1. T.Chhay
viFIKNna
truss edayrUbmnþm:aRTIs
Matrix Formulation of Truss Analysis
rUbmnþm:aRTIssMrab;karviPaK truss sþaTickMNt;KWQrelIviFItMN. eKGacsresrsmIkar
lMnwgBIrEdlÉkraCBIKñasMrab;RKb;tMNrbs; truss EdlpSMeLIgedayRbB½n§kMlaMgRbsBVenAkñúgbøg;.
RbsinebI truss Carcnasm<½n§sþaTickMNt; eKGacedaHRsayrkkMlaMgkñúgrbs;Ggát; truss nImYy²
nigkMlaMgRbtikmμedayviFItMN.
manGñkxøH)anKitfa vaCakarRbesIrCagEdleKKYrKNnakMlaMgRbtikmμmunnwgKNnakMlaMg
rbs;Ggát;. b:uEnþ edIm,IrkSarKMnitrYmrbs;viFI/ kMlaMgRbtikmμ nigkMlaMgrbs;Ggát;RtUv)anKitCa
GBaØatdUcKña. KMnitrYmenHmanRbeyaCn_sMrab;karGnuvtþenAkñúgkMuBüÚT½r. sMrab;eRKOgbgÁúMsþaTicmin
kMNt; kMlaMgelIsGacCakMlaMgGgát; b¤kMlaMgRbtikmμ. dUcenH eKminKYrKNnakMlaMgRbtikmμdac;
edayELkBIrUbmnþviPaKeT.
edIm,IeFVIeGayviFImanlkçN³TUeTA cUrBicarNacMENkén truss sþaTickMNt;EdlbgðajenA
kñúg Fig. 3.10a. tMN i CacMNucTMr ehIyRbtikmμ Ri RtUv)anbgðajkñúgTMrg;CabgÁúMkMlaMg Rix nig
Riy . kMlaMgxageRkAEdlbgðajmanGMeBIenAelItMN i nig j . Pi manbgÁúMkMlaMgBIrKW Pix nig
P ehIy Pj mankMlaMgbgÁúMBIrEdrKW Pjx nig Pjy . Ggát; ij RtUv)ansnμt;CaGgát;rgkarTajCamYy
iy
nwgkMlaMgmanGMeBIRtg;tMN i nigtMN j EdlRtUv)anbgðajenAkñúgrUb. kMlaMgTaMgGs;mantMél
viC¢mantamTisEdlbgðaj.
Fig. 3.10b bgðajBIGgát; ij dac;edayELkCamYynwgkUGredaencMnuccugenAkñúgRbB½n§kUGr-
edaen xy . enARtg;cMnuccug i / eKGackMlaMg Fij edaybgÁúMkMlaMg X ij nig Yij rbs;vatamTisedA
x nig y erogKñadUcxageRkam³
x j − xi
X ij = Fij cos α = Fij = Fij lij
Lij
(1)
y j − yi
Yij = Fij cos β = Fij = Fij mij
Lij
tMél lij nig mij RtUv)aneKehAfa kUsIunUsR)ab;Tis (direction cosines) eRBaHvaeGay
nUvtMélkUsIunUsrbs;muMEdlvas;tamTisbRBa©asTisRTnicnaLikaBIG½kS x nigBIG½kS y eTAGgát;
ij erogKña. enARtg;cug j / eKk¾kMNt;kMlaMg F ji edaybgÁúMkMlaMg X ji nig Y ji rbs;vatamTis
edA x nig y .
-1- Matrix Formulation of truss analysis

2. NPIC
xi − x j
X ji = F ji cos γ = F ji = F ji l ji
Lij
(2)
yi − y j
Y ji = F ji cos δ = F ji = F ji m ji
Lij
Edl l ji nig m ji CakUsIunusR)ab;TissMrab;Ggát; ji . b:uEnþedaysarEtGaMgtg;sIuetrbs;kM
laMgGgát;mantMéldUcKñaenAcugrbs;vanImYy² Edl Fij = F ji . BIsmIkar (1) nig (2) bgðajfa
lij = −l ji nig mij = − m ji . dUcenH eyIg)an
-2- Matrix Formulation of truss analysis

3. T.Chhay
( )
X ij = Fij lij
Yij = Fij (mij )
X ji = Fij (− lij )
(3)
Y ji = Fij (− mij )
edayGnuvtþsmIkarlMnwgRtg;tMN i nig j eyIgTTYl)ansmIkardUcxageRkam³
X ig + X ih + X ij + X ik + Pix + Rix = 0
Yig + Yih + Yij + Yik + Piy + Riy = 0
(4)
X jh + X ji + X jk + X jf + Pjx = 0
Y jk + Y ji + Y jk + Y jf + Pjy = 0
b¤edayeRbITMnak;TMngEdlesñIeLIgedaysmIkar (3) eKGacsresrsmIkarlMnwgTaMgenHCa
Fig lig + Fih lih + Fij lij + Fik lik + Pix + Rix = 0
Fig mig + Fih mih + Fij mij + Fik mik + Piy + Riy = 0
(5)
F jh l jh − Fij lij + F jk l jk + F jf l jf + Pjx = 0
F jk l jk − Fij lij + F jk l jk + F jf l jf + Pjy = 0
RbsinebIrcnasm<½n§mantMNcMnYn n enaHeKGacsresrsmIkarlMnwgcMnYn 2n . eKGacsresr
smIkarTaMgenHkñúgTMrg;m:aRTIsdUcxageRkam³
⎡. . . . . . . . . . . . . . . ⎤⎧ . ⎫ ⎧ . ⎫
⎢. . . . . . . . . . . . . . . ⎥⎪ . ⎪ ⎪ . ⎪
⎢ ⎥⎪ ⎪ ⎪ ⎪
⎢. . − lig − lih − lik − lij . . . . . −1 . . . ⎥ ⎪ Fig ⎪ ⎪ Pix ⎪
⎢ ⎥⎪ ⎪ ⎪ ⎪
⎢. . − mig − mih − mik − mij . . . . . . − 1 . . ⎥ ⎪ Fih ⎪ ⎪ Piy ⎪
⎢. . . . . lij − l jh − l jk − l jf . . . . . . ⎥ ⎪ Fik ⎪ ⎪ Pjx ⎪
⎢ ⎥⎪ ⎪ ⎪ ⎪
⎢. . . . . mij − m jh − m jk − m jf . . . . . . ⎥ ⎪ Fij ⎪ ⎪ Pjy ⎪
⎢. . . . . . . . . . . . . . . ⎥ ⎪ F jh ⎪ ⎪ . ⎪
⎢ ⎥⎪ ⎪ ⎪ ⎪
⎪ ⎪ ⎪ ⎪
⎢. ⎥ ⎨ F jk ⎬ = ⎨ . ⎬ (6)
⎢. ⎥ ⎪F ⎪ ⎪ . ⎪
⎢ ⎥ ⎪ jf ⎪ ⎪ ⎪
⎢. ⎥⎪ . ⎪ ⎪ . ⎪
⎢ ⎥⎪ ⎪ ⎪ ⎪
⎢. ⎥⎪ . ⎪ ⎪ . ⎪
⎢. ⎥ ⎪ Rix ⎪ ⎪ . ⎪
⎢ ⎥⎪ ⎪ ⎪ ⎪
⎢. ⎥ ⎪ Riy ⎪ ⎪ . ⎪
⎢. ⎥⎪ . ⎪ ⎪ . ⎪
⎢ ⎥⎪ ⎪ ⎪ ⎪
⎢.
⎣ . . . . . . . . . . . . . . ⎥⎪ . ⎪ ⎪ . ⎪
⎦⎩ ⎭ ⎩ ⎭
2n × 2n 2n × 1 2n × 1
kñúgTMrg;enH eyIgeXIjy:agc,as;favamanGBaØtiEdlCakMlaMgkñúgrbs;Ggát; nigRbtikmμsrubTaMgGs;Et
2 n . RbsinebIvamanGBaØtieRcInCag 2 n rcnasm<½n§CasþaTicminkMNt;
-3- Matrix Formulation of truss analysis

4. NPIC
pÞúymkvijRbsinebIvamanGBaØti ticCag 2n rcnasm<½n§KμanlMnwg mann½yfavaminmanGgát;
b¤RbtikmμRKb;RKan;edIm,IbMeBjtMrUvkarlMnwg.
eKGacsresrsmIkar (6) kñúgTMrg;xøIdUcxageRkam
{F }
[C ]⎧ ⎫ = {P}
⎨ ⎬ (7)
⎩{R}⎭
Edl [C ] CaemKuNm:aRTIsénkUsIunUsR)ab;Tis b¤m:aRTIssþaTicTaMgmUl/ {F } nig {R} CaviuT½rénGBaØti
kMlaMgkñúgGgát; nigbgÁúMRbtikmμerogKña ehIy {P}CaviuT½rbnÞúk EdlrYmbBa©ÚlbgÁúMénbnÞúkGnuvtþn_. eKGac
sresrkñúgTMrg;kan;EtxøIdUcxageRkam
[C ]{Q} = {P} (8)
Edl {Q} CaviucT½rénGBaØtikMlaMgkñúgGgát; nigkMlaMgRbtikmμ. dMeNaHRsayGacRtUv)ansMEdgeday
{Q} = [C ]−1{P} = [b]QP {P} (9)
Edl [b]QP P¢ab;TMnak;TMngrvagkMlaMgGgát; nigkMlaMgRbtikmμeTAnwgkMlaMgGnuvtþn_. BIsmIkar (9)
eyIgeXIjfaPaBesμIKñaéncMnYnsmIkar nigcMnYnGBaØtiminEmnmann½yfaeRKOgbgÁúMmanesßrPaBeT. Rb
sinebIeRKOgbgÁúMKμanesßrPaB edaysarm:aRTIs [C ] minman inverse ehIysmIkar (9) k¾minGacman
b¤sEdr.
eKminGaceRbIrUbmnþm:aRTIsedIm,IviPaKeRKOgbgÁúMedayéd)aneT elIkElgsMrab;EteRKOgbgÁúM
samBaØbMput. dUcenH viFIenHRtUv)anbegáIteLIgedIm,IviPaKeRKOgbgÁúMedaykMuBüÚT½rdMbUgbMput. b:uEnþ edIm,I
bgðajBIbec©keTskñúgkareRbIR)as;viFIenH eyIgnwgyk]TahrN_y:agsamBaØmkbgðaj.
]TahrN_³ edaHRsaykMlaMgkñúgGgát;rbs; truss xageRkamedayeRbIrUbmnþma:RTIs.
-4- Matrix Formulation of truss analysis

5. T.Chhay
dMeNaHRsay³ kUGredaenrbs;tMN nigTisedAsnμt;rbs;Rbtikmμ nigkMlaMgkñúgGgát;RtUv)anbgðajenA
kñúgrUb.
kUsIunUsR)ab;TisedA³ lij = (x jL− xi ) / mij = (y jL− yi ) / Lij = (x j − xi )2 + (y j − yi )2
ij ij
Ggát; ij (x j − xi ) (y j − yi ) Lij
lij mij
m m m
ab 4−0= 4 4 − 6 = −2 4.47 + 0.895 − 0.447
ba 0 − 4 = −4 6 − 4 = −2 ,, − 0.895 + 0.447
ad 0−0 = 0 0 − 6 = −6 6.00 0 −1
da 0−0=0 6−0 = 6 ,, 0 +1
bd 0 − 4 = −4 0 − 4 = −4 5.66 − 0.707 − 0.707
db 4−0= 4 4−0= 4 ,, + 0.707 + 0.707
be 4−4=0 0 − 4 = −4 4.00 0 −1
eb 4−4=0 4−0= 4 ,, 0 +1
bc 12 − 4 = 8 0 − 4 = −4 8.94 + 0.895 − 0.447
cb 4 − 12 = −8 4−0= 4 ,, − 0.895 + 0.447
ce 4 − 12 = −8 0−0 = 0 8.00 −1 0
ec 12 − 4 = 8 0−0 = 0 ,, +1 0
ed 0 − 4 = −4 0−0 = 0 4.00 −1 0
de 4−0= 4 0−0 = 0 ,, +1 0
smIkarlMnwg³ sresrsmIkarlMnwgsMrab;tMNnImYy² nigsMEdgkñúgTMrg;énsmIkar (8)³
[C ]{Q} = {P}
∑ Pxa ⎡ − .895 0 ⎤ ⎧ Fab ⎫ ⎧+ 150⎫
∑ Pya ⎢+ 447 +1 ⎥ ⎪F ⎪ ⎪ 0 ⎪
⎢ . ⎥ ⎪ ad ⎪ ⎪ ⎪
∑ Pxb ⎢ + .895 + .707 0 − .895 ⎥ ⎪ Fbd ⎪ ⎪ 0 ⎪
⎢ ⎥⎪ ⎪ ⎪ ⎪
∑ Pyb ⎢ − .447 + .707 + 1 + .447 ⎥ ⎪ Fbe ⎪ ⎪ 0 ⎪
∑ Pxc ⎢ + 8.95 + 1 ⎥ ⎪ Fbc ⎪ ⎪ 0 ⎪
⎪ ⎪ ⎪ ⎪
∑ Pyc
⎢ ⎥⎨ ⎬=⎨ ⎬
⎢ − .447 0 − 1⎥ ⎪ Fce ⎪ ⎪ 0 ⎪
∑ Pxd ⎢ 0 −1 +1 ⎥ ⎪ Fed ⎪ ⎪ 0 ⎪
⎢ ⎥⎪ ⎪ ⎪ ⎪
∑ Pyd ⎢ −1 0 0 ⎥ ⎪ Rdx ⎪ ⎪− 120⎪
∑ Pxe ⎢ 0 − .707 −1 −1 ⎥⎪R ⎪ ⎪ 0 ⎪
⎢ ⎥ ⎪ dy ⎪ ⎪ ⎪
∑ Pye ⎢
⎣ − 1 − .707 0 − 1 ⎥ ⎪ Rcy ⎪ ⎪ 0 ⎪
⎦⎩ ⎭ ⎩ ⎭
dMeNaHRsaysMrab;kMlaMgr)ar nigRbtikmμ
-5- Matrix Formulation of truss analysis

6. NPIC
⎧ Fab ⎫ ⎧ − 167.6 ⎫
⎪F ⎪ ⎪ ⎪
⎪ ad ⎪ ⎪ + 75.0 ⎪
⎪ Fbd ⎪ ⎪ − 113.2 ⎪
⎪ ⎪ ⎪ ⎪
⎪ Fbe ⎪ ⎪ + 120.0 ⎪
⎪ F ⎪ ⎪− 257.0⎪
{Q} = [C ]−1{P} = [b]QP {P} = ⎪ bc ⎪ = ⎪
⎨ ⎬ ⎨
⎪
⎬ xñatTaMgGs;KitCa kN
⎪ Fce ⎪ ⎪+ 230.0⎪
⎪ Fed ⎪ ⎪+ 230.0⎪
⎪ ⎪ ⎪ ⎪
⎪ Rdx ⎪ ⎪ − 150.0 ⎪
⎪ Rdy ⎪ ⎪ + 5.0 ⎪
⎪ ⎪ ⎪ ⎪
⎪ Rcy ⎪ ⎪ + 115.0 ⎪
⎩ ⎭ ⎩ ⎭
BIkñúg]TahrN_enH eyIgeXIjfaFatuenAkñúgemKuNma:RTIs [C ] minmantMéleBjeT. cMnYnsmIkarTaMg
Gs;mancMnYn 10 EdleyIgRtUvedaHRsayedIm,IrkGBaØti 10 . eKGackMNt;kMlaMgRbtikmμ Rdx / Rdy
nig Rcy edaysmIkarsþaTic. bnÁÞab;mkeKGacedaHRsaysmIkarTI 1 nigTI 2 edIm,Irk Fab nig Fad .
eRkayeBlsÁal; Fad eKGackMNt; Fbd nig Fed BIsmIkarTI 9 nigTI 10. eRkaymkeTot eyIg
Gacrk Fce nig Fbe BIsmIkarTI 7 nigTI 8 ehIycugeRkay edaHRsaysmIkarTI 6 edIm,Irk Fbc . GVI
Edl)anerobrab;enH CalMnaMedaHRsaykMlaMgkñúgGgát;edayviFItMN.
-6- Matrix Formulation of truss analysis