• Share
  • Email
  • Embed
  • Like
  • Save
  • Private Content
15. beam analysis using the stiffness method
 

15. beam analysis using the stiffness method

on

  • 2,135 views

 

Statistics

Views

Total Views
2,135
Views on SlideShare
2,037
Embed Views
98

Actions

Likes
0
Downloads
0
Comments
0

1 Embed 98

http://civilnpic.wordpress.com 98

Accessibility

Categories

Upload Details

Uploaded via as Adobe PDF

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

    15. beam analysis using the stiffness method 15. beam analysis using the stiffness method Document Transcript

    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa !%> karviPaKFñwmedayeRbIviFIPaBrwgRkaj (Beam analysis using the stiffness method) emeronenHnwgbgðajBIeKalKMnitEdlerobrab;enAkñúgemeronmunedIm,IviPaKFñwm. eyIgeXIjfa enA eBlEdleKbegáItm:aRTIsPaBrwgRkajsRmab;Ggát; nigm:aRTIsbMElg dMeNIrkarsRmab;karGnuvtþmanlkçN³ RsedogKñanwgkarGnuvtþsRmab; truss. emeronenHnwgbgðajBIkrNIBiesssRmab;sRmutDIepr:g;Esül nigsItuNðPaB. !%>!> kMNt;cMNaMbzm (preliminary remarks) munnwgeyIgbgðajBIrebobEdlviFIPaBrwgRkajGnuvtþeTAelIFñwm dMbUgeyIgnwgerobrab;BIeKalKMnit bzm nigniymn½yEdlerobrab;eTAnwgGgát;TaMgenH. karkMNt;sMKal; node nigGgát;³ edIm,IGnuvtþviFIPaBrwgRkajeTAelIFñwm dMbUgeyIgRtUvEtkMNt;BIrebobbMEbk FñwmCabgÁúMFatuGnnþtUc (finite element). CaTUeTA FatunImYy²minRtUvTak;TgnwgbnÞúkeT ehIyRtUvmanmux kat;efr. sRmab;mUlehtuenH node rbs;FatuRtUvEtsßitenARtg;TMr b¤Rtg;cMNucEdlGgát;P¢ab;Kña Rtg;bnÞúk Gnuvtþ Rtg;RkLaépÞmuxkat;pøas;bþÚrPøam² b¤Rtg;cMNucEdleKRtUvkMNt;bMlas;TImMu. ]TahrN_ eKmanFñwm dUcbgðajenAkñúgrUbTI 15-1a. edayeRbIviFIdUcKñasRmab;karviPaK truss eKRtUvkMNt; node bYnCaelxeday dak;kñúgrgVg;. dUcKña cMNaMfa cugCit nigcugq¶ayrbs;Ggát;nImYy²RtUv)ankMNt;sMKal;edaysBaØaRBYjtam beNþayGgát;nImYy². kUGredaenskl nigkUGredaen³ eyIgnwgkMNt;RbB½n§kUGredaenskledayeRbIG½kS x, y, z EdleKalrbs; vasßitenARtg; node EdleFVIeGaykUGredaenrbs;cMNucdéTeTotenAelIFñwmviC¢man ¬rUbTI 15-1a¦. kUGr- Beam analysis using the stiffness method T.Chhay -501
    • Department of Civil Engineering NPIC edaentMbn; nigkUGredaensRmab;Ggát; x' , y' , z' maneKalrbs;vasßitenAcugCitrbs;Ggát;nImYy² G½kS viC¢man x' manTisedAeTArkcugq¶ay. rUbTI 15-1b bgðajkUGredaenTaMgenHsRmab;Ggát;elx @. enAkñúg krNITaMgBIr eyIg)aneRbIRbB½n§kUGredaenédsþaM dUcenHRbsinebIeyIgbgVilédsþaMBIG½kS x b¤ x' eTAG½kS y b¤ y ' eyIgnwgTTYl)anG½kS z b¤ z ' viC¢man EdlmanTisedAecjBIRkdasesovePA. cMNaMfa sRmab;Ggát;Fñwm nImYy²G½kS x nig x' nwgsßitenAelIG½kSEtmYy ehIykUGredaenskl nigkUGredaensRmab;Ggát;TaMgGs;suT§ EtRsbKña. dUcenH ¬mindUckrNIsRmab; truss¦ enATIenH eyIgnwgmincaM)ac;begáItm:aRTIsbMElgrvagRbB½n§ kUGredaenTaMgenH. PaBminkMNt;sIueNm:aTic³ eRkayeBlGgát; nig node RtUv)ankMNt; nigeRkayeBlRbB½n§kUGredaenskl RtUv)anbegáIt eKGackMNt; degree of freedom nigPaBminkMNt;sIueNm:aTicsRmab;Fñwm. RbsinebIeyIg KitTaMgT§iBlm:Um:g;Bt; nigkmøaMgkat; enaH node nImYy²enAelIFñwmGaclman degree of freedom BIreBal KW bMlas;TItamTisQr nigmMurgVil. dUcenAkñúgkrNIrbs; truss eKGackMNt;bMlas;TITaMgenHedayelxkUd. eyIgeRbIelxkUdtUcbMputedIm,IeFVIkMNt;sMKal;bMlas;TIEdlCaGBaØat ¬degree of freedom Edlminmankar Tb;¦ ehIyeyIgeRbIelxFMbMputedIm,IeFVIkMNt;sMKal;bMlas;TIEdleyIgsÁal; ¬degree of freedom Edlman karTb;¦. rMlwkfa mUlehtukñúgkareRCIserIsviFIénkareFVIkMNt;sMKal; eyIgRtUveFVIkarEbgEckm:aRTIsPaBrwg RkajsRmab;rcnasm½<n§eGay)anRsYl dUcenHeyIgGacKNnabM;las;TIEdlCaGBaØattamrebobedaypÞal; bMput. edIm,Ibgðaj]TahrN_kñúgkardak;elxkUd eyIgnwgBicarNaFñwmCab;mþgeTotenAkñúgrUbTI 15-1a. enA TIenH FñwmCarcnasm<½n§minkMNt;edaysIueNm:aTicdWeRkTIbYn. eKman degree of freedom cMnYnR)aMbI Edl elxkUd ! dl;elxkUd $ tMNageGaybMlas;TIEdlCaGBaØat ehIyelxkUd % dl;elxkUd * tMNageGay bMlas;TIEdleKsÁal; ¬EdlenAkñúgkrNIenHvasuT§EtmantémøesμIsUnü¦. ]TahrN_epSgeTot eKGacbMEbk FñwmenAkñúgrUb TI 15-2a CabIGgát; nigbYn node. cMNaMfa snøak;enARtg; node elx # dabdUcGgát;elx @ karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -502
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa nigelx # b:uEnþmMurgVilenARtg;cugGgát;nImYy²mantémøxusKña. enATIenH eKman degree of freedom cMnYn R)aMbYn EdlenAkñúgenaH degree of freedom R)aMCaGBaØat ¬dUcbgðajenAkñúgrUbTI 15-2b¦ ehIy degree of freedom cMnYnbYnRtUv)ansÁal; ¬vaTaMgGs;suT§EtesμIsUnü¦. cugeRkay BicarNakarrGil ¬slider mechanism¦ EdleRbIenAelIFñwmenAkñúgrUbTI 15-3a. enATIenH PaBdabrbs;FñwmRtUv)anbgðajenAkñúgrUbTI 15-3b dUc enHeKmanbgÁúMPaBdabEdlCaGBaØatcMnUnR)aMEdlRtUv)ankMNt;elxerogCamYynwgelxkUdtUcbMput. FñwmCa eRKOgbgÁúMminkMNt;edaysIeu Nm:aTicdWeRkTIR)aM. karbegáItviFIPaBrwgRkajsRmab;FñwmmanlkçN³RsedogeTAnwgdMeNIrkarkñúgkarbegáItPaBrwgRkaj EdleRbIsRmab; truss. dMbUg eyIgRtUvbegáItm:aRTIsPaBrwgRkajsRmab;Ggát;nImYy² ehIybnÞab;mkpÁúMm:aRTIs TaMgenHedIm,IbegáItm:aRTIsPaBrwgRkajsRmab;Fñwm b¤m:aRTIsPaBrwgRkajsRmab;rcnasm½<n§. edayeRbIsmIkar m:aRTIssRmab;rcnasm<½n§ eyIgGacbnþkarKNnaedIm,IkMNt;bM;las;TIEdlCaGBaØatenARtg; node ehIyBIkar kMNt;enH eyIgGackMNt;RbtikmμenAelIFñwm nigkmøaMgkat; nigm:Um:g;enARtg; node. !%>@> m:aRTIsPaBrwgRkajsMrab;Ggát;Fñwm (Beam-member stiffness matrix) enAkñúgkfaxNÐenH eyIgnwgbegáItm:aRTIsPaBrwgRkajsRmab;FñwmEdlmanRkLaépÞmuxkat;efr nig manRbB½n§kUGredaentMbn; x' , y' , z' ¬rUbTI 15-4¦. eKRtUvdak;eKalrbs;kUGredaenenARtg;cugCit N ehIyG½kSviC¢manrbs; x' manTisedAeTArkcugq¶ay F . eKmankmøaMgRbtikmμBIrenARtg;cugnImYy²rbs;Ggát; EdlmankmøaMgkat; q N nig q F nigm:Um:g;Bt; q N nig q F . bnÞúkTaMgenHeFVIGMeBItamTisedAkUGredaenviC¢- z' y' z' z' man. m:Um:g; q N nig q F viC¢manvilRcasTisRTnicnaLika tamviFanédsþaMviucT½rm:Um:g;manTistambeNþay z' z' G½kSviC¢man z' EdlmanTisecjBIRkdasesovePA. Beam analysis using the stiffness method T.Chhay -503
    • Department of Civil Engineering NPIC bMlas;TIlIenEG‘r nigbMlas;TImMuEdlpSMCamYynwgbnÞúkTaMgenHk¾GnuvtþtamkarkMNt;sBaØaviC¢mandUc Kña. enATIenH eyIgnwgdak;bMlas;TITaMgenHdac;edayELkBIKña ehIybnÞab;mkkMNt;bnÞúkEdlmanGMeBIenAelI Ggát;EdlbNþalBIbMlas;TInImYy². bMlas;TI y' ³ enAeBleKeFVIeGaymanbMlas;TIviC¢man d N xN³eBlEdleKGackarBarbMlas;TIEdlGac y' ekItman eKGacbegáItkmøaMgkat; nigm:Um:g;Bt;RtUv)anbgðajenAkñúgrUbTI 15-5a. m:Um:g;RtUv)anbegáItenAkñúg kfaxNÐ 11-2 dUcsmIkar 11-5. dUcKña enAeBleKeFVIeGayman d F eyIgTTYl)ankmøaMgkat; nigm:Um:g;dUc y' bgðajenAkñúgrUbTI 15-5b. mMurgVil z' ³ RbsinebIeKeFVIeGaymanmMurgVil d N xN³bMlas;TIEdlGacekItmanTaMgGs;RtUv)ankarBar z' kmøaMgkat; nigm:Um:g;EdlcaM)ac;edIm,IeFVIeGaymankMhUcRTg;RTayRtUv)anbgðajenAkñúgrUbTI 15-6a. m:Um:g; RtUv)anbegáItenAkñúgkfaxNÐ 11-2 dUcsmIkar 11-1 nig 11-2. dUcKña enAeBleKeFVIeGayekItman d F bnÞúk z' pÁÜbRtUv)anbgðajenAkñúgrUbTI 15-6b. karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -504
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa edayviFItRmYtpl RbsinebIeKbUklT§plenAkñúgrUbTI 15-5 nig 15-6 bBa©ÚlKña eyIgGacsresr TMnak;TMngrvagbnÞúk nigbMlas;TITaMgbYnsRmab;Ggát;enAkñúgTRmg;m:aRTIsCa N y' N z' F y' Fz' ⎡ 12 EI 6 EI 12 EI 6 EI ⎤ ⎢ 3 2 − 3 ⎥ ⎡q N y ' ⎤ ⎢ 6L L L L2 ⎥ d ⎢ q ⎥ ⎢ EI 4 EI 6 EI 2 EI ⎥ ⎡ N y ' ⎤ − 2 ⎢ ⎥ ⎢ N z ' ⎥ = ⎢ L2 L L L ⎥⎢d Nz' ⎥ (15-1) ⎢ q Fy ' ⎥ ⎢ 12 EI 6 EI 12 EI 6 EI ⎥ ⎢ d ⎥ ⎢ ⎥ ⎢− 3 − 2 − 2 ⎥ ⎢ Fy ' ⎥ ⎣ q Fz ' ⎥ ⎢ L ⎢ ⎦ L L3 L ⎥⎢ d F ⎥ ⎢ 6 EI 2 EI 6 EI 4 EI ⎥ ⎣ z ' ⎦ − 2 ⎢ L2 ⎣ L L L ⎥ ⎦ eyIgk¾GacsresrsmIkarTaMgenHkñúgTRmg;kat;dUcxageRkam q = kd (15-2) m:aRTIssIuemRTI k enAkñúgsmIkar15-1 Cam:aRTIsPaBrwgRkajsRmab;Ggát;. emKuNT§iBl kij TaMg 16 EdlpSMCam:aRTIs k )anKitbBa©ÚlbMlas;TIEdlbNþalBIkmøaMgkat; nigm:Um:g;Bt;rbs;Ggát;. tamrUbviTüa emKuNTaMgenHCabnÞúkenAelIGgát; enAeBlGgát;rgbM;las;TIÉktþa. ]TahrN_ RbsinebI d N = 1 ¬rUbTI 15 y' -5a¦ xN³EdlbMlas;TIdéTeTotesμIsUnü Ggát;nwgrgEtbnÞúkbYnEdlbgðajenAkñúgCYrQrTImYyrbs;m:aRTIs k . tamrebobdUcKña CYrQrdéTeTotrbs;m:aRTIs k CabnÞúkenAelIGgát;sRmab;bMlas;TIÉktþaEdlkMNt; edayelxkUd degree of freedom RtUv)anbgðajBIxagelICYrQr. BIkarbegáItenH eKRtUvbMeBjlkçxNÐ lMnwg niglkçxNÐRtUvKña. dUcKña eyIgRtUvcMNaMfam:aRTIsenHdUcKñasRmab;kUGredaenskl nigkUGredaentMbn; edaysarG½kS x' , y' , z' RsbnwgG½kS x, y, z dUcenHeKminRtUvkarm:aRTIsbMElgrvagkUGredaenTaMgBIreT. !%>#> m:aRTIsPaBrwgRkajsMrab;eRKOgbgÁúMFñwm (Beam-structure stiffness matrix) eRkayeBlrkma:RTIsPaBrwgRkajsRmab;Ggát; eyIgRtUvpÁúMvaeTAkñúgm:aRTIsPaBrwgRkajsRmab; eRKOgbgÁúM K . dMeNIrkarenHGaRs½ynwgkarsÁal;TItaMgrbs;FatunImYy²enAkñúgm:aRTIsPaBrwgRkajsRmab; Ggát;CadMbUg. enATIenH CYredk nigCYrQrrbs;m:aRTIs k nImYy² ¬smIkar 15-1¦ RtUv)ankMNt;edayelx kUdBIrRtg;cugCitrbs;Ggát; ¬ N y' , N z' ¦ EdlbnþedayelxkUdBIrRtg;cugq¶ay ¬ Fy' , Fz' ¦. dUcenH enA eBlpÁúMm:aRTIs eKRtUvdak;FatunmYy²enATItaMgdUcKñarbs;m:aRTIs K . tamviFIenH K RtUvmanlMdan;esμInwgelx I kUdFMbMputEdlRtUv)ankMNt;eTAelIFñwm edaysarvaCaelxsrubén degree of freedom. dUcKña enATItaMg EdlGgát;CaeRcInP¢ab;Kñanwg node, emKuNT§iBlPaBrwgRkajsRmab;Ggát; (member stiffness influence Beam analysis using the stiffness method T.Chhay -505
    • Department of Civil Engineering NPIC coefficients) nwgmanTItaMgdUcKñaenAkñúgm:aRTIs K dUcenHvaRtUv)anbUkbBa©ÚlKñatamlkçN³nBVnþedIm,IkMNt; emKuNT§iBlPaBrwgRkajén node sRmab;rcnasm<½n§. eKcaM)ac;RtUveFVIEbbenH edaysaremKuNnImYy²Ca ersIusþg;rbs; node énrcnasm<½n§enAkñúgTisedA ¬ y' b¤ z' ¦ enAeBlekItmanbMlas;TIÉktþa ¬ y' b¤ z' ¦ enA Rtg; node dEdl b¤Rtg; node déT. ]TahrN_ K 23 CabnÞúktamTis nigenARtg;TItaMgénelxkUd@ enAeBl bMlas;TIÉktþaekItmantamTis nigenARtg;TItaMgénelxkUd #. !%>$> karGnuvtþénviFIPaBrwgRkajsMrab;karviPaKFñwm (Application of the stiffness method for bema analysis) eRkayeBleKkMNt;m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§rYcehIy eKGacP¢ab;TMnak;TMngrvagbnÞúk enARtg; node rbs;FñwmeTAnwgbMlas;TIedayeRbIsmIkarPaBrwgRkajsRmab;rcnasm<½n§ Q = KD enATIenH Q nig D Cam:aRTIsCYrQrEdltMNageGaybnÞúk nigbMlas;TIEdlCaGBaØat nigEdlsÁal;. karbMEbkm:aRTIsPaBrwgRkajeGayeTACaFatuénbnÞúk nigbMlas;TIEdlsÁal; nigGBaØat eyIgTTYl)an ⎡Qk ⎤ ⎡ K11 K12 ⎤ ⎡ Du ⎤ ⎢Q ⎥ = ⎢ K ⎥⎢ ⎥ ⎣ u ⎦ ⎣ 21 K 22 ⎦ ⎣ Dk ⎦ enAeBleyIgBnøatm:aRTIsxagelI eyIgnwgTTYl)ansmIkarBIrKW Qk = K11 Du + K12 Dk (15-3) Qu = K 21 Du + K 22 Dk (15-4) eKGackMNt;bMlas;TIEdlCaGBaØatBIsmIkarTImYy. edayeRbIsmIkarTaMgenH eyIgGacKNnakmøaMgRbtikmμ TMr Qu BIsmIkarTIBIr. bnÞúkEdlenAkNþal³ sRmab;karGnuvtþ eKcaM)ac;eGayFaturbs;FñwmminrgbnÞúktambeNþayFñwm edaysar m:aRTIsPaBrwgRkajsRmab;Ggát;RtUv)anbegáItsRmab;bnÞúkEdlGnuvtþEtenAxagcugrbs;va ¬emIlrUbTI 15- 4¦. b:uEnþ eBlxøHFñwmRtUvRTbnÞúkBRgay dUcenHeKRtUvkarEktRmUvlkçxNÐenHedIm,IGnuvtþkarviPaKeday m:aRTIs. edIm,IedaHRsaykrNIenH eyIgnwgeRbIeKalkarN_tRmYtpltamrebobRsedogKñaeTAnwgGIVEdl)an eRbIsRmab; truss Edl)anerobrab;enAkñúgkfaxNÐ 14-8. edIm,IbgðajBIkarGnuvtþrbs;va BicarNaFaturbs; FñwmEdlmanRbEvg L dUcbgðajenAkñúgrUbTI 15-7a EdlrgbnÞúkBRgayesμI w . dMbUg eyIgnwgGnuvtþkmøaMg karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -506
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa bgáb;cug nigm:Um:g;bgáb;cugeTAelIGgát; EdlvaRtUv)aneKeRbIenAkñúgviFIPaBrwgRkaj ¬rUbTI 15-7b¦. eyIgnwg sresrbnÞúkTaMgenHCam:aRTIsCYrQr q0 . bnÞab;mkeTot eyIgGnuvtþbnÞúkBRgayesμI nigkmøaMgRbtikmμ q0 rbs;va ¬rUbTI 15-7c¦. eKkMNt;bnÞúkCak;EsþgenAkñúgFñwmedayeFVIplbUkeTAelIlT§plTaMgBIr. kmøaMg Rbtikmμbgáb;cugsRmab;krNIbnÞúkdéTeTotRtUv)aneGayenAkñúgemeronTI11. bEnßmelIkaredaHRsaycMeNaT EdlBak;B½n§nwgbnÞúkxagdUckrNIenH eyIgk¾GaceRbIviFIenHedIm,IedaHRsaycMeNaTEdlBak;B½n§nwgbERmbRmYl sItuNðPaB b¤kMhuskñúgplitkmμ. kmøaMgkñúgGgát;³ eKGackMNt;kmøaMgkat; nigm:Um:g;Bt;enARtg;cugrbs;Ggát;FñwmnImYy²edayeRbIsmIkar 15-2 ehIybUkbEnßmkmøaMgRbtikmμbgáb;cug q0 RbsinebIGgát;rgbnÞúkenAkNþal. eyIgman q = kd + q0 (15-5) RbsinebIlT§plGviC¢man vabgðajfabnÞúkeFVIGMeBIpÞúynwgTisedAEdl)anbgðajenAkñúgrUbTI 15-4. dMeNIrkarkñúgkarviPaK (Procedure for analysis) viFIxageRkampþl;nUvmeFüa)ayedIm,IkMNt;bMlas;TI RbtikmμTMr kmøaMgkñúgrbs;Ggát; b¤FatuGnnþtUc rbs;FñwmkMNt;edaysþaTic b¤FñwmminkMNt;edaysrsþaTic. kareFVIkMNt;sMKal;³ bMEbkFñwmCaFatuGnnþtUc ehIykMNt;elxerogeGayFatunImYy² nig node nImYy²rbs;va. eRbIelx Beam analysis using the stiffness method T.Chhay -507
    • Department of Civil Engineering NPIC EdlsresrkñúgrgVg;sRmab; node nigelxkñúgkaersRmab;Ggát; b¤Fatu. CaTUeTAGgát; b¤FatusßitenA cenøaHTMr b¤enAcenøaHbnÞúkcMcMNuc b¤enAcenøaHtMN b¤enAcenøaHcMNucEdleKRtUvkarkMNt;kmøaMgkñúg b¤bMlas;TI. kMNt;cugCit nigcugq¶ayrbs;Ggát;edaysBaØaRBYjEdlsßitenAelIGgát; ehIymanTisedAeTArkcug q¶ay. enARtg; node nImYy² kMNt;elxkUdtamTis y nigtamTis z . enARKb;krNITaMgGs;eRbIelxkUdtUc bMputedIm,IsMKal; degree of freedom EdlminmankarTb; ehIybnþedayelxbnþbnÞab; b¤elxEdl FMbMputedIm,IsMKal; degree of freedom EdlmankarTb;. begáItm:aRTIsbMlas;TIEdlsÁal; Dk nigm:aRTIsbnÞúkEdlsÁal; Qk . KitbBa©Úlm:aRTiscRmas;én bnÞúkbgáb;cug RbsinebIGgát;rgbnÞúk. m:aRTIsPaBrwgRkajsRmab;rcnasm½<n§³ GnuvtþsmIkar 15-1 edIm,IkMNt;m:aRTIsPaBrwgRkajsRmab;FatunImYy²edayeRbIkUGredaenskl. eRkayeBlkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát; ehIyCYredk nigCYrQrRtUv)ankMNt;CamYynwg elxkUdsmRsb pÁúMm:aRTIsedIm,ITTYl)anm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§ K . sRmab;kar epÞógpÞat;edayEpñk m:aRTIsPaBrwgRkajsRmab;Ggát; nigm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§RtUv EtCam:aRTIssIuemRTI. bMlas;TI nigbnÞúk³ bMEbksmIkarPaBrwgRkajsRmab;rcnasm<½n§CaRkum ehIyeFVIRbmaNviFIKuNm:aRTIsedIm,IkMNt; bMlas;TIEdlCaGBaØat Du nigkmøaMgRbtikmμTMr Qu . eKGackMNt;kmøaMgkat; nigm:Um:g; q enARtg;cugrbs;Ggát;FñwmnImYy²BIsmIkar 15-5 edayKitbBa©Úl TaMgbnÞúkbgáb;cug. ]TahrN_ 15-1³ kMNt;kmøaMgRbtikmμenARtg;TMrrbs;FñwmEdlbgðajenAkñúgrUbTI 15-8a. EI CacMnYnefr. karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -508
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ kareFVIkMNt;smÁal;³ FñwmmanFatuBIr nig node bI EdlRtUv)ankMNt;dUcbgðajenAkñúgrUbTI 15-8b. eKeRbI elxkUd 1 dl;elxkUd$ sRmab;kMNt; degree of freedom EdlminrgkarTb;. m:aRTIsbnÞúkEdlsÁal; nigbMlas;TIEdlsÁal;KW ⎡ 0 ⎤1 ⎢ − 5⎥ 2 ⎡0 ⎤ 5 Qk = ⎢ ⎥ Dk = ⎢ ⎥ ⎢ 0 ⎥3 ⎣0 ⎦ 6 ⎢ ⎥ ⎣ 0 ⎦4 m:aRTIsPaBrwgRkajsRmab;Ggát;³ eKkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;TaMgBIrBIsmIkar 15-1. eyIg RtUvcMNaMBIrebobbegáItelxkUdsRmab;CYredk nigCYrQr. 6 4 5 3 5 3 2 1 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 6 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 5 ⎢ − 1.5 1 ⎥ 4 ⎢ − 1.5 1 ⎥ 3 k1 = EI ⎢ 1.5 2 ⎥ k 2 = EI ⎢ 1.5 2 ⎥ ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 5 ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 2 ⎢ ⎥ ⎢ ⎥ ⎣ 1.5 1 − 1.5 2 ⎦ 3 ⎣ 1.5 1 − 1.5 2 ⎦ 1 bMlas;TI nigbnÞúk³ eyIgGacpÁúMFatuTaMgenHeTAkñúgm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§. ]TahrN_ Fatu K11 = 0 + 2 = 2, K 55 = 1.5 + 1.5 = 3 .l. dUcenH Q = KD 1 2 3 4 5 6 ⎡0⎤ ⎡ 2 − 1.5 1 0 1.5 0 ⎤ ⎡ D1 ⎤ ⎢ − 5⎥ ⎢− 1.5 1.5 − 1.5 0 0 ⎥ ⎢ D2 ⎥ − 1.5 ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ 0 ⎥ = EI ⎢ 1 − 1.5 4 1 0 1.5 ⎥ ⎢ D3 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢0⎥ ⎢ 0 0 1 2 − 1.5 1.5 ⎥ ⎢ D4 ⎥ ⎢Q5 ⎥ ⎢ 1.5 − 1.5 0 − 1.5 3 − 1.5⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q6 ⎥ ⎣ ⎦ ⎢ 0 ⎣ 0 1.5 1.5 − 1.5 1.5 ⎥ ⎢ 0 ⎥ ⎦⎣ ⎦ m:aRTIsRtUv)anbMEbkCaRkumdUcbgðaj. edayedaHRsayplKuNsRmab;CYredkbYndMbUg eyIgTTYl)an Beam analysis using the stiffness method T.Chhay -509
    • Department of Civil Engineering NPIC 0 = 2 D1 − 1.5D2 + D3 + 0 5 − = −1.5D1 + 1.5D2 − 1.5 D3 + 0 EI 0 = D1 − 1.5 D2 + 4 D3 + D4 0 = 0 + 0 + D3 + 2 D2 edayedaHRsayRbB½n§smIkarxagelI eyIgTTYl)an 16.67 D1 = − EI 26.67 D2 = − EI 6.67 D3 = − EI 3.33 D4 = EI edayeRbIlT§plTaMgenH ehIyedayKuNCYrQrTaMgBIrxageRkay eyIg)an ⎛ 16.67 ⎞ ⎛ 26.67 ⎞ ⎛ 3.33 ⎞ Q5 = 1.5 EI ⎜ − ⎟ − 1.5 EI ⎜ − ⎟ + 0 − 1.5 EI ⎜ ⎟ ⎝ EI ⎠ ⎝ EI ⎠ ⎝ EI ⎠ = 10kN ⎛ 6.67 ⎞ ⎛ 3.33 ⎞ Q6 = 0 + 0 + 1.5EI ⎜ − ⎟ + 1.5 EI ⎜ ⎟ ⎝ EI ⎠ ⎝ EI ⎠ = −5kN ]TahrN_ 15-2³ kMNt;kmøaMgkat; nigm:Um:g;enAkñúgGgát;elx ! rbs;FñwmEdlbgðajenAkñúgrUbTI 15-9a. EI CacMnYnefr. karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -510
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ kareFVIkMNt;smÁal;³ enAkñúgkrNIenH Fñwmman degree of freedom EdlCaGBaØatcMnUnEtBIrb:ueNÑaHEdlman bgðajelxkUd ! nig @ ¬rUbTI 15-9b¦. cMNaMfa bnÞúk M 0 manTMhMGviC¢man. m:aRTIsbnÞúk nigm:aRTIs bMlas;TIEdlCasmμtikmμKW ⎡0 ⎤ 3 ⎢0 ⎥ 4 ⎡ 0 ⎤1 Qk = ⎢ ⎥ Dk = ⎢ ⎥ ⎣− M 0 ⎦ 2 ⎢0 ⎥ 5 ⎢ ⎥ ⎣0 ⎦ 6 m:aRTIsPaBrwgRkajsRmab;Ggát;³ edayGnuvtþsmIkar 15-1 eTAelIGgát;nImYy² edayeyageTAelIelxkUd EdlbgðajenAkñúgrUbTI 15-9b eyIg)an 5 6 4 1 4 1 3 2 ⎡ 12 6 12 6 ⎤5 ⎡ 12 6 12 6 ⎤4 ⎢ 3 2 − 3 ⎥ ⎢ 3 − 3 ⎥ ⎢ L L L L2 ⎥ ⎢ L L2 L L2 ⎥ ⎢ 6 4 − 2 6 2 ⎥ 6 ⎢ 6 4 − 2 6 2 ⎥ 1 k1 = EI ⎢ L2 L L L ⎥ k 2 = EI ⎢ L2 L L L ⎥ ⎢ 12 6 12 6⎥ ⎢ 12 6 12 6⎥ ⎢− 3 − 2 − 2 ⎥4 ⎢− 3 − 2 − 2 ⎥3 ⎢ L L L3 L ⎥ ⎢ L L L3 L ⎥ ⎢ 6 2 − 2 6 4 ⎥ ⎢ 6 2 − 2 6 4 ⎥ ⎢ L2 ⎣ L L L ⎥1 ⎦ ⎢ L2 ⎣ L L L ⎥2 ⎦ bnÞúk nigbMlas;TI³ eKbegáItm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§edaypÁúMm:aRTIsPaBrwgRkajsRmab;Ggát; nImYy². edayGnuvtþsmIkarm:aRTIssRmab;rcnasm<½n§ eyIg)an Q = KD 1 2 3 4 5 6 ⎡ 8 2 −6 6 2 ⎤ ⎡ D1 ⎤ 1 ⎢ L 0 L L2 L2 L ⎥⎢ ⎥ ⎢ 2 4 6 6 ⎥⎢ ⎥ 1⎡ 0 ⎤ ⎢ − 2 0 0 ⎥ ⎢ D2 ⎥ 2 ⎢ L L2 ⎥⎢ ⎥ 2 ⎢− M 0 ⎥ ⎢ 6 L L ⎢ ⎥ 6 12 12 ⎥⎢ ⎥ 3 ⎢ Q3 ⎥ = ⎢− L2 − L2 L3 − L3 0 0 ⎥⎢ 0 ⎥ 3 ⎢ ⎥ ⎢ ⎥ 6 ⎥⎢ ⎥ 4 ⎢ Q4 ⎥ ⎢ 0 6 12 24 − 3 12 − 3 − 2 ⎢ 0 ⎥4 5 ⎢ Q5 ⎥ ⎢ L2 L L3 L L ⎥⎢ ⎥ ⎢ ⎥ ⎢ 6 12 12 6 ⎥⎢ ⎥ 6 ⎢ Q6 ⎥ ⎢ 2 ⎣ ⎦ 0 0 − 3 ⎥⎢ 0 ⎥ 5 ⎢ L L L3 L2 ⎥ ⎢ ⎥ ⎢ 2 0 0 − 2 6 6 4 ⎥⎢ ⎥ ⎢ L ⎣ L L2 L ⎥⎢ 0 ⎥ 6 ⎦⎣ ⎦ eFVIplKuNeTAelICYredkdMbUgTaMgBIredIm,IkMNt;bMlas;TI eyIg)an Beam analysis using the stiffness method T.Chhay -511
    • Department of Civil Engineering NPIC 8EI 2 EI 0= D1 + D2 L L 2 EI 4 EI − M0 = D1 + D2 L L dUenH M L D1 = 0 14 EI 2M 0 L D2 = − 7 EI dUcenAkñúg]TahrN_elIkmun eyIgTTYl)ankmøaMgRbtikmμBIkareFVIplKuNénCYredkEdlenAsl;. ]TahrN_ kmøaMgRbtikmμenAxagsþaMgédKW 6 EI ⎛ M 0 L ⎞ 6 EI ⎛ 2M 0 L ⎞ 9M 0 Q3 = − ⎜ ⎟− ⎜− ⎟= L2 ⎝ 14 EI ⎠ L2 ⎝ 7 EI ⎠ 7L eKkMNt;kmøaMgkñúgenARtg; node ! nig @ BIsmIkar 15-2. eyIg)an q = k1d 5 6 4 1 ⎡ q5 ⎤ ⎡ 12 6 12 6 ⎤⎡ 0 ⎤ 5 ⎢ ⎥ ⎢ 3 − 3 ⎥⎢ ⎥ ⎢ ⎥ ⎢ L L2 L L2 ⎥ ⎢ ⎥ ⎢q6 ⎥ ⎢ 6 4 6 2 ⎥⎢ 0 ⎥ − 2 6 ⎢ ⎥ = EI ⎢ L2 L L L ⎥⎢ ⎥ ⎢ ⎥ ⎢ 12 6 12 6 ⎥⎢ ⎥ ⎢q 4 ⎥ ⎢ − 3 − 2 − 2 ⎥⎢ 0 ⎥ 4 ⎢ ⎥ ⎢ L L L3 L ⎥⎢ ⎥ ⎢ ⎥ ⎢ 6 2 6 4 ⎥⎢ M L ⎥ − 2 L ⎥ ⎢14 EI ⎥ 1 0 ⎢ ⎥ ⎢ L2 ⎣ ⎦⎣ ⎣ q1 ⎦ L L ⎦ 6 EI ⎛ M L ⎞ 3M 0 q5 = 2 ⎜ 0 ⎟ = L ⎝ 14 EI ⎠ 7L 2 EI ⎛ M 0 L ⎞ M 0 q6 = ⎜ ⎟= L ⎝ 14 EI ⎠ 7 6 EI ⎛ M 0 L ⎞ 3M 0 q4 = − ⎜ ⎟=− L ⎝ 2 14 EI ⎠ 7L 4 EI ⎛ M 0 L ⎞ 2 M 0 q1 = ⎜ ⎟= L ⎝ 14 EI ⎠ L lT§plTaMgenHRtUv)anbgðajenAkñúgrUbTI 15-9c. ]TahrN_ 15-3³ FñwmenAkñúgrubTI 15-10a rgm:Um:g;BIr. RbsinebITMrkNþal ②Rsut 1.5mm cUgkMNt; kmøaMgRbtikmμTMrenARtg;TMr. yk E = 200GPa nig I = 22(10 −6 )m 4 . karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -512
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ kareFVIkMNt;smÁal;³ FñwmmanFatuBIr nig degree of freedom EdlCaGBaØatcMnYnbI. BYkvaRtUv)aneKkMNt; edayelxkUdtUcCageK ¬rUbTI 15-10b¦. enATIenH m:aRTIsbnÞúk nigm:aRTIsbMlas;TIEdlCasmμtikmμKW³ ⎡4 ⎤ 1 ⎡0 ⎤4 Qk = ⎢0 ⎥ 2 ⎢ ⎥ ⎢− 0.0015⎥ 5 Dk = ⎢ ⎥ ⎢ − 4⎥ 3 ⎣ ⎦ ⎢0 ⎣ ⎥6 ⎦ m:aRTIsPaBrwgRkajsRmab;Ggát;³ eKkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;edayeRbIsmIkar 15-1 eday eyageTAtamelxkUd nigTisrbs;Ggát;EdlbgðajenAkñúgrUbTI 15-10b. eyIg)an 6 3 5 2 5 2 4 1 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 6 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 5 ⎢ − 1.5 1 ⎥ 3 ⎢ − 1.5 1 ⎥ 2 k1 = EI ⎢ 1.5 2 ⎥ k 2 = EI ⎢ 1.5 2 ⎥ ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 5 ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 4 ⎢ ⎥ ⎢ ⎥ ⎣ 1.5 1 − 1.5 2 ⎦ 2 ⎣ 1.5 1 − 1.5 2 ⎦ 1 bMlas;TI nigbnÞúk³ karpÁúMm:aRTIsPaBrwgRkajsRmab;rcnasm<n§ nigkarsresrsmIkarPaBrwgRkajsRmab; ½ rcnasm<½n§ eyIg)an 1 2 3 4 5 6 ⎡ 4⎤ ⎡ 2 1 0 − 1.5 1.5 0 ⎤ ⎡ D1 ⎤ ⎢0⎥ ⎢ 1 4 1 − 1.5 0 1.5 ⎥ ⎢ D2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢− 4⎥ = EI ⎢ 0 1 2 0 − 1.5 1.5 ⎥ ⎢ D3 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q4 ⎥ ⎢− 1.5 − 1.5 0 1.5 − 1.5 0 ⎥ ⎢ 0 ⎥ ⎢Q5 ⎥ ⎢ 1.5 0 − 1.5 − 1.5 3 − 1.5⎥ ⎢− 0.0015⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎦ ⎣Q6 ⎥ ⎢ 0 ⎣ 1.5 1.5 0 − 1.5 1.5 ⎥ ⎢ 0 ⎦⎣ ⎥ ⎦ edaHRsayrkbMlas;TIEdlCaGBaØat = 2 D1 + D2 + 0 D3 − 1.5(0) + 1.5(− 0.0015) + 0 4 EI 0 = 1D1 + 4 D2 + 1D3 − 1.5(0 ) + 0 + 0 Beam analysis using the stiffness method T.Chhay -513
    • Department of Civil Engineering NPIC −4 = 0 D1 + 1D2 + 2 D3 + 0 − 1.5(− 0.0015) + 0 EI edayCMnYs EI = 200(10 6 )(22)(10 −6 ) nigedaHRsay eyIg)an D1 = 0.001580rad D2 = 0 D3 = −0.001580rad edayeRbIlT§plTaMgenH dUcenHkmøaMgRbtikmμTMrKW Q4 = 200( 6 )22( −6 )[− 1.5(0.001580 ) − 1.5(0 ) + 0 + 1.5(0 ) − 1.5(− 0.0015) + 0] = −0.525kN 10 10 Q5 = 200(10 6 )22(10 −6 )[1.5(0.001580 ) + 0 − 1.5(− 0.001580 ) − 1.5(0 ) + 3(− 0.0015) − 1.5(0 )] = 1.05kN Q6 = 200(10 6 )22( −6 )[0 + 1.5(0 ) + 1.5(− 0.001580 ) + 0 − 1.5(− 0.0015) + 1.5(0 )] = −0.525kN 10 ]TahrN_ 15-4³ kMNt;m:Um:g;EdlekItmanenARtg;TMr A rbs;FñwmEdlbgðajenAkñúgrUbTI 15-11a. yk E = 200GPa nig I = 216(10 6 )mm 4 . dMeNaHRsay³ kareFVIkMNt;smÁal;³ enATIenH Fñwmman degree of freedom EdlCaGBaØatcMnYnBI EdlkMNt;elxerogeday elxkUd ! nig @. karviPaKedaym:aRTIsTamTareGaybnÞúkGnuvtþenARtg; node dUcenHeKRtUvCMnYsbnÞúkBRgay nig bnÞúkcMcMNucedaym:Um:g;bgáb;cugsmmUlrbs;vaEdlRtUv)ankMNt;BItaragenAkñúgemeron !! ¬emIl]TahrN_ 11-2¦. cMNaMfa minmanbnÞúkxageRkAmanGMeBIenARtg; ① eT edaysarkmøaMgRbtikmμenARtg;elxkUd # karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -514
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa nig $ CaGBaØatenAkñúgm:aRTIsbnÞúk. edayeRbIviFItRmYtpl lT§plénkarviPaKm:aRTIssRmab;bnÞúkenAkñúgrUb TI 15-11b RtUv)anEktRmUvenAeBleRkayedaybnÞúkEdlbgðajenAkñúgrUbTI 15-11c. BIrUbTI 15-11b m:aRTIsbMlas;TI nigm:aRTIsbnÞúksmμtikmμKW ⎡0 ⎤ 4 ⎡12 ⎤ 1 Dk = ⎢0 ⎥ 5 ⎢ ⎥ Qk = ⎢ ⎥ ⎢0 ⎥ 6 ⎣84⎦ 2 ⎣ ⎦ m:aRTIsPaBrwgRkajsRmab;Ggát;³ eKkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;BIsmIkar 15-1. Ggát;elx !³ 12EI = 12(2003)(216) = 2400 3 L 6 6 EI 6(200 )(216) = = 7200 L2 62 4 EI 4(200 )(216 ) = = 28800 L 6 2 EI 2(200 )(216) = = 14400 L 6 4 3 5 2 ⎡ 2400 7200 − 2400 7200 ⎤ 4 ⎢ ⎥ k1 = ⎢ 7200 28800 − 7200 14400 ⎥ 3 ⎢− 2400 − 7200 2400 − 7200⎥ 5 ⎢ ⎥ ⎣ 7200 14400 − 7200 28800 ⎦ 2 12 EI 12(200 )(216 ) Ggát;elx @³ = = 64800 L3 23 6 EI 6(200 )(216) = = 64800 L2 22 4 EI 4(200 )(216) = = 86400 L 2 2 EI 2(200 )(216 ) = = 43200 L 2 5 2 6 1 ⎡ 64800 64800 − 64800 64800 ⎤ 5 ⎢ ⎥ k 2 = ⎢ 64800 86400 − 64800 43200 ⎥ 2 ⎢− 64800 − 64800 64800 − 64800⎥ 6 ⎢ ⎥ ⎣ 64800 43200 − 64800 86400 ⎦ 1 bMlas;TI nigbnÞúk³ eyIgtMrUveGay Q = KD Beam analysis using the stiffness method T.Chhay -515
    • Department of Civil Engineering NPIC 1 2 3 4 5 6 ⎡ 12 ⎤ ⎡ 86400 43200 0 0 64800 − 64800⎤ ⎡ D1 ⎤ ⎢ 84 ⎥ ⎢ 43200 115200 14400 7200 57600 − 64800⎥ ⎢ D ⎥ ⎢ ⎥ ⎢ ⎥⎢ 2 ⎥ ⎢Q3 ⎥ = ⎢ 0 14400 28800 7200 − 7200 0 ⎥⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q4 ⎥ ⎢ 0 7200 7200 2400 − 2400 − 64800⎥ ⎢ 0 ⎥ ⎢Q5 ⎥ ⎢ 64800 57600 − 7200 − 2400 67200 − 64800⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q6 ⎦ ⎢− 64800 − 64800 ⎣ ⎥ ⎣ 0 0 − 64800 64800 ⎥ ⎢ 0 ⎥ ⎦⎣ ⎦ edayedaHRsaytamrebobFmμtar 12 = 86400 D1 + 43200 D2 84 = 43200 D1 + 115200 D2 ( ) D1 = −0.2778 10 −3 m D2 = 0.8333( 10 )m −3 dUcenH Q3 = 0 + 14400(0.8333)(10 −3 )m = 12kN .m m:Um:g;Cak;EsþgenARtg; A RtUvEtrYmbBa©ÚlkmøaMgRbtikmμTMrbgáb; + 96kN.m EdlbgðajenAkñúgrUbTI 15-11c CamYynwglT§plEdl)anKNnasRmab; Q3 . dUcenH M AB = 12kN .,+96kN .m = 108kN .m lT§plenHdUcKñanwglT§plEdlKNnaenAkñúg]TahrN_ 11-2. eTaHbICaeKminRtUvkarKNnam:Um:g; nigkmøaMgkat;enARtg;cMNuc B enATIenHk¾eday EteKGacKitGac KNnava)an ]TahrN_ Ggát;elx !/ node @ ¬rUbTI 15-11b¦. lT§plTamTarkarBnøat q1 = k1d + (q0 )1 4 3 5 2 ⎡ q 4 ⎤ ⎡ 2400 7200 − 2400 7200 ⎤ ⎡ 0 ⎤ ⎡ 6 ⎤ ⎢ q ⎥ ⎢− 7200 28800 − 7200 14400 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 3⎥ = ⎢ ⎥⎢ ⎢ q5 ⎥ ⎢− 2400 − 7200 67200 57600 ⎥ ⎢ 0 ⎥ ( ) ⎥ 10 −3 + ⎢ 96 ⎥ ⎢ 6 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣q 2 ⎦ ⎣ 7200 14400 57600 115200⎦ ⎣0.833⎦ ⎣− 96⎦ ]TahrN_ 15-5³ kMNt;PaBdabenARtg;① nigkmøaMgRbtikmμTMrenAelIFñwmEdlbgðajenAkñúgrUbTI 15-12a. EI CacMnYnefr. karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -516
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ kareFVIkMNt;smÁal;³ FñwmRtUv)anEbgEckCaBIrFatu ehIy node nigGgát;RtUv)ankMNt;CamYynwgTisBIcug CiteTAcugq¶ay ¬rUbTI 15-12b¦. PaBdabEdlCaGBaØatRtUv)anbgðajenAkñúgrUbTI 15-12c. cMNaMfa bMlas;TImMu D4 minekItmaneT edayfakarTb;edayTMrrGilxag. m:aRTIsPaBrwgRkajsRmab;Ggát;³ edaysar EI CacMnYnefr ehIyGgát;manRbEvgesμIKña enaHm:aRTIsPaBrwg RkajsRmab;Ggát;RtUvEtdUcKña. edayeRbIelxkUdedIm,IkMNt;elxerogCYredk nigCYrQrnImYy²edayeyag eTAtamsmIkar 15-1 nigrUbTI 15-12b eyIg)an 3 4 1 2 1 2 5 6 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 3 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 1 ⎢ − 1.5 1 ⎥ 4 ⎢ − 1.5 1 ⎥ 2 k1 = EI ⎢ 1.5 2 ⎥ k 2 = EI ⎢ 1.5 2 ⎥ ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 1 ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 5 ⎢ ⎥ ⎢ ⎥ ⎣ 1.5 1 − 1.5 2 ⎦ 2 ⎣ 1.5 1 − 1.5 2 ⎦ 6 bMlas;TI nigbnÞúk³ pÁúMm:aRTIsPaBrwgRkajsRmab;Ggát;eTAkñúgm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§ eyIg)an Q = KD 1 2 3 4 5 6 ⎡− P ⎤ ⎡ 3 0 − 1.5 − 1.5 − 1.5 1.5 ⎤ ⎡ D1 ⎤ ⎢ 0 ⎥ ⎢ 0 4 1.5 1 − 1.5 1 ⎥ ⎢ D2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ 0 ⎥ = EI ⎢− 1.5 1.5 1.5 1.5 0 0 ⎥ ⎢ D3 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ Q4 ⎥ ⎢− 1.5 1 1.5 2 0 0 ⎥⎢ 0 ⎥ ⎢ Q5 ⎥ ⎢− 1.5 − 1.5 0 0 1.5 − 1.5⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ Q6 ⎥ ⎣ ⎦ ⎢ 1.5 ⎣ 1 0 0 − 1.5 2 ⎥ ⎢ 0 ⎥ ⎦⎣ ⎦ Beam analysis using the stiffness method T.Chhay -517
    • Department of Civil Engineering NPIC edayedaHRsayrkbMlas;TI eyIg)an P − = 3D1 + 0 D2 − 1.5D3 EI 0 = 0 D1 + 4 D2 + 1.5D3 0 = −1.5D1 + 1.5D2 + 1.5D3 1.667 P D1 = − EI P D2 = EI 2.667 P D3 = − EI cMNaMfa sBaØarbs;lT§plTaMgenHRtUvEtRtUvKñanwgTisedArbs;PaBdabEdlbgðajenAkñúgrUbTI 15-12c. kareRbIlT§plTaMgenH kmøaMgRbtikmμKW ⎛ 1.667 P ⎞ ⎛ P ⎞ ⎛ 2.667 P ⎞ Q4 = −1.5EI ⎜ − ⎟ + 1EI ⎜ ⎟ + 1.5EI ⎜ − ⎟ = −0.5P ⎝ EI ⎠ ⎝ EI ⎠ ⎝ EI ⎠ ⎛ 1.667 P ⎞ ⎛ P ⎞ ⎛ 2.667 P ⎞ Q5 = −1.5 EI ⎜ − ⎟ − 1.5EI ⎜ ⎟ + 0⎜ − ⎟=P ⎝ EI ⎠ ⎝ EI ⎠ ⎝ EI ⎠ ⎛ 1.667 P ⎞ ⎛ P ⎞ ⎛ 2.667 P ⎞ Q6 = 1.5EI ⎜ − ⎟ + 1EI ⎜ ⎟ + 0⎜ − ⎟ = −1.5P ⎝ EI ⎠ ⎝ EI ⎠ ⎝ EI ⎠ karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -518
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa cMeNaT !%>! kMNt;kmøaMgRbtikmμTMr. snμt;②CaTMrkl;. !%>% kMNt;m:Um:g;Bt;enAkñúgFñwmenARtg; ②nig③. EI CacMnYnefr. snμt;②CaTMrkl; ehIy③ nig④CaTMrsnøak;. EI CacMnYnefr. !%>@ kMNt;m:Um:g;Bt;enAkñúgFñwmenARtg; ① nig②. !%>^ kMNt;kmøaMgRbtikmμTMr. snμt;②CaTMrsnøak; snμt;②CaTMrkl; ehIy③CaTMrsnøak;. EI CacMnYn ehIy① nig③ CaTMrkl . EI CacMnYnefr. efr. !%>& kMNt;kmøaMgRbtikmμTMr. EI CacMnYnefr. !%># kMNt;kmøaMgRbtikmμTMr. EI CacMnYnefr. !%>* kMNt;m:Um:g;Bt;enARtg; ① nig③. snμt;② !%>$ kMNt;m:Um:g;Bt;enARtg;TMr. snμt;②CaTMrkl;. CaTMrkl; ehIy① nig③CaTMrbgáb;. EI CacMnYn EI CacMnYnefr. efr. Problems T.Chhay -519
    • Department of Civil Engineering NPIC !%>( kMNt;m:Um:g;Bt;enARtg; ① nig③ RbsinebI TMr②Rsut 30mm ¬cMeNaT !%>(¦. snμt; ②Ca TMrkl; ehIy① nig③CaTMrbgáb;. EI = 5000kN .m 2 . !%>!0 kMNt;kmøaMgRbtikmμTMr. EI CacMnYnefr. !%>!! kMNt;kmøaMgRbtikmμTMr. eKmanTMrrGiltam TisbBaÄrenARtg;① dUcbgðaj. EI CacMnYnefr. cMeNaT T.Chhay -520