12. displacement method of analysis moment distribution

mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa

!@> karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g;
Displacement method of analysis: Moment distribution

viFIEbgEckm:Um:g;CakarviPaKtamviFIbMlas;TIEdlgayRsYlkñúgkarGnuvtþenAeBlEdleKkMNt;
efreGLasÞic (elastic constant). CMBUkenHnwgbgðajCadMbUgnUvniymn½y nigeKalKMnitsMxan;² sRmab;
viFIEbgEckm:Um:g; ehIybnÞab;mkGnuvtþviFIenHedIm,IedaHRsaycMeNaTEdlBak;B½n§nwgFñwm nig eRKagmin
kMNt;edaysþaTic. karGnuvtþcMeBaHeRKageRcInCab;RtUv)anbgðajenAkñúgCMBUkcugeRkay.

!@>!> eKalkarN_TUeTA nigniymn½y (General principles and definitions)
karsikSaviPaKeRKOgFñwm nigeRKagEdleRbIviFIEbgEckm:Um:g;EdlbegáIteLIgedayhaDIRkUs
(Hardy Cross) enAkñúgqñaM1930. enAeBlenaH viFIenHTTYlkarcab;GarmμN_Pøam²enAeBlEdleKpSBV

pSayvaCaelIkdMbUg ehIyeKTTYlsÁal;favaCaviFIEdlmankareCOnelOneTAmuxKYreGaykt;sMKal;kñúg
karviPaKeRKOgbgÁúMenAkñúgstvtSTI20.
viFIEbgEckm:Um:g;CaviFIRbhak;RbEhlEdleKGacTTYl)ankMritsuRkiteTAtamtMrUvkarEdleKcg;
)an. viFIenHcab;epþImedaysnμt;tMNnImYy²rbs;eRKagCatMNbgáb;. bnÞab;mkedaykarRsay nigkar
Tb;tMNnImYy²bnþbnÞab;Kña enaHm:Um:g;Bt;enARtg;tMNRtUv)anEbgEck nigRtUv)aneFIVeGaymantMélesμIKña
rhUtdl;tMNenaHvildl; b¤esÞIrdl;TItaMgcugeRkay. eyIgnwgeXIjfaviFIenHmanlkçN³sarcuHsareLIg
niggayRsYlkñúgkarGnuvtþ. b:uEnþ munnwgBnül;BIbec©keTsénviFIEbkEckm:Um:g; eyIgRtUvdwgBIniymn½y
nigeKalKMnitxøH.
karkMNt;sBaØa³ eyIgnwgbegáItkarkMNt;sBaØadUcKñanwgkarkMNt;sBaØasMrab;smIkar slope-deflection
Edlm:Um:g;viC¢manCam:Um:g;EdleFVVIeGayGgát;vilRsbRTnicnaLika ehIym:Um:g;GviC¢manCam:Um:g;EdleFVI
eGayGgát;vilRcasRTnicnaLika ¬rUbTI 12-1¦.

m:Um:g;bgáb;cug (FEMs)³ m:Um:g;enARtg;CBa¢aMg b¤enARtg;tMNbgáb;énGgát;rgbnÞúkRtUv)aneKehAfam:Um:g;
bgáb;cug (fixed-end moment). eKGackMNt;m:Um:g;TaMgenHBItaragenAkñúgemeronTI11 m:Um:g;bgáb;cug
Displacement method of analysis: Moment distribution T.Chhay -415

Department of Civil Engineering NPIC

GaRs½yeTAnwgRbePTénkardak;bnÞúk. Ca]TahrN_ FñwmEdlrgbnÞúkdUcbgðajenAkñúgrUbTI 12-2 man
m:Um:g;bgáb;cug FEM = PL / 8 = 800(10) / 8 = 1000 N .m . eKRtUvcMNaMGMBIGMeBIrbs;m:Um:g;TaMgenHmkelI
Fñwm nigkarkMNt;sBaØarbs;eyIg dUcenHeyIgeXIjfa M AB = −1000 N .m ehIy M BA = +1000 N .

emKuNkMrajsMrab;Ggát; (member stiffness factor)³ eKmanFñwmdUcbgðajenAkñúgrUbTI 12-3 Edl
cugmçagRtUv)anRTedayTMrsnøak; nigmçageTotRTedayTMrbgáb;. GMeBIrbs;m:Um:g; M eFVIeGaycug A vil
)anmMurgVil θ A . enAkñúgemeronTI11 eyIg)anP¢ab;TMnak;TMngrvag M nig θ A edayeRbIviFIFñwmnimitþ Edl
M = (4 EI / L )θ A enAkñúgsmIkar 11-1. tYenAkñúgrg;Rkck
4 EI
K= (12-1)
L
cugbgáb;q¶ay
RtUv)aneKeGayeQμaHfaemKuNkMrajenARtg; A . emKuNkMrajCabrimaNénm:Um:g; M EdlcaM)ac;edIm,I
eFVIeGaycug A rbs;Fñwmvil)an θ A = 1rad .

emKuNkMrajsMrab;tMN (joint stiffness factor)³ RbsinebIGgát;CaeRcInRtUv)antP¢ab;edaytMNbgáb;
ehIycugq¶ayrbs;vaCaTMrbgán; enaHedayeKalkarN_tMrYtpl emKuNkMralsrubenARtg;tMNCaplbUkén
emKuNkMrajsMrab;Ggát;enARtg;tMN eBalKW KT = ∑ K . Ca]TahrN_ cUrBicarNatMNrbs;eRKagRtg;
A enAkñúgrUbTI 12-4a. eKkkMNt;témøCaelxrbs;emKuNkMrajsMrab;Ggát;nImYy²BIsmIkar 12-1

ehIybgðajvaenAelIrUb. edayeRbItémøTaMgenH emKuNkMrajsrubéntMN A KW KT = ∑ K = 4000 +
5000 + 1000 = 10000 . TMhMenHCabrimaNénm:Um:g;EdlRtUvkaredIm,IbgViltMNeday)anmMu 1rad .

emKuNEbgEck (Distribution factor DF)³ RbsinebIeKGnuvtþm:Um:g; M eTAelItMNbgáb; enaHGgát;
karviPaKtamviFIbMlas;TI³ karEbgEckm:Um:g; T.Chhay -416


EdltP¢ab;eTAnwgtMNenaHnwgpþl;nUvcMENkénm:Um:g;Tb;caM)ac;edIm,IbMeBjlMnwgrbs;m:Um:g;enARtg;tMN.
cMENkénm:Um:g;Tb;srubEdlpþl;eGayedayGgát;RtUv)aneKeGayeQμaHfaemKuNEbgEck. edIm,ITTYl
)antémøTaMgenH BinitüemIltMNEdlP¢ab;eday n Ggát;. RbsinebIm:Um:g; M eFVIeGaytMNvil)an θ enaH
Ggát; i nImYy²nwgviledaybrimaNdUcKñaenH. RbsinebIemKuNkMrajrbs;Ggát;TI i Ca K i enaHm:Um:g;Edl
cUlrYmedayGgát;enaHKW M i = K iθ . edaysarsmIkarlMnwgRtUvkar M = M1 + M n = K1θ + K nθ =
θ ∑ K i enaHemKuNEbgEcksRmab;Ggát;TI i KW
Mi K iθ
DFi = =
M θ ∑ Ki
edaysRmYltY θ eyIgeXIjfaemKuNEbgEcksRmab;Ggát;esμInwgplEckénemKuNkRmajrbs;Ggát;elI
emKuNkRmajsrubsMrab;tMN EdlCaTUeTAeKsresr
K
DF = (12-2)
∑K
Ca]TahrN_ emKuNEbgEcksMrab;Ggát; AB / AC nig AD enARtg;tMN A ¬enAkñúgrUbTI 12-4a¦ KW
DFAB = 4000 / 10000 = 0.4
DFAC = 5000 / 10000 = 0.5

DFAD = 1000 / 10000 = 0.1
CalT§pl RbsinebI M = 2000 N .m eFVIGMeBIenARtg;cMNuc A ¬rUbTI 12-4b¦ m:Um:g;lMnwgEdlGnuvtþ
edayGgát;mkelItMN ¬enAkñúgrUbTI 12-4c¦ KW
M AB = 0.4(2000 ) = 800 N .m
M AC = 0.5(2000) = 1000 N .m

M AD = 0.1(2000 ) = 200 N .m

emKuNkRmajeFobrbs;Ggát; (member relative-stiffness factor)³ eBlxøH Fñwm b¤eRKagCab;RtUv
)aneFVIBIsmÖar³EtmYy dUcenHGgát;rbs;vaTaMgGs;RtUvEtmanm:UDuleGLasÞic E dUcKña. kñúgkrNIEbbenH



eKGacminKitemKuN 4E EdlmanenAkñúgsmIkar 12-1 BIPaKykénsmIkar 12-2 enAeBlkMNt;emKuN
EbgEcksMrab;tMN. dUcenH eKmanlkçN³gayRsYledayRKan;EtkMNt;emKuNkRmajeFobrbs;Ggát;
I
KR = (12-3)
L
cugbgáb;q¶ay
ehIyeRbIvasRmab;KNnaemKuNEbgEck.
emKuNdMENl (carry-over factor)³ BicarNaFñwmenAkñúgrUbTI 12-3 mþgeTot. enAkñúgemeronTI 11
eyIgeXIjfa M AB = (4EI / L)θ A ¬smIkar 11-1¦ ehIy M BA = (2EI / L)θ A ¬smIkar 11-2¦.
edayedaHRsayrk θ A ehIydak;eGaysmIkarTaMgenHesμIKñaeyIg)an M BA = M AB / 2 . eyIgGac
niyayfa m:Um:g; M enARtg;TMrsnøak;eFVIeGaymanm:Um:g; M ' = M / 2 enARtg;CBa¢aMg. emKuNdMENlCa
cMENkén M EdlRtUv)anepÞrBIsnøak;eTACBa¢aMg. dUcenH enAkñúgkrNIrbs;FñwmEdlmancugq¶ayCaTMrbgáb;
emKuNdMENlKW + 1 / 2 . sBaØabUkmann½yfam:Um:g;TaMgBIreFVIkarkñúgTisedAdUcKña.

!@>@> karEbgEckm:Um:g;sRmab;Fñwm (Moment distribution for beams)
karEbkEckm:Um:g;KWQrelIeKalkarN_Rsay nigTb;tMNrbs;rcnasm<½n§bnþbnÞab;KñaedIm,IGnuBaØat
eGaymankarEbgEckm:Um:g; nigeFVIeGaym:Um:g;mantulüPaBenARtg;tMN. meFüa)ayd¾l¥kñúgkarBnül;viFI
KWkarBnül;edayeRbI]TahrN_.
eKmanFñwmmYyEdlmanm:UDuleGLasÞic E efr nigmanTMhM nigbnÞúkdUcbgðajenAkñúgrUbTI 12-6a.
munnwgeyIgcab;epþIm dMbUgeyIgRtUvKNnaemKuNEbgEckenARtg;cugTaMgBIrrbs;ElVgnImYy².
edayeRbIsmIkar 12-1 ¬ K = 4EI / L ¦ emKuNkRmajenAEpñksgxagrbs; B KW
4 E (120)( 6 )
= 4 E (40)( 6 )mm 4 / m
10
K BA = 10
3
4 E (240 )(10 6 )
K BC = = 4 E (60)(106 )mm 4 / m
4
dUcenH edayeRbIsmIkar 12-2 DF = K / ∑ K sRmab;cugEdltP¢ab;eTAniwgtMN B eyIg)an
4 E (40)
DFBA = = 0.4
4 E (40) + 4 E (60 )
4 E (60)
DFBC = = 0.6
4 E (40 ) + 4 E (60)
enARtg;CBa¢aMg ¬tMN A nigtMN C ¦ emKuNEbgEckGaRs½ynwgemKuNkRmajrbs;Ggát; nigemKuN
kRmajrbs;CBa¢aMg. edaysarenAkñúgRTwsþI eKRtUvkarm:Um:g;EdlmanTMhMGnnþedIm,IbgVilCBa¢aMgeGay)an


mYyra:düg; ¬emKuNkRmajCBa¢aMgKWGnnþ¦. dUcenHsRmab;tMN A nig C eyIgman
4 E (40)
DFAB = =0
∞ + 4 E (40)
4 E(60)
DFCB = =0
∞ + 4 E (60 )
cMNaMfa eKGacTTYl)anlT§plxagelIRbsinebIeKeRbIemKuNkRmajeFob K R = I / L ¬smIkar 12-3¦
sRmab;karKNna. elIsBIenH plbUkénemKuNEbgEckRtg;tMNEtgEtesμInwgmYy elIkElgEtRtg;TMr
bgáb;.

eRkayeBlKNnaemKuNEbgEck eyIgnwgkMNt;m:Um:g;bgáb;cug. manEtElVg BC b:ueNÑaHEdlrg
bnÞúk ehIyedayeRbItaragenAkñúgemeronTI 11 eyIg)an
wL2 6000(4 )2
(FEM )BC =− =− = −8000 N .m
12 12
2
6000(4 )2
(FEM )CB = wL = = 8000 N .m
12 12
eyIgcab;epþImedaysnμt;fatMN B RtUv)anTb; b¤bgáb;. m:Um:g;bgáb;cugenARtg;cMNuc B RTElVg BC kñúgTI


taMgbgáb; b¤TItaMgTb;dUcbgðajenAkñúgrUbTI 12-5b. CakarBit enHvaminEmnCasßanPaBlMnwgCak;EsþgenA
Rtg;tMN B edaysarm:Um:g;enAelIEpñknImYy²rbs;tMNenHRtUvEtesμIKña nigmanTisedApÞúyKña. edIm,IEk
sMrYlva eyIgRtUvGnuvtþm:Um:g; 8000 N.m EdlmanTisedApÞúyKñaeTAelItMN nigGnuBaØateGaytMNvileday
esrI ¬rUbTI 12-5c¦. CalT§pl Epñkénm:Um:g;enHRtUv)anEbgEckeTAElVg BC nig BA edayeKarBtam
emKuNEbgEck ¬b¤emKuNkRmaj¦ rbs;ElVgenARtg;tMN. m:Um:g;enAkñúg BA KW 0.4(8000) = 3200 N.m
ehIym:Um:g;enAkñúg BC KW 0.6(8000) = 4800 N.m . cugeRkay edaysarkarvilEdlekItmanenARtg; B
m:Um:g;enHRtUv)anepÞr edaysarm:Um:g;EdlekItmanenARtg;cugq¶ayrbs;ElVg. edayeRbIemKuNdMENl
+ 1 / 2 enaHlT§plRtUv)anbgðajenAkñúgrUbTI 12-5d.

]TahrN_enHbgðajfaCMhandMbUgEdlcaM)ac;enAeBlEbgEckm:Um:g;enARtg;tMNKW³ kMNt;m:Um:g;
GtulüPaBEdlmanGMeBIenARtg;tMNEdlbgáb;dMbUg/ RsayPaBbgáb;rbs;tMNenaH/ rYcGnuvtþm:Um:g;
GtulüPaBEdlmanTMhMesμIKña EtTisedApÞúyKñaedIm,IEktRmUvlMnwg/ EbgEckm:Um:g;kñúgcMeNamElVgEdlt
P¢ab;/ ehIyepÞrm:Um:g;kñúgElVgrbs;vaeTAcugmçageTot. CaTUeTA CMhanTaMgenHRtUv)anbgðajenAkñúgtarag
¬rUbTI 12-5e¦. enATIenH Dist, CO bgðajfam:Um:g;RtUv)anEbgEck nigepÞr. enAkñúgkrNIBiessenH eK
RtUvkareFVIkarEbgEckm:Um:g;EtmYyCMub:ueNÑaH edaysarCBa¢aMgenARtg; A nig C RsUbykm:Um:g; ehIymin
mantMNq¶ayÉNaeTotEdleKRtUveFVIeGaymantulüPaB b¤RsaykarTb;edIm,IbMeBjlkçxNÐlMnwgrbs;
tMNeT. eRkayeBlEbgEcktamrebobenH eKeFVIplbUkm:Um:g;Rtg;tMNnImYy²EdleFVIeGayeKTTYl)an
lT§plcugeRkayEdlbgðajenAkñúgbnÞat;xageRkamrbs;taragénrUbTI 12-5e. cMNaMfa LÚvtMN B
sßitkñúgsßanPaBlMnwg. edaysar M BC GviC¢man enaHm:Um:g;enHmanGMeBIelIElVg BC RcasRTnicnaLika
dUcbgðajenAelIdüaRkamGgÁesrIrbs;ElVgFñwmenAkñúgrUbTI 12-5f. edaysareyIgsÁal;m:Um:g;cug eyIg
GacKNnakmøaMgkat;enAxagcugrbs;Ggát;BIsmIkarlMnwgedayGnuvtþeTAelIElVgnImYy².
LÚvBicarNaFñwmdUcKña EtTMr C CaTMrkl; ¬rUbTI 12-6a¦. enAkñúgkrNIenHmanEtGgát;mYyenA
Rtg;tMN C dUcenHemKuNEbgEcksRmab;Ggát; CB enARtg;tMN C KW
4 E (60)
DFCB = =1
4 E (60)
emKuNEbgEckepSgeTot nigm:Um:g;bgáb;cugdUcKñanwgkarKNnaelIkmun. BYkvaRtUv)anbgðajenAkñúgCYr
edkTI1 nigTI2éntaragenAkñúgrUbTI 12-6b. CadMbUg eyIgnwg]bmafatMN B nigtMN C RtUv)anTb;.
eyIgcab;epþImedayRsaytMN C nigedaydak;m:Um:g; − 8000 N.m enARtg;tMNenaH. m:Um:g;TaMgmUlRtUv)an
EbgEckeTAkñúgGgát; CB edaysar (1)(− 8000)N .m = −8000 N .m . sBaØaRBYjenARtg;CYredkTI3bgðaj
fa (− 8000)N .m / 2 = −4000 N .m RtUv)anepÞreTAtMN B edaysareKGnuBaØateGaytMN C vileday


esrI. LÚvtMN C RtUv)anTb;eLIgvij. edaysareKRtUveFVIeGaym:Um:g;srubenARtg; C mantulüPaB
eKdak;bnÞat;enABIeRkamm:Um:g; − 8000 N.m . LÚveyIgnwgBicarNam:Um:g;GtulüPaB − 12000 N.m enARtg;
tMN B . enATIenH edIm,IeGaymanlMnwg eyIgRtUvGnuvtþm:Um:g; + 12000 N.m eTAelI B ehIytMNenHRtUv
)anTb;y:agNaedIm,IeGaycMENkénm:Um:g;RtUv)anEbgEckeTA BA nig BC eBalKW (0.4)(12000) =
4800 N.m nig (0.6 )(12000 ) = 7200 N.m dUcbgðajenAkñúgCYredkTI4. ehIyeKRtUvcgcaMfa + 1 / 2 én

m:Um:g;enHRtUv)anepÞreTACBa¢aMg A nigTMrkl; C edaysartMN B vil. LÚveKRtUvTb;tMN B eLIgvij.
eyIgRtUvRsaytMN C mþgeTot ehIym:Um:g;GtulüPaBenARtg;TMrkl;RtUv)anEbgEckdUckareFVIelIkmun.
lT§plRtUv)anbgðajenAkñúgCYredkTI5. eyIgRtUvTb; nigRsaytMN B nig C edIm,Ikat;bnßyTMhMm:Um:g;
EdlRtUveFVIeGaymantulüPaBrhUtdl;eKGacecalva)anebIeFobnwgm:Um:g;edIm. eKRtUvyl;BIdMeNIrkar
énCMhannImYy²BICYredkTI3 rhUtdl;CYredkTI14. edayeFVIplbUkm:Um:g; lT§plcugeRkayRtUv)an
bgðajenAkñúgCYredkTI15 EdleyIgeXIjfam:Um:g;cugeRkaybMeBjlkçxNÐlMnwgrbs;tMN.

RbesIrCakarGnuvtþdMeNIrkarEbgEckm:Um:g;bnþbnÞab;KñaeTAelItMNnImYy² ¬dUcbgðajxagelI¦ eK
k¾manlT§PaBkñúgkarGnuvtþeTAelItMNTaMgGs;kñúgeBldMNalKñapgEdr. karGnuvtþenHRtUv)anbgðajenA
kñúgtaragénrUbTI 12-6c. enAkñúgkrNIenH eyIgcab;epþImedayTb;tMNTaMgGs; ehIybnÞab;mkEbgEck
m:Um:g;bgáb;cugTaMgenARtg;tMN B nigtMN C nigeFVIeGaym:Um:g;enAtMNmantulüPaB ¬CYredkTI3¦.


edayRsaytMN B nigtMN C kñúgeBldMNalKña ¬tMN A RtUv)anbgáb;rhUt¦ bnÞab;mkm:Um:g;RtUv)anepÞr
eTAcugénElVgnImYy² ¬CYredkTI4¦. tMNRtUv)anTb;mþgeTot ehIym:Um:g;RtUv)anEbgEck nigeFVIeGay
manlMnwg ¬CYredkTI 5¦. karRsaytMNenHmþgeTotGnuBaØateGaymankarepÞrm:Um:g; dUcbgðajenAkñúgCYr
edkTI 6. edaykarKNnaCabnþbnÞab; eyIgTTYl)anlT§plcugeRkay ¬dUckarKNnaelIkmun¦ Edl
bgðajenAkñúgCYredkTI 24. edayeFVIkareRbobeFob viFIenHpþl;nUvcemøIyrYmtUcyWtCagviFIelIkmun b:uEnþ
enAkñúgkrNICaeRcInviFIenHmanRbsiT§PaBkñúgkarGnuvtþCag ehIysMrab;ehtuplEbbenHeyIgnwgeRbIvaenA
kñúg]TahrN_bnþbnÞab;. cugeRkayedayeRbIlT§plenAkñúgrUbTI 12-6b b¤ 12-6c, düaRkamGgÁesrI
rbs;ElVgFñwmnImYy²RtUv)anbgðajenAkñúgrUbTI 12-6d.
eTaHbICa viFIenHBak;B½n§nwgCMhanCaeRcInedIm,ITTYl)anlT§plcugeRkay. viFIenHk¾TamTarnUvviFI
saRsþEdr edaysarvaRtUvkarkarGnuvtþnUves‘rIénCMhanEdlmanlkçN³nBVnþCaCagkaredaHRsaysMnMuén
smIkardUcenAkñúgviFI slope-deflection. b:uEnþ eKRtUvcMNaMfa dMeNIrkarCamUldæanénkarEbgEckm:Um:g;
GnuvtþdUcnwgdMeNIrkarénviFIbMlas;TIdéTeTotEdr. dMeNIrkarKwkarbegáItTMnak;TMngrvagbnÞúk nigbMlas;TI
edaykMNt;nUvbMlas;TImMud¾RtwmRtUvsMrab;tMN ¬lkçxNÐRtUvKña¦. b:uEnþ enATIenH lkçxNÐlMnwg niglkç-
xNÐRtUvKñarbs;mMurgVilenARtg;tMNRtUv)anbMeBjedaypÞal;edayeRbIdMeNIrkarEdleFVIeGaym:Um:g;man
tulüPaBEdlcUlrYmCamYynwgTMnak;TMngrvagbnÞúk nigPaBdab ¬emKuNkRmaj¦. eKmanlT§PaBkñúgkar
sRmYlbnþeTotkñúgkareRbIR)as;karEbgEckm:Um:g;EdleyIgnwgelIkylmkerobrab;enAkñúgkfaxNÐxag
mux.

dMeNIrkarkñúgkarviPaK (Procedure for analysis)
xageRkamCaCMhanEdlpþl;nUvviFITUeTAsRmab;kMNt;m:Um:g;cugenAelIElVgFñwmEdleRbIkarEbgEck
m:Um:g;.
emKuNEbgEck nigm:Um:g;bgáb;cug³ eKRtUvkMNt;emKuNkRmajsRmab;ElVgnImYy²enARtg;tMN.
edayeRbItémøTaMgenH eyIgGacKNnaemKuNEbgEckBI DF = K / ∑ K . cMNaMfa DF = 0 sRmab;cug
bgáb; ehIy DF = 1 sMrab;TMrsnøak; b¤TMrkl;.
eyIgkMNt;mU:m:g;bgáb;cugsMrab;ElVgEdlrgbnÞúknImYy²edayeRbItaragEdleGayenAkñúgemeron
TI11. m:Um:g;bgáb;cugviC¢maneFVIGMeBIRsbTisRTnicnaLikavil ehIy FEM GviC¢maneFVIGMeBIRcasRTnic
naLika. edIm,IPaBgayRsYl eKGacerobcMtémøTaMgenHenAkñúgtaragEdlmanlkçN³dUctaragEdl
bgðajenAkñúgrUbTI 12-6c.


dMeNIrkarEbgEckm:Um:g;³ snμt;fatMNTaMgGs;EdleyIgRtUvkarkMNt;m:Um:g;enAkñúgElVgEdltP¢ab;RtUv
)anTb;. bnÞab;mk
!> kMNt;m:Um:g;EdleFVIeGaytMNnImYy²manlMnwg.
@> RsaytMN ehIyEbgEckm:Um:g;GtulüPaBeTAeGayElVgEdltP¢ab;nwgtMNnImYy².
#> epÞrm:Um:g;enAkñúgElVgnImYy²eTAkan;cugnImYy²rbs;vaedayKuNm:Um:g;TaMgenaHedayemKuN
dMENl (carry-over moment) + 1/ 2 .
edaykarTb; nigkarRsaytMNsarcuHsareLIg eyIgeXIjfakarEktMrUvm:Um:g;RtUv)ankat;bnßy
edaysarFñwmGacQaneTArkrUbragdabcugeRkay. enAeBlEdleyIgTTYl)antémøsMrab;karEktRmUvtUc
lμm eKRtUvbBaÄb;dMeNIrkarsarcuHsareLIgRtg;karepÞrm:Um:g;cugeRkayeK. eKRtUvbUkbBa©ÚlKñanUvtémø
FEM / m:Um:g;Edl)anBIkarEbgEck nigm:Um:g;Edl)anBIkarepÞrtamCYrQrnImYy². RbsinebIeyIgeFVIEbb

enH)anRtwmRtUv eyIgnwgTTYl)ansßanPaBlMnwgénm:Um:g;Rtg;tMNnImYy².

]TahrN_ 12-1³ kMNt;m:Um:g;Bt;enARtg;tMNnImYy²rbs;FñwmEdlbgðajenAkñúgrUbTI 12-7a. EI Ca
cMnYnefr.



dMeNaHRsay³ dMbUg eKRtUvkMNt;emKuNEbgEckenARtg;tMNnImYy². emKuNkRmajsMrab;Ggát;KW *

4EI 4EI 4EI
K AB = K BC = K CD =
12 12 8
dUcenH DFAB = DFDC = 0 DFBA = DFBC =
4 EI / 12
4 EI / 12 + 4 EI / 8
= 0.5
4 EI / 12 4 EI / 8
DFCB = = 0.4 DFCD = = 0.6
4 EI / 12 + 4 EI / 8 4 EI / 12 + 4 EI / 8
m:Um:g;bgáb;cugKW
wL2 20(12 )2
(FEM )BC =− =− = −240kN .m
12 12
2
20(12 )2
(FEM )CB = wL = = 240kN .m
12 12
− PL 250(8)
(FEM )CD = =− = −250kN .m
8 8
PL 250(8)
(FEM )DC = = = 250kN .m
8 8
edaycab;epþImCamYynwg FEM CYredkTI4 ¬rUbTI 12-7b¦ m:Um:g;enARtg;tMN B nig C RtUv)anEbgEck
kñúgeBlEtmYy ¬CYredkTI5¦. bnÞab;mk m:Um:g;TaMgenHRtUv)anepÞreTAcugénElVgerog²xøÜn ¬CYredkTI6¦.
m:Um:g;EdlTTYl)anRtUv)anEbgEck nigepÞrkñúgeBlEtmYymþgeTot ¬CYredkTI 7 nigTI8¦. dMeNIrkarRtUv
)anbnþrhUtdl;m:Um:g;RtUv)ankat;bnßyrhUtdl;TMhMmYysmrmü ¬CYredkTI 13¦. m:Um:g;EdlCalT§pl
RtUv)ankMNt;edayeFVIplbUk ¬CYredkTI14¦.
edaydak;m:Um:g;enAelIFñwmnImYy² nigedayGnuvtþsmIkarlMnwgeyIgTTYl)ankmøaMgkat;enAxagcug
ElVgdUcbgðajenAkñúgrUbTI 12-7c ehIydüaRkamm:Um:g;sMrab;FñwmTaMgmUlRtUv)anbgðajenAkñúgrUbTI 12-
7d.

]TahrN_ 12-2³ kMNt;m:Um:g;Bt;enARtg;TMrnImYy²rbs;FñwmEdlbgðajenAkñúgrUbTI 12-8a. m:Um:g;
niclPaBénElVgnImYy²RtUv)an.
dMeNaHRsay³ enAkñúgcMeNaTenH m:Um:g;mnRtUv)anEbgEckenAkñúgElVgly AB eT dUcenHemKuNEbg
i
Eck (DF )BA = 0 . kRmajrbs;ElVg BC KWEp¥kelI 4EI / L edaysarTMrsnøak;minEmnsßitenAcugq¶ay
rbs;Fñwm. emKuNkRmaj emKuNEbgEck nigm:Um:g;bgáb;cugRtUv)anKNnadUcxageRkam³
4 E (300)(106 ) 4 E (240)( 6 )
= 300(106 )E = 320( 6 )E
10
K BC = K CD = 10
4 3

*
enATIenH eyIgeRbIemKuNkRmah 4EI / L b:uEnþeyIgk¾GaceRbIemKuNkRmajeFob I / L pgEdr.


DFBC = 1 − (DF )BA = 1 − 0 = 1
300 E
DFCB = = 0.484
300 E + 320 E
320 E
DFCD = = 0.516
300 E + 320 E
320 E
DFDC = =0
∞ + 320 E
edaysarFñwmenHCaFñwmly
(FEM )BA = 2000 N (2m ) = 4000 N .m
(FEM ) = − wL = − 1500(4) = −2000 N .m
2 2
BC
12 12
2
1500(4 )2
(FEM )CB = wL = = 2000 N .m
12 12



témøTaMgenHRtUv)anbgðajenAkñúgCYredkTI4rbs;taragkñúgrUbTI 12-8b. ElVgEdllyeFVIeGayekItman
m:Um:g;Bt; − 4000 N.m enAxageqVg B . edIm,IeFVIeGaytMN B mantulüPaB eKRtUvmanm:Um:g;Bt; − 4000 N.m
manGMeBIenAxagsþaM B . dUcbgðajenAkñúgCYredkTI 5 eKRtUvbEnßm − 2000 N.m eTAelI BC edIm,IbMeBj
lkçxNÐ. dMeNIrkarénkarEbgEck nigkarepÞrm:Um:g;RtUv)anbnþtamrebobFmμtadUcbgðaj.
edaysareKsÁal;m:Um:g;Bt; eKGacsg;düaRkamm:Um:g;sRmab;FñwmdUcbgðajenAkñúgrUbTI 12-8c.

!@>#> karEktRmUvemKuNkRmaj (Stiffness-factor modification)
enAkñúg]TarhrN_énkarEbgEckm:Um:g;elIkmun eyIg)anBicarNaElVgénFñwmnImYy²manTMrbgáb;
¬tMNRtUv)anTb;¦ enAxagcugq¶ayrbs;vaenAeBl
EbgEck nigepÞrm:Um:g;. sRmab;ehtuplenH eyIg
)anKNnaemKuNkRmaj emKuNEbgEck nigem
KuNdMENledayQrenAelIkrNIdUcbgðajenAkñúg
rUbTI 12-9. Cak;Esþg enATIenHemKuNkRmajKW
K = 4 EI / L ¬smIkar 12-1¦ ehIyemKuNdMENlKW + 1 / 2 .

enAkñúgkrNIxøH eKmanlT§PaBEkERbemKuNkRmajrbs;ElVgFñwmBiess ehtudUcenHeKGacsMrYl
dl;dMeNIrkarEbgEckm:Um:g;. eyIgnwgBicarNakrNIbIEdlekIteLIgjwkjab;enAkñúgkarGnuvtþ.

Ggát;EdlmanTMrsnøak;enAcugq¶ay³ FñwmminkMNt;xøHmanElVgxageRkAeKbMputRTedayTMrsnøak;dUcenA
kñúgkrNIéntMN B enAkñúgrUbTI 12-10a. enATIenH m:Um:g; M bgVilcug A )an θ . edIm,IkMNt; θ eKRtUv
kMNt;kmøaMgkat;enARtg;cMNuc A' énFñwmnimitþ ¬rUbTI 12-10b¦. eyIgman
1⎛ M ⎞ ⎛2 ⎞
+ ∑ M B' = 0 V A ' ( L ) − ⎜ ⎟ L⎜ L ⎟ = 0
2 ⎝ EI ⎠ ⎝ 3 ⎠
ML
V A' = θ =
3EI
b¤ M =
3EI
L
θ

dUcenH emKuNkRmajsRmab;FñwmenHKW
3EI
K= (12-4)
L
cugq¶ayCaTMrsnøak; b¤TMrkl;



ehIy eKRtUvcgcaMfaemKuNdMENlKWsUnü edaysarsnøak;enARtg; A' minTb;Tl;m:Um:g;eT. tamkar
eRbobeFob RbsinebIcugq¶ayCaTMrbgáb;eKRtUvEktRmUvemKuNkRmaj K = 4EI / L eday 3 / 4 edIm,I
eGayeTACakrNIEdlmancugq¶ayRTedayTMrsnøak;. RbsinebIeKKitkarEktRmUvenH dMeNIrkarénkar
EbgEckm:Um:g;RtUv)ansRmYledaysareKminRtUvkarTb; nigRsayvaCabnþbnÞab;enAeBlEbgEckm:Um:g;eT.
ehIyedaysarEtcugElVgRtUv)anRTedayTMrsnøak; eKkMNt;m:Um:g;bgáb;cugsMrab;ElVgedayeRbItémøenA
kñúgCYrQrxagsþaMéntaragenAkñúgemeronTI11. ]TahrN_ 12-4 bgðajBIrebobGnuvtþkarsRmYlenH.
FñwmsIuemRTIrgbnÞúksIuemRTI³ RbsinebIFñwmmanragsIuemRTI ehIyrgbnÞúksIuemRTI düaRkamm:Um:g;Bt;
sRmab;Fñwmk¾nwgmanlkçN³sIuemRTIEdr. CalT§pl eKGaceFVIkarEktRmUvemKuNkRmajsRmab;ElVg
kNþal dUcenHeKRtUvkarEbgEckm:Um:g;enAkñúgFñwmtamry³tMNEdlsßitenAelIFñwmEtBak;kNþal. edIm,I
EktRmUvemKuNkRmajeGay)anRtwmRtUv eyIgnwgBicarNaFñwmEdlbgðajenAkñúgrUbTI 12-11a. eday
sarPaBsIuemRTI m:Um:g;Bt;enARtg; B nig C RtUvEtesμIKña. edaysnμt;témøenHesμInwg M FñwmnimitþsMrab;
ElVg BC RtUv)anbgðajenAkñúgrUbTI 12-11b. dUcenH mMurgVil θ enARtg;cugnImYy²KW
⎛ M ⎞ ⎛L⎞
+ ∑ M C' = 0 − V B ' ( L ) + ⎜ ⎟ L⎜ ⎟ = 0
⎝ EI ⎠ ⎝ 2 ⎠
ML
VB ' = θ =
2 EI
b¤ M =
2 EI
L
θ

dUcenH emKuNkRmajsRmab;ElVgkNþalKW
2 EI
K= (12-5)
L
FñwmsIuemRTI nigrgbnÞúksIuemRTI
dUcenH eKGacEbgEckm:Um:g;sRmab;EtBak;kNþalFñwmRbsinebIeKKNnaemKuNkRmajsMrab;ElVgkNþal
edayeRbIsmIkar 12-5. tamkareRbobeFob emKuNkRmajkNþalElVgesμInwgBak;kNþalénemKuN
kRmajEdleRbI K = 4EI / L .



FñwmsIuemRTIrgbnÞúkGsIuemRTI³ RbsinebIFñwmmanragsIuemRTIrgbnÞúkGsIuemRTI eKnwgTTYl)andüaRkam
GsIuemRTI. dUcenAkñúgkrNIelIkmun eyIgGacEktRmUvemKuNkRmajrbs;ElVgkNþal dUcenHeyIgnwgKit
FñwmEtBak;kNþalsRmab;karviPaKedaykarEbgEckm:Um:g;. BicarNaFñwmenAkñúgrUbTI 12-12a. Fñwm
nimitþsMrab;ElVgkNþal BC rbs;vaRtUv)anbgðajenAkñúgrUbTI 12-12b. edaysarbnÞúkGsIuemRTI m:Um:g;
Bt;enARtg; B nig C RtUvEtesμIKñaEtmanTispÞúyKña. snμt;fa témøenHesμInwg M enaHmMurgVilenARtg;cug
nImYy²KW³
1 ⎛ M ⎞⎛ L ⎞⎛ 5L ⎞ 1 ⎛ M ⎞⎛ L ⎞⎛ L ⎞
+ ∑ M C' = 0 − VB ' ( L ) + ⎜ ⎟⎜ ⎟⎜ ⎟ − ⎜ ⎟⎜ ⎟⎜ ⎟ = 0
2 ⎝ EI ⎠⎝ 2 ⎠⎝ 6 ⎠ 2 ⎝ EI ⎠⎝ 2 ⎠⎝ 6 ⎠
ML
VB ' = θ =
6 EI
b¤ M =
6 EI
L
θ

dUcenH emKuNkRmajsRmab;ElVgkNþalesμInwg
6 EI
K= (12-6)
L
FñwmsIuemRTI nigrgbnÞúkGsIuemRTI
dUcenH enAeBleKKNnaemKuNkRmajsMrab;ElVgkNþaledayeRbIsmIkar 12-6 eKRtUvkarEbgEck
m:Um:g;enAkñúgEtBak;kNþalFñwmb:ueNÑaH. tamkareRbobeFob emKuNkRmajkNþalElVgesμInwgemKuN
kRmajEdleRbI K = 4EI / L mYydgknøH.



]TahrN_ 12-3³ kMNt;m:Um:g;Bt;enARtg;TMrsRmab;FñwmEdlbgðajenAkñúgrUbTI 12-13a. EI CacMnYn
efr.

dMeNaHRsay³ tamkarGegát Fñwm nigbnÞúkmanlkçN³sIuemRTI. dUcenH eyIgnwgGnuvtþ K = 2EI / L
edIm,IKNnaemKuNkRmajrbs;ElVgkNþal BC ehIyeyIgnwgviPaKEtkMNat;Bak;kNþalxageqVgrbs;
FñwmEtb:ueNÑaH. eKGackat;bnßykarviPaKeGaykat;EtxøIEfmeTotedayeRbI K = 3EI / L sRmab;kar
KNnaemKuNkRmajsMrab;kMNat;Ggát; AB edaysarcugq¶ay A CaTMrsnøak;. elIsBIenH eKGacrMlg
karEbgEckm:Um:g;enARtg;cMNuc A edayeRbIm:Um:g;bgáb;cugsRmab;bnÞúkragRtIekaNmanGMeBIelIElVgEdl
manTMrmçagCaTMrsnøak; nigTMrmçageTotCaTMgbgáb;. dUcenH
K AB =
3EI
3
¬edayeRbIsmIkar 12-4¦
K BC =
2EI
4
¬edayeRbIsmIkar 12-5¦
3EI / 3
DFAB = =1
3EI / 3
3EI / 3
DFBA = = 0.667
3EI / 3 + 2 EI / 4
2 EI / 4
DFBC = = 0.333
3EI / 3 + 2 EI / 4
2
100(3)2
(FEM )BA = wL = = 60kN .m
15 15
2
100(4 )2
(FEM )BC = − wL =− = −133.3kN .m
12 12
TIinñn½yTaMgenHRtUv)anerobcMenAkñúgtarageénrUbTI 12-13b. karKNnaemKuNkRmajdUcbgðajxagelI
BitCakat;bnßykarviPaKy:ageRcIn edaysarmanEttMN B EtmYybueNÑaHEdleKRtUvkareFVIeGayman
lMnwg ehIyeKminRtUvkarepÞrm:Um:g;eTAtMN A nigtMN C eT. eyIgeXIjfatMN C rgm:Um:g; 108.9kN.m
dUctMN B Edr.



]TahrN_ 12-4³ kMNt;m:Um:g;Bt;enARtg;TMrsRmab;FñwmEdlbgðajenAkñúgrUbTI 12-14a. m:Um:g;nicl
PaBrbs;ElVgTaMgBIrRtUv)anbgðajenAkñúgrUb.

dMeNaHRsay³ edaysarFñwmRtUv)anRTedayTMrkl;enAcugq¶ay C kRmajrbs;ElVg BC RtUv)anKNna
edayQrelI K = 3EI / L . eyIg)an
K AB = =
( )
4 EI 4 E (120) 10 6
( )
= 160 10 6 E
L 3

K BC =
3EI 3E (240) 106
=
( ) ( )
= 180 106 E
L 4
dUcenH DFAB =
160 E
∞ + 160 E
=0
160 E
DFBA = = 0.4706
160 E + 180 E
180 E
DFBC = = 0.5294
160 E + 180 E
180 E
DFCB = =1
180 E



eKGacsRmYlviFIEbgEckm:Um:g;sRmab;cMeNaTenHedayRtUvdwgfaeKGaceRbIm:Um:g;bgáb;cugeTalsRmab;
ElVg BC . edayeRbICYrxagsþaMéntaragénemeronTI11 sMrab;ElVgEdlrgbnÞúkrayesμImanTMrmçagCaTMr
bgáb; nigmçageTotCaTMrsnøak; enaHeyIg)an
wL2 6000(4 )2
(FEM )BC =− =− = −12000 N .m
8 8
Tinñn½yTaMgGs;RtUv)anbBa©ÚleTAkñúgtaragenAkñúgrUbTI 12-14b ehIykarEbgEckm:Um:g;RtUv)an
Gnuvtþ. edayeRbobeFobCamYynwgrUbTI 12-6b viFIenHBitCasRmYldl;karEbgEcky:agxøaMg.
edayeRbIlT§plEdlTTYl)an kmøaMgkat;xagcugrbs;Fñwm nigdüaRkamm:Um:g;RtUv)anbgðajenAkñúg
rUbTI 12-14c.

!@>$> karEbgEckm:Um:g;sMrab;eRKagmineyal
(Moment distribution for frames: No sidesway)
karGnuvtþénviFIEbgEckm:Um:g;sRmab;eRKagEdlmineyalmanlkçN³dUcnwgdMeNIrkarsRmab;
viPaKFñwm. edIm,Ikat;bnßy»kasénkMhusqÁg eKENnaMeGayeFVIkarviPaKedayeRbItaragdUcenAkñúg
]TahrN_elIkmun. ehIyeKGackat;bnßykarEbgEckm:Um:g; RbsinebIeKEktRmUvemKuNkRmajdUc
bgðajenAkñúgkfaxNÐmun.

]TahrN_ 12-5³ kMNt;m:Um:g;Bt;enARtg;tMNéneRKagEdlbgðajenAkñúgrUbTI 12-15a. eKmanTMr
snøak;enARtg; E nig D nigmanTMrbgáb;enARtg; A . EI CacMnYnefr.
dMeNaHRsay³ tamkarGegát snøak;enARtg; E nigkarBareRKagkMueGayeyal. eKGacKNnaemKuN
kRmajrbs; CD nig CE edayeRbI K = 3EI / L edaysarcugq¶ayCaTMrsnøak;. ehIy bnÞúk 60kN min
)abbegáItm:Um:g;bgáb;cugeT edaysarvaGnuvtþenARtg;cMN B . dUcenH
4EI 4EI 3EI 3EI
K AB = K BC = K CD = K CE =
5 6 5 4
DFAB = 0
4 EI / 5
DFBA = = 0.545
4 EI / 5 + 4 EI / 6
DFBC = 1 − 0.545 = 0.455
4 EI / L
DFCB = = 0.330
4 EI / 6 + 3EI / 5 + 3EI / 4
3EI / 5
DFCD = = 0.298
4 EI / 6 + 3EI / 5 + 3EI / 4


DFCE = 1 − 0.33 − 0.298 = 0.372

DFDC = 1 DFEC = 1
wL2 − 45(6 )2
(FEM )BC =− = = −135kN .m
12 12
2
45(6 )2
(FEM )CB = wL = = 135kN .m
12 12
Tinñn½yRtUv)anbgðajenAkñúgtaragénrUbTI 12-15b. enATIenH karEbgEckm:Um:g;RtUv)aneFVIbnþbnÞab;Et
eTAelItMN B nig C . m:Um:g;cugeRkayRtUv)anbgðajenAkñúgCYredkcugeRkayeK.
edayeRbITinñn½yTaMgenH eyIgGacsg;düaRkamm:Um:g;sRmab;eRKagEdlbgðajenAkñúgrUbTI 12-
15c.

!@>%> karEbgEckm:Um:g;sMrab;eRKagEdleyal
(Moment distribution for frames: Sidesway)
BIkaxNÐ 11-5 eyIgeXIjehIyfaeRKagEdlmanlkçN³minsIuemRTI b¤eRKagEdlrgbnÞúkminsIu-
emRTInwgeyal. ]TahrN_énkrNIEbbenHRtUv)anbgðajenAkñúgrUbTI 12-16a. enATIenH bnÞúkGnuvtþn_
P nwgbegáItm:Um:g;enARtg; B nig C minesμIKña dUcenHeRKagnwgmankMhUcRTg;RTay Δ eTAsþaM. edIm,IkMNt;

kMhUcRTg;RTayenH nigm:Um:g;Bt;enARtg;tMNedayeRbIkarEbgEckm:Um:g; eyIgnwgeRbIeKalkarN_tRmYt
pl. enAkñúgrUbTI 12-16b dMbUgeKKitfaeRKagRtUv)anTb;mineGayeyaledayGnuvtþtMNTMrnimitþenA



Rtg; C . eKGnuvtþkarEbgEckm:Um:g; ehIybnÞab;mkeKKNnakmøaMgTb; R edaysmIkarsþaTic. kmøaMg
Tb;EdlmanTMhMesμIKña EtTisedApÞúyKñaRtUv)anGnuvtþeTAelIeRKag ¬rUbTI 12-16c¦ ehIym:Um:g;enAkñúg
eRKagRtUv)anKNna. viFImYysRmab;edaHRsayCMhancugeRkayenHTamTarnUvkarsnμt;CadMbUgnUvtémøCa
elxsRmab;m:Um:g;Bt;mYy ¬ M 'BA ¦. edayeRbIkarEbgEckm:Um:g; nigsmIkarsþaTic eyIgGackMNt;kMhUc
RTg;RTay Δ' nigkmøaMgxageRkA R EdlRtUvnwgtémøsnμt; M 'BA . edaysarvaCakMhUcRTg;RTayeGLa-
sÞiclIenEG‘r enaHkmøaMg R' begáItm:Um:g;enAkñúgeRKagEdlsmamaRteTAnwgm:Um:g;EdlbegáIteGayman R .
Ca]TahrN_ RbsinebIeKsÁal; M 'BA nig R' enaHm:Um:g;enARtg; B EdlbegáIteday R Ca M BA = M 'BA
(R / R') . karbEnßmm:Um:g;Rtg;tMNsMrab;krNITaMgBIr ¬rUbTI12-16b nig c¦ nwgeFVIeGayeKTTYl)an
m:Um:g;Cak;EsþgenAkñúgeRKag ¬rUbTI 12-16a¦. karGnuvtþénbec©keTsenHRtUv)anbgðajenAkñúg]Ta-
hrN_ 12-6 dl; 12-8.

eRKageRcInCan;³ CaerOy² rcnasm<½n§EdlmaneRcInCan;nwgmanbMlas;TIÉkraCüeRcIn Cavi)akkarEbg
Eckm:Um:g;edayeRbIbec©keTsxagelInwgTamTarnUvkarKNnaeRcIn. ]TahrN_ BicarNaeRKagBIrCan;Edl
bgðajenAkñúgrUbTI 12-17a. rcnasm<½n§enHGacmanbMlas;TIÉkraCüBIr edaysarcm¶ayeyal Δ1
rbs;Can;TImYyÉkraCüBIbMlas;TI Δ 2 rbs;Can;TIBIr.
CaGkusl bMlas;TITaMgenHminRtUv)andwgenAdMNak;kaldMbUg dUcenHkarviPaKRtUveFVIeLIgeday
QrelIeKalkarN_tRmYtplEdlmanlkçN³dUcKñanwgkarENnaMxagelI. enAkñúgkrNIenH eKRtUvGnuvtþ
kmøaMgTb;BIrKW R1 nig R2 ¬rUbTI 12-17b¦ ehIym:Um:g;bgáb;cugRtUv)anKNna nigEbgEck. edayeRbI
smIkarlMnwg eKGackMNt;témøCaelxrbs; R1 nig R2 . bnÞab;mk eKRtUvRsaykarTb;enARtg;kRmal
énCan;TImYy ehIykRmalenaHRtUv)andak;eGaymanbMlas;TI Δ' . bMlas;TIenHeFVIeGaymanm:Um:g;bgáb;
cugenAkñμúgeRKagEdleKGackMNt;Catémøelx. edayEbgEckm:Um:g;enH nigedayeRbIsmIkarlMnwg eKGac


kMNt;témøCaelxén R'1 nig R'2 . tamrebobdUcKña kRmalénCan;TIBIrRtUv)aneGaymanbMlas;TI Δ' '
¬rUbTI 12-17d¦. edaysnμt;témøCaelxsRmab;m:Um:g;bgáb;cug karEbgEckm:Um:g; nigsmIkarlMnwgnwg
pþl;nUvtémø R"1 nig R"2 . edaysarCMhancugeRkayTaMgBIr ¬rUbTI 12-17c nig d¦ GaRs½ynwgtémø
snμt;énm:Um:g;bgáb;cug eKRtUvGnuvtþemKuNEktMrUv (correction factor) C ' nig C" eTAelIm:Um:g;EdlEbg
Eck. eyagtamkmøaMgTb;enAkñúgrUbTI 12-17c nig 12-17d eyIgRtUvkarGnuvtþkmøaMg R1 nig R2 Edl
manTMhMesμIKñaEtTisedApÞúyKñaeTAelIeRKag Edl
R2 = −C ' R ' 2 +C" R"2

R1 = +C ' R '1 −C" R"1
dMeNaHRsayénRbB½n§smIkarenHeGaynUvtémøén C ' nig C" . bnÞab;mkeKRtUvKuNemKuNTaMgenHCamYy
nwgm:Um:g;Bt;Rtg;tMNEdlkMNt;BIkarEbgEckm:Um:g;enAkñúgrUbTI 12-17c nig 12-17d. bnÞab;mkeK
kMNt;m:Um:g;ersIusþg;edayeFVIplbUkm:Um:g;EktRmUvCamYynwgm:Um:g;EdlTTYl)anenAkñúgrUbTI 12-17b.
eKGacviPaKeRKagRbePTepSgeTotEdlmanbMlas;TIÉkraCüedayeRbIdMeNIkardUcKña b:uEnþeKRtUv
dwgfaviFIxagelIEdlmanerobrab;RtUvkarkarKNnay:ageRcIn. EteTaHCay:agNak¾eday eK)anbegáIt
bec©keTsxøHedIm,Ikat;bnßykarKNna ¬CakarRbesIreKKYredaHRsaycMeNaTRbePTenHedayeRbIkMuBüÚ-
T½r¦ CaTUeTAedayeRbIkarviPaKedaym:aRTIs. bec©keTssRmab;karviPaKEbbenHRtUv)anbgðajenAkñúg
emeronTI 16.

]TahrN_ 12-6³ kMNt;m:Um:g;Bt;enARtg;tMNéneRKagEdlbgðajenAkñúgrUbTI 12-18a. EI CacMnYn
efr.
dMeNaHRsay³ dMbUgeyIgKitfaeRKagRtUv)anTb;mineGayeyaldUcbgðajenAkñúgrUbTI 12-18b.
eyIgman
16(4)2 (1)
(FEM )BC =− = −10.24kN .m
(5)2



(FEM )CB = 16(1) 2(4) = 2.56kN .m
2

(5)
emKuNkRmajrbs;ElVgnImYy²RtUv)anKNnaedayQrelI 4EI / L b¤edayeRbIemKuNkRmajeFob
I / L . emKuNEbgEck nigkarEbgEckm:Um:g;RtUv)anbgðajenAkñúgtarageénrUbTI 12-18d. edayeRbI

lT§plTaMgenH eKGnuvtþsmIkarlMnwgeTAelIdüaRkamGgÁesrIénssredIm,IkMNt; Ax nig Dx ¬rUbTI 12-
18e¦. BIdüaRkamGgÁesrIrbs;eRKagTaMgmUl ¬minRtUv)anbgðaj¦ kmøaMgTb; R enAkñúgrUbTI 12-18
manTMhM
∑ Fx = 0 R = 1.73kN − 0.81kN = 0.92kN
eKRtUvGnuvtþkmøaMgEdlmanTMhMesμIKña EtTisedApÞúyKñanwg R = 0.92kN eTAelIeRKagRtg;tMN C ehIy
KNnam:Um:g;Bt; ¬rUbTI 12-18c¦. edIm,IedaHRsaycMeNaTEdlKNnam:Um:g;TaMgenH ¬eyIgnwgsnμt;
kmøaMg R' GnuvtþenARtg; C ¦ EdleFVIeGayeRKagmanbMlas;TI Δ' dUcbgðajenAkñúgrUbTI 12-18f. enATI
enHtMNenARtg; B nig C RtUv)anTb;CabeNþaHGasnñBIkarvil ehIyCalT§plm:Um:g;bgáb;cugenARtg;cug
rbs;ssrRtUv)anKNnaBIrUbmnþsMrab;bMlas;TIEdlmankñúgemeronTI 10 Edl
6 EIΔ
M =
L2
edaysar TaMg B nig C pøas;TI)an Δ' dUcKña ehIy AB nig DC man E / I nig L dUcKña m:Um:g;bgáb;cugenA
kñúg AB ehIymanm:Um:g;bgáb;cugdUcKñaenAkñúg DC . dUcbgðajenAkñúgrUbTI 12-18f eyIgnwgsnμt;m:Um:g;
bgáb;cugNamYyesμInwg
(FEM ) AB = (FEM )BA = (FEM )CD = (FEM )DC = −100kN .m
eKcaM)ac;eRbIsBaØaGviC¢man edaysarm:Um:g;RtUveFVIGMeBIRcasRTnicnaLikaeTAelIssr edayeFIVeGayssr
manbMlas;TI Δ' eTAsþaM. LÚveKGackMNt;témø R' EdlrYmCamYynwgm:Um:g; − 100kN.m . karEbgEcg
m:Um:g;rbs;m:Um:g;bgáb;cugRtUv)anbgðajenAkñúgrUbTI 12-18g. BIsmIkarlMnwg eKGacKNnakmøaMg
RbtikmμtamTisedkenARtg; A nig D ¬rUbTI 12-18h¦. dUcenH sMrab;;eRKagTaMgmUl eyIgtMrUveGay
∑ Fx = 0 R ' = 28 + 28 = 56.0kN
dUcenH R'= 56.0kN begáItm:Um:g;EdlmanenAkñúgtaragénrUbTI 12-18g. eKGackMNt;m:Um:g;EdlbegáIt
eLIgeday R = 0.92kN smamaRt. dUcenH m:Um:g;ers‘ultg;enAkñúgeRKag ¬rUbTI 12-18a¦ esμInwgpl
bUkénm:Um:g;Edl)anKNnasRmab;eRKagenAkñúgrUbTI 12-18b CamYynwgbrimaNsmamaRténm:Um:g;
sRmab;eRKagenAkñúgrUbTI 12-18c. eyIgman
M AB = 2.88 +
0.92
(− 80) = 1.57kN .m
56.0



M BA = 5.78 +
0.92
(− 60) = 4.79kN .m
56.0
M BC = −5.78 +
0.92
(60) = −4.79kN .m
56.0
M CB = 2.72 +
0.92
(60) = 3.71kN .m
56.0
M CD = −2.72 +
0.92
(− 60) = −3.71kN .m
56.0
M DC = −1.32 +
0.92
(− 80) = −2.63kN .m
56.0

]TahrN_ 12-7³ kMNt;m:Um:g;Bt;enARtg;tMNnImYy²rbs;eRKagEdlbgðajenAkñúgrUbTI 12-19a.
m:Um:g;niclPaBénGgát;nImYy²RtUv)anbgðajenAkñúgrUb.



dMeNaHRsay³ dMbUgeRKagRtUv)anTb;mineGayeyaldUcbgðajenAkñúgrUbTI 12-19b. m:Um:g;niclPaB
RtUv)anKNnaenARtg;tMNdUcbgðajenAkñúgrUbTI 12-19d. enATIenH eKKNnaemKuNkRmajrbs; CD
edayeRbI 3EI / L edaysarvamanTMrsnøak;enARtg; D . karKNnakmøaMgRbtikmμtamTisedkenARtg; A
nig D RtUv)anbgðajenAkñúgrUbTI 12-19e. dUcenH sMrab;eRKagTaMgmUl
∑ Fx = 0 R = 5.784 − 2 = 3.78kN
eKGnuvtþkmøaMgEdlmanTispÞúyKñaeTAelIeRKagdUcbgðajenAkñúgrUbTI 12-19c. dUcenAkñúg]TahrN_
elIkmun eyIgnwgBicarNakmøaMg R' EdlmanGMeBIdUcbgðajenAkñúgrUbTI 12-19f. CalT§pl tMN B nig
C pøas;TI)an Δ ' dUcKña. eyIgKNnam:Um:g;bgáb;cugsRmab; BA BI

(FEM ) = (FEM ) = − 6 EIΔ = − 6 E (1000)(10 )Δ'
6
AB BA
L2 (5)2
b:uEnþ BItaragenAkñúgemeronTI10 sRmab; CD eyIgman
(FEM ) = − 3EIΔ = − 3E (1250)(10 )Δ'
6
CD
L2 (5)2



edaysnμt;fam:Um:g;bgáb;cugsRmab; AB KW − 100kN.m dUcbgðajenAkñúgrUbTI 12-19f enaHeKkMNt;m:Um:g;
bgáb;cugenARtg; C EdleFVIeGaymanbMlas;TI Δ' dUcKña edaykareRbobeFob eBalKW
Δ' = −
(− 100)(5)2 = − (FEM )CD (7.5)2
6 E (1000)( 6 )
10 3E (1250)( 6 )
10
(FEM )CD = −27.78kN .m
karEbgEckm:Um:g;sRmab;m:Um:g;bgáb;cugTaMgenHRtUv)anbgðajkñúgtaragénrUbTI 12-19g. karKNna
kmøaMgRbtikmμenARtg; A nig D RtUv)anbgðajenAkñúgrUbTI 12-19h. dUcenH sRmab;eRKagTaMgmUl
∑ Fx = 0 R ' = 21.98 + 3.11 = 25.1kN
dUcenH m:Um:g;ers‘ultg;enAkñúgeRKagKW
⎛ 3.78 ⎞
M AB = 9.58 + ⎜ ⎟(− 69.91) = −0.948kN .m
⎝ 25.1 ⎠
⎛ 3.78 ⎞
M BA = 19.34 + ⎜ ⎟(− 40.01) = −13.3kN .m
⎝ 25.1 ⎠
⎛ 3.78 ⎞
M BC = −19.34 + ⎜ ⎟(40.01) = 13.3kN .m
⎝ 25.1 ⎠
⎛ 3.78 ⎞
M CB = 15.00 + ⎜ ⎟(23.31) = 18.5kN .m
⎝ 25.1 ⎠
⎛ 3.78 ⎞
M CD = −15.00 + ⎜ ⎟(− 23.31) = −18.5kN .m
⎝ 25.1 ⎠

]TahrN_ 12-8³ kMNt;m:Um:g;Bt;enARtg;tMNnImYy²rbs;eRKagEdlbgðajenAkñúgrUbTI 12-20a.
EI CacMnYnefr.
dMeNaHRsay³ dMbUg kareyalRtUv)ankarBaredaykmøaMgTb; R ¬rUbTI 12-20b¦. m:Um:g;bgáb;cug
sRmab;Ggát; BC KW
(FEM )BC = − 16(5) = −10kN .m (FEM )CB = 16(5) = 10kN .m
8 8
edaysarGgát; AB nig DC manTMrsnøak; emKuNkRmajRtUv)anKNnaedayeRbI 3EI / L . karEbgEck
m:Um:g;RtUv)anbgðajenAkñúgrUbTI 12-20d.
edaylT§plTaMgenH eKkMNt;kmøaMgRbtikmμtamTisedkenARtg; A nig D edayeRbIsmIkar
lMnwgénGgát;nImYy² ¬rUbTI 12-20e¦. edayeFVIplbUkm:Um:g;Rtg;cMNuc B nig C enAelIeCIgnImYy²
eyIg)an³
+ ∑MB = 0 − 5.97 + Ax (4) − 8(3) = 0 Ax = 7.49kN

+ ∑ MC = 0 5.97 − D x (4) + 8(3) = 0 D x = 7.49kN



dUcenH sRmab;eRKagTaMgmUl
∑ Fx = 0 R = 7.49 − 7.49 + 40 = 40kN
LÚveyIgGnuvtþkmøaMgpÞúynwg R eTAelIeRKagdUcbgðajenAkñúgrUbTI 12-20c. edIm,IKNnam:Um:g;
Bt;EdlbegáIteLIgeday R CadMbUgeyIgRtUvBicarNakmøaMg R EdlmanGMeBIdUcbgðajenAkñúgrUbTI 12-
20f. enATIenHbnÞat;dac;²minEmnCarUbragxUcRTg;RTayrbs;Ggát;eRKageT vaRtUv)ansg;CabnÞat;Rtg;
EdlbnøayeTATItaMgcugeRkayrbs; B' nig C ' rbs;tMN B nig C erogKña. edaysarPaBsIuemRTIrbs;
eRKag bMlas;TI BB' = CC ' = Δ' . elIsBIenH bMlas;TITaMgenHeFVIeGay BC vil. cm¶aytamTisQr



rvag B' nig C ' KW1.2Δ' dUcbgðajenAelIdüaRkambMlas;TI ¬rUbTI 12-20g¦. edaysartMNcugrbs;
Ggát;nImYy²rgbMlas;TIEdleFVIeGayGgát;vil m:Um:g;bgáb;cugRtUvekItmanenAkñúgGgát;. eKman (FEM )BA
= (FEM )CD = −3EIΔ' / (5)2 / (FEM )BC = (FEM )CB = 6 EI (1.2Δ') / (5)2 .

cMNaMfa sRmab; BA nig CD m:Um:g;mantémøGviC¢man edaysarkarvilRsbTisRTnicnaLikarbs;
Ggát;eFVIeGaym:Um:g;bgáb;cugvilRcasRTnicnaLika.
RbsinebI eyIgsnμt;témørbs; (FEM )BA = (FEM )CD = −100kN .m ehIydak;eGayesμI Δ' enA
kñúgrUbmnþxagelIeyIgnwgTTYl)an (FEM )BC = (FEM )CB = 240kN .m . eKdak;m:Um:g;TaMgenHeGayman
GMeBIelIeRKag nigeFVIkarEbgEckva ¬rUbTI 12-20h¦. edayeRbIlT§plTaMgenH smIkarlMnwgRtUv)an
bgðajenAkñúgrUbTI 12-20i. sRmab;eCIgnImYy² eyIgman
+ ∑MB = 0 − A' x (4) + 58.72(3) + 146.80 = 0 A' x = 80.74kN

+ ∑ MC = 0 − D' x (4) + 58.72(3) + 146.80 = 0 D' x = 80.74kN
dUcenH sRmab;eRKagTaMgmUl
∑ Fx = 0 R ' = 80.74 + 80.74 = 161.48kN
dUcenH m:Um:g;ers‘ultg;enAkñúgeRKagKW
⎛ 40 ⎞
M BA = 5.97 + ⎜ ⎟(− 146.80) = −30.4kN .m
⎝ 161.48 ⎠
⎛ 40 ⎞
M BC = −5.97 + ⎜ ⎟(146.80) = 30.4kN .m
⎝ 161.48 ⎠
⎛ 40 ⎞
M CB = 5.97 + ⎜ ⎟(146.80) = 42.3kN .m
⎝ 161.48 ⎠
⎛ 40 ⎞
M CD = −5.97 + ⎜ ⎟(− 146.80 ) = −42.3kN .m
⎝ 161.48 ⎠



cMeNaT
!@>! kMNt;m:Um:g;enARtg; A / B nig C bnÞab;mk
sg;düaRkamm:Um:g;sRmab;Fñwm. m:Um:g;niclPaB
rbs;Ggát;nImYy²RtUv)anbgðajenAkñúgrUb. snμt;
faTMrRtg; B CaTMrkl; ehIyenARtg; A nig C CaTMr
bgáb;. E = 200GPa . !@>% kMNt;m:Um:g;enARtg; A / B nig C bnÞab;mk
sg;düaRkamm:Um:g;sRmab;Fñwm. snμt; faTMrRtg;
A nig C CaTMrbgáb; ehIyenARtg; B CaTMrkl;.

EI CacMnYnefr.

!@>@ kMNt;m:Um:g;enARtg; A / B nig C bnÞab;mk
sg;düaRkamm:Um:g;sRmab;Fñwm. snμt; faTMr A nig
C CaTMrbgáb;. EI CacMnYnefr.
!@>^ kMNt;m:Um:g;enARtg; A / B nig C bnÞab;mk
sg;düaRkamm:Um:g;sRmab;Fñwm. snμt; faTMrRtg;
A nig C CaTMrkl; ehIyenARtg; B CaTMrsnøak;.

bnÞúkBRgaysßitenAelIkRmalEdlTMrsamBaØEdl
bBa¢ÚnbnÞúkeTAFñwmkRmal. EI CacMnYnefr.
!@># kMNt;m:Um:g;Bt;enAkñúgFñwmRtg; B bnÞab;mk
sg;düaRkamm:Um:g;. snμt; faTMr C CaTMrsnøak;.
kMNat;Ggát; AB manm:Um:g;niclPaB I AB =
0.75I BC . EI CacMnYnefr.

!@>& kMNt;m:Um:g;Bt;enAkñúgFñwmRtg; B bnÞab;mk
sg;düaRkamm:Um:g;sRmab;Fñwm. snμt; faTMr A Ca
TMrsnøak; ehIyTMr B CaTMrkl; nigTMr C CaTMrbgáb;.
!@>$ kMNt;m:Um:g;enARtg; A / B nig C bnÞab;mk EI CacMnYnefr.

sg;düaRkamm:Um:g;. snμt; faTMrRtg; B CaTMrkl;
ehIyenARtg; A nig C CaTMrbgáb;. EI CacMnYnefr.
Problems T.Chhay -441


!@>* kMNt;m:Um:g;Bt;enAkñúgFñwmRtg; A nig B
bnÞab; mksg;düaRkamm:Um:g;. snμt; faTMr A Ca
TMrbgáb; ehIyTMr B CaTMrkl; nigTMr C CaTMrsnøak;.

!@>!@ kMNt;m:Um:g;EdlmanGMeBIenARtg;cugrbs;
Ggát;nImYy² bnÞab;mksg;düaRkamm:Um:g;. snμt;
!@>( r)arRtUv)antP¢ab;edaytMNsnøak;Edl faTMr A nig D CaTMrsnøak; ehIyTMr B CatMN
bgðajedaycMNuc. RbsinebIeKecalkmøaMgEkg bgáb; nigTMr C CaTMrbgáb;. E = 200GPa / I ABC
enAkñúgr)ar cUrkMNt;kmøaMgRbtikmμtamTisQrenA = 280(106 )mm 4 ehIy I BD = 440(106 )mm 4 .
Rtg;snøak;nImYy². EI CacMnYnefr.

!@>!0 kMNt;m:Um:g;Bt;enAkñúgFñwmRtg;TMr bnÞab;
mksg;düaRkamm:Um:g;. snμt;fa A nig D Ca !@>!# kMNt;m:Um:g;EdlmanGMeBIenARtg;tMN
TMrbgáb;. EI CacMnYnefr. nImYy². snμt; A / D nig E CaTMrsnøak; ehIy
B nig C CatMNbgáb;. m:Um:g;niclPaBrbs;Ggát;

nImYy²RtUv)anbgðajenAkñúgrUb. E = 200GPa .

!@>!! kMNt;m:Um:g;Bt;enARtg; B bnÞab;mksg;
düaRkamm:Um:g;sRmab;Ggát;nImYy²rbs;eRKag.
snμt; faTMr A CaTMrsnøak; ehIyTMr C CaTMrbgáb;.
EI CacMnYnefr.



!@>!$ kMNt;m:Um:g;Bt;enARtg; A / C nig D bnÞab; !@>!& kMNt;m:Um:g;Bt;EdlmanGMeBIenARtg;cug
mksg;düaRkamm:Um:g;sMrab;Ggát;nImYy²rbs; rbs;Ggát;nImYy². snμt;faTMrenARtg; A nig D
eRKag. TMr A nigtMN C nig D CatMNbgáb;. CaTMrbgáb;. m:Um:g;niclPaBrbs;Ggát;nImYy²RtUv
EI CacMnYnefr. )anbgðajenAkñúgrUbj. EI = 200GPa .

!@>!% kMNt;m:Um:g;Bt;enARtg; B bnÞab;mksg; !@>!* kMNt;m:Um:g;enARtg; B nig C bnÞab;mksg;
düaRkamm:Um:g;sMrab;Ggát;nImYy²rbs;eRKag. TMr düaRkamm:Um:g;. snμt; A nig D CaTMrsnøak;
A CaTMrsnøak;. EI CacMnYn efr.
ehIy B nig C CatMNbgáb;. EI CacMnYnefr.

!@>!( kMNt;m:Um:g;EdlmanGMeBIenARtg;cugrbs;G
!@>!^ kMNt;m:Um:g;Bt;EdlmanGMeBIenARtg;cug gát;nImYy². snμt;tMNCatMNbgáb; ehIy A nig
rbs;Ggát;nImYy²rbs;eRKag. EI CacMnYnefr. D CaTMrbgáb;. EI CacMnYnefr.

Problems T.Chhay -443


!@>@0 kMNt;m:Um:g;EdlmanGMeBIenARtg;TMrbgáb; !@>@! kMNt;bgÁúMkmøaMgRbtikmμtamTisedk nig
A nig D rbs;eRKag. EI CacMnYnefr. tamTisQrenARtg;TMrsnøak; A nig D . EI
CacMnYnefr.


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