1. T.Chhay
FñwmsþaTicminkMNt;
Statically Indeterminate Beams
1> esckþIepþIm Introduction
FñwmsþaTicminkMNt; CaRbePTFñwmEdlmanGBaØtiEdlCaRbtikmμTMr eRcInCasmIkarlMnwg dUcenHRbtikmμ
TMrminGackMNt;edayeRbIsmIkarsþaTic ∑ F = 0, ∑ F = 0, ∑ M = 0 )aneT. dUcenHedIm,IedaHRsay
V H
RbtikmμTMr eKcaM)ac;RtUvkarsmIkarbEnßm EdlmandUcCasmIkarEdlTak;TgnwgPaBdab nigmMurgVilrbs;Fñwm.
eTaHbICaFñwmsþaTickMNt;gayRsYledaHRsayCagFñwmsþaTicminkMNt; EtFñwmsþaTicminkMNt;pþl;nUvpl
RbeyaCn¾eRcIn dUcCaesdækic© muxgar nigesaPNÐPaB.
2> RbePTFñwmTMrrwg Restrained beams
FñwmTMrrwgrYmman propped cantilever P1 P2
w
P
beam nig fixed beam. RbePTFñwmTaMgenH A B A B
CaRbePTFñwmsþaTic minkMNt;.
Elastic curve Elastic curve
P1 P2 P
Propped cantilever beam CaRbePT
w
MB MA MB
A B A B
FñwmTMrrwgmçag. FñwmenHmanTMrmçagCaTMrbgáb; RA
l
RB RA
l
RB
nigTMrsmBaØenAcugmçageTot b¤enAcMnucEk,rcug a, Propped cantilever beam b, Fixed beam
r)ar.
Fixed beam CaRbePTFñwmTMrrwgsgxag. FñwmenHmanTMrbgáb;enAsgxagcugr)ar.
3> Fñwm Propped cantilever beam
eKmanviFIsaRsþeRcIny:agkñúgkaredaHRsayrkRbtikmμTMrsMrab;FwñmRbePTenH. viFItMrYtpl CaviFImYy
EdlsamBaØCageK edayeRbIsmIkarPaBdabsMrab;Fñwm cantilever EdleGaykñúgtaragTI1.
viFItMrYtplEdleKeRbIsMrab;karedaHRsayenH QrelIeKalkarN_ PaBdabenAcMnucNamYyrbs;Fñwm
propped cantilever esμInwgplbUkBiCKNiténPaBdabBIrKW³
- PaBdabEdl)anBIRbtikmμEdlCaGBaØtielIs sMrab;FñwmTMrbgáb;EdlKμankMlaMgxageRkA
- PaBdabEdl)anBIkMlaMgxageRkA eRkayedaHTMrsmBaØecj
]TahrN_³ rkRbtikmμTMrrbs;Fñwm propped cantilever EdlmanTMr w
roller Rtg;cMnuc A nigTMrbgáb;Rtg;cMnuc B . A B
l
FñwmenHrgbnÞúkBRgayesμI w ehIymanRbEvg l . FñwmenHman EI
efr.
FñwmsþaTicminkMNt; 128

2. T.Chhay
dMeNaHRsay³ w
CadMbUgeKRtUvedaHTMrRtg;cMnuc A edayCMnYsedayGBaØtiEdlCaRbtikmμTMr R A B
l
A
RA
bnÞab;mkeKrkPaBdabEdl)anBIkMlaMgxageRkA w Rtg;cMnuc A w
4 A
eK)an Δ = 8EI ↓
1
wl
Δ1 B
bnÞab;mkeToteKrkPaBdabEdl)anBIGBaØtiEdlCaRbtikmμTMr R A Δ2
Rtg;cMnuc A A B
3
eK)an Δ = REI ↑
2
3
l A R A
edayRtg;cMnuc A CaTMrenaH vaKμanPaBdabeT dUcenH plbUkénPaBdabEdlekItBIkMlaMgTaMgBIrRtUvesIμsUnü b¤
PaBdabEdlekItBIkMlaMgTaMgBIrenHRtUvEtesμIKña
wl 4 R Al 3
=
8 EI 3EI
3
⇒ R A = wl
8
edaysarGBaØtielIsRtUv)anedaHRsay dUcenHRbtikmμTMrEdlenAsl;GacedaHRsay)anedayeRbIsmI
karlMnwgsþaTic
∑ Y = 0 ⇒ R A + RB = wl
3 5
⇒ RB = wl − wl = wl
8 8
l2 3
∑ M B = 0 ⇒ M B − w + wl 2 = 0
2 8
1 2
⇒ M B = wl
8
]TahrN_³ rkRbtikmμTMrrbs;Fñwm propped cantilever Edl P=10kN
manTMr roller Rtg;cMnuc A nigTMrbgáb;Rtg;cMnuc B . 0.5m 2m
FñwmenHrgbnÞúkcMcMnuc P = 10kN ehIymanRbEvg l = 2.5m . A
2.5m
B
FñwmenHman EI efr.
dMeNaHRsay³
rkPaBdabEdl)anBIkMlaMgxageRkA P = 10kN Rtg;cMnuc A P=10kN
2m
tamtaragTI1 eK)an Δ1 B
Pa 2 10 × 2 2 36.67 2.5m
Δ1 = (3l − a ) = (3 × 2.5 − 2) = ↓
6 EI 6 EI EI
rkPaBdabEdl)anBIGBaØtiEdlCaRbtikmμTMr R Rtg;cMnuc A
A
FñwmsþaTicminkMNt; 129

3. T.Chhay
tamtaragTI1 eK)an Δ2
3
R Al R A B
Δ2 = = 5.208 A ↑
3EI EI
RA
PaBdabEdlekItBIkMlaMgTaMgBIrenHRtUvEtesμIKña
36.67 R
= 5.208 A
EI EI
36.67
⇒ RA = = 7.04kN
5.208
∑ Y = 0 ⇒ R A + RB = P
⇒ RB = 10 − 7.04 = 2.96kN
∑ M B = 0 ⇒ M B − P × 2 + R A × 2.5 = 0
⇒ M B = 2.4kN .m
taragTI1
Case Cantilever Beam and loading Slope at free end Diflection at free end
P
Pl 2 Pl 3
1 Δ θ= Δ=
θ l 2 EI 3EI
P a
Pa 2 Pa 2
2 Δ θ= Δ= (3l − a )
θ l 2 EI 6 EI
w
wl 3 wl 4
3 Δ θ= Δ=
θ l 6 EI 8 EI
a
w
4
Δ
wa 3 wa 2
θ l θ= Δ= (4l − a )
6 EI 24 EI
M
Ml Ml 2
5
Δ
θ l
θ= Δ=
EI 2 EI
w
6 wl 3 wl 4
θ= Δ=
Δ
θ l
24 EI 30 EI
FñwmsþaTicminkMNt; 130

4. T.Chhay
4> Fñwm Fixed beam
Fñwm fixed beam CaFñwmEdlmanmMurgVilRtg;TMresμIsUnü b¤Gacniyaymü:ageTot CaFñwmTMrsamBaØEdl
manm:Um:g;eFVIGMeBI enARtg;TMrTaMgBIr )ann½yfam:Um:g;TaMgBIrenHCaGBaØtielIsEdlRtUvkarrk. viFItMrYtplk¾Gac
eRbIedIm,IedaHRsayFñwm fixed beam Edr EteKeRbIsmIkarmMurgVilCMnYseGaysmIkarPaBdabvij. smIkarmMu
rgVilsMrab;FñwmTMrsmBaØeGaykñúgtaragTI2.
viFItMrYtplEdleKeRbIsMrab;karedaHRsayenH QrelIeKalkarN_ mMurgVilenATMrNamYyrbs;Fñwm fixed
beam esμInwgplbUkBiCKNiténmMurgVilBIresμIsUnüKW³
- mMurgVilEdl)anBIRbtikmμEdlCaGBaØtielIs sMrab;FñwmTMrsmBaØEdlKμankMlaMgxageRkA
- mMurgVilEdl)anBIkMlaMgxageRkA sMrab;FñwmTMrsmBaØ
taragTI2
Case Simple Beam and loading Slope at support Diflection at mid-span
l
P l
2 2
Pl 2 Pl 3
A B θ A = θB = Δ=
1 Δ θ
A
θ
B 16 EI 48 EI
l
Pb (l 2 − b 2 )
A
a P b
B
θA =
θ θ
6lEI
2 Δ A B
Pab ( 2l − b)
l
θB =
6lEI
w
A B
wl 3 5 wl 4
3 Δ θ
A
θ
B θ A = θB = Δ=
384 EI
l 24 EI
M Al
MA
θA =
A θ θ B
A
3 EI M Al 2
4
Δ A
A
B
A Δ=
l M l 16 EI
θB = A
A
6 EI
M Bl
MB
θA = M Bl 2
3 EI Δ=
B
Δ A θ θ B
5 A
B
l
B
B
M l
16 EI
θB = B
B
6 EI
w
θ θ B 5 wl 3 wl 4
6 Δ A A B
θ A = θB = Δ=
l 120 EI
192 EI
FñwmsþaTicminkMNt; 131

5. T.Chhay
P
]TahrN_³ rkRbtikmμTMrrbs;Fñwm fixed beam EdlrgbnÞúkcMcMnuc a b
Rtg;cMnuc C . FñwmenHman EI efr. A C B
dMeNaHRsay³ l
edIm,IedaHRsay)an eKRtUvbþÚrTMrTaMgBIrBITMrbgáb; eTACaTMr a
P
b
smBaØnigbEnßmm:Um:g;Bt;EdlCaGBaØtielIs ehIyGBaØtielIsenH M A A
C B
MB
CaRbtikmμTMr EdlRtUvedaHRsayedaysmIkarbEnßm KWsmIkarmMu l
rgVil.
bnÞab;mkeKedaHRsayrkmMurgVilRtg;TMrTaMgsgxagEdl a P b
)anBIkMlaMgxageRkA P A
θ θ
B
A B
tamtaragTI2 eK)an l
Pb(l − b )
nig θ = Pab6(lEI− b)
2 2
2l
θ =A B
6lEI
edaHRsayrkmMurgVilRtg;TMrTaMgsgxagEdl)anBIm:Um:g;Edl
CaGBaØtielIs M A
θAA θB
tamtaragTI2 eK)an
MA
A
A B
θ =A
M l
3EI
A
nig θ = MEIl
A
6
B
A
A
dUcKñamMurgVilRtg;TMrTaMgsgxagEdl)anBIm:Um:g;EdlCa
CaGBaØtielIs M B
θAB θB
B
MB
θ =A
B
M l
6 EI
nig θ = MEIl
B
3
B
B
B
A B
edayTMrTaMgBIrCaTMrbgáb; dUcenHmMurgVilrbs;vaesμIsUnü
eK)anplbUkBiCKNiténmMurgVilEdl)anBIkMlaMgTaMgGs;RtUvesμIsUnü
θ = θ + θ nig θ = θ + θ
A A
A
A
B
B B
A
B
B
⎧ Pb(l − b ) M Al M B l
2 2
⎪ = +
⎪
⇒ ⎨ 6lEI 3EI 6 EI
⎪ Pab( 2l − b) = M Al + M B l
⎪ 6lEI
⎩ 6 EI 3EI
edaHRsayRbBn§½smIkarxagelI eK)an
M =
Pab
l
A nig M = Pa b
l2
2
B 2
2
edaysarGBaØtielIsRtUv)anedaHRsay dUcenHRbtikmμTMrEdlenAsl;GacedaHRsay)anedayeRbIsmI
karlMnwgsþaTic
∑ Y = 0 ⇒ RA + RB = P
FñwmsþaTicminkMNt; 132

6. T.Chhay
∑ M B = 0 ⇒ M B − M A − Pb + RAl = 0
( M A + M B + Pb)
⇒ RA =
l
Pb 2
⇒ RA = 3 (l + 2a )
l
Pa 2
⇒ RB = 3 (l + 2b)
l
]TahrN_³ rkRbtikmμTMrrbs;Fñwm fixed beam EdlrgbnÞúkBRgay w=20kN
m
minesμI. FñwmenHman EI efr.
A B
dMeNaHRsay³ l=5m
rkmMurgVilRtg;TMrTaMgsgxagEdl)anBIkMlaMgxageRkA w w
tamtaragTI2 eK)an
5wl 3 θ θ
θ A = θB = A B
192 EI
edaHRsayrkmMurgVilRtg;TMrTaMgsgxagEdl)anBIm:Um:g;Edl
CaGBaØtielIs M A
θAA θB
tamtaragTI2 eK)an
MA
A
A B
θ = A
M l
3EIA
nig θ = MEIl
A
6
B
A
A
dUcKñamMurgVilRtg;TMrTaMgsgxagEdl)anBIm:Um:g;EdlCa
CaGBaØtielIs M B
θAB θB
B
MB
θ = A
M l
6 EI
B
nig θ = MEIl
B
3
B
B
B A B
edayTMrTaMgBIrCaTMrbgáb; dUcenHmMurgVilrbs;vaesμIsUnü
eK)anplbUkBiCKNiténmMurgVilEdl)anBIkMlaMgTaMgGs;RtUvesμIsUnü
θ = θ + θ nig θ = θ + θ
A A
A
A
B
B B
A
B
B
⎧ 5wl 3
M Al M B l
⎪192 EI = 3EI + 6 EI
⎪
⇒⎨ 3
⎪ 5wl = M Al + M B l
⎪192 EI 6 EI 3EI
⎩
edaHRsayRbBn§½smIkarxagelI eK)an
5wl 2
MA = MB = = 26.04kN .m
96
edaysarbnÞúkxageRkAmanlkçN³sIuemRTI
eyIg)an R = R = 2wl2 = 204× 5 = 25kN
×
A B
dUcenHRbtikmμTMrTaMgBIrmantMéldUcKña
FñwmsþaTicminkMNt; 133

7. T.Chhay
5> FñwmCab; edayviFItMrYtpl Continuous beams-Superposition
FñwmCab; CaFñwmEdlQrenAelITMrsmBaØeRcInCagBIr. FñwmRbePTenHRtUv)aneKeRbIsMrab;rcnasm<n§½TMenIb
² eRBaHvapþl;nUvlkçN³esdækic©CagebIeRbobeFobnwgFñwmTMrsmBaØEdlmancenøaHElVgdUcKña. FñwmCab;)ankat;
bnßym:Um:g;Bt;GtibrmaBIFñwmTMrsmBaØ EdlCaehtueFVIeGayeKGackat;bnßyTMhMFñwm)an. FñwmCab; CaRbePTFñwm
sþaTicminkMNt; EdlkMlaMgRbtikmμminGacrk)anedayeRbIEtsmIkarlMnwgsþaTic.
P
w
L1 L2 L3
CaTUeTA viFItMrYtplEdleRbIsMrab;edaHRsayFñwmsþaTicminkMNt;EdlmanmYycenøaHElVg minmanPaB
gayRsYlsMrab;karedaHRsayFñwmCab;eT elIkElgEtFñwmCab;enaHmanEtBIrcenøaHElVg EdlkaredaHRsayman
lkçN³dUcKñanwgFñwmsþaTicminkMNt;mYycenøaHElVg.
]TahrN_³ rkRbtikmμTMrrbs;FñwmCab; EdlmanBIrcenøaHElVg w
esμIKña l rgbnÞúkBRgayesμI. FñwmenHman EI efr. A
l
B
l
C
dMeNaHRsay³ w
edaHTMrkNþalecj ehIyCMnYsedaykMlaMgRbtikmμ A
l
B
l
C
EdlCaGBaØtielIs RB
rkPaBdabRtg;cMnuc B Edl)ankMlaMgxageRkA w w
A Δ1 B C
5wL4
Δ1 = L
384 EI
Edl L = 2l Δ2
rkPaBdabRtg;cMnuc B Edl)ankMlaMgRbtikmμ R B
A
l
B
l
C
RB L3 RB
Δ2 = L
48 EI
edayRtg;TMr B KμanPaBdab
⇒ Δ1 = Δ 2
5
⇒ RB = wL
8
edayFñwmmanlkçN³rUb nigbnÞúkqøúH
5
wL − wL
8 3
⇒ RA = RC = = wL
2 16
FñwmsþaTicminkMNt; 134

8. T.Chhay
6> RTwsþIbIm:Um:g; Theorem of three moment
viFIEdlmanlkçN³RsYlCageKkñúgkarrkm:Um:g;Bt;sMrab;FñwmCab; KWCameFüa)ayénkareRbITMnak;TMng
EdlekItmanrvagm:Um:g;Bt;enAelITMrTaMgbIénElVgBIrCab;Kña. EdlTMnak;TMngenHeKeGayeQμaHfa RTwsþIbIm:Um:g;.
eRkayeBlEdlm:Um:g;enAelITMrRtUv)ankMNt; enaHkMlaMgkat;TTwg kMlaMgRbtikmTMr nigm:Um:g;enAcMnucepSg²
eTotk¾RtUv)ankMNt;edayeRbIsmIkarlMnwgsþaTic. w1
w2
eKmanFñwmCab;mYy EteKelIkykEtBIrcenøaHElVg A
L1
B
L2
C
Rtg;kNþalmkniyay KWTMr A B nig C . L CaRbEvgBI 1
cenøaHElVgBITMr A eTATMr B EdgrgnUvbnÞúkBRgayesμI w . nig L CaRbEvgBIcenøaHElVgBITMr B eTATMr C Edgrg 1 2
nUvbnÞúkBRgayesμI w . edaysarEtTMrrbs;FñwmCab;suT§EtCaTMrsamBaØenaHmMurgVilRtg;TMrmantMélxusBIsUnü.
2
eyIgeXIjfa bnÞat;b:HeTAnwgExSeGLasÞicRtg;TMr B EpñkxageqVg nigxagsþaMmantMéldUcKñasBaØapÞúyKñμa eday
sarvaCaFñwmCab;. A
θ'
B
1
B
θ'
B
2
C
L1 L2
θ 'B1 = −θ 'B 2
edaypþac;FñwmCab;FñwmTMrsamBaØrgbnÞúk w nig w 1 2 w1
w2
kMNt;mMurgVilRtg;TMr B EdlekItBIkMlaMgxageRkA A
L1
B B
L2
C
sMrab;ElVg L eK)an θ = 24EI
1
wL
B1
1
3
1
w1
w2
sMrab;ElVg L eK)an θ = 24EI
2
wL
B2
2
3
2
θ
B1 θ
B
2
kMNt;mMurgVilRtg;TMr B EdlekItBIm:Um:g; M Rtg;TMr B B θ
BB1
M
B
M
B
θ
B
B2
sMrab;ElVg L eK)an θ = MEIL
1
3
B
B1
B 1
θ θ
sMrab;ElVg L eK)an θ = 3EI
M MC
A BA BC
M L B 2
2 B
B2
kMNt;mMurgVilRtg;TMr B EdlekItBIm:Um:g; M Rtg;TMr A sMrab;ElVg L A 1
M L
θ BA = A 1
6 EI
kMNt;mMurgVilRtg;TMr B EdlekItBIm:Um:g; M Rtg;TMr C sMrab;ElVg L C 1
M C L2
θ BC =
6 EI
⇒ θ 'B1 = θ B1 + θ B + θ B
B1 A
nig θ ' B2
= θB2 + θB + θB
B2 C
⇒ θ B1 + θ B + θ B = −(θ B 2 + θ B + θ B )
B1 A B2 C
w1 L3 M B L1 M A L1 w 2 L32 M B L2 M C L2
⇒ 1
+ + = −( + + )
24 EI 3EI 6 EI
24 EI 3EI 6 EI
w1 L3 w2 L32
⇒ M A L1 + 2M B ( L1 + L2 ) + M C L2 = − 1
−
4 4
enHCaTMrg;Biessrbs;RTwsþIbIm:Um:g; sMrab;krNIkrNIFñwmrgbnÞúBRgayesμI.
FñwmsþaTicminkMNt; 135

9. T.Chhay
xageRkamCaTMrg;TUeTArbs;RTwsþIm:Um:g;
6 A1 x1 6 A2 x2
M A L1 + 2 M B ( L1 + L2 ) + M C L2 = − −
L1 L2
Edl A nig x CaRkLaépÞ nigTItaMgTIRbCMuTMgn;rbs;düaRkamm:Um:g;. tYr 6 A x nig 6 A x ERbRbYleTA
L L
1 1
1
2
2
2
tamRbePTbnÞúkEdlmanGMeBIelIFñwmenaH EdlmankñúgtaragTI3.
taragTI3
6 A1 x1 6 A2 x2
Case loading
L1 L2
w2
w1
w1 L3 w2 L32
1 1
L1 L2 4 4
P1 P2
a b
P1a 2 P2b 2 2
2 ( L1 − a 2 ) ( L2 − b )
L1 L2
L1 L2
w1 w2
2w1L3 2w2 L32
3 1
L1 L2 15 15
]TahrN_³ eKmanFñwmCab;bIRbelaHElVg rgnUvbnÞúkBRgayesμI. FñwmenHman EI efr.
k> kMNt;m:Um:g;enAelITMr w=5kN/m
w=8kN/m
x> kMNt;RbtikmμTMr w=2kN/m
dMeNaHRsay³ 1 2
4m 5m
3
3m
4
k> kMNt;m:Um:g;enAelITMr
tamRTwsþIbIm:Um:g;eyIg)an
6 A1 x1 6 A2 x2
M A L1 + 2 M B ( L1 + L2 ) + M C L2 = − −
L1 L2
edayFñwmrgbnÞúkBRgayesμI
w1 L3 w2 L32
⇒ M A L1 + 2 M B ( L1 + L2 ) + M C L2 = − 1
−
4 4
sMrab;BIrRbelaHElVgdMbUgeyIgman
M = M = 0 / M = M / M = M / L = 4m / L = 5m / w = 5kN / m / w = 2kN / m
1 A 2 B 3 C 1 2 1 2
5 × 4 3 2 × 53
⇒ 2 M 2 ( 4 + 5) + M 3 5 = − −
4 4
⇒ 18M 2 + 5M 3 = −142.5
sMrab;BIrRbelaHElVgbnÞab;eyIgman
FñwmsþaTicminkMNt; 136

10. T.Chhay
M2 = M A /M 3 = MB /M 4 = M C = 0 L1 = 5m / L = 3m / w = 2kN / m / w = 8kN / m
2 1 2
2× 5 8× 3 3 3
⇒ 5M 2 + 2 M 3 (5 + 3) = − −
4 4
⇒ 5M 2 + 16M 3 = −116.5
edaHRsayRbBn§½smIkareyIgTTYl)an
M = −6.454kN .m nig M = −5.264kN .m
2 3
kalNam:Um:g;elITMrmansBaØaGviC¢man mann½yfasésEpñkxagelIrbs;FñwmrgkMlaMgTaj nigEpñkxageRkam
rgkMlaMgsgát;
x> kMNt;RbtikmμTMr
sMrab;Rbtikmμ R 1
w=5kN/m
M2
1 2
∑ M 2 = 0 ⇒ R1 × 4 − 5 × 4 × 2 + 6.454 = 0 4m
R1 V2
⇒ R1 = 8.387 kN
sMrab;Rbtikmμ R 2
w=5kN/m
w=2kN/m
M3
∑ M 3 = 0 ⇒ 8.387 × 9 − 5 × 4 × 7 + 5R2 − 2 × 5 × 2.5 + 5.264 = 0 1 2 3
4m 5m
⇒ R2 = 16.85kN 8.387kN R2 V3
sMrab;Rbtikmμ R 3
∑ M 4 = 0 ⇒ 8.387 ×12 − 5 × 4 ×10 + 8 ×16.85 − 2 × 5 × 5.5 + 3R3 − 8 × 3 ×1.5 = 0
⇒ R3 = 18.52kN w=5kN/m
w=8kN/m
sMrab;Rbtikmμ R
w=2kN/m
4 1 2 3 4
4m 5m 3m
∑ Y = 0 ⇒ 5 × 4 + 2 × 5 + 8 × 3 − 8.387 − 16.85 − 18.52 − R4 = 0
8.387kN 16.85kN R3 V4
⇒ R4 = 10.243kN
]TahrN_³ eKmanFñwmCab;BIrRbelaHElVg nigmanFñwmlyenARtg;TMr 1 . FñwmenHman EI efr.
kMNt;m:Um:g;enAelITMr 10kN 15kN 20kN 1.5m
5kN
dMeNaHRsay³
1m 2m w=2kN/m
1 2 3
edaysarRtg;TMr 1 manFñwmly 1.5m 4m 5m
dUcenHm:Um:g;elITMr 1 GackMNt;edayeRbIsmIkarsþaTic
M 1 = −10 ×1.5 = −15kN .m
edayTMr 3 CaTMrsamBaØ dUcenH M = 0 . 3
kMNt;m:Um:g;Rtg;TMr 2 RtUv)anedaHRsayedayeRbIRTwsþIbIm:Um:g;
FñwmsþaTicminkMNt; 137

11. T.Chhay
6 A1 x1 6 A2 x2
M A L1 + 2M B ( L1 + L2 ) + M C L2 = − −
L1 L2
eyIgman M A
= M 1 = −15kN .m / L = 4 m / L = 5m
1 2
15 ×1 2 2 20 × 3 2 2 5 × 3.5 2 2 × 53
− 15 × 4 + 2M 2 (5 + 4) = − (4 − 1 ) − (4 − 3 ) − (5 − 3.52 ) −
4 4 5 4
⇒ M 2 = −11.576kN .m
]TahrN_³ eKmanFñwmCab;BIrRbelaHElVg EdlmanTMrbgáb;rwgsgxag. FñwmenHman EI efr.
kMNt;m:Um:g;enAelITMr
w=8kN/m
w=2kN/m
dMeNaHRsay³ 1
5m
2
3m
3
edayTMrelx 1 nigelx 3 CaTMrbgáb; enAeyIgRtUvbþÚrTMrbgáb;eTACaTMrsamBaØedaybEnßmRbelaHElVgEdlman
RbEvgesμIsUnü w=8kN/m
enaHrUbragFñwmxagelI)anbþÚrrUbrageTACa
w=2kN/m
0 1 2 3 4
bnÞab;mkeTIbeyIgGnuvtþRTwsþIm:Um:g;eTAFñWmenaH 0m 5m 3m 0m
6 A1 x1 6 A2 x2
M A L1 + 2 M B ( L1 + L2 ) + M C L2 = − −
L1 L2
sMrab;TMrelx 0 / 1 / 2
eyIgman M = M = 0 / M = M / M = M
A 0 B 1 C 2
L = 0m / L = 5m / w = 0 / w = 2kN / m
1 2 1 2
2×5 3
⇒ 10 M 1 + 5M 2 = −
4
⇒ 2 M 1 + M 2 = −12.5
sMrab;TMrelx 1 / 2 / 3
eyIgman M = M / M = M / M = M
A 1 B 2 C 3
L = 5m / L = 3m / w = 2kN / m / w = 8kN / m
1 2 1 2
2 × 53 8 × 33
⇒ 5M 1 + 16 M 2 + 3M 3 = − −
4 4
⇒ 5M 1 + 16M 2 + 3M 3 = −116.5
sMrab;TMrelx 2 / 3 / 4
eyIgman M = M / M = M / M = M = 0
A 2 B 3 C 4
L = 3m / L = 0m / w = 8kN / m / w = 0kN / m
1 2 1 2
8 × 33
⇒ 3M 2 + 6M 3 = −
4
⇒ M 2 + 2M 3 = −18
FñwmsþaTicminkMNt; 138

12. T.Chhay
⎧ 2 M 1 + M 2 = −12.5
⎪
⇒ ⎨5M 1 + 16 M 2 + 3M 3 = −116.5
⎪ M 2 + 2 M 3 = −18
⎩
edaHRsayRbBn§½smIkarxagelIeyIg)an
M 1 = 3.82kN .m
M 2 = 4.85kN .m
M 3 = 6.57kN .m
FñwmsþaTicminkMNt; 139