16. plane frame analysis using the stiffness method

mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa

!^> karviPaKeRKagkñúgbøg;edayeRbIviFIPaBrwgRkaj
(Plane frame analysis using the stiffness method)

eKalKMnitEdl)anbgðajenAkñúgemeronelIkmunEdlerobrab;GMBIkaeRbIviFIPaBrwgRkajenAelI trusses
nigFñwmRtUv)anBnøat nigGnuvtþeTAelIkarviPaKeRKag. viFIenHnwgbgðajfadMeNIrkarsRmab;edaHRsayman
lkçN³RsedogKñaeTAnwgkaredaHRsaysRmab;Fñwm b:uEnþvaRtUvkareRbIm:aRTIsbMElg edaysarGgát;eRKagsßit
kñúgTisepSg².

!^>!> m:aRTIsPaBrwgRkajrbs;Ggát;eRKag (Frame-member stiffness matrix)
enAkñúgkfaxNÐenH eyIgnwgbegáItm:aRTIsPaBrwgRkajsRmab;Ggát;eRKagEdlmanmuxkat;efr
(prismatic frame member) BIRbB½n§kUGredaentMbn; x' , y ' , z ' ¬rUbTI 16-1¦. enATIenH Ggát;rgkmøaMg

tamG½kS q N , q F kmøaMgkat; q N , q F , nigm:Um:g;Bt; q N , q F enARtg;cugCit nigcugq¶ayrbs;va
x' x' y' y' z' z'

erogKña. bnÞúkTaMgenHsuT§EtmanGMeBItamTiskUGredaenviC¢manCamYynwgbMlas;TIrbs;va. dUcKñakñúgkrNIFñwm
m:Um:g; q N nig q F viC¢manvilRcasTisRTnicnaLika edaysarkareRbIviFanédsþaM viucT½rm:Um:g;manTistam
z' z'

G½kS z' EdlecjBIépÞRkdas.

eyIg)anBicarNaTMnak;TMngrvagbnÞúk nigbM;las;TIEdlbNþalBIbnÞúkTaMgenHenAkñúgemeronelIkmun.
bnÞúktamG½kSRtUv)anerobrab;edayeyageTAelIrUbTI 14-2 kmøaMgkat;eyageTAtamrUbTI 15-5 ehIym:Um:g;
Bt;eyagtamrUbTI 15-6. tamviFItRmYtpl RbsinebIeKbUkbBa©ÚllT§plTaMgGs;enHcUlKña eKGacsresr
TMnak;TMngrvagbMlas;TI nigbnÞúkcMnYnR)aMmYysRmab;Ggát;kñúgTRmg;m:aRTIsdUcxageRkam
Plane frame analysis using the stiffness method T.Chhay -521

Department of Civil Engineering NPIC

N x' N y' N z' Fx' F y' Fz'
⎡q N x' ⎤ ⎡d ⎤
⎢ ⎥ ⎡ AE AE ⎤ ⎢ N x' ⎥
0 0 − 0 0 ⎥
⎢ ⎥ ⎢ L L ⎢ ⎥
⎢q N y ' ⎥ ⎢ 12 EI 6 EI 12 EI 6 EI ⎥ ⎢d N y ' ⎥
⎢ ⎥ ⎢ 0 0 − 3 ⎥⎢ ⎥
⎢ ⎥ ⎢ L3 L2 L L2 ⎥ ⎢ ⎥
⎢q Nz' ⎥ ⎢ 0 6 EI 4 EI
0 − 2
6 EI 2 EI ⎥ ⎢ d ⎥
⎥=⎢ L ⎥⎢
Nz'
⎢ L2 L L ⎥ (16-1)
⎢ ⎥ ⎢ AE AE
⎥⎢ ⎥
⎢ q Fx ' ⎥ ⎢− 0 0 0 0 ⎥⎢ d F ⎥
⎢ ⎥ ⎢ L L ⎥ ⎢ x' ⎥
⎢ ⎥ ⎢ 0 12 EI
− 3 − 2
6 EI
0
12 EI 6 EI ⎥ ⎢
− 2 ⎥ ⎥
⎢ q Fy ' ⎥ ⎢ L L L3 L ⎥ ⎢ d Fy ' ⎥
⎢ ⎥ ⎢ 6 EI 2 EI 6 EI 4 EI ⎥ ⎢ ⎥
⎢ ⎥ ⎢ 0 0 − 2 ⎢ ⎥
⎢ qF ⎥ ⎣ ⎢ L2 L L L ⎥⎢ d ⎥
⎦ F
⎣ z' ⎦ ⎣ z' ⎦
b¤tamTRmg;kat; q = k'd (16-2)

m:aRTIsPaBrwgRkajsRmab;Ggát; k ' pSMeLIgedayemKuNT§iBlcMnYn 36 EdltMNagedaybnÞúkenAelIGgát;
enAeBlGgát;rgbMlas;TIÉktþaCak;lak;NamYy. CaBiess CYrQrnImYy²enAkñúgm:aRTIsnImYy²CabnÞúkkñúg
Ggát;sMrab;bMlas;TIÉktþaEdlkMNt;edayelxkUd degree of freedom EdlmanbgðajenABIxagelICYrQr
nImYy². eRKagEdleKRtUvKNnaRtUvEtbMeBjlkçxNÐlMnwg niglkçxNÐbMlas;TIRtUvKña.

!^>@> m:aRTIsbMElgénbMlas;TI nigm:aRTIsbMElgénkmøaMg
(Displacement and force transformation matrices)
dUcenAkñúgkrNI trusses, eyIgRtUvbMElgbnÞúkkñúgGgát; q nigbMlas;TI d BIkUGredaen x' / y' / z' eTA
CakUGredaenskl x, y, z . sRmab;ehtuplenH eKRtUvkarm:aRTIsbMElg.
m:aRTIsbMElgbMlas;TI (displacement transformation matrix)³ eKmanGgát;eRKagdUcbgðajenAkñúgrUb
TI 16-2a. enATIenH eyIgeXIjfabMlas;TI DN kñúgRbB½n§kUGredaensklbegáIt)anbMlas;TIkñúgkUGredaen
x

tMbn;
d N x ' = D N x cos θ x d N y ' = − D N x cos θ y

dUcKña bMlas;TI DN enAkñúgbMlas;TIskl ¬rUbTI 16-2b¦ begáIt)anbMlas;TIenAkñúgkUGedaentMbn;
y

d N x; = D N y cos θ y d N y ' = D N y cos θ x

cugeRkay edaysarG½kS z' nigG½kS z RtYtsIuKña ¬manTisedAecjBIépÞesovePA¦ mMurgVil DN CMuvijG½kS z eFVI
z

eGaymanmMurgVilRtUvKña D N CMuvijG½kS z' . dUcenH
z'

karviPaKeRKagkñúgbøg;edayeRbIviFIPaBrwgRkaj T.Chhay -522


DN z' = DN z

tamrebobdUcKña RbsinebIeKeFVIeGaymanbMlas;TIskl DF tamTis x / DF tamTis y nigmMurgVil DF
x y z

enAxagcugq¶ayrbs;Ggát; smIkarbMElgKW
d Fx ' = DFx cos θ x d Fy ' = − DFx cos θ y

d Fx ' = DFy cos θ y d Fy ' = DFy cos θ x

d Fz ' = DFz

yk λ x = cosθ x , λ y = cosθ y CakUsIunUsR)ab;Tisrbs;Ggát; eyIgGacsresrrYmpSMKñaénbMlas;TICaTRmg;
m:aRTIsdUcxageRkam
⎡d N x' ⎤ ⎡ λ x λ y 0 0 0 0⎤ ⎡ D N x ⎤
⎢d ⎥ ⎢ ⎢ ⎥
⎢ N y ' ⎥ ⎢− λ y λ x 0 0 0 0⎥ ⎢ D N y ⎥
⎥
⎢d N z' ⎥ ⎢ 0 0 1 0 0 0⎥ ⎢ D N z ⎥
⎢ ⎥=⎢ ⎥⎢ ⎥ (16-3)
⎢ d Fx ' ⎥ ⎢ 0 0 0 λx λ y 0⎥ ⎢ DFx ⎥
⎢dF ⎥ ⎢ 0 0 0 − λy λ x 0⎥ ⎢ DFy ⎥
⎢ y' ⎥ ⎢ ⎥⎢ ⎥
⎢ d Fz ' ⎥ ⎢ 0
⎣ ⎦ ⎣ 0 0 0 0 1⎥ ⎢ D Fz ⎥
⎦⎣ ⎦
b¤ d = TD (16-4)

tamkarGegát m:aRTIs T bMElgbMlas;TI D kñúgkUGredaenskl x, y, z TaMgR)aMmYyeGayeTACabMlas;TI d
kñúgkUGredaentMbn; x' , y' , z' TaMgR)aMmYy. enATIenHm:aRTIs T RtUv)aneKsÁal;Cam:aRTIsbMElgbMlas;TI.

m:aRTIsbMElgkMlaMg³ RbsinebIeyIgGnuvtþbgÁúMkmøaMgnImYy²eTAelIcugCitrbs;Ggát; eyIgGackMNt;BIrebob
bMElgbgÁúMkmøaMgBIkUGredaentMbn;eGayeTACakUGredaenskl. edayGnuvtþ q N ¬rUbTI 16-3a¦ eyIgGacx'

eXIjfa
Q N x = q N x ' cos θ x Q N y = q N x ' cos θ y



RbsinebIeKGnuvtþ q N ¬rUbTI 16-3b¦ enaHbgÁúMkmøaMgrbs;vaKW
y'

Q N x = −q N y ' cos θ y Q N y = q N y ' cos θ x

cugeRkay edaysar q N RtYtsIuCamYynwg QN eyIg)an
z' Z

QN z = q N z '

tamrebobdUcKña bnÞúkenARtg;cugGgát; q F x'
, q Fy ' , q Fz ' nwgpþl;nUvbgÁúMkmøaMgdUcxageRkam³
Q Fx = q Fx ' cos θ x QFy = q Fx ' cos θ y

Q Fx = − q Fy ' cos θ y QFy = q Fy ' cos θ x

QFz = q Fz '

smIkarTaMgbIEdlpÁúMenAkñúgTRmg;m:aRTIsCamYynwg λ x = cosθ x , λ y = cos θ y pþl;nUv
⎡Q N x ⎤ ⎡ λ x − λ y 0 0 0 0⎤ ⎡ q N x ' ⎤
⎢Q ⎥ ⎢ ⎢ ⎥
⎢ N y ⎥ ⎢λ y λ x 0 0 0 0⎥ ⎢ q N y ' ⎥
⎥
⎢Q N z ⎥ ⎢ 0 0 1 0 0 0⎥ ⎢ q N z ' ⎥
⎢ ⎥=⎢ ⎥⎢ ⎥ (16-5)
⎢ Q Fx ⎥ ⎢ 0 0 0 λx − λ y 0⎥ ⎢ q Fx ' ⎥
⎢ QF ⎥ ⎢ 0 0 0 λy λx 0⎥ ⎢ q Fy ' ⎥
⎢ y⎥ ⎢ ⎥⎢ ⎥
⎢ QFz ⎥ ⎢ 0
⎣ ⎦ ⎣ 0 0 0 0 1⎥ ⎢ q Fz ' ⎥
⎦⎣ ⎦
b¤ Q =TTq (16-6)

enATIenH dUckarerobrab; m:aRTIs T T bMElgbnÞúkenAelIGgát;TaMgR)aMmYyEdlsresrenAkñúgkUGredaentMbn;
eGayeTACabnÞúkTaMgR)aMmYyEdlsresrenAkñúgkUGredaenskl.

!^>#> m:aRTIsPaBrwgRkajsklsRmab;Ggát;eRKag
(Frame-Member Global Stiffness Matrix)
eKGacpÁúMlT§plénkfaxNÐelIkmunedIm,IkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;EdlP¢ab;TMnak;TMng
rvagbnÞúkskl Q eTAnwgbMlas;TIskl D . edIm,IeFVIEbbenH eKRtUvCMnYssmIkar 16-4 ¬ d = TD ¦ eTAkñúg
smIkar 16-2 ¬ q = k ' d ¦. eyIg)an
q = k 'TD (16-7)
enATIenH kmøaMgkñúgGgát; q Tak;TgnwgbMlas;TIskl D . edayCMnYslT§plenHeTAkñúgsmIkar 16-6
¬ Q = T T q ¦ eKTTYl)anlT§plcugeRkay
Q = T T k 'TD (16-8)



b¤ Q = kD

Edl k = T T k 'T (16-9)

enATIenH k Cam:aRTIsPaBrwgRkajsklsRmab;Ggát;. eyIgGacTTYltémørbs;vakñúgTRmg;TUeTAedayeRbI
smIkar 16-5/ 16-1 nig 16-3 ehIyedayeFVIRbmaNviFIm:aRTIs eKnwgTTYl)anlT§plcugeRkay

(16-10)
cMNaMfam:aRTIsTMhM 6 × 6 Cam:aRTIssIuemRTI. elIsBIenH eKP¢ab;TItaMgrbs;FatunImYy²eTAnwgkUdenARtg;cug
Cit N x , N y , N z EdlbnþedayelxkUdenARtg;cugq¶ay Fx , Fy , Fz EdlRtUv)anbgðajenAxagelIénCYr
Qr nigtambeNþayCYredk. dUcm:aRTIs k ' CYrQrnImYy²rbs;m:aRTIs k CabnÞúkenAelIGgát;Rtg; node Edl
KRtUvkaredIm,ITb;Tl;nwgbM;las;TIÉktþatamTisEdlkMNt;edayelxkUdrbs;CYrQr. ]TahrN_ CYrQrTI
mYyrbs;m:aRTIs k CabnÞúkenAkñúgkUGredaensklRtg;cugCit nigcugq¶ayEdlbgáeLIgedaybMlas;TIÉktþa
enARtg;cugCittamTis x eBalKW N x .

!^>$> karGnuvtþénviFIPaBrwgRkajsMrab;karviPaKeRKag
(Application of the stiffness method for frame analysis)
enAeBlEdleKbegáItm:aRTIsPaBrwgRkajsRmab;Ggát;rYcehIy eKGacpÁúMBYkvabBa©ÚlKñaeTAkñúgm:aRTIs
PaBrwgRkajsRmab;rcnasm<½n§tamrebobFmμta. edaysresrsmIkarm:aRTIssRmab;rcnasm<½n§ eKGac
kMNt;bM;las;TIenARtg; node EdlminmankarTb; EdlbnþedaykmøaMgRbtikmμ nigkmøaMgkñúgenARtg; node.
eKGacedaHRsaykmøaMgxagEdlmanGMeBIelIGgát; kMhusqÁgedaysarplitkmμ bERmbRmYlsItuNðPaB
kmøaMgTRmeRTt nigkmøaMgTRmxagkñúgtamrebobdUcKñanwgGVIEdl)anerobrab;sRmab; truss nigFñwm.


dMeNIrkarkñúgkarviPaK (Procedure for analysis)
viFIxageRkampþl;nUvmeFüa)ayedIm,IkMNt;bMlas;TI RbtikmμTMr kmøaMgkñúgrbs;Ggát;eRKagkMNt;eday
sþaTic nigeRKagminkMNt;edaysþaTic.
kareFVIkMNt;sMKal;³
EckeRKOgbgÁúMCaFatuGnnþtUc ehIykMNt;elxerogeGayGgát; nig node nImYy²rbs;va. eKEtgEt
BnøatFatuenAcnøaHcMNucrbs;TMr cMNucrbs;bnÞúkRtg;cMNuc RCugEkg b¤tMNEdleKRtUvkarkMNt;
bMlas;TI b¤kmøaMgkñúgrbs;Ggát;.
begáItRbB½n§kUGredaen x, y, z CaTUeTAedIm,IPaBgayRsYlCamYynwgeKalEdlmanTItaMgenARtg;
cMNuc node enAelIFatumYy nigG½kSEdlmanTItaMgy:agNaeGayRKb; node TaMgGs;mankUGredaen
viC¢man.
enARtg;cMNuc node nImYy²rbs;eRKag kMNt;bgÁúMelxkUdbIKW x, y, z . RKb;krNITaMgGs; eKeRbI
elxkUdtUcbMputedIm,IkMNt;elxerogsRmab; degree of freedom EdlminmankarTb; Edlbnþeday
elxkUdEdlenAsl; b¤elxkUdEdlmanelxerogFMedIm,IsMKal; degree of freedom Edlmankar
Tb;.
begáItbMlas;TIEdlsÁal; Dk nigbnÞúkxageRkAEdlsÁal; Qk . enAeBlbegáIt Qk eKRtUvR)akdkñúgkar
bBa©ÚlbnÞúkbgáb;cugRbsinebIGgát;RTbnÞúkenAkNþal.
m:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúM³
GnuvtþsmIkar 16-10 edIm,IkMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;nImYy²EdlsresrenAkñúgRbB½n§
kUGredaenskl. eKkMNt;kUsIunUsR)ab;Tis λ x nig λ y BIkUGredaen x, y éncugrbs;Ggát; ¬smIkar
14-5 nig 14-6¦.

eRkayeBlsresrm:aRTIsPaBrwgRkajsRmab;Ggát;nImYy² nigeRkayeBlkMNt;CYredk nigCYrQr
CamYynwgelxkUdcugCit nigcugq¶ay eKGacRc)ac;m:aRTIsTaMgenHbBa©ÚlKñaedIm,IbegáItm:aRTIsPaBrwg
RkajsRmab;eRKOgbgÁúM K . sRmab;karepÞógpÞat;edayEpñk m:aRTIsPaBrwgRkajsRmab;Ggát; nig
m:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúMKYrEtCam:aRTIssIuemRTI.
bMlas;TI nigkmøaMg³
EbgEckm:aRTIsPaBrwgRkajCaRkumdUcbgðajedaysmIkar 14-18. karBnøatenHeyIgTTYl)an


Qk = K11 Du + K12 Dk

Qu = K 21 Du + K 22 Dk
eKkMNt;bMlas;TIEdlCaGBaØat Du BIsmIkarTImYyénsmIkarTaMgBIrxagelI. edayeRbItémø
TaMgenH eKkMNt;kmøaMgRbtikmμ Qu BIsmIkarTIBIr. cugbBa©b; eKGackMNt;kmøaMgkñúg q enARtg;cug
rbs;Ggát;BIsmIkar 16-7 eBalKW
q = k 'TD
RbsinebIlT§plénGBaØatEdl)anKNnaCaTMhMGviC¢man vabgðajfaBYkvaeFVIGMeBItamTiskUGredaen
GviC¢man.

]TahrN_ 16-1³ kMNt;bnÞúkenARtg;tMNrbs;eRKagGgát;BIrEdlbgðajenAkñúgrUbTI 16-4a. yk I =
( ) /
1800 10 6 mm 4 A = 6000mm 2 ehIy E = 200GPa sRmab;Ggát;TaMgBIr.



dMeNaHRsay³
kareFVIkMNt;sMKal;³ tamkarGegát eRKagmanGgát;cMnYnBIr nig node cMnYnbIEdlRtUv)ankMNt;sMKal;dUc
bgðajenAkñúgrUbTI 16-4b. eKalrbs;RbB½n§kUGredaensklRtUvmanTItaMgenARtg; ①. dMbUgelxkUdenA
Rtg; node RtUv)ankMNt;eday degree of freedom EdlminmankarTb;. BIkarTb;enARtg;①nig③ nigbnÞúk
Gnuvtþn_ eyIg)an
⎡20⎤ 1
⎡0 ⎤ 6 ⎢ 0 ⎥2
⎢0 ⎥ 7 ⎢ ⎥
Dk = ⎢ ⎥ Qk = ⎢ 0 ⎥ 3
⎢0 ⎥ 8 ⎢ ⎥
⎢ ⎥ ⎢ 0 ⎥4
⎣0 ⎦ 9 ⎢ 0 ⎥5
⎣ ⎦
m:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúM³ tYxageRkammanlkçN³dUcKñasRmab;m:aRTIsPaBrwgRkajsRmab;
Ggát;TaMgBIr³
AE 6(10 −3 )(200)(10 6 )
= = 200(103 )kN / m
L 6
12 EI 12(200 )(10 6 )(180 )(10 −6 )
= = 2(10 3 )kN / m
L3 63
6 EI
=
( ) ( ) = 6(10 )kN / m
6(200 ) 10 6 (180 ) 10 −6 3
2 2
L 6
4 EI 4(200 )(10 )(180 )(10 )
= 24(10 )kN / m
6 −6
= 3
L 6

=
6
( ) ( ) ( )
2 EI 2(200 ) 10 (180 ) 10 −6
= 12 10 3 kN / m
L 6
6−0 0−0
Ggát;elx !³ λx =
6
=1 λy =
6
=0

edayCMnYsTinñn½yeTAkñúgsmIkar 16-10 eyIg)an
4 6 5 1 2 3
⎡ 200 0 0 − 200 0 0 ⎤ 4
⎢ 0 2 6 0 − 2 − 6⎥ 6
⎢ ⎥
( )
k1 = 10 3 ⎢ 0
⎢
6 24 0 − 6 12 ⎥ 5
⎥
⎢− 200 0 0 200 0 0 ⎥ 1
⎢ 0 −2 −6 0 2 − 6⎥ 2
⎢ ⎥
⎢ 0
⎣ 6 12 0 − 6 24 ⎥ 3
⎦



CYredk nigCYrQrénm:aRTIs 6 × 6 RtUv)ankMNt;edayelxkUdbI x, y, z CadMbUgenARtg;cugCit ehIybnþ
edaycugq¶ayeBalKW $/ ^/ %/ !/ @/ # erogKña ¬rUbTI 16-4b¦. eKeFVIEbbenHsRmab;karpÁúMFatuelIkeRkay.
Ggát;elx @³ λx = 6 − 6 = 0
6
λy =
−6−0
6
= −1

edayCMnYsTinñn½yeTAkñúgsmIkar 16-10 eyIg)an
1 3 2
7 8 9
⎡2 0 6 −2 0 6 ⎤1
⎢ 0 200 0 0 − 200 0 ⎥2
⎢ ⎥
( )
k 2 = 10 3 ⎢ 6
⎢
0 24 − 6 0 12 ⎥ 3
⎥
⎢− 2 0 −6 2 0 − 6⎥ 7
⎢ 0 − 200 0 0 200 0 ⎥8
⎢ ⎥
⎢6
⎣ 0 12 − 6 0 24 ⎥ 9
⎦
CaFmμta karkMNt;elxerogrbs;CUredk nigCYrQrKWeyageTAtamelxkUdTaMgbItamlMdab; x, y, z sRmab;
cugCit nigcugq¶ay erogKña eBalKW !/ @/ # bnÞab;mk &/ */ ( ¬rUbTI 16-4b¦.
m:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúMRtUv)ankMNt;edaykarpÁúMm:aRTIs k1 nig k 2 . lT§plén
Q = KD EdlbgðajedaykarbMEbkCaRkumKW
1 2 3 4 5 6 7 8 9
⎡ 20 ⎤ ⎡ 202 0 6 − 200 0 0 − 2 0 6 ⎤ ⎡ D1 ⎤
⎢0⎥ ⎢ 0 202 −6 0 − 6 − 2 0 − 200 0 ⎥ ⎢ D2 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢0⎥ ⎢ 6 −6 48 0 12 6 − 6 0 12 ⎥ ⎢ D3 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢0⎥
⎢0⎥ ( ) ⎢− 200
= 10 3 ⎢
0
0
−6
0 200 0 0 0
12 0 24 6 0
0
0
0 ⎥ ⎢ D4 ⎥
0 ⎥ ⎢ D5 ⎥
(1)
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢Q6 ⎥ ⎢ 0 −2 6 0 6 2 0 0 0 ⎥⎢ 0 ⎥
⎢Q ⎥ ⎢ −2 0 −6 0 0 0 2 0 − 6⎥ ⎢ 0 ⎥
⎢ 7⎥ ⎢ ⎥⎢ ⎥
⎢Q8 ⎥ ⎢ 0 − 200 0 0 0 0 0 200 0 ⎥ ⎢ 0 ⎥
⎢Q ⎥ ⎢ 6 0 0 −6 24 ⎥ ⎢ 0 ⎥
⎣ 9⎦ ⎣ 0 12 0 0 ⎦⎣ ⎦
kmøaMg nigbMlas;TI³ edayBnøatedIm,IedaHRsaybMlas;TI eyIgTTYl)an
⎡20⎤ ⎡ 202 0 6 − 200 0 ⎤ ⎡ D1 ⎤ ⎡0⎤
⎢0⎥
⎢ ⎥
⎢ 0
⎢ 202 − 6 0 − 6⎥ ⎢ D2 ⎥ ⎢0⎥
⎥⎢ ⎥ ⎢ ⎥
⎢ ⎥
( )
⎢ 0 ⎥ = 10 3 ⎢ 6
⎢
− 6 48 0 12 ⎥ ⎢ D3 ⎥ + ⎢0⎥
⎥⎢ ⎥ ⎢ ⎥
⎢0⎥ ⎢− 200 0 0 200 0 ⎥ ⎢ D4 ⎥ ⎢0⎥
⎢0⎥
⎣ ⎦ ⎢ 0
⎣ − 6 12 0 24 ⎥ ⎢ D5 ⎥ ⎢0⎥
⎦⎣ ⎦ ⎣ ⎦
edayedaHRsaym:aRTIsxagelI eyIgTTYl)am


( )
⎡ D1 ⎤ ⎡ 17.51 10 − 3 m ⎤
⎢D ⎥ ⎢
( )
⎢ 2 ⎥ ⎢ − 37.47 10 m ⎥
−6 ⎥

( )
⎢ D3 ⎥ = ⎢− 2.505 10 − 3 rad ⎥
⎢ ⎥ ⎢
( ) −3
⎢ D4 ⎥ ⎢ 17.51 10 m ⎥
⎥

⎣ ⎦ ⎣( )
⎢ D5 ⎥ ⎢ 1.243 10 − 3 rad ⎥
⎦
edayeRbIlT§plTaMgenH eKGackMNt;kmøaMgRbtikmμBIsmIkar (1) dUcxageRkam
1 2 3 4 5
( ) ⎡ 17.51 10 - 3 m ⎤
⎡Q6 ⎤
⎢Q ⎥
( )
⎡0
⎢− 2
−2 6
( )0 6⎤ ⎢ −6 ⎥ ⎡0⎤ ⎡− 7.50kN ⎤
⎥ ⎢ − 37.47 10 m ⎥ ⎢0⎥ ⎢ − 20kN ⎥
⎢ 7 ⎥ = 10 −6
( )
3 0 0 0⎥
⎢ ⎢− 2.505 10 − 3 rad ⎥ + ⎢ ⎥ = ⎢ ⎥
⎢Q8 ⎥ ⎢ 0 − 200 0 ⎥⎢ ⎥ ⎢0⎥ ⎢ 7.50kN ⎥
⎢ ⎥ ⎢ ( ) 0 0
⎥ ⎢ 17.51 10 − 3 m ⎥ ⎢ ⎥ ⎢ ⎥
⎣Q9 ⎦ ⎣6 0 12
( ) 0 0⎦ ⎢ −3
⎣ 1.243 10 rad ⎦
⎥ ⎣0⎦ ⎣ 75kN .m ⎦

eKGackMNt;kmøaMgkñúgenAkñúg node ② edayGnuvtþsmIkar 16-7 eTAelIGgát;elx 1. enATIenH k ' RtUv 1

)ankMNt;edaysmIkar 16-1 ehIy T edaysmIkar 16-3. dUcenH
1
4 6 5 1 2 3
⎡ 200 0 0 − 200 0 0 ⎤ ⎡1
⎢ 0 ⎢
( ) ⎤4
0 0 0 0 0⎤ ⎡ 17.5 10 − 3
⎥
⎢ 2 6 0 − 2 6 ⎥ ⎢0
⎥⎢ 1 0 0 0 0⎥ ⎢
⎥ 0 ⎥6
( )
q1 = k1T1 D = 10 3 ⎢ 0 6 24 0 − 6 12 ⎥ ⎢0 ( )
0 1 0 0 0⎥ ⎢ 1.243 10 − 3 ⎥5
⎢
⎢− 200 0 0 200 0 0 ⎥ ⎢0
⎥⎢
( )⎥⎢
0 0 1 0 0⎥ ⎢ 17.51 10 − 3
⎥
⎥1
⎢ 0 −2 −6 0 2 − 6 ⎥ ⎢0 ( )
0 0 0 1 0⎥ ⎢ − 37.47 10 6
⎥⎢
⎥2
⎥
⎢
⎢ 0
⎣ 6 12 0
⎥⎢
− 6 24 ⎥ ⎢0
⎦⎣ ( )
0 0 0 0 1⎥ ⎢− 2.505 10 − 3
⎦⎣ ⎥3
⎦
cMNaMkardMerobd¾RtwmRtUvénFatuenAkñúgm:aRTIsdUcEdl)anbgðajedayelxkUdtamRCugxagrbs;CYrQr
nigCYredk. edaHRsaym:aRTIsxagelI eyIg)an
⎡q 4 ⎤ ⎡ 0 ⎤
⎢ q ⎥ ⎢ − 7.50kN ⎥
⎢ 6⎥ ⎢ ⎥
⎢ q5 ⎥ ⎢ 0 ⎥
⎢ ⎥=⎢ ⎥
⎢ q1 ⎥ ⎢ 0 ⎥
⎢q 2 ⎥ ⎢ 7.50kN ⎥
⎢ ⎥ ⎢ ⎥
⎢ q3 ⎥ ⎢− 45kN .m⎥
⎣ ⎦ ⎣ ⎦
lT§plxagelIRtUv)anbgðajenAkñúgrUbTI 16-4c. TisedArbs;viucT½rTaMgenHRtUvKñanwgTisviC¢manEdlkMNt;
enAkñúgrUbTI 16-1. elIsBIenH eKalrbs;kUGredaen x', y' , z' sßitenARtg;cugCitrbs;Ggát;. tamrebob
dUcKña düaRkamGgÁesrIénGgát;elx @ RtUv)anbgðajenAkñúgrUbTI 16-4d.



]TahrN_ 16-2³ kMNt;bnÞúkenARtg;cugrbs;Ggát;nImYy²éneRKagEdlbgðajenAkñúgrUbTI 16-5a. yk
( ) /
I = 225 10 6 mm 4 A = 7500mm 2 ehIy E = 200GPa sRmab;Ggát;nImYy².

dMeNaHRsay³
kareFVIkMNt;sMKal;³ edIm,IGnuvtþkarviPaKedayviFIm:aRTIs bnÞúkBRgayEdlmanGMeBIenAelIGgát;edkRtUv)an
CMnYsedaym:Um:g;cugsmmUl nigkmøaMgkat;enAxagcugsmmUlEdlRtUv)anKNnaBIsþaTic nigBItaragenAkñúg
emeronTI11. bnÞab;mkedayeRbIviFItRmYtpl lT§plEdlTTYl)ansRmab;eRKagenAkñúgrUbTI 16-5b RtUv
)anEktRmUvsRmab;Ggát;enHedaybnÞúkEdlbgðajenAkñúgrUbTI 16-5c.
dUcbgðajenAkñúgrUbTI 16-5b, node nigGgát;RtUv)andak;elxerog ehIyeKalrbs;RbB½n§kUGredaen
sklRtUv)andak;enAkñúg node ①. tamFmμta dMbUgeKRtUvdak;elxkUdeTAelI degree of freedom Edlmin
mankarTb;. dUcenH



⎡0 ⎤ 4
⎢0 ⎥ 5
⎢ ⎥ ⎡ 0 ⎤1
⎢0 ⎥ 6
Dk = ⎢ ⎥ Qk = ⎢− 150⎥ 2
⎢ ⎥
⎢0 ⎥ 7 ⎢ 150 ⎥ 3
⎢0 ⎥ 8 ⎣ ⎦
⎢ ⎥
⎢0 ⎥ 9
⎣ ⎦
m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§
Ggát;elx !³
EA 7500(10 −6 )(200)( 6 )
= 200(10 3 )kN / m
10
=
L 7.5
12 EI
=
( ) ( ) = 1280kN / m
12(200 ) 10 6 (225) 10 −6
L 3
(7.5) 3

6 EI 6(200)(225)
= = 4800kN
L2 (7.5)2
4(200)(225)
= 24(10 3 )kN .m
4 EI
=
L 7.5
2 EI 2(200 )(225)
L
=
7.5
= 12 10 3 kN .m ( )
6−0 4.5 − 0
λx = = 0.8 λy = = 0.6
7.5 7.5
edayGnuvtþsmIkar 16-10/ eyIg)an
4 6 5 1 2 3
⎡ 128.46 95.39 − 2.88 − 128.46 − 95.39 − 2.88⎤ 4
⎢ 95.39 72.82 3.84 − 95.39 − 72.82 3.84 ⎥ 6
⎢ ⎥
( )
k1 = 10 3 ⎢ − 2.88
⎢
3.84 24 2.88 − 3.84 12 ⎥ 5
⎥
⎢− 128.46 − 95.39 2.88 128.46 95.39 2.88 ⎥ 1
⎢ − 95.39 − 72.84 − 3.84 95.39 72.82 − 3.84⎥ 2
⎢ ⎥
⎢ − 2.88
⎣ 3.84 12 2.88 − 3.84 24 ⎥ 3
⎦
Ggát;elx @³
=
( )
EA 7500 10 −6 (200 ) 10 6 ( )
= 250 10 3 kN / m( )
L 6
12 EI 12(200 )(225)
= = 2500kN / m
L3 (6)3
6 EI 6(200 )(225)
= = 7500kN
L2 (6)2
4 EI 4(200)(225)
L
=
6
= 30 10 3 kN .m ( )

2 EI 2(200 )(225)
L
=
6
= 15 10 3 kN .m ( )
12 − 6 4.5 − 4.5
λx = =1 λy = =0
6 6
dUcenH smIkar 16-10 køayCa
1 2 3 7 8 9
⎡ 250 250 0 − 250 0 0 ⎤1
⎢ 0 7.5 7.5 0 − 2.5 7.5 ⎥ 2
⎢ ⎥
( )
k 2 = 10 3 ⎢ 0
⎢
0 30 0 − 7.5 15 ⎥ 3
⎥
⎢− 250 0 0 250 0 0 ⎥7
⎢ 0 − 2.5 − 7.5 0 2.5 − 7.5⎥ 8
⎢ ⎥
⎢ 0
⎣ 7.5 15 0 − 7.5 30 ⎥ 9
⎦
m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§EdlrYmbBa©ÚlenAkñúg Q = KD køayCa
1 2 3 4 5 6 7 8 9
⎡ 0 ⎤ ⎡ 378.46 95.39 7.88 − 128.46 − 95.39 2.88 − 250 0 0 ⎤ ⎡ D1 ⎤
⎢− 150⎥ ⎢ 95.39 75.32 3.66 − 95.39 − 72.82 − 3.84 0 − 2.5 7.5 ⎥ ⎢ D2 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢− 150⎥ ⎢ 2.88 3.66 54 − 2.88 3.84 12 0 − 7.5 15 ⎥ ⎢ D3 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ Q4 ⎥
( )
3 ⎢
− 128.46 − 95.39 − 2.88 128.46 95.39 − 2.88
⎢ Q5 ⎥ = 10 ⎢ − 95.39 − 72.82 3.84 95.39 72.82 3.84
0
0
0
0
0 ⎥⎢ 0 ⎥
0 ⎥⎢ 0 ⎥
(1)
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ Q6 ⎥ ⎢ 2.88 − 3.84 12 − 2.88 3.84 24 0 0 0 ⎥⎢ 0 ⎥
⎢ Q ⎥ ⎢ − 20 0 0 0 0 0 250 0 0 ⎥⎢ 0 ⎥
⎢ 7 ⎥ ⎢ ⎥⎢ ⎥
⎢ Q8 ⎥ ⎢ 0 − 2.5 − 7.5 0 0 0 0 2.5 − 7.5⎥ ⎢ 0 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎣ Q9 ⎦ ⎣ 0 7.5 15 0 0 0 0 − 7.5 30 ⎦ ⎣ 0 ⎦

bMlas;TI nigbnÞúk³ Bnøatm:aRTIsxagelIedIm,IkMNt;bMlas;TI nigedayedaHRsay eyIg)an
⎡ 0 ⎤ ⎡378.46 95.39 2.88⎤ ⎡ D1 ⎤ ⎡0⎤
⎥ ( )
⎢− 150⎥ = 10 3 ⎢ 95.39 75.32 3.66⎥ + ⎢ D ⎥ + ⎢0⎥
⎢ ⎢ ⎥ ⎢ 2⎥ ⎢ ⎥
⎢− 150⎦
⎣ ⎥ ⎢ 2.88 3.66 54 ⎥ ⎢ D3 ⎥ ⎢0⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎡ D1 ⎤ ⎡ 0.716mm ⎤
⎢ D ⎥ = ⎢ − 2.76mm ⎥
⎢ 2⎥ ⎢ ⎥
⎢ D3 ⎥ ⎢− 0.00261rad ⎥
⎣ ⎦ ⎣ ⎦
edayeRbIlT§plTaMgenH eKGackMNt;kmøaMgRbtikmμTMrBIsmIkar (1) dUcbgðaj³



⎡Q4 ⎤ ⎡− 128.46 − 95.39 − 2.88⎤ ⎡0⎤ ⎡ 178.8kN ⎤
⎢Q ⎥ ⎢ − 95.39 − 72.82 3.84 ⎥ ⎢0⎥ ⎢ 122.7 kN ⎥
⎢ 5⎥ ⎢ ⎥⎡ 0.716 ⎤ ⎢ ⎥ ⎢ ⎥
⎢Q6 ⎥ ⎢ 2.88 − 3.84 12 ⎥ ⎢ ⎥+ ⎢0⎥ ⎢ − 18.7 kN .m ⎥
⎢ ⎥=⎢ ⎥ − 2.76 ⎥ ⎢0⎥ = ⎢ − 179.0kN ⎥
0 ⎥⎢
⎢Q7 ⎥ ⎢ − 250
⎢Q8 ⎥ ⎢ 0
0
− 2.5 − 7.5 ⎥⎣
⎢− 0.00261 10 3 ( ) ⎥ ⎢ ⎥ ⎢
⎦ ⎢0⎥ ⎢ 26.5kN ⎥
⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢Q9 ⎥ ⎢ 0
⎣ ⎦ ⎣ 7.5 15 ⎥⎦ ⎢0⎥ ⎢− 59.9kN .m ⎥
⎣ ⎦ ⎣ ⎦
eKGackMNt;kmøaMgkñúgBIsmIkar 16-7 EdlGnuvtþeTAGgát;elx ! nigelx @. enAkñúgkrNIGgát;elx !/
q = k '1 T1 D Edl k '1 RtUv)ankMNt;BIsmIkar 16-1 ehIy T1 RtUv)ankMNt;BIsmIkar 16-3. dUcenH
4 5 6 1 2 3
⎡q4 ⎤ ⎡ 200 0 0 − 200 0 0 ⎤ ⎡ 0.8 0.6 0 0 0 0⎤ ⎡ 0 ⎤ 4
⎢q ⎥ ⎢ 0 1.28 4.8 0 − 1.28 4.8 ⎥ ⎢− 0.6 0.8 0 0 0 0⎥ ⎢ 0 ⎥ 5
⎢ 5⎥ ⎢ ⎥⎢ ⎥⎢ ⎥
⎢ q6 ⎥ = ⎢ 0 4.8 2.4 0 − 4.8 12 ⎥ ⎢ 0 0 1 0 0 0⎥ ⎢ 0 ⎥ 6
⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥
⎢q7 ⎥ ⎢− 200 0 0 200 0 0 ⎥⎢ 0 0 0 0.8 0 0⎥ ⎢ 0.716 ⎥ 1
⎢ q8 ⎥ ⎢ 0 − 1.28 − 4.8 0 1.28 − 1.8⎥ ⎢ 0 0 0 − 0.6 0.8 0⎥ ⎢− 2.76⎥ 2
⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥
⎢ ⎥ ⎢
⎣ q9 ⎦ ⎣ 0 4.8 12 0 − 4.8 24 ⎥ ⎢ 0
⎦⎣ 0 0 0 0 1⎥ ⎢ − 2.61⎥ 3
⎦⎣ ⎦
enATIenH elxkUdbgðajCYredk nigCYrQrsRmab;cugCit nigcugq¶ayrbs;Ggát; erogKña eBalKW $/ %/ ^
bnÞab;mk !/ @/ # rUbTI 16-5b. dUcenH
⎡q4 ⎤ ⎡ 216.6kN ⎤
⎢ q ⎥ ⎢ − 9.15kN ⎥
⎢ 5⎥ ⎢ ⎥
⎢q6 ⎥ ⎢− 18.7kN .m⎥
⎢ ⎥=⎢ ⎥
⎢ q1 ⎥ ⎢ 216.6kN ⎥
⎢q2 ⎥ ⎢ 9.15kN ⎥
⎢ ⎥ ⎢ ⎥
⎢ q3 ⎥ ⎢ − 50kN .m ⎥
⎣ ⎦ ⎣ ⎦
lT§plTaMgenHRtUv)anbgðajenAkñúgrUbTI 16-5d.
karviPaKdUcKñaRtUv)aneFVIsRmab;Ggát;elx @. lT§plRtUv)anbgðajenAxageqVgkñúgrUbTI 16-5e.
sRmab;Ggát;enH eyIgRtUvdak;bnÞúkénrUbTI 16-5c dUcenHlT§plcugeRkaysRmab;Ggát;elx @ RtUv)an
bgðajenAxagsþaM.



cMeNaT
16>1 kMNt;m:aRTIsPaBrwgRkaj K sRmab;eRKag. 16>4 kMNt;kmøaMgRbtikmμTRmkñúgenARtg; ① nig
snμt; ① nig ③ CaTRmsnøak;. yk E = 200GPa ③ ¬cMeNaT 16>3¦. yk E = 200GPa /
I = 243( 6 )mm 4 , A = 6000mm 2 sRmab;Ggát;
10 I = 300( 6 )mm 4 , A = 21( 3 )mm 2 sRmab;
10 10

nImYy². Ggát;nImYy².
16>5 kMNt;m:aRTIsPaBrwgRkaj K sRmab;eRKag.
yk E = 200GPa I = 250(106 )mm 4 ,
A = 19( 3 )mm 2 sRmab;Ggát;nImYy². snμt; ②
10

nig③ CatMNbgáb;.

16>2 kMNt;kmøaMgkñúgenARtg;cugrbs;Ggát;nImYy²
¬cMeNaT 16>1¦. snμt; ① nig ③ CaTRmsnøak;.
yk E = 200GPa / I = 243(106 )mm 4 ,
A = 6000mm 2 sRmab;Ggát;nImYy².

16>3 kMNt;m:aRTIsPaBrwgRkaj K sRmab;Ggát;
nImYy²rbs;eRKag. snμt; ③Casnøak; nig ①Ca 16>6 kMNt;m:aRTIsPaBrwgRkaj K sRmab;Ggát;
TRmbgáb;. yk E = 200GPa nImYy²rbs;eRKag. yk E = 200GPa
I = 300( 6 )mm 4 , A = 21(10 3 )mm 2 sRmab;
10 I = 280( 6 )mm 4 , A = 18( 3 )mm 2
10 10

Ggát;nImYy². sRmab;Ggát;nImYy².

Problems T.Chhay -535


16>7 kMNt;kmøaMgkñúgenARtg;cugrbs;Ggát;nImYy²
¬cMeNaT 16>6¦. yk E = 200GPa /
I = 280( 6 )mm 4 , A = 18( 3 )mm 2
10 10
sRmab;Ggát;nImYy².
16>8 kMNt;m:aRTIsPaBrwgRkaj K sRmab;eRKag.
yk E = 200GPa I = 250(106 )mm 4 ,
A = 12( 3 )mm 2 sRmab;Ggát;nImYy².
10
16>11 kMNt;mMurgVilkñúgenARtg; ① nig ③ nig
kmøaMgRbtikmμenAkñúgcMeNaT 16>10.
16>12 kMNt;m:aRTIsPaBrwgRkaj K sRmab;
Ggát;nImYy²rbs;eRKag. yk E = 200GPa
I = 270( 6 )mm 4 , A = 6( 3 )mm 2 sRmab;Ggát;
10 10

nImYy².

16>9 kMNt;bgÁúMbMlas;TIRtg;① éncMeNaT 16>8.
yk E = 200GPa I = 250(106 )mm 4 ,
A = 12( 3 )mm 2 sRmab;Ggát;nImYy².
10

eRKag. yk E = 200GPa I = 240(106 )mm 4 ,
A = 6( 3 )mm 2 sRmab;Ggát;nImYy². snμt; ①
10 16>13 kMNt;kmøaMgRbtikmμTRm ① nig ④
nig③ Casnøak; ehIy ②CatMNbgáb;. kñúgcMeNaT 16.13. tMN ① nig ④CatMN
snøak; ehIy② nig③ CatMNbgáb;. yk

cMeNaT T.Chhay -536


( )
E = 200GPa I = 270 10 6 mm 4 ,
A = 9( )mm sRmab;Ggát; nImYy².
10 3 2

eRKagEdlmanGgát;BIr. yk E = 200GPa
I = 350( )mm , A = 20( )mm sRmab;
10 6 4
10 3 2

Ggát;nImYy². tMN ① nig③ CatMNsnøak; ehIy
②CatMNbgáb;.

16>15 kMNt;kmøaMgRbtikmμTRmenARtg; ① nig③
éncMeNaT 16>14.

Problems T.Chhay -537

16. plane frame analysis using the stiffness method

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16. plane frame analysis using the stiffness method