11. displacement method of analysis slope deflection equations

Department of Civil Engineering NPIC

!!> karviPaKtamviFIbMlas;TI³ smIkar slope-deflection
Displacement method of analysis: Slope-deflection equations

emeronenHnwgerobrab;y:agsegçbnUveKalKMnitsMrab;viPaKeRKOgbgÁúMedayeRbIviFIbMlas;TI.
eRkayeBlbgðajBIeKalKMnit eyIgnwgbegáItsmIkarTUeTAén slop deflection ehIybnÞab;mkeyIg
nigeRbIvaedIm,IviPaKFñwm nigeRKagminkMNt;edaysþaTic.

!!>!> viFIsaRsþTUeTAkñúgkarviPaKtamviFIbMlas;TI
(Displacement method of analysis: General procedures)
rcnasm<½n§TaMgGs;RtUvEtbMeBjtMrUvkarsmIkarlMnwg TMnak;TMngrvagbnÞúk nigbMlas;TI nigRtUv
bMeBjtMrUvkarlkçxNÐRtUvKñaénbMlas;TIedIm,IFananUvsuvtßiPaBrbs;va. enAkñúgkfaxNÐ 10.1 )anerobrab;
fa eKmanviFIBIrepSgKñaedIm,IbMeBjtMrUvkarTaMgenHenAeBlviPaKeRKOgbgÁúMminkMNt;edaysþaTic. viFI
kmøaMg ¬emeronTI !0¦ Ep¥kelIkarkMNt;kmøaMgEdlCaGBaØatelIs bnÞab;mkbMeBjtMrUvkarlkçxNÐRtUvKña
rbs;rcnasm<½n§. eKeFVIrebobenHedaysresrsmIkarbMlas;TIeGayTak;TgnwgkmøaMgedayeRbITMnak;TMng
rvagkmøaMg nigbMlas;TI. dMeNaHRsayénsmIkarers‘ultg;CakmøaMgRbtikmμEdlCaGBaØatelIs bnÞab;
mkeKeRbIsmIkarlMnwgedIm,IkMNt;kmøaMgRbtikmμEdlenAsl;EdlmanGMeBIelIeRKOgbgÁúM.
viFIbMlas;TICaviFIEdlmanlkçN³pÞúyeTAnwgviFIkmøaMg. dMbUgvaRtUvbMeBjtMrUvkarsmIkarlMnwg
sMrab;rcnasm<½n§. edIm,IeFVIEbbenH eKRtUvsresrbMlas;TIEdlCaGBaØateGayTak;TgnwgbnÞúkedayeRbI
TMnak;TMngrvagkmøaMg nigbMlas;TI. eRkayeBlTTYl)anbMlas;TIrYcehIy eKGackMNt;kmøaMgEdlCa
GBaØatBIsmIkarlkçxNÐRtUvKñaedayeRbITMnak;TMngrvagkmøaMg nigbMlas;TI. RKb;viFIbMlas;TITaMgGs;
GnuvtþtamviFIsaRsþTUeTAenH.
Degrees of freedom enAeBleRKOgbgÁúMrgkmøaMg cMNucCak;lak;enAelIeRKOgbgÁúMEdleKeGay
eQμaHfa node nwgrgbMlas;TI. bMlas;TITaMgenHRtUv)aneKKitCa degrees of freedom sMrab;eRKOgbgÁúM
ehIyenAkñúgviFIbMlas;TIeKcaM)ac;RtUvkMNt; degrees of freedom TaMgenH edaysarvanwgkøayCaGBaØat
enAeBleKGnuvtþviFIbMlas;TI. cMnYnénGBaØatTaMgenHCadWeRkénPaBminkMNt;edaysarsIueNm:aTicrbs;
eRKOgbgÁúM.
edIm,IkMNt;PaBminkMNt;edaysIueNma:Tic eKGacKitvaeRKOgbgÁúMpÁúMedayes‘rIénGgát;EdltP¢ab;
Kñaedaysar node EdlCaTUeTAvasßitenARtg;tMN TMr enAxagcugrbs;Ggát; b¤enARtg;kEnøgbMlas;bþÚrmux
kat;Pøam²rbs;Ggát;. enAkñúglMhr (three dimensions), node nImYy²GacmanbMlas;TIlIenEG‘rcMnYn
karviPaKtamviFIbMlas;TI³ smIkar slope-deflection T.Chhay -378

mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa

BIr nigbMlas;TImMucMnYnmYYy. elIsBIenH eKGacTb;bMlas;TIrbs; node edayTMr b¤edaykarsnμt;Edl
Ep¥kelIkareFVIkarrbs;eRKOgbgÁúM. Ca]TahrN_ RbsinebIeRKOgbgÁúMCaFñwm ehIyeKBicarNaEtkMhUcRTg;
RTayEdlbNþalBIm:Um:g;Bt; enaHvaminmanbMlas;TIlIenEG‘rtambeNþayG½kSrbs;FñwmeT edaysar
bMlas;TI enHbNþalBIkMhUcRTg;RTayedaysarkmøaMgtamG½kS. edIm,IbBa¢ak;BIeKalKMnitenH eyIgnwg
BicarNa]TahrN_xøH edaycab;epþImCamYynwgFñwmenAkñúgrUbTI 11-1a. enATIenH RKb;bnÞúk P Edl
GnuvtþeTAelI FñwmnwgeFVIeGay node A vilEtb:ueNÑaH ¬edayecalkMhUcRTg;RTaytamG½kS¦ cMENkÉ
node B RtUv)an Tb;mineGaycl½tEtmþg. dUcenH FñwmmanEt degree of freedom EtmYyKW θ A ehIy

vaCaeRKOgbgÁúMmin kMNt;edaysIueNm:aTicdWeRkTImYy. FñwmenAkñúgrUbTI 11-1b man node enARtg; A /
B nig C dUcenHva man degree of freedom cMnYnbYn EdlkMNt;edaymMurgVil θ A / θ B / θ C nigbMlas;

TItamTisQr ΔC . vaCaeRKOgbgÁúMminkMNt;edaysIueNma:TicdWeRkTIbYn. LÚvBicarNaeRKagenAkñúgrUbTI
11-1c. mþgeTot RbsinebIeyIgecalkMhUcRTg;RTaytamG½kSrbs;Ggát; bnÞúk P EdlGnuvtþeTAelI
eRKagGaceFVIeGay node B nig C vil ehIy node TaMgenHGacpøas;TItamTisedkedaybrimaNesμIKña.
dUcenH eRKagman degree of freedom cMnYnbIKW θ B / θC nig Δ B ehIyvaCaeRKOgbgÁúMminkMNt;eday
sIueNm:aTicdWeRkTIbI.
Cakarsegçb karkMNt;PaBminkMNt;edaysIueNma:Tic b¤cMnYnrbs; degree of freedom sMrab;
eRKOgbgÁúMCaCMhancaM)ac;bMputEdleKRtUveFVImuneK enAeBlGnuvtþviFIbMlas;TI. vakMNt;nUvcMnYnGBaØat
EdlQrelIkarsnμt;EdlKitcMeBaHkMhUcRTg;RTayrbs;eRKOgbgÁúM. elIsBIenH enAeBlEdleKsÁal;
bMlas;TIrbs; node enaHeKnwgGackMNt;bMlas;TIrbs;eRKOgbgÁúM ehIyeKGacTTYl)ankMlaMgkñúgrbs;
Ggát;.

Displacement method of analysis: Slope-deflection equations T.Chhay -379


!!>@> smIkar slope-deflection (Slope-Deflection equation)
viFIkmøaMgenAkñúgemeronTI!0 RtUvkarsresrsmIkarEdlTak;TgeTAnwgkmøaMg b¤m:Um:g;EdlCaGBaØat.
EteKminGaceRbIvasMrab;eRKOgbgÁúMminkMNt;EdlmandWeRkx<s;)aneT edaysareKRtUvkarbegáItsmIkar
lkçxNÐRtUvKñaeRcIn ehIyelIsBIenH smIkarnImYy²Tak;TgnwgGBaØatTaMgGs;EdleFVIeGayeKBi)akedaH
RsayRbB½n§smIkar elIkElgEteKeRbIkMuBüÚT½r. edaykareRbobeFob viFI slope-deflection minsμúK
sμajdUcviFIkmøaMgeT. dUcEdleyIg)aneXIj vaTamTarkargarticCag TaMgkarsresrsmIkar nigkar
edaHRsaysmIkaredIm,IrkbMlas;TI nigkmøaMgkñúg. ehIyelIsBIenH eKGacsresrkmμviFIkMuBüÚT½rCamYy
nwgviFIenHy:aggayRsYl ehIyeKGaceRbIvaedIm,IviPaKeRKOgbgÁúMminkMNt;eRcInEbb.
viFI slope-deflection RtUv)anbegáIteLIgdMbUgeday ehnrIc Em:ndWLa (Heinrich Manderla)
nig GUfU m: (Otto Morh) edIm,IsikSakugRtaMgbnÞab;bnSM (secondary stresses) enAkñúg trusses. eRkay
mk enAqñaM1915 CI eG EmnI (G.A. Maney) )aneFVIeGaybec©keTsenHkan;EtRbesIeLIg ehIyeRbIva
kñúgkarviPaKFñwm nigeRKagminkMNt;.
General Case eKeGayeQμaHviFIenHfa viFI slope-deflection edaysarGBaØatrbs;vaCamMurgVil nig
PaBdabEdlekIteLIgedaysarGMeBIrbs;bnÞúkmkelIrcnasm<½n§. edIm,IbegáItTMrg;TUeTAénsmIkar slope-
deflection eyIgnwgBicarNaElVgKMrU AB énFñwmCab;dUcbgðajenAkñúgrUbTI 11-2 EdlElVgenH rgGMeBIén

bnÞúkNamYy ehIyvaman EI efr. eyIgcg;eFVI
eGaym:Um:g;Bt;rbs;Fñwm M AB nig M BA man
TMnak;TMngnwg degree of freedom rbs;vaTaMgbI
eBalKW θ A / θ B nig Δ EdlGaceFVIeGayman
sMrutenAcenøaHTMr. enA kñúgkarbegáItrUbmnþ eyIg
Kitfam:Um:g; nigmMurgVilviC¢manenAeBlvaeFVIGMeBIeTA
elIFñwmtamTisRTnicnaLikavil ¬rUbTI 11-2¦.
elIsBIenH bMlas;TIlIenEG‘r Δ mantMélviC¢mandUcbgðajenAkñúgrUb edaysarbMlas;TIeFVIeGaymMψ rbs;
u
ElVgvilRsbTisRTnicnaLikavil.
eKGacTTYlsmIkar slope-deflection edayeRbIeKalkarN_tMrYtpledayBicarNaBicarNa
dac;edayELkBIKñarvagm:Um:g;EdlekItmanenARtg;TMrEdlbNþalBIbMlas;TInImYy² ¬ θ A / θ B nig Δ ¦
nigbnÞúk.



bMlas;TICamMuenARtg;cMNuc A/ θ ³ eKman node A énGgát;EdlbgðajenAkñúgrUbTI 11-3a manmMu
A

rgVil θ A ÉcMENk node B RtUv)anbgáb;. edIm,IkMNt;m:Um:g; M AB EdlcaM)ac;edIm,IeFIVeGaymanbMlas;TI
enH eyIgnwgeRbIviFIFñwmnimitþ. sMrab;krNIenH FñwmnimitþRtUv)anbgðajenAkñúgrUbTI 11-3b. cMNaMfa
kmøaMgkat;enARtg;cMNuc A' eFVIGMeBIeTAelIFñwmedaymanTisedAcuHeRkam edaysar θ A RsbRTnicnaLika.
PaBdabrbs;FñwmBitenAkñúgrUbTI 11-3a RtUvesμIsUnüenARtg; A nig B dUcenHplbUkm:Um:g;enARtg; A'
nig B' rbs;Fñwmnimitþk¾RtUvEtesμIsUnüEdr dUcenHeK)an
⎡ 1 ⎛ M AB ⎞ ⎤ L ⎡ 1 ⎛ M BA ⎞ ⎤ 2 L
+ ∑ M A' = 0 ⎢ 2 ⎜ EI ⎟ L ⎥ 3 − ⎢ 2 ⎜ EI ⎟ L ⎥ 3 = 0
⎣ ⎝ ⎠ ⎦ ⎣ ⎝ ⎠ ⎦
⎡ 1 ⎛ M BA ⎞ ⎤ L ⎡ 1 ⎛ M AB ⎞ ⎤ 2 L
+ ∑ M B' = 0 ⎢ 2 ⎜ EI ⎟ L ⎥ 3 − ⎢ 2 ⎜ EI ⎟ L ⎥ 3 + θ A L = 0
⎣ ⎝ ⎠ ⎦ ⎣ ⎝ ⎠ ⎦
BIsmIkarTaMgBIxagelI eyIgTTYl)anTMnak;TMngbnÞúk nigbMlas;TIdUcxageRkam
4 EI
M AB = θA (11-1)
L
2 EI
M BA = θA (11-2)
L

bMlas;TICamMuenARtg;cMNuc B/ θB ³ tam
rebobdUcKña RbsinebIcug B rbs;FñwmvileTA
rkTItaMgcug eRkay)an θ B ÉcMENkcug A
RtUv)anbgáb;Cab; ¬rUbTI 11-4¦ eyIgGac
sresrTMnak;TMngrvagm:Um:g; M BA eTAnwgmMu
rgVil θ B nigrvagm:Um:g;Rbtikmμ M AB enARtg;CBa¢aMg. lT§plKW
4 EI
M BA = θB (11-3)
L
2 EI
M AB = θB (11-4)
L



bMlas;TIeFob/ Δ ³ RbsinebI node B rbs;Ggát;pøas;TIeFobeTAnwgcMNuc A dUcenHGgát;vilRsbTis
RTnicnaLika ¬bMlas;TIviC¢man¦ ehIycugTaMgBIGt;vl nigmanm:Um:g; nigkmøaMgkat;EdlmanTMhMdUcKñaEt
i
TisedApÞúyKñaekItmanenAkúñgGgát; ¬rUbTI 11-5a¦. dUcelIkmun eKGacP¢ab;TMnak;TMngrvagm:Um:g; M eTA
nwgbMlas;TI Δ edayeRbIviFIFñwmnimitþ. enAkñúgkrNIenH Fñwmnimitþ ¬rUbTI 11-5b¦ mancugTaMgBIresrI
edaysarFñwm ¬Ggát;¦ BitmanTMrbgáb;. b:uEnþ edaysarbMlas;TIrbs;FñwmBitenARtg;cMNuc B m:Um:g;enA
Rtg;cMNuc B' énFñwmnimitþRtUvEtman Δ dUcbgðaj . edayeFVIplbUkm:Um:g;Rtg;cMNuc B' eyIg)an
*

⎡ 1 ⎛ M ⎞ ⎤ 2L ⎡ 1 ⎛ M ⎞ ⎤ L
+ ∑ M B' = 0 ⎢ 2 ⎜ EI ⎟ L ⎥ 3 − ⎢ 2 ⎜ EI ⎟ L ⎥ 3 − Δ = 0
⎣ ⎝ ⎠ ⎦ ⎣ ⎝ ⎠ ⎦
− 6 EI
M AB = M BA = M = 2 Δ (11-5)
L
tamkarkMNt;sBaØa m:Um:g;enHGviC¢manedaysarvaeFVIGMeBIRcasRTnicnaLikaedIm,IeGayGgát;manlMnwg.

m:Um:g;bgáb;cug (Fixed-End Moment)³ enAkñúgkrNIelIkmun eyIg)anKitBITMnak;TMngrvagbMlas;
TInigm:Um:g;caM)ac; M AB nig M BA EdleFVIGMeBIenARtg; node A nig B erogKña. b:uEnþ CaTUeTAbMlas;TI
lIenEG‘r nigbMlas;TImMuén node ekItBIbnÞúkEdlmanGMeBIenAelIElVgrbs;Ggát; minEmnekItBIm:Um:g;Edl
manGMeBIenARtg; node eT. edIm,IbegáItsmIkar slope-deflection, eyIgRtUvbMElgbnÞúkEdlmanGMeBIenA
elIElVgeGayeTACam:Um:g;smmUlEdlmanGMeBIenARtg; node ehIybnÞab;mkeRbITMnak;TMngrvagbnÞúk nig
bMlas;TIedIm,IedaHRsay. eKGaceFVIvaeTA)anedayrkm:Um:g;RbtikmμTb;Tl;nwgbnÞúkEdlekItmanenARtg;
node. Ca]TahrN_ eKmanGgát;TMrbgáb;dUcbgðajenAkñúgrUbTI 11-6a EdlvargbnÞúkcMcMNuc P enARtg;

kNþalElVg. FñwmnimitþsMrab;krNIenHRtUv)anbgðajenAkñúgrUbTI 11-6b. edaysareyIgRtUvkareGay
mMurgVilenARtg;cugnImYy²esμIsUnü
⎡ 1 ⎛ PL ⎞ ⎤ ⎡ 1 ⎛ M ⎞ ⎤
+ ↑ ∑ Fy = 0 ⎢ 2 ⎜ 4 EI ⎟ L ⎥ − 2⎢ 2 ⎜ EI ⎟ L ⎥ = 0
⎣ ⎝ ⎠ ⎦ ⎣ ⎝ ⎠ ⎦

*
düaRkamm:Um:g;EdlbgðajenAlíFñwmnimitþRtUv)ankMNt;edayviFItMrYtplsMrab;FñwmTImrsamBaØ ¬dUckarBnül;enAkñúgkfaxNÐ4-5¦.

PL
M =
8
m:Um:g;enHRtUv)aneKehAfam:Um:g;bgáb;cug (FEM). cMNaMfa GaRs½ynwgkarkMNt;sBaØa vamantMélGviC¢-
manenARtg; node A ¬RcasRTnicnaLika¦ ehIyvamantMélviC¢manenARtg; node B ¬RsbRTnic
naLika¦. edIm,IgayRsYlkñúgkaredaHRsaycMeNaT eKRtUvKNnam:Um:g;bgáb;cugsMrab;kardak;bnÞúkdéT
eTot ehIyvaRtUv)anerobcMCatarag. edayeFVIplbUk FEM Edl)anKNnasMrab;cMeNaTCak;lak; ¬rUb
TI 11-7¦ eyIg)an
M AB = (FEM ) AB M BA = (FEM )BA (11-6)

smIkar Slope-Deflection³ RbsinebIeKbUkm:Um:g;cugEdlbNþalBIbMlas;TInImYy² ¬smIkar 11-1
rhUtdl; 11-5¦ CamYynwgm:Um:g;cugEdlbNþalBIbnÞúk ¬smIkar 11-6¦ enaHeyIgGacsresrm:Um:g;er-
s‘ultg;enAxagcugdUcxageRkam
⎛ I ⎞⎡ ⎛ Δ ⎞⎤
M AB = 2 E ⎜ ⎟ ⎢2θ A + θ B − 3⎜ ⎟⎥ + (FEM ) AB
⎝ L ⎠⎣ ⎝ L ⎠⎦
(11-7)
⎛ I ⎞⎡ ⎛ Δ ⎞⎤
M BA = 2 E ⎜ ⎟ ⎢2θ B + θ A − 3⎜ ⎟⎥ + (FEM )BA
⎝ L ⎠⎣ ⎝ L ⎠⎦
edaysarsmIkarTaMgBIrenHRsedogKña eKGacsresrsmIkareTalEdlGacbgðajsmIkarTaMgBIr
enH)an. edayykcugmçagrbs;ElVgCacugCit (near edd N) ehIycugmçageTotCacugq¶ay (far end F)
ehIyedayykkMrajrbs;Ggát;Ca k = I / L ehIymMurbs;Ggát;Caψ = Δ / L eyIgGacsresr
M N = 2 EK (2θ N + θ F − 3ψ ) + (FEM )N
sMrab;ElVgxagkñúg b¤ElVgxagEdlmancugq¶ayCacugbgáb; (11-8)



Edl MN = m:Um:g;Bt;enAkñúgcugCitrbs;ElVg m:Um:g;enHviC¢man ¬RsbTisRTnicnaLika¦ enAeBlEdl
vamanGMeBIelIElVg.
E / K = m:UDuleGLasÞicrbs;sMPar³ nigkMrajrbs;ElVg k = I / L .

θ N / θ F = mMurgVil b¤bMlas;TImMurbs;cugCit nigcugq¶ayrbs;ElVgenARtg;TMr mMuenHmanxñatra:düg;
ehIyviC¢manenAeBlvavilRsbTisRTnicnaLika.
ψ = mMurgVilrbs;Ggát;bNþalBIbMlas;TIlIenEG‘r Edl ψ = Δ / L mMuenHmanxñatra:düg; ehIy
viC¢manenAeBlvavilRsbTisRTnicnaLika.
(FEM )N = m:Um:g;bgáb;cugenARtg;TMréncugCit m:Um:g;enHviC¢man ¬RsbTisRTnicnaLika¦ enA
eBlvamanGMeBIelIElVg. m:Um:g;bgáb;cugsMrab;lkçxNÐbnÞúkepSg²RtUv)anerobcMenA
kñúgtaragxagelI.
BIkarbkRsay smIkar11-8RtUv)ankMNt;edaysmIkarlkçxNÐRtUvKña nigTMnak;TMngrvagbnÞúk nig
bMlas;TIEdlBicarNaEteTAelIT§iBlm:Um:g;Bt; edaymin)anKitBIkMhUcRTg;RTayedaysarkmøaMgkat;
nigkMhUcRTg;RTayedaysarkmøaMgtamG½kS dUcenHeKKitvaCasmIkar slope-deflection TUeTA. enAeBl
eRbIvasMrab;edaHRsaycMeNaT eKRtUvGnuvtþsmIkarenHBIrdgsMrab;ElVgGgát; AB nImYy² eBalKWGnuvtþ
BI A eTA B nigBI B eTA A sMrab;ElVg AB enAkñúgrUbTI 11-2.
cugElVgRTedayTMrsnøak;³ eBlxøH cugElVgrbs;Fñwm b¤eRKagRtUv)anRTedayTMrsnøak; b¤TMrkl;enAcug
q¶ay ¬rUbTI 11-8a¦. enAeBlekItmankrNIEbbenH m:Um:g;enARtg;TMrkl; b¤TMrsnøak;RtUvEtesμIsUnü
enaHeKmincaM)ac;kMNt;bMlas;TImMu θ B enARtg;TMrenaHeT eKGacEkERbsmIkar slope-deflection TUeTA
dUcenHeKRtUvGnuvtþvaEtmþgKt;eTAelIElVg minEmnBIrdg. edIm,IeFVIEbbenH eyIgnwgGnuvtþsmIkar 11-8
b¤smIkar 11-7 eTAelIcugnImYy²rbs;FñwmenAkñúgrUbTI 11-8. CalT§pl eKTTYl)ansmIkarBIrdUcxag
eRkam
M N = 2 Ek (2θ N + θ F − 3ψ ) + (FEM )N
(11-9)
0 = 2 Ek (2θ F + θ N − 3ψ ) + 0
enATIenH (FEM )F esμIsUnüedaysarcugq¶ayCaTMrsnøak; ¬rUbTI 11-8b¦. elIsBIenH eKGacTTYl)an
(FEM )N ¬Ca]TahrN_¦ edayeRbItaragmU:m:g;bgáb;cugxagelI. edayKuNsmIkarTImYyxagelInwgBIr
ehIydkvaCamYynwgsmIkarTIBIr edIm,IsMrYl θ F enaHeyIg)an
M N = 3Ek (θ N −ψ ) + (FEM )N
sMrab;Etcugq¶ayrbs;ElVgRtUv)anRTedayTMrsnøak; b¤TMrkl; (11-10)



edaysarm:Um:g;enAcugq¶ayesμIsUnü eKGnuvtþsmIkarenHEtmþgb:ueNÑaHsMrab;cugElVg. vasMrYlkarviPaK
edaysarsmIkarTUeTA ¬smIkar 11-8¦ TamTarkarGnuvtþBIrdgsMrab;ElVgenH edaysarvaBak;B½n§nwg
bMlas;TImMu θ B ¬b¤θ F ¦ enAcugTMr.

edIm,IsegçbkarGnuvtþsmIkar slope-deflection, cUrBicarNaFñwmCab;EdlbgðajenAkñúgrUbTI 11-
9 Edlman degree of freedom cMnYnbYn. enATIenH eKGacGnuvtþsmIkar 11-8 cMnYnBIrdgeTAelIElVg
nImYy² eBalKWBI A eTA B / BI B eTA A / BI B eTA C / BI C eTA B / BI C eTA D / nigBI D eTA C . smIkarTaMg
enHnwgBak;B½n§nwgmMurgVilθ A / θ B / θC nig θ D EdlCaGBaØat. b:uEnþ edaysarm:Um:g;cugenARtg; A nig D
sUnü enaHeKmincaM)ac;kMNt; θ A nig θ D eT. eKnwgTTYl)ankaredaHRsayxøICagRbsinebIeKGnuvtþ
smIkar 11-10 BI B eTA A nigBI C eTA D ehIybnÞab;mkGnuvtþsmIkar 11-8 BI B eTA C nig BI C eTA
B . smIkarTaMgbYnenHnwgBak;B½n§EtnwgmMurgVil θ B nig θ C EdlCaGBaØatEtb:ueNÑaH.

!!>#> viPaKFñwm (Analysis of beams)
dMeNIrkarkñúgkarviPaK (Procedure for analysis)
Degree of Freedom
bg;elx b¤GkSrRKb;TMr nigtMN (node) edIm,IkMNt;GtþsBaØaNénElVgrbs;Fñwm b¤eRKagEdlenAcenøaH
node. edayKUrrUbragdabrbs;rcnasm<½n§ eKmanlT§PaBkMNt;cMnYn degree of freedom. enATIenH

node nImYy²GacmanbMlas;TImMu nigbMlas;TIlIenEG‘r. eKRtUvrkrkSaPaBRtUvKñaenARtg; node edIm,IeFVI



eGayGgát;EdltP¢ab;edaytMNbgáb;enARtg; node rgbMlas;TIdUcKña. RbsinebIbMlas;TITaMgenaHCa
GBaØat eKKYrsnμt;eGayvaeFVIGMeBItamTisviC¢man edayeFVIeGayGgát; b¤tMNvilRsbTisRTnicnaLika
¬rUbTI 11-2¦.
smIkar Slope-Deflection
smIkar slope-deflection P¢ab;TMnak;TMngrvagm:Um:g;EdlCaGBaØatEdlGnuvtþenARtg; node CamYynwg
bMlas;TIrbs; nodes sMrab;ElVgTaMgGs;rbs;eRKOgbgÁúM. RbsinebIeKmanbnÞúkenAelIElVg eKRtUvKNna
FEM edayeRbItaragm:Um:g;cugEdleGayenAkñúgkfaxNÐ 11.2. ehIyRbsinebI node manbMlas;TI

lIenEG‘r Δ eKRtUvKNna ψ = Δ / L sMrab;ElVgEk,r. GnuvtþsmIkar 11-8 eTAelIcugrbs;ElVgnImYy²
edayeRbIsmIkar slope-deflection BIrdgsMrab;ElVgnImYy². b:uEnþ RbsinebIcugElVgénFñwm b¤eRKagCab;
CaTMrsnøak; GnuvtþsmIkar 11-10 EteTAelIcugEdlTb; ehIyeRbIsmIkar slope-deflection Etmþg
b:ueNÑaHsMrab;ElVg.
smIkarlMnwg
sresrsmIkarlMnwgsMrab; degree of freedom EdlCaGBaØatnImYy². eKRtUvsresrsmIkarTaMgenH
edayeRbIm:Um:g;Bt;EdlCaGBaØatdUcEdlkMNt;enAkñúgsmIkar slope-deflection. sMrab;Fñwm nigeRKag
sresrsmIkarlMnwgénm:Um:g;enARtg;TMrnImYy² nigsMrab;eRKagsresrsmIkarlMnwgénm:Um:g;Bt;Rtg;tMN.
RbsinebIeRKageyal b¤manPaBdabtamTisedk eKRtUvP¢ab;TMnak;TMngrvagkmøaMgkat;enAkñúgssreTA
nwgm:Um:g;enAxagcugrbs;ssr ¬nwgmanerobrab;enAkñúgkfaxNÐ 11-5¦.
CMnYssmIkar slope-deflection eTAkñúgsmIkarlMnwg ehIyedaHRsayrkbMlas;TIrbs;tMNEdl
CaGBaØat. bnÞab;mkeKRtUvCMnYslT§plTaMgenHeTAkñúgsmIkar slope-deflection edIm,IkMNt;m:Um:g;Bt;
enAxagcugrbs;Ggát;nImYy². RbsinebIlT§plmantémøGviC¢man mann½yfavavilRcasRTnicnaLika
cMENkÉm:Um:g; nigbMlas;TIviC¢manvilRsbRTnicnaLika.

]TarhN_ 11-1³ sg;düaRkamkmøaMgkat; nigdüaRkamm:Um:gsMrab;FñwmEdlbgðajenAkñúgrUbTI 11-
;
10a. EI Camantémøefr.
dMeNaHRsay³
smIkar slope-deflection
enAkñúg]TahrN_enHeKRtUvBicarNaElVgBIr. edaysarKμanElVgNamYymanTMrsnøak; b¤TMrkl; dUcenHeK
RtUvGnuvtþsmIkar 11-8. edayeRbIrUbmnþsMrab; FEM EdlerobcMenAkñúgtaragsMrab;bnÞúkRtIekaN


wL2 6(6 )2
(FEM )BC =− =− = −7.2kN .m
30 30
2
6(6 )2
(FEM )CB = wL = = 10.8kN .m
20 20
cMNaMfa (FEM )BC GviC¢man edaysarvaeFVIGMeBIRcasRTnicnaLikaenARtg;cMNuc B . ehIy (FEM )AB
= (FEM )BA = 0 edaysarminmanbnÞúkenAelIElVg AB .

edIm,IsÁal;GBaØat ExSekageGLasÞicsMrab;FñwmRtUv)anbgðajenAkñúgrUbTI 11-10b. dUckar
bgðaj eKmanm:Um:g;Bt;EdlCaGBaØatcMnYnbYn. manEtmMurgVil θ B enARtg;cMNuc B b:ueNÑaHEdlCaGBaØat.
edaysar A nig C CaTMrbgáb; enaHθ A = θC = 0 . ehIy edaysarTMrminRsut ¬vaminmanbMlas;TIeLIg
elI b¤cuHeRkam¦ ψ AB = ψ BC = 0 . sMrab;ElVg AB ¬edayKitfa A CacugCit nig B Cacugq¶ay¦
eyIg)an
⎛I⎞
M N = 2 E ⎜ ⎟(2θ N + θ F − 3ψ ) + (FEM )N
⎝L⎠
⎛I⎞
M AB = 2 E ⎜ ⎟[2(0 ) + θ B − 3(0)] + 0 =
EI
θB (1)
⎝8⎠ 4
LÚvKitfa B CacugCit nig A Cacugq¶ay eyIg)an
⎛I⎞
M BA = 2 E ⎜ ⎟[2θ B + 0 − 3(0)] + 0 =
EI
θB (2)
⎝8⎠ 2
tamrebobdUcKña sMrab;ElVg BC eyIg)an

⎛I⎞
M BC = 2 E ⎜ ⎟[2θ B + 0 − 3(0)] − 7.2 =
2 EI
θ B − 7.2 (3)
⎝6⎠ 3
⎛I⎞
M CB = 2 E ⎜ ⎟[2(0 ) + θ B − 3(0 )] + 10.8 =
EI
θ B + 10.8 (4)
⎝6⎠ 3
smIkarlMnwg
smIkarTaMgbYnxagelImanGBaØatcMnYnR)aM. smIkarEdlcaM)ac;TIR)aM)anBIlkçxNÐlMnwgrbs;m:Um:g;enARtg;
TMr B . düaRkamGgÁesrIrbs;kMNat;Ggát;enARtg;cMNuc B RtUv)anbgðajenAkñúgrUbTI 11-10c. enATI
enH M BA nig M BC RtUv)ansnμt;eGayeFVIGMeBIkñúgTisedAviC¢manedIm,IeGayRsbeTAnwgsmIkar slope-
deflection . dUcenH
*

+ ∑MB = 0 M BA + M BC = 0 (5)
edIm,IedaHRsay CMnYssmIkar (2) nig (3) eTAkñúgsmIkar (5) enaHeyIg)an
6.17
θB =
EI
edayCMnYstémøenHeTAkñúgsmIkar (1) dl;smIkar (4) eyIg)an
M AB = 1.54kN .m

M BA = 3.09kN .m
M BC = −3.09kN .m

M CB = 12.86kN .m
témøGviC¢mansMrab; M BC bgðajfam:Um:g;enHeFVIGMeBIRcasRTnicnaLikaenAelIFñwm minEmnRsbRTnicna-
LikadUcbgðajenAkñúgrUbTI 11-10b.
edayeRbIlT§plenH eKkMNt;kMlaMgkat;enARtg;cugElVgBIsmIkarlMnwg ¬rUbTI 11-10d¦.
düaRkamGgÁesrIénFñwmTaMgmUl nigdüaRkamkMlaMgkat; nigdüaRkamm:Um:g;RtUv)anbgðajenAkñúgrUbTI 11-
10e.

]TarhN_ 11-2³ sg;düaRkamkmøaMgkat; nigdüaRkamm:Um:gsMrab;FñwmEdlbgðajenAkñúgrUbTI 11-
;
11a. EI Camantémøefr.
dMeNaHRsay³
enAkñúg]TahrN_enHeKRtUvBicarNaElVgBIr. eKGnuvtþsmIkar 11-8 eTAelIElVg AB . eyIgGaceRbI
*
RsbRTnicnaLikaenAelIFñwm b:uEnþRcasRTnicnaLikaenAelITMr.


smIkar 11-10 sMrab;ElVg BC edaysar C CaTMrkl;. edayeRbIrUbmnþsMrab; FEM EdlerobcMenAkñúg
tarag eyIg)an
2
(FEM ) AB = − wL =−
1
(40)(6)2 = −120kN .m
12 12
2
(FEM )BA = wL = 1 (40)(6)2 = 120kN .m
12 12
(FEM )BC = − 3PL = − 3(60)(2) = −22.5kN .m
16 16
cMNaMfa (FEM )AB nig (FEM )BC GviC¢man edaysarvaeFVIGMeBIRcasRTnicnaLikaenARtg;cMNuc A nig
B erogKña. ehIyedaysarTMrKμansMrut ψ AB = ψ BC = 0 . edayGnuvtþsmIkar 11-8 sMrab;ElVg AB

nigedaydwgfa θ A = 0 eyIg)an
⎛I⎞
M N = 2 E ⎜ ⎟(2θ N + θ F − 3ψ ) + (FEM )N
⎝L⎠
⎛I⎞
M AB = 2 E ⎜ ⎟[2(0) + θ B − 3(0)] − 120
⎝L⎠
M AB = 0.333EIθ B − 120 (1)
⎛I⎞
M BA = 2 E ⎜ ⎟[2θ B + 0 − 3(0)] + 120
⎝L⎠

M BA = 0.667 EIθ B + 120 (2)
edayGnuvtþsmIkar 11-12 CamYynwg B CacugCit nig C Cacugq¶ay eyIg)an
⎛I⎞
M N = 3E ⎜ ⎟(θ N −ψ ) + (FEM )N
⎝L⎠
⎛I⎞
M BC = 3E ⎜ ⎟(θ B − 0) − 22.5
⎝2⎠
M BC = 1.5 EIθ B − 22.5 (3)
cgcaMfa eKminGnuvtþsmIkar 11-10 BI B ¬cugCit¦ eTA B ¬cugq¶ay¦ eT.
smIkarlMnwg
smIkarTaMgbIxagelImanGBaØatcMnYnbYn. smIkarEdlcaM)ac;TIR)aM)anBIlkçxNÐlMnwgrbs;m:Um:g;enARtg;
TMr B . düaRkamGgÁesrIRtUv)anbgðajenAkñúgrUbTI 11-11b. eyIg)an
+ ∑MB = 0 M BA + M BC = 0 (4)
edIm,IedaHRsay CMnYssmIkar (2) nig (3) eTAkñúgsmIkar (4) enaHeyIg)an
45
θB = −
EI
edaysar θ B GviC¢man ¬RcasRTnicnaLika¦ ExSekageGLasÞicsMrab;FñwmRtUv)ansg;y:agRtwmRtUvenAkñúg
rUbTI 11-11a. edayCMnYs θ B eTAkñúgsmIkar (1) –(3) eyIg)an
M AB = −135kN .m

M BA = 90kN .m
M BC = −90kN .m
edayeRbIlT§plénm:Um:g;TaMgenH eyIgGackMNt;kmøaMgkat;enARtg;cugrbs;ElVgFñwmenAkñúgrUbTI 11-
11c. düaRkamkmøaMgkat; nigdüaRkamm:Um:g;RtUv)anbgðajenAkñúgrUbTI 11-11d.

]TarhN_ 11-3³ kMNt;m:Um:g;enARtg;cMNuc A nig B sRmab;FñwmEdlbgðajenAkñúgrUbTI 11-12a.
TMrenARtg; A mansMrut 80mm . yk E = 200GPa nig I = 5(106 )mm 4 .
dMeNaHRsay³
enAkúñgcMeNaTenH eyIgBicarNaEtElVg AB mYyb:ueNÑaH edaysarm:Um:g; M AB EdlbNþalBIFñwmly
EdlGacKNnaBIsþaTic. edaysarminmanbnÞúkenAelIElVg AB enaH FEM esμIsUnü. dUcbgðajenAkñúg
rUbTI 11-12b bMlas;TIcuHeRkamrbs;cMNuc B eFVIeGayElVg AB vilRsbRTnicnaLika. dUcenH



0.08m
ψ AB = ψ BA = = 0.02rad
4
kMrajrbs; AB KW
k=
( ) (
I 5 10 6 mm 4 10 −12 m 4 / mm 4
=
)
( )
= 1.25 10 − 6 m 3
L 4m
edayGnuvtþsmIkar slope-deflection ¬smIkar 11-8¦ eTAelIElVg AB CamYynwgθ A =0 eyIg)an
⎛I⎞
M N = 2 E ⎜ ⎟(2θ N + θ F − 3ψ ) + (FEM )N
⎝L⎠
( ( ) )[ ( ) ]
M AB = 2 200 109 N / m 2 1.25 10 −6 m 3 [2(0 ) + θ B − 3(0.02 )] + 0 (1)
M BA = 2(200(10 )N / m )[ .25(10 )m ][2θ
9
1 2 −6 3
B + 0 − 3(0.02 )] + 0 (2)
smIkarlMnwg
düaRkamGgÁesrIrbs;FñwmenARtg;TMr B RtUv)anbgðajenAkñúgrUbTI 11-12c. eyIg)an
+ ∑MB = 0 M BA − 8000 N (3m ) = 0
edIm,IedaHRsay CMnYssmIkar (2) eTAkñúgsmIkarenH eyIg)an
1(10 6 ) B − 30(103 ) = 24(103 )
θ
θ B = 0.054rad ]
dUcenHBIsmIkar (1) nig (2)
M AB = −3.00kN .m

M BA = 24.0kN .m

]TarhN_ 11-4³ kMNt;m:Um:g;enARtg;TMrrbs;FñwmEdlbgðajenAkñúgrUbTI 11-13a. TMrenARtg; A
mansMrut 30mm . yk E = 200GPa nig I = 600(106 )mm 4 .


dMeNaHRsay³
enAkúñgcMeNaTenH eKmanElVgcMnYnbIEdlRtUvBicarNa. eKeRbIsmIkar 11-8 edaysarTMr A nig D CaTMr
bgáb;. ehIymanEtElVg AB b:ueNÑaHEdlman FEM .
2
(FEM ) AB = − wL 1
(20)(7.2)2 = −86.4kN .m
=
12 12
wL2 1
(FEM )BA = = (20 )(7.2)2 = 86.4kN .m
12 12
dUcbgðajenAkñúgrUbTI 11-13b, sMrut ¬bMlas;TI¦ rbs;TMr C eFVIeGayψ BC viC¢man edaysarGgát; BC
vilRsbRTnicnaLika ehIyψ CD GviC¢man edaysarGgát; CD vilRcasRTnicnaLika. dUcenH
0.03m 0.03m
ψ BC = = 0.005rad ψ CD = − = −0.00667rad
6m 4m
edaybgðajxñatrbs;kMrajCaEm:Rt eyIg)an
600(106 )(10 −12 )
k AB = = 83.33( − 6 )m 3
10
7.2
600( 6 )(10 −12 )
= 100(10 − 6 )m 3
10
k BC =
6

k CD =
( )(
600 10 10 −12
6
( ) )
= 133.33 10 −6 m 3
4.5
cMNaMfa θ = θ = 0 edaysar A nig D CaTMrbgáb; ehIyedayGnuvtþsmIkar slope-deflection
A D

¬smIkar 11-8¦ BIrdgenAelIElVgnImYy² eyIg)an
sMrab;ElVg AB
M = 2[200(10 )[83.33(
AB 10 )]2(0 ) + θ − 3(0 )] − 86.4
6 −6
B
M AB = 33333.3θ B − 86.4 (1)

[ ( )[ ( )]
M BA = 2 200 10 6 83.33 10 − 6 2θ B + 0 − 3(0 ) + 86.4 ]


M BA = 66666.7θ B + 86.4 (2)
sMrab;ElVg BC
[ ( )[ ( )] ]
M BC = 2 200 10 6 100 10 − 6 2θ B + θ C − 3(0.005) + 0
M BC = 80000θ B + 40000θ C − 600 (3)

[ ( )[ ( )] ]
M CB = 2 200 10 6 100 10 − 6 2θ C + θ B − 3(0.005) + 0
M CB = 80000θ C + 40000θ B − 600 (4)
sMrab;ElVg CD
[ ( )[ ( )] ]
M CD = 2 200 10 − 6 133 10 − 6 2θ C + 0 − 3(− 0.00667 ) + 0
M CD = 106666.7θ C + 0 + 1066.7 (5)

[ ( )[ ( )] ]
M DC = 2 200 10 6 133.33 10 − 6 2(0 ) + θ C − 3(− 0.00667 ) + 0
M DC = 53333.3θ C + 1066.7 (6)
smIkarlMnwg
smIkarTaMgR)aMmYymanGBaØatR)aMbI. edaysresrsmIkarlMnwgrbs;m:Um:g;sMrab;TMrenARtg;cMNuc B nig
cMNuc C ¬rUbTI 10-13c¦ eyIg)an
+ ∑MB = 0 M BA + M BC = 0 (7)
+ ∑MC = 0 M CB + M CD = 0 (8)
edIm,IedaHRsay CMnYssmIkar (2) nig (3) eTAkñúgsmIkar (7) ehIyCMnYssmIkar (4) nig (5) eTAkñúg
smIkar (8) enaHeyIg)an
θ C + 3.667θ B = 0.01284
− θ C − 0.214θ B = 0.00250
dUcenH θ B = 0.00444rad θ C = −0.00345rad
témøviC¢mansMrab; θC bgðajkarvilRcasRTnicnaLikarrbs;bnÞat;b:HenARtg;cMNuc C ¬rUbTI 11-13a¦.
edayCMnYstémøTaMgenHeTAkñúgsmIkar (1)-(6) eyIg)an
M AB = 61.6kN .m

M BA = 383kN .m
M BC = −383kN .m

M CB = −698kN .m

M CD = 698kN .m

M DC = 883kN .m



!!>$> viPaKeRKagEdlmineyal (Analysis of Frame: No sidesway)
eRKagnwgmineyal b¤pøas;TIeTAeqVg b¤eTAsþaM RbsinebIeKTb;va)anl¥. ]TahrN_RtUv)anbgðaj
enAkñúgrUbTI 11-14. ehIyeRKagEdlminmankarTb;)anl¥nwgmineyaleT RbsinragFrNImaRt nigkar
rgbnÞúkrbs;vamanlkçN³sIuemRTI dUcbgðajenAkñúgrUbTI 11-15. sMrab;krNITaMgBIr tYψ enAkñúgsmIkar
slope-deflection esμIsUnü edaysarm:Um:g;Bt;min)aneFVIeGaytMNmanbMlas;TIlIenEG‘r.

]TahrN_xageRkambgðajBIkarGnuvtþénsmIkar slope-deflection edayeRbIdMeNIrkarkñúgkar
viPaKEdlmanerobrab;enAkñúgkfaxNÐ !!># sMrab;eRKagRbePTenH.

]TarhN_ 11-5³ kMNt;m:Um:g;enARtg;tMNnImYy²rbs;eRKagEdlbgðajenAkñúgrUbTI 11-16a. EI
mantémøefr.
dMeNaHRsay³
enAkúñgcMeNaTenH eKmanGgát;cMnYnbIEdlRtUvBicarNaKW AB / BC nig CD . edayeRKagenHmanTMrbgáb;
enARtg;cMNuc A nig D eyIgnwgGnuvtþsmIkar 11-8 sMrab;karedaHRsayenH.


BItaragm:Um:g;bgáb;cug/ FEM sMrab; BC KW
5wL2 5(24 )(8)2
(FEM )BC =− =− = −80kN .m
96 96
2
5(24 )(8)2
(FEM )CB = 5wL = = 80kN .m
96 96
cMNaMfa θ A = θ D = 0 ehIyψ AB = ψ BC = ψ CD = 0 edaysareRKagenHnwgmineyal.
edayGnuvtþsmIkar 11-8 eyIg)an
M N = 2 Ek (2θ N + θ F − 3ψ ) + (FEM )N
⎛ I ⎞
M AB = 2 E ⎜ ⎟[2(0) + θ B − 3(0)] + 0
⎝ 12 ⎠
M AB = 0.1667 EIθ B (1)
⎛ I ⎞
M BA = 2 E ⎜ ⎟[2θ B + 0 − 3(0 )] + 0
⎝ 12 ⎠
M BA = 0.333EIθ B (3)
⎛I⎞
M BC = 2 E ⎜ ⎟[2θ B + θ C − 3(0)] − 80
⎝8⎠
M BC = 0.5EIθ B + 0.25EIθ C − 80 (3)
⎛I⎞
M CB = 2 E ⎜ ⎟[2θ C + θ B − 3(0)] + 80
⎝8⎠
M CB = 0.8EIθ C + 0.25EIθ B + 80 (4)


⎛ I ⎞
M CD = 2 E ⎜ ⎟[2θ C + 0 − 3(0)] + 0
⎝ 12 ⎠
M CD = 0.333EIθ C (5)
⎛ I ⎞
M DC = 2 E ⎜ ⎟[2(0) + θ C − 3(0)] + 0
⎝ 12 ⎠
M DC = 0.1667 EIθ C (6)
smIkarlMnwg
smIkarTaMgR)aMmYymanGBaØatR)aMbI. smIkarlMnwgBIreTot)anBIsmIkarlMnwgrbs;m:Um:g;sMrab;TMrenARtg;
cMNuc B nig cMNuc C ¬rUbTI 11-16c¦. eyIg)an
M BA + M BC = 0 (7)
M CB + M CD = 0 (8)
edIm,IedaHRsaysmIkarTaMgR)aMbIenH CMnYssmIkar (2) nig (3) eTAkñúgsmIkar (7) nigCMnYssmIkar (4)
nig (5) eTAkñúgsmIkar (8). eyIg)an
0.833EIθ B + 0.25EIθ C = 80

0.833EIθ C + 0.25EIθ B = −80
edayedaHRsayRbB½n§smIkarenHeyIg)an
137.1
θ B = −θ C =
EI
EdleqøIytbeTAnwgrebobEdleRKagdabdUcbgðajenAkñúgrUbTI 11-16a. edayCMnYsvaeTAkñúgsmIkar
(1)-(6) eyIg)an

M AB = 22.9kN .m

M BA = 45.7 kN .m
M BC = −45.7kN .m

M BC = 45.7 kN .m

M CD = −45.7kN .m

M DC = −22.9kN .m
edayeRbIlT§plTaMgenH eyIgGackMNt;kmøaMgRbtikmμenARtg;cugrbs;Ggát;nImYy²BIsmIkarlMnwg ehIy
eyIgGacsg;düaRkamm:Um:g;sRmab;eRKagdUcbgðajenAkñúgrUbTI 11-16c.

]TarhN_ 11-6³ kMNt;m:Um:g;enARtg;tMNnImYy²rbs;eRKagEdlbgðajenAkñúgrUbTI 11-17a.
m:Um:g;niclPaBsMrab;Ggát;nImYy²RtUv)anbgðajenAkñúgrUb. yk E = 200GPa .


dMeNaHRsay³
enAkúñgcMeNaTenH eKmanGgát;cMnYnbYnEdlRtUvBicarNa. eyIgnwgGnuvtþsmIkar 11-8 cMeBaHGgát; AB
nig BC ehIyGnuvtþsmIkar 11-10 cMeBaHGgát; CD nig CE BIeRBaH D nig E CaTMrsnøak;.
edaykMNt;kMrajrbs;Ggát; eyIg)an
160( 6 )(10 −12 ) 80( 6 )( −12 )
= 35.56(
10 )m = 17.78(10 − 6 )m 3
10 −6 3 10 10
k AB = kCD =
4.8 4.5
320(10 )(10 ) 260(106 )(10 −12 )
= 66.67( − 6 )m 3 = 72.23( − 6 )m 3
6 −12
k BC = 10 kCE = 10
4.8 3.6
FEM EdlbNþalBIbnÞúkKW
30(4.8)
(FEM )BC =−
PL
=− = −18kN .m
8 8
PL 30(4.8)
(FEM )CB = = = 18kN .m
8 8
wL2 50(3.6 )2
(FEM )CE =− =− = −81kN .m
8 8
edayGnuvtþsmIkar 11-8 nigsmIkar 11-10 eTAelIeRKag nigcMNaMfaθ A = 0 /ψ AB = ψ BC = ψ CD =
ψ CE = 0 edaysarminmankareyal dUcenHeyIg)an

M N = 2 Ek (2θ N + θ F − 3ψ ) + (FEM )N

[ ( )] ( )
M AB = 2 200 10 6 (35.56 ) 10 − 6 [2(0 ) + θ B − 3(0 )] + 0

M AB = 14222.2θ B (1)

[ ( )] ( )
M BA = 2 200 10 6 (35.56 ) 10 − 6 [2θ B + 0 − 3(0 )] + 0

M BA = 28444.4θ B (2)



[ ( )] ( )
M BC = 2 200 10 6 (66.67 ) 10 −6 [2θ B + θ C − 3(0 )] − 18

M BC = 53333.3θ B + 26666.7θ C − 18 (3)

[ ( )] ( )
M CB = 2 200 10 6 (66.67 ) 10 − 6 [2θ C + θ B − 3(0)] + 18

M BC = 26666.7θ B + 53333.3θ C + 18 (4)

M N = 3Ek (θ N −ψ ) + (FEM )N

[ ( )] ( )
M CD = 3 200 10 6 (17.78) 10 −6 [θ C − 0] + 0

M CD = 10666.7θ C (5)

[ ( )] ( )
M CE = 3 200 106 (72.22 ) 10 − 6 [θ C − 0] + 81

M CE = 43333.3θ C − 81 (6)

smIkarlMnwg
smIkarTaMgR)aMmYymanGBaØatR)aMbI. eKGacsresrsmIkarlMnwgrbs;m:Um:g;cMnYnBIrsMrab;TMrenARtg;cMNuc
B nigcMNuc C ¬rUbTI 11-17c¦. eyIg)an
M BA + M BC = 0 (7)
M CB + M CD + M CE = 0 (8)
edIm,IedaHRsay CMnYssmIkar (2) nig(3) eTAkñúgsmIkar (7) ehIyCMnYssmIkar (4)-(6) eTAkñúgsmIkar
(8) enaHeyIg)an

81777.7θ Β + 26666.7θ C = 18

26666.7θ B + 107333.3θ C = 63
edaHRsayRbB½n§smIkarenHeyIg)an
θ B = 3.124(10 −5 )rad θ C = 5.792(10 −4 )rad
edaysartémøTaMgenHviC¢man dUcenHvaeFVIeGayeRKagxUcRTg;RTaydUcbgðajenAkñúgrUbTI 11-17a.
edayCMnYstémøTaMgenHeTAkñúgsmIkar (1)-(6) nigedaHRsay eyIg)an
M AB = 0.444kN .m

M BA = 0.888kN .m
M BC = −0.888kN .m

M CB = 49.7kN .m



M CD = 6.48kN .m

M CE = −55.9kN .m

!!>%> viPaKeRKagEdleyal (Analysis of frames: Sidesway)
eRKagnwgeyal b¤cl½teTAxagenAeBlEdlva b¤bnÞúkEdlmanGMeBIelIvamanlkçN³minsIuemRTI.
edIm,IbgðajBIT§iBlenH BicarNaeRKagEdlbgðajenAkñúgrUbTI 11-18. enATIenH bnÞúk P eFVIeGayman
m:Um:g; M BC nig M CB enARtg;cMNuc B nig C erogKña ehIym:Um:g;TaMgBIrenHminesμIKñaeT. m:Um:g; M BC eFVI
eGaycMNuc B cl½teTAsþaM Ém:Um:g; M CB eFVIeGaycMNuc C cl½teTAeqVg. edaysar M BC FMCag M CB
enaHlT§plénbMlas;TIsrub Δ rbs;tMN B nig C KWeTAsþaM dUcbgðajenAkñúgrUb . enAeBlGnuvtþsmIkar
*

slope-deflection eTAelIssrnImYy²rbs;eRKag eyIgRtUvBicarNamMurgVilrbs;ssr ψ ¬edaysar

ψ = Δ / L ¦ CaGBaØatenAkñúgsmIkar. CalT§pl eKRtUvkar
bBa¢ÚlsmIkarlMnwgbEnßmsMrab;karedaHRsay. enAkñúgkfa-
xNÐelIkmun eyIgeXIjfabMlas;TImMu θ EdlCaGBaØatmanTMnak;
TMngenAkñúgsmIkar lMnwgénm:Umg;. tamrebobdUcKña enAeBl
:
ekItmanbMlas;TIlIenEG‘r Δ ¬b¤mMurgVilrbs;ElVg ψ ¦ eyIgRtUv
sresrsmIkarlMnwgénkMlaMgedIm,ITTYl)andMeNaHRsayeBj
elj. b:uEnþ GBaØatenAkñúgsmIkarTaMgenHRtUvEtBak;B½n§nwg
m:Um:g;Bt;Edlman GMeBIenARtg;cugssr edaysarsmIkar slope-
deflection Bak;B½n§eTAnwgm:Um:g;TaMgenH. bec©keTssMrab;edaH

RsaycMeNaTéneRKagEdleyalRtUv)anbgðajenAkñúg]TahrN_.

]TarhN_ 11-7³ kMNt;m:Um:g;enARtg;tMNnImYy²rbs;eRKagEdlbgðajenAkñúgrUbTI 11-19a. EI
mantémøefr.
dMeNaHRsay³
edaysarTMr A nig D CaTMrbgáb; eKGacGnuvtþsmIkar 11-8 sRmab;Ggát;TaMgbIrbs;eRKag. enATIenH
eRKagmanPaBeyal edaysarkardak;bnÞúk nigragFrNImaRtrbs;eRKagmanlkçN³minsIuemRTI. enATI
*
rMlwkfa kMhUcRTg;RTayrbs;Ggát;TaMgbIKWbNþalBIm:Um:g;Bt; edayecalkmøaMgkat;nigkmøaMgtamG½kS.


enH eKdak;bnÞúkedaypÞal;eTAelItMN B dUcenHminman FEM eFVIGMeBIeTAelItMNeT. dUcbgðajenAkñúgrUb
TI 11-19a eKsnμt;eGaytMN B nigtMN C pøas;TIedaybrimaN Δ esμIKña. Cavi)ak ψ AB = Δ / 4 ehIy
ψ DC = Δ / 6 . tYTaMgBIrenHviC¢man edaysarGgát; AB nig CD vilRsbTisRTnicnaLika. edayP¢ab;
TMnak;TMngψ AB eTAnwgψ DC eyIg)anψ AB = (6 / 4)ψ DC . edayGnuvtþsmIkar 11-8 eTAelIeRKag
eyIg)an
⎛ I ⎞⎡ ⎛6 ⎞⎤
M AB = 2 E ⎜ ⎟ ⎢2(0) + θ B − 3⎜ ψ DC ⎟⎥ + 0 = EI (0.5θ B − 2.25ψ DC ) (1)
⎝ 4 ⎠⎣ ⎝4 ⎠⎦
⎛ I ⎞⎡ ⎛6 ⎞⎤
M BA = 2 E ⎜ ⎟ ⎢2θ B + 0 − 3⎜ ψ DC ⎟⎥ + 0 = EI (1.0θ C − 2.25ψ DC ) (2)
⎝ 4 ⎠⎣ ⎝4 ⎠⎦
⎛I⎞
M BC = 2 E ⎜ ⎟[2θ B + θ C − 3(0)] + 0 = EI (0.8θ B + 0.4θ C ) (3)
⎝5⎠
⎛I⎞
M CB = 2 E ⎜ ⎟[2θ C + θ B − 3(0)] + 0 = EI (0.8θ B + 0.4θ C ) (4)
⎝5⎠
⎛I⎞
M CD = 2 E ⎜ ⎟[2θ C + 0 − 3ψ DC ] + 0 = EI (0.667θ C − 1.0ψ DC ) (5)
⎝6⎠
⎛I⎞
M DC = 2 E ⎜ ⎟[2(0) + θ C − 3ψ DC ] + 0 = EI (0.333θ C − 1.0ψ DC ) (6)
⎝6⎠

smIkarlMnwg
smIkarTaMgR)aMmYymanGBaØatR)aMbYn. eKGacsresrsmIkarlMnwgrbs;m:Um:g;cMnYnBIrsMrab;TMrenARtg;
cMNuc B nigcMNuc C ¬rUbTI 11-17c¦. eyIg)an


M BA + M BC = 0 (7)
M CB + M CD = 0 (8)
edaysarvaekItmanbMlas;TItamTisedk Δ / eyIgnwgBicarNarplbUkkmøaMgenAelIeRKagTaMgmUltam
Tis x dUcenHeyIgTTYl)an
+
→ ∑ Fx = 0 200 − V A − VD = 0
eKGacP¢ab;TMnak;TMngkmøaMgRbtikmμtamTisedk b¤kmøaMgkat;enAkñúgssrVA nigVD eTAnwgm:Um:g;Bt;
edayBicarNadüaRkamGgÁesrIénssrdac;edayELkBIKña ¬rUbTI 11-19c¦. eyIgman
M AB + M BA
∑MB = 0 VA = −
4
M + M CD
∑ MC = 0 VD = − DC
6
M AB + M BA M DC + M CD
dUcenH 200 +
4
+
6
=0 (9)

edIm,IedaHRsay CMnYssmIkar (2) nig(3) eTAkñúgsmIkar (7) ehIyCMnYssmIkar (4) nig(5) eTAkñúg
smIkar (8) ehIyCMnYssmIkar (1), (2), (5) nig (6) eTAkñúgsmIkar (9) enaHeyIgTTYl)an
1.8θ Β + 0.4θ C − 2.25ψ DC = 0

0.4θ B + 1.467θ C −ψ DC = 0
800
1.5θ B + 0.667θ C − 5.833ψ DC =
EI
edaHRsayRbB½n§smIkarxagelI eyIgTTYl)an
EIθ B = 243.78 EIθ C = 75.66 EIψ DC = 208.48
cugeRkay edayeRbIlT§plTaMgenH nigedaHRsaysmIkar (1)-(6) eyIg)an
M AB = −347 kN .m

M BA = −225kN .m
M BC = 225kN .m

M CB = 158kN .m

M CD = −158kN .m

M DC = −183kN .m

]TarhN_ 11-8³ kMNt;m:Um:g;enARtg;tMNnImYy²rbs;eRKagEdlbgðajenAkñúgrUbTI 11-20a. TMr
enARtg; A nig D CaTMrbgáb; ehIytMN C RtUv)ansnμt;faCatMNsnøak;. EI mantémøefrsMrab;RKb;
Ggát;TaMgGs;.


dMeNaHRsay³
eKGacGnuvtþsmIkar 11-8 eTAelIGgát; AB edaysarvaRtUv)antP¢ab;edaytMNbgáb;enAcugTaMgsgçag.
eKGacGnuvtþsmIkar11-10 BI B eTA C nigBI D eTA C edaysarsnøak;enARtg;cMNuc C RTm:Um:g;sUnü. dUc
bgðajedaydüaRkamPaBdab ¬rUbTI 11-20b¦ eKmanbM;las;TIlIenEG‘r Δ nigbMlas;TImMuθ B enARtg;
cMNuc B EdlCaGBaØat*. edaysar Δ Ggát; AB nig CD vilRsbRTnicnaLika ψ = ψ AB = ψ DC =
Δ / 4 . edaydwgfa θ A = θ D = 0 nigvaminman FEM sMrab;Ggát; enaHeyIg)an

*
bMlas;TImMu θ nig θ enARtg;tMN C ¬snøak;¦ minRtUv)anrab;bBa©ÚlenAkñúgkarviPaKeT edaysareKeRbIsmIkar 11-10.
CB CD



⎛I⎞
M N = 2 E ⎜ ⎟(2θ N + θ F − 3ψ ) + (FEM )N
⎝L⎠
⎛I⎞
M AB = 2 E ⎜ ⎟[2(0 ) + θ B − 3ψ ] + 0 (1)
⎝4⎠
⎛I⎞
M BA = 2 E ⎜ ⎟(2θ B + 0 − 3ψ ) + 0 (2)
⎝4⎠
⎛I⎞
M N = 3E ⎜ ⎟(θ N − ψ ) + (FEM )N
⎝L⎠
⎛I⎞
M BC = 3E ⎜ ⎟(θ B − 0) + 0 (3)
⎝3⎠
⎛I⎞
M DC = 3E ⎜ ⎟(0 − ψ ) + 0 (4)
⎝4⎠
smIkarlMnwg
smIkarlMnwgrbs;m:Um:g;enARtg;tMN B ¬rUbTI 11-20c¦. eyIg)an
M BA + M BC = 0 (5)
RbsinebIeKbUkkmøaMgsRmab;eRKagTaMgmUltamTisedAedk eyIg)an
+
→ ∑ Fx = 0 10 − V A − VD = 0 (6)
dUcbgðajenAkñúgdüaRkamGgÁesrIrbs;ssrnImYy² ¬rUbTI 11-20d¦ eyIg)an
M AB + M BA
∑MB = 0 VA = −
4
M DC
∑ MC = 0 VD = −
4
dUcenHBIsmIkar (6)
M AB + M BA M DC
10 + + =0 (7)
4 4
edayCMnYssmIkar slope-deflection eTAkñúgsmIkar (5) nig (7) ehIyedaysMrYlvaeyIgTTYl)an
3
θB = ψ
4
EI ⎛ 3 15 ⎞
10 + ⎜ θB − ψ ⎟ = 0
4 ⎝2 4 ⎠
dUcenH θB =
240
21EI
ψ=
320
21EI
edayCMnYstémøTaMgenHeTAkñúgsmIkar (1)-(4) eyIg)an
M AB = −17.1kN .m M BA = −11.4kN .m

M BC = 11.4kN .m M DC = −11.4kN .m
edayeRbIlT§plTaMgenH eKGackmøaMgRbtikmμcugrbs;Ggát;nImYy²BIsmIkarlMnwg ¬rUbTI 11-20e¦.
düaRkamm:Um:g;sMrab;eRKagRtUv)anbgðajenAkñúgrUbTI 11-20f.


]TarhN_ 11-9³ cUrBnül;BIrebobkMNt;m:Um:g;enARtg;tMNnImYy²éneRKagBIrCan;EdlbgðajenAkñúg
rUbTI 11-21a. EI mantémøefr.

dMeNaHRsay³
edaysarTMrenARtg; A nig F CaTMrbgáb; eKGacGnuvtþsmIkar 11-8 sMrab;Ggát;TaMgR)aMmYyrbs;eRKag.
eKminKNna FEM eT edaysarbnÞúkTaMgGs;manGMeBIenARtg;tMN. enATIenH bnÞúkeFVIeGaytMN B nig
tMN E pøas;TIedaybrimaN Δ1 ehIyeFVIeGaytMN C nigtMN D pøas;TIedaybrimaN Δ1 + Δ 2 . Ca
lT§pl Ggát; AB nigGgát; FE rgkarviledaybrimaNψ 1 = Δ1 / 5 ehIy BC nig DE rgkarvileday
brimaN ψ 2 = Δ 2 / 5 .
edayGnuvtþsmikar 11-8 eTAelIeRKageyIg)an
⎛I⎞
M AB = 2 E ⎜ ⎟[2(0) + θ B − 3ψ 1 ] + 0 (1)
⎝5⎠
⎛I⎞
M BA = 2 E ⎜ ⎟[2θ B + 0 − 3ψ 1 ] + 0 (2)
⎝5⎠
⎛I⎞
M BC = 2 E ⎜ ⎟[2θ B + θ C − 3ψ 2 ] + 0 (3)
⎝5⎠



⎛I⎞
M CB = 2 E ⎜ ⎟[2θ C + θ B − 3ψ 2 ] + 0 (4)
⎝5⎠
⎛I⎞
M CD = 2 E ⎜ ⎟[2θ C + θ D − 3(0)] + 0 (5)
⎝7⎠
⎛I⎞
M DC = 2 E ⎜ ⎟[2θ D + θ C − 3(0 )] + 0 (6)
⎝7⎠
⎛I⎞
M BE = 2 E ⎜ ⎟[2θ B + θ E − 3(0)] + 0 (7)
⎝7⎠
⎛I⎞
M EB = 2 E ⎜ ⎟[2θ E + θ B − 3(0 )] + 0 (8)
⎝7⎠
⎛I⎞
M ED = 2 E ⎜ ⎟[2θ E + θ D − 3ψ 2 ] + 0 (9)
⎝5⎠
⎛I⎞
M DE = 2 E ⎜ ⎟[2θ D + θ E − 3ψ 2 ] + 0 (10)
⎝5⎠
⎛I⎞
M FE = 2 E ⎜ ⎟[2(0 ) + θ E − 3ψ 1 ] + 0 (11)
⎝5⎠
⎛I⎞
M EF = 2 E ⎜ ⎟[2θ E + 0 − 3ψ 1 ] + 0 (12)
⎝5⎠
smIkarTaMg 12enHmanGBaØat 18.
smIkarlMnwg
smIkarlMnwgrbs;m:Um:g;enARtg;tMN B / C / D nig E ¬rUbTI 11-21c¦. eyIg)an
M BA + M BE + M BC = 0 (13)
M CB + M CD = 0 (14)
M DC + M DE = 0 (15)
M EF + M EB + M ED = 0 (16)
dUcenAkñúg]TahrN_elIkmun kmøaMgkat;enARtg;Kl;rbs;ssrsMrab;RKb;Can;RtUvEteFVIeGaybnÞúktamTis
edkmantulüPaB ¬rUbTI 11-21c¦. eyIg)an
+
→ ∑ Fx = 0 40 − VBC − VED = 0
M + M CB M ED + M DE
40 + BC + =0 (17)
5 5
+
→ ∑ Fx = 0 40 + 80 − V AB − VFE = 0
M + M BA M EF + M FE
120 + AB + =0 (18)
5 5
dMeNaHRsayTamTarnUvkarCMnYssmIkr (1) - (12) eTAkñúgsmIkar (13) - (18) EdlTTYl)anR)aMmYy
smIkarEdlmanR)aMmYyGBaØatKW ψ 1 / ψ 2 / θ B / θC / θ D nigθ E . eKGacedaHRsayRbB½n§smIkarenH.
eKRtUvCMnYslT§plEdlTTYl)aneTAkñúgsmIkar (1) –(12) EdleFVIeGayeyIgTTYl)anm:Um:g;enARtg;tMN.


]TarhN_ 11-10³ kMNt;m:Um:g;enARtg;tMNnImYy²éneRKagEdlbgðajenAkñúgrUbTI 11-22a. EI
mantémøefrsMrab;Ggát;nImYy².

dMeNaHRsay³
eKGacGnuvtþsmIkar 11-8 eTAelIGgát;TaMgbI. FEM KW
2
30(3.6 )2
(FEM )BC = − wL =− = −32.4kN .m
12 12
wL2 30(3.6 )2
(FEM )CB = = = 32.4kN .m
12 12
Ggát;eRTt AB eFVIeGayeRKageTreTAsþaMdUcbgðajkñúgrUbTI 11-22a. CalT§pl tMN B nig C rgnUv
bMlas;TImMu nigbMlas;TIlIenEG‘r. bMlas;TIlIenEG‘rRtUv)anbgðajenAkñúgrUbTI 11-22b Edl B cl½teTA
B' )anTMhM Δ1 ehIy C cl½teTA C ' )an Δ 3 . bMlas;TITaMgenHeFVIeGayGgát;vilRsbRTnicnaLikaψ 1 /

ψ 3 niigRcasRTnicnaLika − ψ 2 dUcbgðaj . dUcenH *

*
rlWkfa kMhUcRTg;RTayEdlbNþalBIkmøaMgtamG½kSRtUv)anecal ehIybMlas;TI BB' nig CC ' RtUv)anKitCaragbnÞat; eday
sarCak;Esþgψ 1 nigψ 3 mantémøtUc.


Δ1 Δ1 Δ3
ψ1 = ψ2 = − ψ3 =
3 3.6 6
dUcbgðajenAkñúgrUbTI 11-22c bMlas;TITaMgbImanTMnak;TMngKña. ]TahrN_ Δ 2 = 0.5Δ1 ehIy Δ 3 =
0.866Δ1 . dUcenH BIsmIkarxagelI eyIg)an

ψ 2 = −0.417ψ 1 ψ 3 = 0.433ψ 1
edaseRbIlT§plTaMgenH smIkar slope-deflection sMrab;eRKagKW
⎛I⎞
M AB = 2 E ⎜ ⎟[2(0) + θ B − 3ψ 1 ] + 0 (1)
⎝3⎠
⎛I⎞
M BA = 2 E ⎜ ⎟[2θ B + 0 − 3ψ 1 ] + 0 (2)
⎝3⎠
⎛ I ⎞
M BC = 2 E ⎜ ⎟[2θ B + θ C − 3(− 0.417ψ 1 )] − 32.4 (3)
⎝ 3.6 ⎠
⎛ I ⎞
M CB = 2 E ⎜ ⎟[2θ C + θ B − 3(− 0.417ψ 1 )] + 32.4 (4)
⎝ 3.6 ⎠
⎛I⎞
M CD = 2 E ⎜ ⎟[2θ C + 0 − 3(0.433ψ 1 )] + 0 (5)
⎝6⎠
⎛I⎞
M DC = 2 E ⎜ ⎟[2(0) + θ C − 3(0.433ψ 1 )] + 0 (6)
⎝6⎠
smIkarTaMgR)aMmYymanGBaØatR)aMbYn.
smIkarlMnwg
smIkarlMnwgrbs;m:Um:g;enARtg;tMN B nig C . eyIg)an
M BA + M BC = 0 (7)
M CD + M CB = 0 (8)
eKGacTTYl)ansmIkarcaM)ac;TIbIedayeFVIplbUkm:Um:g;enARtg;cMNuc O ¬rUbTI 11-22d¦. smIkarenH
lubbM)at;kmøaMgEkg N A nig N D dUcenH
+ ∑ MO = 0
⎛ M + M BA ⎞ ⎛M + M CD ⎞
M AB + M DC − ⎜ AB ⎟(10.2) − ⎜ DC ⎟(12.24) − 108(1.8) = 0
⎝ 3 ⎠ ⎝ 6 ⎠
− 2.4M AB − 3.4M BA − 2.04 M CD − 1.04M CD − 194.4 = 0 (9)
edayCMnYssmIkar (2) nig (3) eTAkñúgsmIkar (7) smIkar (4) nigsmIkar (5) eTAkñúgsmIkar (8) nig
smIkar (1), (2), (5) nig (6) eTAkñúgsmIkar (9) eyIg)an
9.72
0.733θ B + 0.167θ C − 0.392ψ 1 =
EI
9.72
0.167θ B + 0.533θ C + 0.0784ψ 1 = −
EI
58.32
− 1.840θ B − 0.512θ C + 3.880ψ 1 =
EI


edaHRsayRbB½n§smIkarTaMgenH eyIg)an
EIθ B = 35.51 EIθ C = −33.33 EIψ 1 = 27.47
edayCMnYstémøTaMgenHeTAkñúgsmIkar (1) – (6), eyIg)an
M AB = −31.3kN .m M BC = 7.60kN .m M CD = −34.2kN .m

M BA = −7.60kN .m M CB = 34.2kN .m M DC = −23.0kN .m



cMeNaT
!!>!> kMNt;m:Um:g;enARtg;TMr A nig C bnÞab;mksg; !!>%> kMNt;m:Um:g;enARtg;TMr B bnÞab;mksg;düa-
düaRkamm:Um:g;. snμt;fatMN B CatMNkl;. EI Rkamm:Um:g;sMrab;Fñwm. snμt;TMrenARtg; A nig C
mantémøefr. CaTMrbgáb;. EI mantémø efr.

!!>@> kMNt;m:Um:g;enARtg;TMr A nig B bnÞab;mk !!>^> kMNt;m:Um:g;Bt;enARtg;TMr A / B nig C
sg;düaRkamm:Um:g;. EI mantémøefr. bnÞab;mksg;düaRkamm:Um:g;sMrab;Fñwm. snμt;TMr
enARtg; A CaTMrsnøak; ehIy B nig C CaTMrkl;.
EI mantémøefr.

!!>#> kMNt;m:Um:g;enARtg;TMr A nig B bnÞab;mk
sg;düaRkamm:Um:g;sMrab;Fñwm. EI mantémøefr.

!!>&> kMNt;RbtikmμTMrenARtg; A / B nig C
bnÞab;mksg;düaRkamm:Um:g;sMrab;Fñwm. snμt;TMr
enARtg; A CaTMrsnøak;.
!!>$> kMNt;m:Um:g;enARtg;TMr B nig C bnÞab;mk
sg;düaRkamm:Um:g;sMrab;Fñwm. snμt; A / B nig
C CaTMrkl; ehIy D CaTMrsnøak;. EI mantémø

efr.
!!>*> kMNt;m:Um:g;enARtg; B / C nig D bnÞab;
mksg;düaRkamm:Um:g;sMrab; ABDE . snμt; A
CaTMrsnøak; nig D CaTMrkl; ehIy C CaTMrbgáb;.
EI mantémøefr.

cMeNaT T.Chhay -410


!!>(> kMNt;m:Um:g;enARtg; B bnÞab;mksg;düa-
Rkamm:Um:g;sMrab;Ggát;nImYy²rbs;eRKag. snμt;
A nig C CaTMrsnøak; ehIy B CatMNbgáb;. EI

mantémøefr.
!!>!@> kMNt;m:Um:g;enARtg; B nig C . snμt; B nig
C CaTMrkl; ehIy A nig D CatMNsnøak;. EI

mantémøefr.

!!>!0> kMNt;m:Um:g;enARtg; B nig D bnÞab;mksg; !!>!#> kMNt;kmøaMgRbtikmμtamTisedk nigtam
düaRkamm:Um:g;. snμt; A nig C CaTMrsnøak; ehIy TisQrenARtg; A nig C . snμt; A nig C CaTMr
B nig D CatMN nigTMrbgáb;. EI mantémøefr.
snøak; ehIy B CatMNbgáb;. yk E = 200GPa .

!!>!!> kMNt;m:Um:g;enARtg; B bnÞab;mksg;düa-
Rkamm:Um:g;sMrab;Ggát;nImYy²rbs;eRKag. snμt;
A nig C CaTMrbgáb; ehIy B CatMNbgáb;. EI !!>!$> kMNt;m:Um:g;Bt;enARtg; A nig B bnÞab;mk
mantémøefr. sg;düaRkamm:Um:g;. snμt; B nig C CaTMrkl;.
Problems T.Chhay -411


EI mantémøefr. !!>!*> kMNt;m:Um:g;enARtg;tMNnImYy² nigTMrrbs;
eRKag. tMN nigTMrCatMN nigTMrbgáb;. EI man
témøefr.

!!>!%> kMNt;m:Um:g;Bt;enARtg; A / B nig C bnÞab;
mksg;düaRkamm:Um:g;. snμt; A CaTMrbgáb;. EI
mantémøefr.
!!>!(> eRKagenHeFVIBIbMBg;TIbEdlmantMNbgáb;.
RbsinebIvaRTbnÞúkdUcbgðaj cUrkMNt;m:Um:g;Edl
ekItmanenARtg;tMN nigTMrnImYy². EI man
!!>!^> kMNt;m:Um:g;Bt;enARtg;cugénGgát;rbs; témøefr.
eRKag. TMrenARtg; A nig C TMrbgáb; ehIy B
CatMNbgáb;. EI mantémøefr.

!!>@0> kMNt;m:Um:g;Bt;enARtg;tMNnImYy² nigenA
Rtg;TMrbgáb; bnÞab;mksg;düaRkamGgÁesrI. EI
mantémøefr.

!!>!&> FñwmCab;RTbnÞúkcMcMNucbI. kMNt;m:Um:g;Bt;
GtibrmaenAkñúgFñwm nigbnÞab;mksg;düaRkam
m:Um:g;. EI mantémøefr.
!!>@!> kMNt;m:Um:g;Bt;enARtg;tMN nigTMrnImYy².
eKmantMNbgáb;enARtg; B nig C ehIyeKmanTMr
bgáb;enARtg; A nig D . EI mantémøefr.
cMeNaT T.Chhay -412


EdlmanTMrsamBaØ. snμt;TMrRtg; A nig E CaTMr
snøak; ehIytMNTaMgGs;CatMNbgáb;. EI man
témøefr.

!!>@@> kMNt;m:Um:g;Bt;enARtg; A / B / C nig D
bnÞab;mksg;düaRkamm:Um:g;. Ggát;nImYy²RtUv
)antP¢ab;edaytMNbgáb; nigmanTMrbgáb;. EI
mantémøefr.
!!>@%> edaHRsaycMeNaT !!>@$ edaysnμt;faTMr
A nig E CaTMrbgáb;.

!!>@^> kMNt;m:Um:g;Bt;enARtg;tMNnImYy²rbs;
eRKag. TMr A nig D CaTMrsnøak;. EI mantémø
efr.

!!>@#> épÞxagrbs;eRKagrgbnÞúkGIuRdUsþaTicdUc
bgðaj. kMNt;m:Um:g;Bt;enARtg;tMN nigTMr
nImYy². EI mantémøefr.
!!>@&> kMNt;m:Um:g;Bt;enARtg;tMNnImYy²rbs;
eRKag nigenARtg;TMr A nig D EdlCaTMrbgáb;.
EI mantémøefr.

!!>@$> kMNt;m:Um:g;Bt;enARtg;tMNnImYy²rbs;
eRKag. bnÞúkelIdMbUlRtUv)anepÞreTAédrENg
Problems T.Chhay -413


KMerag !!>!> dMbUlRtUv)anRTedayrnUtEdlQr
elIr:tBIr. eKKitfarnUtnImYy²RtUv)anRTedayTMr
samBaØ ehIyeKKitfar:txagmuxRtUv)anRTeday
ssrEdlCaTMrsnøak;enARtg; A nigCaTMrkl;enA
Rtg; B nig C . snμt;fadMbUlRtUv)anplitBIebtug
EdlmankRmas; 75mm ehIyrnUtnImYy²man
TMgn; 2.5kN . eyagtamkUddMbUlenHrgnUvbnÞúk
RBwlEdlmanTMgn; 1.2kN / m2 . rnUtmanRbEvg
8m . sg;düaRkamkmøaMgkat; nigdüaRkamm:Um:g;

Bt;sMrab;r:t. snμt;fassrmanlkçN³rwg.

cMeNaT T.Chhay -414

11. displacement method of analysis slope deflection equations

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11. displacement method of analysis slope deflection equations