1. T.Chhay
PaBdabrbs;Fñwm edayviFIfamBl
Deflection of beam: Energy method
1> kmμnþxageRkA W E External work
eRkamGMeBIénbnÞúk muxkat;mYyénr)ar)anpøas;TI. dUcKñaenHEdr m:Um:g;Rtg;muxkat;mYyénr)ar)aneFVI
eGaymuxkat;enaHmanmMurgVil.
cMeBaHr)arQr EdlmancugesrIenAxageRkam nigmanKl;bgáb;enAxagelI ehIyrgbnÞúk P Rtg;cugesrI
enaHvaeFVIeGaymansac;lUt Δ . kmμnþEdlekIteLIgedaybnÞúk P KWkmμnþxageRkA W = 1 P.Δ .
E
2
cMeBaHm:Um:g; M EdlGnuvtþmkelIFñwmmYyeFVIeGaymanmMurgVil θ . kmμnþxageRkAEdlbegáIteLIgeday
m:Um:g; M KW W = 1 M .θ .
E
2
2> famBlb:Utg;Esülxagkñúg W I Potential energy
kmμnþxageRkA)anrkSaTukenAkñúgr)ar ehIy)anERbkøayeTACafamBlxagkñúg.
k> krNIr)arrgbnÞúktamGkS½
1
WI = WE = P.Δ
2
tamc,ab;h‘Uk eyIg)an
PL
Δ=
AE
P2L
⇒ WI =
2 EI
x> krNIFñwmrgm:Um:g;Bt; M
famBlxagkñúgEdlrkSaTukenAelIFatuGnnþtUc dx ehIyeFVIeGaymanmMurgVil dθ KW
1
dWI = M .dθ
2
b:uEnþeyIgdwgfa dθ = EI dx
M
1
dWI = M 2 dx
2 EI
2
L M
⇒ WI = ∫ dx
0 2 EI
• cMNaM³
- smIkar W = ∫ 2EI dx ¬sMrab;Fñwmrgm:Um:g;Bt; M ¦/ M KWCam:Um:g;Bt;enARtg;muxkat;samBaØmYy
2
M L
I
0
ehIym:Um:g;enaHbNþalmkBIplbUksrubénbnÞúkxageRkA.
PaBdabrbs;Fñwm edayviFIfamBl 118

2. T.Chhay
- krNI truss Edlr)arrbs;vargEtkMlaMgtamGkS½ famBlb:Utg;EsülTak;TgEtnwgkMlaMgtambeNþay
GkS½r)arenaHEtb:ueNÑaH.
- kñúgFñwmeRkamGMeBIénbnÞúk nigRbtikmμTMr eFVIeGayeKmankMlaMgkat;TTwg nigm:Um:g;Bt; Rtg;muxkat;
samBaØNamYyénmuxkat;. dUcenHtameKalkarN_rkSafamBl famBlb:Utg;EsülEdlrkSaTukkñúg
r)arenaH KWCaGnuKmn_énplrbs;kMlaMgkat;TTwg V nigm:Um:g;Bt; M . b:uEnþeKsnμt;fa b:Utg;Esülxag
kñúgEdlekIteLIgedaykMlaMgkat;TTwg V mantMéltUcEdleKGacpat;ecal)an ebIeFobeTAnwgfam-
Blb:Utg;EsülEdlekIteLIgedaysarm:Um:g;Bt; M .
3> karKNnabMErbMrYlragedayviFIfamBl
tameKalkarN_rkSafamBl eK)an³
WE = W I
WE - CakmμnþxageRkAEdlbegáIteLIgedaykMlaMgxageRkAGnuvtþbnþicmþg²eTAelIrcnasm<n§½.
W - CafamBlb:Utg;EsülxagkñúgEdlrkSaTukkñúgrcnasm<n§½.
I
smIkarxagelIGnuBaØatieGaymanEtGBaØtimYyb:ueNÑaH dUcenHr)arRtUvrgbnÞúkEtmYyEtb:ueNÑaH.
]TahrN_³ rkbMErbMrYlrag ¬PaBdab¦ Rtg;cMnuc C P
sMrab;r)arTMrsamBaØEdlman EI efr ehIyrgbnÞúk
Rtg;cMnuc P Rtg;kNþalElVg. A C B
l/2 l/2
dMeNaHRsay³
kmμnþxageRkAEdlbegáIteLIgedaybnÞúk P KW M
PΔ
WE = Px
2 2
Pl
M Cam:Um:g;EdlekIteLIgedaysarbnÞúk P 4
famBlb:Utg;Esül xagkñúgEdlrkSaTukKW
M2
WI = ∫
L
dx
0 2 EI
Et M =
Px
2
⇒M2 =
P2 x2
4
¬KitcMeBaH 0 ≤ x ≤ 2 ¦l
tamlkçN³qøúHeK)anm:Um:g;srubesμInwgBIrdg m:Um:g;EpñkxageqVg b¤xagsþaM dUcenHeK)an³
l
l
P2 x2 P 2 x3 2 P 2l 3
WI = 2 ∫ 2 dx = =
0 8 EI 12 EI 0
96 EI
PΔ P 2l 3 Pl 3
WE = WI ⇒ = ⇒Δ=
2 96 EI 48 EI
]TahrN_³ rkmMurgVilRtg;cMnuc B sMrab;r)arTMrsamBaØEdlman
PaBdabrbs;Fñwm edayviFIfamBl 119

3. T.Chhay
EI efr ehIyrgm:Um:g; M Rtg;cMnuc B .
B A B
M
l
dMeNaHRsay³
kmμnþxageRkAEdlbegáIteLIgedaym:Um:g; M KW
B
MB
1 M=MlB.x
WE = M B .θ B
2
famBlb:Utg;Esül xagkñúgEdlrkSaTukKW M
M2
WI = ∫
L
dx
0 2 EI
eday M= B
M .x
l
l M .x 1
⇒ WI = ∫ ( B ) 2 dx
0 l 2 EI
M2 l
= 2 B ∫ x 2 dx
2l EI 0
M 2l
⇒ WI = B
6 EI
tameKalkarN_rkSafamBl eK)an
M B .θ B M B l
2
WE = WI ⇒ =
2 6 EI
M Bl
⇒ θB =
3EI
4> kmμnþvaTuyEGl b¤kmμnþminBit (Virtual work)
kmμnþvaTuyEGl (V W) CakmμnþEdlekIteLIgedaykMlaMgminBit ehIyeFVIeGaymanbMlas;TIBit.
Bakü virtual EdleKeRbIkñúgkmμnþ b¤bMlas;TI sMedAeTAelItMélminBit b¤TMhMRsemIRsém. ebIeKGnuvtþmkelI
RbBn§½manlMnwgmYy Q eK)anbegáIteGaymankMlaMg virtual xagkñúg nigbMlas;TI virtual tUc². eKsnμt;faRb
i
Bn§½kMlaMg virtual Q smmUlesμInwg kMlaMg virtual Ékta (1kN ) edIm,IKNnabMlas;TIRtg;cMnucmYyénrcna
i
sm<n§½ tameKalkarN_rkSafamBl kmμnþ virtual xageRkAesμInwgkmμnþ virtual xagkñúg.
We = Wi
5> karKNnabMErbMrYlragedayviFIkmμnþvaTuyEGl (Virtual work method)
edIm,IKNnabMlas;TIRtg;muxkat;samBaØmYyénrcnasm<n§½eRkamGMeBIénRbBn§½kMlaMgxageRkA. eKGnuvtþ
nUvCMba‘anxageRkam³
- snμt;lubbM)at;kMlaMgxageRkATaMgGs;ecj
- dak;Rtg;cMnucEdlcg;rkbMlas;TInUvkMlaMg virtual Ékta (1kN ) tamTisedAénbMlas;TIBit
PaBdabrbs;Fñwm edayviFIfamBl 120

4. T.Chhay
- rkRbtikmμkMlaMg virtual xagkñúg ¬rab;TaMg kMlaMgkat;TTwg kMlaMgbeNþayGkS½ nigm:Um:g;pg¦ Edl
ekIteLIgedaysarbnÞúkÉktaenaH.
- rkRbtikmμTMr/ M , V , N EdlekIteLIgedaysarbnÞúkBit
- tameKalkarN_rkSafamBleK)an W = W e i
k> krNI truss
kmμnþxageRkAEdlekIteLIgedaysarbnÞúkÉkta (1kN )
We = 1× δVC
Edl δ - bMlas;TIBittamTisQrRtg;cMnuc C
VC
famBlb:Utg;EsülEdlrkSaTukenAkñúgr)ar truss
Wi = ∑ p.Δ
Edl p - kMlaMgkñúgrbs;r)ar truss Edl)anBIbnÞúkÉkta (1kN )
Δ - bMErbMrYlragBiténr)ar truss
tameKalkarN_rkSafamBleyIg)an
We = Wi ⇒ δVC = ∑ p.Δ
tamc,ab;h‘Uk eyIg)an
PL
Δ=
EA
∑ p.P.L
⇒ δVC =
EA
Edl P - kMlaMgkñúgr)arekItBIbnÞúkBit
L - RbEvgr)ar
A - muxkat;r)ar
E - m:UDuleGLasÞic
]TahrN_³ kMNt;PaBdabrbs; truss Rtg;cMnuc C P 1 =30KN P 2 =50KN
ebIeKdwgfamuxkat;rbs;r)arnImYy² A = 1500mm
B D
2
2m
nigmanm:UDuleGLasÞic E = 200GPa . A
C
E
dMeNaHRsay³ 4m 4m
edaylubecalkMlaMgxageRkATaMgGs;EdlmanGMeBI B D
mkelI truss edaydak;bnÞúkÉkta (1kN ) Rtg;cMnuc 2m
C manTisedAcuHeRkam CacMnucEdleyIgRtukarrkPaBdab
A E
C
4m 4m
p=1KN
PaBdabrbs;Fñwm edayviFIfamBl 121

5. T.Chhay
kMNt;kMlaMgkñúgrbs;r)arEdlekItBIbnÞúkÉkta nigbnÞúkBit
r)ar RbEvg L(m) kMlaMgkñúgr)ar p(kN ) kMlaMgkñúgr)ar P(kN ) p.P.L
AB 2.83 − 0.707 − 49.5 + 99.04
AC 4 + 0 .5 + 35 + 70
BC 2.83 + 0.707 + 7.07 + 14.15
BD 4 −1 − 40 + 160
CD 2.83 + 0.707 − 7.07 − 14.15
CE 4 + 0 .5 + 45 + 90
DE 2.83 − 0.707 − 63.65 + 127.35
∑ p.P.L + 546.39
∑ p.P.L
δVC =
EA
546.39 × 1000
⇒ δVC = = 1.8mm ↓
1500 × 200
x> krNIFñwm ¬PaBdab¦
cg;rkbMlas;TI δ Rtg;cMnuc C eKeFVIdUc truss Edr edayeKlubkMlaMgBitTaMgLayecal rYcehIyenA
C
enARtg;cMnuc C EdleKcg;rkbMlas;TIenaH eKdak;kMlaMg virtual Ékta (1kN ) . eKrkm:Um:g;Bt; virtual m .
bnÞab;mkeKdak;kMlaMgxageRkATaMgGs;mkelIFñwm rYcehIyeKrkm:Um:g;BitmkelI dx . enAeBlmanm:Um:g; M naM
eGaymanmMurgVil dθ enAmuxkat;sgxag dx . ebI W Cakmμnþ virtual xageRkAEdlekIteLIgedaysarkMlaMg
e
virtual Ékta (1kN ) ehIyeFVIeGaymanbMlas;TI δ Rtg;cMnuc C eK)an³
C
We = 1.δ C
Fatu differential énb:Utg;Esülxagkñúg dW EdlrkSaTukenAxagkñúgPaKkMNat;GnnþtUc
i dx RtUv)an
begáIteLIgedaym:Um:g; virtual m EdleFVIeGaymuxkat;vil)anmMu dθ .
dWi = m.dθ
EteyIgman dθ = EI dx
M
m.M
⇒ dWi = dx
EI
b¤Wi = ∫
L m.M
0 EI
dx
tameKalkarN_rkSafamBl
We = Wi
PaBdabrbs;Fñwm edayviFIfamBl 122

6. T.Chhay
L m.M
⇒ δC = ∫ dx
0 EI
K> krNIFñwm ¬mMurgVil¦
eFVIdUcKñakrNIxagelIEdr EteKRtUvdak;m:Um:g; virtual Ékta (1kN.m) CMnYseGaykMlaMg virtual Ékta
(1kN ) vij. eKrkm:Um:g;Bt; virtual m' . bnÞab;mkeKdak;kMlaMgxageRkATaMgGs;mkelIFñwm rYcehIyeKrkm:Um:g;
BitmkelI dx . ebI W Cakmμnþ virtual xageRkAEdlekIteLIgedaysarm:Um:g; virtual Ékta (1kN.m) ehIyeFVI
e
eGaymanmMurgVil θ Rtg;cMnuc C eK)an³
C
We = 1.θ C
Fatu differential énb:Utg;Esülxagkñúg dW EdlrkSaTukenAxagkñúgPaKkMNat;GnnþtUc
i dx RtUv)an
begáIteLIgedaym:Um:g; virtual m' EdleFVIeGaymuxkat;vil)anmMu dθ .
dWi = m'.dθ
EteyIgman dθ = EI dx
M
m'.M
⇒ dWi = dx
EI
b¤Wi = ∫
0
L m'.M
EI
dx
tameKalkarN_rkSafamBl
We = Wi
L m'.M
⇒ θC = ∫ dx
0 EI
]TahrN_³ eKeGayFñwmsamBaØrgbnÞúkBRgayesμI w = 20 kN
m
C w=20kN
m
nig EI = 40000kN.m . rkPaBdabenAkNþalElVg Δ A
2
C B
4m 4m
nigmMurgVilenATMr A θ . A
dMeNaHRsay³
- rkPaBdabenAkNþalElVg Δ C
1kN
CMnYskMlaMg virtual Ékta (1kN ) Rtg;cMnuc C A
4m
C
4m
B
rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 4m BI A eTA C
x
⇒m=
2
smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 4m BI A eTA C
w.L.x w.x 2
⇒M = −
2 2
2
w.L.x w.x 3
⇒ m.M = −
4 4
PaBdabrbs;Fñwm edayviFIfamBl 123

7. T.Chhay
rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 4m BI B eTA C
x
⇒m=
2
smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 4m BI B eTA C
w.L.x w.x 2
⇒M = −
2 2
2
w.L.x w.x 3
⇒ m.M = −
4 4
plKuNm:Um:g; sMrab; 0 ≤ x ≤ 8m BI A eTA B
w.L.x 2 w.x 3
⇒ m.M = −
2 2
2
w.x 3 1
dUcenH 0
4 w.L.x
ΔC = ∫ (
2
−
2 EI
) dx
4 4
w.L.x 3 wx 4 1
⇒ ΔC = ( − )
6 0 8 0 EI
20 × 8 × 4 3 20 × 4 4 1
⇒ ΔC = ( − ) = 26.7 mm ↓
6 8 40000
- rkmMurgVilenATMr A θ A
A B
CMnYsm:Um:g; virtual Ékta (1kN.m) Rtg;cMnuc A 8m
rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 8m BI A eTA B
x
⇒ m' =
L
smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 8m BI A eTA B
w.L.x w.x 2
⇒M = −
2 2
2
w.x w.x 3
⇒ m'.M = −
2 16
2
w.x 3 1
dUcenH 8 w.x
θA = ∫ (
0 2
− )
16 EI
dx
8 8
w.x 3 wx 4 1
⇒θA = ( − )
6 0 64 0 EI
⇒ θ A = 0.01067 rad
]TahrN_³ eKeGayFñwmTMrbgáb;rgkMlaMgcMcMnucRtg;cug 30kN
r)ar edayman EI = 30000kN .m . kMNt;PaBdab nigmMu
2 A
4m
B
rgVilRtg;cugr)ar.
dMeNaHRsay³
PaBdabrbs;Fñwm edayviFIfamBl 124

8. T.Chhay
- rkPaBdabenAcugr)ar δ A
1kN
CMnYskMlaMg virtual Ékta (1kN ) Rtg;cMnuc A
A B
4m
rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 4m BI A eTA B
⇒ m = −x
smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 4m BI A eTA B
⇒ M = −30.x
⇒ m.M = 30 x 2
dUcenH δ = ∫ (30 x ) EI dx
A
0
4 1 2
4
30 x 3 1
⇒δA = ( )
3 0 EI
1
⇒ δ A = (10 × 4 3 ) = 21.3mm ↓
30000
- rkmMurgVilenAcugr)ar θ A
1kN.m
A B
CMnYsm:Um:g; virtual Ékta (1kN.m) Rtg;cMnuc A
4m
rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 4m BI A eTA B
⇒ m' = −1
smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 4m BI A eTA B
⇒ M = −30 x
⇒ m'.M = 30 x
dUcenH θ = ∫ (30 x) EI dx
A
0
4 1
4
30 x 2 1
⇒θA = ( )
2 0 EI
⇒ θ A = 0.008rad
]TahrN_³ eKeGayFñwmTMrsamBaØrgkMlaMgBRgayesμI w = 20 . kN
m
w=20kN
FñwmenHman E = 200 nig I = 200 ×10 mm .
kN
mm 2
6 4 m
C
kMNt;PaBdabRtg;cMnuc C / mMurgVilRtg;cMnuc C nig A .
A B
3m 8m
dMeNaHRsay³
- rkPaBdabRtg;cMnuc C δ C
1kN
CMnYskMlaMg virtual Ékta (1kN ) Rtg;cMnuc C
C A B
3m 8m
rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 3m BI C eTA A
PaBdabrbs;Fñwm edayviFIfamBl 125

9. T.Chhay
⇒ m = −x
smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 3m BI C eTA A
20 2
⇒M =− .x = −10 x 2
2
⇒ m.M = 10 x 3
rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 8m BI B eTA A
⇒ m = −0.375 x
smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 8m BI B eTA A
20 2
⇒ M = 68.75 x − .x = −10 x 2 + 68.75 x
2
⇒ m.M = 3.75 x 3 − 25.78 x 2
dUcenHδC = ∫
0
L m.M
EI
dx
3
3 10 x 8 3.75 8 25.78
⇒ δC = ∫ dx + ∫ x 3 dx − ∫ x 2 dx
0 EI 0 EI 0 EI
1 ⎛ 10 x 4 25.78 x 3 ⎞
4 8 8
⎜ 3.75 x 4 ⎟
⇒ δC = + −
EI ⎜ 4 0 4 0 3 ⎟
⎝ 0⎠
⇒ δ C = 8.94mm ↑
- rkmMurgVilRtg;cMnuc C θ C
1kN.m
CMnYsm:Um:g; virtual Ékta (1kN .m) Rtg;cMnuc C
C A B
3m 8m
rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 3m BI C eTA A
⇒ m' = −1
smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 3m BI C eTA A
⇒ M = −10 x 2
⇒ m'.M = 10 x 2
rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 8m BI B eTA A
⇒ m' = −0.125 x
smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 8m BI B eTA A
⇒ M = −10 x 2 + 68.75 x
⇒ m'.M = 1.25 x 3 − 8.59 x 2
dUcenHθC = ∫
0
L m'.M
EI
dx
PaBdabrbs;Fñwm edayviFIfamBl 126

10. T.Chhay
1 3 8 8
⇒ θC = ( ∫ 10 x 2 dx + ∫ 1.25 x 3 dx − ∫ 8.59 x 2 dx)
EI 0 0 0
3 8 8
1 10 x 3 1.25 4 8.59 3
⇒ θC = ( + x − x
EI 3 0 4 0 3 0
⇒ θ C = −0.0024 rad
- rkmMurgVilRtg;cMnuc C θ A
1kN.m
CMnYsm:Um:g; virtual Ékta (1kN.m) Rtg;cMnuc A C
3m
A
8m
B
rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 3m BI C eTA A
⇒ m'= 0
smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 3m BI C eTA A
⇒ M = −10 x 2
⇒ m'.M = 0
rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 8m BI B eTA A
⇒ m' = −0.125 x
smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 8m BI B eTA A
⇒ M = −10 x 2 + 68.75 x
⇒ m'.M = 1.25 x 3 − 8.59 x 2
dUcenHθA = ∫
0
L m'.M
EI
dx
1 8 8
⇒θA = ( ∫ 1.25 x 3 dx − ∫ 8.59 x 2 dx)
EI 0 0
8 8
1 1.25 4 8.59 3
⇒θA = ( x − x )
EI 4 0 3 0
⇒ θ A = −0.00465rad
PaBdabrbs;Fñwm edayviFIfamBl 127