Contenu connexe Similaire à 14. truss analysis using the stiffness method (17) 14. truss analysis using the stiffness method1. Department of Civil Engineering NPIC
!$> karviPaK truss edayeRbIviFIPaBrwgRkaj
Truss analysis using the stiffness method
enAkñúgemeronenH eyIgnwgBnül;BIeKalkarN_mUldæanénkareRbIR)as;viFIPaBrwgRkajsRmab;viPaKeRKOg
bgÁúM. viFIenHmanPaBsμúKsμajsRmab;karedaHRsayedayéd EtvasaksmsRmab;eRbICamYynwgkMuBüÚT½r.
enAkñúgemeronenHmanbgðajBI]TahrN_Gnuvtþn_eTAelI truss kñúgbøg;. bnÞab;mk eyIgnwgeRbIviFIenH
sRmab; truss kñúglMh. eyIgnwgerobrab;BIkarGnuvtþviFIenHsRmab;Fñwm nigeRKagenAemeroneRkay.
!$>!> eKalkarN_mUldæanénviFIPaBrwgRkaj (Fundamentals of the stiffness method)
eKmanmeFüa)ayBIrsRmab;viPaKrcnasm<½n§edayeRbIviFIm:aRTIs. viFIPaBrwgRkajEdlRtUveRbI
enAkñúgemeronenH nigemeroneRkayCakarviPaKedayeRbIviFIbMlas;TI. viFIkmøaMg EdleKehAfaviFI
flexibility ¬Edlerobrab;enAkñúgkfaxNÐ 10-1¦ k¾GaceRbIedIm,IviPaKrcnasm<½n§ b:uEnþviFIenHminRtUv)an
bgðajenAkñúgesovePAenHeT. mUlehtucm,gKW eKGaceRbIviFIPaBrwgRkajsRmab;viPaKrcnasm<½n§kMNt;
edaysþaTic nigrcnasm<½n§minkMNt;edaysþaTic cMENkÉviFI flexibility RtUvkardMeNIrkarepSgsRmab;
krNInImYy²énkrNITaMgBIr. ehIyviFIPaBrwgRkajpþl;eGaybMlas;TI nigkmøaMgedaypÞal; cMENkÉ
viFI flexibility minpþl;eGaybMlas;TIedaypÞal;eT. elIsBIenH eKmanPaBgayRsYlsresrrUbmnþ
m:aRTIsEdlcaM)ac;sRmab;RbtþibtþikarkMuBüÚT½redayeRbIviFIPaBrwgRkaj ehIyenAeBleKeFVIvarYc eK
GacviPaKeRKOgbgÁúMedaykMuBüÚT½ry:agmanRbsiT§PaB.
karGnuvtþviFIPaBrwgRkajTamTarnUvkarbMEbkeRKOgbgÁúMCaes‘rIén finite elements ehIyeK
RtUvkMNt;GtþsBaØaNeGaycMNuccugrbs;Ggát;Ca node. sRmab;karviPaK truss, finite element Ca
Ggát;nImYy²EdlpSMCa truss ehIy node CatMN. eKRtUvkMNt;lkçN³kmøaMg nigbMlas;TIrbs;Ggát;
nImYy² ehIyeKRtUveFVITMnak;TMngrvagkmøaMg nigbMlas;TIedayeRbIsmIkarlMnwgkmøaMgEdlsresrenA
Rtg; node. bnÞab;mk eKerobcMTMnak;TMngTaMgenH ¬sRmab;rcnasm<½n§TaMgmUl¦ CaRkumbBa©ÚlKña
EdleKeGayeQμaHfa structure stiffness matrix K. enAeBlEdleKbegáItm:aRTIsrYcehIy eKGac
kMNt;bMlas;TIrbs; node sRmab;bnÞúkenAelIrcnasm<½n§. enAeBlEdleKsÁal;bMlas;TIrYcehIy eK
GackMNt;kmøaMgkñúgrbs;eRKOgbgÁúMedayeRbITMnak;TMngrvagkmøaMg nigbMlas;TIsRmab;Ggát;.
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -464
2. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
munnwgbegáItdMeNIrkarsRmab;GnuvtþviFIPaBrwgRkaj CaCMhandMbUgeyIgcaM)ac;yl;dwgBIniym
n½y nigeKalKMnitmYycMnYn³
karkMNt;GtþsBaØaNrbs;Ggát; nig node³ CMhanmYyénCMhandMbUgkñúgkarGnuvtþviFIPaBrwgRkaj
KWkMNt;GtþsBaØaNGgát;rbs;rcnasm<½n§ nig node rbs;va. eyIgnwgkMNt;Ggát;edaybg;elxEdl
B½T§CMuvijedaykaer ehIyelxEdlB½T§CMuvijedayrgVg;kMNt;eGay node. ehIy eKk¾RtUvkMNt;cugCit
nigcugq¶ayrbs;Ggát;edayeRbIk,alRBYjEdlvacg¥úleTAcugq¶ay. ]TahrN_énkarkMNt;Ggát;
node nigTisedAsMrab; truss RtUv)anbgðajenAkñúgrUbTI 14-1a.
kUGredaenskl nigkUGredaenGgát;³ edaysarbnÞúk nigbMlas;TICaTMhMviucT½r enaHeKcaM)ac;RtUv
begáItRbB½n§kUGredaenedIm,IkMNt;TisedArbs;vaeGay)anRtwmRtUv. enATIenH eyIgnwgeRbIRbB½n§kUGr
edaenBIrRbePTepSgKña. RbB½n§kUGredaenskl b¤RbB½n§kUGredaenrcnasm<½n§ ¬ x, y ¦ RtUveRbIedIm,I
kMNt;TisedAénbnÞúkxageRkA nigTisedAénbgÁúM;bMlas;TIenARtg; node ¬rUbTI 14-1a¦. RbB½n§kUGr
edaentMbn; b¤RbB½n§kUGredaenGgát;RtUv)aneRbIsRmab;Ggát;nImYy²edIm,IkMNt;TisedAénbMlas;TI
rbs;va nigkmøaMgkñúgrbs;va. RbB½n§enHRtUv)ankMNt;edayeRbIG½kS x' , y' CamYynwgKl;enARtg;
node Cit ehIyG½kS x' latsn§wgeq<aHeTArk node q¶ay. ]TahrN_sRmab;Ggát; truss elx $ RtUv
)anbgðajenAkñúgrUbTI 14-1b.
PaBminkMNt;sIueNm:aTic³ dUckarerobrab;enAkñúgkfaxNÐ 11-1, degree of freedom Edlminman
karTb;sRmab; truss CaGBaØatdMbUgénviFIbMlas;TI dUcenHeKRtUvEtkMNt;va. tamc,ab;TUeTA eKman
degree of freedom b¤bMlas;TIEdlGacekItmancMnYnBIr sMrab;tMN (node). sRmab;karGnuvtþ
degree of freedom nImYy²RtUv)ankMNt;enAelI truss edayeRbIelxkUd ¬EdlbgðajenARtg;tMN b¤
Truss analysis using the stiffness method T.Chhay -465
3. Department of Civil Engineering NPIC
node ¦ ehIyeKeRbI k,alRBYjedaysMGageTAelIkUGredaensklviC¢man. ]TahrN_ truss
enAkñúgrUbTI 14-1a man degree of freedom cMnYn8 EdlRtUv)ankMNt;;edayelxkUdBIelx 1 dl;
elx 8 dUcbgðaj. Truss enHminkMNt;edaysIueNm:aTicdWeRkTI5 edaysarbMlas;TIEdlGac
ekItmanTaMg 8enH elx1 dl;elx5 CaGBaØat b¤ degree of freedom EdlminmankarTb; ehIyelx
6 dl;elx8Ca degree of freedom EdlmankarTb;. edaysarmankarTb; bMlas;TIenATIenHRtUvesμI
sUnü. sRmab;karGnuvtþbnþbnÞab; eyIgeRbIelxkUdtUc²sRmab;sMKal;bMlas;TIEdleyIgminsÁal;
¬degree of freedom EdlminTb;¦ ehIyelxkUcFM²sRmab;sMKal;bMlas;TIEdlsÁal; ¬degree of
freedom EdlTb;¦. mUlehtukñúgkareFVIEbbenH edIm,IgayRsYlerobcM structure stiffness matrix
dUcenHeyIgnwgGackMNt;bMlas;TIEdlCaGBaØatedaypÞal;.
eRkayeBleyIgbg;elxeGay truss ehIykMNt;elxkUd eyIgGacKNna structure
stiffness matrix K. edIm,IeFVIEbbenH dMbUgeyIgRtUvbegáIt member stiffness matrix k’ sRmab;
Ggát;nImYy²rbs; truss. eKeRbIm:aRTIsenHedIm,IbgáajTMnak;TMngrvagbnÞúk nigbMlas;TIrbs;Ggát;
edayeRbIkUGredaentMbn;. edaysarGgát;TaMgGs;rbs; truss minmanTisdUcKña eyIgRtUvbMElgTMhM
TaMgenHBIkUGredaentMbn; x' , y' eTACakUGredaenskl x, y edayeRbIm:aRTIsbMElgkmøaMg nig
bMlas;TI (force and displacement transformation matrices). eRkaybegáItrYcehIy eyIgGac
bMElgm:aRTIsPaBrwgRkajrbs;Ggát;BIkUGredaentMbn;eTACakUGredaenskl ehIybnÞab;mkpÁúMva
edIm,IbegáItCam:aRTIsPaBrwgRkajrcnasm<½n§. edayeRbI K ¬dUckarbgðajxagelI¦ dMbUgeyIgGac
kMNt;bMlas;TIrbs; node bnÞab;mkeyIgGackMNt;kmøaMgRbtikmμTMr nigcugeRkayKWkmøaMgkñúgrbs;
Ggát;. eyIgnwgbegáItviFIenH.
!$>@> m:aRTIsPaBrwgRkajrbs;Ggát; (Member stiffness matrix)
enAkñúgkfaxNÐenH eyIgnwgbegáItm:aRTIsPaBrwgRkajsMrab;Ggát;eTalrbs; truss edayeRbI
kUGredaentMbn; x' , y' dUcbgðajenAkñúgrUbTI 14-2. tYenAkñúgm:aRTIsenHCaTMnak;TMngrvagbnÞúk nig
bMlas;TIsRmab;Ggát;.
Ggát;rbs; truss Gacpøas;TI)anEttamG½kS x' rbs;vab:ueNÑaH edaysarbnÞúkGnuvtþtamTis
enH. dUcenH eKGacmanbMlas;TIÉkraCüBIr. enAeBleKeGaycugCitrbs;Ggát;manbMlas;TIviC¢man
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -466
4. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
dN ehIycMENkÉcugq¶ayRtUv)anTb;edaysnøak; ¬rUbTI 14-2a¦ enaHkmøaMgEdlekItmanenARtg;
cugrbs;Ggát;KW
AE AE
q' N = dN q' F = − dN
L L
cMNaMfa q' GviC¢manedaysarsßanPaBlMnwg vaRtUveFVIGMeBItamTisedAGviC¢man x' . dUcKña bMlas;TI
F
viC¢man d enARtg;cugq¶ayedaycugCitenAEtTb;edaysnøak; ¬rUbTI 14-2b¦ pþl;nUvkmøaMgkñúgGgát;
F
AE AE
q' ' N = − dF q' ' F = dF
L L
edaykareFIVtRmYtpl ¬rUbTI 14-2c¦ kmøaMgers‘ultg;EdlbgáedaybMlas;TITaMgBI
AE AE
qN = dN − dF (14-1)
L L
AE AE
qF = dF − dN (14-2)
L L
eKGacsresrsmIkarTMnak;TMngrvagbnÞúk nigkmøaMgkñúgTRmg;m:aRTIs*Ca
⎡q N ⎤ AE ⎡ 1 − 1⎤ ⎡d N ⎤
⎢ q ⎥ = L ⎢− 1 1 ⎥ ⎢ d ⎥
⎣ F⎦ ⎣ ⎦⎣ F ⎦
b¤ q = k ' d (14-3)
Edl k ' = AE ⎡−11 −11⎤
L ⎢ ⎥ (14-4)
⎣ ⎦
m:aRTIs k ' RtUv)aneKeGayeQμaHfam:aRTIsPaBrwgRkajsRmab;Ggát; ehIyvamanTRmg;dUcKñasRmab;
Ggát;nImYy²rbs; truss. tYTaMgbYnEdlbegáItCam:aRTIsenHRtUv)aneKeGayeQμaHfaemKuNT§iBl
kRmajsRmab;Ggát; (member stiffness influence coefficient) k' . k' CakmøaMgenARtg;tMN i
ij ij
enAeBltMN j ekItmanbMlas;TImYyÉktþa. ]TahrN_ RbsinebI i = j = 1 enaH k ' CakmøaMgenA11
Rtg;cugCit enAeBlcugq¶ayRtUv)anTb;edaybgáb; ehIycugCitrgbMlas;TI d = 1 eBalKW
N
AE
q N = k '11 =
L
dUcKña eKkMNt;kmøaMgenAcugq¶ayBI i = 2 / j =1 dUcenH
AE
q F = k ' 21 = −
L
tYTaMgBIrenHCaCYrQrTImYyrbs;m:aRTIsPaBrwgRkajGgát;. tamrebobdUcKña CYrQrTIBIrrbs;m:aRTIs
enHCakmøaMgenAkñúgGgát;enAeBlcugq¶ayrbs;Ggát;rgbMlas;TIÉktþa.
*
]bsm<½n§ A pþl;eGaynUvkarrMlwkBIm:aRTIs.
Truss analysis using the stiffness method T.Chhay -467
5. Department of Civil Engineering NPIC
!$>#> m:aRTIsbMElgénbMlas;TI nigkmøaMg
(Displacement and force transformation matrices)
edaysar truss pSMeLIgedayGgát;eRcIn eyIgnwg
begáItviFIsRmab;bMElgkmøaMgkñúgGgát; q nig bMlas;TI d
EdlkMNt;enAkñúgkUGredaentMbn;eGayeTACakUGredaen
skl. edIm,IPaBgayRsYl eyIgnwgBicarNakUGredaen
sklviC¢man x manTisedAeTAsþaM ehIy y manTisedA
eLIgelI. mMurvagG½kSskl x, y nigG½kStMbn; x' , y'
RtUv)ankMNt;eday θ x ehIy θ y dUcbgðajenAkñúgrUbTI
14-3. eyIgnwgeRbI kUsIunUsénmMuTaMgenHenAkñúgkarviPaK
m:aRTIsdUcteTA. eyIgtag λ x = cosθ x ehIy λ y = cosθ y . eKGackMNt;témøCaelxsRmab; λ x
nig λ y y:aggayedayeRbIkMuBüÚT½r enAeBleKkMNt; kUGredaen x, y éncugCit N nigcugq¶ay F rbs;
Ggát;rYcehIy. ]TahrN_ eKmanGgát; NF dUcbgðajenAkñúgrUbTI 14-4. enATIenH kUGredaenrbs; N
nig F KW (x N , y N ) nig (x F , y F ) erogKña . dUcenH
*
xF − x N xF − xN
λ x = cos θ x = = (14-5)
L (x F − xN ) + (yF − y N )
2 2
yF − yN yF − yN
λ y = cos θ y = = (14-6)
L (x F − xN ) + (yF − yN )
2 2
*
Kl;rbs;kUGredaenGacsßitenATINak¾)an eGayEtmanlkçN³gayRsYl. b:uEnþ CaTUeTA vaeRcInsßitenARtg;TItaMgNaEdl
kUGredaenrbs; node TaMgGs;viC¢man dUcbgðajenAkñúgrUbTI 14-4.
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -468
6. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
sBaØanBVnþenAkñúgsmIkarTUeTATaMgenHnwgKitedaysV½yRbvtþisRmab;Ggát;EdlsßitenAkñúgkaRdg;NamYy
rbs;bøg; xy .
m:aRTIsbMElgbMlas;TI³ enAkñúgkUGredaenskl cugnImYy²rbs;Ggát;Gacman degree of freedom b¤
bMlas;TIÉkraCüBIr eBalKWtMN N man DN nig DN ¬rUbTI 14-5a nig 14-5b¦ ehIytMN N man
x y
D F nig DF ¬rUbTI 14-5c nig14-5d¦. eyIgnwgBicarNabMlas;TITaMgenHdac;edayELkBIKñaedIm,I
x y
kMNt;bgÁúMbMlas;TIrbs;vatambeNþayGgát;. enAeBlcugq¶ayRtUv)anTb;edaysnøak; ehIycugCitmanbM
las;TItamkUGredaenskl DN ¬rUbTI 14-5a¦ bMlas;TI ¬kMhUcRTg;RTay¦ EdlRtUvKñatambeNþay
x
Ggát;KW DN cosθ x *. dUcKña bMlas;TI DN nwgeFVIeGayGgát;pøas;TI DN cosθ y tambeNþayG½kS x'
x y y
¬rUbTI 14-5b¦. T§iBlénbMlas;TIsklTaMgeFVIeGayGgát;pøas;TI.
d N = D N x cos θ x + D N y cos θ y
tamrebobdUcKña bMlas;TIviC¢man DF nig DF erogKña EdlGnuvtþenARtg;cugq¶ay F cMENkÉcugCit
x y
RtUv)anTb;edaysnøak; ¬rUbTI 14-5c nig 14-5d¦ nwgeFVIeGayGgát;pøas;TI
d F = D Fx cos θ x + D Fy cos θ y
edayeGay λ x = cosθ x nig λ y = cosθ y CakUsIunUsR)ab;Tis (direction cosine) sRmab;Ggát;
eyIg)an
d N = DN x λ x + DN y λ y
d F = DFx λ x + DFy λ y
EdleKGacsresrvaCaTRmg;m:aRTIs
⎡ DN x ⎤
⎢ ⎥
⎡ d N ⎤ ⎡λ x λ y 0 0 ⎤ ⎢ D N y ⎥
⎢d ⎥ = ⎢ 0 0 λ λ ⎥⎢ D ⎥ (14-7)
⎣ F⎦ ⎣ x y⎦
⎢ x⎥
F
⎢ D Fy ⎥
⎣ ⎦
*
eKminKitBIbMEbMrYl θ x b¤ θ y edaysarvamantémøtUceBk.
Truss analysis using the stiffness method T.Chhay -469
7. Department of Civil Engineering NPIC
b¤ d = TD (14-8)
⎡λ x λ y 0 0 ⎤
Edl T =⎢ ⎥ (14-9)
⎣ 0 0 λx λ y ⎦
BIkarbMEbkxagelI T bMElgBIbMlas;TI D kñúgkUGredaenskl x, y TaMgbYneGayeTACabMlas;TI d kñúg
kUGredaentMbn; x' cMnYnBIr. dUcenH T Cam:aRTIsbMElgbMlas;TI.
m:aRTIsbMElgkmøaMg³ BicarNakarGnuvtþkmøaMg q eTAelIcugCitrbs;Ggát; cMENkcugq¶ayRtUv)anTb;
N
edaysnøak; ¬rUbTI 14-6a¦. enATIenH bgÁúMkmøaMgsklrbs; q N enARtg; N KW
Q N x = q N cos x Q N y = q N cos θ y
dUcKña RbsinebI q F RtUv)anGnuvtþeTAelIr)ar ¬rUbTI 14-6b¦ bgÁúMkmøaMgsklenARtg; F KW
Q Fx = q F cos θ x Q Fy = q F cos θ y
edayeRbIkUsIunUsR)ab;Tis λ x = cosθ x / λ y = cosθ y smIkarTaMgenHkøayCa
QN x = q N λ x QN y = q N λ y
Q Fx = q F λ x Q Fy = q F λ y
EdleKGacsresrvaCaTRmg;m:aRTIsdUcxageRkam
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -470
8. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
⎡Q N x ⎤ ⎡ λ x 0⎤
⎢Q ⎥ ⎢ ⎥
⎢ N y ⎥ = ⎢λ y 0⎥ ⎡q N ⎤
λx ⎥ ⎢qF ⎥
(14-10)
⎢ QFx ⎥ ⎢ 0 ⎣ ⎦
⎢ ⎥ ⎢ ⎥
⎢ Q Fy ⎥ ⎢ 0
⎣ ⎦ ⎣ λy ⎥
⎦
b¤ Q =TTq (14-11)
⎡λ x 0⎤
⎢λ 0⎥
Edl TT =⎢
⎢0
y ⎥
λx ⎥
(14-12)
⎢ ⎥
⎢0
⎣ λy ⎥
⎦
enAkñúgkrNIenH T T bMElgBIkmøaMg q EdlmanGMeBIenARtg;cugrbs;Ggát;kñúgkUGredaentMbn; x' eGayeTA
CakmøaMg Q EdlmanbgÁúMbYnkñúgkUGredaenskl x, y . tamkareRbobeFob m:aRTIsbMElgkmøaMgCa
m:aRTIs transpose énm:aRTIsbMElgbMlas;TI ¬smIkar 14-9¦.
!$>$> m:aRTIsPaBrwgRkajrbs;Ggát;kñúgkUGredaenskl
(Member global stiffness matrix)
eyIgnwgpþúMlT§plenAkñúgkfaxNÐxagelI ehIykMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;EdlTak;
TgnwgbgÁúMkmøaMgskl Q nigbMlas;TIskl D rbs;Ggát;. RbsinebIeyIgCMnYssmIkar 14-8 ¬ d = TD ¦
eTAkñúgsmIkar 14-3 ¬ q = k ' d ¦ eyIgGackMNt;kmøaMg q rbs;Ggát;CaGnuKmn_énbMlas;TIskl D enA
Rtg;cMNuccugrbs;va eBalKW
q = k ' TD (14-13)
edayCMnYssmIkarenHeTAkñúgsmIkar 14-11 ¬ Q = T T q ¦ enaHeyIgnwgTTYl)anlT§plcugeRkay
Q = T T k ' TD
b¤ Q = KD (14-14)
Truss analysis using the stiffness method T.Chhay -471
9. Department of Civil Engineering NPIC
Edk k = T T k 'T (14-15)
m:aRTIs k Cam:aRTIsPaBrwgRkajsRmab;Ggát;enAkñúgkUGredaenskl. edaysareKsÁal; T T / T nig k '
enaHeyIg)an
⎡λ x 0⎤
⎢λ 0 ⎥ AE ⎡ 1 − 1⎤ ⎡λ x λ y 0 0 ⎤
k=⎢ ⎥
y
⎢0 λ x ⎥ L ⎢− 1 1 ⎥ ⎢ 0 0 λ x λ y ⎥
⎣ ⎦⎣ ⎦
⎢ ⎥
⎢0
⎣ λy ⎥⎦
edayKNnaedaHRsaym:aRTIsxagelI eyIg)an
Nx Ny Fx Fy
⎡ λ2 x λxλ y − λ2
x − λxλ y ⎤ N x
AE ⎢ ⎥
k= ⎢ λx λ y λ2
y − λxλ y − λ2 ⎥ N y
y (14-16)
L ⎢
⎢ − λx
2
− λxλ y λ2
x λ x λ y ⎥ Fx
⎥
⎢− λ x λ y − λ2 λxλ y 2 ⎥F
λy ⎦ y
⎣ y
TItaMgrbs;tYnImYy³enAkñúgm:aRTIssIuemRTITMhM 4 × 4 tMNageGay degree of freedom sklnImYy²Edl
pSMCamYynwgcugCit N nigCamYynwgcugq¶ay F . vaRtUv)anbgðajedaynimitþsBaØénelxkUdEdlenAtam
CYredk nigCYrQr eBalKW N x , N y , Fx , Fy . enATIenH k CaTMnak;TMngrvagkmøaMg nigbMlas;TIsRmab;
Ggát;enAeBlEdlbgÁúMénkmøaMg nigbMlas;TIenAcugrbs;Ggát;sßitenAkñúgkUGredaenskl b¤G½kS x, y . dUc
enHtYnImYy²enAkñúgm:aRTIsCaemKuNT§iBlPaBrwgRkaj (stiffness influence coefficient) K ij Edl
bgðajbgÁúMkmøaMg x b¤ y enARtg; i EdlcaM)ac;edIm,IeFVIeGaymanbgÁúMbMlas;TIÉktþa x b¤ y enARtg; j . Ca
lT§pl CYrQrnImYy²rbs;m:aRTIstMNageGaybgÁúMkmøaMgbYnEdlekItmanenARtg;cugrbs;Ggát;enAeBl
cugGgát;rgbMlas;TIÉktþaEdlTak;TgnwgCYrQrebs;m:aRTIsenaH. ]TahrN_ bMlas;TIÉktþa DN = 1 x
nwgbegáItbgÁúMkmøaMgbYnenAelIGgát;EdlbgðajenAkñúgCYrQrTImYyrbs;m:aRTIs.
!$>%> ma:RTIsPaBrwgRkajsMrab; truss (Truss stiffness matrix)
eRkayeBlbegáItm:aRTIsPaBrwgRkajsRmab;Ggát;enAkñúgkUGredaensklrYcehIy eKcaM)ac;pÁúMBUk
vabBa©ÚlKñatamlMdab;d¾RtwmRtUv dUcenHeKnwgTTYl)anm:aRTIsPaBrwgRkaj K sRmab; truss TaMgmUl. dM-
eNIrkarénkarpÁúMm:aRTIsGgát;TaMgenHGaRs½ynwgkarkMNt;GtþsBaØaNrbs;Ggát;enAkñúgm:aRTIsGgát;nImYy
². dUckarerobrab;enAkñúgkfaxNÐmun eKRtUvtMerobCYredk nigCYrQrrbs;m:aRTIsedayelxkUdbYn N x ,
N y , Fx , Fy EdleRbIedIm,IkMNt;GtþsBaØaN degree of freedom sklBIrEdlGacekItmanenARtg;
cugnImYy²rbs;Ggát; ¬emIlsmIkar 14-16¦. m:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúMnwgmanlMdab;esμInwg
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -472
10. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
elxkUdx<s;bMputEdl)ankMNt;eTAelI truss edaysartMNageGaycMnYn degree of freedom srub
sRmab;eRKOgbgÁúM. enAeBleKpÁúMm:aRTIs k eKRtUvCMnYstYnImYy²enAkñúg k eTAkñúgCYredk nigCYrQrRtUvKña
rbs;m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§ K . enAeBlGgát;BIr b¤Ggát;eRcIntP¢ab;KñaenARtg;tMNEtmYy
eKRtUvdak;tYénm:aRTIsrbs;Ggát; k xøHeTAkñúgTItaMgdEdlrbs;m:aRTIs K . eKRtUvbUkbBa©ÚlKñatamlkçN³
nBVnþnUvtYEdlsßitenAkñúgTItaMgdUcKña. eyIgnwgyl;BImUlehtuenH)anc,as; RbsinebIeyIgdwgfatYnImYy²
rbs;m:aRTIs k CaersIusþg;rbs;Ggát;Tb;Tl;nwgkmøaMgxageRkAEdlGnuvtþenARtg;cugrbs;va. tamviFIEbb
enH karbUkbBa©ÚlKñanUversIusþg;tamTis x nigTis y enAeBlbegáItm:aRTIs K kMNt;nUversIusþg;srubrbs;
tMNnImYy²EdlTb;Tl;nwgbMlas;TIÉktþatamTis x b¤tamTis y .
]TahrN_CaelxcMnYnBIrnwgbgðajBIviFIénkarpÁúMm:aRTIssRmab;Ggát;edIm,IbegáItCam:aRTIsPaBrwg
RkajsRmab;eRKOgbgÁúM. eTaHbIvadMeNIrkarmanlkçN³sμúKsμajsRmab;karKNnaedayédbnþic Etvaman
lkçN³gayRsYlCagsMrab;karbegáItkmμviFIenAelIkMuBüÚT½r.
]TahrN_ 14-1³ kMNt;m:aRTIsPaBrwgRkajsRmab; truss EdlmanGgát;BIrdUcbgðajenAkñúgrUbTI
14-7a. AE mantémøefr.
dMeNaHRsay³ tamkarGegát ②manbgÁúMbMlas;TIEdlCaGBaØatcMnYnBIr cMENkÉ ① nig③RtUv)anTb;
mineGaymanbMlas;TI. Cavi)ak eKRtUvkMNt;elxkUdeGaybgÁúMbMlas;TIenARtg;tMN ② dMbUgeK ehIy
bnþedaytMN ③ nig ① ¬rUbTI14-7b¦. eKalrbs;RbB½n§kUGredaensklGacsßitenAcMNucNak¾)an.
edIm,IPaBgayRsYl eyIgnwgeRCIserIstMN ② dUcbgðaj. eyIgGackMNt;elxerogeGayGgát;tam
rebobNak¾)an ehIyeKRtUvKUssBaØaRBYjtambeNÞayGgát;TaMgBIredIm,IeGaydwgcugCit nigcugq¶ay
bs;Ggát;nImYy². eKGacKNnakUsIunUsR)ab;Tis nigm:aRTIsPaBrwgRkajsRmab;Ggát;nImYy².
Truss analysis using the stiffness method T.Chhay -473
11. Department of Civil Engineering NPIC
Ggát;elx1³ edaysar ②CacugCit ehIy ③Cacugq¶ay enaHtamsmIkar14-5 nig14-6 eyIg)an
3−0 0−0
λx = =1 λy = =0
3 3
edayeRbIsmIkar 14-16 nigedayEcktYnImYy²CamYynwg L = 3m eyIg)an
1 2 3 4
⎡ 0.333 0 − 0.333 0⎤ 1
⎢ 0⎥ 2
k1 = AE ⎢ 0 0 0 ⎥
⎢− 0.333 0 0.333 0⎥ 3
⎢ ⎥
⎣ 0 0 0 0⎦ 4
eyIgGacRtYtBinitükarKNnaedaycMNaMfa k1 Cam:aRTIssIuemRTI. cMNaMfa CYredk nigCYrQrenAkñúg
m:aRTIs k1 RtUv)ankMNt;eday degree of freedom x, y enAcugCit Edlbnþedaycugq¶ay eBalKW 1,
2, 3 nig 4 erogKña sRmab;Ggát;elx1 ¬rUbTI 14-7b¦. eKeFVIEbbenHedIm,IkMNt;tYsRmab;karpÁúMenAkñúg
m:aRTIs K .
Ggát;elx 2³ edaysar ②CacugCit ehIy ①Cacugq¶ay enaHeyIg)an
3−0 4−0
λx = = 0.6 λy = = 0.8
5 3
dUcenHsmIkar 14-16 CamYynwg L = 5m køayCa
1 2 5 6
⎡ 0.072 0.096 − 0.072 − 0.096⎤ 1
⎢ 0.128 − 0.096 − 0.128⎥ 2
k 2 = AE ⎢ 0.096 ⎥
⎢− 0.072 − 0.096 0.072 0.096 ⎥ 5
⎢ ⎥
⎣− 0.096 − 0.128 0.096 0.128 ⎦ 6
enATIenH eKkMNt;CYredk nigCYrQrCa1, 2, 5 nig 6 edaysarelxTaMgenHtMNageGay degree of
freedom tamTis x nig y enARtg;cugCit nigcugq¶ayrbs;Ggát;elx 2 .
m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§³ vaCam:aRTIsTMhM 6 × 6 edaysarvaman degree of freedom
sRmab; truss cMnYn 6 ¬rUbTI 14-7b¦. eKRtUvbUktYEdlRtUvKñaénm:aRTIsTaMgBIrxagelIedIm,IbegáItCa
m:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúM. eKRbEhlCaRsYlemIlCagRbsinebIeKBnøatm:aRTIs k1 nig k 2
eGayeTACam:aRTIs 6 × 6 . enaHeK)an
K = k1 + k 2
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -474
12. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
1 2 3 4 5 6 1 2 3 4 5 6
⎡ 0.333 0 − 0.333 0 0 0⎤ 1 ⎡ 0.072 0.096 0 0 − 0.072 − 0.096⎤ 1
⎢ 0 0 0 0 0 ⎥
0⎥ 2 ⎢ 0.096 0.128 0 0 − 0.096 − 0.128⎥ 2
⎢ ⎢ ⎥
K = AE ⎢− 0.333 0 0.333 0 0 0⎥ 3 + AE ⎢ 0 0 0 0 0 0 ⎥3
⎢ ⎥ ⎢ ⎥
⎢ 0 0 0 0 0 0⎥ 4 ⎢ 0 0 0 0 0 0 ⎥4
⎢ 0 0 0 0 0 0⎥ 5 ⎢− 0.072 − 0.096 0 0 0.072 0.096 ⎥ 5
⎢ ⎥ ⎢ ⎥
⎢ 0
⎣ 0 0 0 0 0⎥ 6
⎦ ⎢− 0.096 − 0.128
⎣ 0 0 0.096 0.128 ⎥ 6
⎦
⎡ 0.405 0.096 − 0.333 0 − 0.072 − 0.096⎤
⎢ 0.096 0.128 0 0 − 0.096 − 0.128⎥
⎢ ⎥
K = AE ⎢ − 0.333 0 0.333 0 0 0 ⎥
⎢ ⎥
⎢ 0 0 0 0 0 0 ⎥
⎢− 0.072 − 0.096 0 0 0.072 0.096 ⎥
⎢ ⎥
⎢− 0.096 − 0.128
⎣ 0 0 0.096 0.128 ⎥
⎦
RbsinebIeKeFVIdMeNIrkarenHCamYynwgkMuBüÚT½r CaTUeTAeKcab;epþImCamYynwgm:aRTIs K EdlmanFatuTaMg
Gs;esμIsUnü bnÞab;mkFatuénm:aRTIsPaBrwgRkajsklsRmab;Ggát;Edl)anKNnarYcehIyRtUv)anCMnYs
edaypÞal;eTAkñúgTItaMgFatuEdlRtUvKñaénm:aRTIs K . kareFVIEbbenHvaRbesIrCagkarbegáItm:aRTIsPaBrwg
RkajsRmab;Ggát; rUcehIyrkSavaTuk bnÞab;mkeTIbpÁúMva.
]TahrN_ 14-2³ kMNt;m:aRTIsPaBrwgRkajsRmab; truss EdlmanGgát;BIrdUcbgðajenAkñúgrUbTI
14-8a. AE mantémøefr.
dMeNaHRsay³ eTaHbICa truss Carcnasm<½n§minkMNt;edaysþaTicdWeRkTImYyk¾eday Etvanwgminbgðaj
BIPaBlM)akkñúgkarTTYl)anm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§eT. eKkMNt;elxerogeGaytMN nig
Truss analysis using the stiffness method T.Chhay -475
13. Department of Civil Engineering NPIC
Ggát;nImYy² ehIyeKbgðajcugCit nigcugq¶ayedayRBYjtambeNþayGgát;. dUcbgðajenAkñúgrUbTI 14-
8b dMbUgeKkMNt;elxerogkUdeGaybMlas;TIEdlminRtUv)anTb;. eKman degree of freedom cMnUn 8
dUcenH K RtUvCam:aRTIsTMhM 8 × 8 . edIm,IrkSaeGaykUGredaenrbs;tMNTaMgGs;viC¢man eKRtUveRCIserIs
eKalrbs;kUGredaensklenARtg; ①. eyIgnwgGnuvtþsmIkar 14-5, 14-6 nig 14-16 eTAelIGgát;
nImYy².
Ggát;elx 1³ enATIenH L = 10m eyIg)an
10 − 0 0−0
λx = =1 λy = =0
10 10
1 2 6 5
⎡ 0.1 0 − 0.1 0⎤ 1
⎢ 0⎥ 2
k1 = AE ⎢ 0 0 0 ⎥
⎢− 0.1 0 1 0⎥ 6
⎢ ⎥
⎣ 0 0 0 0⎦ 5
Ggát;elx 2³ enATIenH L = 10 2m dUcenH
10 − 0 10 − 0
λx = = 0.707 λy = = 0.707
10 2 10 2
1 2 7 8
⎡ 0.035 0.035 − 0.035 − 0.035⎤ 1
⎢ 0.035 0.035 − 0.035 − 0.035⎥ 2
k 2 = AE ⎢ ⎥
⎢− 0.035 − 0.035 0.035 0.035 ⎥ 7
⎢ ⎥
⎣− 0.035 − 0.035 0.035 0.035 ⎦ 8
Ggát;elx 3³ enATIenH L = 10m dUcenH
0−0 10 − 0
λx = =0 λy = =1
10 10
1 2 3 4
⎡0 0 0 0 ⎤1
⎢0 0.1 0 − 0.1⎥ 2
k 3 = AE ⎢ ⎥
⎢0 0 0 0 ⎥3
⎢ ⎥
⎣0 − 0.1 0 0.1 ⎦ 4
Ggát;elx 4³ enATIenH L = 10m eyIg)an
10 − 0 0−0
λx = =1 λy = =0
10 10
3 4 7 8
⎡ 0.1 0 − 0.1 0⎤ 3
⎢ 0⎥ 4
k 4 = AE ⎢ 0 0 0 ⎥
⎢− 0.1 0 1 0⎥ 7
⎢ ⎥
⎣ 0 0 0 0⎦ 8
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -476
14. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
Ggát;elx 5³ enATIenH L = 10 2m dUcenH
10 − 0 0 − 10
λx = = 0.707 λy = = −0.707
10 2 10 2
3 4 6 5
⎡ 0.035 − 0.035 − 0.035 0.035 ⎤ 3
⎢ 0.035 − 0.035⎥ 4
k 5 = AE ⎢− 0.035 0.035 ⎥
⎢− 0.035 0.035 0.035 − 0.035⎥ 6
⎢ ⎥
⎣ 0.035 − 0.035 − 0.035 0.035 ⎦ 5
Ggát;elx 3³ enATIenH L = 10m dUcenH
0−0 10 − 0
λx = =0 λy = =1
10 10
6 5 7 8
⎡0 0 0 0 ⎤6
⎢0 0.1 0 − 0.1⎥ 5
k 6 = AE ⎢ ⎥
⎢0 0 0 0 ⎥7
⎢ ⎥
⎣0 − 0.1 0 0.1 ⎦ 8
m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§³ eKGacpÁúMm:aRTIsTaMg 6 edIm,IbegáItm:aRTIsTMhM 8 × 8 edaykarbUk
bBa©ÚlFatuEdlRtUvKña. ]TahrN_ edaysar (k11 )1 = AE (0.1) / (k11 )2 = AE (0.035) / (k11 )3 =
(k11 )4 = (k11 )5 = (k11 )6 = 0 enaH K11 = AE (0.1 + 0.035) = AE (0.135) . dUcenHlT§plcugeRkayKW
1 3 2 4 5 6 7 8
⎡ 0.135 0 0.0350 0 − 0.1 − 0.035 − 0.035⎤ 1
⎢ 0.035 0 − 0.1
0.135 0 0 − 0.035 − 0.035⎥ 2
⎢ ⎥
⎢ 0 0.135 − 0.035 0.035 − 0.035 − 0.1
0 0 ⎥3
⎢ ⎥
K = AE ⎢ 0 − 0.035 0.135 − 0.035 0.035
− 0.1 0 0 ⎥4
⎢ 0 0 0.035 − 0.035 0.135 − 0.035 0 − 0.1 ⎥ 5
⎢ ⎥
⎢ − 0.1 0 − 0.035 0.035 − 0.035 0.135 0 0 ⎥6
⎢− 0.035 − 0.035 − 0.1 0 0 0 0.135 0.035 ⎥ 7
⎢ ⎥
⎢− 0.035 − 0.035
⎣ 0 0 − 0.1 0 0.135 0.135 ⎥ 8
⎦
!$>^> karGnuvtþénviFIPaBrwgRkajsRmab;karviPaK truss
(Application of the stiffness method for truss analysis)
eRkayeBlbegáItm:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúMrYcehIy eKGaceFVIeGaybgÁúMkmøaMgskl
Q EdlmanGMeBIenAelI truss manTMnak;TMngeTAnwgbMlas;TIskl D rbs;vaedayeRbI
Q = KD (14-17)
Truss analysis using the stiffness method T.Chhay -477
15. Department of Civil Engineering NPIC
eKGacsMKal;smIkarenHCasmIkarPaBrwgRkajsRmab;rcnasm<½n§ (structure stiffness equation).
edaysareyIgEtgEtkMNt;elxkUdtUcbMputedIm,IsmÁal; degree of freedom EdlminRtUv)anTb; dUcenH
vaGnuBaØateGayeyIgGacbMEbksmIkarenHkñúgTRmg;dUcxageRkam ³ *
⎡Qk ⎤ ⎡ K11 K12 ⎤ ⎡ Du ⎤
⎢Q ⎥ = ⎢ K ⎥⎢ ⎥ (14-18)
⎣ u ⎦ ⎣ 21 K 22 ⎦ ⎣ Dk ⎦
Edl bnÞúkxageRkA nigbMlas;TIEdleKsÁal;. enATIenH bnÞúkmanGMeBIenAelI truss CaEpñk
Qk , Dk =
mYyéncMeNaT ehIyCaTUeTAbMlas;TIesμIsUnüedaysarTMrRtUv)anTb; dUcCaTMrsnøak;
b¤TMrkl;.
Qu , Du = bnÞúk nigbMlas;TIEdlCaGBaØat. enATIenH bnÞúkCakmøaMgRbtikmμTMrEdleKminsÁal;
ehIybMlas;TIKWsßitenARtg;tMNEdlminRtUv)anTb;tamTisNamYyeT.
K = m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§ EdlRtUv)anbMEbkedIm,IeGaycuHsRmugCamYynwg
karbMEbkrbs; Q nig D .
edayBnøatsmIkar 14-18 eyIg)an
Qk = K11 Du + K12 Dk (14-19)
Qu = K 21 Du + K 22 Dk (14-20)
CaTUeTA Dk = 0 edaysarTMrminmanbMlas;TI. RbsinebIvaEbbenHEmn enaHsmIkar 14-19 køayCa
Qk = K11 Du
edaysarFatuenAkñúgm:aRTIs K11 CaersIusþg;srubenARtg;tMN truss edIm,ITb;Tl;bMlas;TIÉktþatamTI x
b¤ y enaHsmIkarxagelICakarRbmUlpþúMnUvsmIkarlMnwgkmøaMgEdlGnuvtþeTAelItMNEdlbnÞúkxageRkA
esμIsUnü b¤mantémøEdlsÁal; (Qk ) . edayedaHRsayrk Du eyIg)an
Du = [K11 ]−1 Qk (14-21)
BIsmIkarenH eyIgGacTTYl)andMeNaHRsayedaypÞal;sRmab;bMlas;TIEdlCaGBaØatTaMgGs; bnÞab;mk
edayeRbIsmIkar 14-20 CamYynwg Dk = 0 eyIg)an
Qu = K 21 Du (14-22)
BIsmIkarxagelI eyIgGackMNt;kmøaMgRbtikmμTMr. eKGackMNt;kmøaMgkñúgrbs;Ggát;edayeRbIsmIkar
14-13 eBalKW
q = k ' TD
edayBnøatsmIkarenH eyIg)an
*
eyIgnwg)aneXIjBIviFIbMEbkenHenAkñúg]TahrN_xageRkam.
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -478
16. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
⎡ DN x ⎤
⎢ ⎥
⎡q N ⎤ AE ⎡ 1 − 1⎤ ⎡λ x λ y 0 0 ⎤ ⎢ D Ny ⎥
⎢ q ⎥ = L ⎢− 1 1 ⎥ ⎢ 0 0 λ λ y ⎥ ⎢ DFx ⎥
⎣ F⎦ ⎣ ⎦⎣ x ⎦⎢ ⎥
⎢ DFy ⎥
⎣ ⎦
edaysar q N = −q F edIm,IsßanPaBlMnwg dUcenHeKRtUvkarkMNt;EtkmøaMgmYyb:ueNÑaHkñúgcMeNamkmøaMg
TaMgBIr. enATIenH eyIgnwgkMNt; q F kmøaMgEdlGnuvtþkmøaMgTajeTAelIGgát; ¬rUbTI 14-6b¦.
⎡ DN x ⎤
⎢D ⎥
qF =
AE
L
[
− λx − λ y λx ]
λy ⎢ y ⎥
N
⎢ DFx ⎥
(14-23)
⎢ ⎥
⎢ D Fy ⎥
⎣ ⎦
RbsinebIlT§plEdl)anBIkarKNnamantémøGviC¢man enaHGgát;rgkarsgát;.
dMeNIrkarkñúgkarviPaK (Procedure for analysis)
xageRkamCaCMhanEdlpþl;nUvmeFüa)aysRmab;kMNt;bMlas;TI nigkmøaMgRbtikmμTMrEdlCa
GBaØatsRmab; truss edayeRbIviFIPaBrwgRkaj.
kareFVIkMNt;sMKal;³
begáItRbB½n§kUGredaenskl x, y . CaTUeTAeKalrbs;vasßitenARtg;tMNNaEdleFVIeGay
kUGredaensRmab;tMNdéTeTotviC¢man.
kMNt;elxerogeGaytMN nigGgát;nImYy² ehIykMNt;cugCit nigcugq¶ayrbs;Ggát;nImYy²
edayeRbITisedArbs;sBaØaRBYj Edlk,alRBYjeq<aHeTArkcugq¶ay.
kMNt;elxkUdBIrenARtg;tMNnImYy² edayeRbIelxtUcbMputsRmab;sMKal; degree of freedom
EdlminmanrgkarTb; cMENkelxFMbMputsRmab;sMKal; degree of freedom EdlmankarTb;.
begáIt Dk nig Qk .
m:aRTIsPaBrwgRkajsRmab;eRKOgbgÁúM³
sRmab;Ggát;nImYy² kMNt; λ x nig λ y ehIykMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;edayeRbI
smIkar 14-16.
pÁúMm:aRTIsTaMgenHedIm,IbegáItm:aRTIsPaBrwgRkajsRmab; truss TaMgmUl dUckarBnül;enAkñúg
kfaxNÐ 14-5. edaykarRtYtBinitükarKNnaedayEpñk m:aRTIsPaBrwgRkajsRmab;Ggát; nig
m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§RtUvEtCam:aRTIssIuemRTI.
bMlas;TI nigbnÞúk³
Truss analysis using the stiffness method T.Chhay -479
17. Department of Civil Engineering NPIC
bMEbkm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§ dUcbgðajenAkñúgsmIkar 14-18.
kMNt;bMlas;TIrbs;tMNEdlCaGBaØat Du edayeRbIsmIkar 14-21 kmøaMgRbtikmμTMr Qu eday
eRbIsmIkar 14-22 ehIykmøaMgkñúgrbs;Ggát;nImYy² q F edayeRbIsmIkar 14-23.
]TahrN_ 14-3³ kMNt;kmøaMgkñúgrbs;Ggát;én truss EdlmanGgát;BIrdUcbgðajenAkñúgrUbTI 14-9a.
AE mantémøefr.
dMeNaHRsay³
kareFVIkMNt;smÁal;³ eKal x, y nigkarkMNt;elxerogrbs;tMN nigGgát;RtUv)anbgðajenAkñúgrUbTI
14-9b. dUcKña cugCit nigcugq¶ayRtUv)anbgðajkMNt;edaysBaØaRBYj ehIyeKeRbIelxkUdenARtg;tMN
nImYy². tamkarGegáteyIgeXIjfabMlas;TI D3 = D4 = D5 = D6 = 0 . ehIybnÞúkxageRkAEdl
eyIgsÁal;KW Q1 = 0, Q2 = −2kN . dUcenH
⎡0 ⎤ 3
⎢0 ⎥ 4
⎡ 0 ⎤1
Dk = ⎢ ⎥ Qk = ⎢ ⎥
⎢0 ⎥ 5 ⎣ − 2⎦ 2
⎢ ⎥
⎣0 ⎦ 6
m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§³ edayeRbIkareFVIkMNt;smÁal;dUcKña eyIgGacbegáItm:aRTIsPaBrwg
RkajsRmab;rcnasm<½n§dUcbgðajenAkñúg]TahrN_ 14-1.
bMlas;TI nigbnÞúk³ edaysresrsmIkar 14-17 ¬ Q = KD ¦ sRmab; truss eyIg)an
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -480
18. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
⎡0 ⎤ ⎡ 0.405 0.096 − 0.333 0 − 0.072 − 0.096⎤ ⎡ D1 ⎤
⎢ − 2⎥ ⎢ 0.096 0.128 0 0 − 0.096 − 0.128⎥ ⎢ D2 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ Q3 ⎥ ⎢ − 0.333 0 0.333 0 0 0 ⎥⎢ 0 ⎥
⎢ ⎥ = AE ⎢ ⎥⎢ ⎥ (1)
⎢ Q4 ⎥ ⎢ 0 0 0 0 0 0 ⎥⎢ 0 ⎥
⎢ Q5 ⎥ ⎢− 0.072 − 0.096 0 0 0.072 0.096 ⎥ ⎢ 0 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢Q6 ⎥
⎣ ⎦ ⎢− 0.096 − 0.128
⎣ 0 0 0.096 0.128 ⎥ ⎢ 0 ⎥
⎦⎣ ⎦
BIsmIkarenH eyIgGackMNt; K11 dUcenHeyIgGackMNt; Du . eyIgeXIjfaplKuNm:aRTIs ¬dUc
smIkar 14-19¦ eyIg)an
⎡0 ⎤ ⎡0.405 0.096⎤ ⎡ D1 ⎤ ⎡0⎤
⎢− 2⎥ = AE ⎢0.096 0.128⎥ ⎢ D ⎥ + ⎢0⎥
⎣ ⎦ ⎣ ⎦⎣ 2 ⎦ ⎣ ⎦
enATIenH eyIgGacedaHRsayy:agRsYledayBnøatedaypÞal;
0 = AE (0.405 D1 + 0.096 D2 )
− 2 = AE (0.096 D1 + 0.128 D2 )
tamrUbviTüa smIkarTaMgenHtMNageGay ∑ Fx = 0 nig ∑ Fy = 0 EdlGnuvtþenARtg;tMN ②. eday
edaHRsay eyIg)an
4.505 − 19.003
D1 = D2 =
AE AE
tamkarGegátrUbTI 14-9b eKrMBwgfatMN ②nwgpøas;TIeTAsþaM nigcuHeRkamdUcbgðajedaysBaØabUk nig
sBaØadkéncemøIyenH
edayeRbIlT§plTaMgenH eKGacTTYl)ankmøaMgRbtikmμTMrBIsmIkar (1) EdlRtUv)ansresrkñúgTRmg;én
smIkar 14-20 ¬b¤smIkar 14-22¦ Ca
⎡Q3 ⎤ ⎡ − 0.333 0 ⎤ ⎡0 ⎤
⎢Q ⎥ ⎢ 0 ⎥
0 ⎥ 1 ⎡ 4.505 ⎤ ⎢0⎥
⎢ 4 ⎥ = AE ⎢ +⎢ ⎥
⎢Q5 ⎥ ⎢− 0.072 − 0.096⎥ AE ⎢− 19.003⎥ ⎢0⎥
⎣ ⎦
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣Q6 ⎦ ⎣− 0.096 − 0.128⎦ ⎣0 ⎦
edayBnøat nigedaHRsaykmøaMgRbtikmμ
Q3 = −0.333(4.505) = −1.5kN
Q4 = 0
Q5 = −0.072(4.505) − 0.096(− 19.003) = 1.5kN
Q6 = −0.096(4.505) − 0.128(− 19.003) = 2.0kN
eKGacKNnakmøaMgenAkñúgGgát;nImYy²BIsmIkar 14-23. edayeRbITinñn½ysRmab; λ x nig λ y enAkñúg
smIkar 14-1 eyIg)an
Truss analysis using the stiffness method T.Chhay -481
19. Department of Civil Engineering NPIC
Ggát;elx ! ³ λ x = 1, λ y = 0, L = 3m
⎡ 4.505 ⎤ 1
⎢ ⎥
AE 1 2 3 4 1 ⎢− 19.003⎥ 2
q1 =
3 [− 1 0 1 0] AE ⎢ 0 ⎥ 3
⎢ ⎥
⎣ 0 ⎦4
= [− 4.505] = −1.5kN
1
3
Ggát;elx @ ³ λ x = 0.6, λ y = 0.8, L = 5m
⎡ 4.505 ⎤ 1
AE 1 2 5 6 1 ⎢− 19.003⎥ 2
⎢ ⎥
q2 =
5 [− 0.6 − 0.8 0.6 0.8] AE ⎢ 0 ⎥ 5
⎢ ⎥
⎣ 0 ⎦6
= [− 0.6(4.505) − 0.8(− 19.003)] = 2.5kN
1
5
Cak;EsþgeKGacepÞógpÞat;cemøIyTaMgenHedaysmIkarlMnwgEdlGnuvtþenARtg;tMN ②.
]TahrN_ 14-4³ kMNt;kmøaMgRbtikmμTMr nigkmøaMgkñúgrbs;Ggát;elx@ én truss dUcbgðajenAkñúgrUbTI
14-10a. AE mantémøefr.
dMeNaHRsay³
kareFVIkMNt;smÁal;³tMN nigGgát;RtUv)ankMNt;elxerog ehIyeKalrbs;G½kS x, y RtUv)anbegáItenA
Rtg;tMN ① ¬rUbTI 14-10b¦. ehIysBaØaRBYjRtUv)aneKeRbIedIm,IbgðajcugCit nigcugq¶ayrbs;Ggát;
nImYy². edayeRbIelxkUd EdlelxtUcbMputtMNageGay degree of freedom EdlmanrgkarTb; ¬rUb
TI 14-16b¦ eyIg)an
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -482
20. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
⎡ 0 ⎤1
⎢ 0 ⎥2
⎡0 ⎤ 6 ⎢ ⎥
D k = ⎢0 ⎥ 7
⎢ ⎥ Qk = ⎢ 2 ⎥ 3
⎢ ⎥
⎢0 ⎥ 8
⎣ ⎦ ⎢ − 4⎥ 4
⎢ 0 ⎥5
⎣ ⎦
m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§³ m:aRTIsenHRtUv)ankMNt;enAkñúg]TahrN_ 14-2 edayeRbIkar
eFVIkMNt;smÁal;dUcKñanwgkarbgðajenAkñúg]TahrN_ 14-10b.
bMlas;TI nigbnÞúk³ sRmab;cMeNaTenH Q = KD KW
⎡0⎤ ⎡ 0.135 0.035 0 0 0 − 0.1 − 0.035 − 0.035⎤ ⎡ D1 ⎤
⎢0⎥ ⎢ 0.035 0.135 0 − 0 .1 0 0 − 0.035 − 0.035⎥ ⎢ D2 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ 2⎥ ⎢ 0 0 0.135 − 0.035 0.035 − 0.035 − 0.1 0 ⎥ ⎢ D3 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢− 4⎥ = AE ⎢ 0 − 0.1 − 0.035 0.135 − 0.035 0.035 0 0 ⎥ ⎢ D4 ⎥
(1)
⎢0⎥ ⎢ 0 0 0.035 − 0.035 0.135 − 0.035 0 − 0.1 ⎥ ⎢ D5 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢Q6 ⎥ ⎢ − 0 .1 0 − 0.035 0.035 − 0.035 0.135 0 0 ⎥⎢ 0 ⎥
⎢Q ⎥ ⎢− 0.035 − 0.035 − 0.1 0 0 0 0.135 0.035 ⎥ ⎢ 0 ⎥
⎢ 7⎥ ⎢ ⎥⎢ ⎥
⎢ Q8 ⎥
⎣ ⎦ ⎢− 0.035 − 0.035
⎣ 0 0 − 0 .1 0 0.035 0.135 ⎥ ⎢ 0 ⎥
⎦⎣ ⎦
edayeFVIplKuNdUckarsresrsmIkar 14-18 edIm,IedaHRsaybMlas;TI eyIg)an
⎡0 ⎤ ⎡0.135 0.035 0 0 0 ⎤ ⎡ D1 ⎤ ⎡0⎤
⎢0 ⎥
⎢ ⎥
⎢0.035 0.135
⎢ 0 − 0 .1 0 ⎥ ⎢ D2 ⎥ ⎢0 ⎥
⎥⎢ ⎥ ⎢ ⎥
⎢ 2 ⎥ = AE ⎢ 0 0 0.135 − 0.035 0.035 ⎥ ⎢ D3 ⎥ + ⎢0⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎢ − 4⎥ ⎢ 0 − 0.1 − 0.035 0.135 − 0.035⎥ ⎢ D4 ⎥ ⎢0⎥
⎢0 ⎥
⎣ ⎦ ⎢ 0
⎣ 0 0.035 − 0.035 0.135 ⎥ ⎢ D5 ⎥ ⎢0⎥
⎦⎣ ⎦ ⎣ ⎦
edayBnøat nigedayedaHRsaysmIkarsRmab;bMlas;TI eyIg)an
⎡ D1 ⎤ ⎡ 17.94 ⎤
⎢D ⎥ ⎢− 69.20⎥
⎢ 2⎥ 1 ⎢ ⎥
⎢ D3 ⎥ = ⎢ − 2.06 ⎥
⎢ ⎥ AE ⎢ ⎥
⎢ D4 ⎥ ⎢ − 87.14 ⎥
⎢ D5 ⎥
⎣ ⎦ ⎢− 22.06⎥
⎣ ⎦
edaybegáItsmIkar 14-20 BIsmIkar (1) EdleRbIlT§plEdl)anKNna eyIg)an
⎡ 17.94 ⎤
⎢ − 69.20⎥ 0
⎡Q6 ⎤ ⎡ − 0.1 0 − 0.035 0.035 − 0.035⎤
⎥ ⎡ ⎤
⎢Q ⎥ = AE ⎢− 0.035 − 0.035 − 0.1 1 ⎢
⎢ 7⎥ ⎢ 0 0 ⎥ ⎥
⎢ − 2.06 ⎥ + ⎢0⎥
AE ⎢ ⎥ ⎢ ⎥
⎢ ⎥
⎣Q8 ⎦ ⎢− 0.035 − 0.035
⎣ 0 0 − 0.1 ⎥
⎦ ⎢ − 87.14 ⎥ ⎢0⎥
⎣ ⎦
⎢− 22.06⎥
⎣ ⎦
edayBnøat nigKNnakmøaMgRbtikmμTMr eyIg)an
Truss analysis using the stiffness method T.Chhay -483
21. Department of Civil Engineering NPIC
Q6 = −4.0kN
Q7 = 2.0kN
Q8 = 4.0kN
sBaØadksRmab; Q6 bgðajfakmøaMgRbtikmμrbs;TMrkl;eFVIGMeBIkñúgTis x GviC¢man. eKGackMNt;kmøaMg
enAkñúgGgát;elx@ BIsmIkar 14-23 EdlBIsmIkar 14-2 λ x = 0.707, λ y = 0.707, L = 10 2m
dUcenH
⎡ 17.94 ⎤
⎢− 69.20⎥
q2 =
AE
[− 0.707 − 0.707 0.707 0.707] 1 ⎢ ⎥
10 2 AE ⎢ 0 ⎥
⎢ ⎥
⎣ 0 ⎦
= 2.56kN
]TahrN_ 14-5³ kMNt;kmøaMgkñúgrbs;Ggát;elx@ énrcnasm<½n§dUcbgðajenAkñúgrUbTI 14-11a. Rb
sinebIenARtg;tMN① mansMrut 25mm . yk AE = 8(103 )kN .
dMeNaHRsay³
kareFVIkMNt;smÁal;³edIm,IPaBgayRsYl eKRtUvbegáIteKalrbs;kUGredaensklenARtg;tMN ③ dUc
bgðajenAkñúgrUbTI 14-11b ehIytamFmμta eKeRbIelxkUdtUcCageKedIm,ItMNageGay degree of
freedom EdlminmankarTb;. dUcenH
⎡ 0 ⎤3
⎢− 0.025⎥ 4
⎢ ⎥
⎢ 0 ⎥5 ⎡0 ⎤ 1
Dk = ⎢ ⎥ Qk = ⎢ ⎥
⎢ 0 ⎥6 ⎣0 ⎦ 2
⎢ 0 ⎥7
⎢ ⎥
⎢ 0 ⎥8
⎣ ⎦
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -484
22. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§³ edayeRbIsmIkar 14-16 eyIg)an
Ggát;elx !³ λ x = 0 / λ y = 1 / L = 3m dUcenH
3 4 1 2
⎡0 0 0 0⎤3
⎢0 0.333 0 − 0.333⎥ 4
k1 = AE ⎢ ⎥
⎢0 0 0 0 ⎥1
⎢ ⎥
⎣0 − 0.333 0 0.333 ⎦ 2
Ggát;elx @³ λ x = −0.8, λ y = −0.6, L = 5m dUcenH
1 2 5 6
⎡ 0.128 0.096 − 0.128 − 0.096⎤ 1
⎢ 0.096 0.072 − 0.096 − 0.072⎥ 2
k 2 = AE ⎢ ⎥
⎢ − 0.128 − 0.096 0.128 0.096 ⎥ 5
⎢ ⎥
⎣− 0.096 − 0.072 0.096 0.072 ⎦ 6
Ggát;elx #³ λ x = 1, λ y = 0, L = 4m dUcenH
7 8 1 2
⎡ 0.25 0 − 0.25 0⎤ 7
⎢ 0⎥ 8
k 3 = AE ⎢ 0 0 0 ⎥
⎢− 0.25 0 0.25 0⎥ 1
⎢ ⎥
⎣ 0 0 0 0⎦ 2
edaypÁúMm:aRTIsTaMgenH m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§køayCa
1 2 3 4 5 6 7 8
⎡ 0.378 0.096 0 0 − 0.128 − 0.096 − 0.25 0⎤ 1
⎢ 0.096 0.405 0 − 0.333 − 0.096 − 0.072 0 0⎥ 2
⎢ ⎥
⎢ 0 0 0 0 0 0 0 0⎥ 3
⎢ ⎥
K =⎢ 0 − 0.333 0 0.333 0 0 0 0⎥ 4
⎢ − 0.128 − 0.096 0 0 0.128 0.096 0 0⎥ 5
⎢ ⎥
⎢− 0.096 0.072 0 0 0.096 0.072 0 0⎥ 6
⎢ − 0.25 0 0 0 0 0 0.25 0⎥ 7
⎢ ⎥
⎢ 0
⎣ 0 0 0 0 0 0 0⎥ 8
⎦
bMlas;TI nigbnÞúk³ enATIenH Q = KD eyIg)an
Truss analysis using the stiffness method T.Chhay -485
23. Department of Civil Engineering NPIC
⎡0⎤ ⎡ 0.378 0.096 0 0 − 0.128 − 0.096 − 0.25 0⎤ ⎡ D1 ⎤
⎢0⎥ ⎢ 0.096 0.405 0 − 0.333 − 0.096 − 0.072 0 0 ⎥ ⎢ D2 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢Q3 ⎥ ⎢ 0 0 0 0 0 0 0 0⎥ ⎢ 0 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢Q4 ⎥ = AE ⎢ 0 − 0.333 0 0.333 0 0 0 0⎥ ⎢− 0.025⎥
⎢Q5 ⎥ ⎢ − 0.128 − 0.096 0 0 0.128 0.096 0 0⎥ ⎢ 0 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢Q6 ⎥ ⎢− 0.096 − 0.072 0 0 0.096 0.072 0 0⎥ ⎢ 0 ⎥
⎢Q ⎥ ⎢ − 0.25 0 0 0 0 0 0.25 0⎥ ⎢ 0 ⎥
⎢ 7⎥ ⎢ ⎥⎢ ⎥
⎢Q8 ⎥
⎣ ⎦ ⎢ 0
⎣ 0 0 0 0 0 0 0⎥ ⎢ 0 ⎥
⎦⎣ ⎦
edayedaHRsaysRmab;bMlas;TI ¬smIkar 14-19¦ eyIg)an
⎡ 0 ⎤
⎢− 0.025⎥
⎢ ⎥
⎡0 ⎤ ⎡0.378 0.096⎤ ⎡ D1 ⎤ ⎡0 0 − 0.128 − 0.096 − 0.25 0⎤ ⎢ 0 ⎥
⎢0⎥ = AE ⎢0.096 0.405⎥ ⎢ D ⎥ + AE ⎢0 − 0.333 − 0.096 − 0.072 ⎢
0⎥ ⎢ 0 ⎥
⎥
⎣ ⎦ ⎣ ⎦⎣ 2 ⎦ ⎣ 0 ⎦
⎢ 0 ⎥
⎢ ⎥
⎢ 0 ⎥
⎣ ⎦
EdleyIgTTYl)an
0 = AE [(0.378 D1 + 0.096 D2 ) + 0]
0 = AE [(0.096 D1 + 0.405 D2 ) + 0.00833]
edayedaHRsayRbB½n§smIkarenH eyIg)an
D1 = 0.00556m
D2 = −0.021875m
eTaHbICaeKminRtUvkarKNnakmøaMgRbtikmμTMrk¾eday EtRbsinebIcaM)ac;eKRtUvKNnavaBIkarBnøatEdl
kMNt;edaysmIkar 14-20. edayeRbIsmIkar 14-23 edIm,IkMNt;kmøaMgenAkñúgGgát;elx @ eyIg)an
Ggát;elx @³ λ x = −0.8, λ y = −0.6, L = 5m, AE = 8(103 )kN dUcenH
⎡ 0.00556 ⎤
q2 =
( )
8 10 3
[0.8 0.6 − 0.8 − 0.6]⎢
⎢− 0.02187 ⎥
⎥
5 ⎢ 0 ⎥
⎢ ⎥
⎣ 0 ⎦
=
8 10( )
3
(0.00444 − 0.0131) = −13.9kN
5
edayeRbIdMeNIrkarKNnadUcKña bgðajfakmøaMgenAkñúgGgát;elx ! KW q1 = 8.34kN ehIykmøaMgenAkñúg
Ggát;elx # KW q3 = 11.1kN . lT§plRtUv)anbgðajenAkñúgdüaRkamGgÁesrIrbs;tMN ② ¬rUbTI 14-
11c¦ EdleKGacepÞógpÞat;edaysmIkarlMnwg.
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -486
24. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
!$>&> kUGredaenrbs; node (Nodal coordinates)
enAeBlTMrkl;rbs; truss sßitenAelIbøg;eRTt ehIyeKGackMNt;PaBdabsUnüenARtg;TMreday
eRbIRbB½n§kUGredaenskltamTisedk nigtamTisQrEtmYy. ]TahrN_ eKman truss enAkñúgrUbTI 14-
12a. eKRtUvkMNt;lkçxNÐénbMlas;TIsUnüenARtg;tMN ① tambeNþayG½kS y' ' ehIyedaysarTMr
kl;Gacpøas;TItambeNþayG½kS x' ' dUcenH node enHRtUvmanbgÁúMbMlas;TItamG½kSkUGredaen x, y .
sRmab;mUlehtuenH eyIgminGacrYmbBa©ÚllkçxNÐbMlas;TIsUnüenARtg; node enH enAeBlsresr
smIkarPaBrwgRkajsklsRmab; truss edayeRbIG½kS x, y edaymineFVIeGaymankarEktRmUvdMeNIr
karviPaKm:aRTIs.
edIm,IedaHRsaycMeNaTenH eyIgGacbBa©ÚlvaeTAkñgkarviPaKkMuBüÚT½redayRsYl eyIgnwgeRbIsMnMu
ú
kUGredaenrbs; node x' ' , y' ' enARtg;TMreRTt. eKRtUveFVIeGayG½kSTaMgenHmanTItaMgy:agNaedIm,I
eGaykmøaMgRbtikmμTMr nigbMlas;TIrbs;TMrpøas;TItambeNþayG½kSkUGredaennImYy² ¬rUbTI 14-12a¦.
edIm,IKNnasmIkarPaBrwgRkajsklsRmab; truss enaHeKcaM)ac;begáItm:aRTIsbMElgkmøaMg nigma:RTIs
bMElgbMlas;TIsRmab;Ggát;EdltP¢ab;eTAnwgTMrenaH dUcenHeKGaceFVIplbUklT§plTaMgenHenAkñúgRbB½n§
kUGredaenskl x, y dUcKña. edIm,IbgðajBIrebobénkarGnuvtþ eyIgRtUvBicarNaGgát; truss elx! enA
Truss analysis using the stiffness method T.Chhay -487
25. Department of Civil Engineering NPIC
kñúgrUbTI 14-12b EdlmanRbB½n§kUGredaenskl x, y enARtg;cugCit N ehIyRbB½n§kUGredaenrbs;
node x' ' , y ' ' enARtg;cugq¶ay F . enAeBlbMlas;TI D ekIteLIg dUcenHBYkvamanbgÁúMtambeNþayG½kS
nImYy²dUcbgðajenAkñúgrUbTI 14-12c enaHbMlas;TItamTis x tambeNþaycugGgát;nImYy²køayCa
d N = D N x cos θ x + D N y cos θ y
d F = DFx '' cos θ x '' + DFy '' cos θ y ''
eKGacsresrsmIkarTaMgenHenAkñúgTRmg;m:aRTIs
⎡ DN x ⎤
⎢ ⎥
⎡ d N ⎤ ⎡λ x λ y 0 0 ⎤ ⎢ DN y ⎥
⎢d ⎥ = ⎢ 0 0 λ λ y '' ⎥ ⎢ DFx '' ⎥
⎣ F⎦ ⎣ x '' ⎦⎢ ⎥
⎢ D Fy '' ⎥
⎣ ⎦
dUcKña kmøaMg q enARtg;cugCit nigcugq¶ayrbs;Ggát; ¬rUbTI 14-12d¦ manbgÁúM Q tambeNþayG½kSskl
Q N x = q N cos θ x Q N y = q N cos θ y
Q Fx '' = q F cos θ x '' Q Fy '' = q F cos θ y ''
EdleKGacsresrCa
⎡ Q N x ⎤ ⎡λ x 0 ⎤
⎢Q ⎥ ⎢ ⎥
⎢ N y ⎥ = ⎢λ y 0 ⎥ ⎡ q N ⎤
⎢Q Fx '' ⎥ ⎢ 0 λ x '' ⎥ ⎢ q F ⎥
⎣ ⎦
⎢ ⎥ ⎢ ⎥
⎢Q Fy '' ⎥ ⎢ 0 λ y '' ⎥
⎣ ⎦ ⎣ ⎦
eKeRbIm:aRTIsbMElgbMlas;TI nigm:aRTIskmøaMgenAkñúgsmIkarxagelIedIm,IbegáItm:aRTIsPaBrwgRkaj
sRmab;Ggát;enAkñúgsßanPaBenH. edayGnuvtþsmIkar 14-15 eyIg)an
k = T T k 'T
⎡λ x 0 ⎤
⎢λ ⎥
k= ⎢ y 0 ⎥ AE ⎡ 1 − 1⎤ ⎡λ x λ y 0 0 ⎤
⎢− 1 1 ⎥ ⎢ 0 0 λ
⎢ 0 λ x '' ⎥ L ⎣ ⎥
⎦⎣ x '' λ y '' ⎦
⎢ ⎥
⎢ 0 λ y '' ⎥
⎣ ⎦
edayKNnam:aRTIsxagelI eyIgTTYl)an
⎡ λ2 x λ x λ y − λ x λ x '' − λ x λ y '' ⎤
⎢ ⎥
AE ⎢ λ x λ y λ2 y − λ y λ x '' − λ y λ y '' ⎥
k= (14-24)
L ⎢ − λ x λ x '' − λ y λ x '' λ2 '' λ x '' λ y '' ⎥
⎢ x ⎥
⎢− λ x λ y '' − λ y λ y '' λ x '' λ y '' λ2 '' ⎥
⎣ y ⎦
bnÞab;mkeKGaceRbIm:aRTIsPaBrwgRkajsRmab;Ggát;nImYy²EdlRtUvtP¢ab;eTAnwgTMrkl;EdleRTt ehIy
dMeNIrkarpÁúMm:aRTIsedIm,IbegáItm:aRTIsPaBrwgRkajrcnasm<½n§GnuvtþtamdMeNIrkarbTdæan. ]TahrN_xag
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -488
26. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
eRkambgðajBIkarGnuvtþrbs;va.
]TahrN_ 14-6³ kMNt;kmøaMgRbtikmμTMrsRmab; truss dUcbgðajenAkñúgrUbTI 14-13a.
dMeNaHRsay³
kareFVIkMNt;smÁal;³edaysarTMrkl;enARtg; ② sßitenAelIbøg;eRTt eyIgRtUveRbIkUGredaenrbs; node
enARtg; node enH. eKRtUvkMNt;elxerogeGaytMN nigGgát; ehIybegáItkUGredaen x, y enARtg; node
③ ¬rUbTI 14-13b¦. cMNaMfa elxkUd # nig$ sßitenAtambeNþayG½kS x' ' , y ' ' edIm,IeRbIlkçxNÐ
Edl D4 = 0 .
m:aRTIsPaBrwgRkajsRmab;Ggát;³ eKRtUvbegáItm:aRTIsPaBrwgRkajsRmab;Ggát;elx ! nigelx @ eday
eRbIsmIkar 14-24 edaysarGgát;TaMgenHmanelxkUdtamTisénG½kSskl nigG½kSrbs;kUd. eKRtUv
kMNt;m:aRTIsPaBrwgRkajsRmab;Ggát;elx # tamrebobFmμta.
Ggát;elx !³ rUbTI 14-13c/ λ x = 1, λ y = 0, λ x'' = 0.707, λ y'' = −0.707
Truss analysis using the stiffness method T.Chhay -489
27. Department of Civil Engineering NPIC
5 6 3 4
⎡ 0.25 0 − 0.17675 0.17675⎤ 5
⎢ 0 ⎥6
k1 = AE ⎢ 0 0 0 ⎥
⎢− 0.17675 0 0.125 − 0.125 ⎥ 3
⎢ ⎥
⎣ 0.17675 0 − 0.125 0.125 ⎦ 4
Ggát;elx @³ rUbTI 14-13d/ λ x = 0, λ y = −1, λ x '' = −0.707, λ y '' = −0.707
1 2 3 4
⎡ 0 0 0 0 ⎤1
⎢0 0.333 − 0.2357 − 0.2357⎥
k 2 = AE ⎢ ⎥2
⎢0 − 0.2357 0.1667 0.1667 ⎥ 3
⎢ ⎥
⎣0 − 0.2357 0.1667 0.1667 ⎦ 4
Ggát;elx #³ λ x = 0.8, λ y = 0.6
1 2 3 4
⎡ 0.128 0.096 − 0.128 − 0.096⎤ 5
⎢ 0.096 0.072 − 0.0.96 − 0.072⎥ 6
k 3 = AE ⎢ ⎥
⎢− 0.128 − 0.096 0.128 0.096 ⎥ 1
⎢ ⎥
⎣− 0.096 − 0.072 0.0.96 0.072 ⎦ 2
m:aRTIsPaBrwgRkajsRmab;rcnasm<½n§³ pÁúMm:aRTIsTaMgenHedIm,IkMNt;m:aRTIsPaBrwgRkajsRmab;rcna
sm<½n§ eyIg)an
⎡ 30 ⎤ ⎡ 0.128 0.096 0 0 − 0.128 − 0.096⎤ ⎡ D1 ⎤
⎢0⎥ ⎢ 0.096 0.4053 − 0.2357 − 0.2357 − 0.096 − 0.072⎥ ⎢ D2 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢0⎥ ⎢ 0 − 0.2357 0.2917 0.0417 − 0.17675 0 ⎥ ⎢ D3 ⎥
⎢ ⎥ = AE ⎢ ⎥⎢ ⎥
⎢Q4 ⎥ ⎢ 0 − 0.2357 0.417 0.2917 0.17375 0 ⎥⎢ 0 ⎥
⎢Q5 ⎥ ⎢ − 0.128 − 0.096 − 0.17675 0.17675 0.378 0.096 ⎥ ⎢ 0 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢Q6 ⎥
⎣ ⎦ ⎢− 0.096 − 0.072
⎣ 0 0 0.096 0.072 ⎥ ⎢ 0 ⎥
⎦⎣ ⎦
edayeFVIplKuNm:aRTIsénm:aRTIsEpñkxagelI enaHeyIgGackMNt;bMlas;TI D EdlCaGBaØatBIkaredaH
RsayRbB½n§smIkar eBalKW
352.5
D1 =
AE
− 157.5
D2 =
AE
− 127.3
D3 =
AE
eKGacTTYl)ankmøaMgRbtikmμ Q BIplKuNm:aRTIsénm:aRTIsxageRkamenAkñúgsmIkar (1). edayeRbI
bMlas;TIEdl)anKNna eyIg)an
Q4 = 0(352.5) − 0.2357(− 157.5) + 0.0417(− 127.3)
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -490
28. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
= 31.8kN
Q5 = −0.128(352.5) − 0.096(− 157.5) − 0.17675(− 127.3)
= −7.5kN
Q6 = −0.096(352.5) − 0.072(− 157.5) + 0(− 127.3)
= −22.5kN
!$>*> Trusses EdlmanbMErbMrYlsItuNðaPaB nigkMhusénplitkmμ
(Truss having thermal changes and fabrication errors)
RbsinebIGgát;xøHrbs; truss rgnUvkarlUt b¤rYjEdlbNþalBIbMErbMrYlsItuNðPaB b¤kMhusén
plitkmμ enaHeKcaM)ac;RtUveRbIviFItRmYtpledIm,ITTYl)andMeNaHRsay. vaRtUvkarbICMhan. dMbUg eK
caM)ac;RtUvKNnakmøaMgbgáb;cugEdlkarBarkarcl½trbs; node EdlbNþalBIsItuNðPaB b¤kMhusén
plitkmμ. CMhanTIBIrKWeKRtUvdak;kmøaMgEdlesμIKña b:uEnþmanTispÞúyKñaenAelI truss Rtg; node ehIyeK
RtUvKNnabMlas;TIrbs; node edayeRbIkarviPaKm:aRTIs. cugeRkay eKkMNt;kmøaMgCak;EsþgenAkñúg
Ggát; nigkmøaMgRbtikmμenAelI truss edaykareFVItRmYtplénlT§plTaMgBIrenH. Cak;Esþg eKRtUvkar
dMeNIrkarenH RbsinebI truss Carcnasm<½n§minkMNt;edaysþaTic. RbsinebI truss Carcnasm<½n§kMNt;
edaysþaTic eKGackMNt;bMlas;TIenARtg; node edayviFIenH b:uEnþbMErbMrYlsItuNðPaB nigkMhusén
plitkmμnwgminmanT§iBleTAelIkmøaMgRbtikmμ nigkmøaMgkñúgrbs;Ggát; edaysareKGacEktRmUvbERm
bRmYlRbEvgrbs;Ggát; truss edayesrI.
T§iBlkMedA³ RbsinebIGgát; truss manRbEvg
L rgkMeNInsItuNðPaB ΔT RbEvgrbs;Ggát;
nwgmankMhUcRTg;RTay ΔL = αΔTL Edl α Ca
emKuNrIkedaysarkMedA. kmøaMgsgát; qo Edl
GnuvtþeTAelIGgát;nwgeFVIeGayRbEvgrbs;Ggát;
rYj)anRbEvg ΔL' = qo L / AE . RbsinebIeyIg
dak;eGaybMlas;TITaMgBIresμIKña enaH qo =
AEαΔT . kmøaMgenHnwgTb;Ggát;dUcbgðajenAkñúgrUbTI 14-14 dUcenHeyIg)an
(q N )0 = AEαΔT
(q F )0 = − AEαΔT
Truss analysis using the stiffness method T.Chhay -491
29. Department of Civil Engineering NPIC
eKRtUvdwgfa RbsinekItmankarfykMedA enaH ΔT køayCaGviC¢man ehIykmøaMgTaMgenHnwgbRBa©asTisedA
edIm,IeFVIeGayGgát;sßitenAkñúgsßanPaBlMnwg.
eyIgGacbMElgkmøaMgTaMgBIrenHeTAkñúgkUGredaenskledayeRbIsmIkar 14-10 EdleFVIeGay
(
⎡ QN x )0 ⎤ ⎡λ x 0 ⎤ ⎡ λx ⎤
( )
⎢Q
⎢ Ny
⎥ ⎢
λ
0⎥ = ⎢ y
0⎥ ⎥
⎡1⎤
⎢λ ⎥
⎢ y ⎥
( )
⎢ QF ⎥ ⎢ 0 λ x ⎥ AEαΔT ⎢− 1⎥ = AEαΔT ⎢ − λ x ⎥
⎣ ⎦
(14-25)
( )
⎢ x
⎢ Q Fy
⎣
0⎥
0⎥ ⎢
⎦ ⎣
⎢ ⎥
0 λy ⎥
⎦
⎢
⎢− λ y ⎥
⎣
⎥
⎦
kMhusqÁgkñúgplitkmμ³ RbsinebIeKeFVIeGayGgát;EvgCaRbEvgedImedayTMhM ΔL muxeBlP¢ab;vaeTAnwg
truss enaHkmøaMg qo EdlcaM)ac;edIm,IrkSaGgát;RtwmRbEvgDIsaj L KW qo = AEΔL / L dUcenHsRmab;
Ggát;enAkñúgrUbTI 14-14 eyIg)an
AEΔL
(q N )0 =
L
AEΔL
(q F )0 =−
L
RbsinebIGgát;enHxøICagRbEvgedIm enaH ΔL køayCaGviC¢man ehIykmøaMgTaMgenHnwgbRBa©as.
enAkñúgkUGredaenskl kmøaMgTaMgenHKW
(
⎡ QN x )0 ⎤ ⎡ λx ⎤
⎢Q( )
⎢ Ny
⎥ ⎢λ ⎥
0 ⎥ = AEΔL ⎢ y ⎥
( )
⎢ QF ⎥ L ⎢− λ x ⎥
(14-26)
( )
⎢ x
⎢ Q Fy
⎣
0⎥
0⎦⎥
⎢
⎣
⎥
⎢− λ y ⎥
⎦
karviPaKm:aRTIs³ enAkñúgkrNITUeTA CamYy truss rgkmøaMgGnuvtþ bERmbRmYlsItuNðPaB nigkMhusén
plitkmμ TMnak;TMngrvagkmøaMgkñúg nigbMlas;TIsRmab; truss enaHvakøayCa
Q = KD + Q0 (14-27)
enATIenH Q0 Cam:aRTIsCYrQrsRmab; truss TaMgmUlrbs;kmøaMgbgáb;cugEdlbNþalBIbERmbRmYl
sItuNðPaB nigkMhusénplitkmμrbs;Ggát;EdlkMNt;enAkñúgsmIkar 14-25 nig 14-26. eyIgGacEbg
EcksmIkarenHenAkñúgTRmg;dUcxageRkam
⎡Qk ⎤ ⎡ K11 K12 ⎤ ⎡ Du ⎤ ⎡(Qk )0 ⎤
⎢Q ⎥ = ⎢ K ⎥⎢ ⎥ + ⎢ ⎥
⎣ u ⎦ ⎣ 21 K 22 ⎦ ⎣ Dk ⎦ ⎣(Qu )0 ⎦
edayedaHRsaym:aRTIsenAGgÁxagsþaM eyIgTTYl)an
Qk = K11 Du + K 21 Dk + (Qk )0 (14-28)
Qu = K 21 Du + K 22 Dk + (Qu )0 (14-29)
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -492
30. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
eyagtamdMeNIrkartRmYtplEdlerobrab;xagelI eyIgGackMNt;bMlas;TI Du BIsmIkarTImYyedaydk
K12 Dk nig (Qk )0 BIGgÁTaMgBIr bnÞab;mkeyIgedaHRsay Du . eyIgTTYl)an
Du = K111 (Qk − K12 Dk − (Qk )0 )
−
eRkayeBleyIgTTYl)anbMlas;TIrbs; node enaHeyIgGackMNt;kmøaMgkñúgrbs;Ggát;edayviFItRmYtpl
eBalKW
q = k ' TD + q0
RbsinebIeyIgBnøatsmIkarenHedIm,IkMNt;kmøaMgenAcugq¶ayrbs;Ggát; eyIgTTYl)an
⎡ DN x ⎤
⎢D ⎥
qF =
AE
L
[
− λx − λ y λx ]
λ y ⎢ y ⎥ − (q F )0
N
⎢ DFx ⎥
(14-30)
⎢ ⎥
⎢ D Fy ⎥
⎣ ⎦
lT§plenHRsedogKñaeTAnwgsmIkar 14-23 EtvaxusKñaRtg;enATIenHvamanplbUkéntY (q F )0 EdlCa
kmøaMgbgáb;cugrbs;Ggát;EdlbNþalBIbERmbRmYlsItuNðPaB nig / b¤kMhusénplitkmμdUckMNt;dUcxag
elI. eKRtUvdwgfa RbsinebIlT§plEdlTTYl)anBIsmIkarenHmantémøGviC¢man enaHGgát;nwgrgkmøaMg
sgát;.
]TahrN_TaMgBIrxageRkam nwgbgðajBIkarGnuvtþéndMeNIrkarrbs;viFIenH.
]TahrN_ 14-7³ kMNt;kmøaMgkñúgGgát;elx ! nig
elx @ rbs; truss EdlmanTMrsnøak;dUcbgðajenA
kñúgrUbTI 14-15 RbsinebIeKeFVIeGayGgát;elx @
xøICagmun 0.01 munnwgpÁúMvaeTAkñúg truss. yk AE =
8(10 3 )kN .
dMeNaHRsay³
edaysarGgát;manRbEvgxøI enaH ΔL = −0.01m
dUcenHGnuvtþsmIkar 14-26 eTAelIGgát;elx @ CamYy nwg λ x = −0.8, λ y = −0.6 eyIg)an
⎡ (Q1 )0 ⎤ ⎡ − 0.8 ⎤ ⎡ 0.0016 ⎤ 1
⎢(Q ) ⎥ ⎢ − 0.6 ⎥ ⎢ ⎥
⎢ 2 0 ⎥ = AE (− 0.01) ⎢ ⎥ = AE ⎢ 0.0012 ⎥ 2
⎢(Q5 )0 ⎥ 5 ⎢ 0.8 ⎥ ⎢− 0.0016⎥ 5
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢(Q6 )0 ⎥
⎣ ⎦ ⎣ 0.6 ⎦ ⎣− 0.0012⎦ 6
Truss analysis using the stiffness method T.Chhay -493
31. Department of Civil Engineering NPIC
eK)anbegáItm:aRTIsPaBrwgRkajsRmab;rcnasm<n§enAkñúg]TahrN_ 14-4. edayGnuvtþsmIkar 14-27
½
eyIg)an
⎡0⎤ ⎡ 0.378 0.096 0 0 − 0.128 − 0.096 − 0.25 0⎤ ⎡ D1 ⎤ ⎡ 0.0016 ⎤
⎢0⎥ ⎢ 0.096 0.405 0 − 0.333 − 0.096 − 0.072 0 ⎥⎢D ⎥
0⎥ ⎢ 2 ⎥ ⎢ 0.0012 ⎥
⎢ ⎥ ⎢ ⎢ ⎥
⎢Q3 ⎥ ⎢ 0 0 0 0 0 0 0 0⎥ ⎢ 0 ⎥ ⎢ 0 ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎢Q4 ⎥ = AE ⎢ 0 − 0.333 0 0.333 0 0 0 0⎥ ⎢ 0 ⎥ ⎢ 0 ⎥
+ AE
⎢Q5 ⎥ ⎢ − 0.128 − 0.096 0 0 0.128 0.096 0 0⎥ ⎢ 0 ⎥ ⎢− 0.0016⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎢Q6 ⎥ ⎢− 0.096 − 0.072 0 0 0.096 0.072 0 0⎥ ⎢ 0 ⎥ ⎢− 0.0012⎥
⎢Q ⎥ ⎢ − 0.25 0 0 0 0 0 0.25 0⎥ ⎢ 0 ⎥ ⎢ 0 ⎥
⎢ 7⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎢Q8 ⎥
⎣ ⎦ ⎢ 0
⎣ 0 0 0 0 0 0 0⎥ ⎢ 0 ⎥
⎦⎣ ⎦ ⎢
⎣ 0 ⎥
⎦
edayEbgEckm:aRTIsenHdUcbgðaj nigedayedaHRsayplKuNm:aRTIsedIm,ITTYl)ansmIkarsRmab;
bMlas;TI eyIgTTYl)an
⎡0⎤
⎢0⎥
⎢ ⎥
⎡0 ⎤ ⎡0.378 0.096⎤ ⎡ D1 ⎤ ⎡0 0 − 0.128 − 0.096 − 0.25 0⎤ ⎢0⎥ ⎡0.0016⎤
⎢0⎥ = AE ⎢0.096 0.405⎥ ⎢ D ⎥ + AE ⎢0 − 0.333 − 0.096 − 0.072 ⎥ ⎢0⎥ + AE ⎢0.0012⎥
⎣ ⎦ ⎣ ⎦⎣ 2 ⎦ ⎣ 0 0⎦ ⎢ ⎥ ⎣ ⎦
⎢0⎥
⎢ ⎥
⎢0⎥
⎣ ⎦
EdleGay
0 = AE [0.378 D1 + 0.096 D2 ] + AE [0] + AE [0.0016]
0 = AE [0.096 D1 + 0.405 D2 ] + AE [0] + AE [0.0012]
edaHRsayRbB½n§smIkar eyIgTTYl)an
D1 = −0.003704m
D2 = −0.002084m
eTaHbICaminRtUvkar eKGackMNt;kmøaMgRbtikmμ Q BIkarBnøatsmIkar (1) EdlGnuvtþtamKMrUénsmIkar
14-29.
edIm,IkMNt;kmøaMgenAkñúgGgát;elx ! nigelx @ eyIgRtUvGnuvtþsmIkar 14-30 EdlenAkñúgkrNI
enH eyIg)an
Ggát;elx !³ λ x = 0, λ y = 1, L = 3m, AE = 8(103 )kN dUcenH
⎡ 0 ⎤
q1 =
( )
8 10 3
[0 − 1 0 1]
⎢
⎢ 0 ⎥
⎥ + [0]
3 ⎢− 0.003704⎥
⎢ ⎥
⎣− 0.002084⎦
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -494
32. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
q1 = −5.56kN
Ggát;elx @³ λ x = −0.8, ( ) dUcenH
λ y = −0.6, L = 5m, AE = 8 10 3 kN
⎡− 0.003704⎤
q2 =
( )
8 10 3
⎢− 0.002084⎥
[0.8 0.6 − 0.8 − 0.6]⎢ ⎥ − 8 10 (− 0.01)
3
( )
5 ⎢ 0 ⎥ 5
⎢ ⎥
⎣ 0 ⎦
q 2 = 9.26kN
]TahrN_ 14-8³ Ggát;elx @ rbs; Edl truss
bgðajenA kñúgrUbTI 14-16 rgnUvkMeNInsItuNðPaB
83.3o C . kMNt;kmøaMgEdlekItmanenAkñúgGgát;elx @.
yk α = 11.7(10 −6 )/ o C / E = 200GPa . Ggát;
nImYy²manRkLaépÞmuxkat; A = 484mm 2 .
dMeNaHRsay³
edaysar vamankMeNInsItuNðPaB ΔT = +83.3o C . GnuvtþsmIkar 14-25 eTAelIGgát;elx @ Edl
λ x = 0.707, λ y = 0.707 eyIg)an
⎡ (Q1 )0 ⎤ ⎡ 0.707 ⎤ ⎡ 0.000689325 ⎤ 1
⎢(Q ) ⎥ ⎢ 0.707 ⎥ ⎢ ⎥
⎢ (Q3 )0 ⎥
( )
⎢ 2 0 ⎥ = AE (11.7 ) 10 −6 (83.3)⎢ ⎥ = AE ⎢ 0.000689325 ⎥ 2
⎢− 0.707 ⎥ ⎢− 0.000689325⎥ 7
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣(Q4 )0 ⎦ ⎣− 0.707 ⎦ ⎣− 0.000689325⎦ 8
m:aRTIsPaBrwgRkajsRmab; truss enHRtUv)anbegáItenAkñúg]TahrN_ 14-2.
⎡ 0 ⎤ ⎡ 0.135 0.035 0 0 0 − 0.1 − 0.035 − 0.035⎤ ⎡ D1 ⎤ ⎡ 0.000689325 ⎤ 1
⎢ 0 ⎥ ⎢ 0.035 ⎥ ⎢D ⎥ ⎢ 0.000689325 ⎥ 2
⎢ ⎥ ⎢ 0.135 0 − 0.1 0 0 − 0.035 − 0.035⎥ ⎢ 2 ⎥ ⎢ ⎥
⎢ 0 ⎥ ⎢ 0 0 0.135 − 0.035 0.035 − 0.035 − 0.1 0 ⎥ ⎢ D3 ⎥ ⎢ 0 ⎥3
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎢ 0 ⎥ = AE ⎢ 0 − 0.1 − 0.035 0.135 − 0.035 0.035 0 ⎥ ⎢ D4 ⎥
+ AE ⎢ ⎥4
0 0
⎢ 0 ⎥ ⎢ 0 0 0.035 − 0.035 0.135 − 0.035 0 − 0.1 ⎥ ⎢ D5 ⎥ ⎢ 0 ⎥5
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎢Q 6 ⎥ ⎢ − 0.1 0 − 0.035 0.035 − 0.035 0.135 0 0 ⎥⎢ 0 ⎥ ⎢ 0 ⎥6
⎢Q ⎥ ⎢− 0.035 − 0.035 − 0.1 0 0 0 0.135 0.035 ⎥⎢ 0 ⎥ ⎢− 0.000689325⎥ 7
⎢ 7⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎣Q8 ⎥
⎢ ⎦ ⎢− 0.035
⎣ − 0.035 0 0 − 0.1 0 0.035 0.135 ⎥ ⎢ 0 ⎥
⎦⎣ ⎦ ⎢− 0.000689325⎥ 8
⎣ ⎦
edayBnøatedIm,IkMNt;smIkarbMlas;TIEdlCaGBaØat nigedayedaHRsayRbB½n§smIkarenH eyIg)an
D1 = −0.002027m
D2 = −0.01187 m
Truss analysis using the stiffness method T.Chhay -495
33. Department of Civil Engineering NPIC
D3 = −0.002027m
D4 = −0.009848m
D5 = −0.002027m
edayeRbIsmIkar 14-30 edIm,IkMNt;kmøaMgenAkñúgGgát;elx @ eyIg)an
⎡− 0.002027⎤
⎢ − 0.01187 ⎥
484[200]
q2 =
10 2
[− 0.707 − 0.707 0.707 0.707]⎢
⎢ 0
⎥ − 484(200)11.7 10 −6 (83.3)
⎥
[ ( )]
⎢ ⎥
⎣ 0 ⎦
= −27.09kN
cMNaMfa kMeNInsItuNðPaBénGgát;elx @ nwgmineFVIeGaymankmøaMgRbtikmμenAelI truss eT edaysarva
Ca truss kMNt;edaysþaTic. edIm,IbgðajBIkarBicarNakarBnøatm:aRTIsénsmIkar (1) edIm,IkMNt;kmøaMg
Rbtikmμ. edayeRbIlT§plsRmab;bMlas;TI eyIg)an
Q6 = AE[− 0.1(− 0.002027 ) + 0 − 0.035(− 0.002027 ) + 0.035(− 0.009828) − 0.035(− 0.002027 )]
+ AE [0] = 0
Q7 = AE[− 0.035(− 0.002027 ) − 0.035(− 0.01187 ) − 0.1(− 0.002027 ) + 0 + 0]
+ AE [− 0.000689325] = 0
Q8 = AE[− 0.035(− 0.002027 ) − 0.035(− 0.01187 ) + 0 + 0 − 0.1(− 0.002027 )]
+ AE [− 0.000689325] = 0
!$>(> karviPaK truss kñúglMh (Space-truss analysis)
eKGacviPaK truss kñúglMhkMNt;edaysþaTic nig truss kñúglMhminkMNt;edaysþaTicedayeRbI
dMeNIrkarviPaKdUcKñaEdl)anerobrab;BIelIkmun. b:uEnþ edIm,IKitG½kSTaMgbI eKRtUvbBa©ÚlFatubEnßmeTAkñúg
m:aRTIsbMElg T . edIm,ITTYl)anva eyIgnwgBicarNaGgát; truss EdlbgðajenAkñúgrUbTI 14-17.
m:aRTIsPaBrwgRkajsRmab;Ggát;EdlkMNt;edayeRbIkUGredaentMbn; x' RtUv)aneGayedaysmIkar 14-4.
elIsBIenH tamkarGegátrUbTI 14-17 eKGackMNt;kUsIunUsR)ab;TiscenøaHkUGredaenskl nigkUGr-
edaentMbn;edayeRbIsmIkarRsedogKñanwgsmIkar 14-5 nig 14-6 Edl
xF − xN xF − xN
λ x = cos θ x = = (14-31)
L (x F − x N )2 + ( y F − y N )2 + ( z F − z N )2
yF − yN yF − yN
λ y = cos θ y = = (14-32)
L (x F − x N ) + ( y F − y N ) + (z F − z N )
2 2 2
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -496
34. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
zF − zN zF − zN
λ z = cos θ z = = (14-33)
L (x F − x N )2 + ( y F − y N )2 + ( z F − z N )2
CalT§plénTMhM ¬G½kS¦TIbI m:aRTIsbMElg ¬smIkar 14-9¦ køayCa
⎡λ x λ y λ z 0 0 0 ⎤
T =⎢ ⎥
⎣ 0 0 0 λx λ y λz ⎦
edayCMnYsm:aRTIsenH nigsmIkar 14-4 eTAkñúgsmIkar 14-15 ¬ K = T T k 'T ¦ eyIg)an
⎡λ x 0⎤
⎢λ 0⎥
⎢ y ⎥
⎢λ 0 ⎥ AE ⎡ 1 − 1⎤ ⎡λ x λ y λ z 0 0 0 ⎤
k=⎢ z ⎥ ⎢ ⎥
⎢0 λ x ⎥ L ⎢− 1 1 ⎥ ⎣ 0 0 0 λ x λ y λ z ⎦
⎣ ⎦
⎢0 λy ⎥
⎢ ⎥
⎢0
⎣ λz ⎥⎦
edayedaHRsayplKuNm:aRTIs eyIgnwgTTYl)anm:aRTIssIuemRTI
Nx Ny Nz Fx Fy Fz
⎡ λ2 x λxλ y λxλz − λ2
x − λxλ y − λxλz ⎤ N x
⎢ ⎥
⎢ λ y λx λ2
y λ y λ z − λ y λ x − λ2 − λ y λ z ⎥ N y
y
AE ⎢
k= λz λx λz λ y λ z − λ z λ x − λ z λ y − λ2 ⎥ N z
2
(14-34)
L ⎢
z ⎥
⎢ − λ2 x − λxλ y − λxλz λx2
λ x λ y λ x λ z ⎥ Fx
⎢ ⎥
⎢− λ y λ x − λ2
y − λ y λz λ y λx λ2 y λ y λ z ⎥ Fy
⎢− λ λ − λz λ y − λ2 λ z λ x λ z λ y λ2 ⎥ Fz
⎣ z x z z ⎦
smIkarenHCam:aRTIsPaBrwgRkajsRmab;Ggát;Edl
sresredayeRbIkUGredaenskl. elxkUdtam
beNþayCYredk nigCYrQrtMNageGayTis x, y, z
enARtg;cugCit N x , N y , N z Edlbnþedaycugq¶ay
Fx , F y , Fz .
sRmab;karsresrkmμviFIkMuBüÚT½r CaTUeTAvaman
lkçN³gayRsYlkñúgkareRbIsmIkar 14-34 Cagkar
edaHRsayplKuNm:aRTIs T T k 'T sRmab;Ggát;
nImYy². dUckarerobrab;BIxagedIm dMbUgkMuBüÚT½rnwg rkSam:aRTIsPaBrwgRkajsRmab;rcnasm½<n§ K Edl
manFatuesμIsUnü bnÞab;mkeTotedaysarFatunImYy² énm:aRTIsPaBrwgRkajsRmab;Ggát;RtUv)anbegáIt
vaRtUv)andak;eTAkñúgTItaMgRtUvKñarbs;vaenAkñúgm:aRTIs K . eRkayeBlFaturbs;m:aRTIsPaBrwgRkaj
Truss analysis using the stiffness method T.Chhay -497
35. Department of Civil Engineering NPIC
sRmab;rcnasm<½n§RtUv)anbegáIt eKGacGnuvtþdMeNIrkar Edl)anerobrab;enAkñúgkfaxNÐ 14-6 edIm,I
kMNt;bMlas;TIrbs;tMN kmøaMgRbtikmμ nigkmøaMgkñúgrbs; Ggát;.
karviPaK truss edayeRbIviFIPaBrwgRkaj T.Chhay -498
36. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa
cMeNaT
14>1 kMNt;m:aRTIsPaBrwgRkaj K sRmab;eRKOg 14>6 kMNt;m:aRTIsPaBrwgRkaj K sRmab; truss.
bgÁúM. yk A = 300mm 2 nig E = 200GPa yk A = 0.005m 2 nig E = 200GPa . snμt;
sRmab;Ggát;. tMNTaMgGs;tP¢ab;edaysnøak;.
14>2 kMNt;bMlas;TItamTisedk nigTisQr
enARtg;tMN ③ rbs;eRKOgbgÁúMenAkñúgcMeNaT
14>1.
14>7 kMNt;bMlas;TItamTisQrenARtg;tMN ①
14>3 kMNt;kmøaMgkñúgénGgát;nImYy²rbs;eRKOg
nigkmøaMgkñúgrbs;Ggát;elx @ éncMeNaT 14>6.
bgÁúMenAkñúgcMeNaT 14>1.
14>8 kMNt;m:aRTIsPaBrwgRkaj K sRmab; truss.
14>4 kMNt;m:aRTIsPaBrwgRkaj K sRmab; truss.
yk A = 0.0015m 2 nig E = 200GPa sRmab;
yk A = 300mm 2 nig E = 200GPa sRmab;
Ggát;.
Ggát;.
14>5 kMNt;bMlas;TItamTisQrenARtg;tMN ④
nigkmøaMgkñúgrbs;Ggát;elx $ éncMeNaT 14>4. 14>9 kMNt;kmøaMgkñúgénGgát;elx ^ éncMeNaT
yk A = 0.0015m 2 nig E = 200GPa . 14>8. yk A = 0.0015m 2 nig E = 200GPa
Problems T.Chhay -499
37. Department of Civil Engineering NPIC
sRmab;Ggát;nImYy². 14>13. yk A = 1000mm 2 nig E =
14>10 kMNt;kmøaMgkñúgénGgát;elx ! éncMeNaT 200GPa .
14>8 RbsinebI Ggát;EvgCagmun10mm munnwg 14>15 kMNt;kmøaMgkñúgénGgát;elx @éncMeNaT
P¢ab;vaeTAkñúg truss. edIm,IedaHRsay dkbnÞúk 14>8 RbsinebIsItuNðPaBekIneLIg 55o C .
10kN ecj. yk A = 0.0015m 2 nig E = yk A = 1000mm 2 E = 200GPa nig α =
200GPa sRmab;Ggát;nImYy². 11.7(10 −6 )/ o C .
14>11 kMNt;m:aRTIsPaBrwgRkaj K sRmab; 14>16 kMNt;kmøaMgRbtikmμenAelI truss. AE
truss. AE CacMnYnefr. CacMnYnefr.
14>12 kMNt;kmøaMgkñúgénGgát;elx @ nig
elx % éncMeNaT 14>11. AE CacMnYnefr.
14>13 kMNt;m:aRTIsPaBrwgRkaj K sRmab;
truss. yk A = 1000mm 2 nig E = 200GPa .
14>14 kMNt;bMlas;TItamTisedkenARtg;tMN
① nigkmøaMgkñúgrbs;Ggát;elx @ éncMeNaT
cMeNaT T.Chhay -500