Beam analysis using the stiffness method

mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa

!%> karviPaKFñwmedayeRbIviFIPaBrwgRkaj
(Beam analysis using the stiffness method)

emeronenHnwgbgðajBIeKalKMnitEdlerobrab;enAkñúgemeronmunedIm,IviPaKFñwm. eyIgeXIjfa enA
eBlEdleKbegáItm:aRTIsPaBrwgRkajsRmab;Ggát; nigm:aRTIsbMElg dMeNIrkarsRmab;karGnuvtþmanlkçN³
RsedogKñanwgkarGnuvtþsRmab; truss. emeronenHnwgbgðajBIkrNIBiesssRmab;sRmutDIepr:g;Esül
nigsItuNðPaB.

!%>!> kMNt;cMNaMbzm (preliminary remarks)
munnwgeyIgbgðajBIrebobEdlviFIPaBrwgRkajGnuvtþeTAelIFñwm dMbUgeyIgnwgerobrab;BIeKalKMnit
bzm nigniymn½yEdlerobrab;eTAnwgGgát;TaMgenH.
karkMNt;sMKal; node nigGgát;³ edIm,IGnuvtþviFIPaBrwgRkajeTAelIFñwm dMbUgeyIgRtUvEtkMNt;BIrebobbMEbk
FñwmCabgÁúMFatuGnnþtUc (finite element). CaTUeTA FatunImYy²minRtUvTak;TgnwgbnÞúkeT ehIyRtUvmanmux
kat;efr. sRmab;mUlehtuenH node rbs;FatuRtUvEtsßitenARtg;TMr b¤Rtg;cMNucEdlGgát;P¢ab;Kña Rtg;bnÞúk
Gnuvtþ Rtg;RkLaépÞmuxkat;pøas;bþÚrPøam² b¤Rtg;cMNucEdleKRtUvkMNt;bMlas;TImMu. ]TahrN_ eKmanFñwm
dUcbgðajenAkñúgrUbTI 15-1a. edayeRbIviFIdUcKñasRmab;karviPaK truss eKRtUvkMNt; node bYnCaelxeday
dak;kñúgrgVg;. dUcKña cMNaMfa cugCit nigcugq¶ayrbs;Ggát;nImYy²RtUv)ankMNt;sMKal;edaysBaØaRBYjtam
beNþayGgát;nImYy².

kUGredaenskl nigkUGredaen³ eyIgnwgkMNt;RbB½n§kUGredaenskledayeRbIG½kS x, y, z EdleKalrbs;
vasßitenARtg; node EdleFVIeGaykUGredaenrbs;cMNucdéTeTotenAelIFñwmviC¢man ¬rUbTI 15-1a¦. kUGr-
Beam analysis using the stiffness method T.Chhay -501

Department of Civil Engineering NPIC

edaentMbn; nigkUGredaensRmab;Ggát; x' , y' , z' maneKalrbs;vasßitenAcugCitrbs;Ggát;nImYy² G½kS
viC¢man x' manTisedAeTArkcugq¶ay. rUbTI 15-1b bgðajkUGredaenTaMgenHsRmab;Ggát;elx @. enAkñúg
krNITaMgBIr eyIg)aneRbIRbB½n§kUGredaenédsþaM dUcenHRbsinebIeyIgbgVilédsþaMBIG½kS x b¤ x' eTAG½kS y b¤
y ' eyIgnwgTTYl)anG½kS z b¤ z ' viC¢man EdlmanTisedAecjBIRkdasesovePA. cMNaMfa sRmab;Ggát;Fñwm

nImYy²G½kS x nig x' nwgsßitenAelIG½kSEtmYy ehIykUGredaenskl nigkUGredaensRmab;Ggát;TaMgGs;suT§
EtRsbKña. dUcenH ¬mindUckrNIsRmab; truss¦ enATIenH eyIgnwgmincaM)ac;begáItm:aRTIsbMElgrvagRbB½n§
kUGredaenTaMgenH.
PaBminkMNt;sIueNm:aTic³ eRkayeBlGgát; nig node RtUv)ankMNt; nigeRkayeBlRbB½n§kUGredaenskl
RtUv)anbegáIt eKGackMNt; degree of freedom nigPaBminkMNt;sIueNm:aTicsRmab;Fñwm. RbsinebIeyIg
KitTaMgT§iBlm:Um:g;Bt; nigkmøaMgkat; enaH node nImYy²enAelIFñwmGaclman degree of freedom BIreBal
KW bMlas;TItamTisQr nigmMurgVil. dUcenAkñúgkrNIrbs; truss eKGackMNt;bMlas;TITaMgenHedayelxkUd.
eyIgeRbIelxkUdtUcbMputedIm,IeFVIkMNt;sMKal;bMlas;TIEdlCaGBaØat ¬degree of freedom Edlminmankar
Tb;¦ ehIyeyIgeRbIelxFMbMputedIm,IeFVIkMNt;sMKal;bMlas;TIEdleyIgsÁal; ¬degree of freedom Edlman
karTb;¦. rMlwkfa mUlehtukñúgkareRCIserIsviFIénkareFVIkMNt;sMKal; eyIgRtUveFVIkarEbgEckm:aRTIsPaBrwg
RkajsRmab;rcnasm½<n§eGay)anRsYl dUcenHeyIgGacKNnabM;las;TIEdlCaGBaØattamrebobedaypÞal;
bMput.

edIm,Ibgðaj]TahrN_kñúgkardak;elxkUd eyIgnwgBicarNaFñwmCab;mþgeTotenAkñúgrUbTI 15-1a. enA
TIenH FñwmCarcnasm<½n§minkMNt;edaysIueNm:aTicdWeRkTIbYn. eKman degree of freedom cMnYnR)aMbI Edl
elxkUd ! dl;elxkUd $ tMNageGaybMlas;TIEdlCaGBaØat ehIyelxkUd % dl;elxkUd * tMNageGay
bMlas;TIEdleKsÁal; ¬EdlenAkñúgkrNIenHvasuT§EtmantémøesμIsUnü¦. ]TahrN_epSgeTot eKGacbMEbk
FñwmenAkñúgrUb TI 15-2a CabIGgát; nigbYn node. cMNaMfa snøak;enARtg; node elx # dabdUcGgát;elx @

karviPaKFñwmedayeRbIviFIPaBrwgRkaj T.Chhay -502


nigelx # b:uEnþmMurgVilenARtg;cugGgát;nImYy²mantémøxusKña. enATIenH eKman degree of freedom cMnYn
R)aMbYn EdlenAkñúgenaH degree of freedom R)aMCaGBaØat ¬dUcbgðajenAkñúgrUbTI 15-2b¦ ehIy degree of
freedom cMnYnbYnRtUv)ansÁal; ¬vaTaMgGs;suT§EtesμIsUnü¦. cugeRkay BicarNakarrGil ¬slider mechanism¦

EdleRbIenAelIFñwmenAkñúgrUbTI 15-3a. enATIenH PaBdabrbs;FñwmRtUv)anbgðajenAkñúgrUbTI 15-3b dUc
enHeKmanbgÁúMPaBdabEdlCaGBaØatcMnUnR)aMEdlRtUv)ankMNt;elxerogCamYynwgelxkUdtUcbMput. FñwmCa
eRKOgbgÁúMminkMNt;edaysIeu Nm:aTicdWeRkTIR)aM.

karbegáItviFIPaBrwgRkajsRmab;FñwmmanlkçN³RsedogeTAnwgdMeNIrkarkñúgkarbegáItPaBrwgRkaj
EdleRbIsRmab; truss. dMbUg eyIgRtUvbegáItm:aRTIsPaBrwgRkajsRmab;Ggát;nImYy² ehIybnÞab;mkpÁúMm:aRTIs
TaMgenHedIm,IbegáItm:aRTIsPaBrwgRkajsRmab;Fñwm b¤m:aRTIsPaBrwgRkajsRmab;rcnasm½<n§. edayeRbIsmIkar
m:aRTIssRmab;rcnasm<½n§ eyIgGacbnþkarKNnaedIm,IkMNt;bM;las;TIEdlCaGBaØatenARtg; node ehIyBIkar
kMNt;enH eyIgGackMNt;RbtikmμenAelIFñwm nigkmøaMgkat; nigm:Um:g;enARtg; node.

!%>@> m:aRTIsPaBrwgRkajsMrab;Ggát;Fñwm (Beam-member stiffness matrix)
enAkñúgkfaxNÐenH eyIgnwgbegáItm:aRTIsPaBrwgRkajsRmab;FñwmEdlmanRkLaépÞmuxkat;efr nig
manRbB½n§kUGredaentMbn; x' , y' , z' ¬rUbTI 15-4¦. eKRtUvdak;eKalrbs;kUGredaenenARtg;cugCit N
ehIyG½kSviC¢manrbs; x' manTisedAeTArkcugq¶ay F . eKmankmøaMgRbtikmμBIrenARtg;cugnImYy²rbs;Ggát;
EdlmankmøaMgkat; q N nig q F nigm:Um:g;Bt; q N nig q F . bnÞúkTaMgenHeFVIGMeBItamTisedAkUGredaenviC¢-
z' y' z' z'

man. m:Um:g; q N nig q F viC¢manvilRcasTisRTnicnaLika tamviFanédsþaMviucT½rm:Um:g;manTistambeNþay
z' z'

G½kSviC¢man z' EdlmanTisecjBIRkdasesovePA.



bMlas;TIlIenEG‘r nigbMlas;TImMuEdlpSMCamYynwgbnÞúkTaMgenHk¾GnuvtþtamkarkMNt;sBaØaviC¢mandUc
Kña. enATIenH eyIgnwgdak;bMlas;TITaMgenHdac;edayELkBIKña ehIybnÞab;mkkMNt;bnÞúkEdlmanGMeBIenAelI
Ggát;EdlbNþalBIbMlas;TInImYy².

bMlas;TI y' ³ enAeBleKeFVIeGaymanbMlas;TIviC¢man d N xN³eBlEdleKGackarBarbMlas;TIEdlGac
y'

ekItman eKGacbegáItkmøaMgkat; nigm:Um:g;Bt;RtUv)anbgðajenAkñúgrUbTI 15-5a. m:Um:g;RtUv)anbegáItenAkñúg
kfaxNÐ 11-2 dUcsmIkar 11-5. dUcKña enAeBleKeFVIeGayman d F eyIgTTYl)ankmøaMgkat; nigm:Um:g;dUc
y'

bgðajenAkñúgrUbTI 15-5b.

mMurgVil z' ³ RbsinebIeKeFVIeGaymanmMurgVil d N xN³bMlas;TIEdlGacekItmanTaMgGs;RtUv)ankarBar
z'

kmøaMgkat; nigm:Um:g;EdlcaM)ac;edIm,IeFVIeGaymankMhUcRTg;RTayRtUv)anbgðajenAkñúgrUbTI 15-6a. m:Um:g;
RtUv)anbegáItenAkñúgkfaxNÐ 11-2 dUcsmIkar 11-1 nig 11-2. dUcKña enAeBleKeFVIeGayekItman d F bnÞúk z'

pÁÜbRtUv)anbgðajenAkñúgrUbTI 15-6b.



edayviFItRmYtpl RbsinebIeKbUklT§plenAkñúgrUbTI 15-5 nig 15-6 bBa©ÚlKña eyIgGacsresr
TMnak;TMngrvagbnÞúk nigbMlas;TITaMgbYnsRmab;Ggát;enAkñúgTRmg;m:aRTIsCa
N y' N z' F y' Fz'
⎡ 12 EI 6 EI 12 EI 6 EI ⎤
⎢ 3 2
− 3 ⎥
⎡q N y ' ⎤ ⎢ 6L L L L2 ⎥ d
⎢ q ⎥ ⎢ EI 4 EI 6 EI 2 EI ⎥ ⎡ N y ' ⎤
− 2 ⎢ ⎥
⎢ N z ' ⎥ = ⎢ L2 L L L ⎥⎢d Nz' ⎥ (15-1)
⎢ q Fy ' ⎥ ⎢ 12 EI 6 EI 12 EI 6 EI ⎥ ⎢ d ⎥
⎢ ⎥ ⎢− 3 − 2 − 2 ⎥ ⎢ Fy ' ⎥
⎣ q Fz ' ⎥ ⎢ L
⎢ ⎦ L L3 L ⎥⎢ d F ⎥
⎢ 6 EI 2 EI 6 EI 4 EI ⎥ ⎣ z ' ⎦
− 2
⎢ L2
⎣ L L L ⎥ ⎦
eyIgk¾GacsresrsmIkarTaMgenHkñúgTRmg;kat;dUcxageRkam
q = kd (15-2)
m:aRTIssIuemRTI k enAkñúgsmIkar15-1 Cam:aRTIsPaBrwgRkajsRmab;Ggát;. emKuNT§iBl kij TaMg
16 EdlpSMCam:aRTIs k )anKitbBa©ÚlbMlas;TIEdlbNþalBIkmøaMgkat; nigm:Um:g;Bt;rbs;Ggát;. tamrUbviTüa

emKuNTaMgenHCabnÞúkenAelIGgát; enAeBlGgát;rgbM;las;TIÉktþa. ]TahrN_ RbsinebI d N = 1 ¬rUbTI 15
y'

-5a¦ xN³EdlbMlas;TIdéTeTotesμIsUnü Ggát;nwgrgEtbnÞúkbYnEdlbgðajenAkñúgCYrQrTImYyrbs;m:aRTIs
k . tamrebobdUcKña CYrQrdéTeTotrbs;m:aRTIs k CabnÞúkenAelIGgát;sRmab;bMlas;TIÉktþaEdlkMNt;

edayelxkUd degree of freedom RtUv)anbgðajBIxagelICYrQr. BIkarbegáItenH eKRtUvbMeBjlkçxNÐ
lMnwg niglkçxNÐRtUvKña. dUcKña eyIgRtUvcMNaMfam:aRTIsenHdUcKñasRmab;kUGredaenskl nigkUGredaentMbn;
edaysarG½kS x' , y' , z' RsbnwgG½kS x, y, z dUcenHeKminRtUvkarm:aRTIsbMElgrvagkUGredaenTaMgBIreT.

!%>#> m:aRTIsPaBrwgRkajsMrab;eRKOgbgÁúMFñwm (Beam-structure stiffness matrix)
eRkayeBlrkma:RTIsPaBrwgRkajsRmab;Ggát; eyIgRtUvpÁúMvaeTAkñúgm:aRTIsPaBrwgRkajsRmab;
eRKOgbgÁúM K . dMeNIrkarenHGaRs½ynwgkarsÁal;TItaMgrbs;FatunImYy²enAkñúgm:aRTIsPaBrwgRkajsRmab;
Ggát;CadMbUg. enATIenH CYredk nigCYrQrrbs;m:aRTIs k nImYy² ¬smIkar 15-1¦ RtUv)ankMNt;edayelx
kUdBIrRtg;cugCitrbs;Ggát; ¬ N y' , N z' ¦ EdlbnþedayelxkUdBIrRtg;cugq¶ay ¬ Fy' , Fz' ¦. dUcenH enA
eBlpÁúMm:aRTIs eKRtUvdak;FatunmYy²enATItaMgdUcKñarbs;m:aRTIs K . tamviFIenH K RtUvmanlMdan;esμInwgelx
I
kUdFMbMputEdlRtUv)ankMNt;eTAelIFñwm edaysarvaCaelxsrubén degree of freedom. dUcKña enATItaMg
EdlGgát;CaeRcInP¢ab;Kñanwg node, emKuNT§iBlPaBrwgRkajsRmab;Ggát; (member stiffness influence


coefficients) nwgmanTItaMgdUcKñaenAkñúgm:aRTIs K dUcenHvaRtUv)anbUkbBa©ÚlKñatamlkçN³nBVnþedIm,IkMNt;
emKuNT§iBlPaBrwgRkajén node sRmab;rcnasm<½n§. eKcaM)ac;RtUveFVIEbbenH edaysaremKuNnImYy²Ca
ersIusþg;rbs; node énrcnasm<½n§enAkñúgTisedA ¬ y' b¤ z' ¦ enAeBlekItmanbMlas;TIÉktþa ¬ y' b¤ z' ¦ enA
Rtg; node dEdl b¤Rtg; node déT. ]TahrN_ K 23 CabnÞúktamTis nigenARtg;TItaMgénelxkUd@ enAeBl
bMlas;TIÉktþaekItmantamTis nigenARtg;TItaMgénelxkUd #.

!%>$> karGnuvtþénviFIPaBrwgRkajsMrab;karviPaKFñwm
(Application of the stiffness method for bema analysis)
eRkayeBleKkMNt;m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§rYcehIy eKGacP¢ab;TMnak;TMngrvagbnÞúk
enARtg; node rbs;FñwmeTAnwgbMlas;TIedayeRbIsmIkarPaBrwgRkajsRmab;rcnasm<½n§
Q = KD
enATIenH Q nig D Cam:aRTIsCYrQrEdltMNageGaybnÞúk nigbMlas;TIEdlCaGBaØat nigEdlsÁal;.
karbMEbkm:aRTIsPaBrwgRkajeGayeTACaFatuénbnÞúk nigbMlas;TIEdlsÁal; nigGBaØat eyIgTTYl)an
⎡Qk ⎤ ⎡ K11 K12 ⎤ ⎡ Du ⎤
⎢Q ⎥ = ⎢ K ⎥⎢ ⎥
⎣ u ⎦ ⎣ 21 K 22 ⎦ ⎣ Dk ⎦
enAeBleyIgBnøatm:aRTIsxagelI eyIgnwgTTYl)ansmIkarBIrKW
Qk = K11 Du + K12 Dk (15-3)
Qu = K 21 Du + K 22 Dk (15-4)
eKGackMNt;bMlas;TIEdlCaGBaØatBIsmIkarTImYy. edayeRbIsmIkarTaMgenH eyIgGacKNnakmøaMgRbtikmμ
TMr Qu BIsmIkarTIBIr.
bnÞúkEdlenAkNþal³ sRmab;karGnuvtþ eKcaM)ac;eGayFaturbs;FñwmminrgbnÞúktambeNþayFñwm edaysar
m:aRTIsPaBrwgRkajsRmab;Ggát;RtUv)anbegáItsRmab;bnÞúkEdlGnuvtþEtenAxagcugrbs;va ¬emIlrUbTI 15-
4¦. b:uEnþ eBlxøHFñwmRtUvRTbnÞúkBRgay dUcenHeKRtUvkarEktRmUvlkçxNÐenHedIm,IGnuvtþkarviPaKeday
m:aRTIs.
edIm,IedaHRsaykrNIenH eyIgnwgeRbIeKalkarN_tRmYtpltamrebobRsedogKñaeTAnwgGIVEdl)an
eRbIsRmab; truss Edl)anerobrab;enAkñúgkfaxNÐ 14-8. edIm,IbgðajBIkarGnuvtþrbs;va BicarNaFaturbs;
FñwmEdlmanRbEvg L dUcbgðajenAkñúgrUbTI 15-7a EdlrgbnÞúkBRgayesμI w . dMbUg eyIgnwgGnuvtþkmøaMg



bgáb;cug nigm:Um:g;bgáb;cugeTAelIGgát; EdlvaRtUv)aneKeRbIenAkñúgviFIPaBrwgRkaj ¬rUbTI 15-7b¦. eyIgnwg
sresrbnÞúkTaMgenHCam:aRTIsCYrQr q0 . bnÞab;mkeTot eyIgGnuvtþbnÞúkBRgayesμI nigkmøaMgRbtikmμ q0
rbs;va ¬rUbTI 15-7c¦. eKkMNt;bnÞúkCak;EsþgenAkñúgFñwmedayeFVIplbUkeTAelIlT§plTaMgBIr. kmøaMg
Rbtikmμbgáb;cugsRmab;krNIbnÞúkdéTeTotRtUv)aneGayenAkñúgemeronTI11. bEnßmelIkaredaHRsaycMeNaT
EdlBak;B½n§nwgbnÞúkxagdUckrNIenH eyIgk¾GaceRbIviFIenHedIm,IedaHRsaycMeNaTEdlBak;B½n§nwgbERmbRmYl
sItuNðPaB b¤kMhuskñúgplitkmμ.

kmøaMgkñúgGgát;³ eKGackMNt;kmøaMgkat; nigm:Um:g;Bt;enARtg;cugrbs;Ggát;FñwmnImYy²edayeRbIsmIkar 15-2
ehIybUkbEnßmkmøaMgRbtikmμbgáb;cug q0 RbsinebIGgát;rgbnÞúkenAkNþal. eyIgman
q = kd + q0 (15-5)
RbsinebIlT§plGviC¢man vabgðajfabnÞúkeFVIGMeBIpÞúynwgTisedAEdl)anbgðajenAkñúgrUbTI 15-4.

dMeNIrkarkñúgkarviPaK (Procedure for analysis)
viFIxageRkampþl;nUvmeFüa)ayedIm,IkMNt;bMlas;TI RbtikmμTMr kmøaMgkñúgrbs;Ggát; b¤FatuGnnþtUc
rbs;FñwmkMNt;edaysþaTic b¤FñwmminkMNt;edaysrsþaTic.
kareFVIkMNt;sMKal;³
bMEbkFñwmCaFatuGnnþtUc ehIykMNt;elxerogeGayFatunImYy² nig node nImYy²rbs;va. eRbIelx


EdlsresrkñúgrgVg;sRmab; node nigelxkñúgkaersRmab;Ggát; b¤Fatu. CaTUeTAGgát; b¤FatusßitenA
cenøaHTMr b¤enAcenøaHbnÞúkcMcMNuc b¤enAcenøaHtMN b¤enAcenøaHcMNucEdleKRtUvkarkMNt;kmøaMgkñúg
b¤bMlas;TI.
kMNt;cugCit nigcugq¶ayrbs;Ggát;edaysBaØaRBYjEdlsßitenAelIGgát; ehIymanTisedAeTArkcug
q¶ay.
enARtg; node nImYy² kMNt;elxkUdtamTis y nigtamTis z . enARKb;krNITaMgGs;eRbIelxkUdtUc
bMputedIm,IsMKal; degree of freedom EdlminmankarTb; ehIybnþedayelxbnþbnÞab; b¤elxEdl
FMbMputedIm,IsMKal; degree of freedom EdlmankarTb;.
begáItm:aRTIsbMlas;TIEdlsÁal; Dk nigm:aRTIsbnÞúkEdlsÁal; Qk . KitbBa©Úlm:aRTiscRmas;én
bnÞúkbgáb;cug RbsinebIGgát;rgbnÞúk.
m:aRTIsPaBrwgRkajsRmab;rcnasm½<n§³
GnuvtþsmIkar 15-1 edIm,IkMNt;m:aRTIsPaBrwgRkajsRmab;FatunImYy²edayeRbIkUGredaenskl.
eRkayeBlkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát; ehIyCYredk nigCYrQrRtUv)ankMNt;CamYynwg
elxkUdsmRsb pÁúMm:aRTIsedIm,ITTYl)anm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§ K . sRmab;kar
epÞógpÞat;edayEpñk m:aRTIsPaBrwgRkajsRmab;Ggát; nigm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§RtUv
EtCam:aRTIssIuemRTI.
bMlas;TI nigbnÞúk³
bMEbksmIkarPaBrwgRkajsRmab;rcnasm<½n§CaRkum ehIyeFVIRbmaNviFIKuNm:aRTIsedIm,IkMNt;
bMlas;TIEdlCaGBaØat Du nigkmøaMgRbtikmμTMr Qu .
eKGackMNt;kmøaMgkat; nigm:Um:g; q enARtg;cugrbs;Ggát;FñwmnImYy²BIsmIkar 15-5 edayKitbBa©Úl
TaMgbnÞúkbgáb;cug.

]TahrN_ 15-1³
kMNt;kmøaMgRbtikmμenARtg;TMrrbs;FñwmEdlbgðajenAkñúgrUbTI 15-8a. EI CacMnYnefr.



dMeNaHRsay³
kareFVIkMNt;smÁal;³ FñwmmanFatuBIr nig node bI EdlRtUv)ankMNt;dUcbgðajenAkñúgrUbTI 15-8b. eKeRbI
elxkUd 1 dl;elxkUd$ sRmab;kMNt; degree of freedom EdlminrgkarTb;.
m:aRTIsbnÞúkEdlsÁal; nigbMlas;TIEdlsÁal;KW
⎡ 0 ⎤1
⎢ − 5⎥ 2
⎡0 ⎤ 5
Qk = ⎢ ⎥ Dk = ⎢ ⎥
⎢ 0 ⎥3 ⎣0 ⎦ 6
⎢ ⎥
⎣ 0 ⎦4
m:aRTIsPaBrwgRkajsRmab;Ggát;³ eKkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;TaMgBIrBIsmIkar 15-1. eyIg
RtUvcMNaMBIrebobbegáItelxkUdsRmab;CYredk nigCYrQr.
6 4 5 3 5 3 2 1
⎡ 1.5 1.5 − 1.5 1.5 ⎤ 6 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 5
⎢ − 1.5 1 ⎥ 4 ⎢ − 1.5 1 ⎥ 3
k1 = EI ⎢ 1.5 2 ⎥ k 2 = EI ⎢ 1.5 2 ⎥
⎢− 1.5 − 1.5 1.5 − 1.5⎥ 5 ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 2
⎢ ⎥ ⎢ ⎥
⎣ 1.5 1 − 1.5 2 ⎦ 3 ⎣ 1.5 1 − 1.5 2 ⎦ 1
bMlas;TI nigbnÞúk³ eyIgGacpÁúMFatuTaMgenHeTAkñúgm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§. ]TahrN_ Fatu
K11 = 0 + 2 = 2, K 55 = 1.5 + 1.5 = 3 .l. dUcenH

Q = KD
1 2 3 4 5 6
⎡0⎤ ⎡ 2 − 1.5 1 0 1.5 0 ⎤ ⎡ D1 ⎤
⎢ − 5⎥ ⎢− 1.5 1.5 − 1.5 0 0 ⎥ ⎢ D2 ⎥
− 1.5
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ 0 ⎥ = EI ⎢ 1 − 1.5 4 1 0 1.5 ⎥ ⎢ D3 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢0⎥ ⎢ 0 0 1 2 − 1.5 1.5 ⎥ ⎢ D4 ⎥
⎢Q5 ⎥ ⎢ 1.5 − 1.5 0 − 1.5 3 − 1.5⎥ ⎢ 0 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢Q6 ⎥
⎣ ⎦ ⎢ 0
⎣ 0 1.5 1.5 − 1.5 1.5 ⎥ ⎢ 0 ⎥
⎦⎣ ⎦
m:aRTIsRtUv)anbMEbkCaRkumdUcbgðaj. edayedaHRsayplKuNsRmab;CYredkbYndMbUg eyIgTTYl)an


0 = 2 D1 − 1.5D2 + D3 + 0
5
− = −1.5D1 + 1.5D2 − 1.5 D3 + 0
EI
0 = D1 − 1.5 D2 + 4 D3 + D4

0 = 0 + 0 + D3 + 2 D2
edayedaHRsayRbB½n§smIkarxagelI eyIgTTYl)an
16.67
D1 = −
EI
26.67
D2 = −
EI
6.67
D3 = −
EI
3.33
D4 =
EI
edayeRbIlT§plTaMgenH ehIyedayKuNCYrQrTaMgBIrxageRkay eyIg)an
⎛ 16.67 ⎞ ⎛ 26.67 ⎞ ⎛ 3.33 ⎞
Q5 = 1.5 EI ⎜ − ⎟ − 1.5 EI ⎜ − ⎟ + 0 − 1.5 EI ⎜ ⎟
⎝ EI ⎠ ⎝ EI ⎠ ⎝ EI ⎠

= 10kN
⎛ 6.67 ⎞ ⎛ 3.33 ⎞
Q6 = 0 + 0 + 1.5EI ⎜ − ⎟ + 1.5 EI ⎜ ⎟
⎝ EI ⎠ ⎝ EI ⎠

= −5kN

]TahrN_ 15-2³
kMNt;kmøaMgkat; nigm:Um:g;enAkñúgGgát;elx ! rbs;FñwmEdlbgðajenAkñúgrUbTI 15-9a. EI CacMnYnefr.



dMeNaHRsay³
kareFVIkMNt;smÁal;³ enAkñúgkrNIenH Fñwmman degree of freedom EdlCaGBaØatcMnUnEtBIrb:ueNÑaHEdlman
bgðajelxkUd ! nig @ ¬rUbTI 15-9b¦. cMNaMfa bnÞúk M 0 manTMhMGviC¢man. m:aRTIsbnÞúk nigm:aRTIs
bMlas;TIEdlCasmμtikmμKW
⎡0 ⎤ 3
⎢0 ⎥ 4
⎡ 0 ⎤1
Qk = ⎢ ⎥ Dk = ⎢ ⎥
⎣− M 0 ⎦ 2 ⎢0 ⎥ 5
⎢ ⎥
⎣0 ⎦ 6
m:aRTIsPaBrwgRkajsRmab;Ggát;³ edayGnuvtþsmIkar 15-1 eTAelIGgát;nImYy² edayeyageTAelIelxkUd
EdlbgðajenAkñúgrUbTI 15-9b eyIg)an
5 6 4 1 4 1 3 2
⎡ 12 6 12 6 ⎤5 ⎡ 12 6 12 6 ⎤4
⎢ 3 2
− 3 ⎥ ⎢ 3 − 3 ⎥
⎢ L L L L2 ⎥ ⎢ L L2
L L2 ⎥
⎢ 6 4
− 2
6 2 ⎥
6 ⎢ 6 4
− 2
6 2 ⎥
1
k1 = EI ⎢ L2 L L L ⎥ k 2 = EI ⎢ L2 L L L ⎥
⎢ 12 6 12 6⎥ ⎢ 12 6 12 6⎥
⎢− 3 − 2 − 2 ⎥4 ⎢− 3 − 2 − 2 ⎥3
⎢ L L L3 L ⎥ ⎢ L L L3 L ⎥
⎢ 6 2
− 2
6 4 ⎥ ⎢ 6 2
− 2
6 4 ⎥
⎢ L2
⎣ L L L ⎥1
⎦ ⎢ L2
⎣ L L L ⎥2
⎦
bnÞúk nigbMlas;TI³ eKbegáItm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§edaypÁúMm:aRTIsPaBrwgRkajsRmab;Ggát;
nImYy². edayGnuvtþsmIkarm:aRTIssRmab;rcnasm<½n§ eyIg)an
Q = KD
1 2 3 4 5 6
⎡ 8 2 −6 6 2 ⎤ ⎡ D1 ⎤ 1
⎢ L 0
L L2
L2 L ⎥⎢ ⎥
⎢ 2 4 6 6 ⎥⎢ ⎥
1⎡ 0 ⎤ ⎢ − 2 0 0 ⎥ ⎢ D2 ⎥ 2
⎢ L L2 ⎥⎢ ⎥
2 ⎢− M 0 ⎥ ⎢ 6
L L
⎢ ⎥ 6 12 12 ⎥⎢ ⎥
3 ⎢ Q3 ⎥ = ⎢− L2 − L2 L3 − L3 0 0 ⎥⎢ 0 ⎥ 3
⎢ ⎥ ⎢ ⎥
6 ⎥⎢ ⎥
4 ⎢ Q4 ⎥ ⎢ 0 6 12 24
− 3
12
− 3 − 2 ⎢ 0 ⎥4
5 ⎢ Q5 ⎥ ⎢ L2 L L3 L L ⎥⎢ ⎥
⎢ ⎥ ⎢ 6 12 12 6 ⎥⎢ ⎥
6 ⎢ Q6 ⎥ ⎢ 2
⎣ ⎦ 0 0 − 3 ⎥⎢ 0 ⎥ 5
⎢ L L L3 L2 ⎥ ⎢ ⎥
⎢ 2 0 0 − 2
6 6 4 ⎥⎢ ⎥
⎢ L
⎣ L L2 L ⎥⎢ 0 ⎥ 6
⎦⎣ ⎦
eFVIplKuNeTAelICYredkdMbUgTaMgBIredIm,IkMNt;bMlas;TI eyIg)an


8EI 2 EI
0= D1 + D2
L L
2 EI 4 EI
− M0 = D1 + D2
L L
dUenH M L
D1 = 0
14 EI
2M 0 L
D2 = −
7 EI
dUcenAkñúg]TahrN_elIkmun eyIgTTYl)ankmøaMgRbtikmμBIkareFVIplKuNénCYredkEdlenAsl;. ]TahrN_
kmøaMgRbtikmμenAxagsþaMgédKW
6 EI ⎛ M 0 L ⎞ 6 EI ⎛ 2M 0 L ⎞ 9M 0
Q3 = − ⎜ ⎟− ⎜− ⎟=
L2 ⎝ 14 EI ⎠ L2 ⎝ 7 EI ⎠ 7L
eKkMNt;kmøaMgkñúgenARtg; node ! nig @ BIsmIkar 15-2. eyIg)an
q = k1d
5 6 4 1
⎡ q5 ⎤ ⎡ 12 6 12 6 ⎤⎡ 0 ⎤ 5
⎢ ⎥ ⎢ 3 − 3 ⎥⎢ ⎥
⎢ ⎥ ⎢ L L2 L L2 ⎥ ⎢ ⎥
⎢q6 ⎥ ⎢ 6 4 6 2 ⎥⎢ 0 ⎥
− 2 6
⎢ ⎥ = EI ⎢ L2 L L L ⎥⎢ ⎥
⎢ ⎥ ⎢ 12 6 12 6 ⎥⎢ ⎥
⎢q 4 ⎥ ⎢ − 3 − 2 − 2 ⎥⎢ 0 ⎥ 4
⎢ ⎥ ⎢ L L L3 L ⎥⎢ ⎥
⎢ ⎥ ⎢ 6 2 6 4 ⎥⎢ M L ⎥
− 2
L ⎥ ⎢14 EI ⎥ 1
0
⎢ ⎥ ⎢ L2
⎣ ⎦⎣
⎣ q1 ⎦ L L ⎦
6 EI ⎛ M L ⎞ 3M 0
q5 = 2 ⎜ 0 ⎟ =
L ⎝ 14 EI ⎠ 7L
2 EI ⎛ M 0 L ⎞ M 0
q6 = ⎜ ⎟=
L ⎝ 14 EI ⎠ 7
6 EI ⎛ M 0 L ⎞ 3M 0
q4 = − ⎜ ⎟=−
L ⎝
2 14 EI
⎠ 7L
4 EI ⎛ M 0 L ⎞ 2 M 0
q1 = ⎜ ⎟=
L ⎝ 14 EI ⎠ L
lT§plTaMgenHRtUv)anbgðajenAkñúgrUbTI 15-9c.

]TahrN_ 15-3³ FñwmenAkñúgrubTI 15-10a rgm:Um:g;BIr. RbsinebITMrkNþal ②Rsut 1.5mm cUgkMNt;
kmøaMgRbtikmμTMrenARtg;TMr. yk E = 200GPa nig I = 22(10 −6 )m 4 .


dMeNaHRsay³
kareFVIkMNt;smÁal;³ FñwmmanFatuBIr nig degree of freedom EdlCaGBaØatcMnYnbI. BYkvaRtUv)aneKkMNt;
edayelxkUdtUcCageK ¬rUbTI 15-10b¦. enATIenH m:aRTIsbnÞúk nigm:aRTIsbMlas;TIEdlCasmμtikmμKW³
⎡4 ⎤ 1 ⎡0 ⎤4
Qk = ⎢0 ⎥ 2
⎢ ⎥
⎢− 0.0015⎥ 5
Dk = ⎢ ⎥
⎢ − 4⎥ 3
⎣ ⎦ ⎢0
⎣ ⎥6
⎦
m:aRTIsPaBrwgRkajsRmab;Ggát;³ eKkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;edayeRbIsmIkar 15-1 eday
eyageTAtamelxkUd nigTisrbs;Ggát;EdlbgðajenAkñúgrUbTI 15-10b. eyIg)an
6 3 5 2 5 2 4 1
⎡ 1.5 1.5 − 1.5 1.5 ⎤ 6 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 5
⎢ − 1.5 1 ⎥ 3 ⎢ − 1.5 1 ⎥ 2
k1 = EI ⎢ 1.5 2 ⎥ k 2 = EI ⎢ 1.5 2 ⎥
⎢− 1.5 − 1.5 1.5 − 1.5⎥ 5 ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 4
⎢ ⎥ ⎢ ⎥
⎣ 1.5 1 − 1.5 2 ⎦ 2 ⎣ 1.5 1 − 1.5 2 ⎦ 1
bMlas;TI nigbnÞúk³ karpÁúMm:aRTIsPaBrwgRkajsRmab;rcnasm<n§ nigkarsresrsmIkarPaBrwgRkajsRmab;
½
rcnasm<½n§ eyIg)an
1 2 3 4 5 6
⎡ 4⎤ ⎡ 2 1 0 − 1.5 1.5 0 ⎤ ⎡ D1 ⎤
⎢0⎥ ⎢ 1 4 1 − 1.5 0 1.5 ⎥ ⎢ D2 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢− 4⎥ = EI ⎢ 0 1 2 0 − 1.5 1.5 ⎥ ⎢ D3 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢Q4 ⎥ ⎢− 1.5 − 1.5 0 1.5 − 1.5 0 ⎥ ⎢ 0 ⎥
⎢Q5 ⎥ ⎢ 1.5 0 − 1.5 − 1.5 3 − 1.5⎥ ⎢− 0.0015⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ ⎦
⎣Q6 ⎥ ⎢ 0
⎣ 1.5 1.5 0 − 1.5 1.5 ⎥ ⎢ 0
⎦⎣ ⎥
⎦
edaHRsayrkbMlas;TIEdlCaGBaØat
= 2 D1 + D2 + 0 D3 − 1.5(0) + 1.5(− 0.0015) + 0
4
EI
0 = 1D1 + 4 D2 + 1D3 − 1.5(0 ) + 0 + 0



−4
= 0 D1 + 1D2 + 2 D3 + 0 − 1.5(− 0.0015) + 0
EI
edayCMnYs EI = 200(10 6 )(22)(10 −6 ) nigedaHRsay eyIg)an
D1 = 0.001580rad D2 = 0 D3 = −0.001580rad
edayeRbIlT§plTaMgenH dUcenHkmøaMgRbtikmμTMrKW
Q4 = 200( 6 )22( −6 )[− 1.5(0.001580 ) − 1.5(0 ) + 0 + 1.5(0 ) − 1.5(− 0.0015) + 0] = −0.525kN
10 10
Q5 = 200(10 6 )22(10 −6 )[1.5(0.001580 ) + 0 − 1.5(− 0.001580 ) − 1.5(0 ) + 3(− 0.0015) − 1.5(0 )] = 1.05kN

Q6 = 200(10 6 )22( −6 )[0 + 1.5(0 ) + 1.5(− 0.001580 ) + 0 − 1.5(− 0.0015) + 1.5(0 )] = −0.525kN
10

]TahrN_ 15-4³ kMNt;m:Um:g;EdlekItmanenARtg;TMr A rbs;FñwmEdlbgðajenAkñúgrUbTI 15-11a. yk
E = 200GPa nig I = 216(10 6 )mm 4 .

dMeNaHRsay³
kareFVIkMNt;smÁal;³ enATIenH Fñwmman degree of freedom EdlCaGBaØatcMnYnBI EdlkMNt;elxerogeday
elxkUd ! nig @.
karviPaKedaym:aRTIsTamTareGaybnÞúkGnuvtþenARtg; node dUcenHeKRtUvCMnYsbnÞúkBRgay nig
bnÞúkcMcMNucedaym:Um:g;bgáb;cugsmmUlrbs;vaEdlRtUv)ankMNt;BItaragenAkñúgemeron !! ¬emIl]TahrN_
11-2¦. cMNaMfa minmanbnÞúkxageRkAmanGMeBIenARtg; ① eT edaysarkmøaMgRbtikmμenARtg;elxkUd #


nig $ CaGBaØatenAkñúgm:aRTIsbnÞúk. edayeRbIviFItRmYtpl lT§plénkarviPaKm:aRTIssRmab;bnÞúkenAkñúgrUb
TI 15-11b RtUv)anEktRmUvenAeBleRkayedaybnÞúkEdlbgðajenAkñúgrUbTI 15-11c. BIrUbTI 15-11b
m:aRTIsbMlas;TI nigm:aRTIsbnÞúksmμtikmμKW
⎡0 ⎤ 4
⎡12 ⎤ 1
Dk = ⎢0 ⎥ 5
⎢ ⎥ Qk = ⎢ ⎥
⎢0 ⎥ 6 ⎣84⎦ 2
⎣ ⎦
m:aRTIsPaBrwgRkajsRmab;Ggát;³ eKkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;BIsmIkar 15-1.
Ggát;elx !³ 12EI = 12(2003)(216) = 2400
3
L 6
6 EI 6(200 )(216)
= = 7200
L2 62
4 EI 4(200 )(216 )
= = 28800
L 6
2 EI 2(200 )(216)
= = 14400
L 6
4 3 5 2
⎡ 2400 7200 − 2400 7200 ⎤ 4
⎢ ⎥
k1 = ⎢ 7200 28800 − 7200 14400 ⎥ 3
⎢− 2400 − 7200 2400 − 7200⎥ 5
⎢ ⎥
⎣ 7200 14400 − 7200 28800 ⎦ 2
12 EI 12(200 )(216 )
Ggát;elx @³ = = 64800
L3 23
6 EI 6(200 )(216)
= = 64800
L2 22
4 EI 4(200 )(216)
= = 86400
L 2
2 EI 2(200 )(216 )
= = 43200
L 2
5 2 6 1
⎡ 64800 64800 − 64800 64800 ⎤ 5
⎢ ⎥
k 2 = ⎢ 64800 86400 − 64800 43200 ⎥ 2
⎢− 64800 − 64800 64800 − 64800⎥ 6
⎢ ⎥
⎣ 64800 43200 − 64800 86400 ⎦ 1
bMlas;TI nigbnÞúk³ eyIgtMrUveGay
Q = KD



1 2 3 4 5 6
⎡ 12 ⎤ ⎡ 86400 43200 0 0 64800 − 64800⎤ ⎡ D1 ⎤
⎢ 84 ⎥ ⎢ 43200 115200 14400 7200 57600 − 64800⎥ ⎢ D ⎥
⎢ ⎥ ⎢ ⎥⎢ 2 ⎥
⎢Q3 ⎥ = ⎢ 0 14400 28800 7200 − 7200 0 ⎥⎢ 0 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢Q4 ⎥ ⎢ 0 7200 7200 2400 − 2400 − 64800⎥ ⎢ 0 ⎥
⎢Q5 ⎥ ⎢ 64800 57600 − 7200 − 2400 67200 − 64800⎥ ⎢ 0 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢Q6 ⎦ ⎢− 64800 − 64800
⎣ ⎥ ⎣ 0 0 − 64800 64800 ⎥ ⎢ 0 ⎥
⎦⎣ ⎦
edayedaHRsaytamrebobFmμtar
12 = 86400 D1 + 43200 D2

84 = 43200 D1 + 115200 D2

( )
D1 = −0.2778 10 −3 m

D2 = 0.8333(
10 )m −3

dUcenH Q3 = 0 + 14400(0.8333)(10 −3 )m = 12kN .m
m:Um:g;Cak;EsþgenARtg; A RtUvEtrYmbBa©ÚlkmøaMgRbtikmμTMrbgáb; + 96kN.m EdlbgðajenAkñúgrUbTI 15-11c
CamYynwglT§plEdl)anKNnasRmab; Q3 . dUcenH
M AB = 12kN .,+96kN .m = 108kN .m
lT§plenHdUcKñanwglT§plEdlKNnaenAkñúg]TahrN_ 11-2.
eTaHbICaeKminRtUvkarKNnam:Um:g; nigkmøaMgkat;enARtg;cMNuc B enATIenHk¾eday EteKGacKitGac
KNnava)an ]TahrN_ Ggát;elx !/ node @ ¬rUbTI 15-11b¦. lT§plTamTarkarBnøat
q1 = k1d + (q0 )1
4 3 5 2
⎡ q 4 ⎤ ⎡ 2400 7200 − 2400 7200 ⎤ ⎡ 0 ⎤ ⎡ 6 ⎤
⎢ q ⎥ ⎢− 7200 28800 − 7200 14400 ⎥ ⎢ 0 ⎥ ⎢ ⎥
⎢ 3⎥ = ⎢ ⎥⎢
⎢ q5 ⎥ ⎢− 2400 − 7200 67200 57600 ⎥ ⎢ 0 ⎥
( )
⎥ 10 −3 + ⎢ 96 ⎥
⎢ 6 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎣q 2 ⎦ ⎣ 7200 14400 57600 115200⎦ ⎣0.833⎦ ⎣− 96⎦

]TahrN_ 15-5³
kMNt;PaBdabenARtg;① nigkmøaMgRbtikmμTMrenAelIFñwmEdlbgðajenAkñúgrUbTI 15-12a. EI CacMnYnefr.



dMeNaHRsay³
kareFVIkMNt;smÁal;³ FñwmRtUv)anEbgEckCaBIrFatu ehIy node nigGgát;RtUv)ankMNt;CamYynwgTisBIcug
CiteTAcugq¶ay ¬rUbTI 15-12b¦. PaBdabEdlCaGBaØatRtUv)anbgðajenAkñúgrUbTI 15-12c. cMNaMfa
bMlas;TImMu D4 minekItmaneT edayfakarTb;edayTMrrGilxag.
m:aRTIsPaBrwgRkajsRmab;Ggát;³ edaysar EI CacMnYnefr ehIyGgát;manRbEvgesμIKña enaHm:aRTIsPaBrwg
RkajsRmab;Ggát;RtUvEtdUcKña. edayeRbIelxkUdedIm,IkMNt;elxerogCYredk nigCYrQrnImYy²edayeyag
eTAtamsmIkar 15-1 nigrUbTI 15-12b eyIg)an
3 4 1 2 1 2 5 6
⎡ 1.5 1.5 − 1.5 1.5 ⎤ 3 ⎡ 1.5 1.5 − 1.5 1.5 ⎤ 1
⎢ − 1.5 1 ⎥ 4 ⎢ − 1.5 1 ⎥ 2
k1 = EI ⎢ 1.5 2 ⎥ k 2 = EI ⎢ 1.5 2 ⎥
⎢− 1.5 − 1.5 1.5 − 1.5⎥ 1 ⎢− 1.5 − 1.5 1.5 − 1.5⎥ 5
⎢ ⎥ ⎢ ⎥
⎣ 1.5 1 − 1.5 2 ⎦ 2 ⎣ 1.5 1 − 1.5 2 ⎦ 6
bMlas;TI nigbnÞúk³ pÁúMm:aRTIsPaBrwgRkajsRmab;Ggát;eTAkñúgm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§ eyIg)an
Q = KD
1 2 3 4 5 6
⎡− P ⎤ ⎡ 3 0 − 1.5 − 1.5 − 1.5 1.5 ⎤ ⎡ D1 ⎤
⎢ 0 ⎥ ⎢ 0 4 1.5 1 − 1.5 1 ⎥ ⎢ D2 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ 0 ⎥ = EI ⎢− 1.5 1.5 1.5 1.5 0 0 ⎥ ⎢ D3 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ Q4 ⎥ ⎢− 1.5 1 1.5 2 0 0 ⎥⎢ 0 ⎥
⎢ Q5 ⎥ ⎢− 1.5 − 1.5 0 0 1.5 − 1.5⎥ ⎢ 0 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ Q6 ⎥
⎣ ⎦ ⎢ 1.5
⎣ 1 0 0 − 1.5 2 ⎥ ⎢ 0 ⎥
⎦⎣ ⎦


edayedaHRsayrkbMlas;TI eyIg)an
P
− = 3D1 + 0 D2 − 1.5D3
EI
0 = 0 D1 + 4 D2 + 1.5D3

0 = −1.5D1 + 1.5D2 + 1.5D3
1.667 P
D1 = −
EI
P
D2 =
EI
2.667 P
D3 = −
EI
cMNaMfa sBaØarbs;lT§plTaMgenHRtUvEtRtUvKñanwgTisedArbs;PaBdabEdlbgðajenAkñúgrUbTI 15-12c.
kareRbIlT§plTaMgenH kmøaMgRbtikmμKW
⎛ 1.667 P ⎞ ⎛ P ⎞ ⎛ 2.667 P ⎞
Q4 = −1.5EI ⎜ − ⎟ + 1EI ⎜ ⎟ + 1.5EI ⎜ − ⎟ = −0.5P
⎝ EI ⎠ ⎝ EI ⎠ ⎝ EI ⎠
⎛ 1.667 P ⎞ ⎛ P ⎞ ⎛ 2.667 P ⎞
Q5 = −1.5 EI ⎜ − ⎟ − 1.5EI ⎜ ⎟ + 0⎜ − ⎟=P
⎝ EI ⎠ ⎝ EI ⎠ ⎝ EI ⎠
⎛ 1.667 P ⎞ ⎛ P ⎞ ⎛ 2.667 P ⎞
Q6 = 1.5EI ⎜ − ⎟ + 1EI ⎜ ⎟ + 0⎜ − ⎟ = −1.5P
⎝ EI ⎠ ⎝ EI ⎠ ⎝ EI ⎠



cMeNaT
!%>! kMNt;kmøaMgRbtikmμTMr. snμt;②CaTMrkl;. !%>% kMNt;m:Um:g;Bt;enAkñúgFñwmenARtg; ②nig③.
EI CacMnYnefr. snμt;②CaTMrkl; ehIy③ nig④CaTMrsnøak;. EI
CacMnYnefr.

!%>@ kMNt;m:Um:g;Bt;enAkñúgFñwmenARtg; ① nig②. !%>^ kMNt;kmøaMgRbtikmμTMr. snμt;②CaTMrsnøak;
snμt;②CaTMrkl; ehIy③CaTMrsnøak;. EI CacMnYn ehIy① nig③ CaTMrkl . EI CacMnYnefr.
efr.

!%>& kMNt;kmøaMgRbtikmμTMr. EI CacMnYnefr.
!%># kMNt;kmøaMgRbtikmμTMr. EI CacMnYnefr.

!%>* kMNt;m:Um:g;Bt;enARtg; ① nig③. snμt;②
!%>$ kMNt;m:Um:g;Bt;enARtg;TMr. snμt;②CaTMrkl;. CaTMrkl; ehIy① nig③CaTMrbgáb;. EI CacMnYn
EI CacMnYnefr. efr.

Problems T.Chhay -519


!%>( kMNt;m:Um:g;Bt;enARtg; ① nig③ RbsinebI
TMr②Rsut 30mm ¬cMeNaT !%>(¦. snμt; ②Ca
TMrkl; ehIy① nig③CaTMrbgáb;.
EI = 5000kN .m 2 .

!%>!0 kMNt;kmøaMgRbtikmμTMr. EI CacMnYnefr.

!%>!! kMNt;kmøaMgRbtikmμTMr. eKmanTMrrGiltam
TisbBaÄrenARtg;① dUcbgðaj. EI CacMnYnefr.

cMeNaT T.Chhay -520

Beam analysis using the stiffness method

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