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6. influence lines for statically determinate structures
 

6. influence lines for statically determinate structures

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    6. influence lines for statically determinate structures 6. influence lines for statically determinate structures Document Transcript

    • Department of Civil Engineering NPIC ^> ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic Influence lines for statically determinate structures ExST§iBlmansar³sMxan;kñúgkarGnuvtþsMrab;karsikSaKNnaeRKOgbgÁúMEdlTb;Tl;nwgbnÞúkGrefrEdl mantMélFM. CMBUkenHmankarerobrab;BIrebobKUrExST§iBlsMrab;eRKOgbgÁúMkMNt;edaysþaTic. RTwsþI RtUv)anGnuvtþeTAelIbnÞúkBRgay b¤es‘rIénbnÞúkcMcMnuc ehIyk¾pþl;nUvkarGnuvtþCak;lak;eTAelIr:tkMral (floor girder) nig truss s<an. karKNnakMlaMgkat; nigm:Um:g;Gtibrmadac;xatenAkñúgGgát;RtUv)anerob rab;enAcugbBa©b;énCMBUkenH. ^>!> ExST§iBl (Influence line) eyIg)ansikSarYcehIyBIbec©keTssMrab;viPaKkMlaMgenAkñúgGgát;eRKOgbgÁúMEdlekItBIbnÞúkmin cl½t ¬Kμanclna¦. eyIgeXIjfadüaRkamkMlaMgkat; nigdüaRkamm:Um:g;Bt;CameFüa)ayd¾RbesIrkñúgkar bgðajBIbMErbMrYlkMlaMgTaMgenHenAkñúgGgát;. b:uEnþRbsinebIeRKOgbgÁúMrgbnÞúkcl½t ¬manclna¦ ExST§i- Bl (influence line) CameFüa)ayd¾l¥bMputkñúgkarbgðajBIbMErbMrYlkMlaMgkat; nigm:Um:g;Bt;EdlekItman enAkñúgGgát;eRKOgbgÁúM. ExST§iBlbgðajbMErbMrYlrbs;kMlaMgRbtikmμ kMlaMgkat; m:Um:g;Bt; nigPaBdab enARtg;cMnucCak;lak;enAelIFñwmeRkamGMeBIénkarcl½trbs;bnÞúkcMcMnuc. enAeBlEdleKsg;ExST§iBlrYc ehIy eKGacR)ab;BITItaMgrbs;bnÞúkcl½tedIm,IbegáIteGaymanT§iBlFMbMput. elIsBIenH eKGackMNt; TMhMrbs;kMlaMgRbtikmμ kMlaMgkat; m:Um:g;Bt; b¤PaBdabeRkamGMeBIrbs;bnÞúkenARtg;cMnucNamYyBIGreda- en rbs;düaRkamExST§iBl. sMrab;mUlehtuenH ExST§iBledIrtYy:agsMxan;kñúgkarsikSaKNnas<an/ pøÚv rt;rbs;]bkrN_eyag (industrial crane rails, conveyor) nigeRKOgbgÁúMdéTeTotEdlbnÞúkcl½telI ElVgrbs;va. eTaHbICadMeNIrkarsagsg;ExST§iBlmanlkçN³mUldæank¾eday EteKRtUvEbgEckeGaydac;BI Kñarvagkarsg;ExST§iBl nigkarsg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g;Bt;. ExST§iBlbgðajEtBI T§iBlrbs;bnÞúkcl½tenARtg;TItaMgCak;lak;enAelIGgát; EtdüaRkamkMlaMgkat; nigdüaRkamm:Um:g;bgðaj BIT§iBlrbs;bnÞúkGcl½tenARKb;cMnucTaMgGs;tambeNþayG½kSrbs;Ggát;. dMeNIrkarkñúgkarviPaK (Procedure for analysis) xageRkamCaviFIBIrEdlRtUv)aneKeRbIedIm,Isg;ExST§iBlenARtg;cMnucCak;lak; P enAkñúgGgát; ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -168
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa sMrab;kMlaMgRbtikmμ kMlaMgkat; nigm:Um:g;Bt;. sMrab;viFITaMgBIrenHeyIgnwgeRCIserIskMlaMgcl½tEdlman TMhMKμanxñatmYyÉktþa mkeRbIsMrab;sg;ExST§iBl. * CMnYstMélelxkñúgtarag (tabulate values) dak;bnÞúkÉktþaenARtg;TItaMg ¬ x ¦epSg² tambeNþayGgát; ehIyenARtg;TItaMgnImYy²eRbI sþaTicedIm,IkMNt;kMlaMgRbtikmμ kMlaMgkat; m:Um:g;Bt;enARtg;cMnucCak;lak;. RbsinebIeKcg;sg;ExST§iBlsMrab;kMlaMgRbtikmμbBaÄrenARtg;cMnucmYyenAelIFñwm eKRtUvKitfa kMlaMgRbtikmμmanTMhMviC¢manenAeBlvaeFVIGMeBIeLIgelI. RbsinebIeKKUrExST§iBlkMlaMgkat; nigExST§iBlm:Um:g;Bt;sMrab;cMnucmYy ykkMlaMgkat; nig m:Um:g;Bt;viC¢mandUckarkMNt;sBaØaEdleRbIsMrab;KUrdüaRkamkMlaMgkat; nigdüaRkamm:Um:g;Bt; ¬emIlrUbTI 4-1¦. FñwmkMNt;edaysþaTicTaMgGs;manExST§iBlEdlpSMeLIgedaykMNat;bnÞat;Rtg;. eRkayBIkar GnuvtþxøH eKGackat;bnßykarKNnaedayeGaybnÞúkÉktþamanGMeBIEtenAcMnuccugénkMNat;Ggát; nImYy². edIm,IeCosvagkMhusqÁg dMbUgeKENnaMeGayeFVItaragEdlmanBIrCYrQrKW TItaMgrbs;bnÞúkÉktþa x nigtMélEdlKNnasMrab;kMlaMgRbtikmμ b¤kMlaMgkat; b¤m:Um:g;Bt; EdlRtUvnwgTItaMgenaH. eRkayeBleKdak;bnÞúkenaHenAelITItaMgepSg²CaeRcIntambeNþayGgát; taragRtUv)anbMeBj edaytMélelx ehIyeyIgeRbItMélkñúgtaragTaMgenaHedIm,Isg;düaRkamExST§iBl. smIkarExST§iBl (influence-line equation) eKk¾Gacsg;ExST§iBledaydak;bnÞúkÉktþaenATItaMg x elIGgát; rYcKNnatMél R / V b¤ M Ca GnuKmn_ én x . sMrab;krNIenH eKGacsresr)ansmIkarénkMNat;Ggát;CaeRcInEdlbegáItCa ExST§iBl rYceKGacsg;ExST§iBledayeRbIsmIkarTaMgenH. ]TahrN_ 6-1³ sg;ExST§iBlsMrab;kMlaMgRbtikmμbBaÄrrbs;FñwmenARtg; A ¬rUbTI 6-1a¦. dMeNaHRsay³ CMnYstMélelxkñúgtarag³ bnÞúkÉktþaRtUv)andak;enAelIFñwmRtg;cMnuc x EdleRCIserIsnImYy² ehIytMél Ay RtUv)anKNnaedayeFVIplbUkRtg;cMnuc B . ]TahrN_/ enAeBl x = 2.5m nig x = 5m emIlrUbTI * mUlehtuénkareRCIserIsenHRtUv)anbgðajenAkñúgkfaxNÐ 6.2. Influence Lines for Statically Determinate Structures T.Chhay -169
    • Department of Civil Engineering NPIC 6-1b nig 6-1c erogKña. lT§plsMrab; Ay RtUv)andak;bBa©ÚleTAkñúgtarag ¬rUbTI 6-1d¦. düaRkam sMrab;tMélTaMgenH CaExST§iBlsMrab;RbtikmμenARtg;cMnuc A ¬rUbTI 6-1e¦. smIkarExST§iBl³ enAeBleKdak;bnÞúkÉktþacMgayERbRbYl x BIcMnuc A ¬rUbTI 6-1f¦ eKGackMNt; kMlaMgRbtikmμ Ay CaGnuKmn_én x BI + ∑MB = 0 − A y (10) + (10 − x )(1) = 0 1 Ay = 1 − x 10 smIkarbnÞat;enHRtUv)ansg;CadüaRkamenAkñúgrUbTI 6-1e. ]TahrN_ 6-2³ sg;ExST§iBlsMrab;kMlaMgRbtikmμbBaÄrenARtg; B rbs;FñwmenAkñúgrUbTI 6-2a. dMeNaHRsay³ CMnYstMélelxkñúgtarag³ edayeRbIsþaTic epÞógpÞat;fatMélsMrab;Rbtikmμ B y EdlmanenAkñúgtarag ¬rUb ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -170
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa TI 6-2b¦ RtUv)anKNnay:agRtwmRtUvsMrab;TItaMg x énbnÞúkGefr. düaRkamrbs;tMélkñúgtaragCaExS T§iBlsMrab;RbtikmμRtg; B ¬rUbTI 6-2c¦. smIkarExST§iBl³ edayGnuvtþsmIkarm:Um:g;Rtg;cMnuc A ¬enAkñúgrUbTI 6-2d¦ + ∑MA = 0 B y (5) − 1( x ) = 0 1 By = x 5 smIkarbnÞat;enHRtUv)ansg;CadüaRkamenAkñúgrUbTI 6-2c. ]TahrN_ 6-3³ sg;ExST§iBlsMrab;kMlaMgkat;enARtg;cMnuc C rbs;FñwmenAkñúgrUbTI 6-3a. dMeNaHRsay³ CMnYstMélelxkñúgtarag³ enARtg;TItaMg x EdleRCIserIsnImYy²rbs;bnÞúkÉktþa eKeRbIviFImuxkat;edIm,I KNnatMélrbs; VC . cMNaMfa eKRtUvdak;bnÞúkenAxageqVgcMnuc C bnþic ¬ x = 2.5− ¦ nigxagsþaMcMnuc C bnþic ¬ x = 2.5+ ¦ edaysarkMlaMgkat;manlkçN³dac;enARtg;cMnuc C ¬rUbTI 6-3b nig 6-3c¦. Influence Lines for Statically Determinate Structures T.Chhay -171
    • Department of Civil Engineering NPIC düaRkaméntMélenAkñúgrUbTI 6-3d CaExST§iBlsMrab;kMlaMgkat;enARtg;cMnuc C ¬rUbTI 6-3e¦. smIkarExST§iBl³ enATIenH eKmansmIkarBIrEdlRtUvkMNt; edaysareKmankMNat;Ggát;BIrsMrab;ExS T§iBlEdlbNþalBIPaBdac;rbs;kMlaMgkat;enARtg;cMnuc C ¬rUbTI 6-3f¦. smIkarTaMgenHRtUv)anKUr düaRkamenAkñúgrUbTI 6-3e. ]TahrN_ 6-4³ sg;ExST§iBlsMrab;kMlaMgkat;enARtg;cMnuc C rbs;FñwmenAkñúgrUbTI 6-4a. dMeNaHRsay³ CMnYstMélelxkñúgtarag³ edayeRbIsþaTic nigviFImuxkat; epÞógpÞat;fatMélrbs;kMlaMgkat;Vc enARtg; cMnuc C enAkñúgrUbTI 6-4b EdlRtUvKñanwgTItaMg x nImYy²rbs;bnÞúkÉktþaenAelIFñwm. düaRkaméntMél enAkñúgrUbTI 6-4b CaExST§iBlenAkñúgrUbTI 6-4c. smIkarExST§iBl³ BIrUbTI 6-4d epÞógpÞat;fa 1 Vc = − x 0 ≤ x < 4m 8 1 Vc = 1 − x 4m < x ≤ 12m 8 smIkarTaMgenHRtUv)anbgðajCadüaRkamenAkñúgrUbTI 6-4c. ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -172
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ]TahrN_ 6-5³ sg;ExST§iBlsMrab;kMlaMgkat;enARtg;cMnuc C rbs;FñwmenAkñúgrUbTI 6-5a. dMeNaHRsay³ CMnYstMélelxkñúgtarag³ enARtg;TItaMgeRCIserIsnImYy²rbs;bnÞúkÉktþa eKKNnatMélrbs; M C eday eRbIviFImuxkat;. Ca]TahrN_ emIlrUbTI 6-5b sMrab; x = 2.5m . düaRkaméntMélenAkñúgrUbTI 6-5c bgðajCaExST§iBlsMrab;m:Um:g;enARtg;cMnuc C ¬rUbTI 6-5d¦. smIkarExST§iBl³ eKGaceRbIGgát;BIrkMNat;sMrab;kMNt;rksmIkarExST§iBledayeRbI ∑ M C = 0 CamYynwgviFImuxkat;EdlbgðajenAkñúgrUbTI 6-5d. ⎛ 1 ⎞ + ∑MC = 0 M C + 1(5 − x ) − ⎜1 − x ⎟5 = 0 ⎝ 10 ⎠ Influence Lines for Statically Determinate Structures T.Chhay -173
    • Department of Civil Engineering NPIC 1 MC = x 0 ≤ x < 5m 2 ⎛ 1 ⎞ + ∑MC = 0 MC − ⎜1 − x ⎟5 = 0 ⎝ 10 ⎠ 1 MC =5− x 5m < x ≤ 10m 2 ]TahrN_ 6-6³ sg;ExST§iBlsMrab;kMlaMgkat;enARtg;cMnuc C rbs;FñwmenAkñúgrUbTI 6-6a. dMeNaHRsay³ CMnYstMélelxkñúgtarag³ edayeRbIsþaTic nigviFImuxkat; epÞógpÞat;fatMélrbs;m:Um:g; M c enARtg;cMnuc C enAkñúgrUbTI 6-6b EdlRtUvKñanwgTItaMg x nImYy²rbs;bnÞúkÉktþaenAelIFñwm. düaRkaméntMélenAkñúgrUb TI 6-6b CaExST§iBlenAkñúgrUbTI 6-6c. smIkarExST§iBl³ BIrUbTI 6-6d epÞógpÞat;fa 1 MC = x 0 ≤ x < 4m 2 1 MC = 4 − x 4m < x ≤ 12m 2 smIkarTaMgenHRtUv)ansg;CadüaRkamdUcbgðajenAkñúgrUbTI 6-6c. ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -174
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ^>@> ExST§iBlsMrab;Fñwm (Influence line for beams) edaysarFñwm ¬b¤r:t¦CaGgát;rgbnÞúkcMbgénRbB½n§kMral b¤kMrals<an eKcaM)ac;RtUvmanlT§PaB sg;ExST§iBlsMrab;kMlaMgRbtikmμ kMlaMgkat; b¤m:Um:g;enARtg;cMnucCak;lak;enAkñúgFñwm. bnÞúk³ enAeBleKsg;ExST§iBlsMrab;GnuKmn_ ¬kMlaMgRbtikmμ kMlaMgkat; b¤m:Um:g;¦ enaHeKnwgmanlT§- PaBkMNt;TItaMgbnÞúkGefrenAelIFñwmEdlGacbegáItGnuKmn_mantMélGtibrma. eKmanbnÞúkBIrRbePT EdlRtUvBicarNaenATIenH. bnÞúkcMcMnuc³ edaysartMélCaelxénGnuKmn_sMrab;ExST§iBlRtUv)ankMNt;edayeRbIbnÞúkÉktþaKμanxñat enaHsMrab;bnÞúkcMcMnuc F EdlmanGMeBIenAelIFñwmRtg;TItaMg x eKGacrktMélGnuKmn_edayKuNGredaen rbs;ExST§iBlenARtg;TItaMg x edayTMhMrbs; F . Ca]TahrN_ eKmanExST§iBlsMrab;kMlaMgRbtikmμ enARtg; A sMrab;Fñwm AB ¬rUbTI 6-7¦. RbsinebIbnÞúkÉktþasßitenARtg; x = L / 2 enaHkMlaMgRbtikmμ enARtg; A KW Ay = 1 / 2 dUcbgðajenAkñúgExST§iBl. dUcenH RbsinebIkMlaMg F manGMeBIenARtg;TItaMgdUc Kña kMlaMgRbtikmμrbs;vaKW Ay = F / 2 . Cak;Esþg eKGacTTYl)ankMlaMgRbtikmμEdlmantMéldUcKñaBI karKNnaedaysþaTic. eyIgeXIjy:agc,as;fa T§iBlGtibrmaEdlbgáeday F ekItmanenAeBleK dak;vaRtg;TItaMgénkMBUlrbs;ExST§iBl EdlenAkñúgkrNIenH x = 0 ehIyeyIgnwgTTYl)ankMlaMg Rbtikmμ Ay = (1)(F ) . bnÞúkBRgayesμI³ eKmanEpñkrbs;FñwmEdlrgbnÞúkBRgayesμI wo ¬rUbTI 6-8¦. dUckarbgðaj kMNat; dx nImYy²rbs;bnÞúkenHbegáItkMlaMgcMcMnuc dF = wo dx enAelIFñwm. RbsinebI dF sßitenARtg; x Edl GredaenénExST§iBlrbs;FñwmsMrab;GnuKmn_ ¬kMlaMgRbtikmμ kMlaMgkat; m:Um:g;¦xøH KW y enaHtMélén GnuKmn_KW (dF )( y ) = (wo dx )y . T§iBlénkMlaMgcMcMnuc dF TaMgGs;RtUv)ankMNt;edaykareFVIGaMget RkalelIRbEvgTaMgmUlrbs;Fñwm Edl ∫ wo ydx = wo ∫ ydx . ehIyedaysar ∫ ydx smmUleTAnwgRkLa épÞExST§Bl enaHtMélrbs;GnuKmn_EdlbNþalBIbnÞúkBRgayesμInwgplKuNrvagRkLaépÞExST§iBlCa mYynwgGaMgtg;sIuetrbs;bnÞúkBRgayesμI. Ca]TahrN_ enAkñúgkrNIFñwmrgbnÞúkBRgayesμIdUcbgðajenA Influence Lines for Statically Determinate Structures T.Chhay -175
    • Department of Civil Engineering NPIC kñúgrUbTI 6-9 eKGackMNt;kMlaMgRbtikmμ Ay BIExST§iBlCa Ay = (area)(wo ) = [(1)(L) / 2]wo = wo L / 2 . Cak;Esþg eKGackMNt;tMélenHBIsþaTic. ]TahrN_ 6-7³ kMNt;kMlaMgkat;viC¢manGtibrmaEdlGacekItmanenARtg;cMnuc C enAkñúgFñwmdUcbgðaj enAkñúgrUbTI 6-10a EdlbNþalBIbnÞúkcl½tcMcMnuc 4kN nigbnÞúkcl½tBRgay 2kN / m . dMeNaHRsay³ sg;düaRkamExST§iBlsMrab;kMlaMgkat;sMrab;kMlaMgkat;Rtg;cMnuc C enAkñúg]TahrN_ 6-3 ehIyRtUv)an bgðajenAkñúgrUbTI 6-10b. bnÞúkcMcMnuc³ kMlaMgkat;viC¢manGtibrmaenARtg; C nwgekItmanenAeBlkMlaMg 4kN manGMeBIenARtg; x = 2.5+ m edaysarvakMBUlviC¢manrbs;ExST§iBl. GredaensMrab;kMBUlenHKW + 0.75 Edl Vc = 0.75(4kN ) = 3kN ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -176
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa bnÞúkBRgay³ bnÞúkBRgaycl½tbegáItT§iBlviC¢manGtibrmasMrab;VC enAeBlEdlbnÞúkeFVIGMeBIenAelI FñwmcenøaH x = 2.5+ m nig x = 10m edaysarenAkñúgtMbn;enH ExST§iBlmanRkLaépÞviC¢man. TMhMrbs; VC EdlbNþalBIbnÞúkenHKW ⎡1 ⎤ VC = ⎢ (10m − 2.5m )(0.75)⎥ 2kN / m = 5.625kN ⎣2 ⎦ kMlaMgkat;GtibrmasrubenARtg;cMnuc C (VC )max = 3kN + 5.625kN = 8.625kN cMNaMfa enAeBlEdlTItaMgbnÞúkEdlbegáItedayeRbIExST§iBl ¬rUbTI 6-10c¦ tMél (VC )max enHk¾Gac kMNt;edayeRbIsþaTic nigviFImuxkat;. ]TahrN_ 6-8³ eKeRbIeRKOgbgÁúMeRKagdUcbgðajenAkñúgrUbTI 6-11a edIm,IRT]bkrN_sÞÚc. bnÞúkEdl ]brkrN_sÞÚceyagmanbnÞúk 3kN ehIyFñwm CB manm:as 24kg / m . snμt; A CaTMrsnøak; ehIy B CaTMr kl;. kMNt;kMlaMgRbtikmμTMrbBaÄrenARtg; A / B nigm:Um:g;GtibrmaenAkñúgFñwmRtg;cMnuc D . dMeNaHRsay³ kMlaMgGtibrmaenARtg;cMnuc A ³ dMbUgeyIgKUrdüaRkamExST§iBlsMrab; Ay ¬rUbTI 6-11b¦. Cak;Esþg enAeBlbnÞúkÉktþasßitenARtg; A kMlaMgRbtikmμenARtg; A esμInwg1.0 dUcbgðaj. GredaenenARtg; C esμI nwg1.33 . enATIenHtMélGtibrmasMrab; Ay ekItmanenAeBlbnÞúksßitenARtg;cMnuc C . edaysarbnÞúk Gefr ¬TMgn;Fñwm¦ manelIRbEvgFñwmTaMgmUl eyIg)an Influence Lines for Statically Determinate Structures T.Chhay -177
    • Department of Civil Engineering NPIC (Ay )max = 3000(1.33) + 24(9.81)⎡ 1 (4)(1.33)⎤ ⎢2 ⎥ ⎣ ⎦ = 4.63kN kMlaMgGtibrmaenARtg;cMnuc B ³ ExST§iBl ¬b¤Fñwm¦ manragdUcbgðajenAkñúgrUbTI 6-11c. eKkMNt; tMélenARtg; C nig B edaysþaTic. enATIenHbnÞúkRtUvsßitenARtg;cMnuc B . dUcenH (B y )max = 3000(1) + 24(9.81)⎡ 1 (3)(1)⎤ + 24(9.81)⎡ 1 (1)(− 0.333)⎤ ⎢2 ⎥ ⎢2 ⎥ ⎣ ⎦ ⎣ ⎦ = 3.31kN m:Um:g;GtibrmaenARtg;cMnuc D ³ ExST§iBlmanrUbragdUcbgðajenAkñúgrUbTI 6-11d. eKkMNt;tMélenA Rtg; C nig D BIsmIkarsþaTic. enATIenH (M D )max = 3000(0.75) + 24(9.81)⎡ 1 (1)(− 0.5)⎤ + 24(9.81)⎡ 1 (3)(0.75)⎤ ⎢ ⎥ ⎢ ⎥ ⎣2 ⎦ ⎣2 ⎦ = 2.46kN.m ^>#> eKalkarN_muyeyeRbsøÚ (Qualitative influence lines) enAqñaM 1886/ Heinrich Muller Breslau )anbegáItbec©keTssMrab;sg;ExST§iBly:agelOn bMput. eKakarN_muyeyeRbsøÚeBalfaExST§iBlsMrab;GnuKmn_kMlaMg ¬kMlaMgRbtikmμ kMlaMgkat; b¤ m:Um:gBt;¦ manragdUcnwgrUbragdabrbs;FñwmenAeBlFñwmrgGMeBIedayGnuKmn_kMlaMg. edIm,IKUrrUbragdab eGay)anl¥ eKRtUvyklT§PaBrbs;FñwmTb;Tl;GnuKmn_kMlaMg dUcenHFñwmGacdabenAeBlGnuKmn_kMlaMg TaMgenHmanGMeBI. Ca]TahrN_ eKmanFñwmenAkñúgrUbTI 6-12a. RbsinebIeKRtUvkarkMNt;ragrbs;ExS T§iBlsMrab; kMlaMgRbtikmμRtg;cMnuc A enaHeKRtUvCMnYssnøak;edaypøÚvr:UlU (roller guide) dUcbgðajenAkñúgrUbTI 6- 12b. eK RtUvkarpøÚvr:UlU (roller guide) edaysarFñwmRtUvTb;kMlaMgedkenARtg;cMnuc A minEmnkMlaMg ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -178
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa bBaÄr. enAeBleKGnuvtþ kMlaMgviC¢man ¬TisedAeLIgelI¦ Ay enARtg;cMnuc A FñwmxUcRTg;RTaydUcbnÞat; dac; EdlbgðajrUbrag TUeTArbs;ExST§iBlsMrab; Ay ¬rUbTI 6-12c¦. tMélCaelxsMrab;krNICak; * EsþgRtUv)anKNnaenAkñúg]TahrN_ 6-1. RbsinebIeKRtUvkarkMNt;rUbragrbs;ExST§iBlsMrab;kMlaMgkat;enARtg; cMnuc C ¬rUbTI 6-13a¦ eKGaceFVInimitþsBaØapøÚvr:UlU (roller guide) Rtg;cMnuc C dUcbgðajenAkñúgrUbTI 6-13b. ]bkrN_enHnwg Tb;Tb;m:Um:g; nig kMlaMgtamG½kS EtminTb;Tl;nwgkMlaMgkat;eT . edayGnuvtþkMlaMgkat;viC¢man VC eTA † elIFñwmRtg;cMnuc C nigedayGnuBaØateGayFñwmdabdUcTItaMgbnÞat;dac; enaHExST§iBlsMrab;kMlaMgkat; Rtg;cMnuc C manragdUcbgðajenAkñúgrUbTI 6-13c. cugeRkay RbsinebIeKRtUvkarkMNt;ragExST§iBl sMrab;m:Um:g;Rtg;cMnuc C ¬rUbTI 6-14a¦ eKRtUvdak;snøak;enARtg;cMnuc C edaysartMNenHTb;Tl;kMlaMg tamG½kS nigkMlaMgkat; EtminTb;Tl;nwgm:Um:g;eT ¬rUbTI 6-14b¦. edayGnuvtþm:Um:g;viC¢man M C eTAelI FñwmRtg;cMnuc C Fñwmnwg dabdUcTItaMgbnÞat;dac; EdlCaragExST§iBlsMrab;m:Um:g;Bt;Rtg;cMnuc C ¬rUbTI 6- 13d¦. * RKb;karBnül;TaMgGs;rUbragdabRtUv)anKUredaymaRtdæanFMCagedIm,IbgðajBIeKalKMnit. † enATIenHr:UlUCanimitþsBaØaTMrEdlTb;Tl;bnÞúkTaMgkarTaj nigTaMgkarsgát; ¬emIltarag 2-1 TMr (2)¦. Influence Lines for Statically Determinate Structures T.Chhay -179
    • Department of Civil Engineering NPIC eKGacbkRsayeKalkarN_muyeyeRbsøÚedayeRbIeKalkarN_ virtual work. rMlwkfakmμnþ (work) CaplKuNrvagcMgaybMlas;TICamYykMlaMgEdlmanGMeBIRsbtamTisrbs;bMlas;TI b¤Capl KuNrvagmMurgVilnigm:Um:g;EdlmanGMeBIRsbtamTisrbs;mMurgVil. RbsinebIGgÁFaturwg ¬Fñwm¦ sßitkñúgsßan PaBlMnwg plbUkénkMlaMgTaMgGs; nigplbUkm:Um:g;TaMgGs;EdlmanGMeBIelIFñwmRtUvEtesμIsUnü. dUcenH RbsinebIGgÁFatupþl;nUvbMlas;TInimitþ b¤ bMlas;TI virtual/ kmμnþEdlekIteLIgedaykMlaMg nigm:Um:g;TaMg enHk¾RtUvEtesμInwgsUnüEdr. Ca]TahrN_ eKmanFñwmTMrsamBaØ ¬rUbTI 6-15a¦ EdlrgbnÞúkÉktþaEdl dak;enARtg;cMnucNamYytambeNþayRbEvg. RbsinebIFñwmmanbMlas;TInimitþ (virtual displacement) δy enARtg;TMr A ¬rUb 6-15b¦ manEtkMlaMgRbtikmμ Ay nigbnÞúkÉktþabegáItkmμnþnimitþ. CaBiess Ay begáItkmμnþviC¢man Ayδy nigbnÞúkÉktþabegáItkmμnþGviC¢man − 1δy' . TMrenARtg; B mincl½t ehIykMlaMg enARtg; B minbegáItkmμnþ. edaysarFñwmsßitenAkñúgsßanPaBlMnwg dUcenHmincl½tCak;lak; plbUk- kMlaMgnimitþesμIsUnü Edl Ay δy − 1δy ' = 0 RbsinebIeKeGay δy = 1 enaH Ay = δy ' niyaymüa:geTot tMélrbs; Ay CakUGredaenrbs;ExST§iBlenARtg;TItaMgrbs;bnÞúkÉktþa. edaysar tMélenHsmmUleTAnwgbMlas;TI δy' enARtg;TItaMgrbs;bnÞúkÉktþa vabgðajfarUbragrbs;ExST§iBlsMrab; kMlaMgRbtikmμenARtg; A RtUv)anbegáIt. TaMgenHbgðajBIeKalkarN_muyeyeRbsøÚsMrab;kMlaMgRbtikmμ. enAkñúgrebobdUcKña RbsinebIFñwmRtUv)ankat;enARtg; C ehIyFñwmrgbMlas;TI δ y enARtg;cMnucenH ¬rUbTI 6-15c¦ enaHkMlaMgkat;enARtg;cMnuc C ehIybnÞúkÉktþabegáItkmμnþ. dUcenH smIkarkmμnþnimitþKW VC δy − 1δy ' = 0 ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -180
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa mþgeTot RbsinebI δy = 1 enaH VC = δy ' ehIyeK)anbegáItrUbragrbs;ExST§iBlsMrab;kMlaMgkat;enARtg;cMnuc C . cugeRkay snñt;snøak;RtUv)anbegáItenAkñúgFñwmRtg;cMnuc C ¬rUbTI 6-15d¦. RbsinebImMurgVil δφ RtUv)anbegáItenARtg;snøak; kmμnþnimitþRtUv)anbegáItedaym:Um:g;xagkñúg nigbnÞúkÉktþa. dUcenH M C δφ − 1δy ' = 0 edayyk δφ = 1 eyIgeXIjfa M C = δy ' EdlbgðajfaFñwmmanPaBdabdUcragCaExST§iBlsMrab;m:Um:g;xagkñúgenARtg;cMnuc C ¬emIlrUbTI 6-14¦. eyIgeXIjy:agc,as;fa eKalkarN_muyeyeRbsøÚpþl;nUvviFIy:agelOnsMrab;begáItrUbragrbs;ExS T§iBl. enAeBlEdleyIgsÁal;ragrbs;vaehIy eyIgGackMNt;GredaenenARtg;kMBUledayeRbIviFImUl- dæanEdlerobrab;enAkñúgkfaxNÐ 6-1. karsÁal;rUbragTUeTArbs;ExST§iBl eFVIeGayeyIgGacmanlT§- PaBkMNt;TItaMgbnÞúkGefrenAelIFñwm nigbnÞab;mkeyIgGackMNt;tMélGtibrmarbs;GnuKmn_kMlaMgeday eRbIsmIkarsþaTic. ]TahrN_ 6-12 bgðajBIbec©keTsenH. ]TahrN_ 6-9³ sMrab;FñwmnImYy²enAkñúgrUbTI 6-16a rhUtdl; 6-16c KUrExST§iBlsMrab;kMlaMg RbtikmμbBaÄrenARtg; A . Influence Lines for Statically Determinate Structures T.Chhay -181
    • Department of Civil Engineering NPIC dMeNaHRsay³ eKCMnYsTMr A edaypøÚvr:UlU (roller guide) nigGnuvtþkMlaMg Ay sMrab;rUb 6-16a. mþgeTot CMnUspøÚvr:UlU (roller guide) Rtg;cMnuc A ehIyGnuvtþkMlaMg Ay eTAelIva ¬rUbTI 6-16b¦. eKRtUveRbIpøÚvr:UlUBIr (two-roller guide)Rtg;cMnuc A sMrab;krNIrUbTI 6-16c edaysarTMrRbePTenHbBa¢ÚnTaMgm:Um:g; M A nigbnÞúktamG½kS Ax EtminbBa¢Ún Ay . ]TahrN_ 6-10³ sMrab;FñwmnImYy²enAkñúgrUbTI 6-17a rhUtdl; 6-17c KUrExST§iBlsMrab;kMlaMg kat;enARtg;cMnuc B . dMeNaHRsay³ eKdak;pøÚvr:UlU (roller guide) enARtg;cMnuc B nigGnuvtþkMlaMgkat;viC¢man VB ¬rUb 6-17a¦. cMNaMfa kMNat;Ggát;xagsþaMrbs;Ggát;mindabeT edaysarTMrkl;enARtg; A Tb;FñwmmineGayeFVIclnatamTis bBaÄr ¬emIlTMr (2) enAkñúgtarag 2-1¦. edaydak; roller guide Rtg;cMnuc B ehIyGnuvtþkMlaMgkat;viC¢man enARtg;cMnuc B eKTTYl)anragdab rbs;FñwmEdlRtUvKñanwgExST§iBl ¬rUbTI 6-17b¦. eKRtUvdak;pøÚvr:UlURtg;cMnuc B ehIyGnuvtþkMlaMgkat;viC¢man enaHeKnwg)anrUbragxUcRTg;RTayrbs;Fñwm EdlRtUvKñanwgExST§iBl ¬rUbTI 6-17c¦. cMNaMfakMNat;Ggát;xageqVgrbs;Fñwmmindab edaysarTMr bgáb;. ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -182
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ]TahrN_ 6-11³ sMrab;FñwmnImYy²enAkñúgrUbTI 6-18a rhUtdl; 6-18c KUrExST§iBlsMrab;m:Um:g;Bt; enARtg;cMnuc B . dMeNaHRsay³ eKdak;snøak;enARtg;cMnuc B ehIyGnuvtþm:Um:g;viC¢man M B eTAelIFñwm ¬rUb 6-18a¦. ragdabrbs;Fñwm nigExST§iBlRtUv)anbgðajenAkñúgrUbTI 6-18a. edaydak;snøak;enARtg;cMnuc B ehIyGnuvtþm:Um:g;viC¢man M B eTAelIFñwm eKnwgTTYl)anragdabrbs;Fñwm EdlRtUvKñanwgExST§iBl ¬rUbTI 6-18b¦. CamYynwgsnøak; nigm:Um:g;viC¢manenARtg; B rUbragxUcRTg;RTayrbs;FñwmEdlRtUvKñanwgExST§iBlRtUv)an bgðajenAkñúgrUbTI 6-18c. cMNaMfakMNat;Ggát;xageqVgrbs;FñwmRtUv)anTb;mineGayviledaysarTMr bgáb;. ]TahrN_ 6-12³ kMNt;m:Um:g;viC¢manGtibrmaEdlGacekItmanenARtg;cMnuc D enAkñúgFñwmEdlbgðaj enAkñúgrUbTI 6-19a EdlbNþalBIbnÞúkcl½tcMnuc16kN bnÞúkcl½tBRgay 3kN / m nigTMgn;Fñwm 2kN / m . dMeNaHRsay³ eKdak;snøak;enARtg;cMnuc D ehIyGnuvtþm:Um:g;viC¢man M D eTAelIFñwm. rUbragdab nig ExST§iBlEdlRtUvKñaRtUv)anbgðajenAkñúgrUbTI 6-19b. eyIgdwgehIyfa bnÞúkcl½tcMcMnuc16kN begáIt m:Um:g;viC¢manGtibrmaenARtg;cMnuc D enAeBleKdak;vaenARtg;cMnuc D eBalKWenARtg;kMBUlrbs;ExST§i- Bl. ehIybnÞúkcl½tBRgay 3kN / m RtUv)andak;BRgayBIcMnuc C eTAcMnuc E edIm,IRKbelItMbn;Edl RkLaépÞrbs;ExST§iBlmantMélviCman. cugeRkay TMgn;rbs;Fñwm 2kN / m manGMeBIelIRbEvgFñwmTaMg ¢ Influence Lines for Statically Determinate Structures T.Chhay -183
    • Department of Civil Engineering NPIC mUl. bnÞúkEdlmanGMeBIelIFñwmRtUv)anbgðajenAkñúgrUbTI 6-19c. edaysÁal;TItaMgrbs;bnÞúk eyIgGac kMNt;m:Um:g;GtibrmaenARtg; D edayeRbIsmIkarsþaTic. enAkñúgrUbTI 6-19d kMlaMgRbtikmμenAelI BE RtUv)ankMNt;. edaykat;FñwmenARtUv)ancMnuc D nigedayeRbIkMNat;Ggát; DE ¬rUbTI 6-19e¦ eyIg)an ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -184
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa + ∑MD = 0 − M D − 20(2 ) + 19(4 ) = 0 M D = 36kN .m eyIgk¾GacedaHRsaycMeNaTenHedayeRbItMélCaelxsMrab;ExST§iBldUcenAkñúgkfaxNÐ 6-1. edaykarGegátrUbTI 6-19b Cak;Esþg eKRtUvkarKNnaEttMélkMBUl h Rtg; D b:ueNÑaH. dUcenHvaTamTar eGaydak;bnÞúkÉktþaenAelIFñwmRtg;cMnuc D kñúgrUbTI 6-19a nigbnÞab;mkedaHRsaym:Um:g;Bt;enAkñúgFñwm Rtg;cMnuc D . bgðajfatMélEdlTTYl)anKW h = 1.333 . edayRtIekaNdUc h' /(4 − 2) = 1.33 /(6 − 4) b¤ h' = 1.333 . dUcenH CamYynwgbnÞúkenAelIFñwmdUcenAkñúgrUbTI 6-19c edayeRbIRkLaépÞ nigtMélkMBUlén ExST§iBl ¬rUbTI 6-19b¦ eyIg)an ⎡1 ⎤ ⎡1 ⎤ M D = 5⎢ (10 − 4 )(1.333)⎥ + 16(1.333) − 2 ⎢ (4)(1.333)⎥ ⎣2 ⎦ ⎣2 ⎦ = 36kN.m ^>$> ExST§iBlsMrab;r:tRTkMral (Influence lines for floor Girders) eBlxøH RbB½n§kMralRtUv)ansg;dUcbgðajenAkñúgrUbTI 6-20a EdleyIgeXIjfabnÞúkkMral (floor load) RtUv)anepÞrBIkMralxNÐeTAFñwmkMral nigbnÞab;mkepÞreTAr:txag ehIycugeRkayepÞreTA ssrTMr. KMrUm:UEDlénRbB½n§enHRtUv)anbgðajCaKMeXIjkñúgbøg;kñúgrUbTI 6-20b. enATIenH eKsnμt;fa kMralxNÐCakMralmYyTis ehIyRtUv)anbMEbkCaElVgTMrsamBaØEdlQrelIFñwmkMral. ehIy r:tRtUv)an RTedayTMrsamBaØenAelIssr. edaysarr:tCaGgát;rgbnÞúkcMbgenAkñúgRbB½n§enH eBlxøHeKcaM)ac;sg; ExST§iBlsMrab;kMlaMgkat; nigExST§iBlsMrab;m:Um:g;rbs;va sMrab;GKar]sSahkmμEdlrgbnÞúkcMcMnucF¶n;. cMNaMfa bnÞúkÉktþaenAelIkMralxNÐRtUv)anepÞreTAr:tEtenARtg;cMnucRbsBVrvagFñwmkMral nigr:t eBalKW cMMnuc A / B / C nig D . cMnucTaMgenHeKehAfa panel point ehIytMbn;cenøaHcMnucTaMgenHeKeGayeQμaH fa panel dUcCa BC enAkñúgrUbTI 6-20b. eKGackMNt;ExST§iBlsMrab;cMnucCak;lak;enAelIr:tedayeRbIdMeNIrkarsþaTicdUcKñaenAkñúgkfa- xNÐ 6-1 eBalKW dak;bnÞúkÉktþaenAelIcMnuc x NamYyenAelIkMralxNÐ nigkMNt;GnuKmn_kMlaMg ¬kMlaMgkat; b¤m:Um:g;¦ enARtg;cMnucCak;lak; P enAkñúgr:t ¬rUbTI 6-20b¦. sg;düaRkaméntMélTaMgenH GaRs½ynwg x enaHeyIgnwgTTYl)anExST§iBlsMrab;GnuKmn_kMlaMgenARtg; P . CaBiess tMélsMrab; m:Um:g;enAkñúg girder panel EtgGaRs½ynwgkareRCIserIsTItaMgrbs; P sMrab;ExST§iBl edaysarTMhM M P GaRs½ynwgTItaMgrbs;cMnucBIcugrbs;r:t. Ca]TahrN_ RbsinebIbnÞúkÉktþamanGMeBIenAelIkMral xNÐdUcbgðajenAkñúgrUbTI 6-20c dMbUgeKGacrkkMlaMgRbtikmμ FB nig FC enAelIkMral bnÞab;mk Influence Lines for Statically Determinate Structures T.Chhay -185
    • Department of Civil Engineering NPIC KNnakMlaMgRbtikmμ F1 nig F2 enAelIr:t. bnÞab;mkkMNt;m:Um:g;Bt;enARtg; P edayviFImuxkat; ¬rUbTI 6-20d¦. CalT§pleyIgnwgTTYl)an M P = F1d − FB (d − s ) . edayeRbIkarviPaKdUcKña eyIgnwg TTYl)an VP mantMélefrelI panel BC TaMgmUl Edl VP = F1 − FB dUcenHvaminGaRs½ynwgTItaMgCak; lak; d rbs; P enAkñúg panel eT. sMrab;mUlehtuenH ExST§iBlsMrab;kMlaMgkat;rbs;kMralr:tRtUv)an kMNt;sMrab; panel minEmnRtg;cMnucNamYytambeNþayr:teT. kMlaMgkat;enAkñúgr:tRtUv)aneKeGay eQμaHfa panel shear. eKRtUvcMNaMfa edaysarr:trgT§iBledaysarEtbnÞúkEdlepÞredayFñwmkMral CaeKTUeTAeKeRcIndak;bnÞúkÉktþaenARtg;TItaMgFñwmkMralnImYy²edIm,IbegáItTinñn½ycaM)ac;edIm,IKUrExST§i- Bl. ]TarhrN_CatMélelxxageRkamnwgbgðajy:agc,as;BIkarviPaKkMlaMg. ]TahrN_ 6-13³ sg;ExST§iBlsMrab;kMlaMgkat;enAkñúg panel CD sMrab;r:tRTkMralenAkñúgrUbTI 6- 21a. dMeNaHRsay³ CMnYstMélelxkñúgtarag³ eKdak;bnÞúkÉktþaenARtg;TItaMgFñwmkMralnImYy² ehIyKNnakMlaMgkat;enAkñúg panel CD . taragénlT§plRtUv)anbgðajenAkñúgrUbTI 6-21b. karKNnalMGitenAeBl x = 0 nig x = 6m RtUv)aneGayenAkñúgrUbTI 6-21c nig 6-21d erogKña. cMNaMfa dMbUgeKRtUvkMNt;kMlaMgRbtikmμ ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -186
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa rbs;FñwmkMralenAelIr:tkñúgkrNInImYy² bnÞab;mkkMNt;kMlaMgRbtikmμrbs;r:tenARtg;cMnuc F ¬eKmin RtUvkar G y eT¦ ehIycugeRkay eKBicarNakMNat;Ggát;rbs;r:t ehIyKNnakMlaMgkat; VCD enAkñúg panel CD . sMrab;kic©karbEnßm cUrepÞógpÞat;tMél VCD enAeBl x = 3m / 9m nig 12m . ExST§iBl³ enAeBlEdlelxCMnYskñúgtaragRtUv)ansg;CadüaRkamedayP¢ab;BIcMnucmYyeTAcMnucmYy edaybnÞat;Rtg; lT§plénExST§iBlsMrab; VCD RtUv)anbgðajenAkñúgrUbTI 6-21e. ]TahrN_ 6-14³ sg;ExST§iBlsMrab;m:Um:g;enARtg;cMnuc F sMrab;r:tRTkMralenAkñúgrUbTI 6-22a. dMeNaHRsay³ CMnYstMélelxkñúgtarag³ eKdak;bnÞúkÉktþaenARtg; x = 0 nigenARtg; panel point nImYy². tMélRtUvKña sMrab; M F RtUv)anKNna ehIyRtUv)anbgðajenAkñúgtarag ¬rUbTI 6-22b¦. karKNnalMGitsMrab; x = 2m RtUv)anbgðajenAkñúgrUbTI 6-22c. dUcenAkñúg]TahrN_elIkmun dMbUgeKcaM)ac;kMNt;kMlaMgRbtikmμ rbs;FñwmkMralEdlenAelIr:t bnÞab;mkkMNt;kMlaMgRbtikmμ G y rbs;r:t ¬eKmincaM)ac;kMNt; H y eT¦ehIy Influence Lines for Statically Determinate Structures T.Chhay -187
    • Department of Civil Engineering NPIC cugbBa©b; eKBicarNakMNat;Ggát; GF rbs;r:t nigKNnam:Um:g;kñúg M F . sMrab;kic©karbEnßm cUrkMNt; tMélepSgeTotrbs; M F EdlmanenAkñúgtaragkñúgrUbTI 6-22b. ExST§iBl³ düaRkaméntMélkñúgtaragCaExST§iBlsMrab; M F RtUv)anbgðajenAkñúgrUbTI 6-22d. ^>%> ExSTi§BlsMrab; trusses (Influence lines for trusses) eBlxøHeKeRbI truss CaeRKOgbgÁúMRTbnÞúkcMbgsMrab;s<an. dUcenH sMrab;karsikSaKNna eKRtUv manlT§PaBsg;ExST§iBlsMrab;Ggát;rbs;vanImYy². dUcbgðajenAkñúgrUbTI 6-23 bnÞúkenAelIkMral s<anRtUv)anepÞreTA stringer EdlvabBa¢ÚnbnþeTAFñwmkMral nigbnÞab;mkbnÞúkRtUv)anbBa¢ÚneTAtMNénGgát; xageRkam (bottom cord) rbs; truss. edayGgát; truss rgT§iBlEtBIbnÞúkRtg;tMN dUcenHeyIgGac TTYltMélGredaenrbs;ExST§iBlsMrab;Ggát;edaykardak;bnÞúkÉktþaenARtg;tMNnImYy²tambeNþaykMr al ehIyeRbIviFItMN b¤viFImuxkat;edIm,IKNnakMlaMgenAkñúgGgát;. eKGactMerobTinñn½yenAkñúgtarag Edl manbnÞúkÉktþa nigkMlaMgkñúgrbs;Ggát;. edIm,IPaBgayRsYy RbsinebIGgát;rgkMlaMgTaj eKKitfava mantMélviC¢man RbsinebIrgkMlaMgsgát; vamantMélGviC¢man. ExST§iBlsMrab;Ggát;RtUv)ansg;eday bgðajTinñn½yenAkñúgCacMnuc ehIyKUrP¢ab;BIcMnucmYyeTAcMnucmYyedaybnÞat;Rtg;. ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -188
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ]TahrN_xageRkambgðajBIviFIkñúgkarsg;ExST§iBl. ]TahrN_ 6-15³ sg;ExST§iBlsMrab;kMlaMgkñúgGgát; GB sMrab; truss s<anEdlbgðajenAkñúgrUbTI 6-24a. dMeNaHRsay³ CMnYstMélelxkñúgtarag³ enATIenH eKdak;bnÞúkÉktþaenARtg;tMNnImYy²enARtg;Ggát;xageRkambnþbnÞab; Influence Lines for Statically Determinate Structures T.Chhay -189
    • Department of Civil Engineering NPIC rYcKNnakMlaMgenAkñúgGgát; GB edayeRbIviFImuxkat; ¬rUbTI 6-24b¦. Ca]TahrN_ edaydak;bnÞúk ÉktþaenARtg; x = 6m ¬tMN B ¦ dMbUgKNnakMlaMgRbtikmμTMrenARtg; E ¬rUbTI 6-24a¦ bnÞab;mkkat; truss enARtg;Ggát; HG / GB / BC ehIykMNt;kMlaMgenAkñúg GB BIkMNat; truss xagsþaM ¬rUbTI 6-24c¦. tamrebobdUcKña kMNt;tMéldéTepSgeTotEdlmanenAkñúgtarag. ExST§iBl³ edaybgðajtMélkñúgtaragCacMnucenAelIkUGredaen nigedayP¢ab;cMnucnImYy²edaybnÞat;Rtg; eyIg)anExST§iBlsMrab;Ggát; GB ¬rUbTI 6-24d¦. edaysarExST§iBlbnøayelIElVgrbs; truss TaMgRsug enaHeKcat;TukGgát; GB CaGgát;em (primary member). enHmann½yfa GB rgkMlaMgeday minKitTItaMgeFVIGMeBIrbs;bnÞúkmkelIkMrals<aneT. cMnucénkMlaMgsUnü ¬ x = 8m ¦ RtUv)ankMNt;eday RtIekaNdUcrvag x = 6m nig x = 12m Edl (0.354 + 0.707) /(12 − 6) = 0.354 / x' ¬ x' = 2m ¦ dUcenH x = 6 + 2 = 8m . ]TahrN_ 6-16³ sg;ExST§iBlsMrab;kMlaMgkñúgGgát; CG sMrab; truss s<anEdlbgðajenAkñúgrUbTI 6-25a. dMeNaHRsay³ CMnYstMélelxkñúgtarag³ taragénbnÞúkÉktþaenARtg;tMNénGgát;xageRkam nigkMlaMgxagkñúgGgát; CG RtUv)anbgðajenAkñúgrUbTI 6-25b. eyIgGacTTYl)antMélTaMgenHy:agRsYlbMputedaypþac;tMN C ¬rUb TI 6-25c¦. enATIenH eyIgeXIjfaGgát; CG CaGgát;EdlmankMlaMgsUnü elIkElgEtkñúgkrNIbnÞúk ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -190
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ÉktþamanGMeBIenARtg;tMN C EdlenAkñúgkrNIenH FCG = 1(T ) . ExST§iBl³ edaybgðajtMélkñúgtaragCacMnucenAelIkUGredaen nigedayP¢ab;cMnucnImYy²edaybnÞat;Rtg; eyIg)anExST§iBlsMrab;Ggát; CG ¬rUbTI 6-25d¦. cMNaMfa enAeBlbnÞúkÉktþasßitenARtg; x = 9m kMlaMgenAkñúg CG KW FCG = 0.5 . sßanPaBEbbenHTamTareGaydak;bnÞúkÉktþaenAelIkMrals<ancenøaH tMN. karepÞrbnÞúkBIkMraleTA truss RtUv)anbgðajenAkñúgrUbTI 6-25e. BIrUbenH eyIgeXIjfa FCG = 0.5 edaykarviPaKlMnwgéntMN C ¬rUbTI 6-25f¦. edaysarExST§iBlsMrab; CG minlatsn§wgelI ElVgTaMgmUlrbs; truss ¬rUbTI 6-25d¦ eKehAGgát; CG CaGgát;rg (secondary member). ]TahrN_ 6-17³ kMNt;kMlaMgFMbMputEdlGacekItmanenAkñúgGgát; BC rbs; truss s<anEdlbgðaj enAkñúgrUbTI 6-26a EdlbNþalBIbnÞúkcl½t100kN nigbnÞúkBRgaycl½t 7.5kN / m . bnÞúkRtUv)an GnuvtþenAGgát;xagelI. dMeNaHRsay³ CMnYstMélelxkñúgtarag³ taragénTItaMgbnÞúkÉktþa x enARtg;tMNtambeNþayGgát;xagelICamYynwg kMlaMgenAkñúgGgát; BC RtUv)anbgðajenAkñúgrUbTI 6-26b. eKGaceRbIviFImuxkat;sMrab;karKNna. Ca ]TahrN_ enAeBlbnÞúkÉktþaenARtg;cMnuc I Edl x = 8m dUcbgðajenAkñúgrUbTI 6-26a dMbUgeKRtUv kMNt;kMlaMgRbtikmμ E y Edl E y = 0.25 . bnÞab;mk eKkat; truss Rtg;Ggát; BC / HB nig HI ehIy Influence Lines for Statically Determinate Structures T.Chhay -191
    • Department of Civil Engineering NPIC eyIgBinitükMNat; truss xagsþaM ¬rUbTI 6-26c¦. eKTTYl FBC edayeFVIplbUkm:Um:g;Rtg;cMnuc H edIm,I lubbM)at; FHI nig FHB . tamrebobdUcKña kMNt;tMéldéTeTotenAkñúgrUbTI 6-26b. ExST§iBl³ düaRkaméntMélkñúgtaragCaExST§iBl ¬rUbTI 6-26d¦. tamkarGegát Ggát; BC CaGgát; em. ehtuGVI? kMlaMgcMcMnucGefr³ bnÞúkFMbMputenAkñúgGgát; BC ekItmanenAeBleKdak;kMlaMgcl½t100kN enARtg; x = 16m . dUcenH FBC = (1.33)(100) = 133kN bnÞúkBRgayGefr³ eKRtUvdak;bnÞúkBRgayGefrenAelI truss TaMgmUledIm,IbegáIt)ankMlaMgTajFMbMput enAkñúgGgát; BC . dUcenH * ⎡1 ⎤ FBC = ⎢ (32)(1.33)⎥ 7.5 = 160kN ⎣2 ⎦ kMlaMgGtibrmasrub (FBC )max = 1.33kN + 160kN = 293kN ^>^> T§iBlGtibrmaenARtg;cMnucmYyEdlbNþalBIes‘rIénbnÞúkcMcMnuc (Maximum influence at a point due to a series of concentrated loads) enAeBleKsg;ExST§iBlrbs;GnuKmn_kMlaMgsMrab;cMnucmYyenAkñúgeRKOgbgÁúMrUcehIy eKGackM- Nt;T§iBlGtibrmaEdlbNþalBIkMlaMgcMcMnucGefredayKuNGredaenkMBUlrbs;ExST§iBlCamYynwgTM hMrbs;kMlaMg. b:uEnþenAkñúgkrNIxøH eRKOgbgÁúMEtgEtrgnUves‘rIénbnÞúkcMcMnuc dUcCabnÞúkrbs;kg;rfynþdwk TMnij b¤rfePøIg. edIm,IkMNt;T§iBlGtibrmaenAkñúgkrNIEbbenH eKGaceRbIviFIsakl,gEktMrUv (trial- and-error procedure) b¤eRbIviFIEdlQrelIkarbMErbMrYlGnuKmn_kMlaMgEdlKitfabnÞúkcl½t. enATIenH viFInImYy²RtUv)anBnül;sMrab;karGnuvtþeTAelIkMlaMgkat; nigm:Um:g;Bt;. kMlaMgkat;³ eKmanFñwmTMrsamBaØCamYynwgExST§iBlsMrab;kMlaMgkat;enARtg;cMnuc C enAkñúgrUbTI 6-27a. eKGackMNt;kMlaMgkat;viC¢manGtibrmaenARtg;cMnuc C EdlbNþalBIes‘rIénbnÞúkcMcMnucEdlcl½telIFñwmBI sþaMeTAeqVg. bnÞúkeRKaHfñak;ekItmanenAeBleKdak;bnÞúkmYyenAxagsþaMcMnuc C bnþicEdlRtYtsIuKñanwgcMnuc kMBUlrbs;ExST§iBl. eday trial and error eKGaceFVIkarGegátkrNIEdlGacekItmancMnYnbI ¬rUbTI 6- 27b¦. eyIg)an * kMlaMgTajFMbMputenAkñúgGgát; GB én]TahrN_ 6-15 ekItmanenAeBlbnÞúkBRgayeFVIGMeBIenAelI truss BI x = 0m eTA 8m ¬rUbTI 6-24d¦ ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -192
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa krNITI1³ (VC )1 = 4.5(0.75) + 18(0.625) + 18(0.5) = 23.63kN krNITI2³ (VC )2 = 4.5(− 0.125) + 18(0.75) + 18(0.625) = 24.19kN krNITI3³ (VC )3 = 4.5(0) + 18(− 0.125) + 18(0.75) = 11.25kN krNITI2 EdlmankMlaMg 4.5kN sßitenARtg;1.5+ m BITMrxageqVgeFVIeGaytMélVC FMbMput dUcenH vaCakardak;bnÞúkEdleRKaHfñak;bMput. Cak;Esþg karGegátkrNITI3 KWmanlkçN³mincaM)ac; edaysar kartMerobbnÞúkEbbenHeFVIeGaytMélrbs; (VC )3 tUcCag (VC )2 . enAeBlbnÞúkcMcMnucCaeRcIneFVIGMeBIelIElVgFñwm dUckñúgkrNIbnÞúk E-72 énrUbTI 1-11 eKGaceRbI karKNna trial-and-error xagelI)an. eKGackMNt;TItaMgeRKaHfñak;rbs;bnÞúkedayrkbMErbMrYlkMlaMg kat; ΔV EdlekItmanenAeBlbnÞúkcl½tBIkrNITImYyeTAkrNITIBIr bnÞab;mkBIkrNITIBIreTAkrNITIbI nigbnþbnÞab;. enAeBlEdlkarKNna ΔV viC¢man TItaMgfμInwgeFVIeGaymankMlaMgkat;enAkñúgFñwmRtg; C FM CagTItaMgelIkmun. eKRtUveFVIkarGegátrhUtdl;TTYl)anbMErbMrYlkMlaMgkat;GviC¢man. enAeBleyIg TTYl)antMélGviC¢manEbbenH mann½yfaTItaMgrbs;bnÞúkelIkmunpþl;nUvtMéleRKaHfñak;. eKGackMNt; bMErbMrYlkMlaMgkat; ΔV sMrab;bnÞúk P Edlcl½tBITItaMg x1 eTA x2 enAelIFñwmedayKuN P CamYynwgbMEr Influence Lines for Statically Determinate Structures T.Chhay -193
    • Department of Civil Engineering NPIC bMrYlGredaenrbs;ExST§iBl Edl ¬ y2 − y1 ¦. RbsinebICMralrbs;ExST§iBlKW s enaH ( y2 − y1 ) = s ( x2 − x1 ) dUcenH ΔV = Ps(x2 − x1 ) sMrab;ExST§iBlEdlmanCMral (6-1) RbsinebIbnÞúkcl½tqøgcMnucEdlExST§iBlmanPaBdac; dUcbnÞúk C enAkñúgrUbTI 6-27a enaHbMEr bMrYlkMlaMgkat;manlkçN³samBaØbMputKW ΔV = P( y 2 − y1 ) sMrab;ExST§iBlEdlmanPaBdac; (6-2) kareRbIsmIkarxagelICamYynwgFñwmrgbnÞúkbgðajBIExST§iBlsMrab; VC ¬rUbTI 6-27a¦. cMNaMfatMél rbs;CMralénExST§iBlKW s = 0.75 /(12 − 3) = 0.75 / 9 = 0.0833 ehIyPaBdac;enARtg;cMnuc C manTMhM 0.75 + 0.25 = 1 . BicarNabnÞúkénkrNITI1 Edlcl½t 1.5m eTAkrNITI2 ¬rUbTI 6-27b¦. enAeBl ekItmankrNIenH bnÞúk 4.5kN Føak;cuH − 1 ehIybnÞúkTaMgGs;cl½teLIgtamCMralrbs;ExST§iBl. dUc enHbMErbMrYlkMlaMgkat;sMrab;krNIenHKW ΔV1− 2 = 4.5(− 1) + [4.5 + 18 + 18](0.0833)(1.5) = +0.563kN edaysar ΔV1− 2 viC¢man enaHkrNITI2nwgmanVC FMCagkrNITI1. eRbobeFobcMelIy (VC )1 nig (VC )2 Edl)anKNnaBImun EdlCak;Esþg (VC )2 = (VC )1 + 0.563 . eFVIkarGegát ΔV2−3 EdlekItmanenAeBl krNITI2cl½teTAkrNITI3 ¬rUbTI 6-27b¦ eyIgRtUvKitkarFøak;cuHénbnÞúk18kN nigkarcl½ttamTis edk1.5m énbnÞúkTaMgGs;elICMralrbs;ExST§iBl. eyIg)an ΔV2 − 3 = 18(− 1) + [4.5 + 18 + 18](0.0833)(1.5) = −12.94kN edaysar ΔV2−3 GviC¢man enaHkrNITI2CaTItaMgénbnÞúkeRKaHfñak; dUckarkMNt;elIkmun. m:Um:g;Bt;³ eyIgk¾GaceRbIviFIelIkmunedIm,IkMNt;TItaMgeRKaHfñak;énes‘rIénbnÞúkcMcMnucEdlbegáItm:Um:g;Bt;FM bMputenARtg;TItaMgCak;lak;enAelIeRKOgbgÁúM. dMbUg eKcaM)ac;RtUvKUrExST§iBlsMrab;m:Um:g;enARtg;cMnuc nigkMNt;CMral s rbs;kMNat;ExST§iBlrbs;va. sMrab;clnatamTisedk (x2 − x1 ) rbs;bnÞúkcMcMnuc P bMErbMrUlm:Um:g;Bt; ΔM smmUleTAnwgTMhMrbs;bnÞúkKuNnwgbMErbMrYlGredaenrbs;ExST§iBleRkamGMeBI rbs;bnÞúk. ΔM = Ps ( x2 − x1 ) sMrab;ExST§iBlEdlmanCMral (6-3) Ca]TahrN_ BicarNaFñwm bnÞúk nigExST§iBlsMrab;m:Um:g;enARtg;cMnuc C enAkñúgrUbTI 6-28a. RbsinebI eKdak;bnÞúknImYy²énbnÞúkcMcMnucTaMgbIenAelIFñwmeGayRtUvKñanwgkMBUlrbs;ExST§iBl eyIgnwgTTYl)an T§iBlFMbMputBIkMlaMgnImYy². krNIénkardak;bnÞúkbIRtUv)anbgðajenAkñúgrUbTI 6-28b. enAeBlbnÞúk énkrNITI1 cl½t)an1.2m eTAeqVg eyIgnwg)ankrNITI2. eyIgeXIjfa bnÞúk 9kN kat;bnßy ΔM1− 2 ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -194
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa edaysarCMral (2.25 / 3)manTisedAcuHeRkam ¬rUbTI 6-28a¦. dUcKña kMlaMg18kN nig13.5kN eFVI eGay ΔM 1− 2 ekIneLIg edaysarCMral [2.25 /(12 − 3)] manTisedAeLIgelI. eyIgman ⎛ 2.25 ⎞ ⎛ 2.25 ⎞ ΔM 1− 2 = −9⎜ ⎟(1.2 ) + (18 + 13.5)⎜ ⎟(1.2) = 1.35kN .m ⎝ 3 ⎠ ⎝ 12 − 3 ⎠ edaysar ΔM 1− 2 viC¢man eyIgRtUveFVIkarGegátbnþeTotedaycl½tbnÞúk1.8m BIkrNITI2 eTAkrNITI3. ⎛ 2.25 ⎞ ⎛ 2.25 ⎞ ΔM 2 − 3 = −(9 + 18)⎜ ⎟(1.8) + 13.5⎜ ⎟(1.8) = −30.38kN .m ⎝ 3 ⎠ ⎝ 12 − 3 ⎠ enATIenH bMErbMrYlGviC¢man dUcenHm:Um:g;Bt;FMbMputenARtg;cMnuc C nwgekItmanenAeBlFñwmrgbnÞúkdUc bgðajenAkñúgkrNITI2 ¬rUbTI 6-28c¦. dUcenHm:Um:g;GtibrmaenARtg; C KW (M C )max = 9(1.35) + 18(2.25) + 13.5(1.8) = 77.0kN .m ]TahrN_xageRkamnwgbgðajBIbEnßmBIviFIenH. ]TahrN_ 6-18³ kMNt;kMlaMgkat;viC¢manGtibrmaEdlekItmanenARtg;cMnuc B enAkñúgFñwmEdlbgðaj enAkñúgrUbTI 6-29a EdlbNþalBIbnÞúkkg;rbs;rfynþdwkTMnij. dMeNaHRsay³ ExST§iBlsMrab;kMlaMgkat;enARtg;cMnuc B RtUv)anbgðajenAkñúgrUbTI 6-29b. karcl½t 0.9m rbs;bnÞúk18kN ³ KitfabnÞúk18kN eFVIGMeBIenAxagsþaMcMnuc B bnþicEdleFIVeGayeyIg TTYl)anT§iBlviC¢manGtibrma. edaysarkMNat;Fñwm BC manRbEvg 3m enaHbnÞúk 4.5kN minTan;man GMeBIelIFñwmeT. enAeBlrfynþdwkTMnijcl½t 0.9m eTAeqVg bnÞúk18kN Føak;cuHenAelIExST§iBl1Éktþa Influence Lines for Statically Determinate Structures T.Chhay -195
    • Department of Civil Engineering NPIC ehIybnÞúk18kN / 40.5kN nig 67.5kN eFVIeGay ΔVB viC¢manekIneLIg edaysarCMralekIneLIgeTAeqVg. eTaHbICabnÞúk 45kN cl½teTAmux 0.9m k¾eday k¾vaenAminTan;manGMeBIelIFñwmEdr. dUcenH ⎛ 0.5 ⎞ ΔVB = 18(− 1) + (18 + 40.5 + 67.5)⎜ ⎟0.9 = +0.9kN ⎝ 3 ⎠ karcl½t1.8m rbs;bnÞúk 40.5kN ³ enAeBlbnÞúk 40.5kN manGMeBIenAxagsþaMcMnuc B bnþic ehIybnÞab; mkrfynþdwkTMnijcl½t1.8m eTAeqVg eyIgman ⎛ 0.5 ⎞ ⎛ 0.5 ⎞ ΔVB = 40.5(− 1) + (18 + 40.5 + 67.5)⎜ ⎟(1.8) + 4.5⎜ ⎟(1.2) = +6.3kN ⎝ 3 ⎠ ⎝ 3 ⎠ cMNaMfakñúgkarKNnabnÞúk 45kN cl½t)an1.2m enAelIFñwm. karcl½t1.8m rbs;bnÞúk 67.5kN ³ RbsinebIbnÞúk 67.5kN manTItaMgenAxagsþaMcMnuc B bnþic ehIybnÞab; mkrfynþdwkTMnijcl½teTAeqVg)ancMgay1.8m enaHbnÞúk18kN enAsl;EtcMgay 0.3m eTot vanwgecj BIFñwm dUcKñaRbsinbnÞúk 40.5kN cl½tEfm1.2m vak¾nwgcakecjBIFñwmEdr. dUcenH ⎛ 0.5 ⎞ ⎛ 0.5 ⎞ ⎛ 0.5 ⎞ ΔVB = 67.5(− 1) + 18⎜ ⎟(0.3) + 40.5⎜ ⎟(1.2) + (67.5 + 45)⎜ ⎟(1.8) = −24.8kN ⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3 ⎠ edaysar ΔVB GviC¢man TItaMgd¾RtwmRtUvrbs;bnÞúkekItmanenAeBlbnÞúk 67.5kN sßitenAxagsþaMcMnuc B bnþic ¬rUbTI 6-29c¦. dUcenH (VB )max = 18(− 0.05) + 40.5(− 0.2) + 37.5(0.5) + 45(0.2) = 33.8kN kñúgkarGnuvtþ eKk¾GacBicarNaclnarbs;rfynþdwgTMnijBIeqVgeTAsþaM ehIybnÞab;mkeRCIserIstMél GtibrmarvagsßanPaBTaMgBIrenH. ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -196
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa ]TahrN_ 6-19³ kMNt;m:Um:g;Bt;viC¢manGtibrmaEdlekItmanenARtg;cMnuc B enAkñúgFñwmEdlbgðaj enAkñúgrUbTI 6-30a EdlbNþalBIbnÞúkkg;rbs;]bkrN_eyag (crane). dMeNaHRsay³ ExST§iBlsMrab;m:Um:g;Bt;enARtg;cMnuc B RtUv)anbgðajenAkñúgrUbTI 6-30b. karcl½t 2m rbs;bnÞúk 3kN ³ RbsinebIeKsnμt;fabnÞúk 3kN eFVIGMeBIenARtg;cMnuc B ehIybnÞab;mkva cl½t 2m eTAxagsþaM ¬rUbTI 6-30b¦ enaHbMErbMrYlm:Um:g;Bt;KW ⎛ 1.20 ⎞ ⎛ 1.20 ⎞ ΔM B = −3⎜ ⎟(2) + 8⎜ ⎟(2) = 7.20kN .m ⎝ 3 ⎠ ⎝ 2 ⎠ ehtuGIVeKminKitbBa©ÚlbnÞúk 4kN kñúgkarKNna? karcl½t 3m rbs;bnÞúk 8kN ³ RbsinebIeKsnμt;fabnÞúk 8kN eFVIGMeBIenARtg;cMnuc B ehIybnÞab;mkva cl½t 3m eTAxagsþaM enaHbMErbMrYlm:Um:g;Bt;KW ⎛ 1.20 ⎞ ⎛ 1.20 ⎞ ⎛ 1.20 ⎞ ΔM B = −3⎜ ⎟(3) − 8⎜ ⎟(3) + 4⎜ ⎟(2) = −8.40kN .m ⎝ 3 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ cMNaMfa enATIenHbnÞúk 4kN sßitenAxageRkAFñwm1m dUcenHvacl½telIFñwmEt 2m b:ueNÑaH. edaysar ΔM B pøas;bþÚrsBaØa TItaMgBitR)akdrbs;bnÞúksMrab;m:Um:g;viC¢manGtibrmaenARtg; B ekItmanenAeBlEdlkMlaMg 8kN sßitenARtg; B ¬rUbTI 6-30b¦. dUcenH (M B )max = 8(1.20) + 3(0.4) = 10.8kN .m ]TahrN_ 6-20³ kMNt;kMlaMgsgát;GtibrmaEdlekItmanenAkñúgGgát; BG én truss xagenAkñúgrUbTI 6-31a EdlbNþalBIbnÞúkkg;xagsþaMrbs;rfynþ nigrfseNþag. snμt;fabnÞúkGnuvtþedaypÞal;eTAelI truss nigcl½teTAEtsþaM. Influence Lines for Statically Determinate Structures T.Chhay -197
    • Department of Civil Engineering NPIC dMeNaHRsay³ ExST§iBlsMrab;kMlaMgenAkñúgGgát; BG RtUv)anbgðajenAkñúgrUbTI 6-31b. enATIenH eyIgnwgeRbIviFI trial-and error sMrab;karedaHRsaycMeNaTenH. edaysareyIgcg;)ankMlaMgGviC¢man ¬sgát;¦FMbMput enAkñúgGgát; BG eyIgcab;epþImedaHRsaydUcxageRkam³ bnÞúk1.5kN enARtg;cMnuc C ³ enAkñúgkrNIenH ⎛ 0.3125 ⎞ FBG = 1.5kN (− 0.625) + 4(0) + 2kN ⎜ ⎟(1m ) = −0.729kN ⎝ 3m ⎠ bnÞúk 4kN enARtg;cMnuc C ³ edayGegátkrNIenH eyIgeXIjfavamanlkçN³smRsbCagkrNIelIkmun ⎛ − 0.625 ⎞ FBG = 4kN (− 0.625) + 1.5kN ⎜ ⎟(4m ) + 2kN (0.3125) = −2.50kN ⎝ 6m ⎠ bnÞúk 2kN enARtg;cMnuc C ³ enAkñúgkrNIenHbnÞúkTaMgGs;nwgbegáItkMlaMggsgát;enAkñúgGgát; BC . ⎛ − 0.625 ⎞ ⎛ − 0.625 ⎞ FBC = 2kN (− 0.625) + 4kN ⎜ ⎟(3m ) + 1.5kN ⎜ ⎟(1m ) = −2.66kN ⎝ 6m ⎠ ⎝ 6m ⎠ edaysarlT§plénkrNIcugeRkaypþl;nUvcMelIyFMbMput bnÞúkeRKaHfñak;ekItmanenAeBlbnÞúk 2kN enA Rtg;cMnuc C . ^>&> kMlaMgkat; nigm:Um:g;Gtibrmadac;xat (Absolute maximum shear and moment) enAkñúgkfaxNÐ 6-6 eyIg)anbegáItviFIsMrab;KNnakMlaMgkat; nigm:Um:g;Bt;GtibrmaenARtg;cMnuc Cak;lak;enAelIFñwmEdlbNþalBIes‘rIénbnÞúkcl½tcMcMnuc. cMeNaTEdlmanlkçN³TUeTACagEdlBak;B½n§ nwgkarkMNt;TItaMgrbs;cMnucenAelIFñwm nigTItaMgrbs;bnÞúkenAelIFñwmEdlGaceGayeKTTYl)ankMlaMg kat;Gtibrmadac;xat nigm:Um:g;Bt;Gtibrmadac;xatEdlekItBIbnÞúkxageRkA. RbsinebIFñwmCaFñwmkugs‘ul b¤FñwmTMrsamBaØ eKGacedaHRsaydUcxageRkam³ ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -198
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa kMlaMgkat;³ sMrab;Fñwmkugs‘ulkMlaMgkat;Gtibrmadac;xatnwgekItmanenARtg;cMnucEdlmanTItaMgEk,r TMrbgáb;. eKGacrkkMlaMgkat;GtibrmaedayviFImuxkat; CamYynwgbnÞúkEdlmanTItaMgNamYyenAelIElVg Fñwm ¬rUbTI 6-32¦. sMrab;FñwmTMrsamBaØkMlaMgkat;Gtibrmadac;xatnwgekItmanenAEk,rTMrNamYy. enAkñúgkrNIenH eKdak;bnÞúky:agNaeGaybnÞúkmYyenAEk,rTMrCabnþbnÞab; dUcenAkñúgrUbTI 6-33. m:Um:g;Bt;³ m:Um:g;Bt;Gtibrmadac;xatsMrab;Fñwmkugs‘ulekItmanenARtg;cMnucdUcKñanwgkarekItmankMlaMg kat;Gtibrmadac;xat eTaHbICaenAkñúgkrNIenHbnÞúkcMcMnucmanTItaMgenAcugq¶ayBITMrk¾eday dUcenAkñúg rUbTI 6-34. CaFmμta sMrab;FñwmTMrsamBaØeKminGackMNt;TItaMgeRKaHfñak;rbs;bnÞúk nigm:Um:g;Gtibrmadac;xat edaykarGegát)aneT. b:uEnþeyIgGackMNt;TItaMgedaykarviPaK. ]bmaeKmanFñwmrgkMlaMg F1 / F2 / F3 EdlbgðajenAkñúgrUbTI 6-35a. edaysardüaRkamm:Um:g;sMrab;es‘rIénkMlaMgcMcMnucpSMeLIgeday kMNat;bnÞat;Rtg;EdlmankMBUlenARtg;kMlaMgnImYy² m:Um:g;Gtibrmadac;xatnwgekItmaneRkamGMeBIén kMlaMgmYyénkMlaMgTaMgbI. snμt;fam:Um:g;GtibrmaenHekItmaneRkamGMeBI F2 . TItaMgénbnÞúk F1 / F2 / F3 enAelIFñwmRtUv)ankMNt;edaycMgay x Edlvas;BI F2 eTAcMnuckNþalrbs;FñwmdUcbgðaj. edIm,IkMNt; tMélCak;lak;rbs; x dMbUgeyIgRtUvkMNt;kMlaMgpÁÜbrbs;RbB½n§kMlaMg FR nigcMgay x' Edlvas;BI F2 . bnÞab;mkeTot eFVIplbUkm:Um:g;Rtg;cMnuc B eyIgnwg)ankMlaMgRbtikmμ Ay xageqVgrbs;Fñwm Edl ∑MB = 0 Ay = 1 ⎢2 ( ) (FR )⎡ L − x'− x ⎤ ⎥ L ⎣ ⎦ Influence Lines for Statically Determinate Structures T.Chhay -199
    • Department of Civil Engineering NPIC RbsinebIeKkat;FñwmenAxageqVg F2 bnþic enaHdüaRkamGgÁesrIrbs;vaRtUv)anbgðajenAkñúgrUbTI 6-35b. dUcenH m:Um:g; M 2 eRkamGMeBI F2 KW ⎛L ⎞ ∑M = 0 M 2 = Ay ⎜ − x ⎟ − F1d1 ⎝2 ⎠ ⎡L ( ) ⎤⎛ L ⎞ = (FR )⎢ − x'− x ⎥⎜ − x ⎟ − F1d1 1 L ⎣2 ⎦⎝ 2 ⎠ 2 F L F x' F x F x x' = R − R − R + R − F1d1 4 2 2 L edIm,IeGay M 2 Gtibrma eKTamTareGay dM 2 2 F x F x' =− R + R =0 dx L L b¤ x= x' 2 dUcenH eyIgGacsnñidæanfam:Um:g;Gtibrmadac;xatenAkñúgFñwmTMrsamBaØekItmaneRkamGMeBIénkMlaMgcMcMnuc mYy EdlkMlaMgenHmanTItaMgenAelIFñwmEbbNaEdlva nigkMlaMgpÁÜbénRbB½n§kMlaMgmancMgayesμIKñaBI cMnuckNþalFñwm. edaysareKmanes‘rIénkMlaMgenAelIElVg ¬]TahrN_ F1 / F2 / F3 kñúgrUbTI 6-35a¦ eKRtUvGnuvtþeKalkarN_eTAelIbnÞúknImYy²énes‘rIbnÞúk nigKNnam:Um:g;GtibrmaEdlRtUvnwgkMlaMgenaH. edaykareRbobeFob m:Um:g;EdlmanTMhMFMCageKCam:Um:g;Gtibrmadac;xat. CaTUeTA m:Um:g;Gtibrmadac; xateRcInekItmaneRkamGMeBIkMlaMgEdlFMCageKEdlsßitenAEk,rkMlaMgpÁÜbrbs;RbB½n§kMlaMg. kBa©b;éntMélExST§iBlGtibrma³ c,ab; b¤rUbmnþsMrab;kMNt;kMlaMgkat; b¤m:Um:g;Gtibrmadac;xatmankar Bi)aksMrab;FñwmEdlxusBIFñwmkugs‘ul nigFñwmTMrsamBaØEdl)anerobrab;enATIenH. b:uEnþ edIm,IedaHRsay cMeNaTEbbenH eKRtUvsg;ExST§iBlsMrab;kMlaMgkat; nigm:Um:g;Bt;enARtg;cMnuceRCIserIstambeNþayFñwm ehIybnÞab;mkKNnakMlaMgkat; nigm:Um:g;Bt;sMrab;cMnucnImYy²edayeRbIviFIenAkñúgkfaxNÐ 6-6. eRkay eBlsg;düaRkaméntMélTaMgenH eKnwg)ankBa©b;éntMélGtibrmaBItMélkMlaMgkat; b¤m:Um:g;Bt;Gtibrma dac;xat ehIyeKGackMNt;TItaMgrbs;va. CakarBit eKENnaMeGayeRbIkmμviFIkMuBüÚT½rsMrab;edaHRsay ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -200
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa cMeNaTsμúKsμaj edaysarkarviPaKedayédmanlkçN³saMjúaM. ]TahrN_ 6-21³ kMNt;m:Um:g;Gtibrmadac;xatenAkñúgkMrals<anTMrsamBaØEdlbgðajenAkñúgrUbTI 6- 36a. dMeNaHRsay³ dMbUgeKRtUvkMNt;TMhM nigTItaMgrbs;kMlaMgpÁÜbénRbB½n§ ¬rUbTI 6-36a¦. eyIgman + ↓ FR = ∑ F FR = 8 + 6 + 4 = 18kN + M RC = ∑ M C 18 x = 6(3) + 4(4.5) x = 2m dMbUgeyIgsnμt;fam:Um:g;Gtibrmadac;xatekItmaneRkamGMeBIénbnÞúk 6kN . eKdak;bnÞúk nigbnÞúkpÁÜbenA cMgayesμIKñaBIcMnuckNþalrbs;Fñwm ¬rUbTI 6-36b¦. dMbUg KNna Ay ¬rUbTI 6-36b¦ eyIgman + ∑MB = 0 − Ay (9) + 18(5) = 0 Ay = 10kN edayeRbImuxkat;xageqVgrbs;Fñwm ¬rUbTI 6-36c¦ eyIg)an + MS = 0 − 10(5) + 8(3) + M S = 0 M S = 26.0kN .m m:Um:g;Gtibrmadac;xatGacmanlT§PaBekItmaneRkamGMeBIénbnÞúk 8kN edaysar 8kN > 6kN ehIy edaysar FR sßitenAcenøaHbnÞúk 8kN nig 6kN . edIm,IGegátkrNIenH eKRtUvdak;bnÞúk 8kN nig FR Influence Lines for Statically Determinate Structures T.Chhay -201
    • Department of Civil Engineering NPIC cMgayesμIKñaBIcMnuckNþalrbs;Fñwm ¬rUbTI 6-36d¦. bgðajfa Ay = 7kN dUcbgðajenAkñúgrUbTI 6- 36e nigbgðajfa M S = 24.5kN .m edaykareRbobeFob m:Um:g;Gtibrmadac;xagKW M S = 26kN .m EdlekItmaneRkamGMeBIénbnÞúk 6kN enAeBlEdlbnÞúkmanTItaMgenAelIFñwmdUcbgðajenAkñúgrUbTI 6- 36b. ExST§iBlsMrab;rcnasm<½n§kMNt;edaysþaTic T.Chhay -202
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa cMeNaT 6>1> KUrExST§iBlsMrab; (a) kMlaMgRbtikmμbBaÄr 6>6> edaHRsaycMeNaT 6>5> edayeRbIeKal- enARtg;cMnuc A (b) kMlaMgkat;enARtg;cMnuc C nig karN_muyeyeRbsøÚ. (c) m:Um:g;Bt;enARtg;cMnuc C . edaHRsaycMeNaT 6>7> KUrExST§iBlsMrab; (a) kMlaMgkat;enARtg; enHedayeRbIviFImUldæanénkfaxNÐ 6-1. TMrbgáb; A nig (c) m:Um:g;Bt;enARtg;cMnuc B . 6>8> edaHRsaycMeNaT 6>7> edayeRbIeKal- 6>2> edaHRsaycMeNaT 6>1> edayeRbIeKal- karN_muyeyeRbsøÚ. karN_muyeyeRbsøÚ. 6>9> KUrExST§iBlsMrab; (a) kMlaMgRbtikmμbBaÄr 6>3> KUrExST§iBlsMrab; (a) kMlaMgRbtikmμbBaÄr enARtg;cMnuc B (b) m:Um:g;Bt;enARtg;cMnuc C nig enARtg;cMnuc A (b) m:Um:g;Bt;enARtg;cMnuc A nig (c) kMlaMgkat;enARtg;cMnuc C . snμt;fa A CaTMr (c) kMlaMgkat;enARtg;cMnuc B . snμt;faTMrenA snøak; ehIy B CaTMrkl;. edaHRsaycMeNaT Rtg;cMnuc A CaTMrbgáb;. edaHRsaycMeNaT enHedayeRbIviFImUldæanénkfaxNÐ 6-1. enHedayeRbIviFImUldæanénkfaxNÐ 6-1. 6>4> edaHRsaycMeNaT 6>3> edayeRbIeKal- 6>10> edaHRsaycMeNaT 6>9> edayeRbIeKal- karN_muyeyeRbsøÚ. karN_muyeyeRbsøÚ. 6>5> KUrExST§iBlsMrab; (a) kMlaMgRbtikmμbBaÄr 6>11> KUrExST§iBlsMrab; (a) kMlaMgRbtikmμ enARtg;cMnuc A (b) kMlaMgkat;enARtg;cMnuc C nig bBaÄrenARtg;cMnuc B (b) kMlaMgkat;enARtg; (c) m:Um:g;Bt;enARtg;cMnuc C . edaHRsaycMeNaT cMnuc C nig (c) m:Um:g;Bt;enARtg;cMnuc C . edaH enHedayeRbIviFImUldæanénkfaxNÐ 6-1. RsaycMeNaTenHedayeRbIviFImUldæanénkfaxNÐ 6-1. Problems T.Chhay -203
    • Department of Civil Engineering NPIC elxsMrab;kMBUl. FñwmenHmansnøak;enARtg;TMr B ehIy A CaTMrsnøak;. edaHRsaycMeNaTenH edayeRbIviFI mUldæanénkfaxNÐ 6-1. 6>12> edaHRsaycMeNaT 6>10> edayeRbI eKalkarN_muyeyeRbsøÚ. 6>13> KUrExST§iBlsMrab; (a) m:Um:g;Bt;enARtg; cMnuc A nig(b) m:Um:g;Bt;enARtg;cMnuc B . snμt;fa A CaTMrbgáb;. edaHRsaycMeNaT enHedayeRbI 6>18> edaHRsaycMeNaT 6>17> edayeRbI viFImUldæanénkfaxNÐ 6-1. eKalkarN_muyeyeRbsøÚ. 6>19> FñwmmYyRTbnÞúkefrBRgayesμI 500 N / m nigkMlaMgcMcMnucGefreTal 3000 N . kMNt; (a) m:Um:g;viC¢manGtibrmaEdlGacekItmanenARtg; 6>14> edaHRsaycMeNaT 6>13> edayeRbI cMnuc C nig(b) kMlaMgkat;viC¢manGtibrmaEdl eKalkarN_muyeyeRbsøÚ. GacekItmanenARtg;cMnuc C . snμt; A CaTMrsnøak; 6>15> KUrExST§iBlsMrab; (a) m:Um:g;Bt;enARtg; nig B CaTMrkl;. cMnuc C (b) kMlaMgRbtikmμbBaÄrenARtg;cMnuc A nig (c) kMlaMgkat;bBaÄrenARtg;cMnuc B . cMnuc E Casnøak;. edaHRsaycMeNaTenHedayeRbIviFI mUldæanénkfaxNÐ 6-1. 6>20> bnÞúkGefrBRgayesμI10.5kN / m nigbnÞúk cMnucGefreTal 45kN RtUv)andak;elIFñwm. kMNt; (a) kMlaMgRbtikmμbBaÄrGefrviC¢manGtibrmaenA 6>16> edaHRsaycMeNaT 6>15> edayeRbI Rtg;TMr A (b) kMlaMgkat;GefrviC¢manGtibrmaenA eKalkarN_muyeyeRbsøÚ. Rtg;cMnuc C (c) m:Um:g;GefrviC¢manGtibrmaenA 6>17> KUrExST§iBlsMrab; (a) kMlaMgkat;enARtg; Rtg;cMnuc C . snμt; A CaTMrkl; nig B CaTMrsnøak;. cMnuc C (b) m:Um:g;Bt;enARtg;cMnuc C nig (c) kMlaMgkat;bBaÄrenARtg;cMnuc D . bgðajtMélCa cMeNaT T.Chhay -204
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 6>21> KUrExST§iBlsMrab; (a) kMlaMgRbtikmμ 6>25> bnÞúkGefrBRgayesμI 7.5kN / m nig bBaÄrenARtg;cMnuc B (b) m:Um:g;Bt;enARtg;cMnuc E . kMlaMgcMcMnuceTal 90kN manGMeBIenAelIFñwm. snμt;TMr B nig D CaTMrkl; (roller). enARtg; kMNt; (a) kMlaMgRbtikmμbBaÄrviC¢manGtibrma cMnuc C Casnøak;. edaHRsaycMeNaTenHeday enARtg;TMr B (b) kMlaMgkat;GrefrviC¢manGti- eRbIviFImUldæanénkfaxNÐ 6-1. brmaenARtg;cMnuc C nig (c) m:Um:g;GefrviC¢man GtibrmaenARtg;cMnuc C . snμt;fa A CaTMrsnøak; nig B CaTMrkl;. 6>22> edaHRsaycMeNaT 6>21> edayeRbI eKalkarN_muyeyeRbsøÚ. 6>23> FñwmsmasrgbnÞúkefrBRgayesμI 2kN / m 6>26> KUrExST§iBlsMrab;kMlaMgTajkñúgkab BC . kMras;rbs;FñwmRtUv)anecal. etIkMlaMg nigbnÞúkGefrBRgayesμI1.5kN / m . kMNt; (a) m:Um:g;Bt;GviC¢manGtibrmaEdlekItenARtg;cMnuc TajGtibrmakñúgkabEdlbNþalBIbnÞúk 30kN A nig (b) kMlaMgkat;viC¢manGtibrmaenARtg; manTMhMb:unμan? cMnuc D . snμt; B CatMNsnøak; nig C CaTMrkl;. 6>24> FñwmrgbnÞúkefrBRgayesμI 500 N / m nig kMlaMgcMcMnucGefreTal 3000 N . kMNt; (a) m:Um:g;Bt;GefrGtibrmaEdlGacekItmanenARtg;cM 6>27> FñwmrgbnÞúkGefrBRgayesμI 0.6kN / m nig nuc C nig (b) kMlaMgkat;viC¢manGtibrmaEdl kMlaMgcMcMnucGefr 0.8kN . kMNt; (a) m:Um:g;Bt; GacekItmanenARtg;cMnuc C . snμt; B CaTMr GefrviC¢manGtibrmaEdlGacekItmanenARtg; snøak; nig A CaTMrkl;. cMnuc B nig (b) kMlaMgRbtikmμbBaÄr enARtg; Problems T.Chhay -205
    • Department of Civil Engineering NPIC cMnuc D . snμt; C CaTMrsnøak; nig D CaTMrkl;. kat;GefrFMbMputenARtg;cMnuc D ? kMNt;tMélén m:Um:g;Bt; nigkMlaMgkat;. snμt; A CaTMrbgáb; B Ca tMNsnøak; nig C CaTMrkl;. 6>31> KUrExST§iBlsMrab; (a) kMlaMgkat;kñúg panel BC rbs;r:t (b) m:Um:g;Bt;enARtg;cMnuc A . 6>28> FñwmsmasrgbnÞúkefrBRgayesμI snμt; B CaTMrkl; nig D CaTMrsnøak;. 1.5kN / m nig kMlaMgcMcMnucefreTal 10kN . kMNt; (a) m:Um:g; Bt; GviC¢manGtibrmaEdlGac ekItmanenARtg; cMnuc A nig (b) kMlaMgkat; viC¢manenARtg;cMnuc B . snμt; A CaTMrbgáb; B Ca 6>32> bnÞúkBRgayGefr 7.5kN / m nigbnÞúk tMNsnøak; nig C CaTMrkl;. cMcMnucGefr 9kN manGMeBIenAelIkMralxNÐ. kMNt; (a) kMlaMgkat;GefrviC¢manGtibrmaenAkñúg panel DE nig (b) m:Um:g;Bt;GefrGviC¢manGti- brmaenARtg;cMnuc C . 6>29> eKRtUvdak;bnÞúkGefreTal 2kN enAelIFñwm Rtg;TItaMgNaedIm,IeGayvabegáItm:Um:g;Bt;GefrFM bMputenARtg;cMnuc D ? etIm:Um:g;Bt;enHmanTMhM 6>33> KUrExST§iBlsMrab; (a) kMlaMgkat;kñúg b:unμan? snμt; A CaTMrbgáb; B Ca tMNsnøak; panel DE rbs;r:t (b) m:Um:g;Bt;enARtg;cMnuc C . nig C CaTMrkl;. 6>34> KUrExST§iBlsMrab; (a) kMlaMgkat;kñúg panel CD rbs;r:t (b) m:Um:g;Bt;enARtg;cMnuc E . 6>30> etIeKRtUvdak;bnÞúkGefrBRgayesμI 6kN / m Rtg;kEnøgNaenAelIFñwm ABC én cMeNaT 6>29 edIm,I TTYl)an (a) m:Um:g;Bt; GrefrFMbMputenARtg;cMnuc A nig (b) kMlaMg cMeNaT T.Chhay -206
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 6>35> bnÞúkBRgayGefr 6kN / m nigbnÞúkGrfr snμt;fa D CaTMrkl; nig G CaTMrsnøak;. cMcMnuc 9kN manGMeBIelIkMralxNÐ. kMNt; (a) kMlaMgkat;GefrGtibrmaenAkñúg panel CD nig (b) m:Um:g;GefrGtibrmaenARtg;cMnuc B . 6>39> KUrExST§iBlsMrab; (a) m:Um:g;Bt;enAkñúgr:t Rtg;cMnuc D (b) kMlaMgkat;kñúg panel BC rbs; r:t. 6>36> bnÞúkBRgayGefr 27kN / m nigbnÞúk GrfrcMcMnuc 60kN manGMeBIelIkMralxagelIFñwm. RbsinebIFñwm k¾RTbnÞúkBRgayefr 5.25kN / m Edr enaH cUr kMNt; (a) kMlaMgkat;GtibrmaenAkñúg 6>40> KUrExST§iBlsMrab;m:Um:g;Bt;enAkñúgr:tRtg; panel BC rbs;r:t nig (b) m:Um:g;Bt;Gtibrma cMnuc B . kMNt;m:Um:g;Bt;GefrviC¢manGtibrmaenA enAkñúg r:tRtg;cMnuc C . kñúgr:tRtg;cMnuc B RbsinebIkMlaMgGefrcMcMnuc eTal 50kN cl½telIkMralxagelIFñwm. snμt;TMr rbs;FñwmTaMg enHGacGnuvtþkMlaMgEdlmanTis eLIgelI nig kMlaMgEdlmanTiscuHeRkam eTAelIFñwm. 6>37> KUrExST§iBlsMrab; (a) kMlaMgkat;kñúg panel BC rbs;r:t (b) m:Um:g;Bt;enARtg;cMnuc C . snμt; B CaTMrsnøak; nig D CaTMrkl;. 6>41> KUrExST§iBlsMrab;m:Um:g;Bt;enAkñúgr:tRtg; cMnuc F . kMNt;m:Um:g;Bt;GefrviC¢manGtibrmaenA kñúgr:tRtg;cMnuc F RbsinebIkMlaMgGefrcMcMnuc 6>38> bnÞúkBRgayGefr 5kN / m nigbnÞúkGrfr eTal 8kN cl½telIkMralxagelI. snμt;TMrrbs; cMcMnuc18kN manGMeBIelIkMral. kMNt; (a) FñwmTaMg enHGacGnuvtþkMlaMgEdlmanTiseLIgelI kMlaMgRbtikmμbBaÄrenARtg;cMnuc D (b) kMlaMg nig kMlaMgEdlmanTiscuHeRkameTAelIFñwm. kat;viC¢manGtibrmaenAkñúg panel EF rbs;r:t nig (c) m:Um:g;Bt;viC¢manGtibrmaenARtg;cMnuc F . Problems T.Chhay -207
    • Department of Civil Engineering NPIC kMBUlrbs;ExSekag. Ggát;TaMgGs;manRbEvgesμI Kña. 6>42> KUrExST§iBlsMrab;kMlaMgkat;enAkñúg panel CD rbs;r:t. kMNt;kMlaMgkat;Gefr GviC¢manGtibrmaenAkñúg panel CD Edl bNþalBIbnÞúkBRgayGefr 10kN / m Edlman 6>46> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; FC GMeBIenAelIkMralxagelIFñwm. én Warren truss éncMeNaT 6>45 . bgðaj tMélCaelxsMrab;kMBUlrbs;ExSekag. Ggát;TaMg Gs;manRbEvgesμIKña. 6>47> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; 6>43> KUrExST§iBlsMrab;m:Um:g;Bt;enARtg;cMnuc DE én Warren truss éncMeNaT 6>45 . E rbs;r:téncMeNaT 6>41. kMNt;m:Um:g;Bt; bgðaj tMélCaelxsMrab;kMBUlrbs;ExSekag. GefrviC¢manGtibrmaenAkñúgr:tRtg;cMnuc E Rbsin Ggát;TaMgGs;manRbEvgesμIKña. ebIkMlaMgGefrcMcMnuc 40kN cl½tenAelIkMral 6>48> KUrExST§iBlsMrab;kMlaMgenAkñúg (a) Ggát; xagelIFñwm. EH nig (b) Ggát; JE . 6>44> bnÞúkBRgayGefr 5kN / m nigbnÞúkGrfr cMcMnuc18kN manGMeBIelIkMral. kMNt; (a) kMlaMgRbtikmμbBaÄrenARtg;cMnuc D (b) kMlaMg kat;GefrGtibrmaenAkñúg panel DE rbs;r:t nig 6>49> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; JI (c) m:Um:g;Bt;viC¢manGtibrmaenARtg;cMnuc E . én truss kñúgcMeNaT 6>48. snμt; D CaTMrkl; nig G CaTMrsnøak;. 6>50> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; AL én truss kñúgcMeNaT 6>48. 6>51> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; HG bnÞab;mkkMNt;kMlaMgGefrGtibrma ¬kMlaMgTaj 6>45> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; b¤kMlaMgsgát;¦ EdlGacekItmanenAkñúgGgát;Edl GF én Warren truss. bgðajtMélCaelxsMrab; bNþalBIbnÞúkBRgayGefr16kN / m Edlman cMeNaT T.Chhay -208
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa GMeBIelIkMrals<antambeNþayGgát;xageRkam 6>57> eKeRbIdMbUl truss sMrab;BüÜrpøÚvén]brkN_ rbs; truss. eyagEdlP¢ab;eTAnwgGgát;xageRkamrbs; truss dUcbgðaj. kMNt;kMlaMgGefrGtibrma ¬kMlaMg Taj b¤kMlaMgsgát;¦ EdlGacekItmanenAkñúg Ggát; GF EdlbNþalBIbnÞúk]bkrN_eyag 60kN . kMNt;TItaMg x rbs;bnÞúk. snμt; truss 6>52> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; HC RtUv)anRTedayTMrsnøak;Rtg;cMnuc A nigRtg;cMnuc bnÞab;mkkMNt;kMlaMgGefrGtibrma ¬kMlaMgTaj E edayTMrkl;. ehIysnμt;faGgát;TaMgGs;RtUv b¤kMlaMgsgát;¦ EdlGacekItmanenAkñúgGgát;Edl )antP¢ab;KñaRtg;tMNeday gusset plate. bNþalBIbnÞúkBRgayGefr16kN / m Edlman GMeBIelIkMrals<antambeNþayGgát;xageRkam rbs; truss éncMeNaT 6>51. 6>53> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; AH bnÞab;mkkMNt;kMlaMgGefrGtibrma ¬kMlaMgTaj b¤kMlaMgsgát;¦ EdlGacekItman 6>58> kMNt;m:Um:g;Bt;GefrGtibrmaenARtg;cMnuc enAkñúgGgát;Edl bNþalBIbnÞúkBRgayGefr A enAelIr:teTalEdlekIteLIgedaysarkUn 16kN / m Edlman GMeBIelIkMrals<antam reTHEdlmanm:as 2Mg ehIyTIRbCMuTMgn;enARtg; beNþayGgát;xageRkam rbs; truss éncMeNaT G . snμt;fa A CaTMrkl;. 6>51. 6>54> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; RN én Baltimore truss. 6>59> rfynþdwkTMnijTMgn; 54kN GnuvtþkMlaMgkg; dUcbgðajeTAelIr:ts<an. kMNt; (a) kMlaMgkat; 6>55> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; GefrFMbMputEdlekItmanenARtg;tMN C nig (b) NG én truss kñúgcMeNaT 6>54. m:Um:g;Bt;FMbMputenARtg;tMN C . snμt;farfynþcr 6>56> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; tamTisNak¾)antambeNþayG½kSkMral dUcenH OC én truss kñúgcMeNaT 6>54. bnÞúkrbs;vaRtUv)anepÞrBak;kNþaleTAr:txagTaMg BIr. snμt;fatMN C CatMNbgáb; dUcenHvaGac Problems T.Chhay -209
    • Department of Civil Engineering NPIC Tb;Tl;)anTaMgm:Um:g;Bt; nigTaMgkMlaMgkat;. kñúgr:txagRtg; C enAeBlEdlvaqøgkat;s<an. snμt;farfynþcrtamTisNak¾)antambeNþay G½kSkMral dUcenHbnÞúkrbs;vaRtUv)anepÞrBak; kNþaleTAr:txagTaMgBIr. 6>60> rfynþdwkTMnijTMgn; 40.5kN GnuvtþkMlaMg kg;dUcbgðajeTAelIr:ts<an. kMNt; (a) kMlaMg kat;GefrFMbMputEdlekItmanenARtg;tMN C nig 6>63> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; IH (b) m:Um:g;Bt;FMbMputenARtg;tMN C . snμt;eGay rbs; truss s<an. kMNt;kMlaMgGefrGtibrma rfynþcrtamTisNak¾)antambeNþayG½kS ¬kMlaMgTaj b¤kMlaMgsgát;¦ EdlGacekItmanenA kMral dUcenHbnÞúkrbs;vaRtUv)anepÞrBak;kNþal kñúgGgát;enHedaysarrfynþdwkTMnijEdlman eTAr:txagTaMgBIr. snμt;fatMN C CatMNbgáb; TMgn; 83.25kN EdlmanbnÞúkkg;dUcbgðaj. snμt; dUcenHvaGac Tb;Tl;)anTaMgm:Um:g;Bt; nigTaMg eGayrfynþcrtamTisNak¾)antambeNþay kMlaMgkat;. G½kSkMral dUcenHbnÞúkrbs;vaRtUv)anepÞrBak; kNþaleTAr:txagTaMgBIr. ehIysnμt;faGgát; TaMgGs;RtUv)antP¢ab;KñaRtg;tMNeday gusset plate. 6>61> kMNt;m:Um:g;GefrGtibrmaenARtg;cMnuc C enAelIs<anEdlbNþalBIbnÞúkcl½t. 6>64> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; CD rbs; truss s<ankñúgcMeNaT 6>63. kMNt;kMlaMg GefrGtibrma ¬kMlaMgTaj b¤kMlaMgsgát;¦ Edl GacekItmanenAkñúgGgát;enHedaysarrfynþdwk 6>62> rfynþmanTMgn; 21kN nigTIRbCMuTMgn;Rtg; TMnijEdlman TMgn; 83.25kN EdlmanbnÞúkkg; G . kMNt;m:Um:g;GefrGtibrmaEdlekItmanenA dUcbgðaj. snμt; eGayrfynþcrtamTisNa k¾ cMeNaT T.Chhay -210
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa )antambeNþayG½kSkMral dUcenHbnÞúkrbs;va kMral dUcenHbnÞúkrbs;vaRtUv)anepÞrBak;kNþal RtUv)anepÞrBak; kNþaleTAr:txagTaMgBIr. ehIy eTAr:txagTaMgBIr. ehIysnμt;faGgát;TaMgGs; snμt;faGgát; TaMgGs;RtUv)antP¢ab;KñaRtg;tMN RtUv)antP¢ab;KñaRtg;tMNeday gusset plate. eday gusset plate. 6>65> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; BC rbs; truss s<ankñúgcMeNaT 6>63. kMNt;kMlaMg GefrGtibrma ¬kMlaMgTaj b¤kMlaMgsgát;¦ Edl GacekItmanenAkñúgGgát;enHedaysarrfynþdwk TMnijEdlman TMgn; 83.25kN EdlmanbnÞúkkg; 6>68> kMNt;m:Um:g;GefrGtibrmaenARtg;cMnuc C dUcbgðaj. snμt; eGayrfynþcrtamTisNa k¾ EdlbNþalBIbnÞúkcl½t. )antambeNþayG½kSkMral dUcenHbnÞúkrbs;va RtUv)anepÞrBak; kNþaleTAr:txagTaMgBIr. ehIy snμt;faGgát; TaMgGs;RtUv)antP¢ab;KñaRtg;tMN eday gusset plate. 6>66> kMNt;cMgay a rbs;FñwmlyedIm,IeGay bnÞúkcl½tGacbegáItm:Um:g;Bt;GtibrmaenARtg;TMr 6>69> kMNt;kMlaMgkat;GefrGtibrmaenARtg; esμIKñanwgm:Um:g;Bt;enAkNþalElVg. snμt; A CaTMr cMnuc C EdlbNþalBIbnÞúkcl½t ¬rUbéncMeNaT snøak; nig B CaTMrkl;. 6>68¦. 6>70> kMNt;m:Um:g;GefrGtibrmaenARtg;cMnuc C EdlbNþalBIbnÞúkcl½t. 6>67> KUrExST§iBlsMrab;kMlaMgenAkñúgGgát; IH rbs; truss s<an. kMNt;kMlaMgGefrGtibrma 6>71> kMNt;kMlaMgkat;GefrGtibrmaenARtg; ¬kMlaMgTaj b¤kMlaMgsgát;¦ EdlGacekItmanenA cMnuc C EdlbNþalBIbnÞúkcl½t ¬rUbéncMeNaT kñúgGgát;enHedaysarrfynþdwkTMnijEdlman 6>70¦. TMgn; 25kN EdlmanbnÞúkkg;dUcbgðaj. snμt; 6>72> kMNt;kMlaMgkat; nigm:Um:g;Bt;Gtibrmadac; eGayrfynþcrtamTisNak¾)antambeNþayG½kS xatenAkñúgFñwm AB EdlbNþalBIbnÞúk10kN . Problems T.Chhay -211
    • Department of Civil Engineering NPIC lkçxNÐRBMEdnTamTar 0.1m ≤ x ≤ 3.9m . 6>76> bnúÞkkg;GtibrmasMrab;kg;]bkrN_sÞÚc EdleRbIenAkñúgGKar]sSahkmμRtUv)aneGay. ]bkrN_sÞÚccrtambeNþaypøÚvrfEdlRtUv)anRT edayTMrsamBaØenAelIssr. kMNt; (a) kMlaMg 6>73> kMNt;m:Um:g;Bt;Gtibrmadac;xatenAkñúgr:t kat;Gtibrmadac;xatenAkñúgr:tkNþal AB nig s<anEdlbNþalBIbnÞúkrfynþdUcbgðaj. bnÞúk (b) m:Um:g;Gtibrmadac;xatenAkñúgr:t. RtUv)andak;edaypÞal;enAelIr:t. 6>77> kMNt;m:Um:g;Bt;Gtibrmadac;xatenAkñúgr:t 6>74> kMNt;m:Um:g;Bt;Gtibrmadac;xatenAkñúgr:t s<anEdlbNþalBIbnÞúkdUcbgðaj. s<anEdlbNþalBIbnÞúkdUcbgðaj. bnÞúkRtUv)an dak;edaypÞal;enAelIr:t. KMerag6>1> ]bkrN_sÞÚccl½ttambeNþaydgsÞÚc ¬ 0.1m < x < 3.4m ¦ ehIyGaceyag)an 28kN . 6>75> kMNt;m:Um:g;Bt;Gtibrmadac;xatenAkñúgr:t edayeRbIemKuNTgÁic 0.3 cUrkMNt;m:Um:g;Bt;Gti- s<anEdlbNþalBIbnÞúkrfynþdwkTMnijdUcbgðaj. brmadac;xatenAkñúgdgsÞÚc nigkMlaMgGtibrma EdlekItmanenAkñúgExScMNg BC .dgsÞÚcRtUv)n cMeNaT T.Chhay -212
    • mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa P¢ab;edaytMNsnøak;eTAnwgssrCBa¢aMgenAcug xageqVgrbs;va ¬ A ¦. KMerag6>2> s<ansMrab;GñkefμIreCIgEdlmanTMr samBaØRtUv)aneKsg;enAkñúg]TüanTIRkugmYy ehIyGñksikSaKNnaBIrrUb)anesñIKMeragdUcbgðaj enAkñúgkrNI a nig b. Ggát;rbs; truss RtUv)an sg;BIeQI. kMralpSMeLIgedaybnÞHeQIRbEvg 1.5m Edlmanm:as; 20kg / m 2 . kUdkñúgtMbn; bgðajfabnÞúkGefrenAelIkMralEdlRtUvkarKW 5kPa CamYynwgemKuNTgÁic 0.2 . KitfakMral RtUv)anRTedayTMrsamBaØenAelI stringer. Fñwm kMralepÞrbnÞúkeTAtMNxageRkamrbs; truss ¬emIlrUbTI 6-23¦ enAkñúgkrNInImYy² cUrrk Ggát;EdlrgkMlaMgTajFMCageK nigGgát;Edlrg kMlaMgsgát;FMCageK ehIyR)ab;BImUlehtukñúgkar eRCIserIskarKNnamYykñúgcMeNamsMeNITaMgBIr. TMgn;rbs;Ggát; truss RtUv)anecal. Problems T.Chhay -213