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Department of Civil Engineering                                                                NPIC




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mhaviTüal½ysMNg;sIuvil                                                viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                            NPIC



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mhaviTüal½ysMNg;sIuvil                                                  viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                                          NPIC




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mhaviTüal½ysMNg;sIuvil                                                   viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                                   NPIC



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mhaviTüal½ysMNg;sIuvil                                                  viTüasßanCatiBhubec©keTskm<úCa
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Department of Civil Engineering                                                                   NPIC



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mhaviTüal½ysMNg;sIuvil                                                    viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                              NPIC



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mhaviTüal½ysMNg;sIuvil                                                                viTüasßanCatiBhubec©keTskm<úCa
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Department of Civil Engineering                                                              NPIC



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mhaviTüal½ysMNg;sIuvil                                                               viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                              NPIC



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mhaviTüal½ysMNg;sIuvil                                                      viTüasßanCatiBhubec©keTskm<úCa
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Department of Civil Engineering                                                             NPIC



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mhaviTüal½ysMNg;sIuvil                                                             viTüasßanCatiBhubec©keTskm<úCa
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Department of Civil Engineering                                                           NPIC



Ggát;elx ! ³ λ x = 1,   ...
mhaviTüal½ysMNg;sIuvil                                                   viTüasßanCatiBhubec©keTskm<úCa
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Department of Civil Engineering                                                             NPIC



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mhaviTüal½ysMNg;sIuvil                                                          viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                                 NPIC


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mhaviTüal½ysMNg;sIuvil                                                 viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                               NPIC



kñúgrUbTI 14-12b Edl...
mhaviTüal½ysMNg;sIuvil                                              viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                                        NPIC


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mhaviTüal½ysMNg;sIuvil                                                    viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                             NPIC



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mhaviTüal½ysMNg;sIuvil                                                         viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                                   NPIC



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mhaviTüal½ysMNg;sIuvil                                                                  viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                                               NPIC



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mhaviTüal½ysMNg;sIuvil                                                                        viTüasßanCatiBhubec©keTskm<ú...
Department of Civil Engineering                                                           NPIC



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mhaviTüal½ysMNg;sIuvil                                            viTüasßanCatiBhubec©keTskm<úCa

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Department of Civil Engineering                                                        NPIC



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14. truss analysis using the stiffness method

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Transcript of "14. truss analysis using the stiffness method"

  1. 1. Department of Civil Engineering NPIC !$> karviPaK truss edayeRbIviFIPaBrwgRkaj Truss analysis using the stiffness method enAkñúgemeronenH eyIgnwgBnül;BIeKalkarN_mUldæanénkareRbIR)as;viFIPaBrwgRkajsRmab;viPaKeRKOg bgÁúM. viFIenHmanPaBsμúKsμajsRmab;karedaHRsayedayéd EtvasaksmsRmab;eRbICamYynwgkMuBüÚT½r. enAkñúgemeronenHmanbgðajBI]TahrN_Gnuvtþn_eTAelI truss kñúgbøg;. bnÞab;mk eyIgnwgeRbIviFIenH sRmab; truss kñúglMh. eyIgnwgerobrab;BIkarGnuvtþviFIenHsRmab;Fñwm nigeRKagenAemeroneRkay. !$>!> eKalkarN_mUldæanénviFIPaBrwgRkaj (Fundamentals of the stiffness method) eKmanmeFüa)ayBIrsRmab;viPaKrcnasm<½n§edayeRbIviFIm:aRTIs. viFIPaBrwgRkajEdlRtUveRbI enAkñúgemeronenH nigemeroneRkayCakarviPaKedayeRbIviFIbMlas;TI. viFIkmøaMg EdleKehAfaviFI flexibility ¬Edlerobrab;enAkñúgkfaxNÐ 10-1¦ k¾GaceRbIedIm,IviPaKrcnasm<½n§ b:uEnþviFIenHminRtUv)an bgðajenAkñúgesovePAenHeT. mUlehtucm,gKW eKGaceRbIviFIPaBrwgRkajsRmab;viPaKrcnasm<½n§kMNt; edaysþaTic nigrcnasm<½n§minkMNt;edaysþaTic cMENkÉviFI flexibility RtUvkardMeNIrkarepSgsRmab; krNInImYy²énkrNITaMgBIr. ehIyviFIPaBrwgRkajpþl;eGaybMlas;TI nigkmøaMgedaypÞal; cMENkÉ viFI flexibility minpþl;eGaybMlas;TIedaypÞal;eT. elIsBIenH eKmanPaBgayRsYlsresrrUbmnþ m:aRTIsEdlcaM)ac;sRmab;RbtþibtþikarkMuBüÚT½redayeRbIviFIPaBrwgRkaj ehIyenAeBleKeFVIvarYc eK GacviPaKeRKOgbgÁúMedaykMuBüÚT½ry:agmanRbsiT§PaB. karGnuvtþviFIPaBrwgRkajTamTarnUvkarbMEbkeRKOgbgÁúMCaes‘rIén finite elements ehIyeK RtUvkMNt;GtþsBaØaNeGaycMNuccugrbs;Ggát;Ca node. sRmab;karviPaK truss, finite element Ca Ggát;nImYy²EdlpSMCa truss ehIy node CatMN. eKRtUvkMNt;lkçN³kmøaMg nigbMlas;TIrbs;Ggát; nImYy² ehIyeKRtUveFVITMnak;TMngrvagkmøaMg nigbMlas;TIedayeRbIsmIkarlMnwgkmøaMgEdlsresrenA Rtg; node. bnÞab;mk eKerobcMTMnak;TMngTaMgenH ¬sRmab;rcnasm<½n§TaMgmUl¦ CaRkumbBa©ÚlKña EdleKeGayeQμaHfa structure stiffness matrix K. enAeBlEdleKbegáItm:aRTIsrYcehIy eKGac kMNt;bMlas;TIrbs; node sRmab;bnÞúkenAelIrcnasm<½n§. enAeBlEdleKsÁal;bMlas;TIrYcehIy eK GackMNt;kmøaMgkñúgrbs;eRKOgbgÁúMedayeRbITMnak;TMngrvagkmøaMg nigbMlas;TIsRmab;Ggát;. karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -464
  2. 2. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa munnwgbegáItdMeNIrkarsRmab;GnuvtþviFIPaBrwgRkaj CaCMhandMbUgeyIgcaM)ac;yl;dwgBIniym n½y nigeKalKMnitmYycMnYn³ karkMNt;GtþsBaØaNrbs;Ggát; nig node³ CMhanmYyénCMhandMbUgkñúgkarGnuvtþviFIPaBrwgRkaj KWkMNt;GtþsBaØaNGgát;rbs;rcnasm<½n§ nig node rbs;va. eyIgnwgkMNt;Ggát;edaybg;elxEdl B½T§CMuvijedaykaer ehIyelxEdlB½T§CMuvijedayrgVg;kMNt;eGay node. ehIy eKk¾RtUvkMNt;cugCit nigcugq¶ayrbs;Ggát;edayeRbIk,alRBYjEdlvacg¥úleTAcugq¶ay. ]TahrN_énkarkMNt;Ggát; node nigTisedAsMrab; truss RtUv)anbgðajenAkñúgrUbTI 14-1a. kUGredaenskl nigkUGredaenGgát;³ edaysarbnÞúk nigbMlas;TICaTMhMviucT½r enaHeKcaM)ac;RtUv begáItRbB½n§kUGredaenedIm,IkMNt;TisedArbs;vaeGay)anRtwmRtUv. enATIenH eyIgnwgeRbIRbB½n§kUGr edaenBIrRbePTepSgKña. RbB½n§kUGredaenskl b¤RbB½n§kUGredaenrcnasm<½n§ ¬ x, y ¦ RtUveRbIedIm,I kMNt;TisedAénbnÞúkxageRkA nigTisedAénbgÁúM;bMlas;TIenARtg; node ¬rUbTI 14-1a¦. RbB½n§kUGr edaentMbn; b¤RbB½n§kUGredaenGgát;RtUv)aneRbIsRmab;Ggát;nImYy²edIm,IkMNt;TisedAénbMlas;TI rbs;va nigkmøaMgkñúgrbs;va. RbB½n§enHRtUv)ankMNt;edayeRbIG½kS x' , y' CamYynwgKl;enARtg; node Cit ehIyG½kS x' latsn§wgeq<aHeTArk node q¶ay. ]TahrN_sRmab;Ggát; truss elx $ RtUv )anbgðajenAkñúgrUbTI 14-1b. PaBminkMNt;sIueNm:aTic³ dUckarerobrab;enAkñúgkfaxNÐ 11-1, degree of freedom Edlminman karTb;sRmab; truss CaGBaØatdMbUgénviFIbMlas;TI dUcenHeKRtUvEtkMNt;va. tamc,ab;TUeTA eKman degree of freedom b¤bMlas;TIEdlGacekItmancMnYnBIr sMrab;tMN (node). sRmab;karGnuvtþ degree of freedom nImYy²RtUv)ankMNt;enAelI truss edayeRbIelxkUd ¬EdlbgðajenARtg;tMN b¤ Truss analysis using the stiffness method T.Chhay -465
  3. 3. Department of Civil Engineering NPIC node ¦ ehIyeKeRbI k,alRBYjedaysMGageTAelIkUGredaensklviC¢man. ]TahrN_ truss enAkñúgrUbTI 14-1a man degree of freedom cMnYn8 EdlRtUv)ankMNt;;edayelxkUdBIelx 1 dl; elx 8 dUcbgðaj. Truss enHminkMNt;edaysIueNm:aTicdWeRkTI5 edaysarbMlas;TIEdlGac ekItmanTaMg 8enH elx1 dl;elx5 CaGBaØat b¤ degree of freedom EdlminmankarTb; ehIyelx 6 dl;elx8Ca degree of freedom EdlmankarTb;. edaysarmankarTb; bMlas;TIenATIenHRtUvesμI sUnü. sRmab;karGnuvtþbnþbnÞab; eyIgeRbIelxkUdtUc²sRmab;sMKal;bMlas;TIEdleyIgminsÁal; ¬degree of freedom EdlminTb;¦ ehIyelxkUcFM²sRmab;sMKal;bMlas;TIEdlsÁal; ¬degree of freedom EdlTb;¦. mUlehtukñúgkareFVIEbbenH edIm,IgayRsYlerobcM structure stiffness matrix dUcenHeyIgnwgGackMNt;bMlas;TIEdlCaGBaØatedaypÞal;. eRkayeBleyIgbg;elxeGay truss ehIykMNt;elxkUd eyIgGacKNna structure stiffness matrix K. edIm,IeFVIEbbenH dMbUgeyIgRtUvbegáIt member stiffness matrix k’ sRmab; Ggát;nImYy²rbs; truss. eKeRbIm:aRTIsenHedIm,IbgáajTMnak;TMngrvagbnÞúk nigbMlas;TIrbs;Ggát; edayeRbIkUGredaentMbn;. edaysarGgát;TaMgGs;rbs; truss minmanTisdUcKña eyIgRtUvbMElgTMhM TaMgenHBIkUGredaentMbn; x' , y' eTACakUGredaenskl x, y edayeRbIm:aRTIsbMElgkmøaMg nig bMlas;TI (force and displacement transformation matrices). eRkaybegáItrYcehIy eyIgGac bMElgm:aRTIsPaBrwgRkajrbs;Ggát;BIkUGredaentMbn;eTACakUGredaenskl ehIybnÞab;mkpÁúMva edIm,IbegáItCam:aRTIsPaBrwgRkajrcnasm<½n§. edayeRbI K ¬dUckarbgðajxagelI¦ dMbUgeyIgGac kMNt;bMlas;TIrbs; node bnÞab;mkeyIgGackMNt;kmøaMgRbtikmμTMr nigcugeRkayKWkmøaMgkñúgrbs; Ggát;. eyIgnwgbegáItviFIenH. !$>@> m:aRTIsPaBrwgRkajrbs;Ggát; (Member stiffness matrix) enAkñúgkfaxNÐenH eyIgnwgbegáItm:aRTIsPaBrwgRkajsMrab;Ggát;eTalrbs; truss edayeRbI kUGredaentMbn; x' , y' dUcbgðajenAkñúgrUbTI 14-2. tYenAkñúgm:aRTIsenHCaTMnak;TMngrvagbnÞúk nig bMlas;TIsRmab;Ggát;. Ggát;rbs; truss Gacpøas;TI)anEttamG½kS x' rbs;vab:ueNÑaH edaysarbnÞúkGnuvtþtamTis enH. dUcenH eKGacmanbMlas;TIÉkraCüBIr. enAeBleKeGaycugCitrbs;Ggát;manbMlas;TIviC¢man karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -466
  4. 4. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dN ehIycMENkÉcugq¶ayRtUv)anTb;edaysnøak; ¬rUbTI 14-2a¦ enaHkmøaMgEdlekItmanenARtg; cugrbs;Ggát;KW AE AE q' N = dN q' F = − dN L L cMNaMfa q' GviC¢manedaysarsßanPaBlMnwg vaRtUveFVIGMeBItamTisedAGviC¢man x' . dUcKña bMlas;TI F viC¢man d enARtg;cugq¶ayedaycugCitenAEtTb;edaysnøak; ¬rUbTI 14-2b¦ pþl;nUvkmøaMgkñúgGgát; F AE AE q' ' N = − dF q' ' F = dF L L edaykareFIVtRmYtpl ¬rUbTI 14-2c¦ kmøaMgers‘ultg;EdlbgáedaybMlas;TITaMgBI AE AE qN = dN − dF (14-1) L L AE AE qF = dF − dN (14-2) L L eKGacsresrsmIkarTMnak;TMngrvagbnÞúk nigkmøaMgkñúgTRmg;m:aRTIs*Ca ⎡q N ⎤ AE ⎡ 1 − 1⎤ ⎡d N ⎤ ⎢ q ⎥ = L ⎢− 1 1 ⎥ ⎢ d ⎥ ⎣ F⎦ ⎣ ⎦⎣ F ⎦ b¤ q = k ' d (14-3) Edl k ' = AE ⎡−11 −11⎤ L ⎢ ⎥ (14-4) ⎣ ⎦ m:aRTIs k ' RtUv)aneKeGayeQμaHfam:aRTIsPaBrwgRkajsRmab;Ggát; ehIyvamanTRmg;dUcKñasRmab; Ggát;nImYy²rbs; truss. tYTaMgbYnEdlbegáItCam:aRTIsenHRtUv)aneKeGayeQμaHfaemKuNT§iBl kRmajsRmab;Ggát; (member stiffness influence coefficient) k' . k' CakmøaMgenARtg;tMN i ij ij enAeBltMN j ekItmanbMlas;TImYyÉktþa. ]TahrN_ RbsinebI i = j = 1 enaH k ' CakmøaMgenA11 Rtg;cugCit enAeBlcugq¶ayRtUv)anTb;edaybgáb; ehIycugCitrgbMlas;TI d = 1 eBalKW N AE q N = k '11 = L dUcKña eKkMNt;kmøaMgenAcugq¶ayBI i = 2 / j =1 dUcenH AE q F = k ' 21 = − L tYTaMgBIrenHCaCYrQrTImYyrbs;m:aRTIsPaBrwgRkajGgát;. tamrebobdUcKña CYrQrTIBIrrbs;m:aRTIs enHCakmøaMgenAkñúgGgát;enAeBlcugq¶ayrbs;Ggát;rgbMlas;TIÉktþa. * ]bsm<½n§ A pþl;eGaynUvkarrMlwkBIm:aRTIs. Truss analysis using the stiffness method T.Chhay -467
  5. 5. Department of Civil Engineering NPIC !$>#> m:aRTIsbMElgénbMlas;TI nigkmøaMg (Displacement and force transformation matrices) edaysar truss pSMeLIgedayGgát;eRcIn eyIgnwg begáItviFIsRmab;bMElgkmøaMgkñúgGgát; q nig bMlas;TI d EdlkMNt;enAkñúgkUGredaentMbn;eGayeTACakUGredaen skl. edIm,IPaBgayRsYl eyIgnwgBicarNakUGredaen sklviC¢man x manTisedAeTAsþaM ehIy y manTisedA eLIgelI. mMurvagG½kSskl x, y nigG½kStMbn; x' , y' RtUv)ankMNt;eday θ x ehIy θ y dUcbgðajenAkñúgrUbTI 14-3. eyIgnwgeRbI kUsIunUsénmMuTaMgenHenAkñúgkarviPaK m:aRTIsdUcteTA. eyIgtag λ x = cosθ x ehIy λ y = cosθ y . eKGackMNt;témøCaelxsRmab; λ x nig λ y y:aggayedayeRbIkMuBüÚT½r enAeBleKkMNt; kUGredaen x, y éncugCit N nigcugq¶ay F rbs; Ggát;rYcehIy. ]TahrN_ eKmanGgát; NF dUcbgðajenAkñúgrUbTI 14-4. enATIenH kUGredaenrbs; N nig F KW (x N , y N ) nig (x F , y F ) erogKña . dUcenH * xF − x N xF − xN λ x = cos θ x = = (14-5) L (x F − xN ) + (yF − y N ) 2 2 yF − yN yF − yN λ y = cos θ y = = (14-6) L (x F − xN ) + (yF − yN ) 2 2 * Kl;rbs;kUGredaenGacsßitenATINak¾)an eGayEtmanlkçN³gayRsYl. b:uEnþ CaTUeTA vaeRcInsßitenARtg;TItaMgNaEdl kUGredaenrbs; node TaMgGs;viC¢man dUcbgðajenAkñúgrUbTI 14-4. karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -468
  6. 6. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa sBaØanBVnþenAkñúgsmIkarTUeTATaMgenHnwgKitedaysV½yRbvtþisRmab;Ggát;EdlsßitenAkñúgkaRdg;NamYy rbs;bøg; xy . m:aRTIsbMElgbMlas;TI³ enAkñúgkUGredaenskl cugnImYy²rbs;Ggát;Gacman degree of freedom b¤ bMlas;TIÉkraCüBIr eBalKWtMN N man DN nig DN ¬rUbTI 14-5a nig 14-5b¦ ehIytMN N man x y D F nig DF ¬rUbTI 14-5c nig14-5d¦. eyIgnwgBicarNabMlas;TITaMgenHdac;edayELkBIKñaedIm,I x y kMNt;bgÁúMbMlas;TIrbs;vatambeNþayGgát;. enAeBlcugq¶ayRtUv)anTb;edaysnøak; ehIycugCitmanbM las;TItamkUGredaenskl DN ¬rUbTI 14-5a¦ bMlas;TI ¬kMhUcRTg;RTay¦ EdlRtUvKñatambeNþay x Ggát;KW DN cosθ x *. dUcKña bMlas;TI DN nwgeFVIeGayGgát;pøas;TI DN cosθ y tambeNþayG½kS x' x y y ¬rUbTI 14-5b¦. T§iBlénbMlas;TIsklTaMgeFVIeGayGgát;pøas;TI. d N = D N x cos θ x + D N y cos θ y tamrebobdUcKña bMlas;TIviC¢man DF nig DF erogKña EdlGnuvtþenARtg;cugq¶ay F cMENkÉcugCit x y RtUv)anTb;edaysnøak; ¬rUbTI 14-5c nig 14-5d¦ nwgeFVIeGayGgát;pøas;TI d F = D Fx cos θ x + D Fy cos θ y edayeGay λ x = cosθ x nig λ y = cosθ y CakUsIunUsR)ab;Tis (direction cosine) sRmab;Ggát; eyIg)an d N = DN x λ x + DN y λ y d F = DFx λ x + DFy λ y EdleKGacsresrvaCaTRmg;m:aRTIs ⎡ DN x ⎤ ⎢ ⎥ ⎡ d N ⎤ ⎡λ x λ y 0 0 ⎤ ⎢ D N y ⎥ ⎢d ⎥ = ⎢ 0 0 λ λ ⎥⎢ D ⎥ (14-7) ⎣ F⎦ ⎣ x y⎦ ⎢ x⎥ F ⎢ D Fy ⎥ ⎣ ⎦ * eKminKitBIbMEbMrYl θ x b¤ θ y edaysarvamantémøtUceBk. Truss analysis using the stiffness method T.Chhay -469
  7. 7. Department of Civil Engineering NPIC b¤ d = TD (14-8) ⎡λ x λ y 0 0 ⎤ Edl T =⎢ ⎥ (14-9) ⎣ 0 0 λx λ y ⎦ BIkarbMEbkxagelI T bMElgBIbMlas;TI D kñúgkUGredaenskl x, y TaMgbYneGayeTACabMlas;TI d kñúg kUGredaentMbn; x' cMnYnBIr. dUcenH T Cam:aRTIsbMElgbMlas;TI. m:aRTIsbMElgkmøaMg³ BicarNakarGnuvtþkmøaMg q eTAelIcugCitrbs;Ggát; cMENkcugq¶ayRtUv)anTb; N edaysnøak; ¬rUbTI 14-6a¦. enATIenH bgÁúMkmøaMgsklrbs; q N enARtg; N KW Q N x = q N cos x Q N y = q N cos θ y dUcKña RbsinebI q F RtUv)anGnuvtþeTAelIr)ar ¬rUbTI 14-6b¦ bgÁúMkmøaMgsklenARtg; F KW Q Fx = q F cos θ x Q Fy = q F cos θ y edayeRbIkUsIunUsR)ab;Tis λ x = cosθ x / λ y = cosθ y smIkarTaMgenHkøayCa QN x = q N λ x QN y = q N λ y Q Fx = q F λ x Q Fy = q F λ y EdleKGacsresrvaCaTRmg;m:aRTIsdUcxageRkam karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -470
  8. 8. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ⎡Q N x ⎤ ⎡ λ x 0⎤ ⎢Q ⎥ ⎢ ⎥ ⎢ N y ⎥ = ⎢λ y 0⎥ ⎡q N ⎤ λx ⎥ ⎢qF ⎥ (14-10) ⎢ QFx ⎥ ⎢ 0 ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ Q Fy ⎥ ⎢ 0 ⎣ ⎦ ⎣ λy ⎥ ⎦ b¤ Q =TTq (14-11) ⎡λ x 0⎤ ⎢λ 0⎥ Edl TT =⎢ ⎢0 y ⎥ λx ⎥ (14-12) ⎢ ⎥ ⎢0 ⎣ λy ⎥ ⎦ enAkñúgkrNIenH T T bMElgBIkmøaMg q EdlmanGMeBIenARtg;cugrbs;Ggát;kñúgkUGredaentMbn; x' eGayeTA CakmøaMg Q EdlmanbgÁúMbYnkñúgkUGredaenskl x, y . tamkareRbobeFob m:aRTIsbMElgkmøaMgCa m:aRTIs transpose énm:aRTIsbMElgbMlas;TI ¬smIkar 14-9¦. !$>$> m:aRTIsPaBrwgRkajrbs;Ggát;kñúgkUGredaenskl (Member global stiffness matrix) eyIgnwgpþúMlT§plenAkñúgkfaxNÐxagelI ehIykMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;EdlTak; TgnwgbgÁúMkmøaMgskl Q nigbMlas;TIskl D rbs;Ggát;. RbsinebIeyIgCMnYssmIkar 14-8 ¬ d = TD ¦ eTAkñúgsmIkar 14-3 ¬ q = k ' d ¦ eyIgGackMNt;kmøaMg q rbs;Ggát;CaGnuKmn_énbMlas;TIskl D enA Rtg;cMNuccugrbs;va eBalKW q = k ' TD (14-13) edayCMnYssmIkarenHeTAkñúgsmIkar 14-11 ¬ Q = T T q ¦ enaHeyIgnwgTTYl)anlT§plcugeRkay Q = T T k ' TD b¤ Q = KD (14-14) Truss analysis using the stiffness method T.Chhay -471
  9. 9. Department of Civil Engineering NPIC Edk k = T T k 'T (14-15) m:aRTIs k Cam:aRTIsPaBrwgRkajsRmab;Ggát;enAkñúgkUGredaenskl. edaysareKsÁal; T T / T nig k ' enaHeyIg)an ⎡λ x 0⎤ ⎢λ 0 ⎥ AE ⎡ 1 − 1⎤ ⎡λ x λ y 0 0 ⎤ k=⎢ ⎥ y ⎢0 λ x ⎥ L ⎢− 1 1 ⎥ ⎢ 0 0 λ x λ y ⎥ ⎣ ⎦⎣ ⎦ ⎢ ⎥ ⎢0 ⎣ λy ⎥⎦ edayKNnaedaHRsaym:aRTIsxagelI eyIg)an Nx Ny Fx Fy ⎡ λ2 x λxλ y − λ2 x − λxλ y ⎤ N x AE ⎢ ⎥ k= ⎢ λx λ y λ2 y − λxλ y − λ2 ⎥ N y y (14-16) L ⎢ ⎢ − λx 2 − λxλ y λ2 x λ x λ y ⎥ Fx ⎥ ⎢− λ x λ y − λ2 λxλ y 2 ⎥F λy ⎦ y ⎣ y TItaMgrbs;tYnImYy³enAkñúgm:aRTIssIuemRTITMhM 4 × 4 tMNageGay degree of freedom sklnImYy²Edl pSMCamYynwgcugCit N nigCamYynwgcugq¶ay F . vaRtUv)anbgðajedaynimitþsBaØénelxkUdEdlenAtam CYredk nigCYrQr eBalKW N x , N y , Fx , Fy . enATIenH k CaTMnak;TMngrvagkmøaMg nigbMlas;TIsRmab; Ggát;enAeBlEdlbgÁúMénkmøaMg nigbMlas;TIenAcugrbs;Ggát;sßitenAkñúgkUGredaenskl b¤G½kS x, y . dUc enHtYnImYy²enAkñúgm:aRTIsCaemKuNT§iBlPaBrwgRkaj (stiffness influence coefficient) K ij Edl bgðajbgÁúMkmøaMg x b¤ y enARtg; i EdlcaM)ac;edIm,IeFVIeGaymanbgÁúMbMlas;TIÉktþa x b¤ y enARtg; j . Ca lT§pl CYrQrnImYy²rbs;m:aRTIstMNageGaybgÁúMkmøaMgbYnEdlekItmanenARtg;cugrbs;Ggát;enAeBl cugGgát;rgbMlas;TIÉktþaEdlTak;TgnwgCYrQrebs;m:aRTIsenaH. ]TahrN_ bMlas;TIÉktþa DN = 1 x nwgbegáItbgÁúMkmøaMgbYnenAelIGgát;EdlbgðajenAkñúgCYrQrTImYyrbs;m:aRTIs. !$>%> ma:RTIsPaBrwgRkajsMrab; truss (Truss stiffness matrix) eRkayeBlbegáItm:aRTIsPaBrwgRkajsRmab;Ggát;enAkñúgkUGredaensklrYcehIy eKcaM)ac;pÁúMBUk vabBa©ÚlKñatamlMdab;d¾RtwmRtUv dUcenHeKnwgTTYl)anm:aRTIsPaBrwgRkaj K sRmab; truss TaMgmUl. dM- eNIrkarénkarpÁúMm:aRTIsGgát;TaMgenHGaRs½ynwgkarkMNt;GtþsBaØaNrbs;Ggát;enAkñúgm:aRTIsGgát;nImYy ². dUckarerobrab;enAkñúgkfaxNÐmun eKRtUvtMerobCYredk nigCYrQrrbs;m:aRTIsedayelxkUdbYn N x , N y , Fx , Fy EdleRbIedIm,IkMNt;GtþsBaØaN degree of freedom sklBIrEdlGacekItmanenARtg; cugnImYy²rbs;Ggát; ¬emIlsmIkar 14-16¦. m:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúMnwgmanlMdab;esμInwg karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -472
  10. 10. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa elxkUdx<s;bMputEdl)ankMNt;eTAelI truss edaysartMNageGaycMnYn degree of freedom srub sRmab;eRKOgbgÁúM. enAeBleKpÁúMm:aRTIs k eKRtUvCMnYstYnImYy²enAkñúg k eTAkñúgCYredk nigCYrQrRtUvKña rbs;m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§ K . enAeBlGgát;BIr b¤Ggát;eRcIntP¢ab;KñaenARtg;tMNEtmYy eKRtUvdak;tYénm:aRTIsrbs;Ggát; k xøHeTAkñúgTItaMgdEdlrbs;m:aRTIs K . eKRtUvbUkbBa©ÚlKñatamlkçN³ nBVnþnUvtYEdlsßitenAkñúgTItaMgdUcKña. eyIgnwgyl;BImUlehtuenH)anc,as; RbsinebIeyIgdwgfatYnImYy² rbs;m:aRTIs k CaersIusþg;rbs;Ggát;Tb;Tl;nwgkmøaMgxageRkAEdlGnuvtþenARtg;cugrbs;va. tamviFIEbb enH karbUkbBa©ÚlKñanUversIusþg;tamTis x nigTis y enAeBlbegáItm:aRTIs K kMNt;nUversIusþg;srubrbs; tMNnImYy²EdlTb;Tl;nwgbMlas;TIÉktþatamTis x b¤tamTis y . ]TahrN_CaelxcMnYnBIrnwgbgðajBIviFIénkarpÁúMm:aRTIssRmab;Ggát;edIm,IbegáItCam:aRTIsPaBrwg RkajsRmab;eRKOgbgÁúM. eTaHbIvadMeNIrkarmanlkçN³sμúKsμajsRmab;karKNnaedayédbnþic Etvaman lkçN³gayRsYlCagsMrab;karbegáItkmμviFIenAelIkMuBüÚT½r. ]TahrN_ 14-1³ kMNt;m:aRTIsPaBrwgRkajsRmab; truss EdlmanGgát;BIrdUcbgðajenAkñúgrUbTI 14-7a. AE mantémøefr. dMeNaHRsay³ tamkarGegát ②manbgÁúMbMlas;TIEdlCaGBaØatcMnYnBIr cMENkÉ ① nig③RtUv)anTb; mineGaymanbMlas;TI. Cavi)ak eKRtUvkMNt;elxkUdeGaybgÁúMbMlas;TIenARtg;tMN ② dMbUgeK ehIy bnþedaytMN ③ nig ① ¬rUbTI14-7b¦. eKalrbs;RbB½n§kUGredaensklGacsßitenAcMNucNak¾)an. edIm,IPaBgayRsYl eyIgnwgeRCIserIstMN ② dUcbgðaj. eyIgGackMNt;elxerogeGayGgát;tam rebobNak¾)an ehIyeKRtUvKUssBaØaRBYjtambeNÞayGgát;TaMgBIredIm,IeGaydwgcugCit nigcugq¶ay bs;Ggát;nImYy². eKGacKNnakUsIunUsR)ab;Tis nigm:aRTIsPaBrwgRkajsRmab;Ggát;nImYy². Truss analysis using the stiffness method T.Chhay -473
  11. 11. Department of Civil Engineering NPIC Ggát;elx1³ edaysar ②CacugCit ehIy ③Cacugq¶ay enaHtamsmIkar14-5 nig14-6 eyIg)an 3−0 0−0 λx = =1 λy = =0 3 3 edayeRbIsmIkar 14-16 nigedayEcktYnImYy²CamYynwg L = 3m eyIg)an 1 2 3 4 ⎡ 0.333 0 − 0.333 0⎤ 1 ⎢ 0⎥ 2 k1 = AE ⎢ 0 0 0 ⎥ ⎢− 0.333 0 0.333 0⎥ 3 ⎢ ⎥ ⎣ 0 0 0 0⎦ 4 eyIgGacRtYtBinitükarKNnaedaycMNaMfa k1 Cam:aRTIssIuemRTI. cMNaMfa CYredk nigCYrQrenAkñúg m:aRTIs k1 RtUv)ankMNt;eday degree of freedom x, y enAcugCit Edlbnþedaycugq¶ay eBalKW 1, 2, 3 nig 4 erogKña sRmab;Ggát;elx1 ¬rUbTI 14-7b¦. eKeFVIEbbenHedIm,IkMNt;tYsRmab;karpÁúMenAkñúg m:aRTIs K . Ggát;elx 2³ edaysar ②CacugCit ehIy ①Cacugq¶ay enaHeyIg)an 3−0 4−0 λx = = 0.6 λy = = 0.8 5 3 dUcenHsmIkar 14-16 CamYynwg L = 5m køayCa 1 2 5 6 ⎡ 0.072 0.096 − 0.072 − 0.096⎤ 1 ⎢ 0.128 − 0.096 − 0.128⎥ 2 k 2 = AE ⎢ 0.096 ⎥ ⎢− 0.072 − 0.096 0.072 0.096 ⎥ 5 ⎢ ⎥ ⎣− 0.096 − 0.128 0.096 0.128 ⎦ 6 enATIenH eKkMNt;CYredk nigCYrQrCa1, 2, 5 nig 6 edaysarelxTaMgenHtMNageGay degree of freedom tamTis x nig y enARtg;cugCit nigcugq¶ayrbs;Ggát;elx 2 . m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§³ vaCam:aRTIsTMhM 6 × 6 edaysarvaman degree of freedom sRmab; truss cMnYn 6 ¬rUbTI 14-7b¦. eKRtUvbUktYEdlRtUvKñaénm:aRTIsTaMgBIrxagelIedIm,IbegáItCa m:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúM. eKRbEhlCaRsYlemIlCagRbsinebIeKBnøatm:aRTIs k1 nig k 2 eGayeTACam:aRTIs 6 × 6 . enaHeK)an K = k1 + k 2 karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -474
  12. 12. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 1 2 3 4 5 6 1 2 3 4 5 6 ⎡ 0.333 0 − 0.333 0 0 0⎤ 1 ⎡ 0.072 0.096 0 0 − 0.072 − 0.096⎤ 1 ⎢ 0 0 0 0 0 ⎥ 0⎥ 2 ⎢ 0.096 0.128 0 0 − 0.096 − 0.128⎥ 2 ⎢ ⎢ ⎥ K = AE ⎢− 0.333 0 0.333 0 0 0⎥ 3 + AE ⎢ 0 0 0 0 0 0 ⎥3 ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 0⎥ 4 ⎢ 0 0 0 0 0 0 ⎥4 ⎢ 0 0 0 0 0 0⎥ 5 ⎢− 0.072 − 0.096 0 0 0.072 0.096 ⎥ 5 ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎣ 0 0 0 0 0⎥ 6 ⎦ ⎢− 0.096 − 0.128 ⎣ 0 0 0.096 0.128 ⎥ 6 ⎦ ⎡ 0.405 0.096 − 0.333 0 − 0.072 − 0.096⎤ ⎢ 0.096 0.128 0 0 − 0.096 − 0.128⎥ ⎢ ⎥ K = AE ⎢ − 0.333 0 0.333 0 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 0 ⎥ ⎢− 0.072 − 0.096 0 0 0.072 0.096 ⎥ ⎢ ⎥ ⎢− 0.096 − 0.128 ⎣ 0 0 0.096 0.128 ⎥ ⎦ RbsinebIeKeFVIdMeNIrkarenHCamYynwgkMuBüÚT½r CaTUeTAeKcab;epþImCamYynwgm:aRTIs K EdlmanFatuTaMg Gs;esμIsUnü bnÞab;mkFatuénm:aRTIsPaBrwgRkajsklsRmab;Ggát;Edl)anKNnarYcehIyRtUv)anCMnYs edaypÞal;eTAkñúgTItaMgFatuEdlRtUvKñaénm:aRTIs K . kareFVIEbbenHvaRbesIrCagkarbegáItm:aRTIsPaBrwg RkajsRmab;Ggát; rUcehIyrkSavaTuk bnÞab;mkeTIbpÁúMva. ]TahrN_ 14-2³ kMNt;m:aRTIsPaBrwgRkajsRmab; truss EdlmanGgát;BIrdUcbgðajenAkñúgrUbTI 14-8a. AE mantémøefr. dMeNaHRsay³ eTaHbICa truss Carcnasm<½n§minkMNt;edaysþaTicdWeRkTImYyk¾eday Etvanwgminbgðaj BIPaBlM)akkñúgkarTTYl)anm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§eT. eKkMNt;elxerogeGaytMN nig Truss analysis using the stiffness method T.Chhay -475
  13. 13. Department of Civil Engineering NPIC Ggát;nImYy² ehIyeKbgðajcugCit nigcugq¶ayedayRBYjtambeNþayGgát;. dUcbgðajenAkñúgrUbTI 14- 8b dMbUgeKkMNt;elxerogkUdeGaybMlas;TIEdlminRtUv)anTb;. eKman degree of freedom cMnUn 8 dUcenH K RtUvCam:aRTIsTMhM 8 × 8 . edIm,IrkSaeGaykUGredaenrbs;tMNTaMgGs;viC¢man eKRtUveRCIserIs eKalrbs;kUGredaensklenARtg; ①. eyIgnwgGnuvtþsmIkar 14-5, 14-6 nig 14-16 eTAelIGgát; nImYy². Ggát;elx 1³ enATIenH L = 10m eyIg)an 10 − 0 0−0 λx = =1 λy = =0 10 10 1 2 6 5 ⎡ 0.1 0 − 0.1 0⎤ 1 ⎢ 0⎥ 2 k1 = AE ⎢ 0 0 0 ⎥ ⎢− 0.1 0 1 0⎥ 6 ⎢ ⎥ ⎣ 0 0 0 0⎦ 5 Ggát;elx 2³ enATIenH L = 10 2m dUcenH 10 − 0 10 − 0 λx = = 0.707 λy = = 0.707 10 2 10 2 1 2 7 8 ⎡ 0.035 0.035 − 0.035 − 0.035⎤ 1 ⎢ 0.035 0.035 − 0.035 − 0.035⎥ 2 k 2 = AE ⎢ ⎥ ⎢− 0.035 − 0.035 0.035 0.035 ⎥ 7 ⎢ ⎥ ⎣− 0.035 − 0.035 0.035 0.035 ⎦ 8 Ggát;elx 3³ enATIenH L = 10m dUcenH 0−0 10 − 0 λx = =0 λy = =1 10 10 1 2 3 4 ⎡0 0 0 0 ⎤1 ⎢0 0.1 0 − 0.1⎥ 2 k 3 = AE ⎢ ⎥ ⎢0 0 0 0 ⎥3 ⎢ ⎥ ⎣0 − 0.1 0 0.1 ⎦ 4 Ggát;elx 4³ enATIenH L = 10m eyIg)an 10 − 0 0−0 λx = =1 λy = =0 10 10 3 4 7 8 ⎡ 0.1 0 − 0.1 0⎤ 3 ⎢ 0⎥ 4 k 4 = AE ⎢ 0 0 0 ⎥ ⎢− 0.1 0 1 0⎥ 7 ⎢ ⎥ ⎣ 0 0 0 0⎦ 8 karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -476
  14. 14. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa Ggát;elx 5³ enATIenH L = 10 2m dUcenH 10 − 0 0 − 10 λx = = 0.707 λy = = −0.707 10 2 10 2 3 4 6 5 ⎡ 0.035 − 0.035 − 0.035 0.035 ⎤ 3 ⎢ 0.035 − 0.035⎥ 4 k 5 = AE ⎢− 0.035 0.035 ⎥ ⎢− 0.035 0.035 0.035 − 0.035⎥ 6 ⎢ ⎥ ⎣ 0.035 − 0.035 − 0.035 0.035 ⎦ 5 Ggát;elx 3³ enATIenH L = 10m dUcenH 0−0 10 − 0 λx = =0 λy = =1 10 10 6 5 7 8 ⎡0 0 0 0 ⎤6 ⎢0 0.1 0 − 0.1⎥ 5 k 6 = AE ⎢ ⎥ ⎢0 0 0 0 ⎥7 ⎢ ⎥ ⎣0 − 0.1 0 0.1 ⎦ 8 m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§³ eKGacpÁúMm:aRTIsTaMg 6 edIm,IbegáItm:aRTIsTMhM 8 × 8 edaykarbUk bBa©ÚlFatuEdlRtUvKña. ]TahrN_ edaysar (k11 )1 = AE (0.1) / (k11 )2 = AE (0.035) / (k11 )3 = (k11 )4 = (k11 )5 = (k11 )6 = 0 enaH K11 = AE (0.1 + 0.035) = AE (0.135) . dUcenHlT§plcugeRkayKW 1 3 2 4 5 6 7 8 ⎡ 0.135 0 0.0350 0 − 0.1 − 0.035 − 0.035⎤ 1 ⎢ 0.035 0 − 0.1 0.135 0 0 − 0.035 − 0.035⎥ 2 ⎢ ⎥ ⎢ 0 0.135 − 0.035 0.035 − 0.035 − 0.1 0 0 ⎥3 ⎢ ⎥ K = AE ⎢ 0 − 0.035 0.135 − 0.035 0.035 − 0.1 0 0 ⎥4 ⎢ 0 0 0.035 − 0.035 0.135 − 0.035 0 − 0.1 ⎥ 5 ⎢ ⎥ ⎢ − 0.1 0 − 0.035 0.035 − 0.035 0.135 0 0 ⎥6 ⎢− 0.035 − 0.035 − 0.1 0 0 0 0.135 0.035 ⎥ 7 ⎢ ⎥ ⎢− 0.035 − 0.035 ⎣ 0 0 − 0.1 0 0.135 0.135 ⎥ 8 ⎦ !$>^> karGnuvtþénviFIPaBrwgRkajsRmab;karviPaK truss (Application of the stiffness method for truss analysis) eRkayeBlbegáItm:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúMrYcehIy eKGaceFVIeGaybgÁúMkmøaMgskl Q EdlmanGMeBIenAelI truss manTMnak;TMngeTAnwgbMlas;TIskl D rbs;vaedayeRbI Q = KD (14-17) Truss analysis using the stiffness method T.Chhay -477
  15. 15. Department of Civil Engineering NPIC eKGacsMKal;smIkarenHCasmIkarPaBrwgRkajsRmab;rcnasm<½n§ (structure stiffness equation). edaysareyIgEtgEtkMNt;elxkUdtUcbMputedIm,IsmÁal; degree of freedom EdlminRtUv)anTb; dUcenH vaGnuBaØateGayeyIgGacbMEbksmIkarenHkñúgTRmg;dUcxageRkam ³ * ⎡Qk ⎤ ⎡ K11 K12 ⎤ ⎡ Du ⎤ ⎢Q ⎥ = ⎢ K ⎥⎢ ⎥ (14-18) ⎣ u ⎦ ⎣ 21 K 22 ⎦ ⎣ Dk ⎦ Edl bnÞúkxageRkA nigbMlas;TIEdleKsÁal;. enATIenH bnÞúkmanGMeBIenAelI truss CaEpñk Qk , Dk = mYyéncMeNaT ehIyCaTUeTAbMlas;TIesμIsUnüedaysarTMrRtUv)anTb; dUcCaTMrsnøak; b¤TMrkl;. Qu , Du = bnÞúk nigbMlas;TIEdlCaGBaØat. enATIenH bnÞúkCakmøaMgRbtikmμTMrEdleKminsÁal; ehIybMlas;TIKWsßitenARtg;tMNEdlminRtUv)anTb;tamTisNamYyeT. K = m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§ EdlRtUv)anbMEbkedIm,IeGaycuHsRmugCamYynwg karbMEbkrbs; Q nig D . edayBnøatsmIkar 14-18 eyIg)an Qk = K11 Du + K12 Dk (14-19) Qu = K 21 Du + K 22 Dk (14-20) CaTUeTA Dk = 0 edaysarTMrminmanbMlas;TI. RbsinebIvaEbbenHEmn enaHsmIkar 14-19 køayCa Qk = K11 Du edaysarFatuenAkñúgm:aRTIs K11 CaersIusþg;srubenARtg;tMN truss edIm,ITb;Tl;bMlas;TIÉktþatamTI x b¤ y enaHsmIkarxagelICakarRbmUlpþúMnUvsmIkarlMnwgkmøaMgEdlGnuvtþeTAelItMNEdlbnÞúkxageRkA esμIsUnü b¤mantémøEdlsÁal; (Qk ) . edayedaHRsayrk Du eyIg)an Du = [K11 ]−1 Qk (14-21) BIsmIkarenH eyIgGacTTYl)andMeNaHRsayedaypÞal;sRmab;bMlas;TIEdlCaGBaØatTaMgGs; bnÞab;mk edayeRbIsmIkar 14-20 CamYynwg Dk = 0 eyIg)an Qu = K 21 Du (14-22) BIsmIkarxagelI eyIgGackMNt;kmøaMgRbtikmμTMr. eKGackMNt;kmøaMgkñúgrbs;Ggát;edayeRbIsmIkar 14-13 eBalKW q = k ' TD edayBnøatsmIkarenH eyIg)an * eyIgnwg)aneXIjBIviFIbMEbkenHenAkñúg]TahrN_xageRkam. karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -478
  16. 16. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ⎡ DN x ⎤ ⎢ ⎥ ⎡q N ⎤ AE ⎡ 1 − 1⎤ ⎡λ x λ y 0 0 ⎤ ⎢ D Ny ⎥ ⎢ q ⎥ = L ⎢− 1 1 ⎥ ⎢ 0 0 λ λ y ⎥ ⎢ DFx ⎥ ⎣ F⎦ ⎣ ⎦⎣ x ⎦⎢ ⎥ ⎢ DFy ⎥ ⎣ ⎦ edaysar q N = −q F edIm,IsßanPaBlMnwg dUcenHeKRtUvkarkMNt;EtkmøaMgmYyb:ueNÑaHkñúgcMeNamkmøaMg TaMgBIr. enATIenH eyIgnwgkMNt; q F kmøaMgEdlGnuvtþkmøaMgTajeTAelIGgát; ¬rUbTI 14-6b¦. ⎡ DN x ⎤ ⎢D ⎥ qF = AE L [ − λx − λ y λx ] λy ⎢ y ⎥ N ⎢ DFx ⎥ (14-23) ⎢ ⎥ ⎢ D Fy ⎥ ⎣ ⎦ RbsinebIlT§plEdl)anBIkarKNnamantémøGviC¢man enaHGgát;rgkarsgát;. dMeNIrkarkñúgkarviPaK (Procedure for analysis) xageRkamCaCMhanEdlpþl;nUvmeFüa)aysRmab;kMNt;bMlas;TI nigkmøaMgRbtikmμTMrEdlCa GBaØatsRmab; truss edayeRbIviFIPaBrwgRkaj. kareFVIkMNt;sMKal;³ begáItRbB½n§kUGredaenskl x, y . CaTUeTAeKalrbs;vasßitenARtg;tMNNaEdleFVIeGay kUGredaensRmab;tMNdéTeTotviC¢man. kMNt;elxerogeGaytMN nigGgát;nImYy² ehIykMNt;cugCit nigcugq¶ayrbs;Ggát;nImYy² edayeRbITisedArbs;sBaØaRBYj Edlk,alRBYjeq<aHeTArkcugq¶ay. kMNt;elxkUdBIrenARtg;tMNnImYy² edayeRbIelxtUcbMputsRmab;sMKal; degree of freedom EdlminmanrgkarTb; cMENkelxFMbMputsRmab;sMKal; degree of freedom EdlmankarTb;. begáIt Dk nig Qk . m:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúM³ sRmab;Ggát;nImYy² kMNt; λ x nig λ y ehIykMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;edayeRbI smIkar 14-16. pÁúMm:aRTIsTaMgenHedIm,IbegáItm:aRTIsPaBrwgRkajsRmab; truss TaMgmUl dUckarBnül;enAkñúg kfaxNÐ 14-5. edaykarRtYtBinitükarKNnaedayEpñk m:aRTIsPaBrwgRkajsRmab;Ggát; nig m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§RtUvEtCam:aRTIssIuemRTI. bMlas;TI nigbnÞúk³ Truss analysis using the stiffness method T.Chhay -479
  17. 17. Department of Civil Engineering NPIC bMEbkm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§ dUcbgðajenAkñúgsmIkar 14-18. kMNt;bMlas;TIrbs;tMNEdlCaGBaØat Du edayeRbIsmIkar 14-21 kmøaMgRbtikmμTMr Qu eday eRbIsmIkar 14-22 ehIykmøaMgkñúgrbs;Ggát;nImYy² q F edayeRbIsmIkar 14-23. ]TahrN_ 14-3³ kMNt;kmøaMgkñúgrbs;Ggát;én truss EdlmanGgát;BIrdUcbgðajenAkñúgrUbTI 14-9a. AE mantémøefr. dMeNaHRsay³ kareFVIkMNt;smÁal;³ eKal x, y nigkarkMNt;elxerogrbs;tMN nigGgát;RtUv)anbgðajenAkñúgrUbTI 14-9b. dUcKña cugCit nigcugq¶ayRtUv)anbgðajkMNt;edaysBaØaRBYj ehIyeKeRbIelxkUdenARtg;tMN nImYy². tamkarGegáteyIgeXIjfabMlas;TI D3 = D4 = D5 = D6 = 0 . ehIybnÞúkxageRkAEdl eyIgsÁal;KW Q1 = 0, Q2 = −2kN . dUcenH ⎡0 ⎤ 3 ⎢0 ⎥ 4 ⎡ 0 ⎤1 Dk = ⎢ ⎥ Qk = ⎢ ⎥ ⎢0 ⎥ 5 ⎣ − 2⎦ 2 ⎢ ⎥ ⎣0 ⎦ 6 m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§³ edayeRbIkareFVIkMNt;smÁal;dUcKña eyIgGacbegáItm:aRTIsPaBrwg RkajsRmab;rcnasm<½n§dUcbgðajenAkñúg]TahrN_ 14-1. bMlas;TI nigbnÞúk³ edaysresrsmIkar 14-17 ¬ Q = KD ¦ sRmab; truss eyIg)an karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -480
  18. 18. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ⎡0 ⎤ ⎡ 0.405 0.096 − 0.333 0 − 0.072 − 0.096⎤ ⎡ D1 ⎤ ⎢ − 2⎥ ⎢ 0.096 0.128 0 0 − 0.096 − 0.128⎥ ⎢ D2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ Q3 ⎥ ⎢ − 0.333 0 0.333 0 0 0 ⎥⎢ 0 ⎥ ⎢ ⎥ = AE ⎢ ⎥⎢ ⎥ (1) ⎢ Q4 ⎥ ⎢ 0 0 0 0 0 0 ⎥⎢ 0 ⎥ ⎢ Q5 ⎥ ⎢− 0.072 − 0.096 0 0 0.072 0.096 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q6 ⎥ ⎣ ⎦ ⎢− 0.096 − 0.128 ⎣ 0 0 0.096 0.128 ⎥ ⎢ 0 ⎥ ⎦⎣ ⎦ BIsmIkarenH eyIgGackMNt; K11 dUcenHeyIgGackMNt; Du . eyIgeXIjfaplKuNm:aRTIs ¬dUc smIkar 14-19¦ eyIg)an ⎡0 ⎤ ⎡0.405 0.096⎤ ⎡ D1 ⎤ ⎡0⎤ ⎢− 2⎥ = AE ⎢0.096 0.128⎥ ⎢ D ⎥ + ⎢0⎥ ⎣ ⎦ ⎣ ⎦⎣ 2 ⎦ ⎣ ⎦ enATIenH eyIgGacedaHRsayy:agRsYledayBnøatedaypÞal; 0 = AE (0.405 D1 + 0.096 D2 ) − 2 = AE (0.096 D1 + 0.128 D2 ) tamrUbviTüa smIkarTaMgenHtMNageGay ∑ Fx = 0 nig ∑ Fy = 0 EdlGnuvtþenARtg;tMN ②. eday edaHRsay eyIg)an 4.505 − 19.003 D1 = D2 = AE AE tamkarGegátrUbTI 14-9b eKrMBwgfatMN ②nwgpøas;TIeTAsþaM nigcuHeRkamdUcbgðajedaysBaØabUk nig sBaØadkéncemøIyenH edayeRbIlT§plTaMgenH eKGacTTYl)ankmøaMgRbtikmμTMrBIsmIkar (1) EdlRtUv)ansresrkñúgTRmg;én smIkar 14-20 ¬b¤smIkar 14-22¦ Ca ⎡Q3 ⎤ ⎡ − 0.333 0 ⎤ ⎡0 ⎤ ⎢Q ⎥ ⎢ 0 ⎥ 0 ⎥ 1 ⎡ 4.505 ⎤ ⎢0⎥ ⎢ 4 ⎥ = AE ⎢ +⎢ ⎥ ⎢Q5 ⎥ ⎢− 0.072 − 0.096⎥ AE ⎢− 19.003⎥ ⎢0⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣Q6 ⎦ ⎣− 0.096 − 0.128⎦ ⎣0 ⎦ edayBnøat nigedaHRsaykmøaMgRbtikmμ Q3 = −0.333(4.505) = −1.5kN Q4 = 0 Q5 = −0.072(4.505) − 0.096(− 19.003) = 1.5kN Q6 = −0.096(4.505) − 0.128(− 19.003) = 2.0kN eKGacKNnakmøaMgenAkñúgGgát;nImYy²BIsmIkar 14-23. edayeRbITinñn½ysRmab; λ x nig λ y enAkñúg smIkar 14-1 eyIg)an Truss analysis using the stiffness method T.Chhay -481
  19. 19. Department of Civil Engineering NPIC Ggát;elx ! ³ λ x = 1, λ y = 0, L = 3m ⎡ 4.505 ⎤ 1 ⎢ ⎥ AE 1 2 3 4 1 ⎢− 19.003⎥ 2 q1 = 3 [− 1 0 1 0] AE ⎢ 0 ⎥ 3 ⎢ ⎥ ⎣ 0 ⎦4 = [− 4.505] = −1.5kN 1 3 Ggát;elx @ ³ λ x = 0.6, λ y = 0.8, L = 5m ⎡ 4.505 ⎤ 1 AE 1 2 5 6 1 ⎢− 19.003⎥ 2 ⎢ ⎥ q2 = 5 [− 0.6 − 0.8 0.6 0.8] AE ⎢ 0 ⎥ 5 ⎢ ⎥ ⎣ 0 ⎦6 = [− 0.6(4.505) − 0.8(− 19.003)] = 2.5kN 1 5 Cak;EsþgeKGacepÞógpÞat;cemøIyTaMgenHedaysmIkarlMnwgEdlGnuvtþenARtg;tMN ②. ]TahrN_ 14-4³ kMNt;kmøaMgRbtikmμTMr nigkmøaMgkñúgrbs;Ggát;elx@ én truss dUcbgðajenAkñúgrUbTI 14-10a. AE mantémøefr. dMeNaHRsay³ kareFVIkMNt;smÁal;³tMN nigGgát;RtUv)ankMNt;elxerog ehIyeKalrbs;G½kS x, y RtUv)anbegáItenA Rtg;tMN ① ¬rUbTI 14-10b¦. ehIysBaØaRBYjRtUv)aneKeRbIedIm,IbgðajcugCit nigcugq¶ayrbs;Ggát; nImYy². edayeRbIelxkUd EdlelxtUcbMputtMNageGay degree of freedom EdlmanrgkarTb; ¬rUb TI 14-16b¦ eyIg)an karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -482
  20. 20. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ⎡ 0 ⎤1 ⎢ 0 ⎥2 ⎡0 ⎤ 6 ⎢ ⎥ D k = ⎢0 ⎥ 7 ⎢ ⎥ Qk = ⎢ 2 ⎥ 3 ⎢ ⎥ ⎢0 ⎥ 8 ⎣ ⎦ ⎢ − 4⎥ 4 ⎢ 0 ⎥5 ⎣ ⎦ m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§³ m:aRTIsenHRtUv)ankMNt;enAkñúg]TahrN_ 14-2 edayeRbIkar eFVIkMNt;smÁal;dUcKñanwgkarbgðajenAkñúg]TahrN_ 14-10b. bMlas;TI nigbnÞúk³ sRmab;cMeNaTenH Q = KD KW ⎡0⎤ ⎡ 0.135 0.035 0 0 0 − 0.1 − 0.035 − 0.035⎤ ⎡ D1 ⎤ ⎢0⎥ ⎢ 0.035 0.135 0 − 0 .1 0 0 − 0.035 − 0.035⎥ ⎢ D2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ 2⎥ ⎢ 0 0 0.135 − 0.035 0.035 − 0.035 − 0.1 0 ⎥ ⎢ D3 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢− 4⎥ = AE ⎢ 0 − 0.1 − 0.035 0.135 − 0.035 0.035 0 0 ⎥ ⎢ D4 ⎥ (1) ⎢0⎥ ⎢ 0 0 0.035 − 0.035 0.135 − 0.035 0 − 0.1 ⎥ ⎢ D5 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q6 ⎥ ⎢ − 0 .1 0 − 0.035 0.035 − 0.035 0.135 0 0 ⎥⎢ 0 ⎥ ⎢Q ⎥ ⎢− 0.035 − 0.035 − 0.1 0 0 0 0.135 0.035 ⎥ ⎢ 0 ⎥ ⎢ 7⎥ ⎢ ⎥⎢ ⎥ ⎢ Q8 ⎥ ⎣ ⎦ ⎢− 0.035 − 0.035 ⎣ 0 0 − 0 .1 0 0.035 0.135 ⎥ ⎢ 0 ⎥ ⎦⎣ ⎦ edayeFVIplKuNdUckarsresrsmIkar 14-18 edIm,IedaHRsaybMlas;TI eyIg)an ⎡0 ⎤ ⎡0.135 0.035 0 0 0 ⎤ ⎡ D1 ⎤ ⎡0⎤ ⎢0 ⎥ ⎢ ⎥ ⎢0.035 0.135 ⎢ 0 − 0 .1 0 ⎥ ⎢ D2 ⎥ ⎢0 ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ = AE ⎢ 0 0 0.135 − 0.035 0.035 ⎥ ⎢ D3 ⎥ + ⎢0⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ − 4⎥ ⎢ 0 − 0.1 − 0.035 0.135 − 0.035⎥ ⎢ D4 ⎥ ⎢0⎥ ⎢0 ⎥ ⎣ ⎦ ⎢ 0 ⎣ 0 0.035 − 0.035 0.135 ⎥ ⎢ D5 ⎥ ⎢0⎥ ⎦⎣ ⎦ ⎣ ⎦ edayBnøat nigedayedaHRsaysmIkarsRmab;bMlas;TI eyIg)an ⎡ D1 ⎤ ⎡ 17.94 ⎤ ⎢D ⎥ ⎢− 69.20⎥ ⎢ 2⎥ 1 ⎢ ⎥ ⎢ D3 ⎥ = ⎢ − 2.06 ⎥ ⎢ ⎥ AE ⎢ ⎥ ⎢ D4 ⎥ ⎢ − 87.14 ⎥ ⎢ D5 ⎥ ⎣ ⎦ ⎢− 22.06⎥ ⎣ ⎦ edaybegáItsmIkar 14-20 BIsmIkar (1) EdleRbIlT§plEdl)anKNna eyIg)an ⎡ 17.94 ⎤ ⎢ − 69.20⎥ 0 ⎡Q6 ⎤ ⎡ − 0.1 0 − 0.035 0.035 − 0.035⎤ ⎥ ⎡ ⎤ ⎢Q ⎥ = AE ⎢− 0.035 − 0.035 − 0.1 1 ⎢ ⎢ 7⎥ ⎢ 0 0 ⎥ ⎥ ⎢ − 2.06 ⎥ + ⎢0⎥ AE ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣Q8 ⎦ ⎢− 0.035 − 0.035 ⎣ 0 0 − 0.1 ⎥ ⎦ ⎢ − 87.14 ⎥ ⎢0⎥ ⎣ ⎦ ⎢− 22.06⎥ ⎣ ⎦ edayBnøat nigKNnakmøaMgRbtikmμTMr eyIg)an Truss analysis using the stiffness method T.Chhay -483
  21. 21. Department of Civil Engineering NPIC Q6 = −4.0kN Q7 = 2.0kN Q8 = 4.0kN sBaØadksRmab; Q6 bgðajfakmøaMgRbtikmμrbs;TMrkl;eFVIGMeBIkñúgTis x GviC¢man. eKGackMNt;kmøaMg enAkñúgGgát;elx@ BIsmIkar 14-23 EdlBIsmIkar 14-2 λ x = 0.707, λ y = 0.707, L = 10 2m dUcenH ⎡ 17.94 ⎤ ⎢− 69.20⎥ q2 = AE [− 0.707 − 0.707 0.707 0.707] 1 ⎢ ⎥ 10 2 AE ⎢ 0 ⎥ ⎢ ⎥ ⎣ 0 ⎦ = 2.56kN ]TahrN_ 14-5³ kMNt;kmøaMgkñúgrbs;Ggát;elx@ énrcnasm<½n§dUcbgðajenAkñúgrUbTI 14-11a. Rb sinebIenARtg;tMN① mansMrut 25mm . yk AE = 8(103 )kN . dMeNaHRsay³ kareFVIkMNt;smÁal;³edIm,IPaBgayRsYl eKRtUvbegáIteKalrbs;kUGredaensklenARtg;tMN ③ dUc bgðajenAkñúgrUbTI 14-11b ehIytamFmμta eKeRbIelxkUdtUcCageKedIm,ItMNageGay degree of freedom EdlminmankarTb;. dUcenH ⎡ 0 ⎤3 ⎢− 0.025⎥ 4 ⎢ ⎥ ⎢ 0 ⎥5 ⎡0 ⎤ 1 Dk = ⎢ ⎥ Qk = ⎢ ⎥ ⎢ 0 ⎥6 ⎣0 ⎦ 2 ⎢ 0 ⎥7 ⎢ ⎥ ⎢ 0 ⎥8 ⎣ ⎦ karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -484
  22. 22. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§³ edayeRbIsmIkar 14-16 eyIg)an Ggát;elx !³ λ x = 0 / λ y = 1 / L = 3m dUcenH 3 4 1 2 ⎡0 0 0 0⎤3 ⎢0 0.333 0 − 0.333⎥ 4 k1 = AE ⎢ ⎥ ⎢0 0 0 0 ⎥1 ⎢ ⎥ ⎣0 − 0.333 0 0.333 ⎦ 2 Ggát;elx @³ λ x = −0.8, λ y = −0.6, L = 5m dUcenH 1 2 5 6 ⎡ 0.128 0.096 − 0.128 − 0.096⎤ 1 ⎢ 0.096 0.072 − 0.096 − 0.072⎥ 2 k 2 = AE ⎢ ⎥ ⎢ − 0.128 − 0.096 0.128 0.096 ⎥ 5 ⎢ ⎥ ⎣− 0.096 − 0.072 0.096 0.072 ⎦ 6 Ggát;elx #³ λ x = 1, λ y = 0, L = 4m dUcenH 7 8 1 2 ⎡ 0.25 0 − 0.25 0⎤ 7 ⎢ 0⎥ 8 k 3 = AE ⎢ 0 0 0 ⎥ ⎢− 0.25 0 0.25 0⎥ 1 ⎢ ⎥ ⎣ 0 0 0 0⎦ 2 edaypÁúMm:aRTIsTaMgenH m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§køayCa 1 2 3 4 5 6 7 8 ⎡ 0.378 0.096 0 0 − 0.128 − 0.096 − 0.25 0⎤ 1 ⎢ 0.096 0.405 0 − 0.333 − 0.096 − 0.072 0 0⎥ 2 ⎢ ⎥ ⎢ 0 0 0 0 0 0 0 0⎥ 3 ⎢ ⎥ K =⎢ 0 − 0.333 0 0.333 0 0 0 0⎥ 4 ⎢ − 0.128 − 0.096 0 0 0.128 0.096 0 0⎥ 5 ⎢ ⎥ ⎢− 0.096 0.072 0 0 0.096 0.072 0 0⎥ 6 ⎢ − 0.25 0 0 0 0 0 0.25 0⎥ 7 ⎢ ⎥ ⎢ 0 ⎣ 0 0 0 0 0 0 0⎥ 8 ⎦ bMlas;TI nigbnÞúk³ enATIenH Q = KD eyIg)an Truss analysis using the stiffness method T.Chhay -485
  23. 23. Department of Civil Engineering NPIC ⎡0⎤ ⎡ 0.378 0.096 0 0 − 0.128 − 0.096 − 0.25 0⎤ ⎡ D1 ⎤ ⎢0⎥ ⎢ 0.096 0.405 0 − 0.333 − 0.096 − 0.072 0 0 ⎥ ⎢ D2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q3 ⎥ ⎢ 0 0 0 0 0 0 0 0⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q4 ⎥ = AE ⎢ 0 − 0.333 0 0.333 0 0 0 0⎥ ⎢− 0.025⎥ ⎢Q5 ⎥ ⎢ − 0.128 − 0.096 0 0 0.128 0.096 0 0⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q6 ⎥ ⎢− 0.096 − 0.072 0 0 0.096 0.072 0 0⎥ ⎢ 0 ⎥ ⎢Q ⎥ ⎢ − 0.25 0 0 0 0 0 0.25 0⎥ ⎢ 0 ⎥ ⎢ 7⎥ ⎢ ⎥⎢ ⎥ ⎢Q8 ⎥ ⎣ ⎦ ⎢ 0 ⎣ 0 0 0 0 0 0 0⎥ ⎢ 0 ⎥ ⎦⎣ ⎦ edayedaHRsaysRmab;bMlas;TI ¬smIkar 14-19¦ eyIg)an ⎡ 0 ⎤ ⎢− 0.025⎥ ⎢ ⎥ ⎡0 ⎤ ⎡0.378 0.096⎤ ⎡ D1 ⎤ ⎡0 0 − 0.128 − 0.096 − 0.25 0⎤ ⎢ 0 ⎥ ⎢0⎥ = AE ⎢0.096 0.405⎥ ⎢ D ⎥ + AE ⎢0 − 0.333 − 0.096 − 0.072 ⎢ 0⎥ ⎢ 0 ⎥ ⎥ ⎣ ⎦ ⎣ ⎦⎣ 2 ⎦ ⎣ 0 ⎦ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎣ ⎦ EdleyIgTTYl)an 0 = AE [(0.378 D1 + 0.096 D2 ) + 0] 0 = AE [(0.096 D1 + 0.405 D2 ) + 0.00833] edayedaHRsayRbB½n§smIkarenH eyIg)an D1 = 0.00556m D2 = −0.021875m eTaHbICaeKminRtUvkarKNnakmøaMgRbtikmμTMrk¾eday EtRbsinebIcaM)ac;eKRtUvKNnavaBIkarBnøatEdl kMNt;edaysmIkar 14-20. edayeRbIsmIkar 14-23 edIm,IkMNt;kmøaMgenAkñúgGgát;elx @ eyIg)an Ggát;elx @³ λ x = −0.8, λ y = −0.6, L = 5m, AE = 8(103 )kN dUcenH ⎡ 0.00556 ⎤ q2 = ( ) 8 10 3 [0.8 0.6 − 0.8 − 0.6]⎢ ⎢− 0.02187 ⎥ ⎥ 5 ⎢ 0 ⎥ ⎢ ⎥ ⎣ 0 ⎦ = 8 10( ) 3 (0.00444 − 0.0131) = −13.9kN 5 edayeRbIdMeNIrkarKNnadUcKña bgðajfakmøaMgenAkñúgGgát;elx ! KW q1 = 8.34kN ehIykmøaMgenAkñúg Ggát;elx # KW q3 = 11.1kN . lT§plRtUv)anbgðajenAkñúgdüaRkamGgÁesrIrbs;tMN ② ¬rUbTI 14- 11c¦ EdleKGacepÞógpÞat;edaysmIkarlMnwg. karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -486
  24. 24. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa !$>&> kUGredaenrbs; node (Nodal coordinates) enAeBlTMrkl;rbs; truss sßitenAelIbøg;eRTt ehIyeKGackMNt;PaBdabsUnüenARtg;TMreday eRbIRbB½n§kUGredaenskltamTisedk nigtamTisQrEtmYy. ]TahrN_ eKman truss enAkñúgrUbTI 14- 12a. eKRtUvkMNt;lkçxNÐénbMlas;TIsUnüenARtg;tMN ① tambeNþayG½kS y' ' ehIyedaysarTMr kl;Gacpøas;TItambeNþayG½kS x' ' dUcenH node enHRtUvmanbgÁúMbMlas;TItamG½kSkUGredaen x, y . sRmab;mUlehtuenH eyIgminGacrYmbBa©ÚllkçxNÐbMlas;TIsUnüenARtg; node enH enAeBlsresr smIkarPaBrwgRkajsklsRmab; truss edayeRbIG½kS x, y edaymineFVIeGaymankarEktRmUvdMeNIr karviPaKm:aRTIs. edIm,IedaHRsaycMeNaTenH eyIgGacbBa©ÚlvaeTAkñgkarviPaKkMuBüÚT½redayRsYl eyIgnwgeRbIsMnMu ú kUGredaenrbs; node x' ' , y' ' enARtg;TMreRTt. eKRtUveFVIeGayG½kSTaMgenHmanTItaMgy:agNaedIm,I eGaykmøaMgRbtikmμTMr nigbMlas;TIrbs;TMrpøas;TItambeNþayG½kSkUGredaennImYy² ¬rUbTI 14-12a¦. edIm,IKNnasmIkarPaBrwgRkajsklsRmab; truss enaHeKcaM)ac;begáItm:aRTIsbMElgkmøaMg nigma:RTIs bMElgbMlas;TIsRmab;Ggát;EdltP¢ab;eTAnwgTMrenaH dUcenHeKGaceFVIplbUklT§plTaMgenHenAkñúgRbB½n§ kUGredaenskl x, y dUcKña. edIm,IbgðajBIrebobénkarGnuvtþ eyIgRtUvBicarNaGgát; truss elx! enA Truss analysis using the stiffness method T.Chhay -487
  25. 25. Department of Civil Engineering NPIC kñúgrUbTI 14-12b EdlmanRbB½n§kUGredaenskl x, y enARtg;cugCit N ehIyRbB½n§kUGredaenrbs; node x' ' , y ' ' enARtg;cugq¶ay F . enAeBlbMlas;TI D ekIteLIg dUcenHBYkvamanbgÁúMtambeNþayG½kS nImYy²dUcbgðajenAkñúgrUbTI 14-12c enaHbMlas;TItamTis x tambeNþaycugGgát;nImYy²køayCa d N = D N x cos θ x + D N y cos θ y d F = DFx '' cos θ x '' + DFy '' cos θ y '' eKGacsresrsmIkarTaMgenHenAkñúgTRmg;m:aRTIs ⎡ DN x ⎤ ⎢ ⎥ ⎡ d N ⎤ ⎡λ x λ y 0 0 ⎤ ⎢ DN y ⎥ ⎢d ⎥ = ⎢ 0 0 λ λ y '' ⎥ ⎢ DFx '' ⎥ ⎣ F⎦ ⎣ x '' ⎦⎢ ⎥ ⎢ D Fy '' ⎥ ⎣ ⎦ dUcKña kmøaMg q enARtg;cugCit nigcugq¶ayrbs;Ggát; ¬rUbTI 14-12d¦ manbgÁúM Q tambeNþayG½kSskl Q N x = q N cos θ x Q N y = q N cos θ y Q Fx '' = q F cos θ x '' Q Fy '' = q F cos θ y '' EdleKGacsresrCa ⎡ Q N x ⎤ ⎡λ x 0 ⎤ ⎢Q ⎥ ⎢ ⎥ ⎢ N y ⎥ = ⎢λ y 0 ⎥ ⎡ q N ⎤ ⎢Q Fx '' ⎥ ⎢ 0 λ x '' ⎥ ⎢ q F ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢Q Fy '' ⎥ ⎢ 0 λ y '' ⎥ ⎣ ⎦ ⎣ ⎦ eKeRbIm:aRTIsbMElgbMlas;TI nigm:aRTIskmøaMgenAkñúgsmIkarxagelIedIm,IbegáItm:aRTIsPaBrwgRkaj sRmab;Ggát;enAkñúgsßanPaBenH. edayGnuvtþsmIkar 14-15 eyIg)an k = T T k 'T ⎡λ x 0 ⎤ ⎢λ ⎥ k= ⎢ y 0 ⎥ AE ⎡ 1 − 1⎤ ⎡λ x λ y 0 0 ⎤ ⎢− 1 1 ⎥ ⎢ 0 0 λ ⎢ 0 λ x '' ⎥ L ⎣ ⎥ ⎦⎣ x '' λ y '' ⎦ ⎢ ⎥ ⎢ 0 λ y '' ⎥ ⎣ ⎦ edayKNnam:aRTIsxagelI eyIgTTYl)an ⎡ λ2 x λ x λ y − λ x λ x '' − λ x λ y '' ⎤ ⎢ ⎥ AE ⎢ λ x λ y λ2 y − λ y λ x '' − λ y λ y '' ⎥ k= (14-24) L ⎢ − λ x λ x '' − λ y λ x '' λ2 '' λ x '' λ y '' ⎥ ⎢ x ⎥ ⎢− λ x λ y '' − λ y λ y '' λ x '' λ y '' λ2 '' ⎥ ⎣ y ⎦ bnÞab;mkeKGaceRbIm:aRTIsPaBrwgRkajsRmab;Ggát;nImYy²EdlRtUvtP¢ab;eTAnwgTMrkl;EdleRTt ehIy dMeNIrkarpÁúMm:aRTIsedIm,IbegáItm:aRTIsPaBrwgRkajrcnasm<½n§GnuvtþtamdMeNIrkarbTdæan. ]TahrN_xag karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -488
  26. 26. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa eRkambgðajBIkarGnuvtþrbs;va. ]TahrN_ 14-6³ kMNt;kmøaMgRbtikmμTMrsRmab; truss dUcbgðajenAkñúgrUbTI 14-13a. dMeNaHRsay³ kareFVIkMNt;smÁal;³edaysarTMrkl;enARtg; ② sßitenAelIbøg;eRTt eyIgRtUveRbIkUGredaenrbs; node enARtg; node enH. eKRtUvkMNt;elxerogeGaytMN nigGgát; ehIybegáItkUGredaen x, y enARtg; node ③ ¬rUbTI 14-13b¦. cMNaMfa elxkUd # nig$ sßitenAtambeNþayG½kS x' ' , y ' ' edIm,IeRbIlkçxNÐ Edl D4 = 0 . m:aRTIsPaBrwgRkajsRmab;Ggát;³ eKRtUvbegáItm:aRTIsPaBrwgRkajsRmab;Ggát;elx ! nigelx @ eday eRbIsmIkar 14-24 edaysarGgát;TaMgenHmanelxkUdtamTisénG½kSskl nigG½kSrbs;kUd. eKRtUv kMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;elx # tamrebobFmμta. Ggát;elx !³ rUbTI 14-13c/ λ x = 1, λ y = 0, λ x'' = 0.707, λ y'' = −0.707 Truss analysis using the stiffness method T.Chhay -489
  27. 27. Department of Civil Engineering NPIC 5 6 3 4 ⎡ 0.25 0 − 0.17675 0.17675⎤ 5 ⎢ 0 ⎥6 k1 = AE ⎢ 0 0 0 ⎥ ⎢− 0.17675 0 0.125 − 0.125 ⎥ 3 ⎢ ⎥ ⎣ 0.17675 0 − 0.125 0.125 ⎦ 4 Ggát;elx @³ rUbTI 14-13d/ λ x = 0, λ y = −1, λ x '' = −0.707, λ y '' = −0.707 1 2 3 4 ⎡ 0 0 0 0 ⎤1 ⎢0 0.333 − 0.2357 − 0.2357⎥ k 2 = AE ⎢ ⎥2 ⎢0 − 0.2357 0.1667 0.1667 ⎥ 3 ⎢ ⎥ ⎣0 − 0.2357 0.1667 0.1667 ⎦ 4 Ggát;elx #³ λ x = 0.8, λ y = 0.6 1 2 3 4 ⎡ 0.128 0.096 − 0.128 − 0.096⎤ 5 ⎢ 0.096 0.072 − 0.0.96 − 0.072⎥ 6 k 3 = AE ⎢ ⎥ ⎢− 0.128 − 0.096 0.128 0.096 ⎥ 1 ⎢ ⎥ ⎣− 0.096 − 0.072 0.0.96 0.072 ⎦ 2 m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§³ pÁúMm:aRTIsTaMgenHedIm,IkMNt;m:aRTIsPaBrwgRkajsRmab;rcna sm<½n§ eyIg)an ⎡ 30 ⎤ ⎡ 0.128 0.096 0 0 − 0.128 − 0.096⎤ ⎡ D1 ⎤ ⎢0⎥ ⎢ 0.096 0.4053 − 0.2357 − 0.2357 − 0.096 − 0.072⎥ ⎢ D2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢0⎥ ⎢ 0 − 0.2357 0.2917 0.0417 − 0.17675 0 ⎥ ⎢ D3 ⎥ ⎢ ⎥ = AE ⎢ ⎥⎢ ⎥ ⎢Q4 ⎥ ⎢ 0 − 0.2357 0.417 0.2917 0.17375 0 ⎥⎢ 0 ⎥ ⎢Q5 ⎥ ⎢ − 0.128 − 0.096 − 0.17675 0.17675 0.378 0.096 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢Q6 ⎥ ⎣ ⎦ ⎢− 0.096 − 0.072 ⎣ 0 0 0.096 0.072 ⎥ ⎢ 0 ⎥ ⎦⎣ ⎦ edayeFVIplKuNm:aRTIsénm:aRTIsEpñkxagelI enaHeyIgGackMNt;bMlas;TI D EdlCaGBaØatBIkaredaH RsayRbB½n§smIkar eBalKW 352.5 D1 = AE − 157.5 D2 = AE − 127.3 D3 = AE eKGacTTYl)ankmøaMgRbtikmμ Q BIplKuNm:aRTIsénm:aRTIsxageRkamenAkñúgsmIkar (1). edayeRbI bMlas;TIEdl)anKNna eyIg)an Q4 = 0(352.5) − 0.2357(− 157.5) + 0.0417(− 127.3) karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -490
  28. 28. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa = 31.8kN Q5 = −0.128(352.5) − 0.096(− 157.5) − 0.17675(− 127.3) = −7.5kN Q6 = −0.096(352.5) − 0.072(− 157.5) + 0(− 127.3) = −22.5kN !$>*> Trusses EdlmanbMErbMrYlsItuNðaPaB nigkMhusénplitkmμ (Truss having thermal changes and fabrication errors) RbsinebIGgát;xøHrbs; truss rgnUvkarlUt b¤rYjEdlbNþalBIbMErbMrYlsItuNðPaB b¤kMhusén plitkmμ enaHeKcaM)ac;RtUveRbIviFItRmYtpledIm,ITTYl)andMeNaHRsay. vaRtUvkarbICMhan. dMbUg eK caM)ac;RtUvKNnakmøaMgbgáb;cugEdlkarBarkarcl½trbs; node EdlbNþalBIsItuNðPaB b¤kMhusén plitkmμ. CMhanTIBIrKWeKRtUvdak;kmøaMgEdlesμIKña b:uEnþmanTispÞúyKñaenAelI truss Rtg; node ehIyeK RtUvKNnabMlas;TIrbs; node edayeRbIkarviPaKm:aRTIs. cugeRkay eKkMNt;kmøaMgCak;EsþgenAkñúg Ggát; nigkmøaMgRbtikmμenAelI truss edaykareFVItRmYtplénlT§plTaMgBIrenH. Cak;Esþg eKRtUvkar dMeNIrkarenH RbsinebI truss Carcnasm<½n§minkMNt;edaysþaTic. RbsinebI truss Carcnasm<½n§kMNt; edaysþaTic eKGackMNt;bMlas;TIenARtg; node edayviFIenH b:uEnþbMErbMrYlsItuNðPaB nigkMhusén plitkmμnwgminmanT§iBleTAelIkmøaMgRbtikmμ nigkmøaMgkñúgrbs;Ggát; edaysareKGacEktRmUvbERm bRmYlRbEvgrbs;Ggát; truss edayesrI. T§iBlkMedA³ RbsinebIGgát; truss manRbEvg L rgkMeNInsItuNðPaB ΔT RbEvgrbs;Ggát; nwgmankMhUcRTg;RTay ΔL = αΔTL Edl α Ca emKuNrIkedaysarkMedA. kmøaMgsgát; qo Edl GnuvtþeTAelIGgát;nwgeFVIeGayRbEvgrbs;Ggát; rYj)anRbEvg ΔL' = qo L / AE . RbsinebIeyIg dak;eGaybMlas;TITaMgBIresμIKña enaH qo = AEαΔT . kmøaMgenHnwgTb;Ggát;dUcbgðajenAkñúgrUbTI 14-14 dUcenHeyIg)an (q N )0 = AEαΔT (q F )0 = − AEαΔT Truss analysis using the stiffness method T.Chhay -491
  29. 29. Department of Civil Engineering NPIC eKRtUvdwgfa RbsinekItmankarfykMedA enaH ΔT køayCaGviC¢man ehIykmøaMgTaMgenHnwgbRBa©asTisedA edIm,IeFVIeGayGgát;sßitenAkñúgsßanPaBlMnwg. eyIgGacbMElgkmøaMgTaMgBIrenHeTAkñúgkUGredaenskledayeRbIsmIkar 14-10 EdleFVIeGay ( ⎡ QN x )0 ⎤ ⎡λ x 0 ⎤ ⎡ λx ⎤ ( ) ⎢Q ⎢ Ny ⎥ ⎢ λ 0⎥ = ⎢ y 0⎥ ⎥ ⎡1⎤ ⎢λ ⎥ ⎢ y ⎥ ( ) ⎢ QF ⎥ ⎢ 0 λ x ⎥ AEαΔT ⎢− 1⎥ = AEαΔT ⎢ − λ x ⎥ ⎣ ⎦ (14-25) ( ) ⎢ x ⎢ Q Fy ⎣ 0⎥ 0⎥ ⎢ ⎦ ⎣ ⎢ ⎥ 0 λy ⎥ ⎦ ⎢ ⎢− λ y ⎥ ⎣ ⎥ ⎦ kMhusqÁgkñúgplitkmμ³ RbsinebIeKeFVIeGayGgát;EvgCaRbEvgedImedayTMhM ΔL muxeBlP¢ab;vaeTAnwg truss enaHkmøaMg qo EdlcaM)ac;edIm,IrkSaGgát;RtwmRbEvgDIsaj L KW qo = AEΔL / L dUcenHsRmab; Ggát;enAkñúgrUbTI 14-14 eyIg)an AEΔL (q N )0 = L AEΔL (q F )0 =− L RbsinebIGgát;enHxøICagRbEvgedIm enaH ΔL køayCaGviC¢man ehIykmøaMgTaMgenHnwgbRBa©as. enAkñúgkUGredaenskl kmøaMgTaMgenHKW ( ⎡ QN x )0 ⎤ ⎡ λx ⎤ ⎢Q( ) ⎢ Ny ⎥ ⎢λ ⎥ 0 ⎥ = AEΔL ⎢ y ⎥ ( ) ⎢ QF ⎥ L ⎢− λ x ⎥ (14-26) ( ) ⎢ x ⎢ Q Fy ⎣ 0⎥ 0⎦⎥ ⎢ ⎣ ⎥ ⎢− λ y ⎥ ⎦ karviPaKm:aRTIs³ enAkñúgkrNITUeTA CamYy truss rgkmøaMgGnuvtþ bERmbRmYlsItuNðPaB nigkMhusén plitkmμ TMnak;TMngrvagkmøaMgkñúg nigbMlas;TIsRmab; truss enaHvakøayCa Q = KD + Q0 (14-27) enATIenH Q0 Cam:aRTIsCYrQrsRmab; truss TaMgmUlrbs;kmøaMgbgáb;cugEdlbNþalBIbERmbRmYl sItuNðPaB nigkMhusénplitkmμrbs;Ggát;EdlkMNt;enAkñúgsmIkar 14-25 nig 14-26. eyIgGacEbg EcksmIkarenHenAkñúgTRmg;dUcxageRkam ⎡Qk ⎤ ⎡ K11 K12 ⎤ ⎡ Du ⎤ ⎡(Qk )0 ⎤ ⎢Q ⎥ = ⎢ K ⎥⎢ ⎥ + ⎢ ⎥ ⎣ u ⎦ ⎣ 21 K 22 ⎦ ⎣ Dk ⎦ ⎣(Qu )0 ⎦ edayedaHRsaym:aRTIsenAGgÁxagsþaM eyIgTTYl)an Qk = K11 Du + K 21 Dk + (Qk )0 (14-28) Qu = K 21 Du + K 22 Dk + (Qu )0 (14-29) karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -492
  30. 30. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa eyagtamdMeNIrkartRmYtplEdlerobrab;xagelI eyIgGackMNt;bMlas;TI Du BIsmIkarTImYyedaydk K12 Dk nig (Qk )0 BIGgÁTaMgBIr bnÞab;mkeyIgedaHRsay Du . eyIgTTYl)an Du = K111 (Qk − K12 Dk − (Qk )0 ) − eRkayeBleyIgTTYl)anbMlas;TIrbs; node enaHeyIgGackMNt;kmøaMgkñúgrbs;Ggát;edayviFItRmYtpl eBalKW q = k ' TD + q0 RbsinebIeyIgBnøatsmIkarenHedIm,IkMNt;kmøaMgenAcugq¶ayrbs;Ggát; eyIgTTYl)an ⎡ DN x ⎤ ⎢D ⎥ qF = AE L [ − λx − λ y λx ] λ y ⎢ y ⎥ − (q F )0 N ⎢ DFx ⎥ (14-30) ⎢ ⎥ ⎢ D Fy ⎥ ⎣ ⎦ lT§plenHRsedogKñaeTAnwgsmIkar 14-23 EtvaxusKñaRtg;enATIenHvamanplbUkéntY (q F )0 EdlCa kmøaMgbgáb;cugrbs;Ggát;EdlbNþalBIbERmbRmYlsItuNðPaB nig / b¤kMhusénplitkmμdUckMNt;dUcxag elI. eKRtUvdwgfa RbsinebIlT§plEdlTTYl)anBIsmIkarenHmantémøGviC¢man enaHGgát;nwgrgkmøaMg sgát;. ]TahrN_TaMgBIrxageRkam nwgbgðajBIkarGnuvtþéndMeNIrkarrbs;viFIenH. ]TahrN_ 14-7³ kMNt;kmøaMgkñúgGgát;elx ! nig elx @ rbs; truss EdlmanTMrsnøak;dUcbgðajenA kñúgrUbTI 14-15 RbsinebIeKeFVIeGayGgát;elx @ xøICagmun 0.01 munnwgpÁúMvaeTAkñúg truss. yk AE = 8(10 3 )kN . dMeNaHRsay³ edaysarGgát;manRbEvgxøI enaH ΔL = −0.01m dUcenHGnuvtþsmIkar 14-26 eTAelIGgát;elx @ CamYy nwg λ x = −0.8, λ y = −0.6 eyIg)an ⎡ (Q1 )0 ⎤ ⎡ − 0.8 ⎤ ⎡ 0.0016 ⎤ 1 ⎢(Q ) ⎥ ⎢ − 0.6 ⎥ ⎢ ⎥ ⎢ 2 0 ⎥ = AE (− 0.01) ⎢ ⎥ = AE ⎢ 0.0012 ⎥ 2 ⎢(Q5 )0 ⎥ 5 ⎢ 0.8 ⎥ ⎢− 0.0016⎥ 5 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢(Q6 )0 ⎥ ⎣ ⎦ ⎣ 0.6 ⎦ ⎣− 0.0012⎦ 6 Truss analysis using the stiffness method T.Chhay -493
  31. 31. Department of Civil Engineering NPIC eK)anbegáItm:aRTIsPaBrwgRkajsRmab;rcnasm<n§enAkñúg]TahrN_ 14-4. edayGnuvtþsmIkar 14-27 ½ eyIg)an ⎡0⎤ ⎡ 0.378 0.096 0 0 − 0.128 − 0.096 − 0.25 0⎤ ⎡ D1 ⎤ ⎡ 0.0016 ⎤ ⎢0⎥ ⎢ 0.096 0.405 0 − 0.333 − 0.096 − 0.072 0 ⎥⎢D ⎥ 0⎥ ⎢ 2 ⎥ ⎢ 0.0012 ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢Q3 ⎥ ⎢ 0 0 0 0 0 0 0 0⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢Q4 ⎥ = AE ⎢ 0 − 0.333 0 0.333 0 0 0 0⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ + AE ⎢Q5 ⎥ ⎢ − 0.128 − 0.096 0 0 0.128 0.096 0 0⎥ ⎢ 0 ⎥ ⎢− 0.0016⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢Q6 ⎥ ⎢− 0.096 − 0.072 0 0 0.096 0.072 0 0⎥ ⎢ 0 ⎥ ⎢− 0.0012⎥ ⎢Q ⎥ ⎢ − 0.25 0 0 0 0 0 0.25 0⎥ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢ 7⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢Q8 ⎥ ⎣ ⎦ ⎢ 0 ⎣ 0 0 0 0 0 0 0⎥ ⎢ 0 ⎥ ⎦⎣ ⎦ ⎢ ⎣ 0 ⎥ ⎦ edayEbgEckm:aRTIsenHdUcbgðaj nigedayedaHRsayplKuNm:aRTIsedIm,ITTYl)ansmIkarsRmab; bMlas;TI eyIgTTYl)an ⎡0⎤ ⎢0⎥ ⎢ ⎥ ⎡0 ⎤ ⎡0.378 0.096⎤ ⎡ D1 ⎤ ⎡0 0 − 0.128 − 0.096 − 0.25 0⎤ ⎢0⎥ ⎡0.0016⎤ ⎢0⎥ = AE ⎢0.096 0.405⎥ ⎢ D ⎥ + AE ⎢0 − 0.333 − 0.096 − 0.072 ⎥ ⎢0⎥ + AE ⎢0.0012⎥ ⎣ ⎦ ⎣ ⎦⎣ 2 ⎦ ⎣ 0 0⎦ ⎢ ⎥ ⎣ ⎦ ⎢0⎥ ⎢ ⎥ ⎢0⎥ ⎣ ⎦ EdleGay 0 = AE [0.378 D1 + 0.096 D2 ] + AE [0] + AE [0.0016] 0 = AE [0.096 D1 + 0.405 D2 ] + AE [0] + AE [0.0012] edaHRsayRbB½n§smIkar eyIgTTYl)an D1 = −0.003704m D2 = −0.002084m eTaHbICaminRtUvkar eKGackMNt;kmøaMgRbtikmμ Q BIkarBnøatsmIkar (1) EdlGnuvtþtamKMrUénsmIkar 14-29. edIm,IkMNt;kmøaMgenAkñúgGgát;elx ! nigelx @ eyIgRtUvGnuvtþsmIkar 14-30 EdlenAkñúgkrNI enH eyIg)an Ggát;elx !³ λ x = 0, λ y = 1, L = 3m, AE = 8(103 )kN dUcenH ⎡ 0 ⎤ q1 = ( ) 8 10 3 [0 − 1 0 1] ⎢ ⎢ 0 ⎥ ⎥ + [0] 3 ⎢− 0.003704⎥ ⎢ ⎥ ⎣− 0.002084⎦ karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -494
  32. 32. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa q1 = −5.56kN Ggát;elx @³ λ x = −0.8, ( ) dUcenH λ y = −0.6, L = 5m, AE = 8 10 3 kN ⎡− 0.003704⎤ q2 = ( ) 8 10 3 ⎢− 0.002084⎥ [0.8 0.6 − 0.8 − 0.6]⎢ ⎥ − 8 10 (− 0.01) 3 ( ) 5 ⎢ 0 ⎥ 5 ⎢ ⎥ ⎣ 0 ⎦ q 2 = 9.26kN ]TahrN_ 14-8³ Ggát;elx @ rbs; Edl truss bgðajenA kñúgrUbTI 14-16 rgnUvkMeNInsItuNðPaB 83.3o C . kMNt;kmøaMgEdlekItmanenAkñúgGgát;elx @. yk α = 11.7(10 −6 )/ o C / E = 200GPa . Ggát; nImYy²manRkLaépÞmuxkat; A = 484mm 2 . dMeNaHRsay³ edaysar vamankMeNInsItuNðPaB ΔT = +83.3o C . GnuvtþsmIkar 14-25 eTAelIGgát;elx @ Edl λ x = 0.707, λ y = 0.707 eyIg)an ⎡ (Q1 )0 ⎤ ⎡ 0.707 ⎤ ⎡ 0.000689325 ⎤ 1 ⎢(Q ) ⎥ ⎢ 0.707 ⎥ ⎢ ⎥ ⎢ (Q3 )0 ⎥ ( ) ⎢ 2 0 ⎥ = AE (11.7 ) 10 −6 (83.3)⎢ ⎥ = AE ⎢ 0.000689325 ⎥ 2 ⎢− 0.707 ⎥ ⎢− 0.000689325⎥ 7 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣(Q4 )0 ⎦ ⎣− 0.707 ⎦ ⎣− 0.000689325⎦ 8 m:aRTIsPaBrwgRkajsRmab; truss enHRtUv)anbegáItenAkñúg]TahrN_ 14-2. ⎡ 0 ⎤ ⎡ 0.135 0.035 0 0 0 − 0.1 − 0.035 − 0.035⎤ ⎡ D1 ⎤ ⎡ 0.000689325 ⎤ 1 ⎢ 0 ⎥ ⎢ 0.035 ⎥ ⎢D ⎥ ⎢ 0.000689325 ⎥ 2 ⎢ ⎥ ⎢ 0.135 0 − 0.1 0 0 − 0.035 − 0.035⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ 0 0 0.135 − 0.035 0.035 − 0.035 − 0.1 0 ⎥ ⎢ D3 ⎥ ⎢ 0 ⎥3 ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ = AE ⎢ 0 − 0.1 − 0.035 0.135 − 0.035 0.035 0 ⎥ ⎢ D4 ⎥ + AE ⎢ ⎥4 0 0 ⎢ 0 ⎥ ⎢ 0 0 0.035 − 0.035 0.135 − 0.035 0 − 0.1 ⎥ ⎢ D5 ⎥ ⎢ 0 ⎥5 ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢Q 6 ⎥ ⎢ − 0.1 0 − 0.035 0.035 − 0.035 0.135 0 0 ⎥⎢ 0 ⎥ ⎢ 0 ⎥6 ⎢Q ⎥ ⎢− 0.035 − 0.035 − 0.1 0 0 0 0.135 0.035 ⎥⎢ 0 ⎥ ⎢− 0.000689325⎥ 7 ⎢ 7⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣Q8 ⎥ ⎢ ⎦ ⎢− 0.035 ⎣ − 0.035 0 0 − 0.1 0 0.035 0.135 ⎥ ⎢ 0 ⎥ ⎦⎣ ⎦ ⎢− 0.000689325⎥ 8 ⎣ ⎦ edayBnøatedIm,IkMNt;smIkarbMlas;TIEdlCaGBaØat nigedayedaHRsayRbB½n§smIkarenH eyIg)an D1 = −0.002027m D2 = −0.01187 m Truss analysis using the stiffness method T.Chhay -495
  33. 33. Department of Civil Engineering NPIC D3 = −0.002027m D4 = −0.009848m D5 = −0.002027m edayeRbIsmIkar 14-30 edIm,IkMNt;kmøaMgenAkñúgGgát;elx @ eyIg)an ⎡− 0.002027⎤ ⎢ − 0.01187 ⎥ 484[200] q2 = 10 2 [− 0.707 − 0.707 0.707 0.707]⎢ ⎢ 0 ⎥ − 484(200)11.7 10 −6 (83.3) ⎥ [ ( )] ⎢ ⎥ ⎣ 0 ⎦ = −27.09kN cMNaMfa kMeNInsItuNðPaBénGgát;elx @ nwgmineFVIeGaymankmøaMgRbtikmμenAelI truss eT edaysarva Ca truss kMNt;edaysþaTic. edIm,IbgðajBIkarBicarNakarBnøatm:aRTIsénsmIkar (1) edIm,IkMNt;kmøaMg Rbtikmμ. edayeRbIlT§plsRmab;bMlas;TI eyIg)an Q6 = AE[− 0.1(− 0.002027 ) + 0 − 0.035(− 0.002027 ) + 0.035(− 0.009828) − 0.035(− 0.002027 )] + AE [0] = 0 Q7 = AE[− 0.035(− 0.002027 ) − 0.035(− 0.01187 ) − 0.1(− 0.002027 ) + 0 + 0] + AE [− 0.000689325] = 0 Q8 = AE[− 0.035(− 0.002027 ) − 0.035(− 0.01187 ) + 0 + 0 − 0.1(− 0.002027 )] + AE [− 0.000689325] = 0 !$>(> karviPaK truss kñúglMh (Space-truss analysis) eKGacviPaK truss kñúglMhkMNt;edaysþaTic nig truss kñúglMhminkMNt;edaysþaTicedayeRbI dMeNIrkarviPaKdUcKñaEdl)anerobrab;BIelIkmun. b:uEnþ edIm,IKitG½kSTaMgbI eKRtUvbBa©ÚlFatubEnßmeTAkñúg m:aRTIsbMElg T . edIm,ITTYl)anva eyIgnwgBicarNaGgát; truss EdlbgðajenAkñúgrUbTI 14-17. m:aRTIsPaBrwgRkajsRmab;Ggát;EdlkMNt;edayeRbIkUGredaentMbn; x' RtUv)aneGayedaysmIkar 14-4. elIsBIenH tamkarGegátrUbTI 14-17 eKGackMNt;kUsIunUsR)ab;TiscenøaHkUGredaenskl nigkUGr- edaentMbn;edayeRbIsmIkarRsedogKñanwgsmIkar 14-5 nig 14-6 Edl xF − xN xF − xN λ x = cos θ x = = (14-31) L (x F − x N )2 + ( y F − y N )2 + ( z F − z N )2 yF − yN yF − yN λ y = cos θ y = = (14-32) L (x F − x N ) + ( y F − y N ) + (z F − z N ) 2 2 2 karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -496
  34. 34. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa zF − zN zF − zN λ z = cos θ z = = (14-33) L (x F − x N )2 + ( y F − y N )2 + ( z F − z N )2 CalT§plénTMhM ¬G½kS¦TIbI m:aRTIsbMElg ¬smIkar 14-9¦ køayCa ⎡λ x λ y λ z 0 0 0 ⎤ T =⎢ ⎥ ⎣ 0 0 0 λx λ y λz ⎦ edayCMnYsm:aRTIsenH nigsmIkar 14-4 eTAkñúgsmIkar 14-15 ¬ K = T T k 'T ¦ eyIg)an ⎡λ x 0⎤ ⎢λ 0⎥ ⎢ y ⎥ ⎢λ 0 ⎥ AE ⎡ 1 − 1⎤ ⎡λ x λ y λ z 0 0 0 ⎤ k=⎢ z ⎥ ⎢ ⎥ ⎢0 λ x ⎥ L ⎢− 1 1 ⎥ ⎣ 0 0 0 λ x λ y λ z ⎦ ⎣ ⎦ ⎢0 λy ⎥ ⎢ ⎥ ⎢0 ⎣ λz ⎥⎦ edayedaHRsayplKuNm:aRTIs eyIgnwgTTYl)anm:aRTIssIuemRTI Nx Ny Nz Fx Fy Fz ⎡ λ2 x λxλ y λxλz − λ2 x − λxλ y − λxλz ⎤ N x ⎢ ⎥ ⎢ λ y λx λ2 y λ y λ z − λ y λ x − λ2 − λ y λ z ⎥ N y y AE ⎢ k= λz λx λz λ y λ z − λ z λ x − λ z λ y − λ2 ⎥ N z 2 (14-34) L ⎢ z ⎥ ⎢ − λ2 x − λxλ y − λxλz λx2 λ x λ y λ x λ z ⎥ Fx ⎢ ⎥ ⎢− λ y λ x − λ2 y − λ y λz λ y λx λ2 y λ y λ z ⎥ Fy ⎢− λ λ − λz λ y − λ2 λ z λ x λ z λ y λ2 ⎥ Fz ⎣ z x z z ⎦ smIkarenHCam:aRTIsPaBrwgRkajsRmab;Ggát;Edl sresredayeRbIkUGredaenskl. elxkUdtam beNþayCYredk nigCYrQrtMNageGayTis x, y, z enARtg;cugCit N x , N y , N z Edlbnþedaycugq¶ay Fx , F y , Fz . sRmab;karsresrkmμviFIkMuBüÚT½r CaTUeTAvaman lkçN³gayRsYlkñúgkareRbIsmIkar 14-34 Cagkar edaHRsayplKuNm:aRTIs T T k 'T sRmab;Ggát; nImYy². dUckarerobrab;BIxagedIm dMbUgkMuBüÚT½rnwg rkSam:aRTIsPaBrwgRkajsRmab;rcnasm½<n§ K Edl manFatuesμIsUnü bnÞab;mkeTotedaysarFatunImYy² énm:aRTIsPaBrwgRkajsRmab;Ggát;RtUv)anbegáIt vaRtUv)andak;eTAkñúgTItaMgRtUvKñarbs;vaenAkñúgm:aRTIs K . eRkayeBlFaturbs;m:aRTIsPaBrwgRkaj Truss analysis using the stiffness method T.Chhay -497
  35. 35. Department of Civil Engineering NPIC sRmab;rcnasm<½n§RtUv)anbegáIt eKGacGnuvtþdMeNIrkar Edl)anerobrab;enAkñúgkfaxNÐ 14-6 edIm,I kMNt;bMlas;TIrbs;tMN kmøaMgRbtikmμ nigkmøaMgkñúgrbs; Ggát;. karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -498
  36. 36. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa cMeNaT 14>1 kMNt;m:aRTIsPaBrwgRkaj K sRmab;eRKOg 14>6 kMNt;m:aRTIsPaBrwgRkaj K sRmab; truss. bgÁúM. yk A = 300mm 2 nig E = 200GPa yk A = 0.005m 2 nig E = 200GPa . snμt; sRmab;Ggát;. tMNTaMgGs;tP¢ab;edaysnøak;. 14>2 kMNt;bMlas;TItamTisedk nigTisQr enARtg;tMN ③ rbs;eRKOgbgÁúMenAkñúgcMeNaT 14>1. 14>7 kMNt;bMlas;TItamTisQrenARtg;tMN ① 14>3 kMNt;kmøaMgkñúgénGgát;nImYy²rbs;eRKOg nigkmøaMgkñúgrbs;Ggát;elx @ éncMeNaT 14>6. bgÁúMenAkñúgcMeNaT 14>1. 14>8 kMNt;m:aRTIsPaBrwgRkaj K sRmab; truss. 14>4 kMNt;m:aRTIsPaBrwgRkaj K sRmab; truss. yk A = 0.0015m 2 nig E = 200GPa sRmab; yk A = 300mm 2 nig E = 200GPa sRmab; Ggát;. Ggát;. 14>5 kMNt;bMlas;TItamTisQrenARtg;tMN ④ nigkmøaMgkñúgrbs;Ggát;elx $ éncMeNaT 14>4. 14>9 kMNt;kmøaMgkñúgénGgát;elx ^ éncMeNaT yk A = 0.0015m 2 nig E = 200GPa . 14>8. yk A = 0.0015m 2 nig E = 200GPa Problems T.Chhay -499
  37. 37. Department of Civil Engineering NPIC sRmab;Ggát;nImYy². 14>13. yk A = 1000mm 2 nig E = 14>10 kMNt;kmøaMgkñúgénGgát;elx ! éncMeNaT 200GPa . 14>8 RbsinebI Ggát;EvgCagmun10mm munnwg 14>15 kMNt;kmøaMgkñúgénGgát;elx @éncMeNaT P¢ab;vaeTAkñúg truss. edIm,IedaHRsay dkbnÞúk 14>8 RbsinebIsItuNðPaBekIneLIg 55o C . 10kN ecj. yk A = 0.0015m 2 nig E = yk A = 1000mm 2 E = 200GPa nig α = 200GPa sRmab;Ggát;nImYy². 11.7(10 −6 )/ o C . 14>11 kMNt;m:aRTIsPaBrwgRkaj K sRmab; 14>16 kMNt;kmøaMgRbtikmμenAelI truss. AE truss. AE CacMnYnefr. CacMnYnefr. 14>12 kMNt;kmøaMgkñúgénGgát;elx @ nig elx % éncMeNaT 14>11. AE CacMnYnefr. 14>13 kMNt;m:aRTIsPaBrwgRkaj K sRmab; truss. yk A = 1000mm 2 nig E = 200GPa . 14>14 kMNt;bMlas;TItamTisedkenARtg;tMN ① nigkmøaMgkñúgrbs;Ggát;elx @ éncMeNaT cMeNaT T.Chhay -500

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