCB = =1 4 E (60) emKuNEbgEckepSgeTot nigm:Um:g;bgáb;cugdUcKñanwgkarKNnaelIkmun. BYkvaRtUv)anbgðajenAkñúgCYr edkTI1 nigTI2éntaragenAkñúgrUbTI 12-6b. CadMbUg eyIgnwg]bmafatMN B nigtMN C RtUv)anTb;. eyIgcab;epþImedayRsaytMN C nigedaydak;m:Um:g; − 8000 N.m enARtg;tMNenaH. m:Um:g;TaMgmUlRtUv)an EbgEckeTAkñúgGgát; CB edaysar (1)(− 8000)N .m = −8000 N .m . sBaØaRBYjenARtg;CYredkTI3bgðaj fa (− 8000)N .m / 2 = −4000 N .m RtUv)anepÞreTAtMN B edaysareKGnuBaØateGaytMN C vileday karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -420
  7. 7. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa esrI. LÚvtMN C RtUv)anTb;eLIgvij. edaysareKRtUveFVIeGaym:Um:g;srubenARtg; C mantulüPaB eKdak;bnÞat;enABIeRkamm:Um:g; − 8000 N.m . LÚveyIgnwgBicarNam:Um:g;GtulüPaB − 12000 N.m enARtg; tMN B . enATIenH edIm,IeGaymanlMnwg eyIgRtUvGnuvtþm:Um:g; + 12000 N.m eTAelI B ehIytMNenHRtUv )anTb;y:agNaedIm,IeGaycMENkénm:Um:g;RtUv)anEbgEckeTA BA nig BC eBalKW (0.4)(12000) = 4800 N.m nig (0.6 )(12000 ) = 7200 N.m dUcbgðajenAkñúgCYredkTI4. ehIyeKRtUvcgcaMfa + 1 / 2 én m:Um:g;enHRtUv)anepÞreTACBa¢aMg A nigTMrkl; C edaysartMN B vil. LÚveKRtUvTb;tMN B eLIgvij. eyIgRtUvRsaytMN C mþgeTot ehIym:Um:g;GtulüPaBenARtg;TMrkl;RtUv)anEbgEckdUckareFVIelIkmun. lT§plRtUv)anbgðajenAkñúgCYredkTI5. eyIgRtUvTb; nigRsaytMN B nig C edIm,Ikat;bnßyTMhMm:Um:g; EdlRtUveFVIeGaymantulüPaBrhUtdl;eKGacecalva)anebIeFobnwgm:Um:g;edIm. eKRtUvyl;BIdMeNIrkar énCMhannImYy²BICYredkTI3 rhUtdl;CYredkTI14. edayeFVIplbUkm:Um:g; lT§plcugeRkayRtUv)an bgðajenAkñúgCYredkTI15 EdleyIgeXIjfam:Um:g;cugeRkaybMeBjlkçxNÐlMnwgrbs;tMN. RbesIrCakarGnuvtþdMeNIrkarEbgEckm:Um:g;bnþbnÞab;KñaeTAelItMNnImYy² ¬dUcbgðajxagelI¦ eK k¾manlT§PaBkñúgkarGnuvtþeTAelItMNTaMgGs;kñúgeBldMNalKñapgEdr. karGnuvtþenHRtUv)anbgðajenA kñúgtaragénrUbTI 12-6c. enAkñúgkrNIenH eyIgcab;epþImedayTb;tMNTaMgGs; ehIybnÞab;mkEbgEck m:Um:g;bgáb;cugTaMgenARtg;tMN B nigtMN C nigeFVIeGaym:Um:g;enAtMNmantulüPaB ¬CYredkTI3¦. Displacement method of analysis: Moment distribution T.Chhay -421
  8. 8. Department of Civil Engineering NPIC edayRsaytMN B nigtMN C kñúgeBldMNalKña ¬tMN A RtUv)anbgáb;rhUt¦ bnÞab;mkm:Um:g;RtUv)anepÞr eTAcugénElVgnImYy² ¬CYredkTI4¦. tMNRtUv)anTb;mþgeTot ehIym:Um:g;RtUv)anEbgEck nigeFVIeGay manlMnwg ¬CYredkTI 5¦. karRsaytMNenHmþgeTotGnuBaØateGaymankarepÞrm:Um:g; dUcbgðajenAkñúgCYr edkTI 6. edaykarKNnaCabnþbnÞab; eyIgTTYl)anlT§plcugeRkay ¬dUckarKNnaelIkmun¦ Edl bgðajenAkñúgCYredkTI 24. edayeFVIkareRbobeFob viFIenHpþl;nUvcemøIyrYmtUcyWtCagviFIelIkmun b:uEnþ enAkñúgkrNICaeRcInviFIenHmanRbsiT§PaBkñúgkarGnuvtþCag ehIysMrab;ehtuplEbbenHeyIgnwgeRbIvaenA kñúg]TahrN_bnþbnÞab;. cugeRkayedayeRbIlT§plenAkñúgrUbTI 12-6b b¤ 12-6c, düaRkamGgÁesrI rbs;ElVgFñwmnImYy²RtUv)anbgðajenAkñúgrUbTI 12-6d. eTaHbICa viFIenHBak;B½n§nwgCMhanCaeRcInedIm,ITTYl)anlT§plcugeRkay. viFIenHk¾TamTarnUvviFI saRsþEdr edaysarvaRtUvkarkarGnuvtþnUves‘rIénCMhanEdlmanlkçN³nBVnþCaCagkaredaHRsaysMnMuén smIkardUcenAkñúgviFI slope-deflection. b:uEnþ eKRtUvcMNaMfa dMeNIrkarCamUldæanénkarEbgEckm:Um:g; GnuvtþdUcnwgdMeNIrkarénviFIbMlas;TIdéTeTotEdr. dMeNIrkarKwkarbegáItTMnak;TMngrvagbnÞúk nigbMlas;TI edaykMNt;nUvbMlas;TImMud¾RtwmRtUvsMrab;tMN ¬lkçxNÐRtUvKña¦. b:uEnþ enATIenH lkçxNÐlMnwg niglkç- xNÐRtUvKñarbs;mMurgVilenARtg;tMNRtUv)anbMeBjedaypÞal;edayeRbIdMeNIrkarEdleFVIeGaym:Um:g;man tulüPaBEdlcUlrYmCamYynwgTMnak;TMngrvagbnÞúk nigPaBdab ¬emKuNkRmaj¦. eKmanlT§PaBkñúgkar sRmYlbnþeTotkñúgkareRbIR)as;karEbgEckm:Um:g;EdleyIgnwgelIkylmkerobrab;enAkñúgkfaxNÐxag mux. dMeNIrkarkñúgkarviPaK (Procedure for analysis) xageRkamCaCMhanEdlpþl;nUvviFITUeTAsRmab;kMNt;m:Um:g;cugenAelIElVgFñwmEdleRbIkarEbgEck m:Um:g;. emKuNEbgEck nigm:Um:g;bgáb;cug³ eKRtUvkMNt;emKuNkRmajsRmab;ElVgnImYy²enARtg;tMN. edayeRbItémøTaMgenH eyIgGacKNnaemKuNEbgEckBI DF = K / ∑ K . cMNaMfa DF = 0 sRmab;cug bgáb; ehIy DF = 1 sMrab;TMrsnøak; b¤TMrkl;. eyIgkMNt;mU:m:g;bgáb;cugsMrab;ElVgEdlrgbnÞúknImYy²edayeRbItaragEdleGayenAkñúgemeron TI11. m:Um:g;bgáb;cugviC¢maneFVIGMeBIRsbTisRTnicnaLikavil ehIy FEM GviC¢maneFVIGMeBIRcasRTnic naLika. edIm,IPaBgayRsYl eKGacerobcMtémøTaMgenHenAkñúgtaragEdlmanlkçN³dUctaragEdl bgðajenAkñúgrUbTI 12-6c. karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -422
  9. 9. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNIrkarEbgEckm:Um:g;³ snμt;fatMNTaMgGs;EdleyIgRtUvkarkMNt;m:Um:g;enAkñúgElVgEdltP¢ab;RtUv )anTb;. bnÞab;mk !> kMNt;m:Um:g;EdleFVIeGaytMNnImYy²manlMnwg. @> RsaytMN ehIyEbgEckm:Um:g;GtulüPaBeTAeGayElVgEdltP¢ab;nwgtMNnImYy². #> epÞrm:Um:g;enAkñúgElVgnImYy²eTAkan;cugnImYy²rbs;vaedayKuNm:Um:g;TaMgenaHedayemKuN dMENl (carry-over moment) + 1/ 2 . edaykarTb; nigkarRsaytMNsarcuHsareLIg eyIgeXIjfakarEktMrUvm:Um:g;RtUv)ankat;bnßy edaysarFñwmGacQaneTArkrUbragdabcugeRkay. enAeBlEdleyIgTTYl)antémøsMrab;karEktRmUvtUc lμm eKRtUvbBaÄb;dMeNIrkarsarcuHsareLIgRtg;karepÞrm:Um:g;cugeRkayeK. eKRtUvbUkbBa©ÚlKñanUvtémø FEM / m:Um:g;Edl)anBIkarEbgEck nigm:Um:g;Edl)anBIkarepÞrtamCYrQrnImYy². RbsinebIeyIgeFVIEbb enH)anRtwmRtUv eyIgnwgTTYl)ansßanPaBlMnwgénm:Um:g;Rtg;tMNnImYy². ]TahrN_ 12-1³ kMNt;m:Um:g;Bt;enARtg;tMNnImYy²rbs;FñwmEdlbgðajenAkñúgrUbTI 12-7a. EI Ca cMnYnefr. Displacement method of analysis: Moment distribution T.Chhay -423
  10. 10. Department of Civil Engineering NPIC dMeNaHRsay³ dMbUg eKRtUvkMNt;emKuNEbgEckenARtg;tMNnImYy². emKuNkRmajsMrab;Ggát;KW * 4EI 4EI 4EI K AB = K BC = K CD = 12 12 8 dUcenH DFAB = DFDC = 0 DFBA = DFBC = 4 EI / 12 4 EI / 12 + 4 EI / 8 = 0.5 4 EI / 12 4 EI / 8 DFCB = = 0.4 DFCD = = 0.6 4 EI / 12 + 4 EI / 8 4 EI / 12 + 4 EI / 8 m:Um:g;bgáb;cugKW wL2 20(12 )2 (FEM )BC =− =− = −240kN .m 12 12 2 20(12 )2 (FEM )CB = wL = = 240kN .m 12 12 − PL 250(8) (FEM )CD = =− = −250kN .m 8 8 PL 250(8) (FEM )DC = = = 250kN .m 8 8 edaycab;epþImCamYynwg FEM CYredkTI4 ¬rUbTI 12-7b¦ m:Um:g;enARtg;tMN B nig C RtUv)anEbgEck kñúgeBlEtmYy ¬CYredkTI5¦. bnÞab;mk m:Um:g;TaMgenHRtUv)anepÞreTAcugénElVgerog²xøÜn ¬CYredkTI6¦. m:Um:g;EdlTTYl)anRtUv)anEbgEck nigepÞrkñúgeBlEtmYymþgeTot ¬CYredkTI 7 nigTI8¦. dMeNIrkarRtUv )anbnþrhUtdl;m:Um:g;RtUv)ankat;bnßyrhUtdl;TMhMmYysmrmü ¬CYredkTI 13¦. m:Um:g;EdlCalT§pl RtUv)ankMNt;edayeFVIplbUk ¬CYredkTI14¦. edaydak;m:Um:g;enAelIFñwmnImYy² nigedayGnuvtþsmIkarlMnwgeyIgTTYl)ankmøaMgkat;enAxagcug ElVgdUcbgðajenAkñúgrUbTI 12-7c ehIydüaRkamm:Um:g;sMrab;FñwmTaMgmUlRtUv)anbgðajenAkñúgrUbTI 12- 7d. ]TahrN_ 12-2³ kMNt;m:Um:g;Bt;enARtg;TMrnImYy²rbs;FñwmEdlbgðajenAkñúgrUbTI 12-8a. m:Um:g; niclPaBénElVgnImYy²RtUv)an. dMeNaHRsay³ enAkñúgcMeNaTenH m:Um:g;mnRtUv)anEbgEckenAkñúgElVgly AB eT dUcenHemKuNEbg i Eck (DF )BA = 0 . kRmajrbs;ElVg BC KWEp¥kelI 4EI / L edaysarTMrsnøak;minEmnsßitenAcugq¶ay rbs;Fñwm. emKuNkRmaj emKuNEbgEck nigm:Um:g;bgáb;cugRtUv)anKNnadUcxageRkam³ 4 E (300)(106 ) 4 E (240)( 6 ) = 300(106 )E = 320( 6 )E 10 K BC = K CD = 10 4 3 * enATIenH eyIgeRbIemKuNkRmah 4EI / L b:uEnþeyIgk¾GaceRbIemKuNkRmajeFob I / L pgEdr. karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -424
  11. 11. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa DFBC = 1 − (DF )BA = 1 − 0 = 1 300 E DFCB = = 0.484 300 E + 320 E 320 E DFCD = = 0.516 300 E + 320 E 320 E DFDC = =0 ∞ + 320 E edaysarFñwmenHCaFñwmly (FEM )BA = 2000 N (2m ) = 4000 N .m (FEM ) = − wL = − 1500(4) = −2000 N .m 2 2 BC 12 12 2 1500(4 )2 (FEM )CB = wL = = 2000 N .m 12 12 Displacement method of analysis: Moment distribution T.Chhay -425
  12. 12. Department of Civil Engineering NPIC témøTaMgenHRtUv)anbgðajenAkñúgCYredkTI4rbs;taragkñúgrUbTI 12-8b. ElVgEdllyeFVIeGayekItman m:Um:g;Bt; − 4000 N.m enAxageqVg B . edIm,IeFVIeGaytMN B mantulüPaB eKRtUvmanm:Um:g;Bt; − 4000 N.m manGMeBIenAxagsþaM B . dUcbgðajenAkñúgCYredkTI 5 eKRtUvbEnßm − 2000 N.m eTAelI BC edIm,IbMeBj lkçxNÐ. dMeNIrkarénkarEbgEck nigkarepÞrm:Um:g;RtUv)anbnþtamrebobFmμtadUcbgðaj. edaysareKsÁal;m:Um:g;Bt; eKGacsg;düaRkamm:Um:g;sRmab;FñwmdUcbgðajenAkñúgrUbTI 12-8c. !@>#> karEktRmUvemKuNkRmaj (Stiffness-factor modification) enAkñúg]TarhrN_énkarEbgEckm:Um:g;elIkmun eyIg)anBicarNaElVgénFñwmnImYy²manTMrbgáb; ¬tMNRtUv)anTb;¦ enAxagcugq¶ayrbs;vaenAeBl EbgEck nigepÞrm:Um:g;. sRmab;ehtuplenH eyIg )anKNnaemKuNkRmaj emKuNEbgEck nigem KuNdMENledayQrenAelIkrNIdUcbgðajenAkñúg rUbTI 12-9. Cak;Esþg enATIenHemKuNkRmajKW K = 4 EI / L ¬smIkar 12-1¦ ehIyemKuNdMENlKW + 1 / 2 . enAkñúgkrNIxøH eKmanlT§PaBEkERbemKuNkRmajrbs;ElVgFñwmBiess ehtudUcenHeKGacsMrYl dl;dMeNIrkarEbgEckm:Um:g;. eyIgnwgBicarNakrNIbIEdlekIteLIgjwkjab;enAkñúgkarGnuvtþ. Ggát;EdlmanTMrsnøak;enAcugq¶ay³ FñwmminkMNt;xøHmanElVgxageRkAeKbMputRTedayTMrsnøak;dUcenA kñúgkrNIéntMN B enAkñúgrUbTI 12-10a. enATIenH m:Um:g; M bgVilcug A )an θ . edIm,IkMNt; θ eKRtUv kMNt;kmøaMgkat;enARtg;cMNuc A' énFñwmnimitþ ¬rUbTI 12-10b¦. eyIgman 1⎛ M ⎞ ⎛2 ⎞ + ∑ M B' = 0 V A ' ( L ) − ⎜ ⎟ L⎜ L ⎟ = 0 2 ⎝ EI ⎠ ⎝ 3 ⎠ ML V A' = θ = 3EI b¤ M = 3EI L θ dUcenH emKuNkRmajsRmab;FñwmenHKW 3EI K= (12-4) L cugq¶ayCaTMrsnøak; b¤TMrkl; karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -426
  13. 13. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ehIy eKRtUvcgcaMfaemKuNdMENlKWsUnü edaysarsnøak;enARtg; A' minTb;Tl;m:Um:g;eT. tamkar eRbobeFob RbsinebIcugq¶ayCaTMrbgáb;eKRtUvEktRmUvemKuNkRmaj K = 4EI / L eday 3 / 4 edIm,I eGayeTACakrNIEdlmancugq¶ayRTedayTMrsnøak;. RbsinebIeKKitkarEktRmUvenH dMeNIrkarénkar EbgEckm:Um:g;RtUv)ansRmYledaysareKminRtUvkarTb; nigRsayvaCabnþbnÞab;enAeBlEbgEckm:Um:g;eT. ehIyedaysarEtcugElVgRtUv)anRTedayTMrsnøak; eKkMNt;m:Um:g;bgáb;cugsMrab;ElVgedayeRbItémøenA kñúgCYrQrxagsþaMéntaragenAkñúgemeronTI11. ]TahrN_ 12-4 bgðajBIrebobGnuvtþkarsRmYlenH. FñwmsIuemRTIrgbnÞúksIuemRTI³ RbsinebIFñwmmanragsIuemRTI ehIyrgbnÞúksIuemRTI düaRkamm:Um:g;Bt; sRmab;Fñwmk¾nwgmanlkçN³sIuemRTIEdr. CalT§pl eKGaceFVIkarEktRmUvemKuNkRmajsRmab;ElVg kNþal dUcenHeKRtUvkarEbgEckm:Um:g;enAkñúgFñwmtamry³tMNEdlsßitenAelIFñwmEtBak;kNþal. edIm,I EktRmUvemKuNkRmajeGay)anRtwmRtUv eyIgnwgBicarNaFñwmEdlbgðajenAkñúgrUbTI 12-11a. eday sarPaBsIuemRTI m:Um:g;Bt;enARtg; B nig C RtUvEtesμIKña. edaysnμt;témøenHesμInwg M FñwmnimitþsMrab; ElVg BC RtUv)anbgðajenAkñúgrUbTI 12-11b. dUcenH mMurgVil θ enARtg;cugnImYy²KW ⎛ M ⎞ ⎛L⎞ + ∑ M C' = 0 − V B ' ( L ) + ⎜ ⎟ L⎜ ⎟ = 0 ⎝ EI ⎠ ⎝ 2 ⎠ ML VB ' = θ = 2 EI b¤ M = 2 EI L θ dUcenH emKuNkRmajsRmab;ElVgkNþalKW 2 EI K= (12-5) L FñwmsIuemRTI nigrgbnÞúksIuemRTI dUcenH eKGacEbgEckm:Um:g;sRmab;EtBak;kNþalFñwmRbsinebIeKKNnaemKuNkRmajsMrab;ElVgkNþal edayeRbIsmIkar 12-5. tamkareRbobeFob emKuNkRmajkNþalElVgesμInwgBak;kNþalénemKuN kRmajEdleRbI K = 4EI / L . Displacement method of analysis: Moment distribution T.Chhay -427
  14. 14. Department of Civil Engineering NPIC FñwmsIuemRTIrgbnÞúkGsIuemRTI³ RbsinebIFñwmmanragsIuemRTIrgbnÞúkGsIuemRTI eKnwgTTYl)andüaRkam GsIuemRTI. dUcenAkñúgkrNIelIkmun eyIgGacEktRmUvemKuNkRmajrbs;ElVgkNþal dUcenHeyIgnwgKit FñwmEtBak;kNþalsRmab;karviPaKedaykarEbgEckm:Um:g;. BicarNaFñwmenAkñúgrUbTI 12-12a. Fñwm nimitþsMrab;ElVgkNþal BC rbs;vaRtUv)anbgðajenAkñúgrUbTI 12-12b. edaysarbnÞúkGsIuemRTI m:Um:g; Bt;enARtg; B nig C RtUvEtesμIKñaEtmanTispÞúyKña. snμt;fa témøenHesμInwg M enaHmMurgVilenARtg;cug nImYy²KW³ 1 ⎛ M ⎞⎛ L ⎞⎛ 5L ⎞ 1 ⎛ M ⎞⎛ L ⎞⎛ L ⎞ + ∑ M C' = 0 − VB ' ( L ) + ⎜ ⎟⎜ ⎟⎜ ⎟ − ⎜ ⎟⎜ ⎟⎜ ⎟ = 0 2 ⎝ EI ⎠⎝ 2 ⎠⎝ 6 ⎠ 2 ⎝ EI ⎠⎝ 2 ⎠⎝ 6 ⎠ ML VB ' = θ = 6 EI b¤ M = 6 EI L θ dUcenH emKuNkRmajsRmab;ElVgkNþalesμInwg 6 EI K= (12-6) L FñwmsIuemRTI nigrgbnÞúkGsIuemRTI dUcenH enAeBleKKNnaemKuNkRmajsMrab;ElVgkNþaledayeRbIsmIkar 12-6 eKRtUvkarEbgEck m:Um:g;enAkñúgEtBak;kNþalFñwmb:ueNÑaH. tamkareRbobeFob emKuNkRmajkNþalElVgesμInwgemKuN kRmajEdleRbI K = 4EI / L mYydgknøH. karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -428
  15. 15. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ]TahrN_ 12-3³ kMNt;m:Um:g;Bt;enARtg;TMrsRmab;FñwmEdlbgðajenAkñúgrUbTI 12-13a. EI CacMnYn efr. dMeNaHRsay³ tamkarGegát Fñwm nigbnÞúkmanlkçN³sIuemRTI. dUcenH eyIgnwgGnuvtþ K = 2EI / L edIm,IKNnaemKuNkRmajrbs;ElVgkNþal BC ehIyeyIgnwgviPaKEtkMNat;Bak;kNþalxageqVgrbs; FñwmEtb:ueNÑaH. eKGackat;bnßykarviPaKeGaykat;EtxøIEfmeTotedayeRbI K = 3EI / L sRmab;kar KNnaemKuNkRmajsMrab;kMNat;Ggát; AB edaysarcugq¶ay A CaTMrsnøak;. elIsBIenH eKGacrMlg karEbgEckm:Um:g;enARtg;cMNuc A edayeRbIm:Um:g;bgáb;cugsRmab;bnÞúkragRtIekaNmanGMeBIelIElVgEdl manTMrmçagCaTMrsnøak; nigTMrmçageTotCaTMgbgáb;. dUcenH K AB = 3EI 3 ¬edayeRbIsmIkar 12-4¦ K BC = 2EI 4 ¬edayeRbIsmIkar 12-5¦ 3EI / 3 DFAB = =1 3EI / 3 3EI / 3 DFBA = = 0.667 3EI / 3 + 2 EI / 4 2 EI / 4 DFBC = = 0.333 3EI / 3 + 2 EI / 4 2 100(3)2 (FEM )BA = wL = = 60kN .m 15 15 2 100(4 )2 (FEM )BC = − wL =− = −133.3kN .m 12 12 TIinñn½yTaMgenHRtUv)anerobcMenAkñúgtarageénrUbTI 12-13b. karKNnaemKuNkRmajdUcbgðajxagelI BitCakat;bnßykarviPaKy:ageRcIn edaysarmanEttMN B EtmYybueNÑaHEdleKRtUvkareFVIeGayman lMnwg ehIyeKminRtUvkarepÞrm:Um:g;eTAtMN A nigtMN C eT. eyIgeXIjfatMN C rgm:Um:g; 108.9kN.m dUctMN B Edr. Displacement method of analysis: Moment distribution T.Chhay -429
  16. 16. Department of Civil Engineering NPIC ]TahrN_ 12-4³ kMNt;m:Um:g;Bt;enARtg;TMrsRmab;FñwmEdlbgðajenAkñúgrUbTI 12-14a. m:Um:g;nicl PaBrbs;ElVgTaMgBIrRtUv)anbgðajenAkñúgrUb. dMeNaHRsay³ edaysarFñwmRtUv)anRTedayTMrkl;enAcugq¶ay C kRmajrbs;ElVg BC RtUv)anKNna edayQrelI K = 3EI / L . eyIg)an K AB = = ( ) 4 EI 4 E (120) 10 6 ( ) = 160 10 6 E L 3 K BC = 3EI 3E (240) 106 = ( ) ( ) = 180 106 E L 4 dUcenH DFAB = 160 E ∞ + 160 E =0 160 E DFBA = = 0.4706 160 E + 180 E 180 E DFBC = = 0.5294 160 E + 180 E 180 E DFCB = =1 180 E karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -430
  17. 17. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa eKGacsRmYlviFIEbgEckm:Um:g;sRmab;cMeNaTenHedayRtUvdwgfaeKGaceRbIm:Um:g;bgáb;cugeTalsRmab; ElVg BC . edayeRbICYrxagsþaMéntaragénemeronTI11 sMrab;ElVgEdlrgbnÞúkrayesμImanTMrmçagCaTMr bgáb; nigmçageTotCaTMrsnøak; enaHeyIg)an wL2 6000(4 )2 (FEM )BC =− =− = −12000 N .m 8 8 Tinñn½yTaMgGs;RtUv)anbBa©ÚleTAkñúgtaragenAkñúgrUbTI 12-14b ehIykarEbgEckm:Um:g;RtUv)an Gnuvtþ. edayeRbobeFobCamYynwgrUbTI 12-6b viFIenHBitCasRmYldl;karEbgEcky:agxøaMg. edayeRbIlT§plEdlTTYl)an kmøaMgkat;xagcugrbs;Fñwm nigdüaRkamm:Um:g;RtUv)anbgðajenAkñúg rUbTI 12-14c. !@>$> karEbgEckm:Um:g;sMrab;eRKagmineyal (Moment distribution for frames: No sidesway) karGnuvtþénviFIEbgEckm:Um:g;sRmab;eRKagEdlmineyalmanlkçN³dUcnwgdMeNIrkarsRmab; viPaKFñwm. edIm,Ikat;bnßy»kasénkMhusqÁg eKENnaMeGayeFVIkarviPaKedayeRbItaragdUcenAkñúg ]TahrN_elIkmun. ehIyeKGackat;bnßykarEbgEckm:Um:g; RbsinebIeKEktRmUvemKuNkRmajdUc bgðajenAkñúgkfaxNÐmun. ]TahrN_ 12-5³ kMNt;m:Um:g;Bt;enARtg;tMNéneRKagEdlbgðajenAkñúgrUbTI 12-15a. eKmanTMr snøak;enARtg; E nig D nigmanTMrbgáb;enARtg; A . EI CacMnYnefr. dMeNaHRsay³ tamkarGegát snøak;enARtg; E nigkarBareRKagkMueGayeyal. eKGacKNnaemKuN kRmajrbs; CD nig CE edayeRbI K = 3EI / L edaysarcugq¶ayCaTMrsnøak;. ehIy bnÞúk 60kN min )abbegáItm:Um:g;bgáb;cugeT edaysarvaGnuvtþenARtg;cMN B . dUcenH 4EI 4EI 3EI 3EI K AB = K BC = K CD = K CE = 5 6 5 4 DFAB = 0 4 EI / 5 DFBA = = 0.545 4 EI / 5 + 4 EI / 6 DFBC = 1 − 0.545 = 0.455 4 EI / L DFCB = = 0.330 4 EI / 6 + 3EI / 5 + 3EI / 4 3EI / 5 DFCD = = 0.298 4 EI / 6 + 3EI / 5 + 3EI / 4 Displacement method of analysis: Moment distribution T.Chhay -431
  18. 18. Department of Civil Engineering NPIC DFCE = 1 − 0.33 − 0.298 = 0.372 DFDC = 1 DFEC = 1 wL2 − 45(6 )2 (FEM )BC =− = = −135kN .m 12 12 2 45(6 )2 (FEM )CB = wL = = 135kN .m 12 12 Tinñn½yRtUv)anbgðajenAkñúgtaragénrUbTI 12-15b. enATIenH karEbgEckm:Um:g;RtUv)aneFVIbnþbnÞab;Et eTAelItMN B nig C . m:Um:g;cugeRkayRtUv)anbgðajenAkñúgCYredkcugeRkayeK. edayeRbITinñn½yTaMgenH eyIgGacsg;düaRkamm:Um:g;sRmab;eRKagEdlbgðajenAkñúgrUbTI 12- 15c. !@>%> karEbgEckm:Um:g;sMrab;eRKagEdleyal (Moment distribution for frames: Sidesway) BIkaxNÐ 11-5 eyIgeXIjehIyfaeRKagEdlmanlkçN³minsIuemRTI b¤eRKagEdlrgbnÞúkminsIu- emRTInwgeyal. ]TahrN_énkrNIEbbenHRtUv)anbgðajenAkñúgrUbTI 12-16a. enATIenH bnÞúkGnuvtþn_ P nwgbegáItm:Um:g;enARtg; B nig C minesμIKña dUcenHeRKagnwgmankMhUcRTg;RTay Δ eTAsþaM. edIm,IkMNt; kMhUcRTg;RTayenH nigm:Um:g;Bt;enARtg;tMNedayeRbIkarEbgEckm:Um:g; eyIgnwgeRbIeKalkarN_tRmYt pl. enAkñúgrUbTI 12-16b dMbUgeKKitfaeRKagRtUv)anTb;mineGayeyaledayGnuvtþtMNTMrnimitþenA karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -432
  19. 19. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa Rtg; C . eKGnuvtþkarEbgEckm:Um:g; ehIybnÞab;mkeKKNnakmøaMgTb; R edaysmIkarsþaTic. kmøaMg Tb;EdlmanTMhMesμIKña EtTisedApÞúyKñaRtUv)anGnuvtþeTAelIeRKag ¬rUbTI 12-16c¦ ehIym:Um:g;enAkñúg eRKagRtUv)anKNna. viFImYysRmab;edaHRsayCMhancugeRkayenHTamTarnUvkarsnμt;CadMbUgnUvtémøCa elxsRmab;m:Um:g;Bt;mYy ¬ M 'BA ¦. edayeRbIkarEbgEckm:Um:g; nigsmIkarsþaTic eyIgGackMNt;kMhUc RTg;RTay Δ' nigkmøaMgxageRkA R EdlRtUvnwgtémøsnμt; M 'BA . edaysarvaCakMhUcRTg;RTayeGLa- sÞiclIenEG‘r enaHkmøaMg R' begáItm:Um:g;enAkñúgeRKagEdlsmamaRteTAnwgm:Um:g;EdlbegáIteGayman R . Ca]TahrN_ RbsinebIeKsÁal; M 'BA nig R' enaHm:Um:g;enARtg; B EdlbegáIteday R Ca M BA = M 'BA (R / R') . karbEnßmm:Um:g;Rtg;tMNsMrab;krNITaMgBIr ¬rUbTI12-16b nig c¦ nwgeFVIeGayeKTTYl)an m:Um:g;Cak;EsþgenAkñúgeRKag ¬rUbTI 12-16a¦. karGnuvtþénbec©keTsenHRtUv)anbgðajenAkñúg]Ta- hrN_ 12-6 dl; 12-8. eRKageRcInCan;³ CaerOy² rcnasm<½n§EdlmaneRcInCan;nwgmanbMlas;TIÉkraCüeRcIn Cavi)akkarEbg Eckm:Um:g;edayeRbIbec©keTsxagelInwgTamTarnUvkarKNnaeRcIn. ]TahrN_ BicarNaeRKagBIrCan;Edl bgðajenAkñúgrUbTI 12-17a. rcnasm<½n§enHGacmanbMlas;TIÉkraCüBIr edaysarcm¶ayeyal Δ1 rbs;Can;TImYyÉkraCüBIbMlas;TI Δ 2 rbs;Can;TIBIr. CaGkusl bMlas;TITaMgenHminRtUv)andwgenAdMNak;kaldMbUg dUcenHkarviPaKRtUveFVIeLIgeday QrelIeKalkarN_tRmYtplEdlmanlkçN³dUcKñanwgkarENnaMxagelI. enAkñúgkrNIenH eKRtUvGnuvtþ kmøaMgTb;BIrKW R1 nig R2 ¬rUbTI 12-17b¦ ehIym:Um:g;bgáb;cugRtUv)anKNna nigEbgEck. edayeRbI smIkarlMnwg eKGackMNt;témøCaelxrbs; R1 nig R2 . bnÞab;mk eKRtUvRsaykarTb;enARtg;kRmal énCan;TImYy ehIykRmalenaHRtUv)andak;eGaymanbMlas;TI Δ' . bMlas;TIenHeFVIeGaymanm:Um:g;bgáb; cugenAkñμúgeRKagEdleKGackMNt;Catémøelx. edayEbgEckm:Um:g;enH nigedayeRbIsmIkarlMnwg eKGac Displacement method of analysis: Moment distribution T.Chhay -433
  20. 20. Department of Civil Engineering NPIC kMNt;témøCaelxén R'1 nig R'2 . tamrebobdUcKña kRmalénCan;TIBIrRtUv)aneGaymanbMlas;TI Δ' ' ¬rUbTI 12-17d¦. edaysnμt;témøCaelxsRmab;m:Um:g;bgáb;cug karEbgEckm:Um:g; nigsmIkarlMnwgnwg pþl;nUvtémø R"1 nig R"2 . edaysarCMhancugeRkayTaMgBIr ¬rUbTI 12-17c nig d¦ GaRs½ynwgtémø snμt;énm:Um:g;bgáb;cug eKRtUvGnuvtþemKuNEktMrUv (correction factor) C ' nig C" eTAelIm:Um:g;EdlEbg Eck. eyagtamkmøaMgTb;enAkñúgrUbTI 12-17c nig 12-17d eyIgRtUvkarGnuvtþkmøaMg R1 nig R2 Edl manTMhMesμIKñaEtTisedApÞúyKñaeTAelIeRKag Edl R2 = −C ' R ' 2 +C" R"2 R1 = +C ' R '1 −C" R"1 dMeNaHRsayénRbB½n§smIkarenHeGaynUvtémøén C ' nig C" . bnÞab;mkeKRtUvKuNemKuNTaMgenHCamYy nwgm:Um:g;Bt;Rtg;tMNEdlkMNt;BIkarEbgEckm:Um:g;enAkñúgrUbTI 12-17c nig 12-17d. bnÞab;mkeK kMNt;m:Um:g;ersIusþg;edayeFVIplbUkm:Um:g;EktRmUvCamYynwgm:Um:g;EdlTTYl)anenAkñúgrUbTI 12-17b. eKGacviPaKeRKagRbePTepSgeTotEdlmanbMlas;TIÉkraCüedayeRbIdMeNIkardUcKña b:uEnþeKRtUv dwgfaviFIxagelIEdlmanerobrab;RtUvkarkarKNnay:ageRcIn. EteTaHCay:agNak¾eday eK)anbegáIt bec©keTsxøHedIm,Ikat;bnßykarKNna ¬CakarRbesIreKKYredaHRsaycMeNaTRbePTenHedayeRbIkMuBüÚ- T½r¦ CaTUeTAedayeRbIkarviPaKedaym:aRTIs. bec©keTssRmab;karviPaKEbbenHRtUv)anbgðajenAkñúg emeronTI 16. ]TahrN_ 12-6³ kMNt;m:Um:g;Bt;enARtg;tMNéneRKagEdlbgðajenAkñúgrUbTI 12-18a. EI CacMnYn efr. dMeNaHRsay³ dMbUgeyIgKitfaeRKagRtUv)anTb;mineGayeyaldUcbgðajenAkñúgrUbTI 12-18b. eyIgman 16(4)2 (1) (FEM )BC =− = −10.24kN .m (5)2 karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -434
  21. 21. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa (FEM )CB = 16(1) 2(4) = 2.56kN .m 2 (5) emKuNkRmajrbs;ElVgnImYy²RtUv)anKNnaedayQrelI 4EI / L b¤edayeRbIemKuNkRmajeFob I / L . emKuNEbgEck nigkarEbgEckm:Um:g;RtUv)anbgðajenAkñúgtarageénrUbTI 12-18d. edayeRbI lT§plTaMgenH eKGnuvtþsmIkarlMnwgeTAelIdüaRkamGgÁesrIénssredIm,IkMNt; Ax nig Dx ¬rUbTI 12- 18e¦. BIdüaRkamGgÁesrIrbs;eRKagTaMgmUl ¬minRtUv)anbgðaj¦ kmøaMgTb; R enAkñúgrUbTI 12-18 manTMhM ∑ Fx = 0 R = 1.73kN − 0.81kN = 0.92kN eKRtUvGnuvtþkmøaMgEdlmanTMhMesμIKña EtTisedApÞúyKñanwg R = 0.92kN eTAelIeRKagRtg;tMN C ehIy KNnam:Um:g;Bt; ¬rUbTI 12-18c¦. edIm,IedaHRsaycMeNaTEdlKNnam:Um:g;TaMgenH ¬eyIgnwgsnμt; kmøaMg R' GnuvtþenARtg; C ¦ EdleFVIeGayeRKagmanbMlas;TI Δ' dUcbgðajenAkñúgrUbTI 12-18f. enATI enHtMNenARtg; B nig C RtUv)anTb;CabeNþaHGasnñBIkarvil ehIyCalT§plm:Um:g;bgáb;cugenARtg;cug rbs;ssrRtUv)anKNnaBIrUbmnþsMrab;bMlas;TIEdlmankñúgemeronTI 10 Edl 6 EIΔ M = L2 edaysar TaMg B nig C pøas;TI)an Δ' dUcKña ehIy AB nig DC man E / I nig L dUcKña m:Um:g;bgáb;cugenA kñúg AB ehIymanm:Um:g;bgáb;cugdUcKñaenAkñúg DC . dUcbgðajenAkñúgrUbTI 12-18f eyIgnwgsnμt;m:Um:g; bgáb;cugNamYyesμInwg (FEM ) AB = (FEM )BA = (FEM )CD = (FEM )DC = −100kN .m eKcaM)ac;eRbIsBaØaGviC¢man edaysarm:Um:g;RtUveFVIGMeBIRcasRTnicnaLikaeTAelIssr edayeFIVeGayssr manbMlas;TI Δ' eTAsþaM. LÚveKGackMNt;témø R' EdlrYmCamYynwgm:Um:g; − 100kN.m . karEbgEcg m:Um:g;rbs;m:Um:g;bgáb;cugRtUv)anbgðajenAkñúgrUbTI 12-18g. BIsmIkarlMnwg eKGacKNnakmøaMg RbtikmμtamTisedkenARtg; A nig D ¬rUbTI 12-18h¦. dUcenH sMrab;;eRKagTaMgmUl eyIgtMrUveGay ∑ Fx = 0 R ' = 28 + 28 = 56.0kN dUcenH R'= 56.0kN begáItm:Um:g;EdlmanenAkñúgtaragénrUbTI 12-18g. eKGackMNt;m:Um:g;EdlbegáIt eLIgeday R = 0.92kN smamaRt. dUcenH m:Um:g;ers‘ultg;enAkñúgeRKag ¬rUbTI 12-18a¦ esμInwgpl bUkénm:Um:g;Edl)anKNnasRmab;eRKagenAkñúgrUbTI 12-18b CamYynwgbrimaNsmamaRténm:Um:g; sRmab;eRKagenAkñúgrUbTI 12-18c. eyIgman M AB = 2.88 + 0.92 (− 80) = 1.57kN .m 56.0 Displacement method of analysis: Moment distribution T.Chhay -435
  22. 22. Department of Civil Engineering NPIC M BA = 5.78 + 0.92 (− 60) = 4.79kN .m 56.0 M BC = −5.78 + 0.92 (60) = −4.79kN .m 56.0 M CB = 2.72 + 0.92 (60) = 3.71kN .m 56.0 M CD = −2.72 + 0.92 (− 60) = −3.71kN .m 56.0 M DC = −1.32 + 0.92 (− 80) = −2.63kN .m 56.0 ]TahrN_ 12-7³ kMNt;m:Um:g;Bt;enARtg;tMNnImYy²rbs;eRKagEdlbgðajenAkñúgrUbTI 12-19a. m:Um:g;niclPaBénGgát;nImYy²RtUv)anbgðajenAkñúgrUb. karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -436
  23. 23. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ dMbUgeRKagRtUv)anTb;mineGayeyaldUcbgðajenAkñúgrUbTI 12-19b. m:Um:g;niclPaB RtUv)anKNnaenARtg;tMNdUcbgðajenAkñúgrUbTI 12-19d. enATIenH eKKNnaemKuNkRmajrbs; CD edayeRbI 3EI / L edaysarvamanTMrsnøak;enARtg; D . karKNnakmøaMgRbtikmμtamTisedkenARtg; A nig D RtUv)anbgðajenAkñúgrUbTI 12-19e. dUcenH sMrab;eRKagTaMgmUl ∑ Fx = 0 R = 5.784 − 2 = 3.78kN eKGnuvtþkmøaMgEdlmanTispÞúyKñaeTAelIeRKagdUcbgðajenAkñúgrUbTI 12-19c. dUcenAkñúg]TahrN_ elIkmun eyIgnwgBicarNakmøaMg R' EdlmanGMeBIdUcbgðajenAkñúgrUbTI 12-19f. CalT§pl tMN B nig C pøas;TI)an Δ ' dUcKña. eyIgKNnam:Um:g;bgáb;cugsRmab; BA BI (FEM ) = (FEM ) = − 6 EIΔ = − 6 E (1000)(10 )Δ' 6 AB BA L2 (5)2 b:uEnþ BItaragenAkñúgemeronTI10 sRmab; CD eyIgman (FEM ) = − 3EIΔ = − 3E (1250)(10 )Δ' 6 CD L2 (5)2 Displacement method of analysis: Moment distribution T.Chhay -437
  24. 24. Department of Civil Engineering NPIC edaysnμt;fam:Um:g;bgáb;cugsRmab; AB KW − 100kN.m dUcbgðajenAkñúgrUbTI 12-19f enaHeKkMNt;m:Um:g; bgáb;cugenARtg; C EdleFVIeGaymanbMlas;TI Δ' dUcKña edaykareRbobeFob eBalKW Δ' = − (− 100)(5)2 = − (FEM )CD (7.5)2 6 E (1000)( 6 ) 10 3E (1250)( 6 ) 10 (FEM )CD = −27.78kN .m karEbgEckm:Um:g;sRmab;m:Um:g;bgáb;cugTaMgenHRtUv)anbgðajkñúgtaragénrUbTI 12-19g. karKNna kmøaMgRbtikmμenARtg; A nig D RtUv)anbgðajenAkñúgrUbTI 12-19h. dUcenH sRmab;eRKagTaMgmUl ∑ Fx = 0 R ' = 21.98 + 3.11 = 25.1kN dUcenH m:Um:g;ers‘ultg;enAkñúgeRKagKW ⎛ 3.78 ⎞ M AB = 9.58 + ⎜ ⎟(− 69.91) = −0.948kN .m ⎝ 25.1 ⎠ ⎛ 3.78 ⎞ M BA = 19.34 + ⎜ ⎟(− 40.01) = −13.3kN .m ⎝ 25.1 ⎠ ⎛ 3.78 ⎞ M BC = −19.34 + ⎜ ⎟(40.01) = 13.3kN .m ⎝ 25.1 ⎠ ⎛ 3.78 ⎞ M CB = 15.00 + ⎜ ⎟(23.31) = 18.5kN .m ⎝ 25.1 ⎠ ⎛ 3.78 ⎞ M CD = −15.00 + ⎜ ⎟(− 23.31) = −18.5kN .m ⎝ 25.1 ⎠ ]TahrN_ 12-8³ kMNt;m:Um:g;Bt;enARtg;tMNnImYy²rbs;eRKagEdlbgðajenAkñúgrUbTI 12-20a. EI CacMnYnefr. dMeNaHRsay³ dMbUg kareyalRtUv)ankarBaredaykmøaMgTb; R ¬rUbTI 12-20b¦. m:Um:g;bgáb;cug sRmab;Ggát; BC KW (FEM )BC = − 16(5) = −10kN .m (FEM )CB = 16(5) = 10kN .m 8 8 edaysarGgát; AB nig DC manTMrsnøak; emKuNkRmajRtUv)anKNnaedayeRbI 3EI / L . karEbgEck m:Um:g;RtUv)anbgðajenAkñúgrUbTI 12-20d. edaylT§plTaMgenH eKkMNt;kmøaMgRbtikmμtamTisedkenARtg; A nig D edayeRbIsmIkar lMnwgénGgát;nImYy² ¬rUbTI 12-20e¦. edayeFVIplbUkm:Um:g;Rtg;cMNuc B nig C enAelIeCIgnImYy² eyIg)an³ + ∑MB = 0 − 5.97 + Ax (4) − 8(3) = 0 Ax = 7.49kN + ∑ MC = 0 5.97 − D x (4) + 8(3) = 0 D x = 7.49kN karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -438
  25. 25. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dUcenH sRmab;eRKagTaMgmUl ∑ Fx = 0 R = 7.49 − 7.49 + 40 = 40kN LÚveyIgGnuvtþkmøaMgpÞúynwg R eTAelIeRKagdUcbgðajenAkñúgrUbTI 12-20c. edIm,IKNnam:Um:g; Bt;EdlbegáIteLIgeday R CadMbUgeyIgRtUvBicarNakmøaMg R EdlmanGMeBIdUcbgðajenAkñúgrUbTI 12- 20f. enATIenHbnÞat;dac;²minEmnCarUbragxUcRTg;RTayrbs;Ggát;eRKageT vaRtUv)ansg;CabnÞat;Rtg; EdlbnøayeTATItaMgcugeRkayrbs; B' nig C ' rbs;tMN B nig C erogKña. edaysarPaBsIuemRTIrbs; eRKag bMlas;TI BB' = CC ' = Δ' . elIsBIenH bMlas;TITaMgenHeFVIeGay BC vil. cm¶aytamTisQr Displacement method of analysis: Moment distribution T.Chhay -439
  26. 26. Department of Civil Engineering NPIC rvag B' nig C ' KW1.2Δ' dUcbgðajenAelIdüaRkambMlas;TI ¬rUbTI 12-20g¦. edaysartMNcugrbs; Ggát;nImYy²rgbMlas;TIEdleFVIeGayGgát;vil m:Um:g;bgáb;cugRtUvekItmanenAkñúgGgát;. eKman (FEM )BA = (FEM )CD = −3EIΔ' / (5)2 / (FEM )BC = (FEM )CB = 6 EI (1.2Δ') / (5)2 . cMNaMfa sRmab; BA nig CD m:Um:g;mantémøGviC¢man edaysarkarvilRsbTisRTnicnaLikarbs; Ggát;eFVIeGaym:Um:g;bgáb;cugvilRcasRTnicnaLika. RbsinebI eyIgsnμt;témørbs; (FEM )BA = (FEM )CD = −100kN .m ehIydak;eGayesμI Δ' enA kñúgrUbmnþxagelIeyIgnwgTTYl)an (FEM )BC = (FEM )CB = 240kN .m . eKdak;m:Um:g;TaMgenHeGayman GMeBIelIeRKag nigeFVIkarEbgEckva ¬rUbTI 12-20h¦. edayeRbIlT§plTaMgenH smIkarlMnwgRtUv)an bgðajenAkñúgrUbTI 12-20i. sRmab;eCIgnImYy² eyIgman + ∑MB = 0 − A' x (4) + 58.72(3) + 146.80 = 0 A' x = 80.74kN + ∑ MC = 0 − D' x (4) + 58.72(3) + 146.80 = 0 D' x = 80.74kN dUcenH sRmab;eRKagTaMgmUl ∑ Fx = 0 R ' = 80.74 + 80.74 = 161.48kN dUcenH m:Um:g;ers‘ultg;enAkñúgeRKagKW ⎛ 40 ⎞ M BA = 5.97 + ⎜ ⎟(− 146.80) = −30.4kN .m ⎝ 161.48 ⎠ ⎛ 40 ⎞ M BC = −5.97 + ⎜ ⎟(146.80) = 30.4kN .m ⎝ 161.48 ⎠ ⎛ 40 ⎞ M CB = 5.97 + ⎜ ⎟(146.80) = 42.3kN .m ⎝ 161.48 ⎠ ⎛ 40 ⎞ M CD = −5.97 + ⎜ ⎟(− 146.80 ) = −42.3kN .m ⎝ 161.48 ⎠ karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -440
  27. 27. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa cMeNaT !@>! kMNt;m:Um:g;enARtg; A / B nig C bnÞab;mk sg;düaRkamm:Um:g;sRmab;Fñwm. m:Um:g;niclPaB rbs;Ggát;nImYy²RtUv)anbgðajenAkñúgrUb. snμt; faTMrRtg; B CaTMrkl; ehIyenARtg; A nig C CaTMr bgáb;. E = 200GPa . !@>% kMNt;m:Um:g;enARtg; A / B nig C bnÞab;mk sg;düaRkamm:Um:g;sRmab;Fñwm. snμt; faTMrRtg; A nig C CaTMrbgáb; ehIyenARtg; B CaTMrkl;. EI CacMnYnefr. !@>@ kMNt;m:Um:g;enARtg; A / B nig C bnÞab;mk sg;düaRkamm:Um:g;sRmab;Fñwm. snμt; faTMr A nig C CaTMrbgáb;. EI CacMnYnefr. !@>^ kMNt;m:Um:g;enARtg; A / B nig C bnÞab;mk sg;düaRkamm:Um:g;sRmab;Fñwm. snμt; faTMrRtg; A nig C CaTMrkl; ehIyenARtg; B CaTMrsnøak;. bnÞúkBRgaysßitenAelIkRmalEdlTMrsamBaØEdl bBa¢ÚnbnÞúkeTAFñwmkRmal. EI CacMnYnefr. !@># kMNt;m:Um:g;Bt;enAkñúgFñwmRtg; B bnÞab;mk sg;düaRkamm:Um:g;. snμt; faTMr C CaTMrsnøak;. kMNat;Ggát; AB manm:Um:g;niclPaB I AB = 0.75I BC . EI CacMnYnefr. !@>& kMNt;m:Um:g;Bt;enAkñúgFñwmRtg; B bnÞab;mk sg;düaRkamm:Um:g;sRmab;Fñwm. snμt; faTMr A Ca TMrsnøak; ehIyTMr B CaTMrkl; nigTMr C CaTMrbgáb;. !@>$ kMNt;m:Um:g;enARtg; A / B nig C bnÞab;mk EI CacMnYnefr. sg;düaRkamm:Um:g;. snμt; faTMrRtg; B CaTMrkl; ehIyenARtg; A nig C CaTMrbgáb;. EI CacMnYnefr. Problems T.Chhay -441
  28. 28. Department of Civil Engineering NPIC !@>* kMNt;m:Um:g;Bt;enAkñúgFñwmRtg; A nig B bnÞab; mksg;düaRkamm:Um:g;. snμt; faTMr A Ca TMrbgáb; ehIyTMr B CaTMrkl; nigTMr C CaTMrsnøak;. !@>!@ kMNt;m:Um:g;EdlmanGMeBIenARtg;cugrbs; Ggát;nImYy² bnÞab;mksg;düaRkamm:Um:g;. snμt; !@>( r)arRtUv)antP¢ab;edaytMNsnøak;Edl faTMr A nig D CaTMrsnøak; ehIyTMr B CatMN bgðajedaycMNuc. RbsinebIeKecalkmøaMgEkg bgáb; nigTMr C CaTMrbgáb;. E = 200GPa / I ABC enAkñúgr)ar cUrkMNt;kmøaMgRbtikmμtamTisQrenA = 280(106 )mm 4 ehIy I BD = 440(106 )mm 4 . Rtg;snøak;nImYy². EI CacMnYnefr. !@>!0 kMNt;m:Um:g;Bt;enAkñúgFñwmRtg;TMr bnÞab; mksg;düaRkamm:Um:g;. snμt;fa A nig D Ca !@>!# kMNt;m:Um:g;EdlmanGMeBIenARtg;tMN TMrbgáb;. EI CacMnYnefr. nImYy². snμt; A / D nig E CaTMrsnøak; ehIy B nig C CatMNbgáb;. m:Um:g;niclPaBrbs;Ggát; nImYy²RtUv)anbgðajenAkñúgrUb. E = 200GPa . !@>!! kMNt;m:Um:g;Bt;enARtg; B bnÞab;mksg; düaRkamm:Um:g;sRmab;Ggát;nImYy²rbs;eRKag. snμt; faTMr A CaTMrsnøak; ehIyTMr C CaTMrbgáb;. EI CacMnYnefr. karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -442
  29. 29. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa !@>!$ kMNt;m:Um:g;Bt;enARtg; A / C nig D bnÞab; !@>!& kMNt;m:Um:g;Bt;EdlmanGMeBIenARtg;cug mksg;düaRkamm:Um:g;sMrab;Ggát;nImYy²rbs; rbs;Ggát;nImYy². snμt;faTMrenARtg; A nig D eRKag. TMr A nigtMN C nig D CatMNbgáb;. CaTMrbgáb;. m:Um:g;niclPaBrbs;Ggát;nImYy²RtUv EI CacMnYnefr. )anbgðajenAkñúgrUbj. EI = 200GPa . !@>!% kMNt;m:Um:g;Bt;enARtg; B bnÞab;mksg; !@>!* kMNt;m:Um:g;enARtg; B nig C bnÞab;mksg; düaRkamm:Um:g;sMrab;Ggát;nImYy²rbs;eRKag. TMr düaRkamm:Um:g;. snμt; A nig D CaTMrsnøak; A CaTMrsnøak;. EI CacMnYn efr. ehIy B nig C CatMNbgáb;. EI CacMnYnefr. !@>!( kMNt;m:Um:g;EdlmanGMeBIenARtg;cugrbs;G !@>!^ kMNt;m:Um:g;Bt;EdlmanGMeBIenARtg;cug gát;nImYy². snμt;tMNCatMNbgáb; ehIy A nig rbs;Ggát;nImYy²rbs;eRKag. EI CacMnYnefr. D CaTMrbgáb;. EI CacMnYnefr. Problems T.Chhay -443
  30. 30. Department of Civil Engineering NPIC !@>@0 kMNt;m:Um:g;EdlmanGMeBIenARtg;TMrbgáb; !@>@! kMNt;bgÁúMkmøaMgRbtikmμtamTisedk nig A nig D rbs;eRKag. EI CacMnYnefr. tamTisQrenARtg;TMrsnøak; A nig D . EI CacMnYnefr. karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -444

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