More Related Content Similar to 10. analysis of statically indeterminate structures by the force method Similar to 10. analysis of statically indeterminate structures by the force method (8) More from Chhay Teng (20) 10. analysis of statically indeterminate structures by the force method1. Department of Civil Engineering NPIC
!0> viPaKeRKOgbgÁúMminkMNt;edaysþaTictamviFIkmøaMg
Analysis of statically indeterminate structures by the force method
CMBUkenHENnaMBIviFIkmøaMg b¤viFIPøicsIub‘IlIFI (force or flexibility method) edIm,IviFI truss Fñwm
nigeRKagminGackMNt;edaysþaTic. enAcugbBa©b;énemeronmanbgðajBIviFIsRmab;sg;ExST§iBlsRmab;
Fñwm b¤eRKagminGackMNt;edaysþaTic.
!0>!> rcnasm<½n§minkMNt;edaysþaTic (Statically indeterminate structures)
rMlwkBIkfaxNÐ 2.4 eRKOgbgÁúMRKb;RbePTEdlRtUv)ancat;cMNat;fñak;CaeRKOgbgÁúMminkMNt;eday
sþaTicenAeBlEdlcMnYnGBaØatEdlCakmøaMgRbtikmμ b¤kmøaMgkñúgelIsBIcMnYnsmIkarlMnwgEdlGacman
sRmab;karviPaKenaH. enAkñúgkfaxNÐenH eyIgnwgBiPakSaBIviFIénkareRbIeRKOgbgÁúMminkMNt; nigviFImUl-
dæan cMnYnBIrEdleKRtUvkarviPaK. eyIgRtUvdwgfaeRKOgbgÁúMPaKeRcInEdlsikSaKNnasBVéf¶CaeRKOgbgÁúM
minkMNt;edaysþaTic. cMnYnminkMNt;GacekIneLIgedaysarkarbEnßmTMr b¤Ggát; b¤edayTMrg;TUeTArbs;
rcnasm<½n§. Ca]TahrN_ GKarebtugGarem:esÞIrEtTaMgGs;CaeRKOgbgÁúMminkMNt;edaysþaTic edaysar
ssr nigFñwmRtUv)ancak;CaGgát;Cab;Rtg;tMN nigRtg;TMr.
KuNsm,tþi nigKuNvibtþi³ eTaHbICakarviPaKeRKOgbgÁúMminkMNt;edaysþaTicmanlkçN³sμúKsμajCagkar
viPaKeRKOgbgÁúMkMNt;edaysþaTick¾eday k¾eKenAEteRCIserIsvasRmab;karsikSaKNnaEdr edaysarCaTU
eTAeKmanmUlehtuCaeRcIn. GVIEdlsMxan;CaeKKW sRmab;kardak;bnÞúkdUcKñakugRtaMgGtibrma nigPaBdab
sRmab;eRKOgbgÁúMminkMNt;edaysþaTicmantémøtUcCagsRmab;eRKOgbgÁúMkMNt;edaysþaTic. Ca]TahrN_
eRKOgbgÁúMminkMNt;edaysþaTic ¬FñwmTMrbgÁúM¦ enAkñúgrUbTI 10-1a nwgrgnUvm:Um:g;Bt;Gti`brma M max =
PL / 8 b:uEnþFñwmTMrsamBaØ ¬rUbTI 10-1b¦ nwgrgnUvm:Um:g;Bt;eTVdgeBalKW M max = PL / 4 . CalT§pl
FñwmTMrbgáb;manPaBdab;tUcCagbYndg nigmankugRtaMgBt;tUcCagBIrdgénFñwmTMrsamBaØenARtg;kNþal
ElVg.
ehtupld¾sMxan;mYyeTotkñúgkareRCIserIseRKOgbgÁúMminkMNt;edaysþaTicKWedaysarvaGacEbg
EckbnÞúkrbs;vaeTATMrEdlmancMnYnelIs enAkñúgkrNIEdlmankarsikSaKNnamanlkçN³xusqÁg b¤k¾
kardak;bnÞúkelIsGacekItman. enAkñúgkrNITaMgenH rcnasm<½n§rkSasßirPaB nigkarBarkar)ak;rlM. va
manlkçN³sMxan;CaBiess enAeBlEdlbnÞúkxag dUcCabnÞúkxül; b¤bnÞúkrBa¢ÜydImanGMeBIelIeRKOgbgÁúM.
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -334
2. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
edIm,IbgðajBIlkçN³sMxan;enH cUrBicarNaFñwmTMrbgáb;enAkñúgrUbTI 10-1amþgeTot. enAeBlkmøaMg P
ekIneLIg sMPar³rbs;FñwmenARtg;CBa¢aMg nigenARtg;kNþalElVgcab;epþImeFVIkardl;tMbl;yal (yield)
ehIybegáIt)anCasnøak;)aøsÞic EdleFVIeGayFñwmdabdUckrNIFñwmEdltP¢ab;edaysnøak;enARtg;cMNuc
enaH. eTaHbICaPaBdabmantémøFMk¾eday EtCBa¢aMgnwgbegáItkmøaMgRbtikmμtamTisedk nigm:Um:g;Rbti-
kmμEdl nwgTb;FñwmBIkar)ak;TaMgRsug. enAkñúgkrNIFñwmTMrsamBaØ ¬rUbTI 10-1b¦ bnÞúkelIs P nigeFVI
eGaysnøak;)aøsÞicekItmanEtenARtg;kNþalElVgrbs;Fñwm ehIyedaysarPaBdabmantémøFM TMrnwgmin
begáItkmøaMgRbtikmμtamTisedk nigm:Um:g;RbtikmμEdlcaM)ac;sRmab;kar)ak;TaMgRsug.
eTaHbICaeRKOgbgÁúMminkMNt;edaysþaTicGacRTbnÞúkCamYynwgGgát;esþIgCag nigmansßirPaBCag
ebIeRbobeFobnwgrcnasm<½n§kMNt;edaysþaTick¾eday k¾mankrNIxøHKuNsm,tþiTaMgenHGacCMnYseday
KuNvibtþivijEdr. eKRtUveRbobeFobtémøsMPar³EdlsnSMCamYynwgtémøbEnßmEdlcaM)ac;edIm,Isagsg;
rcnasm<½n§enaH edaysareBlxøHeKRtUvkarcMNayfvikarCaeRcInkñúgkarsagsg;TMr nigtMNrbs;eRKOgbgÁúM
minkMNt;edaysþaTicebIeRbobeFobeTAnwgrcnasm<½n§kMNt;edaysþaTic. EtGVIEdlsMxan;CagenH eday
sareRKOgbgÁúMminkMNt;edaysþaTicmankmøaMgRbtikmμEdlCaGBaØat enaHeKRtUvRby½tñkñúgkarkarBarsMrut
DIepr:g;Esülrbs;TMr BIeRBaHT§iBlenHnwgbegáItkugRtaMgxagkñúgenAkñúgeRKOgbgÁúM. Ca]TahrN_ Rbsin
ebICBa¢aMgenAcugmçagénFñwmTMrbgáb;enAkñúgrUbTI 10-1a Rsut ehIykugRtaMgnwgekItmanenAkñúgFñwmeday
sarkmøaMgEdlekItmanedaysarkMhUcRTg;RTayenH. mü:agvijeTot RbsinebIRtUv)anRTedayTMrsamBaØ
b¤Carcnasm<½n§kMNt;edaysþaTic ¬rUbTI 10-1b¦ enaHtMruténcugrbs;vanwgmineFVIeGayFñwmxUcRTg;RTay
ehIyvanwgmineFVIeGaymankugRtaMgekItmanenAkñúgFñwmeT. CaTUeTA RKb;kMhUcRTg;RTayTaMgGs;dUcCa
kMhUcRTg;RTayedaysarEdlbNþalBIbMlas;TIrbs;TMr b¤bMErbMrYlrbs;RbEvgrbs;Ggát;EdlbNþalBI
sItuNðPaB b¤edaysarkMhugqÁgkñúgkarplit nwgbegáItkugRtaMgbEnßmenAkñúgrcnasm<½n§ EdleKcaM)ac;RtUv
KitBicarNaenAeBlsikSaKNnaeRKOgbgÁúMminkMNt;edaysþaTic.
viFIviPaK³ enAeBlviPaKrcnasm<½n§minkMNt;edaysþaTicTaMgGs; eKcaM)ac;RtUvbMeBjtMrUvkarlMnwg
(equilibrium) PaBRtUvKña (compatibility) nigTMnak;TMngkmøaMg nigbMlas;TI (force-displacement)
Analysis of statically indeterminate structures by the force method T.Chhay -335
3. Department of Civil Engineering NPIC
sRmab;rcnasm<½n§. lkçxNÐlMnwgRtUv)anbMeBjenAeBlEdlkmøaMgRbtikmμTb;rcnasm<½n§enAmYykEnøg
ehIylkçxNÐRtUvKñaRtUv)anbMeBjenAeBlEdlkMNat;Ggát;CaeRcInrbs;rcnasm<½n§eFVIkarCamYyKñaeday
minmankardac;xagkñúg b¤KgBIelIKña. tMrUvkarTMnak;TMngkmøaMg nigbMlas;TIGaRs½ynwgkareFVIkarrbs;
sMPar³ enAkñúgGtßbTenHeyIgsnμt;fasMPar³eFVIkarCalkçN³eGLasÞiclIenEG‘r. CaTUeTA eKmanviFIBIr
epSgKñaedIm,IbMeBjtMrUvkarTaMgenHenAeBlviPaKrcnasm<n§minkMNt;edaysþaTicKW viFIkmøaMg b¤viFIPøicsIu-
½
b‘IlIFI (force or flexibility method) nigviFIbMlas;TI b¤viFIPaBrwgRkaj (displacement or stiffness
method).
viFIkmøaMg³ viFIenHRtUv)anbegáIteLIgCadMbUgedayecmkøWkm:akesVl (James Clerk Maxwell) enAkñúgqñaM
1864 nigeRkaymk viFIenHRtUv)anEksMrYleGaymanlkçN³l¥RbesIredayGUtUmr (Otto Mohr) nig
ehnrIcm‘uyeyeRbsøÚ (Heinrich Muller-Breslau). edaysarlkçxNÐRtUvKñabegáIteKalkarN_sRmab;
viFIenH eBlxøHeKGacehAvaCaviFIxMepFIb‘IlIFI (compatibility method) b¤viFIbMlas;TIzitezr (method
of consistent displacement). viFIenHRtUv)anbegáIteLIgedaykarsresrsmIkarEdlbMeBjlkçxNÐ
RtUvKña nigtMrUvkarénTMnak;TMngkmøaMg nigbMlas;TIsRmab;eRKOgbgÁúMedIm,IkMNt;kmøaMgEdlCaGBaØatelIs.
enAeBlEdleKkMNt;kmøaMgTaMgenHrYcehIy eKGackMNt;kmøaMgRbtikmμEdlenAsl;edaybMeBjtMrUvkar
lkçxNÐlMnwg. eKalkarN_mUldæanEdlBak;B½n§nwgkarGnuvtþviFIenHmanlkçN³gayRsYlyl; nigman
lkçN³gayRsYlbegáIt ehIyvak¾RtUv)anelIkykmkBiPakSaenAkñúgCMBUkenHpgEdr.
viFIbMlas;TI³ viFIenHEp¥kelIkarsresrTMnak;TMngrvagkmøaMg nigbMlas;TIsRmab;Ggát;CadMbUg bnÞab;mk
Ep¥kelIkarsresrsmIkarlMnwgsRmab;eRKOgbgÁúM. enAkñúgkrNIenH GBaØatenAkñúgsmIkarKWbMlas;TI. enA
eBlEdleKTTYl)anbMlas;TI eKGackMNt;kmøaMgBIlkçxNÐRtUvKña nigsmIkarTMnak;TMngrvagbMlas;TI nig
kmøaMg. eyIgnwgsikSabec©keTsxøHEdleRbIedIm,IGnuvtþviFIbMlas;TIenAkñúgCMBUk11 nigCMBUk12. rUbmnþ
ma:RTIsénviFIenHmanbgðajenAkñúgCMBUk14/ 15 nig16.
viFInImYy²énviFITaMgBIrEdlbgðajenAkñúgrUbTI 10-2 manKuNsm,tþi nigKuNvibtþiGaRs½yeTAnwg
ragFrNImaRtrbs;eRKOgbgÁúM nigdWeRkénPaBminkMNt;rbs;va.
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -336
4. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
!0>@> dMeNIrkarTUeTAkñúgkarviPaKedayviFIkmøaMg
(Force method of analysis: general procedure)
meFüa)ayd¾l¥bMputkñúgkarbgðajeKalkarN_Edl
Bak;B½n§nwgviFIkmøaMgKWBicarNaFñwmEdlbgðaj enAkñúgrUbTI
10-3a. RbsinebIeKKUrdüaRkamGgÁesrIrbs;va enaHvanwg
mankmøaMgRbtikmμEdlCaGBaØatcMnYcbYn ehIyedaysarEt
vaGacmansmIkarlMnwgcMnYnbIsRmab;karedaHRsay enaHFñwm
enHCarcnasm½<n§dWeRkTImYy. dUcenH eKcaM)ac;RtUvkarsmIkar
bEnßmcMnYnmYysRmab;edaHRsay. edIm,ITTYlsmIkarenH
eyIgnwgeRbIeKalkarN_tMrYtpl nigBicarNalkçN³RtUv
KñaénbMlas;TIenARtg;TMrmçag. eKGacviPaKFñwmenHeday
eRCIserIskmøaMgRbtikmμTMrmYyCaGBaØatelIs ehIydk
vaecjBIFñwmCabeNþaHGasnñ enaHFñwmenHnwgkøayCaeRKOg
bgÁúMkMNt;edaysþaTic nigmansßirPaB. FñwmenHeKeGay
eQμaHfaeRKOgbgÁúMbzm (primary structure). enATIenH
eyIgnwgykkmøaMgRbtikmμrbs;TMrkl;Rtg;cMNuc B ecj.
CalT§pl bnÞúk P nwgeFVIeGaycMNuc B pøas;TIcuHeRkam
edaybrimaN Δ B dUcbgðajenAkñúgrUbTI 10-3b. b:uEnþ
edayviFItMrYtpl kmøaMgRbtikmμEdlCaGBaØatenARtg;cMNuc B eBalKW B y eFVIeGayFñwmenARtg;cMNuc B
pøas;TIeLIgelIedaybrimaN Δ'BB ¬rUbTI 10-3c¦. enATIenHGkSr B dMbUgsMedAelIcMNuc B EdlRtUv
karkMNt;PaBdab ehIyGkSr B TIBIrsMedAelIcMNuc B EdlkmøaMgRbtikmμEdlCaGBaØatmanGMeBI. eday
bMlas;TIviC¢manEdl)ansnμt;manGMeBIeLIgelI enaHBIrUbTI 10-3a rhUtdl;rUbTI 10-3c eyIgGac
sresrsmIkarRtUvKñaEdlcaM)ac;enARtg;TMrkl;Ca
(+ ↑) 0 = −Δ B + Δ' BB
LÚveyIgsMKal;bMlas;TIenARtg;cMNuc B EdlbNþalBIbnÞúkÉktþaEdlmanGMeBItamTisrbs; B y eday
emKuNPøicsIub‘IlIFIlIenEG‘r (linear flexibility coefficient) f BB ¬rUbTI 10-3d¦. edaykareRbInimitþ
sBaØakñúgTMrg;RsedogKña f BB CaPaBdabenARtg;cMNuc B EdlbNþalBIbnÞúkÉktþaenARtg;cMNuc B .
edaysarsMPar³manlkçN³eGLasÞiclIenEG‘r kmøaMg B y EdlmanGMeBIIenARtg;cMNuc B ¬CMnYseGay
Analysis of statically indeterminate structures by the force method T.Chhay -337
5. Department of Civil Engineering NPIC
bnÞúkÉktþa¦ nwgeFVIeGaymankMeNInPaBdabsmamaRtnwg f BB . dUcenHeyIgGacsresr
Δ' BB = B y f BB
BIkarsresrkúñgTMrg;EbbenH eyIgeXIjfaemKuN linear flexibility f BB manxñatKitCaPaBdabkñúgmYy
ÉktþakmøaMg eBalKW m / N b¤ mm / N .l. dUcenHeKGacsresrsmIkarlkçxNÐRtUvKñaxagelIeday
eRbIGBaØat B y Ca
0 = −Δ B + B y f BB
edayeRbIviFIénCMBUk8 nig9 b¤edayeRbItaragPaBdabenAkñúgCMBUk9 eKGacTTYl)anTMnak;TMngrvag
bMlas;TI nigkmøaMgsRmab;PaBdab Δ B ¬rUbTI 10-3b¦ nig flexibility coefficient f BB ¬rUbTI 10-
3d¦ ehIyeKGacedaHRsayrk B y )an Edl B y = Δ B / f BB . eRkayeBledaHRsayrkkmøaMgRbtikmμ
EdlCaGBaØatelIsrYcehIy eKGacedaHRsayrkkmøaMgRbtikmμbIeTotenARtg;CBa¢aMg A BIsmIkarlMnwg.
dUcGVIEdl)anerobrab;BIxagedIm eKeRCIserIs
GBaØatelIsedayminRbkan;. ]TahrN_ eKGackMNt;
m:Um:g;enARtg;cMNuc A ¬rUbTI 10-4a¦ edaypÞal;eday
dklT§PaBrbs;FñwmedIm,ITb;Tl;nwgm:Um:g; Rtg; A ecj
eBalKWCMnYsTMrbgáb;edayTMrsnøak;. dUcbgðajenAkñúg
rUbTI 10-4b mMurgVilenARtg;cMNuc A EdlbNþalBI
bnÞúk P KW θ A ehIymMurgVilenARtg; A EdlbNþalBI
M A EdlCaGBaØatelIenARtg; A KW θ ' AA ¬rUbTI 10-
4c¦. RbsinebIeyIgkMNt;nimitþsBaØarbs; angular
flexibility coefficient Ca α AA Edlmann½yfamMu
rgVilenARtg;cMNuc A EdlekItBIm:Um:g;mYyÉktþaman
GMeBIenARtg;cMNuc A ¬rUbTI 10-4d¦ enaH
θ ' AA = M Aα AA
dUcenH angular flexibility coefficient manxñatKitCa
mMurgVilkñúgmYYyÉktþam:Um:g; dUcenHvamanxñat rad / N .m
b¤ rad / kN .m .l. dUcenHsmIkarlkçxNÐRtUvKñasRmab;mMurgVilenARtg;cMNuc A tMrUveGay
( +) 0 = θ A + M Aα AA
enAkñúgkrNIenH M A = −θ A / α AA mantémøGviC¢man mann½yfa M A manGMeBIpÞúynwgm:Um:g;mYyÉktþa.
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -338
6. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
]TahrN_TIbIEdlbgðajBIkarGnuvtþénviFIkmøaMgRtUv
)aneGayenAkñúgrUbTI 10-5a. enATIenH FñwmCarcnasm<½n§
minkMNt;edaysþaTicdWeRkTIBIr dUcenHeKRtUvkarsmIkar
lkçxNÐRtUvKñacMnYnBIrsRmab;eFVIkar edaHRsay. eyIgnwg
eRCIserIskmøaMgbBaÄrenARtg;TMr B nig C Ca GBaØatelIs.
eRkayeBldkGBaØatelIsecj FñwmkMNt;edaysþaTicman
PaBdabdUcbgðajenAkñúgrUbTI 10-5b. kmøaMgEdlCa
GBaØatelIs nImYy²EdlRtUv)ansnμt;eGaymanGMeBIcuH
eRkameFVIeGayFñwmmanPaBdabdUcbgðajenAkñúgrUbTI 10-5c
nig 10-5d erogKña. enATIenH eKkMNt; flexibility
coefficient * f BB nig f CB BIkmøaMgÉktþamanGMeBIenARtg;
cMNuc B ¬rUbTI 10-5e¦ ehIyeKkMNt; fCC nig f BC
BIkmøaMgÉktþamanGMeBIenARtg;cMNuc C ¬rUbTI 10-5f¦.
edayviFItMrYtpl smIkarlkçxNÐRtUvKñasRmab;PaBdabRtg;
cMNuc B nig C KW
(+ ↓) 0 = Δ B + B y f BB + C y f BC
(+ ↓) 0 = Δ C + B y f CB + C y f CC (10-1)
eRkayeBlbegáItTMnak;TMngbnÞúk nigbMlas;TIedayeRbIviFI
enAkñúgCMBUk 8 nig9 eKGacedaHRsayRbB½n§smIkarenHedIm,I
rk B y nig C y .
edaysarkarGnuvtþviFIkmøaMgRtUv)anbgðajtamry³]TahrN_ eyIgnwgBiPakSakarGnuvtþrbs;va
kñúgn½yTUeTA ehIybnÞab;mkeyIgnwgeRbIvaCamUldæansRmab;edaHRsaycMeNaTEdlTak;Tgnwg truss,
Fñwm nigeRKag. b:uEnþsRmab;krNITaMgGs;enH eyIgRtUvdwgfa edaysarviFIenHGaRs½ynwgkartMrYtplén
bMlas;TI eKcaM)ac;RtUveGaysMPar³enAEteFVIkarkñgEdneGLasÞiclIenEG‘renAeBlEdlvargbnÞúk. ehIy
ú
eyIgk¾RtUvdwgEdrfa eKGackMNt;kmøaMgRbtikmμ nigkmøaMgkñúgTaMgGs;enARtg;cMNucmYyenAkñúgeRKOgbgÁúM
edayyklT§PaBRTRTg;bnÞúkrbs;eRKOgbgÁúMecj ehIybnÞab;mksresrsmIkarlkçxNÐRtUvKñaenARtg;
*
CaPaBdabenARtg;cMNuc B EdlbNþalBIkmøaMgÉktþaenARtg;cMNuc B
f BB f CB CaPaBdabenARtg;cMNuc C EdlbNþalBI
kmøaMgÉktþaenARtg;cMNuc B .
Analysis of statically indeterminate structures by the force method T.Chhay -339
7. Department of Civil Engineering NPIC
cMNucenaH ¬emIl]TahrN_ 10-5¦.
dMeNIrkarkñúgkarviPaK (Procedure for analysis)
CMhanxageRkampþl;nUvviFITUeTAsRmab;kMNt;kmøaMgRbtikmμ b¤kmøaMgkñúgrbs;rcnasm<½n§minkMNt;
edaysþaTicedayeRbIviFIkmøaMg b¤viFIPøicsIub‘IlIFI.
eKalkarN_tMrYtpl
kMNt;cMnYndWeRkminkMNt; n rbs;rcnasm<½n§. bnÞab;mkkMNt;GBaØatelIs n CakmøaMg b¤Cam:Um:g;EdlRtUv
dkecjCabeNþaHGasnñBIrcnasm<½n§edIm,IeFVIeGayvaeTACaeRKOgbgÁúMkMNt;edaysþaTic nigmansßirPaB.
edayeRbIviFItMrYtpl KUreRKOgbgÁúMminkMNt;edaysþaTic nigbgðajfavaesμIeTAnwges‘rIéneRKOgbgÁúMkMNt;
edaysþaTicEdlRtUvKña. rcnasm<½n§dMbUgCaeRKOgbgÁúMkMNt;edaysþaTicEdlRTbnÞúkxageRkAdUceRKOgbgÁúM
minkMNt;edaysþaTic ehIyrcnasm<½n§déTepSgeTotCarcnasm<½n§kMNt;edaysþaTicbEnßmeTAelIrcna
sm<½ndMbUg ehIyfaRTbnÞúk b¤m:Um:g;EdlCaGBaØatelIsdac;edayELkBIKña. ehIyKUrExSekageGLasÞic
enAelIrcnasm<½n§nImYy² nigbgðajnimitþsBaØabMlas;TI b¤nimitþsBaØamMurgVilrbs;kMlaMg b¤m:Um:g;EdlCa
GBaØatelIs.
smIkarlkçxNÐRtUvKña
sresrsmIkarlkçxNÐRtUvKñasRmab;bMlas;TI nigmMurgVilenARtg;cMNucnImYy²EdlmankmøaMg b¤m:Um:g;
EdlCaGBaØatelIs. eKKYrsresrsmIkarTaMgenHedayeRbIGBaØatelIs nig flexibility coefficient Edl
RtUvKñarbs;vaEdl)anBIbnÞúkÉktþa b¤m:Um:gÉktþaEdlRtUvKñanwgkmøaMg b¤m:Um:g;elIs.
;
kMNt;PaBdab nig flexibility coefficient TaMgGs;edayeRbItaragenAkñúgCMBUk9 b¤eRbIviFIenAkñúg
CMBUk8 b¤CMBUk9 . CMnYsTMnak;TMngrvagbMlas;TI nigkmøaMgeTAkñúgsmIkarlkçxNÐRtUvKña ehIyedaHRsay
*
rkGBaØatelIs. RbsinebItémøCaelxrbs;GBaØatelIsGviC¢man vabgðajfaGBaØatelIsmanGMeBIpÞúyeTA
nwgkmøaMgÉktþa b¤m:Um:g;Éktþa.
smIkarlMnwg
sg;düaRkamGgÁesrIrbs;rcnasm<½n§. edaysareyIg)anKNnakmøaMg nig/b¤m:Um:g;EdlCaGBaØatelIsrYc
ehIy eyIgGackMNt;kmøaMgRbtikmμEdlCaGBaØatEdlenAesssl;BIsmIkarlMnwg.
*
eKesñIfa RbsinebIdüaRkam M / EI sRmab;FñwmmanlkçN³samBaØ eKGaceRbIRTwsIRkLaépÞm:Um:g; b¤viFIFñwmnimitþ. eKGackMNt;
FñwmEdlmandüaRkam M / EI sμúKsμajenARKb;cMNucenAelIrcnasm<½n§edayeRbIviFIEdl)anerobrab;sRmab;eRKOgbgÁúMkMNt;eday
sþaTic.
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -340
8. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
eyIgKYrdwgfa enAeBlEdleyIgTTYl)ankmøaMgRbtikmμTMrTaMgGs;rYcehIy eyIgGacsg;
düaRkamkmøaMgkat; nigdüaRkamm:Um:g; ehIyeyIgGackMNt;PaBdabenARKb;cMNucTaMgGs;enAelIeRKOg
bgÁúMedayeRbIviFIEdlmanerobrab;sRmab;rcnasm<½n§kMNt;edaysþaTic.
!0>#> RTwsþIbMlas;TIersIuRbUkrbs;m:akSEvl c,ab;ebFI
Maxwell’s theorem of reciprocal displacement; Betti’s law
enAeBlEdlm:akSEvl)anbegáItviFIkmøaMg Kat;k¾)anpSBVpSayRTwsþIEdlTak;TgeTAnwgeTAnwg
flexibility coefficient énBIrcMNucenAelIrcnasm<½n§eGLasÞicEdlmandUcCa truss, Fñwm b¤eRKag. eK
GacehARTwsþIenHfaRTwsþIbMlas;TIersIuRbUk EdleBaldUcteTA³ bMlas;TIrbs;cMNuc B enAelIeRKOgbgÁúM
EdlbNþalBIbnÞúkÉktþaEdlmanGMeBIenARtg;cMNuc A esμIeTAnwgbMlas;TIrbs;cMNuc A enAeBlEdl
bnÞúkÉktþamanGMeBIenARtg;cMNuc B eBalKW f BA = f AB .
eKbkRsayRTwsþIenHedayRsYledayeRbIeKalkarN_kmμnþnimitþ. Ca]TahrN_ BicarNaFñwmenA
kñúgrUbTI 10-6. enAeBlbnÞúkÉktþaBitmanGMeBIenARtg; A snμt;fam:Um:g;Bt;enAkñúgFñwmtameday m A .
edIm,IkMNt; flexibility coefficient enARtg; B eBalKW f BA eKRtUvdak;bnÞúkÉktþanimitþenARtg;cMNuc B
¬rUbTI 10-7¦ ehIyeKkMNt;m:Um:t;Bt; mB . edayGnuvtþsmIkar 9-18 eyIg)an
mB m A
f BA = ∫ dx
EI
dUcKña RbsinebIeKkMNt; flexibility coefficient f AB enAeBlkmøaMgÉktþaBitmanGMeBIenARtg;
cMNuc B ¬rUbTI10-7¦ enaH mB Cam:Um:gBt;enAkñúgFñwmEdlbNþalBIkmøaMgÉktþaBit. mü:agvijeTot m A
Cam:Um:g;Bt;EdlbNþalBIkmøaMgÉktþanimitþenARtg;cMNuc A ¬rUbTI 10-6¦. dUcenH
m A mB
f AB = ∫ dx
EI
GaMgetRkalTaMgBIrpþl;nUvlT§pldUcKñaEdlRtUvnwgBMenalrbs;RTwsþI. RTwsþIenHk¾GacGnuvtþ
sRmab;mMurgVilersIuRbUkEdr EdlRTwsþIenHeBaldUcteTA³ mMurgVilenARtg;cMNuc B enAelIeRKOgbgÁúMEdl
bNþalBIm:Um:g;ÉktþaEdlmanGMeBIenARtg;cMNuc A esμInwgmMurgVIlenARtg;cMNuc A enAeBlm:Um;g;manGMeBI
enARtg;cMNuc B . elIsBIenH edayeRbIkmøaMg nigm:Um:g;ÉktþaEdlGnuvtþenAelIrcnasm<½n§Rtg;cMNucdac;
Analysis of statically indeterminate structures by the force method T.Chhay -341
9. Department of Civil Engineering NPIC
edayELkBIKña eyIgk¾GaceBaldUcteTA³ mMurgVilKitCara:düg;enARtg;cMNuc B enAelIrcnasm<½n§Edl
bNþalBIbnÞúkÉktþaEdlmanGMeBIenARtg;cMNuc A esμInwgbMlas;TIenARtg;cMNuc A enAeBlm:Um:g;Éktþa
manGMeBIenARtg;cMNuc B .
Cavi)akénRTwsþIenH eKGacsnSMsMéckargar)anxøHenAeBlGnuvtþviFIkmøaMgedIm,IedaHRsaycMeNaT
Edlrcnasm<½n§Carcnasm<<½n§minkMNt;edaysþaTicdWeRkTIBIr b¤eRcInCagenH. ]TahrN_ eKRtUvkarkMNt;
flexibility coefficient EtmYyb:ueNÑaHkñúgcMeNam f BC nig f CB enAkñúgsmIkar 10-1 edaysar f BC
= f CB . elIsBIenH eKGaceRbIRTwsþIbMlas;TIersIuRbUkenAkñúgkarkarviPaKrcnasm<½n§ nigsRmab;sg;ExS
T§iBledayeRbIeKalkarN_m‘uyeyeRbsøÚ ¬emIlkfaxNÐ 10.10¦.
enAeBleKeFIVeGayRTwsþIbMlas;TIersIuRbUkmanTMrg;kan;EtTUeTA eKeGayeQμaHvafac,ab;ebFI.
c,ab;enHeBaly:agsegçbfa³ kmμnþnimitþ δU AB eFVIedayRbB½n§kmøaMg ∑ PB CamYynwgbMlas;TIEdlbNþal
BIRbB½n§kmøaMg ∑ PA esμIeTAnwgkmμnþnimitþ δU BA EdlbNþalBIkmøaMg ∑ PA enAeBlrcnasm<½n§xUcRTg;
RTayedaysarRbB½n§kmøaMg ∑ PB . eKGacniyaymü:ageTotfa δU AB = δU BA . karbkRsaynUv
BMenalenHmanlkçN³dUcKñanwgkarbkRsayRTwsþIbMlas;TIersIuRbUkxagelI.
!0>$> karviPaKFñwmtamviFIkmøaMg (Force method of analysis: Beams)
viFIkmøaMgEdlGnuvtþeTAelIFñwmRtUv)anerobrab;enAkñúgkfaxNÐ 10-2. edayGnuvtþtamkarerob
rab;BIdMeNIrkarviFIEdlmanEcgenAkñúgkfaxNÐ 10-2 eyIgnwgbgðajBI]TahrN_CaeRcInEdlENnaMBIkar
Gnuvtþbec©keTsenH.
]TarhN_ 10-1³ kMNt;RbtikmμTMrenARtg;TMrkl; B rbs;FñwmEdlbgðajenAkñúgrUbTI 10-8a. EI Ca
témøefr.
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -342
10. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
dMeNaHRsay³
eKalkarN_tMrYtpl³ tamkarGegát FñwmCaeRKOgbgÁúMminkMNt;edaysþaTicdWeRkTImYy. eKnwgyk B y Ca
GBaØatelIs dUcenHeKGackMNt;kmøaMgenHedaypÞal;. rUbTI 10-8b bgðajBIkarGnuvtþéneKalkarN_
tMrYtpl. cMNaMfa karykGBaØatelIsecjTamTareGayykTMrkl; b¤karTb;rbs;FñwmtamTis By ecj.
enATIenH eyIgsnμt;fa B y eFVIGMeBIeTAelIFñwmedaymanTisedAeLIgelI.
smIkarlkçxNÐRtUvKña³ edayykbMlas;TIviC¢manmanTisedAeLIgelI ¬rUbTI 10-8b¦ eyIg)an
(+ ↑) 0 = −Δ B + B y f BB (1)
eKGacTTYl)antY Δ B nig f BB edaygayedayeRbItaragenAkñúCMBUk9. cMNaMfa Δ B = ΔC + θC (6m)
dUcenH
P(L / 2)3 P(L / 2)2 ⎛ L ⎞
ΔB = + ⎜ ⎟
3EI 2 EI ⎝ 2 ⎠
=
(50kN )(6m )3 + (50kN )(6m )2 (6m ) = 9000kN .m3 ↓
3EI 3EI EI
PL3 1(12m ) 576m33
f BB = = = ↑
3EI 3EI EI
edayCMnYslT§plTaMgenHeTAkñúgsmIkar (1) eyIg)an
(+ ↑) 0=−
9000 ⎛ 576 ⎞
+ By ⎜ ⎟ B y = 15.6kN
EI ⎝ EI ⎠
RbsinebIeKdak;kmøaMgRbtikmμenHeTAelIdüaRkamGgÁesrIrbs;Fñwm eKGacTTYlkmøaMgRbtikmμenARtg; A
BIsmIkarlMnwgbI ¬rUbTI 10-8c¦.
eRkayeBlkMNt;kmøaMgRbtikmμTaMgGs; eKGacsg;düaRkamm:Um:g;dUcbgðajenAkñúgrUbTI 10-8d.
]TarhN_ 10-2³ kMNt;m:Um:g;enARtg;CBa¢aMgbgáb;sRmab;FñwmEdlbgðajenAkñúgrUbTI 10-9a. EI Ca
témøefr.
dMeNaHRsay³
eKalkarN_tMrYtpl³ eyIgnwgyk M A CaGBaØatelIs edaysareyIgGackMNt;m:Um:g;enHedaypÞal;.
rUbTI 10-8b bgðajBIkarGnuvtþéneKalkarN_tMrYtpl. enATIenH eyIgdklT§PaBrbs;FñwmTb;Tl;nwg
m:Um:g;enARtg;TMr A . eyIgsnμt;fa M A eFVIGMeBIRcasRTnicnaLika.
smIkarlkçxNÐRtUvKña³ edayykmMurgVilviC¢manvilRcasRTnicnaLika ¬rUbTI 10-9b¦ eyIg)an
( +) 0 = θ A + M Aα AA (1)
eKGackMNt;tY θ A nig α AA BItaragenAkñúgtaragkñúgCMBUk9. eyIg)an
Analysis of statically indeterminate structures by the force method T.Chhay -343
11. Department of Civil Engineering NPIC
ML 30kN .m(3m ) 15kN .m 2
θA = = =
6 EI 6 EI EI
ML 1(3m ) 1m
α AA = = =
3EI 3EI EI
edayCMnYseTAkñúgsmIkar (1) eyIg)an
15 ⎛ 1 ⎞
( +) 0= + M A⎜ ⎟ M A = -15kN.m
EI ⎝ EI ⎠
sBaØaGviC¢manbgðajfa M A eFVIGMeBIpÞúyeTAnwgGVIEdl)anbgðajenAkñúgrUbTI 10-9b. enAeBleyIgdak;
kmøaMgRbtikmμenHeTAelIFñwm eyIgGackMNt;kmøaMgRbtikmμdéTeTotBIsmIkarlMnwg ¬rUbTI 10-9c¦.
düaRkamm:Um:g;RtUv)anbgðajenAkñúgrUbTI 10-9d.
]TarhN_ 10-3³ sg;düaRkamkmøaMgkat; nigdüaRkamm:Um:gsRmab;FñwmEdlbgðajenAkñúgrUbTI 10-
;
10a. TMr B mansMrut 40mm . yk E = 200GPa / I = 500(106 )mm4 .
dMeNaHRsay³
eKalkarN_tMrYtpl³ tamkarGegát FñwmenHCaeRKOgbgÁúMminkMNt;edaysþaTicdWeRkTImYy. eyIgeRCIs
erIsTMrkNþal B CaGBaØatelIs dUcenHeyIgRtUvdkTMrkl;enARtg; B ecj ¬rUbTI 10-10b¦. enATIenH
eyIgsnμt;fa B y manTisedAcuHeRkam.
smIkarlkçxNÐRtUvKña³ eyagtamcMNuc B enAkñúgrUbTI 10-10b nigedayeRbIxñatEm:Rt eyIg)an
(+ ↓) 0.04 = Δ B + B y f BB (1)
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -344
12. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
enATIenHeyIgnwgeRbIviFIFñwmRtUvKñaedIm,IKNna Δ B nig f BB edaysardüaRkamm:Um:g;pSMeLIgedaybnÞat;
Rtg;. sRmab; Δ B epÞógpÞat;karKNnaEdlbgðajenAkñúgrUbTI 10-10c. cMNaMfa
+ ∑ M B' = 0 − M B' +
800
(2.667 ) − 1000 (8) = 0
EI EI
5866.67 5867
M B' =− =
EI EI
epÞógpÞat;karKNnaenAkñúgrUbTI 10-10d sRmab;karKNna f BB . cMNaMfa
+ ∑ M B' = 0 − mB ' +
16
(2.667 ) − 16 (8) = 0
EI EI
85.33 85.33
mB ' = − =
EI EI
edayCMnYslT§plTaMgenHeTAkñúgsmIkar (1) eyIg)an
5866.67 ⎛ 85.33 ⎞
0.04 = + By ⎜ ⎟
EI ⎝ EI ⎠
Analysis of statically indeterminate structures by the force method T.Chhay -345
13. Department of Civil Engineering NPIC
edayCMnYs E nig I Caelx eyIg)an
(0.04m )(200)(106 )kN / m 2 ][500(106 )(10−12 )m 4 ] = 5866.67 + B y (85.33)
[
B y = −21.875kN
smIkarlMnwg³ sBaØaGviC¢manbgðajfa By eFVIGMeBIelIFñwmedaymanTisedAeLIgelI. BIdüaRkamGgÁesrI
EdlbgðajenAkñúgrUbTI 10-10e eyIg)an
+∑MA = 0 − 100(4) + 21.875(8) + C y (16) = 0
C y = 14.06kN
+ ↑ ∑ Fy = 0 Ay − 100 + 21.875 + 14.06 = 0
Ay = 64.06kN
edayeRbIlT§plTaMgenH epÞógpÞat;düaRkamkmøaMgkat; nigdüaRkamm:Um:g;Bt;EdlbgðajenAkñúgrUbTI 10-
10f.
]TarhN_ 10-4³ sg;düaRkamkmøaMgkat; nigdüaRkamm:Um:g;sRmab;FñwmEdlbgðajenAkñúgrUbTI 10-
11a. EI mantémøefr. ecalT§iBlrbs;kmøaMgtamG½kS.
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -346
14. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
dMeNaHRsay³
eKalkarN_tMrYtpl³ edaysareyIgecalbnÞúktamG½kS FñwmCarcnasm<½n§minkMNt;edaysþaTicdWeRkTI
mYy. eyIgKitm:Um:g;cugTaMgBIrenARtg;cMNuc A nig B CaGBaØatelIs. eKRtUvdklT§PaBrbs;FñwmEdlTb;
Tl;nwgm:Um:g; edayCMnYsTMrsnøak;enARtg;cMNuc A nigTMrkl;enARtg;cMNuc B . eKalkarN_tMrYtplEdl
GnuvtþeTAelIFñwmRtUv)anbgðajenAkñúgrUbTI 10-11b.
smIkarlkçxNÐRtUvKña³ eyagtamcMNuc A nigcMNuc B ¬rUbTI 10-11b¦ tMrUveGay
( +) 0 = θ A + M Aα AA + M Bα AB (1)
( +) 0 = θ B + M Aα BA + M Bα BB (2)
eyIgGackMNt;mMurgVilEdlRtUvkar nig angular flexibility coefficient edayeRbItaragenAkñúgCMBUk 9.
eyIg)an
3wL3 3(30 )(6 )3 151.9
θA = = =
128 EI 128 EI EI
7 wL3 7(30 )(6 )3 118.1
θB = = =
384 EI 384 EI EI
ML 1(6) 2
α AA = = =
3EI 3EI EI
ML 1(6) 2
α BB = = =
3EI 3EI EI
ML 1(6) 1
α AB = = =
6 EI 6 EI EI
cMNaMfa α AB = α BA CalkçN³sMxan;rbs;RTwsþIbMlas;TIersIuRbUkrbs;m:akSEvl.
edayCMnYsTinñn½yxagelIeTAkñúgsmIkar (1) nig (2) eyIg)an
151.9 ⎛ 2 ⎞ ⎛ 1 ⎞
0= + M A⎜ ⎟ + M B ⎜ ⎟
EI ⎝ EI ⎠ ⎝ EI ⎠
118.1 ⎛ 1 ⎞ ⎛ 2 ⎞
0= + M A⎜ ⎟ + M B ⎜ ⎟
EI ⎝ EI ⎠ ⎝ EI ⎠
edaysMrYl EI nigedayedaHRsayRbB½n§smIkarTaMgenH eyIg)an
M A = −61.9kN .m M B = −28.1kN .m
edayeRbIlT§plTaMgenH eyIgGackMNt;kmøaMgkat;xagcugFñwm ¬rUbTI 10-11c¦ ehIyeyIgGacsg;
düaRkamkmøaMgkat; nigdüaRkamm:Um:g;.
]TarhN_ 10-5³ kMNt;kmøaMgRbtikmμTMrsRmab;FñwmEdlbgðajenAkñúgrUbTI 10-12a. EI mantémø
efr.
Analysis of statically indeterminate structures by the force method T.Chhay -347
15. Department of Civil Engineering NPIC
dMeNaHRsay³
eKalkarN_tMrYtpl³ edaykarGegát FñwmenHCaeRKOgbgÁúMminkMNt;edaysþaTicdWeRkTImYy. enATIenH
eyIgnwgeRCIserIsm:Um:g;Bt;enARtg;TMr B CaGBaØatelIs. Cavi)ak eyIgnwgkat;FñwmenHedaydac;ecjBIKña
ehIydak;TMrsnøak; b¤TMrkl;enARtg;cMNuc B edIm,IdklT§PaBrbs;FñwmEdlTb;Tl;nwgm:Um:g;enARtg;cMNuc
enH ¬rUbTI 10-12b¦. eKeGaym:Um:g;Bt;manGMeBIelIFñwmenAkñúgrUbTI 10-12c.
smIkarlkçxNÐRtUvKña³ BIrUbTI 10-12a eyIgtMrUveGay
( +) 0 = θ B + M Bα BB
Edl θ B = θ ' B +θ "B
nig α BB = α ' BB +α "BB
eKGackMNt;mMurgVil nig angular flexibility coefficient BItaragenAkñúgCMBUk 9 Edl
wL3 2(3.6 )3 3.888kN .m 2
θ 'B = = =
24 EI 24 EI EI
PL2 2.5(3)2 1.406kN .m 2
θ "B = = =
16 EI 16 EI EI
ML 1(3.6 ) 1.2m
α ' BB = = =
3EI 3EI EI
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -348
16. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
ML 1(3) 1m
α "BB = = =
3EI 3EI EI
3.888kN .m 2 1.406kN .m 2 ⎛ 1.2m 1m ⎞
dUcenH EI
+
EI
+ MB⎜
⎝ EI
+ ⎟=0
EI ⎠
M B = −2.406kN .m
sBaØadkbgðajfa M B eFVIGMeBItamTisedApÞúyeTAnwgkarbgðajenAkñúgrUbTI 10-12c. edayeRbIlT§pl
enH eyIgGackMNt;kmøaMgRbtikmμdUcbgðajenAkñúgrUbTI 10-12d. elIsBIenH düaRkamkmøaMgkat; nig
düaRkamm:Um:g;Bt;RtUv)anbgðajenAkñúgrUbTI 10-12e.
!0>%> karviPaKeRKagtamviFIkmøaMg (Force method of analysis: Frames)
viFIkmøaMgmanPaBgayRsYlkñúgkaredaHRsayeRKagminkMNt;edaysþaTicmYyCan; nigmanrag
FrNImaRtminFmμtadUcCa gabled frames. sRmab;edaHRsayeRKagminkMNt;eRcInCan; b¤eRKOgbgÁúMEdl
mandWeRkminkMNt;x<s; eKeRbIviFIsøÚbDIPøicsin ¬slope-deflection) b¤viFIEbgEckm:Um:g; (moment-
distribution) b¤viFIsÞIveNs (stiffness method) Edlnwgerobrab;enAkñúgemeronxagmux.
]TahrN_xageRkambgðajBIkarGnuvtþénviFIkmøaMgedayeRbIdMeNIrviFIdUcerobrab;enAkñúgkfaxNÐ
10-2.
]TarhN_ 10-6³ kMNt;kmøaMgRbtikmμTMrrbs;eRKagEdlbgðajenAkñúgrUbTI 10-13a. EI mantémø
efr.
Analysis of statically indeterminate structures by the force method T.Chhay -349
17. Department of Civil Engineering NPIC
dMeNaHRsay³
eKalkarN_tMrYtpl³ edaykarGegát eRKagCarcnasm<½n§minkMNt;dWeRkTImYy. eyIgnwgeRCIserIskmøaMg
RbtikmμtamTisedkenARtg; B CaGBaØatelIs. dUcenH eyIgCMnYssnøak;enARtg; B edayTMrkl; edaysar
TMrkl;minTb;cMNuc B tamTisedk. dUcenHeKalkarN_tMrYtplEdlGnuvtþeTAelIeRKagRtUv)anbgðajenA
kñúgrUbTI 10-13b.
smIkarlkçxNÐRtUvKña³ eyagtamcMNuc B enAkñúgrUbTI 10-13b tMrUveGay
+
(→) 0 = Δ B + Bx f BB (1)
eKGackMNt;tY Δ B nig f BB edayeRbIviFIkmμnþnimitþ. kUGredaen x rbs;eRKag nigm:Um:g;Bt;RtUv)anbgðaj
enAkñúgrUbTI10-13c nig10-13d. eKcaM)ac;RtUveCIserIskUGredaen x1 nig x2 dUcKñasRmab;krNI
nImYy² ¬TaMgsRmab;bnÞúkBit nigsRmab;bnÞúknimitþ¦. ehIyTisedAviC¢masRmab; M nig m k¾RtUvEtdUcKña.
sRmab; Δ B eyIgtMrUveGaybnÞúkBit ¬rUbTI 10-13c¦manGMeBIenAelIeRKag nigbnÞúkÉktþanimitþ
manGMeBIenARtg;cMNuc B ¬rUbTI 10-13d¦. dUcenH
L Mm 5 (20 x1 − 4 x1 )(0.8 x1 )dx1
2 4 0(1x2 )dx2
ΔB = ∫0 dx =
EI ∫0 EI
+ ∫0 EI
166.7 166.7
= +0=
EI EI
sRmab; f BB eyIgtMrUveGaybnÞúkÉktþaBit nigbnÞúkÉktþanimitþmanGeBIenARtg;cMNuc B ¬rUbTI 10-
13d¦. dUcenH
5 (0.8 x1 ) dx1 4 (1x2 ) dx2
2 2
L mm
f BB = ∫ dx = ∫ +∫
0 EI 0 EI 0 EI
26.7 21.3 48.0
= + =
EI EI EI
edayCMnYsTinñn½yxagelIeTAkñúgsmIkar (1) nigedayedaHRsay eyIgTTYl)an
166.7 ⎛ 48.0 ⎞
0= + Bx ⎜ ⎟ Bx = −3.47kN
EI ⎝ EI ⎠
smIkarlMnwg³ edaybgðaj Bx enAelIdüaRkamGgÁesrIrbs;eRKagedayeRbITisedARtwmRtUv ¬rUbTI 10-
13e¦ nigedayGnuvtþsmIkarlMnwg eyIgTTYl)an
+
→ ∑ Fx = 0 Ax − 3.47 = 0 Ax = 3.47kN
+∑MA = 0 − 40(2.5) + B y (5) − 3.47(4) = 0 B y = 22.8kN
+ ↑ ∑ Fy = 0 Ay − 40 + 22.8 = 0 Ay = 17.2kN
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -350
18. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
]TarhN_ 10-7³ kMNt;m:Um:g;enARtg;TMrbgáb; A rbs;eRKagEdlbgðajenAkñúgrUbTI 10-14a. EI
mantémøefr.
dMeNaHRsay³
eKalkarN_tMrYtpl³ eRKagenHCaeRKOgbgÁúMminkMNt;dWeRkTImYy. eKGacTTYldMeNaHRsaysRmab;
M A edaypÞal;edayeRCIserIsvaCaGBaØatelIs. dUcenHeyIgGacdklT§PaBrbs;eRKagTb;Tl;nwgm:Um:g;
Rtg; A dUcenHeyIgGaceRbITMrsnøak;enARtg;cMNuc A . eKalkarN_tMrYlplEdlGnuvtþeTAelIeRKag
RtUv)anbgðajenAkñúgrUbTI 10-14b.
smIkarlkçxNÐRtUvKña³ eyagtamcMNuc B enAkñúgrUbTI 10-14b tMrUveGay
( +) 0 = θ A + M Aα AA (1)
dUc]TahrN_mun² eKGacKNna θ A nig α AA edayeRbIviFIkmμnþnimitþ. kUGredaen x rbs;eRKag
nigm:Um:g;Bt;RtUv)anbgðajenAkñúgrUbTI 10-14c nig10-14d.
sRmab; θA eyIgTamTarnUvkarGnuvtþrbs;bnÞúkBit ¬rUbTI 10-14c¦
nigm:Um:g;ÉktþanimitþenARtg;cMNuc A ¬rUbTI 10-14d¦. dUcenH
Analysis of statically indeterminate structures by the force method T.Chhay -351
19. Department of Civil Engineering NPIC
MmθL
θA = ∑ ∫ dx
0 EI
=∫
2.4 (0.13 x1 )(1 − 0.278 x1 )dx1
+∫
(
1.5 1.34 x2 − 0.75 x2 (0.222 x2 )dx2
2
)
0 EI 0 EI
0.209 0.124 0.333
= + =
EI EI EI
sRmab; α AA eyIgTamTarnUvkarGnuvtþrbs;m:Um:g;ÉktþaBit nigm:Um:g;ÉktþanimitþEdlmanGMeBIenARtg;
cMNuc A ¬rUbTI 10-14d¦. dUcenH
L mθ mθ
α AA = ∑ ∫ dx
0 EI
=∫
2.4 (1 − 0.278 x2 )2 dx1 + 1. 5 (0.222 x2 )2 dx2
0 EI ∫0 EI
1.155 0.555 1.21
= + =
EI EI EI
edayCMnYslT§plTaMgenHeTAkñúgsmIkar (1) nigedayedaHRsayeyIgTTYl)an
0.333 ⎛ 1.21 ⎞
0= + M A⎜ ⎟ M A = −0.275kN .m
EI ⎝ EI ⎠
sBaØadkbgðajfa M A manGMeBItamTisedApÞúyeTAnwgGVIEdlbgðajenAkñúgrUbTI10-14b.
!0>^> karviPaK truss tamviFIkmøaMg (Force method of analysis: Trusses)
CaTUeTA eKGackMNt;dWeRkminkMNt;rbs; truss edaykarGegát b:uEnþRbsinebIeKmankarlM)ak
kñúgkareFVIEbbenH eKGaceRbIsmIkar 3-1 (b + r > 2 j ) . enATIenH GBaØatRtUv)ansMEdgedaycMnYnkmøaMg
r)ar (b) bUknwgkmøaMgRbtikmμTMr (r ) ehIycMnYnsmIkarlMnwgEdlGacmanKW (2 j ) edaysareKGac
sresrsmIkarBIrsRmab;tMN ( j ) nImYy².
viFIkmøaMgCaviFImYyEdlmanlkçN³smRsbkñúgkarviPaK truss minkMNt;edaysþaTicdWeRkTImYy
b¤TIBIr. ]TahrN_xageRkambgðajBIkarGnuvtþrbs;vFIenHedayeRbIdMeNIrkarviPaKEdlerobrab;enAkñúg
i
kfaxNÐ 10-2.
]TarhN_ 10-8³ kMNt;kmøaMgenAkñúgGgát; AC rbs; truss EdlbgðajenAkñúgrUbTI 10-15a. EI
mantémøefrsRmab;RKb;Ggát;TaMgGs;.
dMeNaHRsay³
eKalkarN_tMrYtpl³ tamkarGegát truss enHCaeRKOgbgÁúMminkMNt;edaysþaTicdWeRkTImYy . edaysar *
*
edayGnuvtþsmIkar (3-1) b + r > 2 j b¤ 6 + 3 > 2(4) / 9 > 8 / 9 − 8 = 1
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -352
20. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
eKkMNt;kmøaMgenAkñúgGgát; AC edayeRCIserIsvaCaGBaØatelIs. eKRtUvkat;Ggát;enHedIm,IkMueGayvaman
kmøaMg ehtudUcenHehIy truss enHkøayCarcnasm<½n§kMNt;edaysþaTic nigmansßirPaB. eKalkarN_
tMrYtEdlGnuvtþeTAelI truss RtUv)anbgðajenAkñúgrUbTI 10-15b.
smIkarlkçxNÐRtUvKña³ edayeyageTAtamGgát; AC enAkñúgrUbTI 10-15b eyIgtMrUveGaybMlas;TI
Δ AC EdlekIteLIgenAxagcugGgát;Edlkat; AC EdlbNþalBIbnÞúk 2kN bUknwgbMlas;TI FAC f ACAC
EdlbNþalBIkmøaMgEdlCaGBaØatelIsEdlmanGMeBIEtÉgesμIsUnü enaH
0 = Δ AC + FAC f ACAC (1)
enATIenH flexibility coefficient f ACAC CabMlas;TIenARtg;cMNuccugkMNt;Ggát;Edlkat;rbs; AC Edl
bNþalBIbnÞúkÉktþaBitEdlmanGMeBIenARtg;cugEdlrgkarkat;énGgát; AC . tYr f ACAC nig Δ AC RtUv
)ankMNt;edayeRbIviFIkmμnþnimitþ. karviPaKkmøaMgedayeRbIviFItMNRtUv)ansegçbenAkñúgrUbTI 10-15c
nig 10-15d.
sRmab; Δ AC eyIgtMrUveGaymankarGnuvtþénkmøaMgBit 2kN ¬rUbTI 10-15c¦ nigkmøaMgÉktþa
nimitþmanGMeBIenARtg;cugrgkarkat;énGgát; AC ¬rUbTI 10-15d¦. dUcenH
Analysis of statically indeterminate structures by the force method T.Chhay -353
21. Department of Civil Engineering NPIC
nNL
Δ AC = ∑
AE
⎡ (− 0.8)(2)(2.4) ⎤ ⎡ (− 0.6 )(0 )(1.8)⎤ ⎡ (− 0.6)(1.5)(1.8) ⎤
= 2⎢ ⎥+⎢ ⎥+⎢ ⎥
⎣ AE ⎦ ⎣ AE ⎦ ⎣ AE ⎦
⎡ (1)(− 2.5)(3)⎤ ⎡ (1)(0)(3) ⎤
+⎢ ⎥ + ⎢ AE ⎥
⎣ AE ⎦ ⎣ ⎦
16.8
=−
AE
sRmab; f ACAC eyIgtMrUveGaymankarGnuvtþkmøaMgÉktþaBit nigkmøaMgÉktþanimitþenARtg;cugkMNt;Edl
rgkarkat;énGgát; AC ¬rUbTI 10-15d¦. dUcenH
n2 L
f ACAC = ∑
AE
⎡ (− 0.8)2 (2.4) ⎤ ⎡ (− 0.6)2 (1.8) ⎤ ⎡ (1)2 (3) ⎤
= 2⎢ ⎥ + 2⎢ ⎥+⎢ ⎥
⎢
⎣ AE ⎥
⎦ ⎢
⎣ AE ⎥ ⎢ AE ⎥
⎦ ⎣ ⎦
10.37
=
AE
edayCMnYsTinñn½yTaMgenHeTAkñúgsmIkar (1) nigedayedaHRsay eyIg)an
16.8 10.37
0=− + FAC
AE AE
FAC = 1.62kN (T )
edaysarlT§plCaelxmantémøviC¢man enaH AC rgkmøaMgTajdUckarsnμt; ¬rUbTI 10-15b¦. eday
eRbIlT§plenH eyIgGackkMNt;kmøaMgenAkñúgGgát;déTedayeRbIviFItMN nigsmIkarlMnwg.
]TarhN_ 10-9³ kMNt;kmøaMgenAkñúgGgát;nImYy²én truss EdlbgðajenAkñúgrUbTI 10-16a Rbsin
ebIeKeRbIekALak;enAelIGgát; AC edIm,IeFVIeGayGgát;enHxøICagGgát; BD RbEvg 12.5mm . Ggát;
nImYy²manmuxkat; 125mm2 nig E = 200GPa .
dMeNaHRsay³
eKalkarN_tMrYtpl³ Truss enHmanragFrNImaRtdUcnwg truss enAkñúg]TahrN_ 10-8. edaysareK
eFVIeGayGgát; AC rYjxøI eyIgnwgeRCIserIsvaCaGBaØatelIs ¬rUbTI 10-16b¦.
smIkarlkçxNÐRtUvKña³ edaysarminmanbnÞúkxageRkAmanGMeBIelIeRKOgbgÁúM (truss) edIm dUcenHvamin
manbMlas;TIenAcenøaHcugkMNat;rgkarkat;rbs;Ggát;edaysarbnÞúkeT eBalKW Δ AC = 0 . Flexibility
coefficient f ACAC RtUv)ankMNt;rUcehIyenAkñúg]TahrN_ 10-8 dUcenH
10.37
f ACAC =
AE
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -354
22. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
edaysnμt;brimaNrYjxøIrbs;Ggát;mantémøviC¢man dUcenHsmIkarlkçxNÐRtUvKñasRmab;Ggát;KW
10.37
0.0125m = 0 + FAC
AE
edaydwgfa f ACAC CabMlas;TIkñúgmYyÉktþakmøaMg eyIg)an
10.37 m
0.0125m = 0 +
(125mm )(200kN / mm ) F
2 2 AC
dUcenH FAC = 30.14kN = 30.1kN (T )
edaysarminmankmøaMgxageRkAmanGMeBIelI truss kmøaMgRbtikmμTMresμIsUnü. dUcenH edayeRbI FAC
nigedayviPaK truss edayviFItMN eyIgTTYl)anlT§pldUcbgðajenAkñúgrUbTI 10-16c.
!0>&> eRKOgbgÁúMkMub:UsIut (Composite structures)
eRKOgbgÁúMkMub:UsIutpSMeLIgedayGgát;xøHEdlrgEtnwgkmøaMgtamG½kS xN³Ggát;déTeTotrgkar
Bt;. RbsinebIvaCarcnasm<½n§minkMNt;edaysþaTic viFIkmøaMgmanlkçN³gayRsYlkñúgkareRbIsRmab;
viPaKva. ]TahrN_xageRkambgðajBIdMeNIrkarviPaKeRKOgbgÁúMEbbenH.
]TarhN_ 10-10³ FñwmEdlbgðajenAkñúgrUbTI 10-17a RtUv)anRTedayTMrsnøak;enARtg;cMNuc A
nigP¢ab;eTAnwgr)arBIrEdlmanTMrsnøak;enARtg;cMNuc B . kMNt;kmøaMgenAkñúgGgát; BD . yk
E = 200GPa / I = 300( 6 )mm 4 sRmab;Fñwm nig A = 1800mm 2 sRmab;r)ardéTeTot .
10
Analysis of statically indeterminate structures by the force method T.Chhay -355
23. Department of Civil Engineering NPIC
dMeNaHRsay³
eKalkarN_tMrYtpl³ tamkarGegát FñwmCaeRKOgbgÁúMminkMNt;edaysþaTicdWeRkTImYy. sRmab;karedaH
Rsay eKeRCIserIskmøaMgenAkñúgGgát; BD CaGBaØatelIs. dUcenH eKRtUvkat;Ggát;enHedIm,IlubbM)at;
lT§PaBTb;Tl;kmøaMgrbs;va. eKalkarN_tMrYtplEdlGnuvtþeTAelIeRKOgbgÁúMRtUv)anbgðajenAkñúgrUbTI
¬rUbTI 10-17b¦.
smIkarlkçxNÐRtUvKña³ edayeyagtambMlas;TIénGgát; BD Edlrgkarkat; ¬rUbTI 10-17b¦ eyIg
tMrUveGay
0 = Δ BD + FBD f BDBD (1)
eyIgGaceRbIviFIkmμnþnimitþedIm,IKNna Δ BD nig f BDBD . karviPaKkmøaMgEdlcaM)ac;RtUv)anbgðajenA
kñúgrUbTI 10-17c nig10-17d.
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -356
24. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
sRmab; Δ BD eyIgtMrUveGaymankarGnuvtþkmøaMgBit ¬rUbTI 10-17c¦ nigkmøaMgnimitþÉktþa
GnuvtþenARtg;cugrgkarkat;rbs;Ggát; BD ¬rUbTI 10-17d¦. enATIenH eyIgnwgBicarNaEtfamBlbMEr
bMrYlrageFobEdlbNþalBIkarBt; nigbNþalBIkmøaMgtamG½kSenAkñúgFñwmEtb:ueNÑaH. dUcenH
Mm
L nNL
Δ BD = ∫ dx + ∑
0 EI AE
=∫
( ) (
3 30 x − 3.33 x (0 )dx (− 69.3)(− 0.816 ) 1.8 / cos 30 o
3
+ +
) (
(0)(1) 1.8 / cos 45o )
0 EI AE AE
117.5
=0+ + 0 = 0.0003264m = 0.326mm
1800(20)
sRmab; f BDBD eyIgtMrUveGaymankarGnuvtþrbs;bnÞúkBitÉktþa nigbnÞúkÉktþanimitþenARtg;cug
kMNat;Ggát;rgkarkat;rbs;Ggát; BD ¬rUbTI 10-17d¦. dUcenH
L m 2 dx n2 L
f BDBD = ∫ +∑
0 EI AE
=∫
3 (0)2 dx + (− 0.816)2 (1.8 / cos 30o ) + (1)2 (1.8 / cos 45o )
0
EI AE AE
=
3.93
( )
1800(200 )
= 1.092 10 − 5 m / kN
edayCMnYsTinñn½yEdl)anrkBIxagelIeTAkñúgsmIkar (1) eyIg)an
0 = Δ BD + FBD f BDBD
0 = 0.0003264 + FBD (1.092 ) 10 −5 ( )
FBD = −29.9kN = 29.9kN (C )
edayeRbIlT§plTaMgenH KUrdüaRkamGgÁesrIsRmab;Fñwm ehIybgðaj FBC = 44.9kN (C ) /
Ax = 1.31kN nig Ay = 30.0kN .
!0>*> kMNt;sMKal;bEnßmelIkarviPaKtamviFIkmøaMg
(Additional remarks on the force method of analysis)
enAeBlenHedaysareKalKMniténviFIkmøaMgRtUv)anGPivDÄ eyIgnwgbnþeFVIeGaykarGnuvtþrbs;va
manlkçN³TUeTA ehIyeyIgnwgBiPakSaBIplRbeyaCn_rbs;va.
enAeBlKNna flexibility coefficient fij b¤ αij sRmab;rcnasm<½n§ eKRtUvcMNaMfavaGaRs½y
Et nwglkçN³sMPar³ niglkçN³FrNImaRtrbs;Ggát; nigminGaRs½ynwgkardak;bnÞúkelIeRKOgbgÁúMdMbUg
(primary structure) eT. dUcenH enAeBlEdltémøTaMgenHRtUv)anKNna eKGaceRbIvaedIm,IKNna
kmøaMgsRmab;RKb;kardak;bnÞúkTaMgGs;.
Analysis of statically indeterminate structures by the force method T.Chhay -357
25. Department of Civil Engineering NPIC
sRmab;rcnasm<½n§EdlmanGBaØat Rn elIscMnYn n eyIgGacsresrsmIkarlkçxNÐRtUvKñacMnYn
n Edl
Δ1 + f11R1 + f12 R2 + L + f1n Rn = 0
Δ 2 + f 21R1 + f 22 R2 + L + f 2n Rn = 0
M
Δ n + f n1R1 + f n 2 R2 + L + f nn Rn = 0
enATIenH bMlas;TI Δ1 / >>>/ Δ n bNþalmkBIbnÞúkBitEdlmanGMeBIenAelI primary structure nigbNþalBI
sMrutrbs;TMr b¤bNþalBIbMErbMrYlsItuNðPaB b¤kMhusqÁgkñúgkarplitGgát;. edIm,IsMrYldl;karKNna
rcnasm<½n§EdlmandWeRkminkMNt;x<s; eKGacsresrsmIkarxagelICaragm:aRTIsdUcxageRkam
⎡ f11 f12 L f1n ⎤ ⎧ R1 ⎫ ⎧ Δ1 ⎫
⎢f ⎥⎪R ⎪
L f 2n ⎥ ⎪ 2 ⎪ ⎪Δ ⎪
⎢ 21 f 22 ⎪ 2⎪
⎨ ⎬ = −⎨ ⎬ (10-2)
⎢ M ⎥⎪ M ⎪ ⎪ M ⎪
⎢ ⎥⎪ ⎪ ⎪Δ n ⎪
⎣ f1n f 2 n L f nn ⎦ ⎩ Rn ⎭ ⎩ ⎭
b¤ f R = −Δ
CaBiess eKRtUvcMNaMfa fij = f ji ( f12 = f 21, etc.) tamRTwsþIm:akSEvlénbMlas;TIrIsIuRbUk
¬b¤c,ab;ebFI¦. dUcenH flexibility matrix Cam:aRTIssIuemRTI ehIyTMrg;enHmanRbeyaCn_enAeBledaH
RsaysMnMusmIkarlIenEG‘rFM² dUcenAkñúgkrNIrcnasm<½n§EdlmandWeRkminkMNt;x<s;.
enAkñúgCMBUkenH eyIg)ankMNt; flexibility coefficient edayeRbIviFIkmμnþnimitþ edaysarvaGnuvtþ
eTAelIeRKOgbgÁúMTaMgmUl. b:uEnþ eKGacTTYlemKuNsRmab;Ggát;nImYy²sRmab;eRKOgbgÁúMTaMgmUleday
eRbIsmIkarbMElg (transformation equation) edIm,ITTYltémøTaMgenH sRmab;eRKOgbgÁúMTaMgmUl. viFIenH
RtUv)anerobrab;enAkñúgesovePAEdlniyayGMBIkarviPaKeRKOgbgÁúMedaym:aRTIs nigminRtUv)anerobrab;enA
kñúgesovePAenHeT*.
eTaHbIkarlMGitsRmab;karGnuvtþviFIkmøaMgedayeRbIviFIkMuBüÚT½rk¾minRtUv)anelIkykmkniyayenA
kñúgesovePAenHk¾eday k¾eyIgGaceFIVkarGegátTUeTA nigeFVImUlvicarenAeBleRbIviFIenHedIm,IedaHRsaycM-
eNaTEdlmandWeRkminkMNt;x<s;. Tak;TgnwgbBaðaenH edIm,ITTYl)andMeNaHRsaykan;EtsuRkiteKRtUv
eGay flexibility coefficient EdlenAEk,rGgát;RTUgrbs;m:aRTIs f FMCag flexibility coefficient
EdlenAq¶ay. edIm,ITTYl)anEbbenH eKRtUvBicarNakñúgkareRCIserIs primary structure. edIm,IsMrYl
kúñgkarKNna fij eKENnaMeGayeRCIserIs primary structure y:agNaeGayvaCaeRKOgbgÁúMsIuemRTI.
kareFVIEbbenHnwgeGayeyIgTTYl)an flexibility coefficient xøHdUcKña b¤k¾GacesμIsUnü. cugeRkay rUb
*
emIl H.C. Martin, Introduction to Matrix Methods of Structural Analysis, McGraw-Hill, New York Ca]TahrN_.
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -358
26. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
ragxUcRTg;RTayrbs; primary structure nwgRsedognwgrUbragxUcRTg;RTayrbs;rcnasm<½n§Cak;Esþg.
RbsinebIeKGaceFVIEbbenH)anEmn enaHeKRtUvkarEksMrYlGBaØatEtbnþicbnÞÜcb:ueNÑaHeTAelI primary
structure EdleFVIeGayeKTTYl)anlT§plEdl)anBIsmIkar 10-2 kan;EtsuRkit.
!0>(> eRKOgbgÁúMsIuemRTI (Symmetric structures)
karviPaKeRKOgbgÁúMminkMNt;lMdab;x<s;k¾dUckarviPaKeRKOgbgÁúMkMNt;edaysþaTic GñksikSaKNna
RtUvdwgfaeRKOgbgÁúMenaHCaeRKOgbgÁúMEdlmanTMr b¤bnÞksIuemRTI b¤minsIuemRTI. kñúgn½yTUeTA eKGaccat;cM-
ú
Nat;eRKOgbgÁúMCaeRKOgbgÁúMsIuemRTIRbsinebIkmøaMgkñúg nigPaBdabrbs;vamanlkçN³qøúHKñaeFobnwgG½kS
kNþalrbs;va. CaTUeTA PaBsIuemRTITamTareGaysMPar³ ragFrNImaRt TMr nigbnÞúkRtUvEtdUcKñaenAsg
xageRKOgbgÁúM. b:uEnþ vaminEmnTal;EtdUecñaHTaMgGs;eT. cMNaMfa sRmab;sßirPaBtamTisedk eKRtUvkar
TMr snøak;edIm,IRTFñwm nig truss ¬rUbTI 10-18a nig 10-18b¦. enATIenH kmøaMgRbtikmμtamTisedk
enA Rtg;snøak;esμIsUnü dUcenHrcnasm<½n§enHnwgmankmøaMgkñúg nigPaBdabqøúHKña. CalT§pl eKGaccat;
cMNat;fñak;vaCarcnasm<½n§sIuemRTI. eKRtUvdwgfavaminEmnCakrNIsRmab;eRKageT ¬rUbTI 10-
18c¦Rbsin ebIeKCMnYsTMrbgáb;enARtg;cMNuc A edayTMrsnøak; enaHrUbragdab nigkmøaMgkñúgenAxagsþaM
nigxageqVg nwgmindUcKñaeT.
enAeBlxøH eRKOgbgÁúMsIuemRTIRTbnÞúkminsIuemRTI dUcbgðajenAkñúgrUbTI 10-19. kñúgkrNIrcna
sm<½n§manlkçN³sIuemRTI nigbnÞúkrbs;vamanlkçN³sIuemRTI b¤minsIuemRTI enaHeKnwgeFVIkarviPaKeRKOg
bgÁúMEtmYykMNat;b:ueNÑaH edaysarlT§pldUcKña b¤pÞúyKñanwgekItmanEtenAelIrcnasm<½n§mYykMNat;.
Analysis of statically indeterminate structures by the force method T.Chhay -359
27. Department of Civil Engineering NPIC
RbsinebIrcnasm<½n§sIuemRTIrgbnÞúkminsIuemRTI
enaHeKRtUvbMElgvaeGayeTACabgÁúMbnÞúksIuem-
RTI nigbgÁúMbnÞúkminsIuemRTI. edIm,IeFVI EbbenH
dMbUgeKRtUvEckbnÞúknwgBIr bnÞab;mkeTot eK
dak;eGayvamanlkçN³sIuemRTI nigminsIuemRTI.
Ca]TahrN_ eKRtUvEckbnÞúk enAelIFñwmkñúgrUb
TI 10-20a nwgBIr rYcdak;eGayva qøúHKñaeFob
nwgG½kSsIuemRTI. bnÞab;mkeKTTYl)anbgÁúM
bnÞúksIuemRTI nigminsIuemRTI dUcbgðajenAkñúg
rUbTI 10-20b. enAeBlEdleKbUkbgÁúMbnÞúk
TaMgBIrenHbBa©ÚlKña eKnwgTTYl)anbnÞúkedIm.
eKGaceFVIkarviPaKrcnasm<½n§dac;edayELk
eTAelIbgÁúMbnÞúksIuemRTI nigbgÁúMbnÞúkminsIuemRTI
ehIyeKRtUveFVIplbUkénlT§pl TaMgenHedIm,ITTYl)anlkçN³eFVIkaredImrbs; rcnasm<½n.
!0>!0> ExST§iBlsRmab;FñwmminkMNt;edaysþaTic
(Influence lines for statically indeterminate beams)
enAkñúgkfaxNÐ 6-3 eyIg)ansikSarYcmkehIyBIkareRbIeKalkarN_m‘uyey:eRbsøÚedIm,IKUrExS
T§iBlsRmab;kmøaMgRbtikmμ kmøaMgkat; nigm:Um:g;Bt;enARtg;cMNucmYyenAelIFñwmkMNt;edaysþaTic. enA
kñúgkfaxNÐenH eyIgnwgBnøatviFIenH nigeRbIvasRmab;FñwmminkMNt;edaysþaTic.
rMlwkfa sRmab;Fñwm eKalkarN_m‘uyey:eRbsøÚeBalfaExST§iBlsRmab;GnuKmn_ ¬kmøaMgRbtikmμ
kmøaMgkat; b¤m:Um:g;Bt;¦ manrUbragdabrbs;FñwmmanlkçN³dUcKñaenAeBlFñwmrgGMeBIedayGnuKmn_RtUvKña.
edIm,IKUrrUbragdabeGay)anRtwmRtUv eKRtUvdklT§PaBTb;Tl;nwgkarGnuvtþrbs;GnuKmn_ enaHFñwmGacdab
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -360
28. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
enAeBlGnuKmn_manGMeBIelIFñwm. sRmab;FñwmkMNt;edaysþaTic rUbragdab ¬b¤ExST§iBl¦ Caes‘rIénkM-
Nat;bnÞat;Rtg;. sRmab;FñwmminkMNt;edaysþaTic ExST§iBlnwgmanlkçN³CaExSekag. eyIgnwgsikSa
BI sMNg;énRbePT nImYy²rbs;ExST§iBlTaMgbiI ¬kmøaMgRbtikmμ kmøaMgkat; nigm:Um:g;Bt;¦ sRmab;rcna
sm<½n§minkMNt;edaysþaTic. enAkñúgkrNInImYy² eyIgnwgbgðajBIPaBeRbIkar)anéneKalkarN_
m‘uyey:eRbsøÚ edayeRbIRTwsþIbMlas;TIersIuRbUkrbs;m:akSEvl.
RbtikmμenARtg;cMNucA³ edIm,IkMNt;ExST§iBlsRmab;kmøaMgRbtikmμenARtg;cMNuc A enAkñúgrUbTI 10-
21a eKdak;bnÞúkÉktþaenAelIFñwmRtg;cMNucbnþbnÞab; ehIyenARtg;cMNucnImYy²eKRtUvkMNt;kmøaMg
RbtikmμenARtg;cMNuc A . düaRkaménlT§plTaMgenHCaExST§Bl. Ca]TahrN_ enAeBlbnÞúkenARtg;
cMNuc D ¬rUbTI 10-21a¦ eKGackMNt;kmøaMgRbtikmμenARtg;cMNuc A EdlCaGredaenrbs;ExST§iBl
enARtg; D edayeRbIviFIkmøaMg. edIm,IeFIVy:agenH eKRtUvGnuvtþeKalkarN_tMrYtpldUcbgðajenAkñúgrUbTI
10-21a rhUtdl;rUbTI 10-21c. dUcenH smIkarlkçxNÐRtUvKñasRmab;cMNuc A KW 0 = f AD + Ay f AA
b¤ Ay = − f AD / f AA b:uEnþtamRTwsþIbMlas;TIersIiuRbUkrbs;m:akSEvl f AD = − f DA ¬rUbTI 10-21d¦
dUcenHeyIgk¾GackMNt; Ay ¬b¤Gredaenrbs;ExST§iBlenARtg;cMNuc D ¦ edayeRbIsmIkar
⎛ 1 ⎞
⎜ f ⎟ f DA
Ay = ⎜ ⎟
⎝ AA ⎠
edayeFVIkareRbobeFob eKalkarN_m‘uyey:eRbsøÚtMrUveGayykTMrenARtg; A ecj ehIyGnuvtþ
bnÞúkÉktþabBaÄrenARtg;cMNucenaH. ExSekagPaBdabEdlTTYl)an ¬rUbTI 10-21d¦ manTMrg;dUcKñanwg
rUbragrbs;ExST§iBlsRmab; Ay . b:uEnþ BIsmIkarxagelIeyIgeXIjfaemKuNmaRtdæan (scale factor) KW
1 / f AA .
Analysis of statically indeterminate structures by the force method T.Chhay -361
29. Department of Civil Engineering NPIC
kmøaMgkat;enARtg; E³ RbsinebIeKRtUvkarkMNt;
ExST§iBlsRmab;kmøaMgkat;enARtg;cMNuc E rbs;
FñwmenA kñúgrUbTI 10-22a tameKalkarN_
m‘uyey:eRbsøÚ eKRtUvkat;FñwmRtg;cMNucenaH ehIy
s‘k sliding device enARtg;cMNucenaH ¬rUbTI
10-22b¦. ]bkrN_enHnwgbBa¢Únm:Um:g;Bt; nig
kmøaMgEkg b:uEnþminbBa¢ÚnkmøaMgkat;eT. enAeBl
Fñwm dabedaysarkmøaMgkat;ÉktþaviC¢manEdlman
GMeBIenARtg;cMNuc E mMurgVilenAsgxag]bkrN_
mantémødUcKña ehIyExSekagPaBdabCaExS
T§iBlsRmab;kmøaMgkat;enARtg;cMNuc E rUbTI
10-22c. edayGnuvtþviFIeKalsRmab;sg;ExST§iBlsRmab;kmøaMgkat;enARtg;cMNuc E enaHeKcaM)ac;
GnuvtþkmøaMgÉktþaenARtg;cMNucnImYy² dUcCacMNuc D ehIykMNt;kmøaMgkat;enARtg;cMNuc E ¬rUbTI
10-22a¦. témø VE CaGredaenrbs;ExST§iBlenARtg;cMNuc D . edayeRbIviFIkmøaMg nigRTwsþIbMlas;TI
ersIuRbUkrbs;m:akSEvl ¬dUckrNIelIkmun¦ eyIgTTYl)an
⎛ 1 ⎞
VE = ⎜
⎜f ⎟ f DE
⎟
⎝ EE ⎠
kmøaMgkat;ÉktþaviC¢manEdlGnuvtþeTAelIFñwmenARtg;cMNuc E ¬rUbTI 10-22c¦ nwgeFVIeGayFñwmdabkñúg
TMrg;CaExST§iBl sRmab;kmøaMgkat;enARtg;cMNuc E . enATIenHemKuNmaRtdæanKW (1 / f EE ) .
m:Um:g;Bt;enARtg; E³ eKGackMNt;ExST§iBlsRmab;m:Um:g;Bt;enARtg;cMNuc E enAkñúgrUbTI 10-23a
edayCMnYssnøak;enARtg;cMNucenaH edaysartMNEbbenHbBa¢ÚnkmøaMgEkg nigkmøaMgkat; b:uEnþminGacTb;
Tl; nwgm:Um:g;)an ¬rUbTI 10-23b¦. edayGnuvtþm:Um:g;ÉktþaviC¢man enaHFñwmnwgdabdUcTItaMgbnÞat;dac;²
¬rUb TI 10-23c¦ Edlpþl;nUvExST§iBl. edayeRbIviFIkmøaMg nigRTwsþIersIuRbUkrbs;m:akSEvl eyIgGac
bgðajfa³
⎛ 1 ⎞
⎜ α ⎟ f DE
ME = ⎜ ⎟
⎝ AA ⎠
emKuNmaRtdæanenATIenHKW (1/ α EE ) .
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -362
30. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
dMeNIrkarkñúgkarviPaK (Procedure for analysis)
CMhanxageRkampþl;nUvviFIsRmab;sg;ExST§iBlsRmab;kmøaMgRbtikmμ kmøaMgkat; nigm:Um:g;Bt;enA
Rtg;cMNucmYyenAelIFñwmedayeRbIbec©keTsm‘uyey:eRbsøÚ.
ragExSekagT§iBl
RbsinebIeKcg;kMNt;ExST§iBlrbs;cMNucmYyenAelIFñwm eKRtUvdklT§PaBTb;Tl;rbs;FñwmeTAnwgGnuKmn_
rbs;ExST§iBl edayCMnYsmkvijnUvtMN. RbsinebI GnuKmn_CakmøaMgRbtikmμbBaÄr eRbI vertical roller
guide, RbsinebIGnuKmn_CakmøaMgkat; eRbI sliding device, RbsinebIGnuKmn_Cam:Um:g;Bt; eRbIsnøak;. dak;
bnÞúkÉktþaenARtg;tMNedayeGayvamanGMeBIelIFñwmkñúgTisedAviC¢man. sg;ExSekagPaBdabrbs;Fñwm.
ExSekagenHCarUbragrbs;ExST§iBlsRmab;Fñwm.
témøExSekagT§iBl
RbsinebIeKRtUvkarkMNt;témøCaelxrbs;ExST§iBl eKRtUvKNnabMlas;TIéncMNucbnþbnÞab;tambeNþay
Fñwm enAeBlEdlFñwmrgbnÞúkÉktþaEdlmanGMeBIenARtg;tMNEdl)aneBalBIxagelI. EcktémøénbMlas;
TInImYy²edaybMlas;TIEdl)ankMNt;enARtg;cMNucEdlbnÞúkÉktþamanGMeBI. edayGnuvtþemKuNmaRt-
dæan eKnwgTTYl)antémøEdlCaGredaenrbs;ExST§iBl.
!0>!!> eKalkarN_mu‘yeyeRbsøÚsRmab;eRKag (Qualitative influence line for frames)
eKalkarN_m‘uyey:eRbsøÚCaviFIy:agelOnkñúgkarsg;ragTUeTArbs;ExST§iBlsRmab;eRKagsMNg;.
enAeBlEdleKsÁal;ragExST§iBl eKGackMNt;TItaMgrbs;bnÞúkGefrEdleFVIeGaymanT§iBlénGnuKmn_
¬kmøaMgRbtikmμ kmøaMgkat; b¤m:Um:g;Bt;¦ FMCageKenAelIeRKagPøam². Ca]TahrN_ rUbragrbs;ExST§iBl
sRmab;m:Um:g;viC¢manenARtg;cMNuckNþal I rbs;r:t FG rbs;eRKagenAkñúgrUbTI 10-24a Edlbgðaj
edaybnÞat;dac;. dUcenH eKGacdak;bnÞúkrayesμIEtenAelIr:t AB / CD nig FG edIm,IeFVIeGaym:Um:g;viC¢-
manenARtg;cMNuc I FMCageK. CamYynwgeRKagrgbnÞúkEbbenH ¬rUbTI 10-24b¦ eKGacviPaKeRKagedIm,I
KNnam:Um:g;eRKaHfñak;enARtg;cMNuc I .
Analysis of statically indeterminate structures by the force method T.Chhay -363
31. Department of Civil Engineering NPIC
]TarhN_ 10-11³ sg;ExST§iBlsRmab;kmøaMgRbtikmμbBaÄrenARtg;cMNuc A sRmab;FñwmenAkñúgrUbTI
10-25a. EI Catémøefr. sg;tMélCaelxral; 2m .
dMeNaHRsay³
yklT§PaBrbs;FñwmTb;Tl;nwgkmøaMgRbtikmμ Ay . eKGaceFVIvaeTA)anedayeRbI vertical roller device
EdlbgðajenAkñúgrUbTI 10-25b. edayGnuvtþbnÞúkÉktþabBaÄrenARtg;cMNuc A eKTTYl)anrUbragExS
T§iBldUceXIjenAkñúgrUbTI 10-25c.
edIm,IkMNt;Gredaenrbs;ExST§iBl eyIgnwgeRbIviFIFñwmnimitþ (conjugate beam method).
kmøaMgRbtikmμenARtg;cMNuc A nigcMNuc B enAelIFñwmBit enAeBlrgbnÞúkÉktþaenARtg;cMNuc A RtUv)an
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -364
32. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
bgðajenAkñúgrUbTI 10-25b. FñwmnimitþEdlRtUvKñaRtUv)anbgðajenAkñúgrUbTI 10-25d. cMNaMfaTMrenA
Rtg; A' enAEtrkSadUcTMrenARtg; A enAkñúgrUbTI 10-25b BIeRBaH vertical roller device enAelIFñwm
nimitþTb;Tl;m:Um:g; b:uEnþminTb;Tl;kmøaMgkat; EdlRtUvKñaeTAnwgbMlas;TI b:uEnþminEmnmMurgVilenARtg;TMr A
énFñwmBit ¬rUbTI 10-25c¦. kmøaMgRbtikmμenARtg;TMrrbs;FñwmnimitþRtUv)anKNna nigRtUv)anbgðajenA
kñúgrUbTI 10-25d. bMlas;TIéncMNucenAelIFñwmBit ¬rUbTI 10-25b¦ k¾RtUv)anKNna.
sRmab; B' edaysarminmanm:Um:g;Bt;enAelIFñwmnimitþRtg;cMNuc B' ¬rUbTI 10-25d¦ enaH
Δ B = M B' = 0
sRmab; D' / rUbTI 10-25e
∑ M D' = 0 Δ D = M D' =
18
(2) − 1 ⎛ 2 ⎞(2)⎛ 2 ⎞ = 34.67
⎜ ⎟ ⎜ ⎟
EI 2 ⎝ EI ⎠ ⎝ 3 ⎠ EI
sRmab; C ' / rUbTI 10-25f
∑ M C' = 0 ΔC = M C' =
18
(4) − 1 ⎛ 4 ⎞(4)⎛ 4 ⎞ = 61.33
⎜ ⎟ ⎜ ⎟
EI 2 ⎝ EI ⎠ ⎝ 3 ⎠ EI
sRmab; A' / rUbTI 10-25d
72
Δ A = M A' =
EI
edaysarkmøaMgbBaÄr 1kN manGMeBIenARtg;cMNuc A enAelIFñwmenAkñúgrUbTI 10-25a eFVIeGayman
kmøaMgRbtikmμ 1kN enARtg;cMNuc A . bMlas;TIenARtg;cMNuc A ¬ Δ A = 72 / EI ¦ KYrRtUvnwgtémø 1
sRmab;GredaenExST§iBlenARtg;cMNuc A . dUcenH edayEckbMlas;TIEdl)anKNnaedayemKuNenH
eyigTTYl)an
x Ay
A 1
C 0.852
D 0.481
B 0
düaRkaméntémøTaMgenHCaExST§iBlEdlbgðajenAkñúgrUbTI 10-25g.
]TarhN_ 10-12³ sg;ExST§iBlsRmab;kmøaMgkat;enARtg;cMNuc D sRmab;FñwmenAkñúgrUbTI 10-
26a. EI Catémøefr. sg;tMélCaelxral; 3m .
Analysis of statically indeterminate structures by the force method T.Chhay -365
33. Department of Civil Engineering NPIC
dMeNaHRsay³
yklT§PaBrbs;FñwmTb;Tl;nwgkmøaMgkat;enARtg;cMNuc D . eKGaceFVIvaeTA)anedayeRbI roller device
EdlbgðajenAkñúgrUbTI 10-26b. edayGnuvtþbnÞúkkmøaMgkat;ÉktþaenARtg;cMNuc D eKTTYl)anrUbrag
ExST§iBldUceXIjenAkñúgrUbTI 10-26c.
kmøaMgRbtikmμenARtg; A / B nig C enAelIFñwmBit enAeBlrgkmøaMgkat;ÉktþaenARtg;cMNuc D
RtUv)anbgðajenAkñúgrUbTI 10-26b. FñwmnimitþEdlRtUvKñaRtUv)anbgðajenAkñúgrUbTI 10-26d. enATIenH
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -366
34. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
eKRtUvGnuvtþm:Um:g; M D' enARtg;cMNuc D' edIm,IeFVIeGaym:Um:g;Bt;epSgKñaenAxageqVg nigxagsþaMcMNuc D' .
m:Um:g;Bt;TaMgenHRtUvKñanwgbMlas;TIxageqVg nigxagsþaMéncMNuc D enAelIFñwmBit ¬rUbTI 10-26c¦.
kmøaMgRbtikmμenARtg;TMr A' / B' / C ' nigm:Um:g; M D' enAelIFñwmnimitþRtUv)anKNna nigRtUv)anbgðaj
enAkñúgrUbTI 10-26e. cUrepÞógpÞat;karKNnaeLIgvij.
edaysarPaBdac;énm:Um:g;enARtg;cMNuc D' eKRtUvKNnam:Um:g;Bt;EdlenAxagsþaM nigxageqVg
cMNuc D' bnþic. sRmab;enAxageqVgcMNuc D' ¬rUbTI 10-26f¦ eyIg)an
∑ M D' L = 0 Δ DL = M D' L =
4.5
(1) − 30 (3) = − 85.5
EI EI EI
enAxagsþaMgcMNuc D' ¬rUbTI 10-26g¦ eyIgman
∑ M D' R = 0 Δ DR = M D' R =
4.5
(1) − 30 (3) + 144 = 58.5
EI EI EI EI
BIrUbTI 10-26e
Δ A = M A' = 0 Δ B = M B' = 0 ΔC = M C' = 0
sMrab;cMNuc E ¬rUbTI10-26b¦ edayeRbIviFImuxkat;enARtg;cMNuc E' EdlRtUvKñaenAelIFñwmnimitþ ¬rUbTI
10-26h¦ eyIgman
∑ M E' = 0 Δ E = M E' =
4.5
(1) − 6 (3) = − 13.5
EI EI EI
eKTTYl)anGredaenrbs;ExST§iBledayEcktémøxagelInwgemKuNmaRtdæan M D ' = 144 / EI . eyIg
man
x VD
A 0
DL − 0.594
DR 0.406
B 0
E − 0.0938
C 0
düaRkaméntMélTaMgenHCaExST§iBlEdlbgðajenAkñúgrUbTI 10-26i.
]TarhN_ 10-13³ sg;ExST§iBlsRmab;m:Um:g;Bt;enARtg;cMNuc D sRmab;FñwmenAkñúgrUbTI 10-27a.
EI Catémøefr. sg;tMélCaelxral; 3m .
Analysis of statically indeterminate structures by the force method T.Chhay -367
35. Department of Civil Engineering NPIC
dMeNaHRsay³
eKdak;snøak;enARtg;cMNuc D edIm,IdklT§PaBrbs;FñwmEdlTb;Tl;nwgm:Um:g;enARtg;cMNucenH ¬rUbTI 10-
27b¦. edaydak;m:Um:g;ÉktþaviC¢manenARtg; D eyIgTTYl)anExST§iBldUcbgðajenAkñúgrUbTI 10-27c.
kmøaMgRbtikmμenARtg;cMNuc A / B nig C enAelIFñwmBit enAeBlrgm:Um:g;enARtg;cMNuc D RtUv
)anbgðajenAkñúgrUbTI 10-27b. FñwmnimitþRtUvKña nigkmøaMgRbtikmμrbs;vaRtUv)anbgðajenAkñúgrUbTI 10-
27d. eKesñIeGayepÞógpÞat;kmøaMgRbtikmμenAkñúgkrNITaMgBIr. BIrUbTI 10-27d cMNaMfa
Δ A = M A' = 0 Δ B = M B' = 0 ΔC = M C' = 0
sRmab;cMNuc D' / rUbTI 10-27e
∑ M D' = 0 Δ D = M D' =
1.5
(1) + 6 (3) = 19.5
EI EI EI
sRmab;cMNuc E ' / rUbTI 10-27f
∑ M E' = 0 Δ E = M E' =
1.5
(1) − 2 (3) = 4.5
EI EI EI
viPaKeRKOgbgÁúMminGackMNt;edaysþaTictamviFIkmøaMg T.Chhay -368
36. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
bMlas;TImMu (angular displacement) α DD enARtg; D énFñwmBitenAkñúgrUbTI 10-27c RtUv)ankMNt;
edaykmøaMgRbtikmμenARtg; D' enAelIFñwmnimitþ. edIm,ITTYl)anGredaenrbs;ExST§iBl eKEcktémø
Edl)anKNnaxagelICamYynwgemKuN D' y = 16 / EI Edl
x MD
A 0
D 1.219
B 0
E − 0.281
C 0
düaRkaméntMélTaMgenHCaExST§iBlEdlbgðajenAkñúgrUbTI 10-27g.
Analysis of statically indeterminate structures by the force method T.Chhay -369
37. Department of Civil Engineering NPIC
cMeNaT
!0>!> kMNt;kmøaMgRbtikmμenARtg;TMr nigbnÞab;mk !0>%> kMNt;kmøaMgRbtikmμenARtg;TMr A nig B .
sg;düaRkamm:Um:g;sMrab;Fñwm. snμt;faTMrenA A Ca EI CatMélefr.
TMrbgáb; nigTMr B CaTMrkl;. EI CatMélefr.
!0>@> kMNt;kmøaMgRbtikmμenARtg;TMr. snμt;faTMr !0>^> kMNt;kmøaMgRbtikmμenARtg;TMr bnÞab;mk
enA A CaTMrbgáb; nigTMr B CaTMrkl;. yk E = sg;düaRkamkmøaMgkat; nigdüaRkamm:Um;g;Bt;.
200GPa . m:Um:g;niclPaBsMrab;kMNat;Ggát; EI CatMélefr.
nImYy²RtUv)anbgðajenAkñúgrUb.
!0>&> FñwmRtUv)anRTedayTMrsnøak;enARtg;cMNuc
!0>#> kMNt;kmøaMgRbtikmμenARtg;TMr A / B nig A RTedayrWus½rEdlmankMraj (stiffness) k enA
C bnÞab;mksg;düaRkamkMlaMgkat; nigm:Um:g;Bt; Rtg; B nigRTedayTMrkl;enARtg;cMNu C . kMNt;
sMrab;Fñwm. EI CatMélefr. kmøaMgrWus½rGnuvtþeTAelIFñwm. EI CatMélefr.
!0>$> kMNt;kmøaMgRbtikmμenARtg;TMr A nigTMr B . !0>*> kMNt;kmøaMgRbtikmμTMr. snμt; B CaTMr
snμt;TMrenARtg;cMNuc A Gnuvtþm:Um:g;Bt;eTAelIFñwm. snøak; ehIy A nig C CaTMrkl;. EI CatMélefr.
EI CatMélefr.
!0>(> kMNt;kmøaMgRbtikmμenARtg;TMr bnÞab;mk
sg;düaRkamm:Um;g;Bt;. snμt; A CaTMrsnøak; ehIy
cMeNaT T.Chhay -370
38. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
B nig C CaTMrkl;. EI CatMélefr.
!0>!0> kMNt;kmøaMgRbtikmμTMr. snμt;TMr A CaTMr
snøak; ehIy B nig C CaTMrkl;. EI CatMélefr. !0>!$> kMNt;kmøaMgRbtikmμTMrenAelIFñwm. CBa¢aMg
enARtg;cMNuc A cl½teLIgelI 30mm . snμt;
fa A CaTMrsnøak; ehIyTMr B nig C CaTMrkl; .
yk E = 200GPa nig I = 90(106 )mm4 .
!0>!!> kMNt;kmøaMgRbtikmμTMr. snμt;TMr A CaTMr
snøak; ehIy B nig C CaTMrkl;. EI CatMélefr.
!0>!%> kMNt;kmøaMgRbtikmμTMr bnÞab;sg;düaRkam
!0>!@> kMNt;PaBdabenARtg;cug B rbs;r)arEdk. m:Um:g;. snμt;faTMr A CaTMrsnøak; ehIy B nig C
rWus½rmankMraj k = 2 N / mm . r)arEdkmanTTwg CaTMr. EI mantémøefr.
5mm nigkMBs; 10mm . ehIy cUrsg;düaRkam
kmøaMgkat; nigm:Um:g;Bt;sMrab;r)arEdkenH. yk
E = 200GPa .
!0>!^> sg;düaRkamm:Um:g;sMrab;Fñwm. EI mantémø
efr. snμt;faTMr A CaTMrbgáb; ehIy B nig C Ca
!0>!#> kMNt;kmøaMgRbtikmμTMr bnÞab;mksg; TMr.
düaRkamm:Um:g;. m:Um:g;niclPaBsMrab;kMNat;Ggát;
nImYy²RtUv)anbgðajenAkñúgrUb. snμt;fa A nig
C CaTMrkl; nig B CaTMrsnøak;. yk E =
200GPa .
Problems T.Chhay -371
39. Department of Civil Engineering NPIC
!0>!&> kMNt;kmøaMgRbtikmμenARtg;TMrbgáb; A !0>@!> kMNt;kmøaMgRbtikmμTMr bnÞab;mksg;
nig B . EI mantMélefr. düaRkamm:Um:g;sMrab;Ggát;nImYy². snμt; A nig
B CaTMrsnøak; ehIytMNenARtg;cMNuc C CatMN
bgáb;. EI mantMélefr.
!0>!*> sg;düaRkamm:Um:g;Bt;sMrab;FñwmTMrbgáb;.
EI mantMélefr.
!0>@@> kMNt;kmøaMgRbtikmμTMr bnÞab;mksg;
düaRkamm:Um:g;sMrab;Ggát;nImYy². EI mantMél
!0>!(> kMNt;kmøaMgRbtikmμTMr. EI mantMél efr.
efr.
!0>@#> kMNt;kmøaMgRbtikmμTMr bnÞab;mksg;
!0>@0> kMNt;kmøaMgRbtikmμTMr. EI mantMél düaRkamm:Um:g;sMrab;Ggát;nImYy². EI mantMél
efr. efr.
cMeNaT T.Chhay -372
40. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
!0>@$> kMNt;kmøaMgRbtikmμTMr RbsinebITMrRtg; C !0>@&> kMNt;kmøaMgkñúgGgát;nImYy²rbs; truss .
RtUv)anrujeLIgelI 4mm . yk E = 200GPa RkLaépÞmuxkat;rbs;Ggát;nImYy²RtUv)anbgðaj
/ I = 250(106 )mm4 . enAkñúgrUb. yk E = 200GPa . snμt;Ggát;
RtUv)antP¢ab;edaysnøak;.
!0>@%> kMNt;kmøaMgRbtikmμTMr A nig D . m:Um:g;
niclPaBrbs;kMNat;Ggát;RtUv)anbgðajenAkñúgrUb.
yk E = 200GPa . !0>@*> kMNt;kmøaMgkñúgGgát;nImYy²rbs; truss
EdltP¢ab;edaytMNsnøak;. EA mantMélefr.
!0>@^> kMNt;kmøaMgkñúgGgát;nImYy²rbs; truss . !0>@(> kMNt;kmøaMgkñúgrbs;Ggát; HG rbs;
RkLaépÞmuxkat;rbs;Ggát;nImYy²RtUv)anbgðaj truss . EA mantMélefr.
enAkñúgrUb. snμt;Ggát;RtUv)antP¢ab;edaysnøak;.
yk E = 200GPa .
!0>#0> kMNt;kmøaMgkñúgrbs;Ggát; HB rbs; truss
éncMeNaT !0>@(. EA mantMélefr.
Problems T.Chhay -373
41. Department of Civil Engineering NPIC
!0>#!> kMNt;kmøaMgkñúgrbs;Ggát; HG . EA man
tMélefr.
!0>#@> kMNt;kmøaMgkñúgrbs;Ggát; BE . EA
mantMélefr. !0>#%> eRKOgbgÁúMRTbnÞúkdUcbgðajenAkñúgrUb.
cUrsg;düaRkamm:Um:g;Bt;sMrab;FñwmnImYy². yk
I = 100(10 6 )mm 4 sMrab;Fñwm nig A = 200mm 2
énsMrab;Edkr)ar. Ggát;TaMgGs;eFVIGMBIEdkEdl
man E = 200GPa .
!0>##> kMNt;kmøaMgkñúgrbs;Ggát; DF én truss .
EA mantMélefr.
!0>#^> eKeRbI queen-post trussed beam edIm,I
RTbnÞúkrayesμI 60kN / m . kMNt;kmøaMgEdlekIt
!0>#$> eKBüÜrcugFñwmkugs‘uledayEdkr)ar AC mankñúgr)arEdkTaMgR)aM. eKminKitkMras;rbs;Fñwm
EdlmanGgát;p©it 25mm . ehIyFñwmenHmanTMr nigsnμt;faGgát; truss RtUv)antP¢ab;eTAnwgFñwm
bgáb;enARtg;cMNuc B . kMNt;kmøaMgkñúgrbs;Edk edaytMNsnøak;. ehIyminKitT§iBlkmøaMgsgát;
r)arEdlbNþalBIbnÞúkrayesμI 60kN / m . E = tamG½kS nigkmøaMgkat;enAkñúgFñwm. RkLaépÞ
200GPa sMrab;Fñwm nigEdkr)ar. muxkat;rbs;Edkr)arnImYy²KW 1800mm2 ehIy
sMrab;Fñwm I = 235(106 )mm 4 . yk E =
cMeNaT T.Chhay -374
42. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
200GPa . kat;enAkñúgFñwm. Fñwm I b = 200(106 )mm4 ehIy
sMrab;Edkr)arnImYy² A = 100mm 2 . yk
E = 200GPa .
!0>#&> eKeRbI king-post trussed beam edIm,I
RTbnÞúkcMNuc 200kN EdlmanGMeBIenAkNþal
ElVg. kMNt;kmøaMgkñúgrbs;r)arEdkTaMgbI. Edk
r)arnImYy²manRkLaépÞmuxkat; 1250mm 2 .
snμt;faBYkvaRtUv)antP¢ab;edaytMNsnøak;. eK
minKitkMras;rbs;Fñwm nigT§iBlkmøaMgsgát; tam
G½kSenAkñúgFñwmeT. yk E = 200GPa sMrab;Fñwm !0>$!> sg;ExST§iBlsMrab;kmøaMgkat;enARtg;
nigEdkr)ar. ehIy I AB = 150(106 )mm 4 . cMNuc C . bgðajCatémøelxral; 5m . snμt; TMr
B CaTMrkl;. EI mantémøefr.
!0>$@> sg;ExST§iBlsMrab;kmøaMgRbtikmμenARtg;
!0>#*> kMNt;m:Um:g;GtibrmaenAkñúgFñwmén cMNuc C . bgðajCatémøelxral; 5m . EI man
cMeNaT !0>#&. témøefr.
!0>#(> kMNt;kmøaMgRbtikmμenARtg;TMr C . EI
mantMélefrsMrab;FñwmTaMgBIr.
!0>$#> sg;ExST§iBlsMrab;kmøaMgkat;enARtg;
cMNuc B . bgðajCatémøelxral; 5m . snμt;TMr
enARtg; A CaTMrkl; nig C CaTMrbgáb;. EI man
témøefr.
!0>$0> eKRTFñwmkugs‘ul AB CamYynwgEdkr)ar
bEnßmcMnYnBIr. kMNt;kmøaMgenAkñúgEdkr)ar
nImYy². eKminKitkmøaMgsgát;tamG½kS nigkmøaMg
Problems T.Chhay -375
43. Department of Civil Engineering NPIC
!0>$$> sg;ExST§iBlsMrab;kmøaMgRbtikmμenARtg; bnÞúkGefrrayesμIEdleFVIeGayekItmanGnuKmn_
cMNuc A éncMeNaT !0>$#. viC¢manGtibrma. snμt;faFñwmmanTMrbgáb;enARtg;
!0>$%> KUrRBagnUvExST§iBlsMrab;m:Um:g;Bt;enA cMNuc F .
Rtg;cMNuc D edayeRbIeKalkarN_m‘uyey:eRbsøÚ.
kMNt;m:Um:g;viC¢manGtibrmaenARtg;cMNuc D Edl
bNþalBIbnÞúkGefrrayesμI 5kN / m . EI man
témøefr. snμt; A CaTMrsnøak; ehIy B nig C !0>$(> eRbIeKalkarN_m‘uyey:eRbsøÚedIm,Isg;rag
CaTMrkl;. TUeTAénExST§iBlsMrab; (a) m:Um:g;Bt;enARtg;
cMNuc A nig (b) kmøaMgkat;enARtg;cMNuc B .
!0>$^> KUrRBagnUvExST§iBlsMrab;kmøaMgkat;enA
Rtg;cMNuc B . RbsinebIFñwmenHrgbnÞúkGefrray
!0>%0> eRbIeKalkarN_m‘uyey:eRbsøÚedIm,Isg;rag
esμI 6kN / m cUrkMNt;kmøaMgkat;viC¢manGtibrma
TUeTAénExST§iBlsMrab; (a) m:Um:g;Bt;enARtg;
enARtg;cMNuc B . snμt; A CaTMrsnøak;. EI man
cMNuc A nig (b) kmøaMgkat;enARtg;cMNuc B .
témøefr.
!0>$&> KUrRBagnUvExST§iBlsMrab;m:Um:g;Bt;enA
Rtg;cMNuc B éncMeNaT !0>$^. RbsinebIFñwmenH
!0>%!> eRbIeKalkarN_m‘uyey:eRbsøÚedIm,Isg;rag
rgbnÞúkGefrray esμI 6kN / m cUrkMNt;m:Um:g;Bt;
TUeTAénExST§iBlsMrab; (a) m:Um:g;Bt;enARtg;
viC¢manGtibrma enARtg;cMNuc B . snμt; A Ca
cMNuc A nig (b) kmøaMgkat;enARtg;cMNuc B .
TMrsnøak;. EI man témøefr.
!0>$*> KUrRBagnUvExST§iBlsMrab; (a) kmøaMg
RbtikmμbBaÄrenARtg;cMNuc C (b) m:Um:g;Bt;enA
Rtg;cMNuc B nig (c) kmøaMgkat;enARtg;cMNuc E .
enAkñúgkrNInImYy² cUrbgðajTItaMgEdlRtUvdak;
cMeNaT T.Chhay -376
44. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
!0>%@> eRbIeKalkarN_m‘uyey:eRbsøÚedIm,Isg;rag !0>%$> eRbIeKalkarN_m‘uyey:eRbsøÚedIm,Isg;rag
TUeTAénExST§iBlsMrab; (a) m:Um:g;Bt;enARtg; TUeTAénExST§iBlsMrab; (a) m:Um:g;Bt;enARtg;
cMNuc A nig (b) kmøaMgkat;enARtg;cMNuc B . cMNuc A nig (b) kmøaMgkat;enARtg;cMNuc B .
!0>%#> eRbIeKalkarN_m‘uyey:eRbsøÚedIm,Isg;rag
TUeTAénExST§iBlsMrab; (a) m:Um:g;Bt;enARtg;
cMNuc A nig (b) kmøaMgkat;enARtg;cMNuc B .
Problems T.Chhay -377