8. deflection

Department of Civil Engineering NPIC

*> PaBdab
(Deflection)

CMBUkenHnwgbgðajBIrebobkMNt;PaBdabeGLasÞicrbs;FñwmedayeRbIviFIGaMgetRkalDub nigviFI
FrNImaRty:agsMxan;cMnYnBIrKW RTwsþIRkLaépÞm:Um:g; (moment-area method) nigviFI conjugate beam.
eKeRbIviFIGaMgetRkalDubedIm,ITTYl)ansmIkarEdlkMNt;CMral b¤mMurgVil (slope) nigExSekageGLasÞic.
viFIFrNImaRtpþl;nUvmeFüa)ayedIm,ITTYlmMurgVil nigPaBdabenARtg;cMnucCak;lak;NamYyenAelIFñwm.
viFITaMgenHmanKuNsm,tþi nigKuNvibtþrbs;va EdlRtUv)anerobrab;enAeBlENnaMBIviFInImYy².

*>!> düaRkamPaBdab nigExSekageGLasÞic
(Deflection diagram and the elastic curve)
PaBdabrbs;eRKOgbgÁúMGacekItBIktþaepSg²CaeRcIndUcCa
bnÞúk sItuNðPaB kMhuskñúgkarplit b¤sMrut. enAkñúgkarsikSa
KNna eKRtUvkMritEdnkMNt;eGayPaBdabedIm,IkarBarsñameRbH
EdlGacekItmanelIsMPar³EdlmanlkçN³RsYydUcCaebtug
b¤kMe)arbUk. elIsBIenH eRKOgbgÁúMminRtUvj½r b¤dabxøaMgeT edIm,I
bgðajsuvtßiPaBdl;GñkEdleRbIR)as;GKarenaH. eTaHbICay:ag
Nak¾eday enAeBlEdleKviPaK rcnasm<½n§minkMNt;edaysþaTic
eKRtUvkMNt;PaBdabRtg;cMnucCak;lak;enAelIeRKOgbgÁúM.
PaBdabEdlBicarNaenAkñúgesovePAenHGnuvtþEteTA
elIrcnasm<½n§EdlmansMPar³CaeGLasÞic (linear elastic
material). eRkamlkçxNÐenH eRKagbgÁúMEdlrgbnÞúknwgRtLb;

eTArksßanPaBedImvij eRkayeBlEdleKdkbnÞúkecjmkvij.
PaBdabrbs;rcnasm<½n§RtUv)anbgáeLIgedaybnÞúkxagkñúgrbs;
vadUcCakMlaMgEkg (normal force), kMlaMgkat; (shear force)
b¤m:Um:g;Bt; (bending moment). b:uEnþ sMrab;Fñwm nigeRKag PaB
dabFMbMputRtUv)anbgáeLIgedaym:Um:g;Bt; b:uEnþkMlaMgtamG½kS
eFVIeGayman PaBdabrbs; truss.

PaBdab T.Chhay -242

mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa

munnwgKNnamMurgVil b¤PaBdabrbs;cMnucenAelIFñwm b¤eRKag eKKYrKUrragrbs;eRKOgbgÁúMEdlrg
bnÞúk eRBaHvaCaEpñkmYyEdlCYyeyIgkñúgkarBinitülT§plEdl)anKNna. düaRkamPaBdabenHtMNag
eGayExSekageGLasÞicsMrab;cMnucenARtg;TIRbCMuTMgn;énRkLaépÞmuxkat;tambeNþayGgát;nImYy².
sMrab;cMeNaTPaKeRcIn eKminBi)akkñúgkarKUrExSekageGLasÞicenHeT. b:uEnþ enAeBlKUrExSekageGLa-
sÞic eKcM)ac;RtUvdwgBIlkçxNÐénkarekItmanmMurgVil nigPaBdabenARtg;TMr b¤enARtg;tMN. eyagtam
taragTI 8-1 TMrEdlTb;Tl;nwgkMlaMgdUcCa TMrsnøak;minGnuBaØateGaymanbMlas;TIenARtg;TMreT ehIy
TMrEdlTb;Tl;nwgm:Um:g; dUcCaTMrbgáb; minGnuBaØateGaymanmMurgVileT. cMNaMfa PaBdabénGgát;rbs;
eRKOgbgÁúMEdlmantMNbgÁb; (4) eFVIeGayGgát;enARtg;tMNenaHmanmMurgVilesμIKña. mü:agvijeTot Rbsin
ebIvatP¢ab;edaytMNsnøak; (5) mMurgVilrbs;Ggát;Rtg;tMNenaHnwgmantMélxusKña edaysartMNsnøak;
minGacTb;Tl;nwgm:Um:g;eT.

RbsinebIeKBi)aksg;ExSekageGLasÞic dMbUgeKesñIeGaysg;düaRkamm:Um:g;Bt;sMrab;Fñwm nig
eRKag. edayeRbIkarkMNt;sBaØasMrab;m:Um:g;Bt;Edl)anbegáIteLIgenAkñúgCMBUkTI 4 m:Um:g;Bt;viC¢mancg;eFVI
eGayFñwm b¤Ggát;edkpt ¬rUbTI 8-1¦. dUcKña m:Um:g;Bt;GviC¢mancg;eFVIeGayFñwm b¤Ggát;edke)a:g ¬rUbTI
8-2¦ dUcenH RbsinebIeKsÁal;düaRkamm:Um:g;eKGacsg;ExSekageGLasÞicedaygay. ]TahrN_ eK
manFñwmdUcbgðajenAkñúgrUbTI 8-3 CamYynwgdüaRkamm:Um:g;rbs;va. edaysarTMrsnøak; nigTMrkl;
bMlas;TIenARtg;TMr A nigTMr D RtUvesμInwgsUnü. enAkñúgtMbn;m:Um:g;GviC¢man ExSekageGLasÞicnwge)a:g
ehIyenAkñúgtMbn;m:Um:g;viC¢man ExSekageGLasÞicnwgpt. vaRtUvmancMnucrbt;enARtg;cMnucEdlExSekagbþÚr
PaBekag edaysarRtg;cMnucenHCacMnucm:Um:g;sUnü. edayeRbIeKalkarN_dUcKña cMNaMBIrebobKUrExSekag
eGLasÞicsMrab;FñwmenAkñúgrUbTI 8-4 EdlQrelIdüaRkamm:Um:g;Bt;rbs;va. ]TahrN_xageRkambgðajBI
rebobKUrExSekagPaBdabsMrab;Fñwm nigsMrab;eRKagedayeRbIeKalkarN_TaMgenH.

Deflection T.Chhay -243


]TahrN_ 8-1³ sg;rUbragdabrbs;FñwmnImYy²EdlbgðajenAkñúgrUbTI 8-5.

PaBdab T.Chhay -244


dMeNaHRsay³
enAkñúgrUbTI 8-5a TMrkl;enARtg;cMnuc A GnuBaØateGaymanmMurgVilEdlminGnuBaØateGaymanPaBdabeT
xN³EdlTMrbgáb;enARtg; B Tb;mineGaymanTaMgmMurgVilTaMgPaBdab. rUbragdabRtUv)anbgðajedayExS
dit. enAkñúgrUbTI 8-5b minGacmanmMurgVil nigPaBdabekItmanenARtg;TMr A nig B eT. enAkñúgrUbTI 8-
5c m:Um:g;nwgeFVIeGaycugTMenr A vil. m:Um:g;enHnwgeFVIeGaycugsgçagrbs;FñwmbMlas;TI EtenARtg;TMr B
nig C minmanPaBdabeT. cMNaMfa kMNat;Ggát; CD enArkSaragminxUcRTg;RTay ¬bnÞat;Rtg;¦ edaysar
KμankMlaMgxagkñúgGnuvtþeTAelIva. enAkñúgrUbTI 8-5d snøak;enARtg;cMnuc B GnuBaØateGaymankarvil
edayesrI ehIymMurgVilrbs;ExSekagPaBdabnwgbþÚry:agelOnenARtg;cMnucenH xN³EdlFñwmRtUv)anTb;
edayTMrrbs;va. enAkñúgrUbTI 8-5e FñwmpSMdabdUcbgðaj. mMurgVilbþÚry:agxøaMgenAsgçagsnøak;Rtg;cMnuc B .
cugeRkay sMrab;rUbTI 8-5f ElVg BC nwgdabedayptedaysarbnÞúk. edaysarvaCaFñwmCab; ElVgcug
nwgdabedaye)a:g.

]TahrN_ 8-2³ sg;rUbragdabrbs;eRKagnImYy²EdlbgðajenAkñúgrUbTI 8-6.



dMeNaHRsay³
enAkñúgrUbTI 8-6a enAeBlbnÞúk P rujtMN B nigtMN C eTAsþaM vanwgeFVIeGayssrnImYy²vilRsbRTnic
naLika. CalT§pl tMN B nigtMN C RtUvEtvilRsbRTnicnaLika. edaysarGgát;RtUvrkSamMurbs;tMN
EdlmanTMhM 90o dUcenHFñwmnwgxUcRTg;RTajdUcbgðajedaykMeNagrbs;vapøas;bþÚrBIpteTAe)a:g. cMNaM
fakarpøas;bþÚkMeNagEbbenHeFVIeGayekItmancMnucrbt;enAkñúgFñwm. enAkñúgrUbTI 8-6b bnÞúk P eFVIeGay
tMN A / B nig C pøas;TIeTAsþaM EdleFVIeGayssrnImYy²ekagdUcbgðaj. tMNbgáb;RtUvEtrkSamMuEkg
rbs;va dUcenHGgát; BC nig CD RtUvEtpøas;bþÚrPaBekagedaymancMnucrbt;enAEk,rkNþalGgát;. enA
kñúgrUbTI 8-6c bnÞúkbBaÄrEdlmanGMeBIenAelIeRKagsIuemRTInwgeFVIeGayFñwm CD pt EdleFVIeGaytMN
C vilRsbRTnicnaLika ehIyeFVIeGaytMN D vilRcasRTnicnaLika. edaysarvaRtUvrkSamMuEkgrbs;va

eGayenAdEdl enaHssrRtUvekagdUcbgðaj. dUcenHGgát; BC nig DE RtUvEte)a:g EdleFVIeGaytMN
B vilRcasRTnicnaLika ehIytMN E vilRsbRTnicnaLika. cugeRkay enAkñúgrUbTI 8-6d bnÞúk P ruj

tMN B nigtMN C eTAsþaM EdleFVIeGayssrekagdUcbgðaj. tMNbgáb;Rtg; B RtUvrkSamMuEkgrbs;va
b:uEnþvaminmankarTb;kñúgkarvilrvagGgát;EdltP¢ab;enARtg;cMnuc C eT edaysartMN C CatMNsnøak;.
Cavi)ak manEtFñwm CD minmankarpøas;bþÚrkMeNag.

*>@> RTwsþIFñwmeGLasÞic (Elastic-curve theory)
enAkñúgkfaxNÐenH eyIgnwgbegáItsmIkard¾sMxan;cnYnBIredayP¢ab;TMnak;TMngrvagm:Um:g;Bt;enAkñúg
M
FñwmeTAnwgbMlas;TI nigmMurgVilénExSekageGLasÞicrbs;va. smIkarTaMgenHbegáItCaeKalkarN_sMrab;viFI
KNnaPaBdabEdlerobrab;enAkñúgCMBUkenH ehIysMrab;mUlehtuenH eKRtUvyl;BIkarsnμt; nigEdnkMNt;
EdleRbIenAkñúgkarbegáItviFIenHeGay)anc,as;.
edIm,IbegáItTMnak;TMngenH eyIgnwgkMritkarviPaKeGayenAkñúgkrNITUeTAbMputEdlFñwmxUcRTg;RTay
edaybnÞúkmanGMeBIEkgeTAnwgG½kS x rbs;Fñwm ehIyvasßitenAkñúgbøg; x −ν müa:geTotmuxkat;rbs;Fñwm
manlkçN³sIuemRTI ¬rUbTI 8-7a¦. edaysarbnÞúk kMhUcRTg;RTayrbs;FñwmnwgekIteLIgedaysar
kMlaMgkat; nigm:Um:g;Bt;. RbsinebIFñwmmanRbEvgFMCagkMBs;rbs;vaxøaMgenaH kMhUcRTg;RTajEdlFMCageK
nwgbgáeLIgedaym:Um:g;Bt; dUcenHeyIgnwgykcitþTukdak;EteTAnwgT§iBlrbs;m:Um:g;Bt;b:ueNÑaHsMrab;Fñwm
enAeBlenH. PaBdabEdlbgáedaykMlaMgkat;nwgRtUv)anerobrab;enAeBleRkayenAkñúgemeronenH.
enAeBlm:Um:g;Bt; M eFVIeGayFatuénFñwmxUcRTg;RTay mMurvagmuxkat;køayCa dθ ¬rUbTI 8-7b¦.
RbEvgFñÚ dx EdltMNageGaycMENkrbs;ExSekageGLasÞicRbsBVCamYynwgG½kSNWtsMrab;muxkat;nImYy
PaBdab T.Chhay -246


². eKkMNt;kaMkMeNagsMrab;FñÚenHedaycMgay ρ EdlRtUv)anvas;BIp©iténkMeNag O' eTA dx . FñÚenAelI
Ggát;eRkABI dx rgnUvbMErbMrYlrageFobEkg (normal strain). Ca]TahrN_ bMErbMrYlrageFobenAkñúg
kMNat;FñÚ ds ¬EdlsßitenAcMgay y BIG½kSNWt¦ KW ε = (ds'−ds ) / ds . b:uEnþ ds = dx = ρdθ ehIy
ds ' = (ρ − y )dθ dUcenH

ε=
(ρ − y )dθ − ρdθ b¤ ρ = − εy
1
ρdθ
RbsinebIvaCasMPar³sac;mYy (homogeneous) ehIyeFVIkarCalkçN³eGLasÞic enaHeKGacGnuvtþc,ab;
h‘Uk (Hooke’s law) ε = σ / E . ehIymüa:geTot eKmansmIkarkugRtaMgm:Um:g;Bt; σ = −My / I .
edaypSMsmIkarTaMgenH ehIyCMnYsvaeTAkñúgsmIkarxagelI eyIg)an
1 M
= (8-1)
ρ EI
Edl kaMrbs;kMeNagenARtg;cMnucCak;lak;enAelIExSekageGLasÞic ¬eKehA1 / ρ CakMeNag¦
ρ=
M = m:Um:g;Bt;enAkñúgFñwmenARtg;cMnucEdleKRtUvkarkMNt; ρ

E = m:UDuleGLasÞicrbs;sMPar³

I = m:Um:g;niclPaBrbs;FñwmEdlKNnaeFobG½kSNWt.

plKuN EI enAkñúgsmIkar 8-1 eKeGayeQμaHfaPaBrwgTb;nwgkarBt;begáag (flexural rigidity)
ehIyvamantMélviC¢manCaTUeTA. edaysar dx = ρdθ enaHBIsmIkar 8-1
M
dθ = dx (8-2)
EI
RbsinebIeyIgeRCIserIsG½kS v viC¢manenAeBlvamanTisedAeLIgelI ¬rUbTI 8-7a¦ ehIyRbsin
ebIeyIgsresrsmIkarkMeNag1 / ρ CaGnuKmn_én x nig v enaHeyIgGackMNt;ExSekageGLasÞicsMrab;
Fñwm. enAkñúgesovePA calculus PaKeRcInbgðajfa
1 d 2ν / dx 2
=
ρ [1 + (dν / dx) ]
2 3/ 2

d 2ν / dx 2
dUcenH M
=
[ ]
(8-3)
EI 1 + (dν / dx )2 3/ 2

smIkar 8-3 CasmIkarDIepr:g;Esül nonlinear dWeRkTIBIr. dMeNaHRsayrbs;va ν = f (x ) eGaynUv
ragBitR)akdénExSekageGLasÞic ¬edaysnμt;fa PaBdabrbs;FñwmekIteLIgedaysarEtm:Um:g;Bt;
b:ueNÑaH¦. edIm,IsMrYlkñúgkaredaHRsaynUvcMeNaTy:ageRcIn eKsMrYlsmIkar 8-3 eGaymanlkçN³gay
RsYl. edaysarmMurgVilrbs;ExSekageGLasÞicsMrab;rcnasm<½n§PaKeRcInmantMéltUcEmnETn eyIgnwg



eRbIRTwsþIPaBdabtUc (small deflection theory) ehIysnμt;fa dv / dx ≈ 0 . Cavi)ak kaeréntMél
dv / dx Gacecal)anRbsinebIeRbobeFobnwgtMélÉktþa dUcenHsmIkar 8-3 kat;bnßymkRtwm
d 2v M
= (8-4)
dx 2 EI
edaysarkarsnμt; dv / dx ≈ 0 enaHRbEvgedImrbs;G½kSFñwm x nigkMNat;FñÚrbs;ExSekageGLasÞicnwg
manRbEvgRbEhlKña. eyIgGacniyaymüa:geTotfa ds enAkñúgrUbTI 8-7b manRbEvgRbEhlnwg dx
ds = dx 2 + dν 2 = 1 + (dν / dx )2 dx ≈ dx
lT§plenHbgðajfacMnucenAelIExSekagmanbMlas;TIEttamTisQr nigmanbMlas;TItamTisedkeT.

*>#> viFIGaMgetRkalDub (The double integration method)
enAeBlEdleKKNnasmIkarm:Um:g:Bt; M CaGnuKmn_énTItaMg x enaHGaMgetRkalbnþbnÞab;én
smIkar 8-4 pþl;nUvPaBdabrbs;Fñwm θ ≈ tan θ = dν / dx = ∫ (M / EI )dx ¬smIkar 8-2¦ ehIysmIkar
rbs;ExSekageGLasÞicν = f (x ) = ∫∫ (M / EI )dx erogKña. sMrab;kareFVIGaMgetRkalnImYy² eKcaM)ac;
RtUvbBa©Úl “tMélefrrbs;GaMgetRkal” ehIybnÞab;mkedaHRsayrktMélefrenaHedIm,ITTYl)andMeNaH
Rsay. BIkfaxNÐ 4-2 RbsinebIbnÞúkenAelIFñwmminEmnCaGnuKmn_Cab; ¬vamanes‘rIénbnÞúkcMcMnuc nig
bnÞúkBRgay¦ enaHeKRtUvsresrsmIkarGnuKmn_m:Um:gBt;CaeRcIn EdlsmIkarnImYy²mann½yEtenAkñúg
;
tMbn;EdlsßitenAcenøaHPaBdac;. Ca]TahrN_ eKmanFñwmdUcbgðajenAkñúgrUbTI 8-8. eKRtUvsresr
smIkarGnuKmn_m:Um:g;Bt;enAkñúgtMbn; AB / BC nig CD CaGnuKmn_eTAnwgGab;sIus x1 / x2 nig x3 . enA
eBleKeFVIGaMgetRkalGnuKmn_TaMgenHtamkarGnuvtþénsmIkar 8-4 CamYynwgkarkMNt;tMélefrrbs;
PaBdab T.Chhay -248


GaMgetRkal GnuKmn_nwgeGaynUvmMurgVil nigPaBdab ¬ExSekageGLasÞic¦ sMrab;tMbn;nImYy²rbs;Fñwm.

karkMNt;sBaØa³ enAeBlGnuvtþsmIkar 8-4 eKcaM)ac;RtUveRbIsBaØad¾RtwmRtUvsMrab;m:Um:g;Bt; M Edl)an
begáItedaykarkMNt;sBaØaEdleRbIenAkñúgkarbMEbksmIkarenH ¬rUbTI 8-9a¦. elIsBIenH rMlwkfaPaB
dabν viC¢man manTisedAeLIgelI ehIyCalT§pl mMurgVil θ viC¢manRtUv)anvas;edayRcasRTnicnaLika
BIG½kS x . ehtuplsMrab;karkMNt;sBaØaenHRtUv)anbgðajenAkñúgrUbTI 8-9b. enATIenH kMeNIn dx nig
dν enAelIG½kS x nigν mantMélviC¢man eFVIeGaykMeNIn dθ vilRcasRTnicnaLika. ehIyedaysarEtmMu

rgVil θ mantMéltUc enaHeKGackMNt;tMélrbs;vaEdlKitCara:düg;edaypÞal;BIθ ≈ tan θ = dν / dx .
lkçxNÐCab; niglkçxNÐRBMEdn (boundary and continuity condition)³ eKkMNt;tMélrbs;GaMget-
RkaledaykMNt;GnuKmn_sMrab;mMurgVil nigGnuKmn_sMrab;bMlas;TIenARtg;cMnucCak;lak;enAelIFñwm EdleK
GacdwgtMélrbs;GnuKmn_. tMélTaMgenHRtUv)aneKehAfalkçxNÐRBMEdn. Ca]TahrN_ RbsinebIFñwmRtUv
)anRTedayTMrkl; b¤TMrsnøak; enaHeKTamTareGaybMlas;TIrbs;vaesμIsUnüenARtg;cMnucenaH. ehIy enA
Rtg;TMrbgáb; TaMgmMurgVil nigbMlas;TIesμInwgsUnü.
RbsinebIeKminGaceRbIEtkUGredaen x edIm,IedaHRsaysmIkarsMrab;mMurgVilrbs;Fñwm b¤ExSekag
eGLasÞic enaHeKRtUveRbIlkçxNÐCab;edIm,IkMNt;tMélefrrbs;GaMgetRkalxøH. BicarNaFñwmenAkñúgrUbTI
8-10. enATIenH kUGredaen x1 nig x2 mann½yEtenAkñúgtMbn; AB nig BC erogKña. enAeBleKTTYl)an
GnuKmn_sMrab;mMurgVil nigGnuKmn_sMrab;PaBdabenARtg;cMnuc B ¬ x1 = x2 = a ¦ dUcenHExSekageGLa
sÞicmanlkçN³Cab;enARtg;cMnucenH. edayKittamn½yKNitviTüa vaTamTareGayθ1 (a ) = θ 2 (a ) ehIy
ν 1 (a ) = ν 2 (a ) . eKGaceRbIsmIkarTaMgenHsMrab;kMNt;tMélefrrbs;GaMgetRkalcMnYnBIr.



dMeNIrkarkñúgkarviPaK (Procedure for analysis)
CMhanxageRkampþl;nUvviFIsMrab;kMNt;mMurgVil nigPaBdabrbs;FñwmedayeRbIviFIGaMgetRkalDub.
eKKYrdwgfaviFIenHsaksmsMrab;EtPaBdabeGLasÞicsMrab;FñwmEdlmanmMurgViltUcb:ueNÑaH. elIsBIenH
viFIenHBicarNaEtPaBdabEdlbNþalBIm:Um:g;Bt;b:ueNÑaH. PaBdabbEnßmEdlbNþalBIkMlaMgkat;CaTUeTA
esμInwgBIr b¤bIPaBryénPaBdabedaym:Um:g;Bt;b:ueNÑaH dUcenHCaFmμtavaRtUv)aneKecalsMrab;karGnuvtþ.
ExSekageGLasÞic
sg;KMehIjénExSekageGLasÞicrbs;Fñwm. rMlwkfacMnucénmMurgVil nigPaBdabsUnüekItmanenA
Rtg;TMrbgáb; ehIyPaBdabsUnüekItmanenARtg;TMrkl; nigTMrsnøak;.
sg;G½kSkUGredaen x nig v . G½kS x RtUvEtRsbeTAnwgG½kSFñwmEdlminTan;xUcRTg;RTay ehIy
Kl;rbs;vaenAEpñkxageqVgrbs;Fñwm EdlTisedATissþaMCaTisedAviC¢man.
RbsinebImanbnÞúkEdlmanCab;eRcIn eKRtUvbegáItkUGredaen x Edlmann½ysMrab;tMbn;nImYy²
rbs;FñwmrvagPaBminCab;enaH.
enARKb;krNITaMgGs; G½kSν mantMélviC¢manenAeBlEdlvamanTisedAeLIgelI
GnuKmn_bnÞúk b¤GnuKmn_m:Um:g;Bt;
sMrab;tMbn;nImYy²EdlenAkñúgenaHmankUGredaen x eKRtUvsresrsmIkarm:Um:g;Bt; M CaGnuKmn_
én x .
eKEtgEtsnμt;fam:Um:g;Bt; M eFVIGMeBIkñúgTisedAviC¢manenAeBlEdlGnuvtþsmIkarlMnwgedIm,IkMN
t; M = f (x ) .
mMurgVil nigExSekageGLasÞic
eday EI CatMélefr GnuvtþsmIkarm:Um:g; EId 2ν / dx 2 = M (x ) EdlTamTarGaMgetRkalcMnYnBIr
dg. sMrab;kareFVIGaMgetRkalnImYy² eKcaM)ac;RtUvEtbBa©ÚltMélefrrbs;GaMgetRkal. eK
kMNt;tMélTaMgenHedayeRbIlkçxNÐRBMEdnsMrab;TMr ehIyeKeRbIlkçxNÐCab;sMrab;GnuKmn_BIr
CYbKña.
enAeBlEdleKkMNt;efrGaMgetRkalehIy eKRtUvCMnYsvaeTAkñúgsmIkarmMurgVil nigsmIkarPaB
dabeLIgvij. cugeRkay eKGackMNt;mMurgVil nigPaBdabenARtg;cMnucCak;lak;enAelIExSekag
eGLasÞic)an. eKGacepÞógpÞat;lT§plEdlTTYledaylkçN³RkaPicedayeRbobeFobvaCa
mYynwgExSekageGLasÞicEdleyIg)anKUrBIxagelI.
tMélviC¢mansMrab;mMurgVilKWRcasRTnicnaLika ehIyPaBdabviC¢manmanTisedAeLIgelI.
PaBdab T.Chhay -250


]TahrN_ 8-3³ Fñwmkugs‘ulEdlbgðajenAkñúgrUbTI 8-11a rgm:Um:g; M enAcugTMenrrbs;va.
0

kMNt;smIkarExSekageGLasÞic. EI CatMélefr.

dMeNaHRsay³
ExSekageGLasÞic³ bnÞúkxageRkAeFVIFñwmxUcRTg;RTaydUcbgðajenAkñúgrUbTI 8-11a. tamkarGegát
eKGackMNt;m:Um:g;Bt;enAkñúgFñwmTaMgmUledayeRbIEtkUGredaen x .
GnuKmn_m:Um:g;Bt;³ BIdüaRkamGgÁesrI CamYynwgm:Um:g;Bt; M EdlmanGMeBIenAkñúgTisedAviC¢man ¬rUbTI 8-
11b¦ eyIg)an
M = M0
mMurgVil nigExSekageGLasÞic³ edayGnuvtþsmIkar 8-4 nigedayeFVIGaMgetRkalBIrdgenaH
d 2ν
EI = M0 (1)
dx
dν
EI = M 0 x + C1 (2)
dx
M x2
EIν − 0 + C1 x + C 2 (3)
2
edayeRbIlkçxNÐRBMEdn dν / dx = 0 enARtg; x = 0 nigν = 0 enARtg; x = 0 enaH C1 = C2 = 0 .
edayCMnYslT§plTaMgenHeTAkñúgsmIkar (2) nig (3) CamYynwg θ = dν / dx eyIg)an
M0x
θ=
EI
M x2
ν= 0
2 EI
mMurgVilGtibrma nigPaBdabGtibrma ekItmanenARtg;cMnuc A Edl x = L
M 0L
θA = (4)
EI
M L2
νA = 0 (5)
2 EI



lT§plviC¢mansMrab; θ A bgðajfacMnuc A xUcRTg;RTayedayvilRcasRTnicnaLika ehIylT§plviC¢man
sMrab;ν A bgðajfaPaBdabenARtg;cMnuc A manTisedAeLIgelI. lT§plenHRtUvKñanwglT§plEdl)anKUr
RBagenAkñúgrUbTI 8-11a.
edIm,ITTYl)anKMnitxøHCaTMhMCak;EsþgsMrab;mMurgVil nigPaBdabenARtg;cMnuc A eKeGayFñwmenA
kñúgrUbTI 8-11a manRbEvg 3.6m Tb;Tl;nwgm:Um:g; M 0 = 20kN .m ehIyFñwmenHRtUv)ansg;BIEdkEdl
man Est = 200GPa . RbsinebIeKsikSaKNnaFñwmenHedaymineRbIemKuNsuvtßiPaBedaysnμt;fa kug
RtaMgEkgGnuBaØatesμInwgkugRtaMgyal (yield stress) σ aloow = 250 N / mm 2 enaHeKrkeXIjfamuxkat;
W 6 × 9 ¬ I = 6.8 ⋅ 10 6 mm 4 ¦ manlkçN³RKb;RKan;kñúgkarTb;Tl;nwgbnÞúkxageRkAenH. BIsmIkar (4)

nig (5) eyIgTTYl)an
20kN .m(3.6m )
θA =
[200 ⋅10 kN / m ][6.8 ⋅10
6 2 6
× 10 −12 m 4 ] = 0.0529rad
20kN .m(3.6m )2
νA =
[ ][
2 200 ⋅ 10 6 kN / m 2 6.8 ⋅ 10 6 × 10 −12 m 4 ] = 95.3mm
edaysar θ A = 0.00280(10 −6 ) << 1 enaHvabgðajfakareRbIR)as;smIkar 8-4 manlkçN³RbesIrCag
2

karGnuvtþsmIkar 8-3 sMrab;karkMNt;PaBdabrbs;Fñwm. ehIy edaysarkarGnuvtþCaelxenHGnuvtþeTA
elIFñwmkugs‘ul dUcenHeyIgTTYl)anθ max nigν max mantMélFMCagθ max nigν max sMrab;FñwmTMrsamBaØ.

]TahrN_ 8-4³ FñwmenAkñúgrUbTI 8-12a rgbnÞúk P enAxagcugTMenrrbs;va. kMNt;bMlas;TIenARtg; C .
EI CatMélefr.

dMeNaHRsay³
ExSekageGLasÞic³ FñwmxUcRTg;RTaykñúgTMrg;dUcbgðajenAkñúgrUbTI 8-12a. edaysarEtbnÞúk eKRtUv
BicarNakUGredaen x BIIr.
PaBdab T.Chhay -252


GnuKmn_m:Um:g;Bt;³ edayeRbIdüaRkamGgÁesrIEdlbgðajenAkñúgrUbTI 8-12b eyIg)an
P
M1 = − x1 0 ≤ x1 ≤ 2a
2
P
M 2 = − x2 +
3P
( x 2 − 2a )
2 2
= Px 2 − 3Pa 2a ≤ x 2 ≤ 3a
mMurgVil nigExSekageGLasÞic³ edayGnuvtþsmIkar 8-4
d 2ν 1
sMrab; x1 / EI 2 = − P x1 2
dx1
dν P 2
EI 1 = − x1 + C1 (1)
dx1 4
P 3
EIν 1 = − x1 + C1 x1 + C 2 (2)
12
d ν2
2
sMrab; x2 / EI 2
= Px 2 − 3Pa
dx 2
dν 2 P 2
EI = x 2 − 3Pax 2 + C3 (3)
dx 2 2
P 3 3
EIν 2 = x 2 − Pax 2 + C3 x 2 + C 4
2
(4)
6 2
eKkMNt;efrGaMgetRkalTaMgbYnedayeRbIlkçxNÐRBMEdnbI eBalKWν 1 = 0 enARtg; x1 = 0 / ν 1 = 0 enA
Rtg; x1 = 2a nigν 2 = 0 enARtg; x2 = 2a nigsmIkarénPaBCab;mYy. enATIenH PaBCab;rbs;mMurgVil
enARtg;TMrkl;TamTareGay dν 1 / dx1 = dν 2 / dx2 enARtg; x1 = x2 = 2a . cMNaMfaPaBCab;énbMlas;
TIenARtg;cMnuc B RtUv)aneKKitenAkñúglkçxNÐRBMEdnrYcehIy edaylkçN³minpÞal;edaysar ν 1 = ν 2 =
0 enARtg; x1 = x 2 = 2a . edayGnuvtþlkçxNÐTaMgbYnenH eyIg)an

ν 1 = 0 enARtg; x1 = 0 0 = 0 + 0 + C2

ν 1 = 0 enARtg; x1 = 2a 0 = − (2a )3 + C1 (2a ) + C 2
P
12
ν 2 = 0 enARtg; x 2 = 2a 0 = (2a )3 − Pa(2a )3 + C3 (2a ) + C 4
P 3
6 2
dν 1 (2a ) dν 2 (2a )
= −
P
(2a )2 + C1 = P (2a )2 − 3Pa(2a ) + C3
dx1 dx 2 4 2
edayedaHRsaysmIkarxagelI eyIg)an
Pa 2 10 2
C1 = C2 = 0 C3 = Pa C 4 = −2 Pa 3
3 3
edayCMnYs C3 nig C4 eTAkñúgsmIkar (4) eyIg)an
P 3 3 Pa 2 10 Pa 2 2 Pa 3
ν2 = x2 − x2 + x2 −
6 EI 2 EI 3EI EI



eKGackMNt;PaBdabenARtg;cMnuc C edayCMnYs x2 = 3a . eyIg)an
Pa 3
νc = −
EI

*>$> RTwsþIRkLaépÞm:Um:g; (Moment-Area Theorems)
KMnitedImcMnYnBIrsMrab;RTwsþIRkLaépÞm:Um:g;RtUv)anbegáIteLIgeday Otto Mohr nigeRkaymk
RtUv)anbkRsayCapøÚvkareday Charles E. Greene enAkñúgqñaM 1873. RTwsþITaMgenHpþl;nUvbec©keTs
RkaPicBak;kNþal (semigraphical technique) sMrab;kMNt;mMurgVilénExSekageGLasÞic nigPaBdab
rbs;vaEdlbNþalBIm:Um:g;Bt;. vamanKuNsm,tþiBiessenAeBleKeRbIvasMrab;edaHRsaycMeNaTEdl
Tak;TgnwgFñwm. CaBiessFñwmEdlrges‘rIénbnÞúkcMcMnuc b¤EdlkMNat;rbs;vamanm:Um:g;niclPaBxusKña.

eKeyageTAelIFñwmEdlrgbnÞúkdUcbgðajenAkñúgrUbTI 8-13a edIm,IbegáItRTwsþITaMgenH. Rbsin
ebIeyIgKUrdüaRkamm:Um:g;Bt;sMrab;Fñwm ehIybnÞab;mkEckvanwgPaBrwgTb;nwgkarBt;begáag (flexural
rigidity) EI enaHeKTTYl)andüaRkam M / EI dUcEdlbgðajenAkñúgrUbTI 8-13b. tamsmIkar 8-2
⎛M ⎞
dθ = ⎜ ⎟dx
⎝ EI ⎠
dUcenH eyIgeXIjfabMErbMrYlmMurgVil dθ énbnÞat;b:HenAelIRCugsgxagrbs;Fatu dx esμInwgRkLaépÞqUtén
ExSekag M / EI ¬BIcMnuc A eTAcMnuc B énrUbTI 8-13¦. kareFVIGaMgetRkalBIcMnuc A enAelIExSekag
eGLasÞiceTAcMnuc B ¬rUbTI 8-13c¦ eyIg)an
BM
θB/ A = ∫ dx (8-5)
A EI
smIkarenHbegáIt)anCaeKalkarN_sMrab;RTwsþIRkLaépÞm:Um:g;TImYy.
PaBdab T.Chhay -254


RTwsþITI1³ bMErbMrYlmMurgVilrvagcMnucBIrenAelIExSekageGLasÞicesμInwgRkLaépÞrbs;düaRkam M / EI
cenøaHcMnucTaMgBIrenaH.
nimitþsBaØaθ B / A CamMuénbnÞat;b:HenARtg;cMnuc B Edlvas;eFobeTAnwgbnÞat;b:HenARtg;cMnuc A . BI
karRsaybBa¢ak; eyIgeXIjfaeKvas;mMuenHBIbnÞat;b:HRtg;cMnuc A eTAbnÞat;b:HRtg;cMnuc B RbsinebIRkLa
épÞéndüaRkam M / EI mantMélviC¢man ¬rUbTI 8-13c¦. pÞúymkvij RbsinebIRkLaépÞenHGviC¢man b¤
enABIeRkamG½kS x enaHmMuθ B / A RtUv)anvas;RsbRTnicnaLikaBIbnÞat;b:HRtg;cMnuc A eTAbnÞat;b:HRtg;cMnuc
B . elIsBIenH BIkarKitxñaténsmIkar 8-5 θ B / A RtUv)anKitCara:düg;.

RTwsþIRkLaépÞm:Um:g;Bt;TIBIrKWEp¥kelIlMgakeFobénbnÞat;b:Hrbs;ExSekageGLasÞic. dUceXIjenA
kñúgrUbTI 8-14c dt CalMgakbBaÄrrbs;bnÞat;b:HenAelIRCugmçagrbs;FatuGnnþtUc dx . eKvas;lMgakenH
tambeNþayExSQrEdlkat;tamcMnuc A . edaysareKsnμt;famMurgVilrbs;ExSekageGLasÞic nigPaB
dabrbs;vamantMéltUc dUcenHeKGacKitfaRbEvgbnÞat;b:HesμInwgRbEvg x ehIyRbEvgFñÚ ds' esμInwg dt .
edayeRbIrUbmnþkMNat;FñÚrbs;rgVg; s = θr eday r CaRbEvg x dUcenHeyIgGacsresr dt = xdθ . eday
eRbIsmIkar 8-2 ¬ dθ = (M / EI )dx ¦ enaHeKGackMNt;lMgaktamTisQrrbs;bnÞat;b:HRtg;cMnuc A Edl
eFobnwgbnÞat;b:HRtg;cMnuc B edayGaMgetRkal EdlenAkñúgkrNIenH
B M
t A/ B = ∫ x dx (8-6)
A EI
rMlwkBIemeronsþaTic TIRbCMuTMgn;rbs;RkLaépÞRtUv)ankMNt;BI x ∫ dA = ∫ xdA . edaysar ∫ (M / EI )dx
CaRkLaépÞrbs;düaRkam M / EI enaHeyIgGacsresr


v BM
t A/ B = x∫ dx (8-7)
A EI

enATIenH x CacMgayBIG½kSQrEdlkat;tamcMnuc A eTATIRbCMuTMgn;rbs;RkLaépÞcenøaHcMnuc A nig B ¬rUb
TI 8-14b¦.
eKGacbkRsayRTwsþIRkLaépÞm:Um:g;TIBIrdUcxageRkam³
RTwsþITI2³ RbEvglMgakbBaÄrrbs;bnÞat;b:HRtg;cMnuc A enAelIExSekageGLasÞiceFobnwgbnÞat;b:HEdl
bnøayBIcMnuc B esμInwgm:Um:g;énRkLaépÞrbs;düaRkam M / EI cenøaHcMnucBIr ¬ A nig B ¦.
m:Um:g;enHRtUv)aneKKNnaRtg;cMnuc A ¬cMnucenAelIExSeGLasÞic¦ EdlCaTItaMgEdleKRtUvkar
kMNt;lMgak t A / B .
kñúgkrNIEdlm:Um:g;énRkLaépÞ M / EI BIcMnuc A eTAcMnuc B mantMélviC¢man dUcenAkñúgrUbTI 8-
14b enaHmann½yfabnÞat;b:HRtg;cMnuc A sßitenABIelIbnÞat;b:HeTAnwgExSekagEdlbnøayBIcMnuc B ¬rUbTI
8-14c¦. dUcKña m:Um:g;énRkLaépÞ M / EI EdlmanTMhMGviC¢man bgðajfabnÞat;b:HRtg;cMnuc A sßitenA
BIeRkambnÞat;b:HEdlbnøayBIcMnuc B . cMNaMfa CaTUeTA t A / B minesμInwg t B / A eT dUcEdlbgðaj enAkñúg
rUbTI 8-14d. CaBiess edIm,IkMNt; t A / B eKRtUvrkm:Um:g;énRkLaépÞrbs;düaRkam M / EI enAcenøaH
cMnuc A nigcMnuc B Rtg;cMnuc A ¬rUbTI 8-14b¦ ehIyedIm,IKNna t B / A eKRtUvrkm:Um:g;énRkLaépÞrbs;
düaRkam M / EI enAcenøaH cMnuc A nigcMnuc B Rtg;cMnuc B .
eKRtUvdwgfa eKGaceRbIRTwsþIRkLaépÞm:Um:g;edIm,IkMNt;mMu niglMgakcenøaHbnÞat;b:HenAelIExSekag
eGLasÞicrbs;FñwmRtg;BIrcMnuc. CaTUeTA vaminpþl;nUvdMeNaHRsaysMrab;mMurgVil nigPaBdabRtg;cMnucmUy
enAelIFñwmedaypÞal;eT. dMbUgeKRtUveFVIeGayGBaØatTaMgenHTak;TgeTAnwgmMu b¤lMgakbBaÄrrbs;bnÞat;
b:H eTAnwgExSekageGLasÞic. CaFmμta eKRtUvKUrbnÞat;b:HRtg;TMredaysarcMnucTaMgenHminmanPaBdab
nig/b¤manmMurgVilsUnü. krNIBiesskñúgkarbegáItTMnak;TMngEdlmanlkçN³FrNImaRtmanbgðajenAkñúg
cMeNaT]TahrN_.

CMhanxageRkampþl;nUvviFIEdlGaceRbIedIm,IkMNt;PaBdab nigmMurgVilRtg;cMnucmYyenAelIExS
ekageGLasÞicrbs;FñwmedayeRbIRTwsþIRkLaépÞm:Um:g;.
düaRkam M / EI
kMNt;RbtikmμTMr nigsg;düalRkam M / EI rbs;Fñwm.

PaBdab T.Chhay -256


RbsinebIFñwmrgbnÞúkcMcMnuc enaHdüaRkam M / EI nwgpSMeLIgedayes‘rIénkMNat;bnÞat;Rtg; dUc
enHeKnwggayRsYlkñúgkarKNnaRkLaépÞ nigm:Um:g;énRkLaépÞsMrbs;RTwsþIRkLaépÞm:Um:g;.
RbsinebIFñwmrgnUves‘rIénbnÞúkcMcMnuc nigbnÞúkBRgay eKGacKNnaRkLaépÞ M / EI nigm:Um:g;én
RkLaépÞ M / EI edaysg;düaRkam M / EI CaEpñk edayeRbIviFItMrYtpldUckarENnaMenAkñúg
kfaxNÐ 4-5. enAkñúgkNIEdldüaRkam M / EI manrag)a:ra:bUl b¤ExSekagEdlmandWeRkx<s;
eKENnaMeGayeRbItaraglkçN³FrNImaRtrbs;RkLaépÞ.
ExSekageGLasÞic
sg;ExSekageGLasÞicrbs;Fñwm. rMlwkfamMurgVilsUnüekItmanenARtg;TMrbgáb; ehIybMlas;TIsUnü
ekItmanenARtg;TMrbgáb; b¤TMrsnøak; b¤k¾TMrkl;.
RbsinebIeKBi)aksg;ragTUeTArbs;ExSekageGLasÞic cUreRbIdüaRkamm:Um:g; b¤k¾düaRkam M / EI .
eKRtUvdwgfa enAeBlEdlFñwmrgm:Um:g;viC¢man Fñwmpt cMENkÉFñwmrgm:Um:g;GviC¢man Fñwme)a:g.
elIsBIenH cMnucrbt; b¤karpøas;bþÚrkMeNagekItmanenAeBlm:Um:g;enAkñúgFñwmesμIsUnü.
eKRtUvbgðajbMlas;TI nigmMurgVilEdlRtUvkarKNnaenAelIExSekag. edaysarRTwsþIRkLaépÞ
m:Um:g;Gnuvtþ)anEtenAcenøaHbnÞat;b:HBIr eKRtUvRby½tñkñúgkarsg;bnÞat;b:HedayeFVIy:agNaeGaymMu
niglMgakrvagBYkvaGacedaHRsayrkPaBdab nigmMurgVi)an. cMeBaHbBaðaenH eKRtUvBicarNakñúg
karsg;bnÞat;b:HenARtg;cMnucEdlRtUvrkmMurgVil nigPaBdab nigenARtg;TMr edaysarCaTUeTAFñwm
manbMlas;TIsUnü nig/b¤mMurgVilsUnüenARtg;TMr.
RTwsþIRkLaépÞm:Um:g;
GnuvtþRTwsþITImYyedIm,IkMNt;mMurvagbnÞat;b:HBIr ehIyRTwsþITI2edIm,IkMNt;lMgakbBaÄrrvagbnÞat;
b:HTaMgenaH.
eKRtUvdwgfaCaTUeTARTwsþITI2minpþl;nUvbMlas;TIrbs;cMnucenAelIExSekageGLasÞiceT. enAeBl
EdleKGnuvtþ)anRtwmRtUv vanwgpþl;nUgcMgaybBaÄr b¤lMgakénbnÞat;b:HenARtg;cMnuc A enAelI
ExSekageGLasÞicBIbnÞat;b:HRtg;cMnuc B .
bnÞab;BIGnuvtþRTwsþITI1 b¤RTwsþITI2 eKGacepÞógpÞat;sBaØarbs;cMeLIyBImMu niglMgakEdlbgðajenA
elIExSekageGLasÞic.



PaBdab T.Chhay -258


]TahrN_ 8-5³ kMNt;mMurgVilenARtg;cMnuc B nigcMnuc C rbs;FñwmEdlbgðajenAkñúgrUbTI 8-15a. yk
E = 200GPa nig I = 360(10 6 )mm 4 .

dMeNaHRsay³
düaRkam M / EI ³ düaRkamenHRtUv)anbgðajenAkñúgrUbTI 8-15b. vamanlkçN³gayRsYlCagkñúgkar
edaHRsaycMeNaTenHedayeRbItY EI nigCMnYsCatMélelxenACMhancugeRkay.
ExSekageGLasÞic³ bnÞúk10kN eFVIeGayFñwmxUcRTg;RTaydUcbgðajenAkñúgrUbTI 8-15c. FñwmxUcRTg;
RTayedaylkçN³e)a:g edaysar M / EI GviC¢man. enATIenH bnÞat;b:HRtg;cMnuc A ¬TMr¦EtgEtmanTis
edk. bnÞat;b:HRtg;cMnuc B nigcMnuc C manlkçN³dUcbgðajenAkñúgrUb. eyIgRtUvkarkMNt;rkθ B nig θ C .
edaykarsg;bnÞat;b:H mMurvagbnÞat;b:HRtg; A nigbnÞat;b:HRtg; B eBalBIθ B / A smmUlnwg θ B .
θB = θB/ A
ehIy θC = θC / A
RTwsþIRkLaépÞm:Um:g;³ edayGnuvtþRTwsþITI1 θ B / A esñInwgRkLaépÞrbs;düaRkam M / EI cenøaHcMnuc A nig
cMnuc E Edl
⎛ 50kN .m ⎞ 1 ⎛ 100kN .m 50kN .m ⎞
θ B = θ B / A = −⎜ ⎟(5m ) − ⎜ − ⎟(5m )
⎝ EI ⎠ 2 ⎝ EI EI ⎠
375kN .m 2
=−
EI
edayCMnYstMélCaelxsMrab; E nig I eyIg)an
− 375kN .m 2
[200(10 )kN / m ][360(10 )(10 )m ] = −0.00521rad
6 2 6 −12 4

sBaØaGviC¢manbgðajfamMuRtUv)anvas;RsbTisRTnicnaLikaBIcMnuc A ¬rUbTI 8-15c¦.


tamrebobdUcKña RkLaépÞrbs;düaRkam M / EI cenøaHcMnuc A nigcMnuc C esμInwgθ C / A .
eyIg)an
2
1 ⎛ 100kN .m ⎞
θC = θC / A = ⎜ − ⎟ (10m ) = − 500kN .m
2⎝ EI ⎠ EI
edayCMnYsCatMélelxsMrab; EI / eyIg)an
− 500kN .m 2
[200(10 )kN / m ][360(10 )(10 )m ] = −0.00694rad
6 2 6 −12 4

]TahrN_ 8-6³ kMNt;PaBdabenARtg;cMnuc B nigcMnuc C rbs;FñwmEdlbgðajenAkñúgrUbTI 8-16a.
tMélénm:Um:g;niclPaBrbs;Ggát;nImYy²RtUv)anbgðajenAkñúgrUb. yk E = 200GPa .

dMeNaHRsay³
düaRkam M / EI ³ edaykarGegát düaRkamm:Um:g;sMrab;FñwmmanragCactuekaN. enATIenH eyIgnwgsg;
düaRkam M / EI EdlTak;TgeTAnwg I BC edaydwgfa I AB = 2I BC ¬rUbTI 8-16b¦. tMélCaelxsMrab;
EI nwgRtUv)anCMnYsenACMhancugeRkay.

ExSekageGLasÞic³ m:Um:g;xageRkAenARtg;cMnuc C eFVIeGayFñwmxUcRTg;RTaydUcbgðajenAkñúgrUbTI 8-16c.
bnÞat;b:HRtg;cMnuc A ¬TMr¦/ Rtg;cMnuc B nigcMnuc C RtUv)anbgðaj. eKtMrUveGayeyIgrk Δ B nig Δ C .
bMlas;TITaMgenHGacTak;TgedaypÞal;eTAnwglMgakcenøaHbnÞat;b:H. BIkñúgrUbeyIgeXIjfa Δ B esμInwg
lMgakrbs;bnÞat;b:HRtg;cMnuc B eFobeTAnwgbnÞat;b:HRtg;cMnuc A Edl
Δ B = tB / A
ehIy Δ C = tC / A

PaBdab T.Chhay -260


RTwsþIRkLaépÞm:Um:g;³ edayGnuvtþRTwsþITI2 t B / A esñInwgm:Um:g;RkLaépÞrbs;düaRkam M / EI BC cenøaHcMnuc
A nigcMnuc B EdlKitRtg;cMnuc B edaysarvaCacMnucEdleKRtUvkarkMNt;lMgakrbs;bnÞat;b:H. dUcenH

BIrUbTI 8-16b
⎡ 250 N .m ⎤ 3
Δ B = tB / A = ⎢ (4m )⎥(2m ) = 2000 N .m
⎣ EI BC ⎦ EI BC

edayCMnYstMélCaelx enaHeyIg)an
2000 N .m 3
ΔB =
[200(10 )N / m ]⎡⎢⎣4(10 )mm ⎛⎜⎝1m /(10 ) mm
9 2 6 4 4 3 4 4 ⎞⎤
⎟⎥
⎠⎦

= 0.0025m = 2.5mm
dUcKña sMrab; tC / A eyIgRtUvEtKNnam:Um:g;éndüaRkam M / EI BC BIcMnuc A eTAcMnuc C Rtg;cMnuc C .
eyIg)an
⎡ 250 N .m ⎤ ⎡ ⎤
Δ C = tC / A = ⎢ (4m )⎥(5m ) + ⎢ 500 N .m (3m )⎥(1.5m )
⎣ EI BC ⎦ ⎣ EI BC ⎦
3 3
7250 N .m 7250 N .m
=
EI BC
=
[ ( ) ][ ( )(
200 10 N / m 2 4 10 6 10 −12 m 4
6
) ]
= 0.00906m = 9.06mm
edaysarcMeLIyTaMgBIrmantMélviC¢man enaHcMnuc B nigcMnuc C RtUvsßitenABIxagelIbnÞat;b:HRtg;cMnuc A .

]TahrN_ 8-7³ kMNt;mMurgVilenARtg;cMnuc C rbs;FñwmEdlbgðajenAkñúgrUbTI 8-17a. yk E =
/ ( )
200GPa I = 6 10 6 mm 4 .

dMeNaHRsay³
düaRkam M / EI ³ rUbTI 8-17b.


ExSekageGLasÞic³ edaysarbnÞúkEdlGnuvtþmkelIFñwmmanlkçN³sIuemRTI enaHExSekageGLasÞick¾RtUv
manlkçN³sIuemRTIEdr dUcbgðajenAkñúgrUbTI 8-17c. eKTamTareGayeyIgrk θ C . eKRtUvdwgfabnÞat;
b:HRtg;cMnuc D CabnÞat;edk dUcenHBIsMNg;bnÞat;b:H mMuθ D / C cenøaHbnÞat;b:HRtg;cMnuc C nigbnÞat;b:HRtg;
cMnuc D esμInwg θ C . dUcenH
θC = θ D / C
RTwsþIRkLaépÞm:Um:g;³ edayeRbIRTwsþITI1 θ D / C esñInwgRkLaépÞqUtrbs;düaRkam M / EI BC cenøaHcMnuc C
nigcMnuc D . eyIg)an
⎛ 30kN .m ⎞ 1 ⎛ 60kN .m 30kN .m ⎞
θ C = θ D / C = 3m⎜ ⎟ + (3m )⎜ − ⎟
⎝ EI ⎠ 2 ⎝ EI EI ⎠
135kN .m 2
=
EI
135kN .m 2
dUcenH θC =
[ ( ) ][ ( )( ) ]
200 10 6 kN / m 2 6 10 6 10 −12 m 4
= 0.112rad

]TahrN_ 8-8³ kMNt;mMurgVilenARtg;cMnuc C rbs;FñwmEdlbgðajenAkñúgrUbTI 8-18a. yk E =
/
200GPa I = 360 10 6 mm 4 ( ) .

dMeNaHRsay³
ExSekageGLasÞic³ ExSekageGLasÞicRtUv)anbgðajenAkñúgrUbTI 8-18c. eKtMrUveGayeyIgkMNt; θ C .
sg;bnÞat;b:HRtg;cMnuc A / B ¬TMr¦ nigRtg;cMnuc C ehIycMNaMfaθ C / A CamMurvagbnÞat;b:HRtg;cMnuc A nig
PaBdab T.Chhay -262


cMnuc C . ehIyeKGackMNt;mMu φ enAkñúgrUbTI 8-18c edayeRbI φ = t B / A / L AB . smIkarenHmann½y
edaysar t B / A mantMéltUc ehtudUcenHeKGacyk t B / A esμInwgRbEvgkMNat;FñÚEdlkMNt;edaykaM
L AB = 8m nigmMu φ ¬rMlwkfa s = rθ ¦. BIrUbFrNImaRténrUbTI 8-18c eyIg)an
tB / A
θC = φ − θC / A = − θC / A (1)
8
RTwsþIRkLaépÞm:Um:g;³ edayeRbIRTwsþITI1 θ C / A esñInwgRkLaépÞrbs;düaRkam M / EI cenøaHcMnuc A nigcMnuc
C . eyIg)an
2
θC / A =
1
(2m )⎜ 20kN .m ⎞ = 20kN .m
⎛
⎟
2 ⎝ EI ⎠ EI
edayGnuvtþRTwsþI2 t B / A esμInwgm:Um:g;RkLaépÞrbs;düaRkamm:Um:g;cenøaHcMnuc B nigcMnuc A Rtg;cMnuc B
edaysarvaCacMnucEdleKRtUvkarrklMgakrbs;bnÞat;b:H. eyIg)an
⎡ ⎤⎡1 ⎛ 60kN .m ⎞⎤ 2 ⎡1 ⎛ 60kN .m ⎞⎤
t B / A = ⎢2m + (6m )⎥ ⎢ (6m )⎜ ⎟⎥ + (2m )⎢ (2m )⎜
1
⎟⎥
⎣ 3 ⎦⎣ 2 ⎝ EI ⎠⎦ 3 ⎣2 ⎝ EI ⎠⎦
800kN .m 3
=
EI
edayCMnYslT§plTaMgenHeTAkñúgsmIkar (1) eyIg)an
800kN .m 3 20kN .m 2 80kN .m 2
θC = − =
(8m )EI EI EI
80kN .m 2
dUcenH θC =
[200(10 )kN / m ][360(10 )(10 )m ] = 0.0011rad
6 2 6 −12 4

]TahrN_ 8-9³ kMNt;PaBdabenARtg;cMnuc C rbs;FñwmEdlbgðajenAkñúgrUbTI 8-19a. yk E =
/
200GPa I = 4 10 6 mm 4 ( ) .



dMeNaHRsay³
ExSekageGLasÞic³ enATIenH eKtMrUveGayeyIgrk Δ C ¬rUbTI 8-19c¦. vaminEmnCaPaBdabGtibrma
rbs;FñwmeT edaysarbnÞúk nigExSekageGLasÞicminmanlkçN³sIuemRTIeT. GIVEdlbgðajenAkñúgrUbTI 8-
19c CabnÞat;b:HenARtg;cMnuc A / cMnuc B ¬TMr¦ nigcMnuc C . RbsinebIeKkMNt; t A / B enaHeKGacrk Δ' BI
RtIekaNdUc Edl Δ' / 3 = t A / B / 6 b¤ Δ' = t A / B / 2 . BIsMNg;bnÞat;b:HenAkñúgrUbTI 8-19c eyIg)an
t A/ B
ΔC = − tC / B (1)
2
RTwsþIRkLaépÞm:Um:g;³ eyIgnwgGnuvtþRTwsþITI2 edIm,Irk t A / B nig tC / B . enATIenH t A / B Cam:Um:g;RkLaépÞ
rbs;düaRkam M / EI cenøaHcMnuc A nigcMnuc B Rtg;cMnuc A . eyIg)an
⎤⎡1 ⎛ 5kN .m ⎞⎤ 30kN .m
3
⎡1
t A / B = ⎢ (6m )⎥ ⎢ (6m )⎜ ⎟⎥ =
⎣3 ⎦⎣ 2 ⎝ EI ⎠⎦ EI
ehIy tC / B Cam:Um:g;RkLaépÞ rbs;düaRkam M / EI cenøaHcMnuc C nigcMnuc B Rtg;cMnuc C . eyIg)an
⎤⎡1 ⎛ 2.5kN .m ⎞⎤ 3.75kN .m
3
⎡1
tC / B = ⎢ (3m )⎥ ⎢ (3m )⎜ ⎟⎥ =
⎣3 ⎦⎣ 2 ⎝ EI ⎠⎦ EI
edayCMnYslT§plTaMgenHeTAkñúgsmIkar (1) eyIg)an
1 ⎛ 30kN .m 3 ⎞ 3.75kN .m 3 11.25kN .m 3
⎜ ⎟−
ΔC = =
2⎜⎝ EI ⎟
⎠ EI EI

CMnYsCatMélelxeyIg)an
11.25kN .m 3
ΔC =
[200(10 )kN / m ][4(10 )(10 )m ]
6 2 6 −12 4

= 0.0141m = 14.1mm

]TahrN_ 8-10³ kMNt;PaBdabenARtg;cMnuc C rbs;FñwmEdlbgðajenAkñúgrUbTI 8-20a. yk E =
/
200GPa I = 250 10 6 mm 4 ( ) .
dMeNaHRsay³
düaRkam M / EI ³ dUcbgðajenAkñúgrUbTI 8-20b düaRkamenHpSMeLIgedayRtIekaN nig)a:ra:bUl.
ExSekageGLasÞic³ bnÞúkbgáeGayFñwmxUcRTg;RTaydUcbgðajenAkñúgrUbTI 8-20c. eKtMrUveGayeyIgrk
Δ C . edaysg;bnÞat;b:HenARtg;cMnuc A / cMncu B ¬TMr¦ nigcMnuc C eyIgeXIjfa Δ C = t C / A − Δ' . b:uEnþ

eKGacP¢ab;TMnak;TMng Δ' eTAnwg t B / A edayRtIekaNdUc Edl Δ' / 16 = t B / A / 8 b¤ Δ' = 2t B / A dUcenH
Δ C = t C / A − 2t B / A (1)

PaBdab T.Chhay -264


RTwsþIRkLaépÞm:Um:g;³ eyIgnwgGnuvtþRTwsþITI2 edIm,Irk tC / A nig t B / A . enATIenH tC / A Cam:Um:g;RkLaépÞ
rbs;düaRkam M / EI cenøaHcMnuc C nigcMnuc A Rtg;cMnuc C . eyIg)an
⎡3 ⎤⎡1 ⎛ 192kN .m ⎞⎤ ⎡ 1 ⎤⎡1 ⎛ 192kN .m ⎞⎤
t C / A = ⎢ (8m )⎥ ⎢ (8m )⎜ − ⎟⎥ + ⎢ (8m ) + 8m⎥ ⎢ (8m )⎜ − ⎟
⎣4 ⎦⎣3 ⎝ EI ⎠⎦ ⎣ 3 ⎦⎣ 2 ⎝ EI ⎠⎥ ⎦
11264kN .m 3
=−
EI
ehIy t B / A Cam:Um:g;RkLaépÞ rbs;düaRkam M / EI cenøaHcMnuc B nigcMnuc A Rtg;cMnuc B . eyIg)an
⎡1 ⎤⎡1 ⎛ 192kN .m ⎞⎤ 2048kN .m 3
= ⎢ (8m )⎥ ⎢ (8m )⎜ − ⎟ =−
EI ⎠⎥
tB / A
⎣3 ⎦⎣ 2 ⎝ ⎦ EI
ehtuGIV)anCatYTaMgenHmantMélGviC¢man? edayCMnYslT§plTaMgenHeTAkñúgsmIkar (1) eyIg)an
11264kN .m 3 ⎛ 2048kN .m 3 ⎞
ΔC = − − 2⎜ − ⎟
EI ⎜ EI ⎟
⎝ ⎠
3
7168kN .m
=−
EI
7168kN .m 3
dUcenH ΔC = −
[ ( ) ][ ( )( ) ]
200 10 6 kN / m 2 250 10 6 10 −12 m 4

= −0.143m

]TahrN_ 8-11³ kMNt;mMurgVilenARtg;cMnuc B rbs;FñwmlysgxagdUcbgðajenAkñúgrUbTI 8-21a.
yk E = / ( )
200GPa I = 18 10 6 mm 4 .



dMeNaHRsay³
düaRkam M / EI ³ eKGacsMrYldüaRkam M / EI edayKUrvaCaEpñk nigBicarNadüaRkam M / EI sMrab;
bnÞúkbI EdlbnÞúknImYy²manGMeBIenAelIFñwmkugs‘ulEdlmanTMrbgáb;enARtg;cMnuc D ¬ rUbTI 8-21b¦.
eKminKitbnÞúk10kN eT edaysarvaminbegáItm:Um:g;enARtg;cMnuc D .
ExSekageGLasÞic³ RbsinebIeKKUrbnÞat;b:HRtg;cMnuc B nigRtg;cMnuc C ¬rUbTI 8-21c¦ eKGackMNt;mMu
rgVil B edayrk tC / B nigsMrab;mMutUc
tC / B
θB = (1)
2m
RTwsþIRkLaépÞm:Um:g;³ edIm,Irk t B / C eyIgGnuvtþRTwsþIRkLaépÞm:Um:g;edayrkm:Um:g;RkLaépÞrbs;düaRkam
M / EI cenøaH BC Rtg;cMnuc C . m:Um:g;RkLaépÞEdleyIg)anrkCam:Um:g;énRkLaépÞEdlqUtrbs;düaRkam

BIrenAkñúgrUbTI 8-21b. dUcenH
⎡ ⎛ 30kN .m ⎞⎤ ⎛ 2m ⎞ ⎡ 1 ⎛ 10kN .m ⎞⎤
t C / B = (1m )⎢(2m )⎜ − ⎟⎥ + ⎜ ⎟ ⎢ (2m )⎜ ⎟⎥
⎣ ⎝ EI ⎠⎦ ⎝ 3 ⎠ ⎣ 2 ⎝ EI ⎠⎦
53.33kN .m 3
=
EI
edayCMnYslT§plxagelIeTAkñúgsmIkar (1) eyIg)an

PaBdab T.Chhay -266


53.33kN .m 3
θB =
(2m )[200(10 6 )kN / m 3 ][18(10 6 )(10 −12 )m 4 ]
= 0.00741rad

*>%> viFIFñwmnimitþ (Conjugate-beam method)
vIFIFñwmnimitþRtUv)anbegáItedaym‘uyeyeRbsøÚ enAkñúgqñaM 1865. CaBiess vatMrUveGaykar
KNnamanTMhMesμIKñanwgRTwsþIRkLaépÞm:Um:g;edIm,IkMNt;mMurgVil b¤PaBdabrbs;Fñwm b:uEnþviFIenHk¾QrelI
eKalkarN_sþaTic EdleyIgGacedaHRsayedaygay.

eKalkarN_sMrab;TTYl)anviFIenHRsedogKñanwgsmIkar 4-1 nigsmIkar 4-2 nigBIsmIkar 8-2
dl;smIkar 8-4. edIm,IbgðajnUvPaBdUcKña eyIgnwgsresrsmIkarTaMgenHdUcxageRkam
dV
=w d 2M
=w
dx dx 2
dθ M d 2v M
= =
dx EI dx 2 EI
edayeFVIGaMgetRkal eyIgTTYl)an
V = ∫ wdx [
M = ∫ ∫ wdx dx ]
b b b b
⎛M ⎞ ⎡ ⎛M ⎞ ⎤
θ = ∫ ⎜ ⎟dx ν = ∫ ⎢ ∫ ⎜ ⎟dx ⎥dx
⎝ EI ⎠ ⎣ ⎝ ⎠ ⎦
EI

enATIenH kMlaMgkat; V manlkçN³RsedognwgmMurgVilθ m:Um:g;Bt; M manlkçN³RsedognwgbMlas;Tν I
ehIybnÞúkxageRkAmanlkçN³RsedognwgdüaRkam M / EI . edIm,Iyl;BIkarRbdUcenH eyIgnwgBicarNa
FñwmEdlmanRbEvgdUcKñanwgFñwmBit EdleKeGayeQμaHfaFñwmnimitþ (conjugate beam) enAkñúg rUbTI 8-



22. FñwmnimitþrgnUvbnÞúkéndüaküam M / EI EdldUcKñanwgbnÞúk w EdlmanGMeBIenAelIFñwmBit. BIkarRb
dUcxagelI eyIgGaceBalnUvRTwsþIBIrEdlTak;TgeTAnwgFñwmnimitþKW
RTwsþITI1³ mMurgVilenARtg;cMnucmYyenAelIFñwmBitmantMélCaelxesμInwgkMlaMgkat;enARtg;cMnucRtUvKñaenA
elIFñwmnimitþ.
RTwsþITI2³ bMlas;TIrbs;cMnucmYyenAelIFñwmBitmantMélCaelxesμInwgm:Um:g;Bt;enARtg;cMnucRtUvKñaenAelI
Fñwmnimitþ.

TMrrbs;Fñwmnimitþ³ enAeBlKUrFñwmnimitþeKcaM)ac;RtUveFVIeGaykMlaMgkat; nigm:Um:g;Bt;EdlekItmanenARtg;
TMrrbs;FñwmnimitþRtUvKñanwgmMurgVil nigPaBdabrbs;FñwmBitenARtg;TMrrbs;vaEdr edayeyagtamRTwsþITI1
nigRTwsþITI2. Ca]TahrN_ dUcEdl)aneXIjenAkñúgtarag 8-2 TMrsnøak; nigTMrkl;enARtg;cugrbs;Fñwm
Bitpþl;nUvbMlas;TIsUnü b:uEnþmMurgVilrbs;vaminsUnü. Cavi)ak BIRTwsþITI1 nigRTwsþITI2 FñwmnimitþRtUvEtRT
PaBdab T.Chhay -268


edayTMrsnøak; nigTMrkl;; edaysarTMrTaMgenHmanm:Um:g;sUnü EtmankMlaMgkat; b¤kMlaMgRbtikmμ. enA
eBlFñwmBitRtUv)anRTedayTMrbgáb; (3) TaMgmMurgVil nigTaMgbMlas;enARtg;TMresμIsUnü. enATIenH Fñwm
nimitþmancugesrI ¬KμanTMr¦ edaysarenAxagcugmankMlaMgkat;sUnü nigm:Um:g;Bt;sUnü. TMrRtUvKñarvagFñwm
Bit nigFñwmnimitþsMrab;krNIepSgeTotmanbgðajenAkñúgtarag. ]TahrN_énFñwmBit nigFñwmnimitþRtUv)an
bgðajenAkñúgrUbTI 8-23. cMNaMfa tamk,Ünc,ab; ¬edayecalkMlaMgtamG½kS¦ FñwmBitkMNt;eday
sþaTicmanFñwmnimitþkMNt;edaysþaTic. cMENkÉFñwmBitminkMNt;edaysþaTicEdlbgðajenAkñúgkrNIcug
eRkayénrUbTI 8-23 køayCaFñwmnimitþKμanesßrPaB. eTaHbICakrNIenHekIteLIgk¾eday k¾bnÞúk M / EI
nwgpþl;nUvlMnwgedIm,IeFIVeGayFñwmmansißrPaB.

CMhanxageRkampþl;nUvviFIEdlGaceRbIedIm,IkMNt;bMlas;TI nigmMurgVilRtg;cMnucmYyenAelIExS
ekageGLasÞicrbs;FñwmedayeRbIviFIFñwmnimitþ.
Fñwmnimitþ
KUrFñwmnimitþsMrab;FñwmBit. FñwmenHRtUvmanRbEvgdUcKñanwgFñwmBit nigmanTMrRtUvKñadUcbgðajenAkñúg
tarag 8-2.
CaTUeTA RbsinebITMrrbs;FñwmBitmanmMurgVil enaHTMrrbs;FñwmnimitþRtUvmankMlaMgkat; ehIyebITMr
rbs;FñwmBitmanPaBdab enaHTMrrbs;FñwmnimitþRtUvEtmanm:Um:g;Bt;.



FñwmnimitþRtUvrgnUvbnÞúkEdlCadüaRkam M / EI rbs;FñwmBit. eKsnμt;bnÞúkenHBRgayelIFñwm
nimitþ nigmanTisedAeLIgelIenAeBl M / EI viC¢man ehIymanTisedAcuHeRkamenAeBl M / EI
GviC¢man. eKGacniyaymüa:geTotfa bnÞúkmanGMeBIecjBIFñwm.
smIkarlMnwg
edayeRbIsmIkarlMnwg kMNt;kMlaMgRbtikmμenARtg;TMrrbs;Fñwmnimitþ.
kat;FñwmnimitþenARtg;cMnucEdleKRtUvkarrkmMurgVilθ nigbMlas;TI Δ rbs;FñwmBit. enARtg;muxkat;
eKRtUvbgðajkMlaMgkat;V ' nigm:Um:g;Bt; M ' EdlCaGBaØatEdleFVIkarkñúgTisedAviC¢man.
kMNt;kMlaMgkat; nigm:Um:g;Bt;edayeRbIsmIkarlMnwg. V ' nig M ' esμInwgθ nig Δ erogKña sMrab;
FñwmBit. RbsinebItMélTaMgenHviC¢man mMurgVilKWvilRcasRTnicnaLika ehIybMlas;TImanTis
eLIgelI.

]TahrN_ 8-12³ kMNt;mMurgVil nigPaBdabenARtg;cMnuc B rbs;FñwmEdkdUcbgðajenAkñúgrUbTI 8-24a.
kMlaMgRbtikmμRtUv)anKNnaeGay. yk E = 200GPa / I = 475(106 )mm 4 .

dMeNaHRsay³
Fñwmnimitþ³ FñwmnimitþRtUv)anbgðajenAkñúgrUbTI 8-24b. TMrenARtg; A' nig B' RtUvKñanwgTMr A nigTMr B enA
elIFñwmBit ¬tarag 8-2¦. eKcaM)ac;RtUvdwgBImUlehturbs;va. düaRkam M / EI GviC¢man dUcenHbnÞúk
BRgayeFVIGMeBIcuHeRkam eBalKWecjBIFñwm.
smIkarlMnwg³ edaysareKRtUvkarkMNt; θ B nig Δ B dUcenHeyIgRtUvKNnaVB' nig M B' enAkñúgFñwmnimitþ
¬rUbTI 8-24c¦.
PaBdab T.Chhay -270


250kN .m 2
+ ↑ ∑ Fy = 0 − − VB ' = 0
EI
250kN .m 2
θ B = VB ' = −
EI
− 250kN .m 2
=
[200(10 )kN / m ][475(10 )(10 )m ]
6 2 6 −12 4

= −0.00263rad
250kN .m 2
+ ∑ M B' = 0 (8.33m ) + M B ' = 0
EI
2083kN .m 3
Δ B = M B' = −
EI
− 2083kN .m 3
=
[ ( ) ][ ( )(
200 10 6 kN / m 2 475 10 6 10 −12 m 4 ) ]
= −0.0219m = −21.9mm
sBaØaGviC¢manbgðajfamMurgVilrbs;FñwmRtUv)anvas;RsbRTnicnaLika ehIybMlas;TImanTiscuHeRkam
¬rUbTI 8-24d¦.

]TahrN_ 8-13³ kMNt;PaBdabGtibrmarbs;FñwmEdkdUcbgðajenAkñúgrUbTI 8-25a. kMlaMgRbtikmμ
RtUv)anKNnaeGay. yk E = 200GPa / I = 60(106 )mm 4 .

dMeNaHRsay³
Fñwmnimitþ³ FñwmnimitþRtUvrgbnÞúkéndüaRkam M / EI dUcEdlbgðajenAkñúgrUbTI 8-25b. edaysardüa-
Rkam M / EI viC¢man bnÞúkBRgayeFVIGMeBIeLIgelI ¬ecjBIFñwm¦.


smIkarlMnwg³ dMbUg eKRtUvkMNt;kMlaMgRbtikmμxageRkAenAelIFñwmnimitþ nigbgðajvaenAelIdüaRkam
GgÁesrIenAkñúgrUbTI 8-25c. PaBdabGtibrmarbs;FñwmBitekItmanenARtg;cMnucEdlmMurgVilrbs;FñwmesμI
sUnü. vaRtUvKñanwgcMnucenAelIFñwmnimitþEdlmankMlaMgesμIsUnü. edaysnμt;facMnucmanGMeBIenAkñúgtMbn;
0 ≤ x ≤ 9m BIcMnuc A' s eyIgGackat;Fñwm nigBinitüEpñkxagsþaMénFñwmdUcbgðajenAkñúgrUbTI 8-25d.

cMNaMfa eKRtUv KNnakMBUlrbs;bnÞúkBRgayBIRtIekaNdUc Edl w / x = (18 / EI ) / 9 . eyIgtMrUveGay
V ' = 0 Edl
45 1 ⎛ 2 x ⎞
+ ↑ ∑ Fy = 0 − + ⎜ ⎟x = 0
EI 2 ⎝ EI ⎠
x = 6.71m (0 ≤ x ≤ 9m ) O.K.
edayeRbItMélenHsMrab; x PaBdabGtibrmaenAkñúgFñwmBitRtUvnwgm:Um:g;Bt; M ' . dUcenH
+ ∑M = 0
45
(6.71) − ⎡ 1 ⎛ 2(6.71) ⎞6.71⎤ 1 (6.71) + M ' = 0
⎢ 2 ⎜ EI ⎟ ⎥3
EI ⎣ ⎝ ⎠ ⎦
201.2kN .m 3
Δ max = M ' = −
EI
− 201.2kN .m 3
=
[ ( ) ⎡
⎣
] ( ) ⎜
⎝
( )
200 10 6 kN / m 2 ⎢60 10 6 mm 4 ⎛1m 4 / 10 3 mm 4 ⎞⎥
4
⎟
⎠⎦
⎤

= −0.0168m = −16.8mm
sBaØaGviC¢manbgðajfaPaBdabcuHeRkam.

]TahrN_ 8-14 ³ r:tenAkñúgrUbTI 8-26a Car:tCab; nigeFVIBIebtug ehIyenAkNþalElVgRtUv)anGb
edaybnÞHEdkedaysarm:Um:g;Bt;enARtg;kEnøgenHmantMélFM. kMNat;Ggát;cugsgxagEdlmanRbEvg
4m manm:Um:g;niclPaB I = 270( 6 )mm 4 ehIykMNat;Ggát;EdlenAkNþalman I = 540( 6 )mm 4 .
10 10

kMNt;PaBdabenARtg;kNþalElVg. yk E = 200GPa . kMlaMgRbtikmμRtUv)anKNnaeGay.
dMeNaHRsay³
Fñwmnimitþ³ dMbUgeKRtUvkMNt;düaRkamm:Um:g;sMrab;Fñwm dUcbgðajenAkñúgrUbTI 8-26b. edaysar M / EI
nimitþRtUvrgbnÞúkéndüaRkam I ' = 2I edIm,ICakargayRsYl eyIgGacsMEdgbnÞúkenAelIFñwmnimitþeday
eRbItY EI efr dUcbgðajenAkñúgrUbTI 8-26c.
smIkarlMnwg³ eKGacKNnakMlaMgRbtikmμenAelIFñwmnimitþedayPaBsIuemRTIrbs;bnÞúk b¤edayeRbIsmIkar
lMnwg. lT§plRtUv)anbgðajenAkñúgrUbTI 8-26d. edaysareKRtUvkarkMNt;PaBdabenARtg;cMnuc C

PaBdab T.Chhay -272


dUcenHeyIgRtUvKNnam:Um:g;Bt;enARtg;cMnuc C ' . edayeRbIviFImuxkat; eKRtUvpþac;kMNat;Ggát; A' C ' ehIy
eKRtUvkMNt;kMlaMgpÁÜbénkMlaMgBRgay nigTItaMgrbs;va ¬rUbTI 8-26e¦. dUcenH

+ ∑ MC = 0
620
(6) − 400 (3.333) − 200 (1) − 20 (0.667 ) + M C ' = 0
EI EI EI EI
3
2173kN .m
M C' = −
EI
edayCMnYstMélCaelxén EI eyIgTTYl)an
− 2173kN .m 3
Δ C = M C' =
[200(10 )kN / m ][270(10 )(10 )m ] = −0.0402m = −40.2mm
6 2 6 −12 4

sBaØaGviC¢manbgðajfaPaBdabmanTiscuHeRkam.


]TahrN_ 8-15 ³ kMNt;bMlas;TIrbs;snøak;enARtg;cMnuc B nigmMurgVilrbs;kMNat;FñwmnImYy²Edl
)anP¢ab;enARtg;snøak;énFñwmpSMEdlbgðajenAkñúgrUbTI 8-27a.

dMeNaHRsay³
Fñwmnimitþ³ ExSekageGLasÞicsMrab;FñwmRtUv)anbgðajenAkñúgrUbTI 8-27b edIm,IdwgBIbMlas;TI Δ B nig
mMurgVil (θ B )L nig (θ B )R enAxagsþaM nigenAxageqVgrbs;snøak;. edayeRbItarag 8-2 FñwmnimitþRtUv)an
bgðajenAkñúgrUbTI 8-27c. edIm,IsMrYldl;karKNna eKKUrdüaRkam M / EI CaEpñkedayeRbIeKal-
karN_tMrYtpl dUckatBN’naenAkñúgkfaxNÐ4-5. cMeBaHkrNIenH eKKitFñwmBitCaFñwmkugs‘ulEdlmanTMr
bgáb;enAcugxageqVg A . düaRkamm:Um:g;sMrab;bnÞúk 40kN kMlaMgRbtikmμ C y = 10kN nigkMlaMgm:Um:g;
50kN .m RtUv)aneGay. cMNaMfa tMbn;GviC¢manrbs;düaRkamenHbegáItbnÞúkBRgaycuHeRkam ehIytMbn;

viC¢manmanbnÞúkBRgayEdlmanGMeBIeLIgelI.

PaBdab T.Chhay -274


smIkarlMnwg³ dMbUgeKRtUvKNnakMlaMgRbtikmμenARtg;cMnuc B' nig C ' ehIylT§plRtUv)anbgðajenAkñúg
rUbTI 8-27d. edIm,IkMNt; (θ B )R eKRtUvkat;FñwmnimitþenAxagsþaMcMnuc B' bnþic ehIyKNnakMlaMgkat;
(V B ' ) ¬rUbTI 8-27e¦. dUcenH
+ ↑ ∑ Fy = 0 (V B ' )R + 125 − 250 − 2
=0
EI EI EI
127kN .m 2
(θ B )R = (V B ' )R =
EI
127kN .m 2
=
[200(10 )kN / m ][18(10 )(10 )m ] = 0.0353rad
6 2 6 −12 4

m:Um:g;Bt;enARtg;cMnuc B' eGayCabMlas;TIénsnøak;. dUcenH
+ ∑ M B' = 0 − M B' +
125
(1.67 ) − 250 (2.5) − 2 (5) = 0
EI EI EI
3
427kN .m 427 kN .m 3
Δ B = M B' = −
EI
=−
[ ( ) ][ ( )(
200 10 6 kN / m 2 18 10 6 10 −12 m 4 ) ]
= −0.119m = −119mm
eKGacrkmMurgVilBIkarkat;FñwmenAxageqVgcMnuc B' bnþic ¬rUbTI 8-27f¦. dUcenH
+ ↑ ∑ Fy = 0 (V B ' )L + 127 + 125 − 250 − 2
=0
EI EI EI EI
(θ B )L = (V B ' )L = 0
eyIgeXIjy:agc,as;fa Δ B = M B' sMrab;kMNat;Ggát;enHmantMélesμInwgtMélEdl)anKNnaBIelIkmun
edaysarédXñas;rbs;vaenAkñúgrUbTI 8-27e nigenAkñúgrUbTI 8-27f xusKñaEtbnþicb:ueNÑaH.



cMeNaT
8>1> kMNt;smIkarExSekageGLasÞicrbs;Fñwm
edayeRbIkUGredaen x Edlmann½ysMrab;
0 ≤ x < L / 2 . kMNt;mMurgVilenARtg;cMnuc A nig

PaBdabGtibrmarbs;Fñwm. EI CatMélefr.
8>5> kMNt;smIkarExSekageGLasÞicrbs;Fñwmkug
s‘ulEdlrgkMlaMgm:Um:g; M 0 . ehIykMNt;mMurgVil
Gtibrma nigPaBdabGtibrmarbs;Fñwm. EI Ca
tMélefr.
edayeRbIkUGredaen x1 nig x2 . EI CatMélefr.

8>6> kMNt;PaBdabGtibrmarbs;Fñwm nigmMurgVil
Rtg;cMnuc A edayeRbIviFIGaMgetRkalDub. EI Ca
tMélefr.

edayeRbIkUGredaen x1 nig x2 . kMNt;PaBdab
Gtibrmarbs;Fñwm. EI CatMélefr. 8>7> kMNt;smIkarExSekageGLasÞicedayeRbI
kUGredaen x ehIykMNt;mMurgVilRtg;cMnuc A nig
PaBdabRtg;cMnuc C . EI Ca tMélefr.

edayeRbIkUGredaen x1 nig x2 . kMNt;mMurgVilRtg;
8>8> kMNt;smIkarExSekageGLasÞicrbs;FñwmTMr
cMnuc A nigPaBdab Gtibrmarbs;Fñwm. EI Ca
samBaØEdlrgkMlaMgm:Um:g; M 0 . ehIykMNt;
tMélefr.
PaBdab T.Chhay -276


mMurgVilGtibrma nigPaBdabGtibrmarbs;Fñwm.
EI CatMélefr.

8>13> FñwmrgbnÞúkdUcbgðaj. eRbIRTwsþIRkLaépÞ
m:Um:g;edIm,IkMNt;mMurgVilRtg;cMnuc A nigbMlas;TI
8>9> kMNt;mMurgVilGtibrma nigPaBdabGtibrma
enARtg;cMnuc B . snμt; A CaTMrsnøak; nig B CaTMr
rbs;FñwmTMr samBaØEdlrgkMlaMgm:Um:g; M 0 eday
kl;. EI CatMélefr.
eRbIviFIGaMgetRkalDub. EI CatMélefr.

8>10> eRbIRTwsþIRkLaépÞm:Um:g;edIm,IkMNt;mMurgVil
nigPaBdabRtg;cMnuc B . EI CatMélefr. 8>14> FñwmrgbnÞúk P dUcbgðaj. eRbIRTwsþIRkLa
épÞm:Um:g;edIm,IkMNt;TMhMkMlaMg F EdlRtUvdak;enA
Rtg;cugénFñwmlyRtg;cMnuc C edIm,IeFVIeGaybM-
las;TIRtg;cMnuc C esμIsUnü. EI CatMélefr.
8>11> kMNt;mMurgVilRtg;cMnuc B nigPaBdabGti-
brmarbs;Fñwm. snμt; A CaTMrkl; nig B CaTMr
snøak;. yk E = 200GPa / I = 195(106 )mm 4 .
eRbIRTwsþIRkLaépÞm:Um:g;.
8>15> sMrab;FñwmkñúgcMeNaT 8>15 RbsinebI
F = P kMNt;bMlas;TIRtg;cMnuc D edayeRbI

RkLaépÞm:Um:g;. EI CatMélefr.
8>16> kMNt;mMurgVilRtg;cMnuc B nigPaBdab
8>12> FñwmrgbnÞúkBIr. eRbIRTwsþIRkLaépÞm:Um:g; Gtibrmarbs;Fñwm. yk E = 200GPa /
edIm,IkMNt;mMurgVil nigbMlas;TIenARtg;cMnuc A nig I = 200( 6 )mm 4 . eRbIRTwsþIRkLaépÞm:Um:g;.
10

cMnuc B . EI CatMélefr.
Problems T.Chhay -277


enARtg;cMnuc B nigPaBdabenARtg;cMnuc C . EI
CatMélefr.

8>17> etIeKRtUvdak;TMr A nigTMr B enARtg;cMgay
a y:agNaedIm,IeGayPaBdabenAkNþalElVg

rbs;FñwmesμInwgPaBdabenARtg;xagcugrbs;va? 8>21> eRbIRTwsþIRkLaépÞm:Um:g;edIm,IkMNt;PaB
eRbIRTwsþIRkLaépÞm:Um:g;. TMrGnuvtþEtkMlaMg dabRtg;cMnuc C nigmMurgVilrbs;FñwmRtg;cMnuc A /
RbtikmμbBaÄrelIFñwmb:ueNÑaH. EI CatMélefr. B nig C . EI CatMélefr.

8>18> FñwmrgbnÞúkdUcbgðaj. eRbIRTwsþIRkLaépÞ Rtg;cMnuc C nigPaBdabRtg;cMnuc B . EI Ca
m:Um:g;edIm,IkMNt;mMurgVilenARtg;cMnuc B nigPaBdab tMélefr.
enARtg;cMnuc C . EI CatMélefr.

8>23> eRbIRTwsþIRkLaépÞm:Um:g;edIm,IkMNt;tMél
8>19> FñwmrgbnÞúkdUcbgðaj. RbsinebITMrenARtg; rbs; a EdleFVIeGaymMurgVilRtg; A esμIsUnü.
A nig B GnuvtþEtkMlaMgRbtikmμbBaÄreTAelIFñwm EI Ca tMélefr.

cUrkMNt;mMurgVilenARtg;cMnuc A nigbMlas;TIenA
Rtg;cMnuc C . eRbIRTwsþIRkLaépÞm:Um:g;. EI Ca
tMélefr.

cMeNaT T.Chhay -278


Rtg;cMnuc C nigbMlas;TIRtg;cMnuc B . EI Ca
tMélefr.
8>29> eRbIviFIFñwmnimitþedIm,IkMNt;bMlas;TIRtg;
C nigmMurgVilRtg; A / B nig C . EI CatMélefr.

Rtg;cMnuc B nigbMlas;TIRtg;cMnuc C . Ggát;enHeFVI
GMBIEdk A-36 Edlman I = 32(106 )mm 4 .
8>30> eRbIviFIFñwmnimitþedIm,IkMNt;mMurgVilRtg;
B nig bMlas;TIRtg; C . EI CatMélefr.

8>26> eRbIRTwsþIRkLaépÞm:Um:g;edIm,IkMNt;bMlas;
TIRtg;cMnuc B nig mMurgVilRtg;cMnuc A . EI Ca
tMélefr. 8>31> eRbIviFIFñwmnimitþedIm,IkMNt;mMurgVilnig
bMlas;TIRtg; C . E = 200GPa /
I = 70(10 6 )mm 4 .

8>27> eRbIRTwsþIRkLaépÞm:Um:g;edIm,IkMNt;bMlas;
TIRtg;cMnuc C . yk E = 200GPa / I =
500( 6 )mm 4 .
10

8>32> eRbIviFIFñwmnimitþedIm,IkMNt;mMurgVilnig
bMlas;TIRtg; C . snμt; A CaTMrsnøak; nig B Ca
TMrkl;. EI Ca tMélefr.
8>28> eRbIviFIFñwmnimitþedIm,IkMNt;mMurgVil nig
bMlas;TIRtg; A . EI CatMélefr.

Problems T.Chhay -279


8>33> eRbIviFIFñwmnimitþedIm,IkMNt;mMurgVilnig
bMlas;TIRtg; C . E = 200GPa /
D nigmMurgVilRtg; C . snμt;faTMrRtg; A CaTMr
I = 400( 6 )mm 4 .
10
bgáb; nig C CaTMrkl;. EI CatMélefr.

A . snμt; B CaTMrkl;. E = 200GPa /
8>38> eRbIviFIFñwmnimitþedIm,IkMNt;mMurgVilEdlenA
I = 80(10 6 )mm 4 .
xagsþaM nigxageqVgrbs; B bnþic. ehIykMNt;
PaBdabenARtg; D . snμt;faFñwmenHmanTMrbgáb;
enARtg; A nig C CaTMrkl;. EI CatMélefr.

8>35> eRbIviFIFñwmnimitþedIm,IkMNt;mMurgVilRtg; C
nigbMlas;TIRtg; B . EI CatMélefr.

B nigmMurgVilRtg; A . snμt;faTMrRtg; A CaTMr

snøak; nig C CaTMrkl;. EI CatMélefr.

cMeNaT T.Chhay -280

8. deflection

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8. deflection