4.internal loading developed in structural members

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4.internal loading developed in structural members

  1. 1. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa $> bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁúM Internal loading developed in structural members munnwgeyIgGacKitsmamaRtmuxkat;rbs;Ggát;eRKOgbgÁúM eyIgcaM)ac;RtUvkMNt;kMlaMg nigm:Um:g; manGMeBIenAxagkñúgva. enAkñúgCMBUkenH eyIgnwgbegáItviFIsMrab;rkbnÞúkTaMgenHenARtg;TItaMgCak;lak;tam beNþayG½kSrbs;Ggát; nigsMrab;KUrdüaRkamkMlaMgkat; nigdüaRkamm:Um:g;. karGnuvtþRtUv)aneFVIeTAelI Fñwm nigeRKag. $>!> bnÞúkxagkñúgenARtg;cMnucCak;lak; (Internal loading at a specified point) dUcerobrab;enAkñúgkfaxNÐ 2-3 eKGackMNt;bnÞúkxagkñúgedayeRbIviFImuxkat;. CaTUeTA sMrab; eRKOgbgÁúMkñúgbøg; bnÞúkTaMgenHrYmmankMlaMgEkg (normal force) N / kMlaMgkat; (shear force) V nig m:Um:g;Bt; (bending moment) M . b:uEnþeKRtUvdwgfa Cak;EsþgbnÞúkTaMgenHtageGayers‘ultg;énkug * RtaMgBRgayEdlmanGMeBIelIépÞmuxkat;rbs;Ggát;enARtg;muxkat;Edlkat;. enAeBlEdleKdwgers‘ultg; bnÞúkxagkñúg enaHeKGackMNt;TMhMkugRtaMgRbsinebIeKsÁal;épÞmuxkat;EdlkugRtaMgRtUvBRgayelI. karkMNt;sBaØa (Sign Convention): munnwgbgðajBIviFIsMrab;rkkMlaMgEkg kMlaMgkat; nigm:Um:g; Bt;eyIgcaM)ac;RtUvkMNt;sBaØaeGayva . eTaHbICakareRCIserIsmanlkçN³buKÁlk¾eday EtkarkMNt; † * rcnasm<½n§bITMhMk¾Gacnwgrgm:Um:g;rmYl EdlmYlGgát;CMuvijG½kSbeNþayrbs;va. † karkMNt;sBaØaenHnwgpþl;PaBgayRsYlsMrab;kfaxNÐ 4-2 nig 4-3 EdleyIgnwgbgðaj V nig M CaGnuKmn_eTAnwg x rYceyIg nwgKUrdüaRkaménGnuKmn_enH. karkMNt;sBaØamanlkçN³RsedognwgkarkMNt;eGayTisedA x mansBaØabUkenAeBlvamanTis edAeTAsþaM nigTisedAeLIgelIsMrab; y enAeBlEdleyIgKUrdüaRkaménGnuKmn_ y=f(x). Internal loading developed in structural members T.Chhay -109
  2. 2. Department of Civil Engineering NPIC sBaØaenATIenHRtUv)anvisVkreRKOgbgÁúMTTYlsÁal;CaTUeTA ehIyvaRtUv)anbgðajenAkñúgrUbTI 4-1a. enA elIépÞxageqVgédrbs;Ggát;Edlkat;kMlaMgEkg N manGMeBIeTAsþaM kMlaMgkat; V manGMeBIcuHeRkam ehIy m:Um:g;Bt;eFVIGMeBIbRBa©as;RTnicnaLika. eyagtamc,ab;TIbIrbs;jÚtun muxkat;xagsþaMédrbs;Ggát;nwgrg kMlaMgEkg kMlaMgkat; nigm:Um:g;Bt;EdlmanTMhMb:unKña EtTisedApÞúyKña. viFIEdlgayRsYlkñúgkarcgcaM karkMNt;sBaØaTaMgenHKWpþac;Ggát;CakMNt;d¾tUc ehIycMNaMfakMlaMgEkgviC¢manKWTajkMNat;Ggát; ¬rUbTI 4-1b¦ kMlaMgkat;viC¢manKWeFVIeGayGgát;vilRsbTisRTnicnaLika ¬rUbTI 4-1c¦ ehIym:Um:g;Bt;viC¢man KWBt;kMNat;Ggát;ekageLIgelIdUcfñk;Twk ¬rUbTI 4-1d¦. dMeNIrkarkñúgkarviPaK (Procedure for analysis) CMhanxageRkamenHpþl;nUvmeFüa)aysMrab;karGnuvtþviFImuxkat;edIm,IkMNt;kMlaMgEkg kMlaMg kat; nigm:Um:g;Bt;enARtg;TItaMgCak;lak;enAkñúgGgát;eRKOgbgÁúM. kMlaMgRbtikmμTMr (Support reactions) muxnwgkat;Ggát; eKcaM)ac;RtUvkMNt;RbtikmμTMrrbs;Ggát;edIm,IeGayeKGaceRbIEtsmIkarlMnwg edIm,IedaHRsayrkMlaMgkñúgenAeBlEdlGgát;RtUv)ankat;. RbsinebIGgát;mantMNsnøak; eKGackMNt;klaMgRbtikmμsnøak;edayeRbIviFIenAkñúgkfaxNÐ 2-5. düaRkamGgÁesrI (Free-body diagram) rkSabnÞúkBRgay m:Um:g; nigkMlaMgTaMgGs;EdlmanGMeBIenAelIGgát;eGayenATItaMgdEdl bnÞab; mkkat;Ggát;eGayEkgnwgG½kSrbs;Ggát;enARtg;cMnucEdleKRtUvkarkMNt;bnÞúkxagkñúg. eRkayBIkat;Ggát;ehIy KUrdüaRkamGgÁesrIrbs;kMNat;Ggát;EdlmancMnYnbnÞúkticCageK. enA Rtg;muxkat;bgðajers‘ultg;EdlCaGBaØat N / V nig M EdlmanGMeBItamTisedAviC¢man ¬rUbTI 4-1a¦. smIkarlMnwg (Equation of Equilibrium) eKKYreFVIplbUkm:Um:g;enARtg;mxkat;CMuvijG½kSEdlkat;tamTIRbCMuTMgn;rbs;épÞmuxkat;rbs;Ggát; u edIm,IlubbM)at;GBaØat N nig V ehIyeKnwgTTYl)andMeNaHRsaysMrab; N edaypÞal;. RbsinebIdMeNaHRsayénsmIkarlMnwgEdlTTYl)anCaTMhMGviC¢man enaHTisedArbs;kMlaMgKW pÞúyeTAnwgGIVEdl)anbgðajenAkñúgdüaRkamGgÁesrI. bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁMú T.Chhay -110
  3. 3. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ]TahrN_ 4-1³ kMNt;kMlaMgkat; nigm:Um:g;Bt;EdlmanGMeBIenAkñúgFñwmkugs‘ul (cantilever beam) EdlbgðajenAkñúgrUbTI 4-2a enARtg;muxkat;Edlkat;Rtg;cMnuc C nig D . dMeNaHRsay³ düaRkamGgÁesrI³ RbsinebIeyIgBicarNadüaRkamGgÁesrIrbs;kMNat;Ggát;EdlenAxagsþaMédrbs;Ggát; enaHeKmincaM)ac;KNnakMlaMgRbtikmμTMrenARtg;cMnuc A eT. düaRkamsMrab;kMNat;Ggát; CB nig DB RtUv )anbgðajenAkñúgrUbTI4-2b nig4-2c. cMNaMfabnÞúkxagkñúgsnμt;eFVIGMeBItamTisviC¢man. smIkarlMnwg³ kMNat; CB / rUbTI 4-2b/ + ↑ ∑ Fy = 0 VC − 5 − 5 − 5 = 0 VC = 15kN + ∑ MC = 0 − M C − 5(1) − 5(2) − 5(3) − 20 = 0 M C = −50kN .m kMNat; DB / rUbTI 4-2c/ + ↑ ∑ Fy = 0 VD − 5 − 5 − 5 = 0 VD = 20kN + ∑MD = 0 − M D − 5(1) − 5(2 ) − 5(3) − 20 = 0 M D = −50kN .m ]TahrN_enHbgðajvakMlaMgkat;mantMélxusKñaenARtg;épÞmuxkat; TaMgsgxagEdlrgkMlaMgcMcMnuc xN³Edlm:Um:g;Bt;mantMéldUcKña. dUcenHeKRtUvkMNt;kMlaMgkat;cMnYnBIrenARtg;muxkat;rgkMlaMgcMcMnuc. Internal loading developed in structural members T.Chhay -111
  4. 4. Department of Civil Engineering NPIC ]TahrN_ 4-2³ kMNt;kMlaMgkat; nigm:Um:g;Bt;EdlmanGMeBIenARtg;muxkat;Edlkat;Rtg;cMnuc C enA kñúgFñwmEdlbgðajenAkñúgrUbTI 4-3a. dMeNaHRsay³ RbtikmμTMr³ edayCMnYsbnÞúkBRgayedaykMlaMgpÁÜbrbs;va nigedayKNnaRbtikmμTMr enaHeyIgTTYl lT§pldUcbgðajenAkñúgrUbTI 4-3b. düaRkamGgÁesrI³ eyIgnwgBicarNakMNat;GgÁt; AC edaysareyIgGacTTYl)an dMeNaHRsayy:aggayRsYl ¬rUbTI 4-3c¦. eyIgGacKNna GaMgtg;sIuetbnÞúkBRgayenARtg;cMnuc C edaysmamaRtdUcxag eRkam wC = (2m / 6m )(45kN / m ) = 15kN / m smIkarlMnwg³ + ↑ ∑ Fy = 0 45 − 15 − VC = 0 VC = 30kN + ∑MC = 0 − 45(2 ) + 15(0.667 ) + M C = 0 M C = 80kN .m ]TahrN_enHbgðajBIsar³sMxan;kñúgkarrkSabnÞúkBRgayeGayenAelIFñwmrhUtdl;eRkayeBlEdlFñwm RtUv)ankat;. RbsinebIFñwmenARtg;rUbTI 4-3b RtUv)ankat;enARtg;cMnuc C enaHT§iBlrbs;bnÞúkBRgay enAelIkMNat;Ggát; AC nwgminRtUv)anKit ehIylT§plVC = 45kN nig M C = 90kN .m EdlTTYl)an nwgxus. ]TahrN_ 4-3³ kMlaMg 45kN enAkñúgrUbTI 4-4a RtVv)anRTedaykMral DE EdleKRkalvaenAelIrnUt edayTMrsamBaØ. rnUtenHepÞrbnÞúkrbs;vaeTAFñwmTMrsamBaØ AB . kMNt;kMlaMgkat; nigm:Um:g;EdleFVIGMeBI enARtg;cMnuc C enAkñúgFñwm. bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁMú T.Chhay -112
  5. 5. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ RbtikmμTMr³ lMnwgrbs;kMral rnUt nwgFñwmRtUv)anbgðajenAkñúgrUbTI 4-4b. eyIgKYrEtRtYtBinitülT§pl TaMgenH. düaRkamGgÁesrI³ eKeRbIdüaRkamGgÁesrIrbs;kMNat;Ggát; AC rbs;Fñwm edaysareKGacTTYl)an dMeNaHRsayy:agsamBaØbMput ¬rUbTI 4-4c¦. smIkarlMnwg³ + ↑ ∑ Fy = 0 18.75 − 30 − Vc = 0 VC = 11.25kN + ∑MC = 0 − 18.75(3.75) + 30(0.75) + M C = 0 M C = 47.8kN .m $>@> GnuKmn_kMlaMgkat; nigGnuKmn_m:Um:g; (Shear and Moment Functions) karsikSaKNnaFñwmTamTarcMeNHdwglMGitBIbMErbMrYlkMlaMgkat;V nigbMErbMrYlm:Um:g; M Edl manGMeBIenARtg;cMnucnImYy²tambeNþayG½kSrbs;Fñwm. CaTUeTA kMlaMgEkgminRtUv)anBicarNaeday mUlehtuBIr³ ¬1¦ enAkñúgkrNICaeRcIn bnÞúkEdlmanGMeBIeTAelIFñwmeRcInEtEkgeTAnwgG½kSrbs;Fñwm dUc enHehIyvanwgbegáIt)anEtkMlaMgkat; nigm:Um:g;Bt;b:ueNÑaH nig¬2¦ sMrab;eKalbMNgénkarsikSaKNna ersIusþg;rbs;FñwmTb;Tl;nwgkMlaMgkat; nigCaBiessTb;Tl;nwgkMlaMgkat;mansar³sMxan;CaglT§PaBrbs; FñwmkñúgkarTb;Tl;nwgkMlaMgEkg. b:uEnþkMlaMgEkgRtUv)anBicarNa enAeBlFñwmrgkMlaMg sgát;tamG½kS edaysareKRtUveFVIkarGegátelI buckling b¤GsßirPaBEdlGacekItman. eKGacTTYl)anbMErbMrYlrbs; V nig M CaGnuKmn_eTAnwgTItaMg x rbs;cMnucNamYytambeNþay FñwmedayeRbIviFImuxkat;Edl)anerobrab;enAkñúgkfaxNÐ 4-1. b:uEnþenATIenH eKcaM)ac;kat;FñwmenARtg;TI Internal loading developed in structural members T.Chhay -113
  6. 6. Department of Civil Engineering NPIC taMg x NamYyBIcugmçagrbs;FñwmRbesIrCakat;enARtg;Cak;ElkNamYy. CaTUeTA GnuKmn_kMlaMgkat; nigGnuKmn_m:Um:g;CaGnuKmn_dac; b¤k¾CMral (slope) rbs;vaCaCMral dac;enARtg;cMnucNaEdlRbePT b¤TMhMrbs;bnÞúkBRgaypøas;bþÚr b¤enARtg;kEnøgNaEdlmankMlaMgcMcMnuc b¤m:Um:g; (couple moment) eFVIGMeBI. edaysarmUlehtuenH eKRtUvkMNt;GnuKmn_kMlaMgkat; nigGnuKmn_ m:Um:g;eTAtamtMbn;rbs;FñwmEdlmanTItaMgenAcenøaHPaBdac;énbnÞúkBIr. ]TahrN_ eKeRbIkUGredaen x1 / x2 nig x3 edIm,IBN’nabMErbMrYlrbs; V nig M elIRbEvgrbs;FñwmenAkñúgrUbTI 4-5a. kUGredaenHTaMgenH mann½yEtenAkñúgtMbn;BI A eTA B sMrab; x1 / BI B eTA C sMrab; x2 nigBI C eTA D sMrab; x3 . eTaHbICakU- Gredaenrbs;vaenATIenHmaneKaldUcKñak¾eday k¾vaKμanbBaðaEdr. b:uEnþvaCakargayRsYlCagkñúgkar begáItGnuKmn_kMlaMgkat; nigGnuKmn_m:Um:g;edayeRbIkUGredaen x1 / x2 nig x3 EdlmaneKalenARtg;cMnuc A / B nig D dUcbgðajenAkñúgrUbTI 4-5b. enATIenH x1 nig x2 viC¢maneTAsþaM ehIy x3 viC¢maneTAeqVg. dMeNIrkarkñúgkarviPaK (Procedure for analysis) CMhanxageRkampþl;nUvviFIsMrab;kMNt;bMErbMrYlkMlaMgkat; nigm:Um:g;enAkñúgFñwmCaGnuKmn_eTAnwg TItaMg x . RbtikmμTMr (Support reactions) kMNt;kMlaMgRbtikmμTMrenAelIFñwm nigbMEbkRKb;kMlaMgxageRkATaMgGs;eGayeTACabgÁúMkMlaMg EdleFVIGMeBIEkg nigRsbeTAnwgG½kSrbs;Fñwm. GnuKmn_kMlaMgkat; nigGnuKmn_m:Um:g; (Shear and Moment Functions) kMNt;kUGredaen x EdlmaneKaldac;edayELk nigkUGredaen x EdlmaneKalrYm Edllat sn§wgeTAkñúgtMbn;rbs;FñwmEdlenAcenøaHkMlaMgcMcMnuc nig/b¤cenøaHm:Um:g; b¤EdlmanPaBminCab; énbnÞúBRgay. kat;FñwmEkgeTAnwgG½kSrbs;vaenARtg;cMgay x nImYy² nigBIdüaRkamGgÁesrIénkMNat;Ggát;mYy kMNt;GBaØatV nig M enARtg;muxkat;CaGnuKmn_én x . enAelIdüaRkamGgÁesrI eyIgeXIjfa V nig M kñúgTisedAviC¢manrbs;vaEdleKarBtamkarkMNt;sBaØaEdleGayenAkñúgrUbTI 4-1. bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁMú T.Chhay -114
  7. 7. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa eKTTYl)an V BI ∑ Fy = 0 ehIyeKTTYl)an M edayeFVIplbUkm:Um:g;Rtg;cMnuc S EdlmanTItaMg enARtg;muxkat;/ ∑ M S = 0 . eKGacepÞógpÞat;lT§pledaycMNaMfa dM / dx = V nig dV / dx = w . TMnak;TMngenHnwgbk RsayenAkñúgkfaxNÐ 4-3. ]TahrN_ 4-4³ kMNt;kMlaMgkat; nigm:Um:g;enAkñúgFñwmEdlbgðajenAkñúgrUbTI 4-6a CaGnuKmn_én x . dMeNaHRsay³ RbtikmμTMr³ sMrab;eKalbMNgénkarKNnaRbtikmμTMr eKRtUvCMnYsbnÞúkBRgayedaykMlaMgpÁÜb135kN ¬rUbTI 4-6b¦. b:uEnþeKcaM)ac;RtUvcaMfabnÞúkpÁÚbminEmnCabnÞúkCak;EsþgenAelIFñwmeT. GnuKmn_kMlaMgkat; nigGnuKmn_m:Um:g;³ düaRkamGgÁesrIénkMNat;FñwmRbEvg x RtUv)anbgðajenAkñúgrUbTI 4-6c. cMNaMfa eKkMNt;GaMgtg;sIuetrbs;bnÞúkRtIekaNenARtg;muxkat;edaysmamaRt Edl w / x = 10 / 3 b¤ w = 10 x / 3 . CamYynwgkarsÁal;bnÞúkGaMgtg;sIuet eKGackMNt;kMlaMgpÁÜbénbnÞúkBRgayenA kñúgrebobFmμtaRtUv)anbgðajenAkñúgrUb. dUcenH 1 ⎛ 10 x ⎞ + ↑ ∑ Fy = 0 135 − ⎜ ⎟x − V = 0 V = 135 − 1.667 x 2 2⎝ 3 ⎠ ⎡ 1 ⎛ 10 x ⎞ ⎤ x + ∑MS = 0 810 − 135 x + ⎢ ⎜ ⎟ x⎥ − M = 0 M = 810 + 135 x − 0.556 x 3 ⎣2 ⎝ 3 ⎠ ⎦ 3 cMNaMfa dM / dx = V nig dV / dx = − 10 x 3 =wEdlsMrab;karRtYtBinitülT§pl. Internal loading developed in structural members T.Chhay -115
  8. 8. Department of Civil Engineering NPIC ]TahrN_ 4-5³ kMNt;kMlaMgkat; nigm:Um:g;enAkñúgFñwmEdlbgðajenAkñúgrUbTI 4-7a CaGnuKmn_én x . dMeNaHRsay³ kMlaMgRbtikmμTMr³ kMlaMgRbtiikmμenARtg;TMrbgáb;KW V = 480kN nig M = 2274kN .m ¬rUbTI 4-7b¦. GnuKmn_kMlaMgkat; nigGnuKmn_m:Um:g;³ edaysarvamanPaBminCab;énbnÞúkBRgayenARtg; x = 3.6m eK RtUvBicarNatMbn;BIrén x edIm,IBiBN’naGnuKmn_kMlaMgkat; nigGnuKmn_m:Um:g;sMrab;FñwmTaMgmUl. enATIenH x1 sMrab;RbEvg 3.6m xageqVg nigeKGaceRbI x2 sMrab;kMNat;Ggát;EdlenAsl;. sMrab; 0 ≤ x1 ≤ 3.6m ¬cMNaMfaV nig M EdleXIjenAkñúgrUbTI 4-7c manTisedAviC¢man¦ + ↑ ∑ Fy = 0 480 − 50 x1 − V = 0 V = 480 − 50 x1 ⎛x ⎞ + ∑MS = 0 2274 − 480 x1 + 50 x1 ⎜ 1 ⎟ + M = 0 M = −2274 + 480 x1 − 25 x1 2 ⎝2⎠ sMrab; 3.6m ≤ x2 ≤ 6m / rUbTI 4-7d + ↑ ∑ Fy = 0 480 − 180 − V = 0 V = 300 + ∑MS = 0 2274 − 480 x2 + 180(x2 − 1.8) + M = 0 M = 300 x2 − 1950 eKGacRtYtBinitülT§plenHedaycMNaMfaenAeBl x2 = 6m enaHV = 300kN nig M = −150kN .m . ehIycMNaMfa dM / dx = V nig dV / dx = w . ]TahrN_ 4-6³ kMNt;kMlaMgkat; nigm:Um:g;enAkñúgFñwmEdlbgðajenAkñúgrUbTI 4-8a CaGnuKmn_én x . dMeNaHRsay³ kMlaMgRbtikmμTMr³ edIm,IkMNt;RbtikmμTMr eKEbgEckbnÞúkBRgayCabnÞúkRtIekaN nigbnÞúkctuekaN ehIyeKCMnYsbnÞúkTaMgenHedaykMlaMgpÁÜbrbs;va. kMlaMgRbtikmμRtUv)anKNna nigbgðajenAelIdüa- bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁMú T.Chhay -116
  9. 9. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa RkamGgÁesrIrbs;Fñwm ¬rUbTI 4-8b¦. GnuKmn_kMlaMgkat; nigGnuKmn_m:Um:g;³ düaRkamGgÁesrIrbs;kMNat;Ggát;Edlkat;RtUv)anbgðajenAkñúg rUbTI 4-8c. dUceXIjkñúgrUbxagelI eyIgCMnYsbnÞúkctuekaNBñayedaybnÞúkBRgayctuekaN nigbnÞúk BRgayRtIekaN. cMNaMfa eKGackMNt;GaMgtg;sIuetbnÞúkBRgayRtIekaNenARtg;muxkat;edaysma- maRt. kMlaMgpÁÜbénbnÞúkBRgaynImYy² nigTItaMgrbs;vaRtUv)anbgðaj. edayGnuvtþsmIkarlMnwg eyIgman ⎡1 ⎛ x⎞ ⎤ + ↑ ∑ Fy = 0 75 − 10 x − ⎢ (20)⎜ ⎟ x ⎥ − V = 0 V = 75 − 10 x − 1.11x 2 ⎣2 ⎝9⎠ ⎦ ⎛ x ⎞ ⎡1 ⎛x⎞ ⎤ x + ∑MS = 0 − 75 x + (10 x )⎜ ⎟ + ⎢ (20)⎜ ⎟ x ⎥ + M = 0 M = 75 x − 5 x 2 − 0.370 x 3 ⎝ 2 ⎠ ⎣2 ⎝9⎠ ⎦3 $>#> düaRkamkMlaMgkat; nigdüaRkamm:Um:g;Bt;sMrab;Fñwm (Shear and Moment Diagram for beams) RbsinebIeKKUrbMErbMrYlrbs;V nig M CaGnuKmn_én x EdlTTYl)anenAkñúgkfaxNÐ 4-2 enaHeK eGayeQμaHvaCadüaRkamkMlaMgkat; nigdüaRkamGgÁesrI erogKña. enAeBlEdlFñwmrgkMlaMgbnÞúkcMcMnuc m:Um:g; nigbnÞúkBRgayeRcIn karsg;düaRkamV nig M eFobnwg x manlkçN³sμúKsμaj edaysareKRtUv karsg;düaRkamedayeRbIGnuKmn_eRcIn. enAkñúgkfaxNÐenH eyIgnwgerobrab;BIviFId¾samBaØ ¬viFIEdlQr elITMnak;TMngDIepr:g;EsülEdlekItmanrvagbnÞúk kMlaMgkat; nigm:Um:g;¦ edIm,Isg;düaRkamTaMgenH. Internal loading developed in structural members T.Chhay -117
  10. 10. Department of Civil Engineering NPIC edIm,IbegáItTMnak;TMngTaMgenH eKRtUvBicarNaFñwmenAkñúgrUbTI 4-9a EdlrgnUvbnÞúkBRgay W = w(x ) niges‘rIénbnÞúkcMcMnuc nigm:Um:g;. sMrab;karerobrab;xageRkamenH eKKitfabnÞúkBRgayviC¢manenA eBlbnÞúkeFVIGMeBIeLIgelIdUcbgðaj. eyIgnwgBicarNadüaRkamGgÁesrIsMrab;kMNat;Ggát;tUcrbs;FñwmEdl manRbEvg Δx ¬rUbTI 4-9b¦. edaysar eKeRCIserIskMNat;Ggát;enHenARtg;cMnuc x tambeNþayFñwm EdlminrgkMlaMgcMcMnuc b¤m:Um:g; eKnwgTTYl)anlT§plelIkMNat;EdlminrgkMlaMgcMcMnuc b¤m:Um:g;. eK snμt;eGaykMlaMgkat; nigm:Um:g;Bt;EdlbgðajenAelIdüaRkamGgÁesrImanGMeBItamTisviC¢manEdlGaRs½y eTAtamkarkMNt;sBaØa ¬rUbTI 4-1¦. cMNaMfa TaMgkMlaMgkat; nigm:Um:g;EdleFVIGMeBIenAelIépÞxagsþaMekIn eLIgedaytMéltictYc edIm,IrkSalMnwgrbs;kMNat;Ggát;enH. eKCMnYsbnÞúkBRgayedaykMlaMgcMcMnuc w(x )Δx EdleFVIGMeBIenAelIcMgay ε (Δx ) BIcugxagsþaM Edl 0 < ε < 1 ¬Ca]TahrN_ RbsinebI w(x ) Ca bnÞúkBRgay b¤bnÞúkefr enaH w(x )Δx nigekItmanenARtg; 12 Δx dUcenH ε = 12 ¦. edayGnuvtþsmIkar lMnwg eyIg)an + ↑ ∑ Fy = 0 V + w(x )Δx − (V + ΔV ) = 0 ΔV = w( x )Δx + ∑MS = 0 − VΔx − M − w( x )Δxε (Δx ) + (M + ΔM ) = 0 ΔM = VΔx + w( x )ε (Δx )2 edayEcksmIkarTaMgBIrxagelIeday Δx ehIyeGaylImItén Δx → 0 enaHsmIkarTaMgenHnwgkøayCa = w(x ) dV (4-1) dx CMralrbs;düaRkamkMlaMgkat; = GaMgtg;sIurbs;bnÞúkBRgay dM =V (4-2) dx CMralrbs;düaRkamm:Um:g;Bt; = GaMgtg;sIurbs;bnÞúkBRgay bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁMú T.Chhay -118
  11. 11. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa eKRtUveFVIkarkt;cMNaMfa smIkar 4-1 bgðajBIPaBesμIKñaénCMralrbs;düaRkamkMlaMgkat; (dV / dx ) nig GaMgtg;sIuetrbs;bnÞúkBRgay w(x ) . dUcKña smIkar 4-2 bgðajfaCMralrbs;düaRkamm:Um:g; (dM / dx ) esIμnwgGaMgtg;sIuetrbs;kMlaMgkat;. eKGaceFVIGaMgetRkalsmIkar 4-1 nig 4-2 BIcMnucmYyeTAcMnucmYyeTotrvagbnÞúkcMcMnuc b¤m:Um:g; ¬dUcCa BIcMnuc B eTAcMnuc C enAkñúgrUbTI 4-9a¦ EdlenAkñúgkrNIenH ΔV = ∫ w( x )dx (4-3) bMErbMrYlkMlaMgkat; = RkLaépÞéndüaRkambnÞúkBRgay nig ΔM = ∫ V (x )dx (4-4) bMErbMrYlm:Um:g;Bt; = RkLaépÞéndüaRkamkMlaMgkat; eKRtUveFVIkarkt;cMNaMfa smIkar 4-3 bgðajBIPaBesμIKñaénbMErbMrYlkMlaMgkat;rvagBIrcMnucCamYynwgRkLa épÞéndüaRkambnÞúkBRgayrvagcMnucenaH. dUcKña smIkar 4-4 bgðajfabMErbMrYlm:Um:g;Bt;rvagBIrcMnucesμI nwgRkLaépÞéndüaRkamkMlaMgkat;enAcenøaHcMnucenaH. RbsinebIeKgayRsYlKNnaRkLaépÞrbs;düa- RkambnÞúk nigdüaRkamkMlaMgkat; enaHsmIkar 4-3 nig4-4 pþl;nUvviFIsMrab;kMNt;tMélrbs;kMlaMgkat; nig m:Um:g;Bt;CatMélelxenARtg;cMnucepSg²enAtambeNþayFñwm. eyIgRtUvcMNaMfa eKminGaceRbIsmIkar 4-1 nig 4-3 sMrab;bnÞúkcMcMnuceT edaysarsmIkarTaMgenH min)anKitBIbMErbMrYlkMlaMgkat;Pøam²enARtg;cMnucTaMgenH. dUcKña edaysarPaBminCab;rbs;m:Um:g; eKmin GaceRbIsmIkar 4-2 nig 4-4 enARtg;cMnucEdlrgGMeBIm:Um:g;eT. edIm,IKitkrNITaMgBIrenH eKRtUvBicarNa düaRkamGgÁesrIrbs;FatuGnnþtUcrbs;FñwmenAkñúgrUbTI 4-9a EdlmanTItaMgenARtg;kEnøgEdlbnÞúkcMcMnuc nigm:Um:g;manGMeBI. ]TahrN_énFatuenHRtUv)anbgðajenAkñúgrUbTI 4-10a nig 4-10b erogKña. BIrUbTI 4-10a eKeXIjfasmIkarlMnwgkMlaMgTamTareGaybMErbMrYlkMlaMgkat;køayCa + ↑ ∑ Fy = 0 ΔV = − F (4-5) Internal loading developed in structural members T.Chhay -119
  12. 12. Department of Civil Engineering NPIC dUcenH enAeBlEdl F eFVIGMeBIedaymanTisedAcuHeRkammkelIFñwm enaH ΔV GviC¢man dUcenHdüaRkam kMlaMgkat;manTiscueRkam. dUcKña RbsinebI F manTisedAeLIgelI enaH ΔV nwgeLIgelI. BIrUbTI 4- 10b edayyk Δx → 0 smIkarlMnwgm:Um:g;Bt;TamTareGaybMErbMrYlm:Um:g;køayCa + ∑ MO = 0 ΔM = M ' (4-6) enAkñúgkrNIenH RbsinebIm:Um:g;xageRkA M ' manGMeBIRsbTisRTnicnaLika enaH ΔM viC¢man dUcenH düaRkammanTiseLIgelI ehIyenAeBl M ' eFVIGMeBIRcasTisRTnicnaLika enaH ΔM RtUvEtmanTiscuH eRkam. tarag 4-1 bgðajBIkarGnuvtþénsmIkar 4-1, 4-2, 4-5, nig 4-6 eTAelIkrNIbnÞúkFmμtaeday snμt;faV nig M enAEtrkSatMélviC¢man. CMralenARtg;cMnucepSg²enAlIExSekagnImYy²RtUv)anbgðaj. eKmincaM)ac;cgcaMlT§plTaMgenHeT EteKRtUvsikSaeGay)anhμt;ct;dUcenHeKGacyl;BIrebobsg;düaRkam bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁMú T.Chhay -120
  13. 13. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa kMlaMgkat; nigdüaRkamm:Um:g;Bt;edayEp¥kelIkaryl;dwgBIbMErbMrYlénCMralBIdüaRkambnÞúk nigdüaRkam kMlaMgkat; erogKña. dMeNIrkarkñúgkarviPaK (Procedure for analysis) CMhanxageRkampþl;nUvviFIsMrab;sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g;sMrab;Fñwm edayeRbI smIkar 4-1 rhUtdl; 4-6. kMlaMgRbtikmμTMr (Support Reactions) kMNt;RbtikmμTMr nigbMEbkkMlaMgEdleFVIGMeBIenAelIFñwmCabgÁúMkMlaMgEdlEkg nigRsbeTAnwg G½kSrbs;Fñwm. düaRkamkMlaMgkat; (Shear diagram) begáItG½kSV nig x ehIybgðajtMélrbs;kMlaMgkat;enAcugTaMgBIrrbs;Fñwm. edaysar dV / dx = w enaHCMralrbs;düaRkamkMlaMgkat;enARKb;cMnucesμInwgGaMgtg;sIuetrbs; bnÞúkBRgayenARtg;cMnucenH. ¬cMNaMfa w vIC¢manenAeBlEdlvaeFVIGMeBIeLIgelI¦. RbsinebIeKkMNt;tMélCaelxrbs;kMlaMgkat;enARtg;cMnuc eKGackMNt;tMélenHedayeRbIviFImux kat;Edlerobrab;enAkñúgkfaxNÐ 4-1 b¤edaysmIkar 4-3 EdlbgðajfabMErbMrYlkMlaMgkat;esμI nwgRkLaépÞdüaRkambnÞúkBRgay. edaysareKeFVIGaMgetRkaleTAelI w(x ) edIm,ITTYl)an V enaHRbsinebI w(x ) CaExSekagdWeRkTI n enaH V ( x ) nwgkøayCaExSekagdWeRkTI n + 1 . Ca]TahrN_ RbsinebI w(x ) BRgayesñI enaH V (x ) CaExSekagbnÞat;. düaRkamm:Um:g;Bt; (Moment diagram) begáItG½kS M nig x ehIybgðajtMélrbs;m:Um:g;enAcugTaMgBIrrbs;Fñwm. edaysar dM / dx = V enaHCMralrbs;düaRkamm:Um:g;enARKb;cMnucesμInwgGaMgtg;sIuetrbs;kMlaMg kat;enARtg;cMnucenH. enARtg;cMnucEdlkMlaMgkat;esμIsUnü ¬ dM / dx = 0 ¦ dUcenHvaCacMnucm:Um:g;Gtibrma b¤Gb,brma. RbsinebIeKkMNt;tMélCaelxrbs;m:Um:g;enARtg;cMnuc eKGackMNt;tMélenHedayeRbIviFImuxkat; Edlerobrab;enAkñúgkfaxNÐ 4-1 b¤edaysmIkar 4-4 EdlbgðajfabMErbMrYlm:Um:g;esμInwgRkLaépÞ düaRkamkMlaMgkat;. edaysareKeFVIGaMgetRkaleTAelI V (x) edIm,ITTYl)an M enaHRbsinebIV (x) CaExSekagdWeRkTI Internal loading developed in structural members T.Chhay -121
  14. 14. Department of Civil Engineering NPIC n enaH M (x ) nwgkøayCaExSekagdWeRkTI n + 1 . Ca]TahrN_ RbsinebI V (x) CaExSekagbnÞat; enaH M (x ) CaExSekag)a:ra:bUl. ]TahrN_ 4-7³ KUrdüaRkamkMlaMgkat; nigdüaRkamm:Um:g;sMrab;FñwmenAkñúgrUbTI 4-11a. dMeNaHRsay³ kMlaMgRbtikmμTMr³ kMlaMgRbtikmμRtUv)anbgðajenAelIdüaRkamGgÁesrIrbs;Fñwm ¬rUbTI 4-11b¦. düaRkamkMlaMgkat;³ dMbUgsg;kMlaMgkat;enAcMnuccug x = 0 / V = +30kN nig x = 9m / V = −60kN . edaysarbnÞúk w GviC¢man ehIyekIneLIgCalkçN³bnÞat; enaHCMralrbs;düaRkamkMlaMgkat;ekIneLIglkçN³bnÞat; EttMélfycuH ¬ dV / dx = w ¦. eKRtUvkMNt;cMnucEdlkMlaMgkat;esμIsUnüedayeRbIviFI muxkat;BIkMNat;FñwmelIRbEvg x ¬rUbTI 4-11e¦. eyIgTamTareGayV = 0 dUcenH 1 ⎡ ⎛ x ⎞⎤ + ↑ ∑ Fy = 0 30 − ⎢20⎜ ⎟⎥ x = 0 2 ⎣ ⎝ 9 ⎠⎦ x = 5.20m düaRkamm:Um:g;³ BIdüaRkamkMlaMgkat; ¬rUbTI 4-11c¦ sMrab; 0 < x < 5.20m tMélrbs;kMlaMgkat;viC¢manb:uEnþfycuH dUcenHCMralrbs;düaRkamm:Um:g; M (x ) fycuH EttMélrbs;vaekIneLIg ¬ dM / dx = V ¦. enARtg;cMnuc x = 5.20m / dM / dx = 0 . dUcKñasMrab; 5.20m < x < 9m kMlaMgkat; nigCMralrbs; düaRkamm:Um:g;ekIneLIg EttMélrbs;m:Um:g;fycuH. tMélGtibrmarbs;m:Um:g;KWenARtg; x = 5.20m eday sar dM / dx = V = 0 enARtg;cMnucenH ¬rUbTI 4-11d¦. BIdüaRkamGgÁesrIenAkñúgrUbTI 4-11e enaH bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁMú T.Chhay -122
  15. 15. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 1 ⎡ ⎛ 5.20 ⎞⎤ ⎛ 5.20 ⎞ + ∑MS = 0 − 30(5.20) + ⎢20⎜ ⎟⎥ (5.20)⎜ ⎟+M =0 M = 104kN .m 2 ⎣ ⎝ 9 ⎠⎦ ⎝ 3 ⎠ ]TahrN_ 4-8³ KUrdüaRkamkMlaMgkat; nigdüaRkamm:Um:g;sMrab;FñwmenAkñúgrUbTI 4-12. dMeNaHRsay³ RbtikmμTMr³ RbtikmμTMrRtUv)anKNna nigbgðajenAelI düaRkamGgÁesrIrbs;FñwmnImYy². düaRkamkMlaMgkat;³ BIkarkMNt;sBaØa ¬rUbTI 4-1¦ dMbUg eKRtUvbgðajkMlaMgkat;enAxagcugrbs;FñwmnImYy². eday sar dV / dx = w eKGackMNt;CMralrbs;düaRkamkMlaMg kat;rvagcMnuccugTaMgBIr ehIyeKGacKUrdüaRkamV GaRs½y nwg x )an. düaRkamm:Um:g;³ BIkarkMNt;sBaØamþgeTot ¬rUbTI 4-1¦ dMbUg eyIgRtUvbgðajm:Um:g;enAcugnImYy²rbs;Fñwm. edaysar dM / dx = V eKGackMNt;CMralrbs;düaRkamm:Um:g;rvagcMnuc TaMgenH ehIyeKGacKUrdüaRkam M GaRs½ynwg x )an. Internal loading developed in structural members T.Chhay -123
  16. 16. Department of Civil Engineering NPIC ]TahrN_ 4-9³ KUrdüaRkamkMlaMgkat; nigdüaRkamm:Um:g;sMrab;FñwmenAkñúgrUbTI 4-13a. dMeNaHRsay³ RbtikmμTMr³ RbtikmμTMrRtUv)anKNna nigbgðajenAelI düaRkamGgÁesrIrbs;FñwmnImYy². düaRkamkMlaMgkat;³ eKRtUvbgðajtMélrbs;kMlaMgkat; enARtg;cug A ¬V A = 0.5kN ¦ ehIyRtg;cMnuc B ¬VB = −0.25kN ¦. enARtg;cMnuckNþalrvag A nig C CMralrbs;düaRkamkMlaMgkat;KWsUnüedaysar dV / dx = w = 0 . dUcenHkMlaMgkat;enAEtrkSatMél 0.5kN rbs;vadEdlenAkñúgtMbn;enH. enARtg;cMnuc C düaRkam kMlaMgkat;manlkçN³dac; ¬minCab;¦edaysarkMlaMg cMcMnucxageRkA 3kN enATIenaH. eKGacrktMélrbs;kMlaMg kat;enAxagsþaMcMnuc C Etbnþic edaykat;FñwmenARtg;cMnuc enaH. düaRkamGgÁesrIrbs;vaRtUv)anbgðajkñúgsßanPaB lMnwgkñúgrUbTI 4-13e. cMnucenH ¬V = −2.5kN ¦ RtUv)an bgðajenAelIdüaRkamkMlaMgkat;. dUcelIkmun w = 0 BI C eTA B dUcenHCMral dV / dx = 0 . düaRkambBa©b;eday tMél − 2.5kN enARtg;cMnuc B dUcbgðaj. cMNaMfa vamin manPaBdac;énkMlaMgkat;enARtg;cMnuc D EdlCacMnucm:Um:g; 6kN .m manGMeBI ¬rUbTI 4-13b¦. eKKYrkt;cMNaMfa eKGacsg;düaRkamkMlaMgkat; y:agelOnedayGnuvtþtamdüaRkambnÞúkrbs;düaRkam GgÁesrI ¬rUbTI 4-13b¦. edaycab;epþImBIcMnuc A CamYy nwgkMlaMg 0.5kN manGMeBIeLIgelI dUcenHV A = +0.5kN . edayKμanbnÞúkeFVIGMeBIenAcenøaH A nig C dUcenHkMlaMgkat; enArkSatMélefr. enARtg;cMnuc C kMlaMg 3kN manGMeBI edayTisedAcuHeRkam dUcenHkMlaMgkat;RtUvcuHeRkam 3kN ¬BI 0.5kN mk − 2.5kN ¦. mþgeTot kMlaMg kat;efr ¬KμanbnÞúkxageRkA¦ ehIybBa©b;enARtg;cMnuc B CamYynwg − 2.5kN ehIycugeRkaydüaRkam enHbiTedayvart;BI − 2.5kN eTAsUnüenARtg;cMnuc B ¬TisedAeLIgelI¦. bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁMú T.Chhay -124
  17. 17. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa düaRkamm:Um:g;Bt;³ m:Um:g;Bt;enAcugsgxagrbs;FñwmKWsUnü ¬rUbTI 4-13d¦. CMralrbs;düaRkamm:Um:g; Bt;BI A eTA C mantMélefredaysar dM / dx = V = +0.5 . eKGackMNt;tMélrbs;m:Um:g;Bt;enARtg; cMnuc C edayviFImuxkat; ¬rUbTI 4-13e¦ b¤edaykMNt;RkLaépÞrbs;düaRkamkMlaMgkat;rvagcMnuc A nig C . edaysar M A = 0 ΔM AC = M C − 0 = (0.5kN )(0.3m ) M C = 1.5kN .m CMralBIcMnuc C eTA D KW dM / dx = V = −2.5kN ¬rUbTI 4-13c¦ edaysar M C = 1.5kN.m m:Um:g;Bt; Rtg;cMnuc D KW ΔM CD = M D − 1.5kN .m = (− 2.5kN .m )(1.5m ) M D = 2.25kN .m düaRkamm:Um:g;RtUv)ankat;pþac;Rtg;cMnuc D edaysarm:Um:g; 6kN .m . viFImuxkat; ¬rUbTI 4-13f¦ eGay tMélm:Um:g;Bt; 3.75kN .m xagsþaMcMnuc D Etbnþic. BIcMnucenH CMral dM / dx = −2.5kN enArkSarhUtdl; düaRkamenHbiTCamYynwgtMélsUnüenARtg;cMnuc B ¬rUbTI 4-13d¦. ]TahrN_ 4-10³ FñwmEdlbgðajenAkñúgrUbftRtUv)aneRbIedIm,IRTsMy:absMrab;pøÚvcUlGKar. KMrUm:UEDl rbs;FñwmCamYynwgbnÞúkEdlmanGMeBIelIvaRtUv)anbgðajenAkñúgrUbTI 4-14a. KUrdüaRkamkMlaMgkat; nig düaRkamm:Um:g;Bt;sMrab;Fñwm. Internal loading developed in structural members T.Chhay -125
  18. 18. Department of Civil Engineering NPIC dMeNaHRsay³ RbtikmμTMr³ eKKNnakMlaMgRbtikmμtamrebobFmμta. lT§plRtUv)anbgðajenAkñúgrUbTI 4-14b. düaRkamkMlaMgkat;³ dMbUgeKRtUvsg;kMlaMgkat;enAxagcugrbs;Fñwm rYmmanV A = 0 nigVC = −2.19kN ¬rUbTI 4-14c¦. CMralrbs;düaRkamkMlaMgkat;GviC¢manelIRbEvgFñwmTaMgmUl dV / dx = w = −10kN . edIm,IrkkMlaMgkat;enAxageqVgcMnuc B eKeRbIviFImuxkat;sMrab;kMNat;Ggát; AB b¤k¾KNnaRkLaépÞbnÞúk BRgay eBalKW ΔV = VB − 0 = −10(0.75) enaHVB = −7.5kN . kMlaMgRbtikmμeFVIeGaykMlaMgkat;bþÚrBI − 7.5kN eTA − 7.5 + 15.31 = 7.81kN . bnÞab;mk CMraldüaRkameq<aHeTArk − 2.19kN . eKGackMNt; cMnucEdlmankMlaMgkat;sUnüBICMral − 10kN / m b¤tamRtIekaNdUc 7.81/ x = 2.19 /(1 − x ) enaHeyIg )an x = 0.781m . düaRkamm:Um:g;³ dMbUgeKRtUvsg;m:Um:g;enAcugFñwm M A = M C = 0 ¬rUbTI 4-14d¦. karGegátelIdüaRkam kMlaMgkat;bgðajfadüaRkamm:Um:g;Bt;tMélGviC¢man ehIyCMralrbs;vaekIneLIg. enARtg;cMnuc B ekIt mankarpøas;bþÚrCMralGviC¢man − 7.50 eTACMralviC¢man 7.81 . ehIybnÞab;mkCMralfycuHeTArksUnü ¬rUbTI 4-14c¦ nigkøayCaGviC¢manrhUtdl;CMralrbs;vamantMélesμInwg − 2.19 . eKGacKNnatMél − 2.81 nig 0.239 enAelIdüaRkamm:Um:g;edayviFImuxkat; b¤edayrkRkLaépÞdüaRkamkMlaMgkat;. Ca]TahrN_ ΔM = M B − 0 = 12 (− 7.50)(0.75) = −2.81 enaH M B = −2.81kN .m . eyIgeFVIdUcKñaedIm,I bgðajfam:Um:g;viC¢manGtibrmaKW 0.239kN .m . ]TahrN_ 4-11³ KUrdüaRkamkMlaMgkat; nigm:Um:g;Bt;sMrab;FñwmpSM (compound beam) EdlbgðajenA kñúgrUbTI 4-15a. snμt;faTMrenARtg; A nig C CaTMrkl; (roller) ehIy B nig E CatMNsnøak;. dMeNaHRsay³ RbtikmμTMr³ enAeBlEdleKpþac;FñwmRtg;snøak; B eKGacKNnaRbtikmμTMrdUcbgðajenAkñúgrUbTI 4-15b. düaRkamkMlaMgkat;³ CaFmμta ehIycab;epþImedaykarsg;kMlaMgkat;enAxagcugFñwmRtg; A nig E ¬rUbTI 4-15c¦. ExSekagRtUv)anbgðajedayeRbIsmIkar dV / dx = w . cMNaMfa CMralGviC¢manefr dV / dx bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁMú T.Chhay -126
  19. 19. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa = −20kN / m BIcMnuc A eTA B / CMralsUnüBIcMnuc B eTAcMnuc C / CMralGviC¢manekIneLIgBIcMnuc C eTA cMnuc D ¬ 0 eTA 30kN / m ¦ ehIycugeRkayCMralGviC¢manfycuHBIcMnuc D eTAcMnuc E ¬ 30kN / m eTA 0 ¦. sakl,grktMélkMlaMgkat;edayeRbIRkLaépÞdüaRkambnÞúk ¬ExSekag w ¦ edIm,IrkbMErbMrYlkMlaMg kat;. eKGacTItaMgkMlaMgkat;esμIsUnü ¬ x = 1m ¦ edayeRbIRtIekaNdUc b¤edayeRbIsþaTic dUcenAkñúgrUbTI 4-11e én]TahrN_ 4-7. düaRkamm:Um:g;Bt;³ dMbUgeKRtUvsg;m:Um:g;enAxagcugFñwm M A = 150kN .m nig M E = 0 ¬rUbTI 4-15d¦. sikSadüaRkam nigcMNaMBIrebobkñúgkarsg;ExSekagepSg²edayeRbI dM / dx = V . epÞógpÞat;tMélCaelx sMrab;kMBUlExSekagedayeRbIsþaTic b¤edayKNnaRkLaépÞdüaRkamkMlaMgkat;edIm,IrkbMErbMrYlm:Um:g;. ]TahrN_ 4-12³ KUrdüaRkamkMlaMgkat; nigm:Um:g;Bt;sMrab;FñwmnImYy²EdlbgðajenAkñúgrUbTI 4-16. dMeNaHRsay³ enAkñúgkrNInImYy² kMlaMgRbtikmμTMrRtUv)anKNna ehIyRtUv)anbgðajenAkñúgrUb. Gnuvtþtambec©keTs Edlerobrab;enAkñúg]TahrN_elIkmun düaRkamkMlaMgkat; nigdüaRkamm:Um:g;RtUv)anbgðaj. eKRtUveFVI karkt;cMNaMfadüaRkamTaMgenHRtUv)ansg;edayEp¥kelICMral nigm:Um:g; Edl dV / dx = w nig dM / dx = V . eKKNnaTMhMrbs;kMlaMgkat; nigm:Um:g;Bt;edayeRbIviFImuxkat; b¤edayrkRkLaépÞdüaRkambnÞúk nigRkLaépÞdüaRkamkMlaMgkat;. Internal loading developed in structural members T.Chhay -127
  20. 20. Department of Civil Engineering NPIC $>$> düaRkamkMlaMgkat; nigdüaRkamm:Um:g;Bt;sMrab;eRKag (Shear and Moment Diagram for Frames) rMlwkfaeRKagCarcnasm<½n§EdlGgát;CaeRcInRtUv)antP¢ab;edaytMNbgáb; nigtMNsnøak;enAxag cugrbs;va. karsikSaKNnaBIeRKOgbgÁúMenATamTarkarKUrdüaRkamkMlaMgkat; nigdüaRkamm:Um:g;sMrab; Ggát;nImYy². edIm,IviPaKRKb;cMeNaTTaMgGs; eyIgGaceRbIdMeNIrkarsMrab;karviPaKEdlmanerobrab;enA kñúgkfaxNÐ 4-3. dMbUg vaTamTarkarkMNt;kMlaMgRbtikmμenARtg;TMreRKag. bnÞab;mkedayeRbIviFImux kat; eyIgGacrkkMlaMgtamG½kS kMlaMgkat; nigm:Um:g;Bt;EdlmanGMeBIenARtg;cugrbs;Ggát;nImYy². bnÞúkxageRkATaMgGs;RtUv)anbMEbkeGayeFVIGMeBIEkg nigRsbnwgG½kSrbs;Ggát; bnÞab;mkeKGacKUrdüa RkamkMlaMgkat; nigdüaRkamm:Um:g;sMrab;GgátnImYy²dUckarerobrab;enAkñúgkfaxNÐelIkmun. ; enAeBlKUrdüaRkamm:Um:g; eKeRcIneRbIkarkMNt;sBaØamYykñúgcMeNamBIrenAkñúgkarGnuvtþ. CaTUeTA RbsinebIeRKagplitBIebtugGarem: GñksikSaKNnaniymKUrdüaRkamm:Um:g;viC¢manenAEpñkxagTajrbs; eRKag. müa:gvijeTot RbsinebIm:Um:g;begáItkugRtaMgTajenAépÞxageRkArbs;eRKag. edaysarebtugman ersIusþg;TajtUc enaHeKGacdwgBIEpñkEdlRtUvkarEdkBRgwgedayRKan;EteXIjdüaRkamm:Um:g;. b:uEnþenA kñúgesovePAenH eyIgnwgeRbIkarkMNt;sBaØapÞúyBIGIVEdlerobrab;xagelI edayeyIgnwgKUrdüaRkamm:Um:g; bUkenAEpñkrgkugRtaMgsgát;. karkMNt;sBaØaRtUv)anGnuvtþsMrab;FñwmEdlerobrab;enAkñúgkfaxNÐ 4-1. ]TahrN_xageRkambgðajBIdMeNIrkarGnuvtþCaelx. bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁMú T.Chhay -128
  21. 21. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ]TahrN_ 4-13³ KUrdüaRkamm:Um:g;Bt;sMrab;eRKagEdlbgðajenAkñúgrUbTI 4-17a. snμt;TMr B CaTMr snøak;. dMeNaHRsay³ kMlaMgRbtikmμTMr³RbtikmμTMrRtUv)anbgðajenAkñúgdüaRkamGgÁesrIrbs;eRKagTaMgmUl ¬rUbTI 4-17b¦. bnÞab;BIkat;eRKagenHCaBIrGgát; eyIgGackMNt;kMlaMgkñúgenAcugTMredayeRbIlT§plTaMgenH ¬rUbTI 4- 17c¦. cMNaMfa eKbgðajEtbnÞúkxageRkA 25kN enAelIdüaRkamGgáesrIrbs;tMN C Etb:ueNÑaH. düaRkamm:Um:g;³ edayeKarBtamkarkMNt;sBaØaEdlmantMélviC¢man nigedayeRbIbec©keTsEdlerobrab; enAkñúgkfaxNÐ 4-3 düaRkamm:Um:g;sMrab;Ggát;eRKagRtUv)anbgðajenAkñúgrUbTI 4-17d. ]TahrN_ 4-14³ KUrdüaRkamkMlaMgkat; nigdüaRkamm:Um:g;sMrab;eRKagEdlbgðajenAkñúgrUbTI 4-18a. snμt; A CaTMrsnøak; C CaTMrkl; ehIy B CatMNbgáb; (fixed joint). dMeNaHRsay³ kMlaMgRbtikmμTMr³ düaRkamGgÁesrIrbs;eRKagTaMgmUlRtUv)anbgðajenAkñúgrUbTI 4-18b. enATIenHbnÞúk BRgayEdltMNagedaybnÞúkxül;RtUv)anCMnYsedaykMlaMgpÁÜbrbs;va ehIyeKKNnakMlaMgRbtikmμ. bnÞab;mkeyIgkat;eRKagRtg;tMN B ehIyKNnakMlaMgkñúgenARtg;tMN B ¬rUbTI 4-18c¦. tamkar RtYtBinitü smIkarlMnwgRtUv)anbMeBjenARtg;tMN B ehIyk¾RtUv)anbgðajenAkñúgrUb. Internal loading developed in structural members T.Chhay -129
  22. 22. Department of Civil Engineering NPIC düaRkamkMlaMgkat; nigdüaRkamm:Um:g;³ bgÁúMkMlaMg ¬kMub:Usg;¦énbnÞúkBRgay (72kN ) /(5m) = 14.4kN / m ehIy (96kN ) /(5m) = 19.2kN / m RtUv)anbgðajenAelIGgát; AB ¬rUbTI 4-18d¦. düaRkamkMlaMgkat; nigdüaRkamm:Um:g;sMrab;Ggát;nImYy²RtUv)anbgðajenAkñúgrUbTI 4-18d nig 4-18e. bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁMú T.Chhay -130
  23. 23. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ]TahrN_ 4-15³ sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g;sMrab;eRKagEdlbgðajenAkñúgrUbTI 4- 19a. snμt; A CaTMrsnøak; C CaTMrkl; ehIy B CatMNbgáb; (fixed joint). ecalkMras;rbs;Ggát;. dMeNaHRsay³ kMlaMgRbtikmμTMrenAelIeRKagTaMgmUlRtUv)anKNna nigbgðajenAelIdüaRkamGgÁesrIrbs;va ¬rUbTI 4- 19b¦. BIdüaRkamenH eKKUrdüaRkamGgÁesrIrbs;GgátnImYy² ¬rUbTI 4-19¦. bgÁúMkMlaMgénkMlaMgpÁÜb ; enAelIGgát; BC KW (9kN )cos 45o = (9kN )sin 45o = 6.364kN . dUcenHbgÁúMkMlaMgénbnÞúkBRgayelI Ggát; BC KW 6.364kN / 4.243m = 1.5kN / m dUcbgðaj. edayeRbIlT§plTaMgenH düaRkamkMlaMgkat; nigdüaRkamm:Um;g;RtUv)anbgðajenAkñúgrUbTI 4-19c. Internal loading developed in structural members T.Chhay -131
  24. 24. Department of Civil Engineering NPIC $>%> karsg;düaRkamm:Um:g;Bt;edayeRbIviFItMrYtpl (Moment diagrams constructed by the method of superposition) edaysareKeRbIFñwmedIm,ITb;Tl;nwgkugRtaMgBt; enaHdMeNaHRsayéndüaRkamm:Um:g;RtUv)aneKeRbI CacaM)ac;sMrab;karsikSaKNnamuxkat;rbs;Ggát;. enAkñúgkfaxNÐ 4-3 eKsg;düaRkamm:Um:g;Bt; edaydM- bUgeKRtUvdüaRkamkMlaMgkat;. b:uEnþRbsinebIeyIgeRbIeKalkarN_tMrYtpl enaHeyIgGacKitkMlaMgEdl manGMeBIelIFñwmdac;edayELkBIKña bnÞab;mkeyIgGacsg;düaRkamm:Um:g;énGgát;nImYy²EdlbNþalBI kMlaMgGnuvtþn_nImYy². eyIgnwgeXIjBIGtßRbeyaCn_rbs;viFIenH enAeBleyIgGnuvtþviFIPaBdabeday FrNImaRt (geometric deflection method) edIm,IkMNt;PaBdabrbs;Fñwm nigkMlaMgRbtikmμenAelIFñwm minkMNt;edaysþaTic. sMrab;karviPaKeRKOgbgÁúM bnÞúkPaKeRcInEdlmanGMeBIenAelIFñwmeRcInpSMeLIgedaybnÞúkEdl bgðajenAkñúgrUbTI 4-20. karsg;düaRkamm:Um:g;sMrab;eFVIkarbUkbBa©ÚlKñamanbgðajenAkñúg]TahrN_ 4- 8. edIm,Iyl;BIrebobeRbIviFItMrYlplsMrab;sg;düaRkamm:Um:g; eyIgniwgBicarNaFñwmTMrsamBaØenAxagelI eKbMputénrUbTI 4-21a. enATIenHkMlaMgRbtikmμRtUv)anKNna dUcenHehIyRbB½n§kMlaMgenAelIFñwm begáIt)ankMlaMgpÁÜb nigm:Um:g;pÁÜbesμIsUnü. düaRkamm:Um:g;sMrab;krNIenHRtUv)anbgðajenAkñúgrUbxagelI bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁMú T.Chhay -132
  25. 25. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa eKbMputénrUbTI 4-21b. cMNaMfa eKsg;düaRkamm:Um:g;sMrab;Fñwmkugs‘ul (cantilevered beam) enAeBl vargnUvRbB½n§kMlaMgsmmUledaysþaTiceTAnwgbnÞúkEdlmanGMeBIelIFñwmsamBaØ. RbesIrCagkarBicarNa bnÞúkTaMgGs;EdlGMeBIelIFñwmkñúgeBlEtmYysMrab;karsg;düaRkamm:Um:g; eyIgGaceFVIkarbUkbBa©ÚlKñanUv lT§plénbnÞúknImYy²EdlmanGMeBIelIFñwmkugs‘ulbIdac;edayELkBIKñadUcbgðajenAkñúgrUbTI 4-21a. dUcenH RbsinebIdüaRkamm:Um:g;sMrab;Fñwmkugs‘ulnImYy²RtUv)anKUr ¬rUbTI 4-21b¦ kareFVItMrYtpléndüa- RkamTaMgenHnaMeGayeyIgTTYl)andüaRkamm:Um:g;pÁÜbsMrab;FñwmTMrsamBaØ. Ca]TahrN_ BIdüaRkamdac; edayELknImYy² m:Um:g;enAxagcug A KW M A = −500 − 750 + 1250 = 0 dUckarepÞógpÞat;edaydüaRkam m:Um:g;enAxagelIbMputénrUbTI 4-21b. enAkñúgkrNIxøH karsg;düaRkamm:Um:g;edayeRbIviFItMrYtplman lkçN³gayRsYlCag. tamrebobdUcKña eyIgGacsMrYlkarsagsg;düaRkamm:Um:g;sMrab;FñwmedayeRbIviFItMrYtplénFñwm TMrsamBaØ. Ca]TahrN_ bnÞúkenAelIFñwmEdlbgðajenAxagelIbMputénrUbTI 4-22a smmUleTAnwgFñwm rgbnÞúkEdlbgðajenAxageRkamva. dUcenH eKGaceRbIdüaRkamm:Um:g;dac;edayELksMrab;FñwmTaMgbIsMrab; Internal loading developed in structural members T.Chhay -133
  26. 26. Department of Civil Engineering NPIC sg;düaRkamm:Um:g;rYmEdlbgðajenAkñúgrUbTI 4-22b RbesIrCakarsg;düaRkamm:Um:g;edayeRbIbnÞúkCa eRcInmanGMeBIelIFñwmEtmYy. ]TahrN_ 4-16³ sg;düaRkamm:Um:g;sMrab;FñwmEdlbgðajenAkñúgrUbxagelIbMputénrUbTI 4-23a edayviFItMrYtpl. BicarNFñwmCaFñwmkugs‘ulBITMrenARtg;cMnuc B . dMeNaHRsay³ RbsinebIFñwmRtUv)anRTCalkçN³Fñwmkugs‘ulenARtg;cMnuc B vaRtUvrgbnÞúksmmUledaysþaTicEdlbgðaj enAkñúgrUbTI 4-23a. karbUkbBa©ÚlKñanUvFñwmkugs‘ulTaMgbIRtUv)anbgðajenAxageRkamva edayrYmCamYy nwgdüaküamm:Um:g;énFñwmnImYy²kñúgrUbTI 4-23b. CMnYykñúgkarsg;düaRkamTaMgenH manbgðajenAkñúgrUb TI 4-20. eTaHbICamincaM)ac;enATIenHk¾eday eKKYrKUrdüaRkamm:Um:g;énbnÞúksrub. sMrab;karGnuvtþ sak l,gsg;düaRkamenH nigRtYtBinitülT§plrbs;va. bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁMú T.Chhay -134
  27. 27. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ]TahrN_ 4-17³ sg;düaRkamm:Um:g;sMrab;FñwmEdlbgðajenAkñúgrUbxagelIbMputénrUbTI 4-24a edayviFItMrYtpl. BicarNFñwmCaFñwmkugs‘ulBITMrenARtg;cMnuc A . dMeNaHRsay³ karbUkbBa©ÚlKñaénFñwmkugs‘ulRtUv)anbgðajenAkñúgrUbTI 4-24 rYmCamYynwgdüaRkam:Um:g;EdlRtUvKñanwg Fñwmkugs‘ulEdlrgbnÞúkdac;edayELkBIKñaRtUv)anbgðajenAkñúgrUbTI 4-24b. CMnYykñúgkarKUrdüaRkam TaMgenHmanbgðajenAkñúgrUbTI 4-12. cMNaMfa eKminKitkMlaMgRbtikmμenARtg;snøak;EdlmanTMhM 100.7 kN eT edaysarvamin)anbegáItm:Um:g;eT. eKGaceRbIviFIenAkñúgkfaxNÐ 4-3 edIm,IepÞógpÞat;düa- Rkamm:Um:g;srubEdleGayenAkñúgrUbxagelIeKbMputénrUbTI 4-24b. Internal loading developed in structural members T.Chhay -135
  28. 28. Department of Civil Engineering NPIC bnÞúkxagkñúgEdlekItmanenAkñúgGgát;eRKOgbgÁMú T.Chhay -136
  29. 29. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa cMeNaT 4>1> kMNt;kMlaMgkat; kMlaMgtamG½kS nigm:Um:g; 4>6> kMNt;kMlaMgkat; kMlaMgtamG½kS nigm:Um:g; Bt;enARtg;cMnuc C nig D . snμt;faTMrRtg;cMnuc Bt;enARtg;cMnuc C nig D . snμt;faTMrRtg;cMnuc B CaTMrkl;. cMnuc C EdlmanTItaMgenAxagsþaM A CaTMrkl; nig B CaTMrsnøak;. bnÞúk 40kN Etbnþic. 4>2> KUrdüaRkamkMlaMgkat; nigdüaRkamm:Um:g;Bt; 4>7> kMNt;kMlaMgkat; kMlaMgtamG½kS nigm:Um:g; sMrab;FñwmenAkñúgcMeNaT 4>1. Bt;enARtg;cMnuc C EdlsßitenAxagsþaMédTMrkl; 4>3> kMNt;kMlaMgkat; kMlaMgtamG½kS nigm:Um:g; Rtg; A . nigRtg;cMnuc D EdlenAxageqVgbnÞúk 15kN . Bt;enAkñúgFñwmRtg;cMnuc B . 4>4> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; 4>8> kMNt;kMlaMgkat; kMlaMgtamG½kS nigm:Um:g; Bt;sMrab;FñwmenAkñúgcMeNaT 4>3. Bt;enAkñúgFñwmRtg;cMnuc B . 4>5> kMNt;kMlaMgkat; kMlaMgtamG½kS nigm:Um:g; Bt;enAkñúgFñwmRtg;cMnuc C nig D . snμt;TMr B Ca TMrkl;. cMnuc D EdlmanTItaMgenAxagsþaMbnÞúk 50kN Etbnþic. 4>9> kMNt;kMlaMgkat; kMlaMgtamG½kS nigm:Um:g; Bt;enARtg;cMnuc C . snμt;faTMrRtg;cMnuc A Ca TMrsnøak; nig B CaTMrkl;. Problems T.Chhay -137
  30. 30. Department of Civil Engineering NPIC 4>10> kMNt;kMlaMgkat; nigm:Um:g;Bt;enAkñúgr:t 4>15> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; kMralCaGnuKmn_én x Edl 4m < x < 8m . snμt; Bt;sMrab;r:tkMralenAkñúgcMeNaT 4>14. faTMrRtg;cMnuc A CaTMrsnøak; nig B CaTMrkl;. 4>16> kMNt;kMlaMgkat; kMlaMgEkg nig m:Um:g;Bt; enARtg;cMnuc E nig D énFñwmpSM. 4>11> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; 4>17> RKwHebtugCab;RTssrBIrEdlrgbnÞúk. Rb Bt;sMrab;r:tkMralenAkñúgcMeNaT 4>10. sinebIeKsnμt;fasMBaFdIxageRkamRKwHBRgayesμI 4>12> kMNt;kMlaMgkat; kMlaMgtamG½kS nig cUrkMNt;GaMgtg;sIuet w rbs;va nigcMgayKMlat m:Um:g;Bt;enAkñúgFñwmRtg;cMnuc C nig D . snμt; d rbs;ssrenARtg; B . KUrdüaRkamkMlaMgkat; faTMrRtg;cMnuc A CaTMrsnøak; nig B CaTMrkl;. nigdüaRkamm:Um:g;Bt;rbs;RKwHenH. 4>13> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;FñwmenAkñúgcMeNaT 4>12. 4>18> kMNt;kMlaMgkat; kMlaMgEkg nigm:Um:g;Bt; 4>14> kMNt;kMlaMgkat; kMlaMgtamG½kS nig enARtg;cMnuc C rbs;Fñwm. m:Um:g;Bt;enAkñúgFñwmRtg;cMnuc C . snμt;faTMr Rtg;cMnuc A CaTMrsnøak; nig B CaTMrkl;. cMeNaT T.Chhay -138
  31. 31. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 4>19> kMNt;cMgay a rvagTMrCaGnuKmn_énRbEvg 12kN Etbnþic. snμt;faTMr B CaTMrsnøak;. rbs;Fñwm L edIm,IeGaym:Um:g;Bt;esμIsUnüenARtg;cM- nuckNþalrbs;Fñwm. GaMgtg;sIuetrbs;bnÞúk BRgayenARtg;kNþalFñwmKW wo . 4>25> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;FñwmenAkñúgcMeNaT 4>24. 4>26> kMNt;kMlaMgkat; nigm:Um:g;Bt;enAkñúgFñwmCa 4>20> kMNt;kMlaMgkat; nigm:Um:g;Bt;enAkñúgFñwmCa GnuKmn_én x . GnuKmn_én x . snμt;faTMrRtg;cMnuc B CaTMrkl;. 4>21> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; 4>27> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;FñwmenAkñúgcMeNaT 4>20. Bt;sMrab;Fñwm. 4>22> kMNt;kMlaMgkat; nigm:Um:g;Bt;CaGnuKmn_ én x Edl 2m < x < 4m . 4>23> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; 4>28> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;FñwmenAkñúgcMeNaT 4>22. Bt;sMrab;Fñwmkugs‘ul. 4>24> kMNt;kMlaMgkat; kMlaMgEkg nigm:Um:g;Bt; enARtg; (a) cMnuc C EdlsßitenAxageqVgTMrkl; bnþic nig(b) cMnuc D EdlsßitenAxagsþaMbnÞúkcMnuc Problems T.Chhay -139
  32. 32. Department of Civil Engineering NPIC 4>29> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Fñwm nigkMNt;kMlaMgkat; nigm:Um:g;Bt; enAkñúgFñwmCaGnuKmn_én x . 4>33> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;r:tkMralenAkñúgcMeNaT 4>32. 4>34> kMNt;kMlaMgkat; nigm:Um:g;Bt;enAkñúgFñwmCa 4>30> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; GnuKmn_én x . Bt;sMrab;Fñwm nigkMNt;kMlaMgkat; nigm:Um:g;Bt; enAkñúgFñwmCaGnuKmn_én x . 4>35> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Fñwmkugs‘ulenAkñúgcMeNaT 4>34. 4>31> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; 4>36> kMNt;kMlaMgkat; nigm:Um:g;Bt;enAkñúgFñwm Bt;sMrab;Fñwmkugs‘ul. kugs‘ulsþÜc (tapered cantilever beam) Ca GnuKmn_én x . 4>32> kMNt;kMlaMgkat; nigm:Um:g;Bt;enAkñúgr:t kMralCaGnuKmn_ én x Edl1m < x < 2m . 4>37> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; snμt;faTMrenARtg;cMnuc A CaTMrkl; nig B Ca TMr Bt;sMrab;Fñwmkugs‘ulenAkñúgcMeNaT 4>36. snøak;. kMralRtUv)anRTedayTMrsamBaØenAelI 4>38> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; rnUtRtg; C / D / E / F nig G . Bt;sMrab;Fñwm ehIykMNt;kMlaMgkat; nigm:Um:g;Bt; enAkñúgFñwmCaGnuKmn_én x . cMeNaT T.Chhay -140
  33. 33. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 4>42> kMNt;kMlaMgkat; nigm:Um:g;Bt;enAkñúgFñwm CaGnuKmn_én x nigsg;düaRkamkMlaMgkat; nig düaRkamm:Um:g;Bt;sMrab;Fñwm. 4>39> eKeRbIr:ts<an (pier) ebtugGarem:edIm,IRT FñwmsMrab;RTkMrals<an. sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g;sMrab;r:ts<anenAeBlvargbnÞúkBI FñwmdUcbgðaj. snμt;ssrenARtg;cMnuc A nig B 4>43> Fñwmrgm:Um:g;BRgayesμI m ¬m:Um:g; / RbEvg¦. GnuvtþEtkMlaMgRbtikmμbBaÄreTAelIr:tenaH. sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g;Bt; sMrab;Fñwm. 4>44> RbB½n§kMralsMrab;GKarmYypSMeLIgeday r:tEdlRTrnUtTTwg ehIyrnUtenaHRTkMralbnþ 4>40> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; eTotedayTMrsamBaØ. sg;düaRkamkMlaMgkat; Bt;sMrab;Fñwm. TMrRtg; A nig B GnuvtþEtkMlaMg nigdüaRkamm:Um:g;Bt;sMrab;r:t. snμt;r:tRTeday RbtikmμbBaÄreTAelIFñwm. TMrsamBaØ. 4>41> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; 4>45> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Fñwm. Bt;sMrab;Fñwm. Problems T.Chhay -141
  34. 34. Department of Civil Engineering NPIC 4>46> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; 4>50> FñwmebtugRTCBa¢aMgEdlGnuvtþbnÞúkBRgay Bt;sMrab;Fñwm. snμt;faTMrRtg;cMnuc B CaTMrsnøak; mkelIFñwmdUcbgðaj. FñwmenHmanTMhMmuxkat; nig A CaTMrkl;. 300mm × 650mm nigeFVIBIebtugEdlmandg;sIu- et γ = 23.6kN / m3 . sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g;sMrab;Fñwm ehIykMNt;m:Um:g;Gti- brma nigm:Um:g;Gb,brmaenAkñúgFñwm. ecalTMgn; rbs;EdkBRgwgnAkñúgFñwm. 4>47> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Fñwm. snμt;faTMrRtg;cMnuc B CaTMr snøak;;. 4>51> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Fñwm. 4>48> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Fñwm. 4>52> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Fñwm. 4>49> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Fñwm. cMnuc C Casnøak;. 4>53> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Fñwm. cMeNaT T.Chhay -142
  35. 35. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 4>54> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;FñwmpSM. kMNat;FñwmRtUv)antP¢ab;eday snøak;enARtg;tMN B nig D . 4>58> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;FñwmpSM. kMNat;FñwmRtUv)antP¢ab;eday snøak;enARtg;tMN B . 4>55> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Fñwm. 4>59> kMNt;kMlaMgkat; kMlaMgEkg nigm:Um:g;Bt; 4>56> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; enAkñúgFñwmRtg;cMnuc D nigcMnuc E . cMnuc E sßit Bt;sMrab;FñwmpSM. vaRtUv)anRTedaybnÞHrelagenA enAxagsþaMbnÞúk 20kN bnþic. snμt;faTMrRtg;cMnuc A CaTMrkl;/ B CatMNsnøak;; nig C CaTMrbgáb;. Rtg; A EdlrGilenAkñúgcg¥Úr dUcenHehIyvamin GacRTbnÞúkbBaÄr b:uEnþvaGacTb;Tl;m:Um:g; nig bnÞúkedk. 4>60> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Fñwm CDE . snμt;fa A CaTMrkl; ehIy B TMrsnøak;. D nig E CatMNbgáb;. 4>57> bnÞHEdk ABC nig BCD RtUv)anP¢ab; bBa©ÚlKñaedayb‘ULúgy:agrlugdUcbgðaj. Rbsin ebIb‘ULúgenHGnuvtþEtRbtikmμbBaÄreTAelIbnÞH cUr kMNt;kMlaMgRbtikmμenARtg;TMr nigsg;düaRkam kMlaMgkat; nigdüaRkamm:Um:g;sMrab;bnÞHnImYy². Problems T.Chhay -143
  36. 36. Department of Civil Engineering NPIC 4>61> FñwmlyRtUv)anplitedaymanGgát;bBaÄr BD enABIelI. sg;düaRkamkMlaMgkat; nigdüa- Rkamm:Um:g;Bt;sMrab;Fñwm ABC RbsinebIvaRTbnÞúk 3.6kN . CMnYy³ eKCMnYsbnÞúkenAkñúgGgát; DE edaybnÞúksmmUlenARtg;cMnuc B enAelIG½kSrbs; Fñwm. 4>64> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Ggát;nImYy²rbs;eRKag. snμt;fa A Ca TMrsnøak;/ C CaTMrkl;;. tMN B CatMNbgáb;. bnÞúkxül;RtUv)anepÞreTAGgát;tamry³rnUt nig r)arTTwg. 4>62> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Ggát;nImYy²rbs;eRKag. snμt;fa A Ca TMrsnøak;; ehIy D TMrkl;. 4>65> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Ggát;nImYy²rbs;eRKag. snμt;fa A / 4>63> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; C / D CaTMrsnøak; ehIy B CatMNbgáb;. Bt;sMrab;Ggát;nImYy²rbs;eRKag. snμt;fa A Ca TMrbgáb;/ B CatMNsnøak;; ehIy C TMrkl;. cMeNaT T.Chhay -144
  37. 37. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 4>69> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;FñwmedayviFItMrYtpl. KitfaFñwmCaFñwm kugs‘ulBITMrsnøak; A . 4>70> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;FñwmedayviFItMrYtpl. 4>66> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;Ggát;nImYy²rbs;eRKag. snμt;fa A / nig C CaTMrsnøak; ehIy B CatMNsnøak;. 4>71> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;FñwmedayviFItMrYtpl. KitfaFñwmCaFñwm kugs‘ulBITMrsnøak; A . 4>67> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; Bt;sMrab;FñwmedayviFItMrYtpl. KitFñwmCaFñwmTMr samBaØ. snμt; A CaTMrsnøak; nig B CaTMrkl;. 4>68> edaHRsay 4>67 eLIgvij edayKitfaFñwm CaFñwmkugs‘ulBITMrsnøak; A . Problems T.Chhay -145
  38. 38. Department of Civil Engineering NPIC KMerag-1> y:mYymanTItaMgenACan;TIbIrbs;m:UETl mandg;sIuet16kN / m3 . bnÞúkGefrEdlbgá dUceXIjkñúgrUbft. eKsg;vaedayeRbIkMralebtug edaykarkkrwgénRBilRtUv)ansnμt;famanragctu kMras;100mm edayQrenAelIFñwmkMralTMr ekaNBñayCamYynwgbnÞúk 3kN / m enAcugxagsþaM samBaØbYn EdlkñúgenaHmanr:tkugs‘ulxagBIr AB nig1kN / m enAcugxageqVg. snμt;fakMralebtug nig HG / nigr:tmuxeRkay. bøg;m:UEDlrbs;eRKag RtUv)anRTedayTMrsamBaØcenøaHr)arrnUt. sg; CamYynwgTMhMrbs;vaRtUv)anbgðajenAkñúgrUbxag düaRkamkMlaMgkat; nigdüaRkamm:Um:g;sMrab;r:t eRkam. eyagtamkUdtMbn; bnÞúkGefrEdlman AB . ecalTMgn;rbs;va. GMeBIelIy:KW 2kN / m 2 . KUrdüaRkamkMlaMgkat; nigdüaRkamm:Um:g;sMrab;r:txagmux BG nigr:txag AB . snμt;r:txagmuxCaEdkEdlmanragGkSr C EdlmanTMgn; 0.36kN / m ehIyr:txagmanmux kat;GkSr I EdlmanTMgn; 0.66kN / m . ecal TMgn;rbs;FñwmkMral nigbgáan;édxagmux. KMerag-2> sMyabEdleXIjenAkñúgrUbftpþl;Ca KMerag-3> bøg;eRKagm:EDlsMrab;RbB½n§kMralxNÐ møb;sMrab;pøÚvcUlGKar. BicarNaRKb;Ggát;TaMg EdlmanTItaMgenAkñúgrebogTTYlePJógénGKarkari Gs;CaTMrsamBaØ. r)arrnUtenARtg; C / D / E / yal½yRtUv)anbgðajkñúgrUb. kMralRtUv)anplit F manTMgn; 0.6kN ehIymanRbEvg 6m . dMbUl BIebtugBRgwgedayEdkEdlmankMras; 100mm . mankMras;100mm nigeFVIBIebtugTMgn;RsalEdl RbsinebICBa¢aMgrbs;CeNþIrynþeyageFVIBIbøúktan; cMeNaT T.Chhay -146
  39. 39. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ebtugTMgn;RsalkMras; 100mm nigmankMBs; 3m kMNt;m:Um:g;GtibrmaenAkñúgFñwm AB . ecal TMgn;rbs;Ggát;. Problems T.Chhay -147

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