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# 16. plane frame analysis using the stiffness method

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### 16. plane frame analysis using the stiffness method

1. 1. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa !^> karviPaKeRKagkñúgbøg;edayeRbIviFIPaBrwgRkaj (Plane frame analysis using the stiffness method) eKalKMnitEdl)anbgðajenAkñúgemeronelIkmunEdlerobrab;GMBIkaeRbIviFIPaBrwgRkajenAelI trusses nigFñwmRtUv)anBnøat nigGnuvtþeTAelIkarviPaKeRKag. viFIenHnwgbgðajfadMeNIrkarsRmab;edaHRsayman lkçN³RsedogKñaeTAnwgkaredaHRsaysRmab;Fñwm b:uEnþvaRtUvkareRbIm:aRTIsbMElg edaysarGgát;eRKagsßit kñúgTisepSg². !^>!> m:aRTIsPaBrwgRkajrbs;Ggát;eRKag (Frame-member stiffness matrix) enAkñúgkfaxNÐenH eyIgnwgbegáItm:aRTIsPaBrwgRkajsRmab;Ggát;eRKagEdlmanmuxkat;efr (prismatic frame member) BIRbB½n§kUGredaentMbn; x' , y ' , z ' ¬rUbTI 16-1¦. enATIenH Ggát;rgkmøaMg tamG½kS q N , q F kmøaMgkat; q N , q F , nigm:Um:g;Bt; q N , q F enARtg;cugCit nigcugq¶ayrbs;va x' x' y' y' z' z' erogKña. bnÞúkTaMgenHsuT§EtmanGMeBItamTiskUGredaenviC¢manCamYynwgbMlas;TIrbs;va. dUcKñakñúgkrNIFñwm m:Um:g; q N nig q F viC¢manvilRcasTisRTnicnaLika edaysarkareRbIviFanédsþaM viucT½rm:Um:g;manTistam z' z' G½kS z' EdlecjBIépÞRkdas. eyIg)anBicarNaTMnak;TMngrvagbnÞúk nigbM;las;TIEdlbNþalBIbnÞúkTaMgenHenAkñúgemeronelIkmun. bnÞúktamG½kSRtUv)anerobrab;edayeyageTAelIrUbTI 14-2 kmøaMgkat;eyageTAtamrUbTI 15-5 ehIym:Um:g; Bt;eyagtamrUbTI 15-6. tamviFItRmYtpl RbsinebIeKbUkbBa©ÚllT§plTaMgGs;enHcUlKña eKGacsresr TMnak;TMngrvagbMlas;TI nigbnÞúkcMnYnR)aMmYysRmab;Ggát;kñúgTRmg;m:aRTIsdUcxageRkam Plane frame analysis using the stiffness method T.Chhay -521
2. 2. Department of Civil Engineering NPIC N x' N y' N z' Fx' F y' Fz' ⎡q N x' ⎤ ⎡d ⎤ ⎢ ⎥ ⎡ AE AE ⎤ ⎢ N x' ⎥ 0 0 − 0 0 ⎥ ⎢ ⎥ ⎢ L L ⎢ ⎥ ⎢q N y ' ⎥ ⎢ 12 EI 6 EI 12 EI 6 EI ⎥ ⎢d N y ' ⎥ ⎢ ⎥ ⎢ 0 0 − 3 ⎥⎢ ⎥ ⎢ ⎥ ⎢ L3 L2 L L2 ⎥ ⎢ ⎥ ⎢q Nz' ⎥ ⎢ 0 6 EI 4 EI 0 − 2 6 EI 2 EI ⎥ ⎢ d ⎥ ⎥=⎢ L ⎥⎢ Nz' ⎢ L2 L L ⎥ (16-1) ⎢ ⎥ ⎢ AE AE ⎥⎢ ⎥ ⎢ q Fx ' ⎥ ⎢− 0 0 0 0 ⎥⎢ d F ⎥ ⎢ ⎥ ⎢ L L ⎥ ⎢ x' ⎥ ⎢ ⎥ ⎢ 0 12 EI − 3 − 2 6 EI 0 12 EI 6 EI ⎥ ⎢ − 2 ⎥ ⎥ ⎢ q Fy ' ⎥ ⎢ L L L3 L ⎥ ⎢ d Fy ' ⎥ ⎢ ⎥ ⎢ 6 EI 2 EI 6 EI 4 EI ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 − 2 ⎢ ⎥ ⎢ qF ⎥ ⎣ ⎢ L2 L L L ⎥⎢ d ⎥ ⎦ F ⎣ z' ⎦ ⎣ z' ⎦ b¤tamTRmg;kat; q = k'd (16-2) m:aRTIsPaBrwgRkajsRmab;Ggát; k ' pSMeLIgedayemKuNT§iBlcMnYn 36 EdltMNagedaybnÞúkenAelIGgát; enAeBlGgát;rgbMlas;TIÉktþaCak;lak;NamYy. CaBiess CYrQrnImYy²enAkñúgm:aRTIsnImYy²CabnÞúkkñúg Ggát;sMrab;bMlas;TIÉktþaEdlkMNt;edayelxkUd degree of freedom EdlmanbgðajenABIxagelICYrQr nImYy². eRKagEdleKRtUvKNnaRtUvEtbMeBjlkçxNÐlMnwg niglkçxNÐbMlas;TIRtUvKña. !^>@> m:aRTIsbMElgénbMlas;TI nigm:aRTIsbMElgénkmøaMg (Displacement and force transformation matrices) dUcenAkñúgkrNI trusses, eyIgRtUvbMElgbnÞúkkñúgGgát; q nigbMlas;TI d BIkUGredaen x' / y' / z' eTA CakUGredaenskl x, y, z . sRmab;ehtuplenH eKRtUvkarm:aRTIsbMElg. m:aRTIsbMElgbMlas;TI (displacement transformation matrix)³ eKmanGgát;eRKagdUcbgðajenAkñúgrUb TI 16-2a. enATIenH eyIgeXIjfabMlas;TI DN kñúgRbB½n§kUGredaensklbegáIt)anbMlas;TIkñúgkUGredaen x tMbn; d N x ' = D N x cos θ x d N y ' = − D N x cos θ y dUcKña bMlas;TI DN enAkñúgbMlas;TIskl ¬rUbTI 16-2b¦ begáIt)anbMlas;TIenAkñúgkUGedaentMbn; y d N x; = D N y cos θ y d N y ' = D N y cos θ x cugeRkay edaysarG½kS z' nigG½kS z RtYtsIuKña ¬manTisedAecjBIépÞesovePA¦ mMurgVil DN CMuvijG½kS z eFVI z eGaymanmMurgVilRtUvKña D N CMuvijG½kS z' . dUcenH z' karviPaKeRKagkñúgbøg;edayeRbIviFIPaBrwgRkaj T.Chhay -522
3. 3. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa DN z' = DN z tamrebobdUcKña RbsinebIeKeFVIeGaymanbMlas;TIskl DF tamTis x / DF tamTis y nigmMurgVil DF x y z enAxagcugq¶ayrbs;Ggát; smIkarbMElgKW d Fx ' = DFx cos θ x d Fy ' = − DFx cos θ y d Fx ' = DFy cos θ y d Fy ' = DFy cos θ x d Fz ' = DFz yk λ x = cosθ x , λ y = cosθ y CakUsIunUsR)ab;Tisrbs;Ggát; eyIgGacsresrrYmpSMKñaénbMlas;TICaTRmg; m:aRTIsdUcxageRkam ⎡d N x' ⎤ ⎡ λ x λ y 0 0 0 0⎤ ⎡ D N x ⎤ ⎢d ⎥ ⎢ ⎢ ⎥ ⎢ N y ' ⎥ ⎢− λ y λ x 0 0 0 0⎥ ⎢ D N y ⎥ ⎥ ⎢d N z' ⎥ ⎢ 0 0 1 0 0 0⎥ ⎢ D N z ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ (16-3) ⎢ d Fx ' ⎥ ⎢ 0 0 0 λx λ y 0⎥ ⎢ DFx ⎥ ⎢dF ⎥ ⎢ 0 0 0 − λy λ x 0⎥ ⎢ DFy ⎥ ⎢ y' ⎥ ⎢ ⎥⎢ ⎥ ⎢ d Fz ' ⎥ ⎢ 0 ⎣ ⎦ ⎣ 0 0 0 0 1⎥ ⎢ D Fz ⎥ ⎦⎣ ⎦ b¤ d = TD (16-4) tamkarGegát m:aRTIs T bMElgbMlas;TI D kñúgkUGredaenskl x, y, z TaMgR)aMmYyeGayeTACabMlas;TI d kñúgkUGredaentMbn; x' , y' , z' TaMgR)aMmYy. enATIenHm:aRTIs T RtUv)aneKsÁal;Cam:aRTIsbMElgbMlas;TI. m:aRTIsbMElgkMlaMg³ RbsinebIeyIgGnuvtþbgÁúMkmøaMgnImYy²eTAelIcugCitrbs;Ggát; eyIgGackMNt;BIrebob bMElgbgÁúMkmøaMgBIkUGredaentMbn;eGayeTACakUGredaenskl. edayGnuvtþ q N ¬rUbTI 16-3a¦ eyIgGacx' eXIjfa Q N x = q N x ' cos θ x Q N y = q N x ' cos θ y Plane frame analysis using the stiffness method T.Chhay -523
4. 4. Department of Civil Engineering NPIC RbsinebIeKGnuvtþ q N ¬rUbTI 16-3b¦ enaHbgÁúMkmøaMgrbs;vaKW y' Q N x = −q N y ' cos θ y Q N y = q N y ' cos θ x cugeRkay edaysar q N RtYtsIuCamYynwg QN eyIg)an z' Z QN z = q N z ' tamrebobdUcKña bnÞúkenARtg;cugGgát; q F x' , q Fy ' , q Fz ' nwgpþl;nUvbgÁúMkmøaMgdUcxageRkam³ Q Fx = q Fx ' cos θ x QFy = q Fx ' cos θ y Q Fx = − q Fy ' cos θ y QFy = q Fy ' cos θ x QFz = q Fz ' smIkarTaMgbIEdlpÁúMenAkñúgTRmg;m:aRTIsCamYynwg λ x = cosθ x , λ y = cos θ y pþl;nUv ⎡Q N x ⎤ ⎡ λ x − λ y 0 0 0 0⎤ ⎡ q N x ' ⎤ ⎢Q ⎥ ⎢ ⎢ ⎥ ⎢ N y ⎥ ⎢λ y λ x 0 0 0 0⎥ ⎢ q N y ' ⎥ ⎥ ⎢Q N z ⎥ ⎢ 0 0 1 0 0 0⎥ ⎢ q N z ' ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ (16-5) ⎢ Q Fx ⎥ ⎢ 0 0 0 λx − λ y 0⎥ ⎢ q Fx ' ⎥ ⎢ QF ⎥ ⎢ 0 0 0 λy λx 0⎥ ⎢ q Fy ' ⎥ ⎢ y⎥ ⎢ ⎥⎢ ⎥ ⎢ QFz ⎥ ⎢ 0 ⎣ ⎦ ⎣ 0 0 0 0 1⎥ ⎢ q Fz ' ⎥ ⎦⎣ ⎦ b¤ Q =TTq (16-6) enATIenH dUckarerobrab; m:aRTIs T T bMElgbnÞúkenAelIGgát;TaMgR)aMmYyEdlsresrenAkñúgkUGredaentMbn; eGayeTACabnÞúkTaMgR)aMmYyEdlsresrenAkñúgkUGredaenskl. !^>#> m:aRTIsPaBrwgRkajsklsRmab;Ggát;eRKag (Frame-Member Global Stiffness Matrix) eKGacpÁúMlT§plénkfaxNÐelIkmunedIm,IkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;EdlP¢ab;TMnak;TMng rvagbnÞúkskl Q eTAnwgbMlas;TIskl D . edIm,IeFVIEbbenH eKRtUvCMnYssmIkar 16-4 ¬ d = TD ¦ eTAkñúg smIkar 16-2 ¬ q = k ' d ¦. eyIg)an q = k 'TD (16-7) enATIenH kmøaMgkñúgGgát; q Tak;TgnwgbMlas;TIskl D . edayCMnYslT§plenHeTAkñúgsmIkar 16-6 ¬ Q = T T q ¦ eKTTYl)anlT§plcugeRkay Q = T T k 'TD (16-8) karviPaKeRKagkñúgbøg;edayeRbIviFIPaBrwgRkaj T.Chhay -524
5. 5. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa b¤ Q = kD Edl k = T T k 'T (16-9) enATIenH k Cam:aRTIsPaBrwgRkajsklsRmab;Ggát;. eyIgGacTTYltémørbs;vakñúgTRmg;TUeTAedayeRbI smIkar 16-5/ 16-1 nig 16-3 ehIyedayeFVIRbmaNviFIm:aRTIs eKnwgTTYl)anlT§plcugeRkay (16-10) cMNaMfam:aRTIsTMhM 6 × 6 Cam:aRTIssIuemRTI. elIsBIenH eKP¢ab;TItaMgrbs;FatunImYy²eTAnwgkUdenARtg;cug Cit N x , N y , N z EdlbnþedayelxkUdenARtg;cugq¶ay Fx , Fy , Fz EdlRtUv)anbgðajenAxagelIénCYr Qr nigtambeNþayCYredk. dUcm:aRTIs k ' CYrQrnImYy²rbs;m:aRTIs k CabnÞúkenAelIGgát;Rtg; node Edl KRtUvkaredIm,ITb;Tl;nwgbM;las;TIÉktþatamTisEdlkMNt;edayelxkUdrbs;CYrQr. ]TahrN_ CYrQrTI mYyrbs;m:aRTIs k CabnÞúkenAkñúgkUGredaensklRtg;cugCit nigcugq¶ayEdlbgáeLIgedaybMlas;TIÉktþa enARtg;cugCittamTis x eBalKW N x . !^>\$> karGnuvtþénviFIPaBrwgRkajsMrab;karviPaKeRKag (Application of the stiffness method for frame analysis) enAeBlEdleKbegáItm:aRTIsPaBrwgRkajsRmab;Ggát;rYcehIy eKGacpÁúMBYkvabBa©ÚlKñaeTAkñúgm:aRTIs PaBrwgRkajsRmab;rcnasm<½n§tamrebobFmμta. edaysresrsmIkarm:aRTIssRmab;rcnasm<½n§ eKGac kMNt;bM;las;TIenARtg; node EdlminmankarTb; EdlbnþedaykmøaMgRbtikmμ nigkmøaMgkñúgenARtg; node. eKGacedaHRsaykmøaMgxagEdlmanGMeBIelIGgát; kMhusqÁgedaysarplitkmμ bERmbRmYlsItuNðPaB kmøaMgTRmeRTt nigkmøaMgTRmxagkñúgtamrebobdUcKñanwgGVIEdl)anerobrab;sRmab; truss nigFñwm. Plane frame analysis using the stiffness method T.Chhay -525
6. 6. Department of Civil Engineering NPIC dMeNIrkarkñúgkarviPaK (Procedure for analysis) viFIxageRkampþl;nUvmeFüa)ayedIm,IkMNt;bMlas;TI RbtikmμTMr kmøaMgkñúgrbs;Ggát;eRKagkMNt;eday sþaTic nigeRKagminkMNt;edaysþaTic. kareFVIkMNt;sMKal;³ EckeRKOgbgÁúMCaFatuGnnþtUc ehIykMNt;elxerogeGayGgát; nig node nImYy²rbs;va. eKEtgEt BnøatFatuenAcnøaHcMNucrbs;TMr cMNucrbs;bnÞúkRtg;cMNuc RCugEkg b¤tMNEdleKRtUvkarkMNt; bMlas;TI b¤kmøaMgkñúgrbs;Ggát;. begáItRbB½n§kUGredaen x, y, z CaTUeTAedIm,IPaBgayRsYlCamYynwgeKalEdlmanTItaMgenARtg; cMNuc node enAelIFatumYy nigG½kSEdlmanTItaMgy:agNaeGayRKb; node TaMgGs;mankUGredaen viC¢man. enARtg;cMNuc node nImYy²rbs;eRKag kMNt;bgÁúMelxkUdbIKW x, y, z . RKb;krNITaMgGs; eKeRbI elxkUdtUcbMputedIm,IkMNt;elxerogsRmab; degree of freedom EdlminmankarTb; Edlbnþeday elxkUdEdlenAsl; b¤elxkUdEdlmanelxerogFMedIm,IsMKal; degree of freedom Edlmankar Tb;. begáItbMlas;TIEdlsÁal; Dk nigbnÞúkxageRkAEdlsÁal; Qk . enAeBlbegáIt Qk eKRtUvR)akdkñúgkar bBa©ÚlbnÞúkbgáb;cugRbsinebIGgát;RTbnÞúkenAkNþal. m:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúM³ GnuvtþsmIkar 16-10 edIm,IkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;nImYy²EdlsresrenAkñúgRbB½n§ kUGredaenskl. eKkMNt;kUsIunUsR)ab;Tis λ x nig λ y BIkUGredaen x, y éncugrbs;Ggát; ¬smIkar 14-5 nig 14-6¦. eRkayeBlsresrm:aRTIsPaBrwgRkajsRmab;Ggát;nImYy² nigeRkayeBlkMNt;CYredk nigCYrQr CamYynwgelxkUdcugCit nigcugq¶ay eKGacRc)ac;m:aRTIsTaMgenHbBa©ÚlKñaedIm,IbegáItm:aRTIsPaBrwg RkajsRmab;eRKOgbgÁúM K . sRmab;karepÞógpÞat;edayEpñk m:aRTIsPaBrwgRkajsRmab;Ggát; nig m:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúMKYrEtCam:aRTIssIuemRTI. bMlas;TI nigkmøaMg³ EbgEckm:aRTIsPaBrwgRkajCaRkumdUcbgðajedaysmIkar 14-18. karBnøatenHeyIgTTYl)an karviPaKeRKagkñúgbøg;edayeRbIviFIPaBrwgRkaj T.Chhay -526
7. 7. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa Qk = K11 Du + K12 Dk Qu = K 21 Du + K 22 Dk eKkMNt;bMlas;TIEdlCaGBaØat Du BIsmIkarTImYyénsmIkarTaMgBIrxagelI. edayeRbItémø TaMgenH eKkMNt;kmøaMgRbtikmμ Qu BIsmIkarTIBIr. cugbBa©b; eKGackMNt;kmøaMgkñúg q enARtg;cug rbs;Ggát;BIsmIkar 16-7 eBalKW q = k 'TD RbsinebIlT§plénGBaØatEdl)anKNnaCaTMhMGviC¢man vabgðajfaBYkvaeFVIGMeBItamTiskUGredaen GviC¢man. ]TahrN_ 16-1³ kMNt;bnÞúkenARtg;tMNrbs;eRKagGgát;BIrEdlbgðajenAkñúgrUbTI 16-4a. yk I = ( ) / 1800 10 6 mm 4 A = 6000mm 2 ehIy E = 200GPa sRmab;Ggát;TaMgBIr. Plane frame analysis using the stiffness method T.Chhay -527
8. 8. Department of Civil Engineering NPIC dMeNaHRsay³ kareFVIkMNt;sMKal;³ tamkarGegát eRKagmanGgát;cMnYnBIr nig node cMnYnbIEdlRtUv)ankMNt;sMKal;dUc bgðajenAkñúgrUbTI 16-4b. eKalrbs;RbB½n§kUGredaensklRtUvmanTItaMgenARtg; ①. dMbUgelxkUdenA Rtg; node RtUv)ankMNt;eday degree of freedom EdlminmankarTb;. BIkarTb;enARtg;①nig③ nigbnÞúk Gnuvtþn_ eyIg)an ⎡20⎤ 1 ⎡0 ⎤ 6 ⎢ 0 ⎥2 ⎢0 ⎥ 7 ⎢ ⎥ Dk = ⎢ ⎥ Qk = ⎢ 0 ⎥ 3 ⎢0 ⎥ 8 ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥4 ⎣0 ⎦ 9 ⎢ 0 ⎥5 ⎣ ⎦ m:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúM³ tYxageRkammanlkçN³dUcKñasRmab;m:aRTIsPaBrwgRkajsRmab; Ggát;TaMgBIr³ AE 6(10 −3 )(200)(10 6 ) = = 200(103 )kN / m L 6 12 EI 12(200 )(10 6 )(180 )(10 −6 ) = = 2(10 3 )kN / m L3 63 6 EI = ( ) ( ) = 6(10 )kN / m 6(200 ) 10 6 (180 ) 10 −6 3 2 2 L 6 4 EI 4(200 )(10 )(180 )(10 ) = 24(10 )kN / m 6 −6 = 3 L 6 = 6 ( ) ( ) ( ) 2 EI 2(200 ) 10 (180 ) 10 −6 = 12 10 3 kN / m L 6 6−0 0−0 Ggát;elx !³ λx = 6 =1 λy = 6 =0 edayCMnYsTinñn½yeTAkñúgsmIkar 16-10 eyIg)an 4 6 5 1 2 3 ⎡ 200 0 0 − 200 0 0 ⎤ 4 ⎢ 0 2 6 0 − 2 − 6⎥ 6 ⎢ ⎥ ( ) k1 = 10 3 ⎢ 0 ⎢ 6 24 0 − 6 12 ⎥ 5 ⎥ ⎢− 200 0 0 200 0 0 ⎥ 1 ⎢ 0 −2 −6 0 2 − 6⎥ 2 ⎢ ⎥ ⎢ 0 ⎣ 6 12 0 − 6 24 ⎥ 3 ⎦ karviPaKeRKagkñúgbøg;edayeRbIviFIPaBrwgRkaj T.Chhay -528
9. 9. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa CYredk nigCYrQrénm:aRTIs 6 × 6 RtUv)ankMNt;edayelxkUdbI x, y, z CadMbUgenARtg;cugCit ehIybnþ edaycugq¶ayeBalKW \$/ ^/ %/ !/ @/ # erogKña ¬rUbTI 16-4b¦. eKeFVIEbbenHsRmab;karpÁúMFatuelIkeRkay. Ggát;elx @³ λx = 6 − 6 = 0 6 λy = −6−0 6 = −1 edayCMnYsTinñn½yeTAkñúgsmIkar 16-10 eyIg)an 1 3 2 7 8 9 ⎡2 0 6 −2 0 6 ⎤1 ⎢ 0 200 0 0 − 200 0 ⎥2 ⎢ ⎥ ( ) k 2 = 10 3 ⎢ 6 ⎢ 0 24 − 6 0 12 ⎥ 3 ⎥ ⎢− 2 0 −6 2 0 − 6⎥ 7 ⎢ 0 − 200 0 0 200 0 ⎥8 ⎢ ⎥ ⎢6 ⎣ 0 12 − 6 0 24 ⎥ 9 ⎦ CaFmμta karkMNt;elxerogrbs;CUredk nigCYrQrKWeyageTAtamelxkUdTaMgbItamlMdab; x, y, z sRmab; cugCit nigcugq¶ay erogKña eBalKW !/ @/ # bnÞab;mk &/ */ ( ¬rUbTI 16-4b¦. m:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúMRtUv)ankMNt;edaykarpÁúMm:aRTIs k1 nig k 2 . lT§plén Q = KD EdlbgðajedaykarbMEbkCaRkumKW 1 2 3 4 5 6 7 8 9 ⎡ 20 ⎤ ⎡ 202 0 6 − 200 0 0 − 2 0 6 ⎤ ⎡ D1 ⎤ ⎢0⎥ ⎢ 0 202 −6 0 − 6 − 2 0 − 200 0 ⎥ ⎢ D2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢0⎥ ⎢ 6 −6 48 0 12 6 − 6 0 12 ⎥ ⎢ D3 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢0⎥ ⎢0⎥ ( ) ⎢− 200 = 10 3 ⎢ 0 0 −6 0 200 0 0 0 12 0 24 6 0 0 0 0 ⎥ ⎢ D4 ⎥ 0 ⎥ ⎢ D5 ⎥ (1) ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q6 ⎥ ⎢ 0 −2 6 0 6 2 0 0 0 ⎥⎢ 0 ⎥ ⎢Q ⎥ ⎢ −2 0 −6 0 0 0 2 0 − 6⎥ ⎢ 0 ⎥ ⎢ 7⎥ ⎢ ⎥⎢ ⎥ ⎢Q8 ⎥ ⎢ 0 − 200 0 0 0 0 0 200 0 ⎥ ⎢ 0 ⎥ ⎢Q ⎥ ⎢ 6 0 0 −6 24 ⎥ ⎢ 0 ⎥ ⎣ 9⎦ ⎣ 0 12 0 0 ⎦⎣ ⎦ kmøaMg nigbMlas;TI³ edayBnøatedIm,IedaHRsaybMlas;TI eyIgTTYl)an ⎡20⎤ ⎡ 202 0 6 − 200 0 ⎤ ⎡ D1 ⎤ ⎡0⎤ ⎢0⎥ ⎢ ⎥ ⎢ 0 ⎢ 202 − 6 0 − 6⎥ ⎢ D2 ⎥ ⎢0⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ( ) ⎢ 0 ⎥ = 10 3 ⎢ 6 ⎢ − 6 48 0 12 ⎥ ⎢ D3 ⎥ + ⎢0⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎢0⎥ ⎢− 200 0 0 200 0 ⎥ ⎢ D4 ⎥ ⎢0⎥ ⎢0⎥ ⎣ ⎦ ⎢ 0 ⎣ − 6 12 0 24 ⎥ ⎢ D5 ⎥ ⎢0⎥ ⎦⎣ ⎦ ⎣ ⎦ edayedaHRsaym:aRTIsxagelI eyIgTTYl)am Plane frame analysis using the stiffness method T.Chhay -529
10. 10. Department of Civil Engineering NPIC ( ) ⎡ D1 ⎤ ⎡ 17.51 10 − 3 m ⎤ ⎢D ⎥ ⎢ ( ) ⎢ 2 ⎥ ⎢ − 37.47 10 m ⎥ −6 ⎥ ( ) ⎢ D3 ⎥ = ⎢− 2.505 10 − 3 rad ⎥ ⎢ ⎥ ⎢ ( ) −3 ⎢ D4 ⎥ ⎢ 17.51 10 m ⎥ ⎥ ⎣ ⎦ ⎣( ) ⎢ D5 ⎥ ⎢ 1.243 10 − 3 rad ⎥ ⎦ edayeRbIlT§plTaMgenH eKGackMNt;kmøaMgRbtikmμBIsmIkar (1) dUcxageRkam 1 2 3 4 5 ( ) ⎡ 17.51 10 - 3 m ⎤ ⎡Q6 ⎤ ⎢Q ⎥ ( ) ⎡0 ⎢− 2 −2 6 ( )0 6⎤ ⎢ −6 ⎥ ⎡0⎤ ⎡− 7.50kN ⎤ ⎥ ⎢ − 37.47 10 m ⎥ ⎢0⎥ ⎢ − 20kN ⎥ ⎢ 7 ⎥ = 10 −6 ( ) 3 0 0 0⎥ ⎢ ⎢− 2.505 10 − 3 rad ⎥ + ⎢ ⎥ = ⎢ ⎥ ⎢Q8 ⎥ ⎢ 0 − 200 0 ⎥⎢ ⎥ ⎢0⎥ ⎢ 7.50kN ⎥ ⎢ ⎥ ⎢ ( ) 0 0 ⎥ ⎢ 17.51 10 − 3 m ⎥ ⎢ ⎥ ⎢ ⎥ ⎣Q9 ⎦ ⎣6 0 12 ( ) 0 0⎦ ⎢ −3 ⎣ 1.243 10 rad ⎦ ⎥ ⎣0⎦ ⎣ 75kN .m ⎦ eKGackMNt;kmøaMgkñúgenAkñúg node ② edayGnuvtþsmIkar 16-7 eTAelIGgát;elx 1. enATIenH k ' RtUv 1 )ankMNt;edaysmIkar 16-1 ehIy T edaysmIkar 16-3. dUcenH 1 4 6 5 1 2 3 ⎡ 200 0 0 − 200 0 0 ⎤ ⎡1 ⎢ 0 ⎢ ( ) ⎤4 0 0 0 0 0⎤ ⎡ 17.5 10 − 3 ⎥ ⎢ 2 6 0 − 2 6 ⎥ ⎢0 ⎥⎢ 1 0 0 0 0⎥ ⎢ ⎥ 0 ⎥6 ( ) q1 = k1T1 D = 10 3 ⎢ 0 6 24 0 − 6 12 ⎥ ⎢0 ( ) 0 1 0 0 0⎥ ⎢ 1.243 10 − 3 ⎥5 ⎢ ⎢− 200 0 0 200 0 0 ⎥ ⎢0 ⎥⎢ ( )⎥⎢ 0 0 1 0 0⎥ ⎢ 17.51 10 − 3 ⎥ ⎥1 ⎢ 0 −2 −6 0 2 − 6 ⎥ ⎢0 ( ) 0 0 0 1 0⎥ ⎢ − 37.47 10 6 ⎥⎢ ⎥2 ⎥ ⎢ ⎢ 0 ⎣ 6 12 0 ⎥⎢ − 6 24 ⎥ ⎢0 ⎦⎣ ( ) 0 0 0 0 1⎥ ⎢− 2.505 10 − 3 ⎦⎣ ⎥3 ⎦ cMNaMkardMerobd¾RtwmRtUvénFatuenAkñúgm:aRTIsdUcEdl)anbgðajedayelxkUdtamRCugxagrbs;CYrQr nigCYredk. edaHRsaym:aRTIsxagelI eyIg)an ⎡q 4 ⎤ ⎡ 0 ⎤ ⎢ q ⎥ ⎢ − 7.50kN ⎥ ⎢ 6⎥ ⎢ ⎥ ⎢ q5 ⎥ ⎢ 0 ⎥ ⎢ ⎥=⎢ ⎥ ⎢ q1 ⎥ ⎢ 0 ⎥ ⎢q 2 ⎥ ⎢ 7.50kN ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ q3 ⎥ ⎢− 45kN .m⎥ ⎣ ⎦ ⎣ ⎦ lT§plxagelIRtUv)anbgðajenAkñúgrUbTI 16-4c. TisedArbs;viucT½rTaMgenHRtUvKñanwgTisviC¢manEdlkMNt; enAkñúgrUbTI 16-1. elIsBIenH eKalrbs;kUGredaen x', y' , z' sßitenARtg;cugCitrbs;Ggát;. tamrebob dUcKña düaRkamGgÁesrIénGgát;elx @ RtUv)anbgðajenAkñúgrUbTI 16-4d. karviPaKeRKagkñúgbøg;edayeRbIviFIPaBrwgRkaj T.Chhay -530
11. 11. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ]TahrN_ 16-2³ kMNt;bnÞúkenARtg;cugrbs;Ggát;nImYy²éneRKagEdlbgðajenAkñúgrUbTI 16-5a. yk ( ) / I = 225 10 6 mm 4 A = 7500mm 2 ehIy E = 200GPa sRmab;Ggát;nImYy². dMeNaHRsay³ kareFVIkMNt;sMKal;³ edIm,IGnuvtþkarviPaKedayviFIm:aRTIs bnÞúkBRgayEdlmanGMeBIenAelIGgát;edkRtUv)an CMnYsedaym:Um:g;cugsmmUl nigkmøaMgkat;enAxagcugsmmUlEdlRtUv)anKNnaBIsþaTic nigBItaragenAkñúg emeronTI11. bnÞab;mkedayeRbIviFItRmYtpl lT§plEdlTTYl)ansRmab;eRKagenAkñúgrUbTI 16-5b RtUv )anEktRmUvsRmab;Ggát;enHedaybnÞúkEdlbgðajenAkñúgrUbTI 16-5c. dUcbgðajenAkñúgrUbTI 16-5b, node nigGgát;RtUv)andak;elxerog ehIyeKalrbs;RbB½n§kUGredaen sklRtUv)andak;enAkñúg node ①. tamFmμta dMbUgeKRtUvdak;elxkUdeTAelI degree of freedom Edlmin mankarTb;. dUcenH Plane frame analysis using the stiffness method T.Chhay -531
12. 12. Department of Civil Engineering NPIC ⎡0 ⎤ 4 ⎢0 ⎥ 5 ⎢ ⎥ ⎡ 0 ⎤1 ⎢0 ⎥ 6 Dk = ⎢ ⎥ Qk = ⎢− 150⎥ 2 ⎢ ⎥ ⎢0 ⎥ 7 ⎢ 150 ⎥ 3 ⎢0 ⎥ 8 ⎣ ⎦ ⎢ ⎥ ⎢0 ⎥ 9 ⎣ ⎦ m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§ Ggát;elx !³ EA 7500(10 −6 )(200)( 6 ) = 200(10 3 )kN / m 10 = L 7.5 12 EI = ( ) ( ) = 1280kN / m 12(200 ) 10 6 (225) 10 −6 L 3 (7.5) 3 6 EI 6(200)(225) = = 4800kN L2 (7.5)2 4(200)(225) = 24(10 3 )kN .m 4 EI = L 7.5 2 EI 2(200 )(225) L = 7.5 = 12 10 3 kN .m ( ) 6−0 4.5 − 0 λx = = 0.8 λy = = 0.6 7.5 7.5 edayGnuvtþsmIkar 16-10/ eyIg)an 4 6 5 1 2 3 ⎡ 128.46 95.39 − 2.88 − 128.46 − 95.39 − 2.88⎤ 4 ⎢ 95.39 72.82 3.84 − 95.39 − 72.82 3.84 ⎥ 6 ⎢ ⎥ ( ) k1 = 10 3 ⎢ − 2.88 ⎢ 3.84 24 2.88 − 3.84 12 ⎥ 5 ⎥ ⎢− 128.46 − 95.39 2.88 128.46 95.39 2.88 ⎥ 1 ⎢ − 95.39 − 72.84 − 3.84 95.39 72.82 − 3.84⎥ 2 ⎢ ⎥ ⎢ − 2.88 ⎣ 3.84 12 2.88 − 3.84 24 ⎥ 3 ⎦ Ggát;elx @³ = ( ) EA 7500 10 −6 (200 ) 10 6 ( ) = 250 10 3 kN / m( ) L 6 12 EI 12(200 )(225) = = 2500kN / m L3 (6)3 6 EI 6(200 )(225) = = 7500kN L2 (6)2 4 EI 4(200)(225) L = 6 = 30 10 3 kN .m ( ) karviPaKeRKagkñúgbøg;edayeRbIviFIPaBrwgRkaj T.Chhay -532
13. 13. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 2 EI 2(200 )(225) L = 6 = 15 10 3 kN .m ( ) 12 − 6 4.5 − 4.5 λx = =1 λy = =0 6 6 dUcenH smIkar 16-10 køayCa 1 2 3 7 8 9 ⎡ 250 250 0 − 250 0 0 ⎤1 ⎢ 0 7.5 7.5 0 − 2.5 7.5 ⎥ 2 ⎢ ⎥ ( ) k 2 = 10 3 ⎢ 0 ⎢ 0 30 0 − 7.5 15 ⎥ 3 ⎥ ⎢− 250 0 0 250 0 0 ⎥7 ⎢ 0 − 2.5 − 7.5 0 2.5 − 7.5⎥ 8 ⎢ ⎥ ⎢ 0 ⎣ 7.5 15 0 − 7.5 30 ⎥ 9 ⎦ m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§EdlrYmbBa©ÚlenAkñúg Q = KD køayCa 1 2 3 4 5 6 7 8 9 ⎡ 0 ⎤ ⎡ 378.46 95.39 7.88 − 128.46 − 95.39 2.88 − 250 0 0 ⎤ ⎡ D1 ⎤ ⎢− 150⎥ ⎢ 95.39 75.32 3.66 − 95.39 − 72.82 − 3.84 0 − 2.5 7.5 ⎥ ⎢ D2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢− 150⎥ ⎢ 2.88 3.66 54 − 2.88 3.84 12 0 − 7.5 15 ⎥ ⎢ D3 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ Q4 ⎥ ( ) 3 ⎢ − 128.46 − 95.39 − 2.88 128.46 95.39 − 2.88 ⎢ Q5 ⎥ = 10 ⎢ − 95.39 − 72.82 3.84 95.39 72.82 3.84 0 0 0 0 0 ⎥⎢ 0 ⎥ 0 ⎥⎢ 0 ⎥ (1) ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ Q6 ⎥ ⎢ 2.88 − 3.84 12 − 2.88 3.84 24 0 0 0 ⎥⎢ 0 ⎥ ⎢ Q ⎥ ⎢ − 20 0 0 0 0 0 250 0 0 ⎥⎢ 0 ⎥ ⎢ 7 ⎥ ⎢ ⎥⎢ ⎥ ⎢ Q8 ⎥ ⎢ 0 − 2.5 − 7.5 0 0 0 0 2.5 − 7.5⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ Q9 ⎦ ⎣ 0 7.5 15 0 0 0 0 − 7.5 30 ⎦ ⎣ 0 ⎦ bMlas;TI nigbnÞúk³ Bnøatm:aRTIsxagelIedIm,IkMNt;bMlas;TI nigedayedaHRsay eyIg)an ⎡ 0 ⎤ ⎡378.46 95.39 2.88⎤ ⎡ D1 ⎤ ⎡0⎤ ⎥ ( ) ⎢− 150⎥ = 10 3 ⎢ 95.39 75.32 3.66⎥ + ⎢ D ⎥ + ⎢0⎥ ⎢ ⎢ ⎥ ⎢ 2⎥ ⎢ ⎥ ⎢− 150⎦ ⎣ ⎥ ⎢ 2.88 3.66 54 ⎥ ⎢ D3 ⎥ ⎢0⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡ D1 ⎤ ⎡ 0.716mm ⎤ ⎢ D ⎥ = ⎢ − 2.76mm ⎥ ⎢ 2⎥ ⎢ ⎥ ⎢ D3 ⎥ ⎢− 0.00261rad ⎥ ⎣ ⎦ ⎣ ⎦ edayeRbIlT§plTaMgenH eKGackMNt;kmøaMgRbtikmμTMrBIsmIkar (1) dUcbgðaj³ Plane frame analysis using the stiffness method T.Chhay -533
14. 14. Department of Civil Engineering NPIC ⎡Q4 ⎤ ⎡− 128.46 − 95.39 − 2.88⎤ ⎡0⎤ ⎡ 178.8kN ⎤ ⎢Q ⎥ ⎢ − 95.39 − 72.82 3.84 ⎥ ⎢0⎥ ⎢ 122.7 kN ⎥ ⎢ 5⎥ ⎢ ⎥⎡ 0.716 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢Q6 ⎥ ⎢ 2.88 − 3.84 12 ⎥ ⎢ ⎥+ ⎢0⎥ ⎢ − 18.7 kN .m ⎥ ⎢ ⎥=⎢ ⎥ − 2.76 ⎥ ⎢0⎥ = ⎢ − 179.0kN ⎥ 0 ⎥⎢ ⎢Q7 ⎥ ⎢ − 250 ⎢Q8 ⎥ ⎢ 0 0 − 2.5 − 7.5 ⎥⎣ ⎢− 0.00261 10 3 ( ) ⎥ ⎢ ⎥ ⎢ ⎦ ⎢0⎥ ⎢ 26.5kN ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢Q9 ⎥ ⎢ 0 ⎣ ⎦ ⎣ 7.5 15 ⎥⎦ ⎢0⎥ ⎢− 59.9kN .m ⎥ ⎣ ⎦ ⎣ ⎦ eKGackMNt;kmøaMgkñúgBIsmIkar 16-7 EdlGnuvtþeTAGgát;elx ! nigelx @. enAkñúgkrNIGgát;elx !/ q = k '1 T1 D Edl k '1 RtUv)ankMNt;BIsmIkar 16-1 ehIy T1 RtUv)ankMNt;BIsmIkar 16-3. dUcenH 4 5 6 1 2 3 ⎡q4 ⎤ ⎡ 200 0 0 − 200 0 0 ⎤ ⎡ 0.8 0.6 0 0 0 0⎤ ⎡ 0 ⎤ 4 ⎢q ⎥ ⎢ 0 1.28 4.8 0 − 1.28 4.8 ⎥ ⎢− 0.6 0.8 0 0 0 0⎥ ⎢ 0 ⎥ 5 ⎢ 5⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ q6 ⎥ = ⎢ 0 4.8 2.4 0 − 4.8 12 ⎥ ⎢ 0 0 1 0 0 0⎥ ⎢ 0 ⎥ 6 ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢q7 ⎥ ⎢− 200 0 0 200 0 0 ⎥⎢ 0 0 0 0.8 0 0⎥ ⎢ 0.716 ⎥ 1 ⎢ q8 ⎥ ⎢ 0 − 1.28 − 4.8 0 1.28 − 1.8⎥ ⎢ 0 0 0 − 0.6 0.8 0⎥ ⎢− 2.76⎥ 2 ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎣ q9 ⎦ ⎣ 0 4.8 12 0 − 4.8 24 ⎥ ⎢ 0 ⎦⎣ 0 0 0 0 1⎥ ⎢ − 2.61⎥ 3 ⎦⎣ ⎦ enATIenH elxkUdbgðajCYredk nigCYrQrsRmab;cugCit nigcugq¶ayrbs;Ggát; erogKña eBalKW \$/ %/ ^ bnÞab;mk !/ @/ # rUbTI 16-5b. dUcenH ⎡q4 ⎤ ⎡ 216.6kN ⎤ ⎢ q ⎥ ⎢ − 9.15kN ⎥ ⎢ 5⎥ ⎢ ⎥ ⎢q6 ⎥ ⎢− 18.7kN .m⎥ ⎢ ⎥=⎢ ⎥ ⎢ q1 ⎥ ⎢ 216.6kN ⎥ ⎢q2 ⎥ ⎢ 9.15kN ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ q3 ⎥ ⎢ − 50kN .m ⎥ ⎣ ⎦ ⎣ ⎦ lT§plTaMgenHRtUv)anbgðajenAkñúgrUbTI 16-5d. karviPaKdUcKñaRtUv)aneFVIsRmab;Ggát;elx @. lT§plRtUv)anbgðajenAxageqVgkñúgrUbTI 16-5e. sRmab;Ggát;enH eyIgRtUvdak;bnÞúkénrUbTI 16-5c dUcenHlT§plcugeRkaysRmab;Ggát;elx @ RtUv)an bgðajenAxagsþaM. karviPaKeRKagkñúgbøg;edayeRbIviFIPaBrwgRkaj T.Chhay -534
15. 15. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa cMeNaT 16>1 kMNt;m:aRTIsPaBrwgRkaj K sRmab;eRKag. 16>4 kMNt;kmøaMgRbtikmμTRmkñúgenARtg; ① nig snμt; ① nig ③ CaTRmsnøak;. yk E = 200GPa ③ ¬cMeNaT 16>3¦. yk E = 200GPa / I = 243( 6 )mm 4 , A = 6000mm 2 sRmab;Ggát; 10 I = 300( 6 )mm 4 , A = 21( 3 )mm 2 sRmab; 10 10 nImYy². Ggát;nImYy². 16>5 kMNt;m:aRTIsPaBrwgRkaj K sRmab;eRKag. yk E = 200GPa I = 250(106 )mm 4 , A = 19( 3 )mm 2 sRmab;Ggát;nImYy². snμt; ② 10 nig③ CatMNbgáb;. 16>2 kMNt;kmøaMgkñúgenARtg;cugrbs;Ggát;nImYy² ¬cMeNaT 16>1¦. snμt; ① nig ③ CaTRmsnøak;. yk E = 200GPa / I = 243(106 )mm 4 , A = 6000mm 2 sRmab;Ggát;nImYy². 16>3 kMNt;m:aRTIsPaBrwgRkaj K sRmab;Ggát; nImYy²rbs;eRKag. snμt; ③Casnøak; nig ①Ca 16>6 kMNt;m:aRTIsPaBrwgRkaj K sRmab;Ggát; TRmbgáb;. yk E = 200GPa nImYy²rbs;eRKag. yk E = 200GPa I = 300( 6 )mm 4 , A = 21(10 3 )mm 2 sRmab; 10 I = 280( 6 )mm 4 , A = 18( 3 )mm 2 10 10 Ggát;nImYy². sRmab;Ggát;nImYy². Problems T.Chhay -535
16. 16. Department of Civil Engineering NPIC 16>7 kMNt;kmøaMgkñúgenARtg;cugrbs;Ggát;nImYy² ¬cMeNaT 16>6¦. yk E = 200GPa / I = 280( 6 )mm 4 , A = 18( 3 )mm 2 10 10 sRmab;Ggát;nImYy². 16>8 kMNt;m:aRTIsPaBrwgRkaj K sRmab;eRKag. yk E = 200GPa I = 250(106 )mm 4 , A = 12( 3 )mm 2 sRmab;Ggát;nImYy². 10 16>11 kMNt;mMurgVilkñúgenARtg; ① nig ③ nig kmøaMgRbtikmμenAkñúgcMeNaT 16>10. 16>12 kMNt;m:aRTIsPaBrwgRkaj K sRmab; Ggát;nImYy²rbs;eRKag. yk E = 200GPa I = 270( 6 )mm 4 , A = 6( 3 )mm 2 sRmab;Ggát; 10 10 nImYy². 16>9 kMNt;bgÁúMbMlas;TIRtg;① éncMeNaT 16>8. yk E = 200GPa I = 250(106 )mm 4 , A = 12( 3 )mm 2 sRmab;Ggát;nImYy². 10 16>10 kMNt;m:aRTIsPaBrwgRkaj K sRmab; eRKag. yk E = 200GPa I = 240(106 )mm 4 , A = 6( 3 )mm 2 sRmab;Ggát;nImYy². snμt; ① 10 16>13 kMNt;kmøaMgRbtikmμTRm ① nig ④ nig③ Casnøak; ehIy ②CatMNbgáb;. kñúgcMeNaT 16.13. tMN ① nig ④CatMN snøak; ehIy② nig③ CatMNbgáb;. yk cMeNaT T.Chhay -536
17. 17. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ( ) E = 200GPa I = 270 10 6 mm 4 , A = 9( )mm sRmab;Ggát; nImYy². 10 3 2 16>14 kMNt;m:aRTIsPaBrwgRkaj K sRmab; eRKagEdlmanGgát;BIr. yk E = 200GPa I = 350( )mm , A = 20( )mm sRmab; 10 6 4 10 3 2 Ggát;nImYy². tMN ① nig③ CatMNsnøak; ehIy ②CatMNbgáb;. 16>15 kMNt;kmøaMgRbtikmμTRmenARtg; ① nig③ éncMeNaT 16>14. Problems T.Chhay -537