10. analysis of statically indeterminate structures by the force method

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

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


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.



!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


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.


]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 .


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.


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;


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.



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


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)



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 ⎠



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.



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.


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


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.



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



]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


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


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


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



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



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.



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;.


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_.


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;.


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


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 .



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 ) .



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 .


]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


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 .



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


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 .



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



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.



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


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


!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


!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.


!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


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


!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
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;
viC¢manGtibrma enARtg;cMNuc B . snμt; A Ca
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


!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


10. analysis of statically indeterminate structures by the force method

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10. analysis of statically indeterminate structures by the force method