14. truss analysis using the stiffness method

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

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


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



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.


!$>#> 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.


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.


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


⎡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)



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


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.


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


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

dMeNaHRsay³ eTaHbICa truss Carcnasm<½n§minkMNt;edaysþaTicdWeRkTImYyk¾eday Etvanwgminbgðaj
BIPaBlM)akkñúgkarTTYl)anm:aRTIsPaBrwgRkajsRmab;rcnasm<½n§eT. eKkMNt;elxerogeGaytMN nig


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



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)



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.

⎡ 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


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


⎡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


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

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


⎡ 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


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
⎣ ⎦


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



⎡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.



!$>&> 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


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


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



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)



= 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



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)



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


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⎦



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



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


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



sRmab;rcnasm<½n§RtUv)anbegáIt eKGacGnuvtþdMeNIrkar Edl)anerobrab;enAkñúgkfaxNÐ 14-6 edIm,I
kMNt;bMlas;TIrbs;tMN kmøaMgRbtikmμ nigkmøaMgkñúgrbs; Ggát;.



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


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

14. truss analysis using the stiffness method

Empfohlen

Empfohlen

Weitere ähnliche Inhalte

Was ist angesagt?

Was ist angesagt? (20)

Andere mochten auch

Andere mochten auch (19)

Ähnlich wie 14. truss analysis using the stiffness method

Ähnlich wie 14. truss analysis using the stiffness method (17)

Mehr von Chhay Teng

Mehr von Chhay Teng (20)

14. truss analysis using the stiffness method