9. deflection using energy method

mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa

(> PaBdabedayeRbIviFIfamBl
(Deflection Using Energy method)

CMBUkenHnwgbgðajBIrebobGnuvtþviFIfamBledIm,IedaHRsaybBaðaEdlBak;B½n§nwgmMurgVil nigPaB
dab. CMBUkenHcab;epþImCamYynwgkarBiPakSaBIkmμnþ (work) nigfamBlénbMErbMrYlrageFob (strain
energy) EdlbnþedaykarbegáIteKalkarN_kmμnþ nigeKalkarN_famBl. bnÞab;mkviFIkmμnþnimitþ

(virtual work) nigRTwsþIkasÞIKøIy:aNU (Castigliano’s theorem) RtUv)anbegáIt ehIyviFIenHRtUv)aneRbI

edIm,IkMNt; bMlas;TIrbs;cMnucmYyenAelI truss, Fñwm nig eRKag.

(>!>kmμnþxageRkA nigfamBlbMErbMrYlrageFob (External work and strain energy)
viFIBak;kNþalRkaPicEdlbgðajenAkñúgCMBUkelIkmunKWmanRbsiT§PaBsMrab;rkbMlas;TI nigmMu
rgVilRtg;cMnucmYyenAelIFñwmEdlrgbnÞúkEdlmanlkçN³samBaØ. sMrab;bnÞúkEdlmanlkçN³sμúKsμaj
b¤sMrab;rcnasm<½n§dUcCa truss nigeRKag eKesñIeGayeRbIviFIfamBlsMrab;karKNnabMlas;TI nigmMurgVil.
viFIfamBlPaKeRcInEp¥kelIeKalkarN_rkSafamBl (conservation of energy principle) Edlbgðaj
fakmμnþ U e EdleFVIedaykMlaMgxageRkATaMgGs;EdlmanGMeBIenAelIeRKOgbgÁúMRtUv)anbMElgeTACakmμnþ
xagkñúg b¤famBlbMErbMerYlrageFob U i EdlRtUv)anbegáItenAeBleRKOgbgÁúMxUcRTg;RTay. RbsinebI
kMhUcRTg;RTayminFMCagEdnkMNt;eGLasÞicrbs;sMPar³ enaHfamBlbMErbMrYlrageFobeGLasÞic
(elastic stain energy) nwgeFVIeGayrcnasm<½n§RtLb;eTArksßanPaBedImenAeBlEdleKdkbnÞúkecj

vij. eKGacsMEdgeKalkarN_rkSafamBlCalkçN³KNitviTüadUcxageRkam³
Ue = Ui (9-1)
b:uEnþ munnwgbegáItviFIfamBlEdlQrelIeKalkarN_enH dMbUgeyIgRtUvkMNt;kmμnþxageRkA nig
famBlbMErbMrYlrageFobEdlbgáeLIgedaykMlaMg nigm:Um:g;. karbegáItrUbmnþenHnwgpþl;nUgmUldæan
sMrab;yl;dwgBIviFIkmμnþ nigviFIfamBldUcteTA.
kmμnþxageRkAEdlekItBIkMlaMg³ enAeBlkMlaMg F eFVIeGaymanbMlas;TI dx kñúgTisdUcnwgTisrbs;kMlaMg
kmμnþEdleFVI)anKW dU e = Fdx . RbsinebIbMlas;TIsrubKW x kmμnþnwgkøayCa
x
U e = ∫ Fdx (9-2)
0

BicarNaT§iBlEdlbgáeLIgedaykMlaMgtamG½kSEdlGnuvtþenAxagcugr)arEdlbgðajenAkñúgrUbTI 9-1a.
edaysarTMhMrbs; F ekIneLIgBItMélsUnüeTAtMélkMNt; F = P enaHsac;lUtrbs;r)arnwgkøayCa Δ .
Deflection Using Energy Method T.Chhay -281

Department of Civil Engineering NPIC

RbsinebIsMPar³rbs;r)areKarBtameGLasÞiclIenEG‘r enaH F = (P / Δ )x . edayCMnYsvaeTAkñúgsmIkar
9-2 nigedayeFVIGaMgetRkalBI 0 eTA Δ enaHeyIgnwg)an
1
Ue = PΔ (9-3)
2
EdlCaRkLaépÞRtIekaNEdlqUtenAkñúgrUbTI 9-1a.
eyIgk¾Gacsnñidæan)anfakMlaMgRtUv)anGnuvtþbnþicmþg²eTAelIr)ar ehIyTMhMrbs;vaekIneLIgCa
lkçN³bnÞat;BIsUnüeTAdl; P kmμnþEdleFVI)anesμInwgkMlaMgmFüm ¬ P / 2 ¦ KuNnwgbMlas;TI ¬ Δ ¦.

]bmafaeK)anGnuvtþkMlaMg P eTAelIr)arrYcehIy ehIyeKGnuvtþkMlaMg F ' mYyepSgeTot dUcenH
r)arxUcRTg;RTayEfmeTotedayTMhM Δ' ¬rUbTI 9-1b¦. enaHkmμnþEdleFVIeday P ¬minEmn F ' ¦ enA
eBlr)arrgnUvkMhUcRTg;RTay Δ' EfmeTotKW
U e ' = PΔ' (9-4)
enATIenHkmμnþCaRkLaépÞctuekaNqUtenAkñúgrUbTI 9-1b. enAkñúgkrNIenH P min)anpøas;bþÚrRbEvgrbs;
r)areT eday Δ' RtUv)anbgáeLIgedaysarEt F ' b:ueNÑaH. dUcenH kmμnþrbs;vaesμInwgplKuNrvagkMlaMg
P nigbMlas;TI Δ ' .
PaBdabedayeRbIviFIfamBl T.Chhay -282


Casegçb enAeBleKGnuvtþkMlaMg P eTAelIr)ar ehIybnÞab;mkGnuvtþkMlaMg F ' kMlaMgsrubEdl
eFVI)anedaykMlaMgTaMgBIrKWsMEdgedayRkLaépÞRtIekaN ACE enAkñúgrUbTI 9-1b. RkLaépÞRtIekaN
ABG CakmμnþénkMlaMg P EdlbgáeLIgedaybMlas;TI Δ rbs;kMlaMgenH/ RtIekaN BCD CakmμnþénkMlaMg

F ' edaysarkMlaMgenHbgáeGaymanbMlas;TI Δ ' ehIycugeRkay RkLaépÞctuekaN BDEG Cakmμnþ

bEnßmEdleFVIedaykMlaMg P enAeBlbMlas;TI Δ' bgáeLIgedaykMlaMg F ' .
kmμnþxageRkAEdlekItBIm:Um:g;³ kmμnþénm:Um:g;RtUv)ankMNt;edayplKuNén
TMhMénm:Um:g; M nigmMu dθ Edlviledaysarm:Um:g;enaH Edl dU e = Mdθ /
rUbTI 9-2. RbsinebImMusrubrbs;mMurgVilKWθ ra:düg; enaH kmμnþkøayCa
θ
U e = ∫ Mdθ (9-5)
0

dUcenAkñúgkrNIkMlaMg RbsinebIeKGnuvtþm:Um:g;bnþicmþg²eTAelIrcnasm<½n§
EdlmanlkçN³eGLasÞic lIenEG‘rBIsUnüeTA M enaHkmμnþesμInwg
1
Ue = Mθ (9-6)
2
b:uEnþ RbsinebIeKGnuvtþm:Um:g;eTAelIrcnasm<½n§rYcehIy ehIykarGnuvtþm:Um:g;epSgeToteFVIeGayrcnasm<½n§
xUcRTg;RTayedaybrimaNθ ' enaHkmμnþénm:Um:g; M KW
U e ' = Mθ ' (9-7)
famBlbMErbMrYlrageFobedaysarbnÞúktamG½kS³ enAeBleKGnuvtþkMlaMgtam
G½kS N bnþicmþg²eTAelIr)arenAkñúgrUbTI 9-3 vanwgeFVIeGaysMPar³manbMEr
bMrYlrageFobEdlkmμnþxageRkAEdleFVIeday N Rtuv)anbMElgeTACafamBl
bMErbMrYlrageFobEdlRtUv)anrkSaTukenAkñúgr)ar ¬smIkar 9-1¦. c,ab;h‘Uk
σ = Eε mann½ykñúgkrNIsMPar³manlkçN³eGLasÞiclIenEG‘r ehIyRbsin
ebIr)armanmuxkat; A efr nigRbEvg L ehIykugRtaMgEkgKW σ = N / A nigbMEr
bMrYlrageFobcugeRkayKW ε = Δ / L . Cavi)ak N / A = E (Δ / L) ehIyPaBdab
cugeRkayKW
NL
Δ= (9-8)
AE
edayCMnYsvaeTAkñúgsmIkar 9-3 eday P = N dUcenHfamBlbMErbMrYlrageFobenAkñúgr)arKW
N 2L
Ui = (9-9)
2 AE



famBlbMErbMrYlrageFobedaysarm:Um:g;Bt;³ BicarNaFñwmEdlbgðajenAkñúgrUbTI 9-4a EdlxUcRTg;
RTayedaysarkarGnuvtþbnÞúk P nig w bnþicmþg². bnÞúkTaMgenHbegáItm:Um:g;Bt; M enAkñúgFñwmenARtg;
muxkat;EdlmancMgay x BITMrxageqVg. eKGackMNt;mMurgVilrbs;FatuGnnþtUc dx ¬rUbTI 9-4b¦ BIsmIkar
8-2 Edl dθ = (M / EI )dx . Cavi)ak eKkMNt;famBlbMErbMrYlrageFob b¤kmμnþEdlrkSaTukenAkñúg

Ggát;BIsmIkar 9-6 edaysarm:Um:g;Bt;ekIteLIgbnþicmþg². dUcenH
M 2 dx
dU i = (9-10)
2 EI
eKkMNt;famBlbMErbMrYlrageFobsMrab;FñwmedayeFVIGaMgetRkaleTAelIsmIkar 9-10 elIRbEvg L
rbs;Fñwm. lT§plKW
L M 2 dx
Ui = ∫ (9-11)
0 2 EI

(>@> eKalkarN_kmμnþ nigfamBl (Principle of work and energy)
bnÞab;BIeyIg)ankMNt;rUbmnþsMrab;kmμnþ nigfamBlbMErbMrYlrageFobsMrab;kMlaMg nigm:Um:g; rYc
ehIy eyIgnwgbgðajBIrebobGnuvtþkarrkSafamBl b¤eKalkarN_kmμnþ nigfamBledIm,IkMNt;bMlas;TI
enARtg;cMNucmYyenAelIeRKOgbgÁúM. BicarNakñúgkarkMNt;bMlas;TI Δ enARtg;cMNucEdlkMlaMg P Gnuvtþ
eTAelIFñwmkugs‘ulenAkñúgrUbTI 9-5. BIsmIkar 9-3 kmμnþxageRkAKWU e = PΔ / 2 . edIm,ITTYl)anfam-
BlbMErbMrYlrageFob dMbUgeyIgRtUvkMNt;m:Um:g;Bt;CaGnuKmn_eTAnwgTItaMg x enAkñúgFñwm bnÞab;mkGnuvtþ
smIkar 9-11. enAkñμúgkrNI M = − Px dUcenH
L (− Px ) dx
2
L M 2 dx 1 P 2 L3
Ui = ∫ =∫ =
0 2 EI 0 2 EI 6 EI
edayeGaykmμnþxagkñúgesμInwgfamBlbMErbMrYlrageFobxagkñúgesμIKña ehIyedayedaHRsayrkbMlas;TI
Δ / eyIg)an

Ue = Ui



1 1 P 2 L3
PΔ =
2 6 EI
3
PL
Δ=
3EI
eTaHbICaeyIgGacTTYl)andMeNaHRsayedaypÞal;k¾eday EteKGacGnuvtþviFIenHsMrab;EtcMeNaTeRCIs
erIsxøHb:ueNÑaH. eKRtUvcMNaMfa sMrab;viFIenHeKGacGnuvtþbnÞúk)anEtmYyb:ueNÑaHeTAelIeRKOgbgÁúM BIeRBaH
RbsinebImanbnÞúkeRcInCagmYymanGMeBIelIeRKOgbgÁúM enaHvanwgekItmanbMlas;TIEdlCaGBaØateRkamGMeBI
énbnÞúknImYy² ehIyEdleKmanlT§PaBsresrsmIkarkmμnþEtmYysMrab;Fñwm. elIsBIenH eKGacTTYl
)anEtbMlas;TIeRkamGMeBIbnÞúk edaysarkmμnþxageRkAGaRs½ynwgkMlaMg nigbMlas;TIEdlRtUvnwgkMlaMg
enaH. viFIEtmYyedIm,ICMnHnUvkarkMritenHKWeRbIviFIkmμnþnimitþ b¤RTwsþIkasÞIKøIy:aNU EdleyIgnwgBnül;BIviFI
TaMgBIrenHenAkñúgkfaxNÐbnþbnÞab;.

(>#> eKalkarN_kmμnþnimitþ (Principle of virtual work)
eKalkarN_kmμnþnimitþRtUv)anbegáIteLIgeday cn Eb‘nuyI (John Bernoulli) kñúgqñaM 1717
ehIyEdleBlxøHeKeGayeQμaHvafaviFIkMlaMgÉktþa. vapþl;nUvmeFüa)ayTUeTAedIm,ITTYl)anbMlas;TI
nigmMurgVilenARtg;cMNucCak;lak;enAelIeRKOgbgÁúM EdlGacCaFñwm eRKag b¤ truss.
munnwgbegáIteKalkarN_kmμnþnimitþ eKcaM)ac;RtUveFVIBMenalTUeTAxøHGMeBIeKalkarN_kmμnþ nigfam-
Bl Edl)anerobrab;enAkñúgkfaxNÐelIkmun. RbsinebIeyIgmanrcnasm<½n§EdlGacxUcRTg;RTayEdl
manrUbragTUeTANamYy ehIyGnuvtþes‘rIénbnÞúkxageRkA P eTAelIva vanwgekItmanbnÞúkxagkñúg u enAkñúg
rcnasm<½n§TaMgmUl. eKcaM)ac;eGaybnÞúkxageRkA nigbnÞúkxagkñúgmanTMnak;TMngKñaedaysmIkarlMnwg.
edaysarbnÞúkTaMgenH bMlas;TIxageRkA Δ nwgekItmanRtg;bnÞúk P ehIybMlas;TIxagkñúg δ nwgekItman
enARtg;cMnucnImYy²énbnÞúkxagkñúg u . CaTUeTA bMlas;TITaMgenHmincaM)ac;manlkçN³eGLasÞiceT ehIy
BYkvaGacminmanTMnak;TMngeTAnwgbnÞúkeT b:uEnþbMlas;TIxagkñúg nigbMlas;TIxageRkARtUvEtmanTMnak;TMng
KñaedayPaBRtUvKñaénbMlas;TI (compatibility of displacements). eKGacniyaymü:ageTotfa Rbsin



ebIeKsÁal;bMlas;TIxageRkA enaHeKGackMNt;blas;TIxagkñúgEdlRtUvKña. CaTUeTA eKalkarN_kmμnþ nig
M
famBlsMEdgfa³
∑ PΔ = ∑ uδ (9-12)

kmμnþénkMlaMgxageRkA kmμnþénkMlaMgxagkñúg
edayEp¥kelIKMnitenH eKGacbegáIteKalkarN_kmμnþnimitþ. eyIgnwgBicarNarcnasm<½nþ ¬b¤GgÁ-
Fatu¦ EdlmanragTUeTAdUcbgðajenAkñúgrUbTI 9-6b . ]bmafa eyIgcaM)ac;RtUvkMNt;bMlas;TI Δ rbs;
*

cMNuc A enAelIGgÁFatuEdlbgáeLIgedaybnÞúkBit P1 / P2 nig P3 . eKRtUvdwgfabnÞúkTaMgenHminbgá
eGayTMrcl½t b:uEnþCaTUeTABYkvaGaceFVIeGaysMPar³xUcRTg;RTayputBIEdneGLasÞic. edaysarminman
bnÞúkxageRkAmanGMeBIeTAelIGgÁFatuenARtg;cMnuc A nigkñúgTisén Δ enaHeKGackMNt;bMlas;TIeday
dak;bnÞúknimitþ (virtual load) enAelIGgÁFatuEdlbnÞúk P' enHmanGMeBIenAkñúgTisedAdUcKñanwg Δ ¬rUbTI 9-
6a¦. edIm,IPaBgayRsYl ¬EdleXIjenAeBleRkay¦ eyIgnwgeRCIserIs P' CabnÞúkEdlmanTMhMmYy
Éktþa Edl P'= 1 . BakünimitþEdleRbICamYynwgbnÞúk P' edaysarvaminEmnCabnÞúkBit ehIyvak¾min
EmnCaEpñkénbnÞúkBit. bnÞúkÉktþa P' begáItCabnÞúkkñúgÉktþa u enAkñúgFatuEdlCatMNag b¤srésrbs;
GgÁFatuEdlbgðajenAkñúgrUbTI 9-6a. enATIenHeKtMrUveGay P' nig u manTMnak;TMngeTAnwgsmIkarlMnwg . †

enAeBleKdak;bnÞúknimitþ bnÞab;mkeKGnuvtþbnÞúkBit P1 / P2 nig P3 ¬rUbTI 9-6b¦. cMnuc A eFVI
eGaymanbMlas;TI Δ EdleFVIeGayFatuxUcRTg;RTayedayTMhM dL . CalT§pl kMlaMgnimitþxageRkA P'

*
ragTUeTAmanragCa truss/ Fñwm b¤eRKag.
†
eTaHbICabnÞúkTaMgenHbgáeGaymanbMlas;TInimitk¾eday k¾eyIgnwgminKitBITMhMrbs;vaEdr.
þ


nigbnÞúknimitþxagkñúg u manGMeBItamTisrbs;bMlas;TI Δ nig dL erogKña dUcenHvaeFVIeGaymankmμnþnimitþ
xageRkA 1 ⋅ Δ enAelIGgÁFatu nigeFVIeGaymankmμnþnimitþxagkñúg u ⋅ dL enAelIFatu. edaydwgfakmμnþ
nimitþxageRkAesμInwgkmμnþnimitþxagkñúgEdleFVIenAelIRKb;FatuTaMgGs;rbs;GgÁFatu eyIgGacsresr
smIkarkmμnþnimitþCa³
bnÞúknimitþ
1 ⋅ Δ = ∑ u ⋅ dL (9-13)
bMlas;TIBitR)akd
Edl P' = 1 = bnÞúknimitþÉktþaxageRkAEdlmanGMeBItamTisedArbs; Δ
u = bnÞúknimitþxagkñúgEdlmanGMeBIenAelIFatukñúgTisedArbs; dL

Δ = bMlas;TIxageRkAEdlbgáedaybnÞúkBit

dL = kMhUcRTg;RTayxagkñúgrbs;FatuEdlbgáedaybnÞúkBit

edayeRCIserIs P' = 1 eyIgGaceXIjfadMeNaHRsayrbs; Δ manlkçN³edaypÞal; edaysar
Δ = ∑ udL .

enAkñúgrebobdUcKña RbsinebIeKRtUvkarkMNt;mMurgVil b¤CMralénbnÞat;b:HenARtg;cMNucenAelIrcna
sm<½n§ eKRtUvGnuvtþm:Um:g;nimitþ M ' EdlmanTMhMmYyÉktþa. Cavi)ak m:Um:g;xageRkAbegáItbnÞúknimitþ uθ
enAkñúgFatumYyrbs;GgÁFatu. edaysnμt;fabnÞúkBiteFVIeGayFatuxUcRTg;RTayedaybrimaN dL eKGac
kMNt;mMurgVil θ BIsmIkarkmμnþnimitþ
bnÞúknimitþ
1 ⋅θ = ∑ uθ ⋅ dL (9-14)
bMlas;TIBitR)akd
Edl M ' = 1 = m:Um:g;nimitþÉktþaxageRkAEdlmanGMeBItamTisedArbs; θ
uθ = bnÞúknimitþxagkñúgEdlmanGMeBIenAelIFatukñúgTisedArbs; dL

θ = bMlas;TIxageRkAEdlbgáedaybnÞúkBit
dL = kMhUcRTg;RTayxagkñúgrbs;FatuEdlbgáedaybnÞúkBit

viFIeKalkarN_kmμnþnimitþCaerOy²eKehAfaviFIkMlaMgnimitþ
edaysareKGackMNt;bMlas;TIBitedayeRbIkMlaMgnimitþ. smIkarkmμnþnimitþenAkñúgkrNIenHCatMrUvkar
RtUvKñaénbMlas;TIsMrab;eRKOgbgÁúM. eTaHbICaenATIenH vaminsMxan;k¾eday eyIgRtUvdwgfaeyIgk¾Gac
GnuvtþeKalkarN_kmμnþnimitþCaviFIbMlas;TInimitþk¾)anEdr. enAkñúgkrNIenHeKeGayrcnasm<½nmanbMlas;



TInimitþ xN³EdlvargbnÞúkBit. eKGaceRbIviFIenHedIm,IkMNt;kMlaMgenAelI b¤enAkñúgrcnasm<½n§ EdleK
*

GacsMEdgsmIkarkmμnþnimitþCatMrUvkarénsmIkarlMnwg.
(>$> viFIkmμnþnimitþsMrab; Trusses (Method of virtual work: Trusses)
eyIgGaceRbIviFIkmμnþnimitþedIm,IkMNt;bMlas;TIrbs;tMN truss enAeBl truss rgbnÞúkxageRkA
bMErbMrYlsItuNðPaB b¤kMhusqÁgkñúgkarplit. eyIgnwgsikSaBIsßanPaBnImYy²enATIenH.
bnÞúkxageRkA³ sakl,gBicarNabMlas;TIbBaÄr Δ rbs;tMN
B rbs; truss enAkñúgrUbTI 9-7. enATIenH Ggát;mYyrbs;

truss KMrUmanRbEvg L . RbsinebIbnÞúk P nig P2 manGMeBIelI
1

truss EdleRbIsMPar³EdlmanlkçN³eGLasÞiclIenEG‘r
enaHGgát;enHnwgxUcRTg;RTayedaybrimaN ΔL = NL / AE
Edl N CakMlaMgEkg b¤kMlaMgtamG½kSenAkñúgGgát; EdlbgáedaykMlaMg. edayGnuvtþsmIkar 9-13 dUc
enHsmIkarkmμnþnimitþsMrab; truss KW
nNL
1⋅ Δ = ∑ (9-15)
AE
Edl 1=kMlaMgnimitþÉktþaxageRkAEdlmanGMeBIelItMN truss enAkñúgTisedArbs; Δ .
n = kMlaMgEkgnimitþkñúgrbs;Ggát; truss EdlbgáeLIgedaybnÞúknimitþÉktþaxageRkA.

Δ = bMlas;TIrbs;tMNEdlbgáeLIgedaybnÞúkBitEdlmanGMeBIelI truss.

L = RbEvgrbs;Ggát;.

A = RkLaépÞmuxkat;rbs;Ggát;.

E = m:UDuleGLasÞicrbs;Ggát;.

karbegáItsmIkarenHGnuvtþtamkarerobrab;enAkñúgkfaxNÐ 9-3. enATIenHbnÞúknimitþÉktþabegáIt
kMlaMgnimitþxagkñúg n enAkñúgGgát; truss nImYy². bnÞab;mkkMlaMgBiteFVIeGaytMN truss pøas;TI Δ kñúgTis
edAdUcKñanwgkMlaMgnimitþÉktþa ehIyGgát;nImYy²pøas;TIedayTMhM NL / AE kñúgTisedAdUcKñanwgTisrbs;
kMlaMg n . Cavi)ak kmμnþ 1 ⋅ Δ esμInwgkmμnþnimitþxagkñúg b¤famBlbMErbMrYlrageFob¬nimitþ¦xagkñúgEdl
rkSaenAkñúgGgát; truss TaMgGs; Edl ∑ nNL / EA .
sItuNðPaB³ enAkñúgkrNIxøH Ggát; truss Gacpøas;bþÚrRbEvgrbs;BYkvaedaysarkMedA. RbsinebI α Caem
KuNlUtedaysarkMedAsMrab;Ggát; ehIy ΔT CabMErbMrYlsItuNðPaB enaHbMErbMrYlRbEvgrbs;Ggát;KW
*
eKeRbIrebobenHenAkñúgkfaxNÐ 6.3 CamYynwgeKalkarN_m‘uyeyeRbsøÚ.


ΔL = α ⋅ ΔT ⋅ L. dUcenHeyIgGackMNt;bMlas;TIrbs;tMN truss eRCIserIsNamYyEdlbNþalBIbMEr
bMrYlsItuNðPaBBIsmIkar 9-13 EdlGacsresrCa
1 ⋅ Δ = ∑ nαΔT ⋅ L (9-16)
Edl 1=kMlaMgnimitþÉktþaxageRkAEdlmanGMeBIelItMN truss enAkñúgTisedArbs; Δ .

Δ = bMlas;TIrbs;tMNEdlbgáeLIgedaybMErbMrYlsItuNðPaB.


α = emKuNlUtedaysarkMedA.
ΔT = bMErbMrYlsItuNðPaBrbs;Ggát;.

kMhusqÁgkñúgkarplit nigPaBekag³ eBlxøH kMhusqÁgkñúgkarplitGacekItmaneLIgcMeBaHRbEvgrbs;
Ggát;rbs; truss. ehIykñúgkrNIxøH eKplitGgát;rbs; truss manRbEvgEvg b¤xøICagbnþicedIm,IeFVIeGay
truss manPaBekag. CaerOy² eKRtUvkarPaBekagsMrab; truss s<anEdleKeFVIeGayGgát;xageRkamrbs;

truss ekageLIgelICamYynwgTMhMsmmUleTAnwgPaBdabrbs;vaEdlekItBIbnÞúkGefrrbs;s<an. Rbsin

ebIGgát; truss xøI b¤EvgCagGVIEdlcg;)an eKGackMNt;bMlas;TIrbs;tMN truss BITItaMgrMBwgTukedaypÞal;
edayGnuvtþsmIkar 9-13 EdlGacsresrCa
1 ⋅ Δ = ∑ nΔL (9-17)

Edl 1= kMlaMgnimitþÉktþaxageRkAEdlmanGMeBIelItMN truss enAkñúgTisedArbs; Δ .

Δ = bMlas;TIrbs;tMNEdlbgáeLIgedaykMhusqÁgkñúgkarplit.

ΔL = bMErbMrYlRbEvgrbs;Ggát;BITMhMEdlcg;)anrbs;vaEdlbNþalBIkMhusqÁgkñúgkarplit.

eKcaM)ac;RtUveFVIplbUkénGgÁxagsþaMénsmIkar 9-15 rhUtdl; 9-17 RbsinebIbnÞúkxageRkAmanGMeBI
eTAelI truss ehIyGgát;rbs; truss xøHrgnUvbMErbMrYlsItuNðPaB b¤mankMhusqÁgkñúgkarplit.
dMeNIrkarkñúgkarviPaK (Procedure for analysis)
CMhanxageRkampþl;nUvviFIsMrab;kMNt;bMlas;TIrbs;tMNén truss edayeRbIviFIkmμnþnimitþ
kMlaMgnimitþ n
dak;kMlaMgmYyÉktþaenAelI truss enARtg;tMNEdlRtUvkarkMNt;rkbMlas;TI. bnÞúkenHRtUvman
TisedAdUcnwgTisedArbs;bMlas;TI eBalKWedk b¤Qr.


CamYynwgbnÞúkÉktþaEdldak;elI truss eKRtUvdkbnÞúkBitEdlmanGMeBIelI truss ecjTaMgGs;
rYceRbIviFItMN b¤viFImuxkat;edIm,IKNnakMlaMgkñúg n enAkñúgGgát; truss nImYy². snμt;fakMlaMg
TajviC¢man nigkMlaMgsgát;GviC¢man.
kMlaMgBit N
eRbIviFImuxkat; b¤viFItMNedIm,IkMNt;kMlaMg N enAkñúgGgát;nImYy². kMlaMgTaMgenHbgáeLIgeday
sarEtkMlaMgBitmanGMeBIelI truss Etb:ueNÑaH. dUcKña eKRtUvsnμt;fakMlaMgTajviC¢man ehIy
kMlaMgsgát;GviC¢man.
smIkarkmμnþnimitþ
GnuvtþsmIkarkmμnþnimitþedIm,IkMNt;bMlas;TIEdlcg;)an. eKcaM)ac;rkSaPaBdUcKñaénkarsnμt;
sBaØaBiCKNitrbs;kMlaMg n nigkMlaMg N enAeBlCMnYsvaeTAkñúgsmIkar.
RbsinebI ∑ nNL / EA viC¢man bMlas;TI Δ manTisedAdUcTisedArbs;bnÞúkÉktþa. RbsinebI
lT§plmantMélGviC¢man Δ nwgmanTisedApÞúynwgTisedArbs;bnÞúkÉktþa.
enAeBlGnuvtþ1⋅ Δ = ∑ n ⋅α ⋅ ΔT ⋅ L eKRtUvdwgfaRbsinebIGgát;nImYy²rgnUvkMeNInsItuNðPaB
enaH ΔT nwgviC¢man b:uEnþebIsItuNðPaBEdlfycuHvanwgeFVIeGay ΔT GviC¢man.
sMrab; 1 ⋅ Δ = ∑ n ⋅ ΔL enAeBlEdlkMhusqÁgkñúgkarpliteFVIeGayRbEvgrbs;Ggát;EvgCagmun
ΔL nwgviC¢man b:uEnþebIRbEvgrbs;vaxøICagmun enaH ΔL nwgGviC¢man.

]TahrN_ 9-1³ kMNt;bMlas;TIbBaÄrrbs;tMN C én truss EdkEdlbgðajenAkñúgrUbTI 9-8a. RkLa
épÞmuxkat;rbs;Ggát;nImYy²KW A = 300mm 2 ehIy E = 200GPa .

dMeNaHRsay³
kMlaMgnimitþ n³ eKdak;EtbnÞúk1kN enARtg;tMNC ehIyeKedaHRsaykMlaMgenAkñúgGgát;nImYy²edayeRbI
viFItMN. lT§plRtUv)anbgðajenAkñúgrUbTI 9-8b. elxviC¢manbgðajkMlaMgTaj ehIyelxGviC¢man
bgðajkMlaMgsgát;.


kMlaMgBit N³ kMlaMgBitenAkñúgGgát;RtUv)anKNnaedayeRbIviFItMN. lT§plRtUv)anbgðajenAkñúgrUbTI
9-8c.
smIkarkmμnþnimitþ³ edayerobcMTinñn½yenAkñúgtarag eyIg)an

2
⋅m
dUcenH 1kN ⋅ Δ C = ∑
nNL 369.6kN
AE
=
v
AE
edayCMnYstMélCaelxsMrab; A nig E eyIg)an
396.6kN 2 ⋅ m
1kN ⋅ Δ C v =
[300(10 )m ][200(10 )kN / m ]
−6 2 6 2

Δ C v = 0.00616m = 6.16mm

]TahrN_ 9-2³ RkLaépÞmuxkat;rbs;Ggát;nImYy²rbs; truss EdlbgðajenAkñúgrUbTI 9-9a KW A =
400mm 2 ehIy E = 200GPa . (a) kMNt;bMlas;TItamTisQrrbs;tMN C RbsinebIeKdak;kMlaMg
4kN eTAelI truss Rtg;tMN C. (b) RbsinKμanbnÞúkmanGMeBIelI truss cUrkMNt;bMlas;TItamTisQr

rbs;tMN C RbsinebIGgát; AB manRbEvgxøICamun 5mm .



dMeNaHRsay³
(a)
kMlaMgnimitþ n³ edaysareKRtUvkarrkbMlas;TItamTisQrrbs;tMNC dUcenHeKRtUvdak;kMlaMgnimitþ1kN
enARtg;tMNC edaymanTisQr. xñatrbs;kMlaMgRtUvdUcKñanwgxñatrbs;kMlaMgBit. eKRtUvKNnakMlaMg
RbtikmμenARtg;tMN A nigtMN B nigkMNt;kMlaMg n enAkñúgGgát;nImYy²edayviFItMNdUcbgðajenAkñúg
düaRkamGgÁesrIrbs;tMN A nigtMN B ¬rUbTI 9-9b¦.
kMlaMgBit N³ karviPaKtMN A nigtMN B enAeBleKdak;kMlaMgBit 4kN eTAelI truss RtUv)aneGayenA
kñúgrUbTI 9-9c.
smIkarkmμnþnimitþ³ edaysar AE efr eKGacKNna nigerobtY nNL enAkñúgtarag. enATIenHsBaØaviC¢man
bgðajfakMlaMgTaj nigsBaØaGviC¢manbgðajkMlaMgGviC¢man.

nNL 10.67kN ⋅ m 2
dUcenH 1kN ⋅ Δ C = ∑
AE
= v
AE
edayCMnYstMél A = 400mm = 400(10 )m nig E = 200GPa = 200(106 )kN / m 2 eyIg)an
2 −6 2

10.67 kN 2 ⋅ m
1kN ⋅ Δ C v =
[400(10 )m ][200(10 )kN / m ]
−6 2 6 2

Δ C v = 0.000133m = 0.133mm
(b)
enATIenH eyIgRtUvGnuvtþsmIkar 9-17. edaysareKRtUvkarkMNt;bMlas;TItamTisQrRtg;tMN C eyIg
GaceRbIlT§plrbs;rUbTI 9-7b. manEtGgát; AB b:ueNÑaHEdlrgbMErbMrYlRbEvg eBalKW ΔL = −5mm .
dUcenH
1 ⋅ Δ = ∑ nΔL
1kN ⋅ Δ C v = (0.667 kN )(− 0.005m )

Δ C v = −0.00333m = −3.33mm

sBaØadkbgðajfatMN C bMlas;TIeLIgelI EdlpÞúyBIkMlaMgbBaÄr1kN . cMNaMfa RbsinebIKitTaMgkMlaMg
4kN nigkMhusqÁgkñúgkarplitbBa©ÚlKña enaHbMlas;TIsrubKW Δ C = 0.133 − 3.33 = −3.20mm ¬eLIg
v

elI¦.



]TahrN_ 9-3³ kMNt;bMlas;TIbBaÄrrbs;tMN C én truss EdkEdlbgðajenAkñúgrUbTI 9-10a.
edaysarkMedAEdlPayecjBICBa¢aMg Ggát; AD rgnUvkMeNInsItuNðPaB ΔT = +60o C . yk α =
1.08(10 −5 )/ o C nig E = 200GPa . RkLaépÞmuxkat;rbs;Ggát;nImYy²RtUv)anbgðajenAkñúgrUb.

dMeNaHRsay³
kMlaMgnimitþ n³ eKdak;kMlaMgbBaÄr 1kN eTAelI truss enARtg;tMN C ehIyKNnakMlaMgenAkñúgGgát;
¬rUbTI 9-10b¦.
kMlaMgBit N³ edaysarkMlaMg n enAkñúgGgát; AB nigGgát; BC esμIsUnü enaHeKmincaM)ac;kMNt;kMlaMg
N enAkñúgGgát;TaMgenaHeT. ehtuGVI? EtsMrab;karKNnaeBjelj karviPaKkMlaMgkñúgrbs;Ggát;TaMg

Gs;EdlekItBIkMlaMgBitxageRkARtUv)anbgðajenAkñúgrUbTI 9-10c.
smIkarkmμnþnimitþ³ bnÞúk nigkMedAeFVIeGay truss xUcRTg;RTay dUcenHeKRtUvKitsmIkar 9-15 nigsmIkar
9-18 bBa©ÚlKña. edaybMEbkxñateGay)anRtwmRtUv eyIg)an
nNL
1⋅ Δ Cv = ∑ + ∑ nαΔTL
AE
0.75(600 )(1.8) (1)(400)(2.4)
Δ Cv =
( )[ ( )] ( )[ ( )]
+
1200 10 − 6 200 10 6 1200 10 − 6 200 10 6

( )[ ( )] [ ( )]
+
(− 1.25)(− 500)(3) + (1)1.08 10 − 5 (60)(2.4)
900 10 − 6 200 10 6
Δ C v = 0.00193m = 19.3mm



(>%> viFIkmμnþnimitþsMrab;Fñwm nigeRKag (Method of virtual work: beams and frames)
eKk¾GaceRbiviFIkmμnþnimitþedIm,IkMNt;PaBdabrbs;Fñwm nigeRKag. edaysarbMErbMrYlrageFob
EdlekIteLIgedaysarkarBt;CaktþasMxan;kñúgkareFVIeGayFñwm nigeRKagmanPaBdab dMbUgeyIgniwg
BiPakSaBIT§iBlenH. eyIgnwgBiPakSaBIPaBdabEdlbNþalBIkMlaMgkat; kMlaMgtamG½kS kMlaMgrmYl
nigbMErbMrYlsItuNðPaBenAkñúgkfaxNÐ 9-6.

tamry³eKalkarN_kmμnþnimitþ b¤viFIkMlaMgnimitþ eKGacbegáItrUbmnþsMrab;PaBdabrbs;Fñwm nig
PaBrbs;eRKagedayBicarNaelIFñwmEdl)anbgðajenAkñúgrUbTI 9-11a. enATIenH eKRtUvkarkMNt;
bMlas;TI Δ rbs;cMNuc A . edIm,IkMNt; Δ eKRtUvdak;kMlaMgÉktþaEdlmanGMeBIenAelIFñwmRtg;cMNuc A
tamTisedArbs; Δ ehIyeKRtUvkMNt;m:Um:g;Bt;nimitþ m edayviFImuxkat;enARtg;TItaMg x NamYyBITMrxag
eqVg ¬rUbTI 9-11b¦. enAeBlkMlaMgBitmanGMeBIelIFñwm ¬rUbTI 9-11a¦ cMnuc A pøas;TI)an Δ . kñúg
krNIEdlkMlaMgTaMgenHeFVIeGaysMPar³eFVIkarenAkñúgtMbn;eGLasÞiclIenEG‘r enaHBIsmIkar 8-2 Fatu
dx xUcRTg;RTay b¤vil)an dθ = (M / EI )dx *. enATIenH M Cam:Um:g;Bt;enARtg; x EdlbgáeLIgeday

kMlaMgBit. dUcenH kmμnþnimitþxageRkAEdleFVIedaykMlaMgÉktþaKW 1 ⋅ Δ ehIykmμnþnimitþxagkñúgEdleFVI
edaym:Um:g; m KW m ⋅ dθ = m(M / EI )dx . plbUkénT§iBlenAelIFatu dx TaMgGs;tambeNþayFñwmtMrUv
eGaymankareFVIGaMgetRkal dUcenHsmIkar 9-13 køayCa
L mM
1⋅ Δ = ∫ dx (9-18)
0 EI

Edl 1= kMlaMgnimitþÉktþaxageRkAEdlmanGMeBIenAelIFñwm b¤eRKagenAkñúgTisedArbs; Δ .

*
rMlwkfa RbsinebIsMPar³manbMErbMrYlrageFobhYsEdneGLasÞicrbs;va eKk¾GacenAEteRbIeKalkarN_kmμnþnimitþ)anEdr eTaH
bICaenAkñúgenHeKRtuvkareRbIkarviPaK nonlinear b¤)øasÞick¾eday.


m=m:Um:g;Bt;nimitþenAkñúgFñwm b¤eRKag EdlsresrCaGnuKmn_én x nigEdlbgáeLIgedaybnÞúk
nimitþÉktþaxageRkA.
Δ = bMlas;TIrbs;cMNucEdlbgáeLIgedaybnÞúkBitEdlmanGMeBIelIFñwm b¤eRKag.

M = m:Um:g;Bt;enAkñúgFñwm b¤eRKag EdlRtUv)ansresrCaGnuKmn_én x ehIyvabgáeLIgeday

kMlaMgBit.
I = m:Um:g;niclPaBénRkLaépÞmuxkat; EdlRtUv)anKNnaeFobnwgG½kSNWt.


tamrebobdUcKña RbsinebIeKRtUvkarkMNt;mMurgVil θ enARtg;cMNucNamYyrbs;Fñwm eKRtUvdak;m:Um:g;mYy
ÉktþaenARtg;cMNucenaH ehIyeKRtUvkMNt;m:Um:g;Bt; mθ EdlRtUvKña. edaysarkmμnþénm:Um:g;mYyÉktþaKW
L mθ M
1⋅θ = ∫ dx (9-19)
0 EI
enAeBlGnuvtþsmIkar 9-18 nig 9-19 eKcaM)ac;RtUvdwgfaGaMgetRkalkMNt;énGgÁxagsþaMCabri-
maNénfamBlbMErbMrYlragnimitþEdlrkSaTukenAkñúgFñwm. RbsinebIkMlaMgcMcMNuc b¤m:Um:g;xageRkAeFVIGMeBI
enAelIFñwm b¤k¾bnÞúkBRgaymanlkçN³minCab; eKminGaceFVIGaMgetRkalEtmYyenAelIRbEvgTaMgmUlrbs;
Fñwm)aneT KWeKRtUveRCIserIskUGredaen x dac;edayELkenAkñúgtMbn;EdlKμanbnÞúkEdlmanlkçN³minCab;
ehIykUGredaenrbs; x nImYy²mincaM)ac;mancMnuceKalEtmYyeT b:uEnþkUGredaenrbs; x EdleRCIserIs
sMrab;kMNt;m:Um:g;Bt; M BitRtUvEtdUcKñanwgkUGredaen x EdleRCIserIssMrab;kMNt;m:Um:g;Bt; nimitþ m b¤ mθ
enAkñúgtMbn;dUcKña. ]TahrN_ BicarNaFñwmEdlbgðajenAkñúgrUbTI 9-12. edIm,IkMNt; bMlas;TIrbs; D
eKRtUvBicarNabYntMbn; dUcenHeKRtUvedaHRsayGaMgetRkalEdlmanTMrg; ∫ (mM / EI )dx . eyIgGac
eRbI x1 edIm,IkMNt;famBlbMErbMrYlrageFobenAkñúgtMbn; AB / x2 sMrab;tMbn; BC / x3 sMrab;tMbn; DE nig
x4 sMrab;tMbn; DC . enARKb;krNITaMgGs; eKeRCIserIs x y:agNaedIm,IkMNt; M nig m ¬b¤ mθ ¦ y:ag

gay.



GaMgetRkaledayeRbItarag³ enAeBlrcnasm<½n§rgbnÞúkEdlmanlkçN³samBaØ ehIydMeNaHRsay
sMrab;bMlas;TIminRtUvkarGaMgetRkaleRcIn eKGaceRbIviFItamtaragedIm,IedaHRsayGaMgetRkal. edIm,I
eFVIEbbenH dMbUgeKRtUvKUrdüaRkamm:Um:g;rbs;Ggát;nImYy²sMrab;bnÞúkBit nigbnÞúknimitþ. edaypÁÚpÁgdüa
RkamsMrab; m nig M CamYynwgtaragsMrab;KNna ∫0L mm' dx xageRkam eKGackMNt;GaMgetRkal
∫ mMdx BIrUbmnþsmRsb. ]TahrN_ 9-6 nig 9-8 bgðajBIkarGnuvtþviFIenH.

eKGaceRbICMhanxageRkamedIm,IkMNt;bMlas;TI nig/b¤mMurgVilenARtg;cMNucmYyenAelIExSekag
eGLasÞicrbs;Fñwm b¤eRKagedayeRbIviFIkmμnþnimitþ.
m:Um:g;Bt;nimitþ m b¤ m θ

dak;bnÞúkÉktþaenAelIFñwm b¤eRKagenARtg;cMNucEdleyIgRtUvkarrkbMlas;TIedaymanTisdUcKña.
RbsinebIeKRtUvkarkMNt;mMurgVil eKRtUvdak;m:Umg;xageRkAmYyÉktþaenARtg;cMNucEdlcg;rk.
:
begáItkUGredaen x smRsbEdlmann½yenAkñúgtMbn;rbs;Fñwm b¤eRKagEdlminmanPaBdac;én
bnÞúkBit b¤bnÞúknimitþ.
CamYynwgkardak;bnÞúknimitþ eKRtUvdkbnÞúkBitTaMgGs;ecjBIFñwm b¤eRKag ehIyKNnam:Um:g;Bt;
m b¤ mθ CaGnuKmn_énkUGredaennImYy².

snμt; m b¤ mθ eFVIGMeBIkñúgTisedAviC¢mandUcbgðajenAkñúgrUbTI 4-1.
m:Um:g;Bt;Bit
edayeRbIkUGredaen x dUcKñanwgkarbegáItsMrab;KNna m nig mθ kMNt;m:Um:g;Bt; M EdlbNþal
BIkMlaMgBit.
edaysareKsnμt; m b¤ mθ manGMeBIkñúgTisedAviC¢man eKcaM)ac;RtUveGay M EdlmanTMhMviC¢man
manGMeBIkñúgTisedAdUcKña BIeRBaHkmμnþxagkñúgviC¢man nigkmμnþxagkñúgGviC¢manGaRs½ynwgTisedA
rbs;bnÞúk ¬EdlkMNt;eday ± m b¤ ± mθ ¦ nigTisedArbs;bMlas;TI ¬EdlkMNt;eday
± Mdx / EI ¦.

smIkarkmμnþnimitþ
GnuvtþsmIkarkmμnþnimitþedIm,IkMNt;bMlas; Δ b¤mMurgVilθ Edlcg;)an. eKcaM)ac;RtUvrkSasBaØa
BiCKNitrbs;GaMgetRkalnImYy²EdlKNnaCamYynwgtMbn;kMNt;Cak;lak;rbs;va.



RbsinebIplbUkBiCKNitrbs;GaMgetRkalTaMgGs;sMrab;Fñwm b¤eRKagTaMgmUlviC¢man enaH Δ nig
θ manTisedAdUcKñanwgbnÞúknimitþÉktþa b¤m:Um:g;nimitþÉktþa erogKña. RbsinebIlT§plrbs;va
GviC¢man enaHTisedArbs; Δ b¤ θ nwgpÞúyBITisedArbs;bnÞúknimitþÉktþa nigm:Um:g;nimitþÉktþa.

]TahrN_ 9-4³ kMNt;bMlas;TIrbs;cMNuc B rbs;FñwmEdkEdlbgðajenAkñúgrUbTI 9-13. yk E =
/
200GPa I = 500 10 6 mm 4 ( ) .

dMeNaHRsay³
m:Um:g;nimitþ m³ eKTTYlbMlas;TItamTisbBaÄrrbs;cMNuc B edaydak;bnÞúknimitþÉktþa1kN enARtg;
cMNuc B rUbTI 9-13b. edaykarGegát eyIgeXIjfaminmanPaBdac;énbnÞúkBIt nigbnÞúknimitþenAelI
FñwmeT. dUcenH eKGaceRbIkUGredaenH x EtmYyedIm,IkMNt;famBlbMErbMrYlrageFobnimitþ. eKeRCIserIs
eKalrbs; x Rtg;cMnuc B edaysareKminRtUvkarrkkMlaMgRbtikmμenARtg;cMNuc A edIm,IkMNt;m:Um:g;Bt;
m nig M . edayeRbIviFImuxkat; smIkarrbs;m:Um:g;Bt; m RtUv)anbgðajenAkñúgrUbTI 9-13b.

m:Um:g;Bit M³ edayeRbIkUGredaen x dUcKña smIkarm:Um:g;Bt; M RtUv)anbgðajenAkñúgrUbTI 9-13c.
smIkarkmμnþnimitþ³ bMlas;TItamTisQrrbs; B KW
L mM 10 (− 1x )(− 6 x 2 )dx
1kN ⋅ Δ B = ∫0dx =
EI ∫0 EI

1kN ⋅ Δ B =
( )
15 103 kN 2 .m 3
EI



b¤ ΔB =
( )
15 103 kN .m 3
( )
200 10 6 kN / m 2 (500(10 )mm )⎛10
6 4
⎜
⎝
−12
m 4 / mm ⎞
4
⎟
⎠
= 0.150m = 150mm

]TahrN_ 9-5³ kMNt;mMurgVilenARtg;cMNuc A rbs;FñwmEdkEdlbgðajenAkñúgrUbTI 9-14. yk E =
/
200GPa I = 60 10 6 mm 4 ( ) .

dMeNaHRsay³
m:Um:g;nimitþ m ³ eKTTYl)anmMurgVilRtg;cMNuc A edaydak;m:Um:g;nimitþÉktþa1kN .m enARtg;cMNuc A ¬rUbTI
θ

9-14b¦. eKk¾GaceRbIkUGredaen x EtmYyedIm,IkMNt;famBlbMErbMrYlrageFobnimitþ. kUGredaenenH
maneKalenARtg;cMNuc A edIm,Irkm:Um:g;Bt; mθ nig M . edayeRbIviFImuxkat; smIkarm:Um:g;Bt; mθ RtUv
)anbgðajenAkñúgrUbTI 9-14b.
m:Um:g;Bit M³ edayeRbIkUGredaen x dUcKña smIkarm:Um:g;Bt; M RtUv)anbgðajenAkñúgrUbTI 9-14c.
smIkarkmμnþnimitþ³ mMurgVilenARtg;cMNuc A KW
mθ M
(1kN .m )θ A = ∫0
L
dx
EI
⎛ 3⎞
(− 1)⎜ − x ⎟
⎜ 3 ⎟
=∫
3 ⎝ ⎠ dx
0 EI
1 3 3
3EI ∫0
= x dx
6.75
=
( ) ( ( ) )(
200 10 6 kN / m 2 60 10 6 mm 4 10 −12 m 4 / mm 4 )


b¤ θ A = 0.000563rad

]TahrN_ 9-6³ kMNt;mMurgVilθ enARtg;cMNuc B rbs;FñwmEdkEdlbgðajenAkñúgrUbTI 9-15a. yk
/
E = 200GPa I = 60 10 6 mm 4 ( ) .

dMeNaHRsay³
m:Um:g;nimitþ m ³ eKkMNt;mMurgVilRtg;cMNuc B edaydak;m:Um:g;nimitþÉktþa1kN.m enARtg;cMNuc B ¬rUbTI 9-
θ

15b¦. enATIenH eKRtUveRCIserIskUGredaen x cMnYnBIr edIm,IkMNt;famBlbMErbMrYlrageFobnimitþenAkñúg
Fñwm. eKKNnafamBlbMErbMrYlrageFobenAkñúgGgát; AB nigGgát; BC edayeRbIkUGredaen x1 nig x2
erogKña. eKKNnam:Um:g;Bt; mθ enAkñúgGgát;nImYy²edayeRbIviFImuxkat;dUcbgðajenAkñúgrUbTI 9-15b.
m:Um:g;Bit M³ edayeRbIkUGredaen x1 nig x2 dUcKña enaHsmIkarm:Um:g;Bt; M RtUv)anbgðajenAkñúgrUbTI
9-15c.
smIkarkmμnþnimitþ³ mMurgVilRtg;cMNuc B RtUv)aneGayeday



mθ M
L
1⋅θ B = ∫ dx
0 EI
5 (0 )(− 3 x1 )dx1 5 (1)[− 3(5 + x 2 )]dx 2
=∫ +∫
0 EI 0 EI
− 112.5kN .m 2
θB = (1)
EI
eyIgk¾GacKNnaGaMgetRkal ∫ mθ Mdx tamRkaPicedayeRbItaragsMrab;KNnaGaMgetRkalxagelI
∫0 mm' dx . dMbUgeKcaM)ac;KUrdüaRkamm:Um:g; sMrab;FñwmenAkñúgrUbTI 9-15b nig 9-15c. ehIydüaRkam
L

m:Um:g;TaMgenHRtUv)anbgðajenAkñúgrUbTI 9-15d nig 9-15e erogKña. edaysarvaKμanm:Um:g; m sMrab;
0 ≤ x < 5m eyIgGaceRbIEtRkLaépÞqUtragctuekaN nigRkLaépÞqUtragctuekaNBñayedIm,IKNna

GaMgetRkal. edayrkrUbragsmRsbTaMgenHenAkñúgCYredk nigCYrQrrbs;tarag eyIgman
mθ (M 1 + M 2 )L = (1)(− 15 − 30 )5
10 1 1
∫5 mθ Mdx =
2 2

= −112.5kN 2 .m 3
tMélenHdUcKñanwglT§plEdl)ankMNt;enAkñúgsmIkar (1). dUcenH
− 112.5kN 2 .m 3
(1kN .m )θ B =
( ) [ ( ) ](
200 10 6 kN / m 2 60 10 6 mm 4 10 −12 m 4 / mm 4 )
θ B = −0.00928rad
sBaØaGviC¢manbgðajfaθ B manTisedApÞúyeTAnwgTisedArbs;m:Um:g;nimitþEdlbgðajenAkñúgrUbTI 9-15b.

]TahrN_ 9-7³ kMNt;bMlas;TIenARtg;cMNuc D rbs;FñwmEdkEdlbgðajenAkñúgrUbTI 9-16a. yk
/
E = 200GPa I = 300 10 6 mm 4 ( ) .
dMeNaHRsay³
m:Um:g;nimitþ m³ FñwmrgkMlaMgnimitþÉktþaenARtg;cMNuc D dUcbgðajenAkñúgrUbTI 9-16b. tamkarGegát eK
RtUveRbIkUGredaenbIdUcCa x1 / x2 nig x3 edIm,IRKbdNþb;RKb;tMbn;TaMgGs;rbs;Fñwm. cMNaMfa kUGredaen
TaMgenHRKbdNþb;tMbn;EdlbnÞúkBit b¤bnÞúknimitþEdlmanGMeBIelIFñwmminmanPaBdac;. eKKNnam:Um:g;Bt;
m enAkñúgrUbTI 9-16b edayeRbIviFImuxkat;.

m:Um:g;Bit M³ dMbUgeKRtUvkMNt;Rbtikmμrbs;Fñwm bnÞab;mkedayeRbIIkUGredaen x dUcKñanwgkUGredaenEdl
eRbIsMrab;KNna m eKkMNt;m:Um:g;Bt; M dUcbgðajenAkñúgrUbTI 9-16c.
smIkarkmμnþnimitþ³ edayGnuvtþsmIkarkmμnþnimitþeTAelIFñwmedayeRbITinñn½yenAkñúgrUbTI 9-16b nig 9-
16c eyIg)an


L mM
1⋅ Δ D = ∫ dx
0 EI
4.5 (− 1x1 )(0 )dx1 3 (0.75 x 2 − 4.5)(35 x2 )dx 2 3 (− 0.75 x3 )(120 − 5 x3 )
=∫ +∫ +∫ dx3
0 EI 0 EI 0 EI
0 472.5 371.25 843.75kN .m 3
ΔD = − − =−
EI EI EI EI
− 843.75kN .m 3
b¤ ΔD =
[200(10 kN / m )][300(10 )(10 )m ]
6 2 6 −12 4

= −0.0141m = −14.1mm
sBaØaGviC¢manbgðajfabMlas;TImanTiseLIgelI EdlpÞúyeTAnwgTisedAcuHeRkamrbs;bnÞúkÉktþa ¬rUbTI
9-16b¦. ehIycMNaMfa eKmincaM)ac;KNna m1 edaysar M 1 = 0 .



]TahrN_ 9-8³ kMNt;bMlas;TItamTisedkenARtg;cMNuc C enAelIeRKagEdlbgðajenAkñúgrUbTI 9-
17a. yk E = 200GPa / I = 235(106 )mm 4 sMrab;Ggát;TaMgBIr.

dMeNaHRsay³
m:Um:g;nimitþ m³ edIm,IPaBgayRsYl eyIgnwgeRbIkUGredaen x1 nig x2 enAkñúgrUbTI 9-17a. bnÞúkmYy
ÉktþaRtUv)andak;enARtg;cMnuc C rUbTI 9-17b. ehtuGVI? kMlaMgRbtikmμ nigm:Um:g;Bt;RtUv)anKNnadUc
bgðaj.


m:Um:g;Bit M³ tamrebobdUcKña kMlaMgRbtikmμ nigm:Um:g;Bt; M BitRtUv)anKNna nigbgðajenAkñúgrUbTI 9-
17c.
smIkarkmμnþnimitþ³ edayeRbITinñn½yenAkñúgrUbTI 9-17b nig 9-17c eyIg)an
3 (1x1 )(
180 x1 − 30 x1 )dx1 2.4 (1.25 x 2 )(112.5 x 2 )dx 2
2
L mM
1⋅ Δ C =
h ∫0
dx =
EI ∫0 EI
+ ∫0 EI
1012.5 648 1660.5kN .m 3
Δ Ch = + = (1)
EI EI EI
eKk¾GaceFVIGaMgetRkal ∫ mMdx tamRkaPicedayeRbItaragsMrab;KNnaGaMgetRkal ∫0L mm' dx . düa
Rkamm:Um:g;sMrab;eRKagenAkñúg rUbTI 9-17b nig 9-17c RtUv)anbgðajenAkñúgrUbTI 9-17d nig 9-17e
erogKña. dUcenH edayeRbIrUbmnþ sMrab;ragRsedogKñaenAkñúgtaragsMrab;KNnaGaMgetRkal ∫0L mm' dx
eyIg)an
∫ mMdx = 12 (3)(270)(3) + 3 (3)(270)(2.4)
5 1

= 1012.5 + 648 = 1660.5kN 2 .m 3
lT§plenHdUcKñaeTAnwglT§plEdl)anBIkarKNnaenAkñúgsmIkar (1). dUcenH
1660.5kN .m 3
Δ Ch =
[200(10 )kN / m ][235(10 )(10 )m ]
6 2 6 −12 4

= 0.0353m = 35.3mm

]TahrN_ 9-9³ kMNt;mMurgVilenARtg;cMNuc C rbs;eRKagEdlbgðajenAkñúgrUbTI 9-18a. yk
/
E = 200GPa I = 15 10 6 mm 4 ( ) .



dMeNaHRsay³
m:Um:g;nimitþ m ³ eKRtUveRbIkUGredaen x1 nig x2 EdlbgðajenAkñúgrUbTI 9-18a. eKRtUvdak;m:Um:g;mYy
θ

ÉktþaenARtg;cMNuc C rYcKNnam:Um:g;Bt; mθ rUbTI 9-18b.
m:Um:g;Bit M³ tamrebobdUcKña m:Um:g;Bt; M BitRtUv)anKNnadUcbgðajenAkñúgrUbTI 9-18c.
smIkarkmμnþnimitþ³ edayeRbITinñn½yenAkñúgsmIkar 9-18b nig 9-18c eyIg)an
1⋅θC = ∫
L mθ M
dx = ∫
3 (− 1)(− 2.5 x1 )dx1 + 2(1)(7.5)dx2
0
EI 0 EI ∫0 EI
11.25 15 26.25kN .m 2
θC = + =
EI EI EI
26.25kN .m 2
b¤ θC =
( ) [ ( ) ](
200 10 6 kN / m 2 15 10 6 mm 4 10 −12 m 4 / mm 4 )
= 0.00875rad

(>^> famBlbMErbMrYlrageFobnimitþEdlbNþalBIbnÞúktamG½kS kMlaMgkat; kMlaMgrmYl
nigsItuNðPaB
(Virtual strain energy caused by axial load, shear, torsion and Temperature)
eTaHbICaPaBdabrbs;Fñwm nigeRKagbgáeLIgedayfamBlbMErbMrYlrageFobEdlbNþalBIm:Um:g;
Bt;CacMbgk¾eday k¾enAkñúgrcnasm<½n§xøHfamBlbMErbMrYlrageFobbEnßménkMlaMgtamG½kS kMlaMgkat;
kMlaMgrmYl nigsItuNðPaBGackøayCasMxan;. eyIgnwgsikSaBIT§iBlnImYy²enATIenH.
kMlaMgtamG½kS³ Ggát;rbs;eRKagGacrgkMlaMgtamG½kS ehIyeKGacsresrsmIkarfamBlbMErbMrYlrag
eFobEdlbNþalBIkM;laMgenHdUcenAkñúgkfaxNÐ 9-4. sMrab;Ggát;Edlmanmuxkat;efr eyIgman
nNL
Un = (9-20)
AE

Edl n= kMlaMgtamG½kSnimitþxagkñúgEdlbNþalBIkMlaMgnimitþÉktþaxageRkA.
N = kMlaMgtamG½kSxagkñúgenAkñúgGgát;EdlbNþalBIkMlaMgBit.

E = m:UDuleGLasÞicsMrab;sMPar³.





kMlaMgkat;³ edIm,IkMNt;famBlbMErbMrYlrageFobenAkñúgFñwmEdlbNþalBI
kMlaMgkat; eyIgnwgBicarNa FatuGnnþtUc dx rbs;FñwmEdlbgðajenAkñúgrUbTI
9-19. kMhUcRTg;RTayedaykMlaMgkat; (shearing distortion) dy rbs;
FatuGnnþtUcEdlbNþal BIkMlaMgBitKW dy = γdx . RbsinebIbMErbMrYlrag
eFobedaysarkMlaMgkat; (shearing strain) γ EdlbNþalBIkMlaMgkat;man
GMeBIelIsMPar³eGLasÞiclIenEG‘r enaHeKGacGnuvtþc,ab;h‘Uk γ = τ / G . dUc
enH dy = (τ / G )dx . eyIgGacsresrkugRtaMgkat;Caτ = K (V / A) Edl K CaemKuNrag (form factor)
EdlGaRs½ynwgrUbragénRkLaépÞmuxkat; A rbs;Fñwm. dUcenHeyIgGacsresr dy = K (V / GA)dx .
sMrab;FñwmTaMgmUl eKGackMNt;famBlbMErbMrYlrageFobnimitþedayGaMgetRkal
L ⎛ vV ⎞
Us = ∫ K⎜ ⎟dx (9-21)
0 ⎝ GA ⎠

Edl v=kMlaMgkat;nimitþxagkñúgenAkñúgGgát;EdlRtUv)ansresrCaGnuKmn_én x ehIybNþalBIbnÞúk
nimitþÉktþaxageRkA.
V = kMlaMgkat;xagkñúgenAkñúgGgát;EdlRtUv)ansresrCaGnuKmn_én x ehIybNþalBIkMlaMgBit.


K = emKuNragsMrab;RkLaépÞmuxkat; (form factor for cross-sectional area)

K = 1.2 sMrab;muxkat;ctuekaN

K = 10 / 9 sMrab;muxkat;rgVg;

K ≈ 1.0 sMrab;FñwmGkSr I Edl A CaRkLaépÞrbs;RTnug

G = m:UDuleGLasÞicénkMlaMgkat; (shear modulus of elasticity) sMrab;sMPar³.

kMlaMgrmYl³ CaerOy²rcnasm<½n§bITMhM ¬kñúglMh¦ rgkMlaMgrmYl.
RbsinebIGgát;manmuxkat;mUl enaH muxkat;rbs;vaminman warping
ekIteLIgeT enAeBlEdlvargbnÞúk. CalT§pl eKGacsresrfamBl
bMErbMrYlrageFobenAkñúgGgát;)anedayRsYl edayBicarNaelIFatu
GnnþtUc dx rbs;Ggát;Edlrgm:Um:g; rmYl T ¬rUbTI 9-20¦. kMlaMg
rmYlenHeFVIeGaymanbMErbMrYlrageFobkMlaMgkat; γ = (cdθ ) / dx . kñúg krNIsMPar³manlkçN³
eGLasÞiclIenEG‘r enaH γ = τ / G Edlτ = Tc / J . dUcenH mMurmYl dθ = (γdx )/ c = (τ / Gc)dx =


(T / GJ )dx . RbsinebIeKdak;kMlaMgnimitþÉktþaeTAelIrcnasm<½n§EdleFVIeGay mankMlaMgrmYlxagkñúg t
enAkñúgGgát; enaHeRkayBIkarGnuvtþkMlaMgBit famBlbMErbMrYlrageFobenAkñúg Ggát;elIRbEvg dx KW dU t
= tdθ = tTdx / GJ . edayeFVIGaMgetRkalelIRbEvg L rbs;Ggát; eyIg)an³

tTL
Ut = (9-22)
GJ

Edl kMlaMgrmYlnimitþxagkñúgEdlbNþalBIbnÞúknimitþÉktþaxageRkA.
t=

T = kMlaMgrmYlxagkñúgenAkñúgGgát;EdlbNþalBIkMlaMgBit.

G = m:UDuleGLasÞicénkMlaMgkat; (shear modulus of elasticity) sMrab;sMPar³.

J = m:Um:g;niclPaBb:UElrsMrab;muxkat;/ J = πc 4 / 2 Edl c CakaMrbs;muxkat;.

K = RbEvgrbs;Ggát;.

eKRtUvkMNt;famBlbMErbMrYlrageFobEdlbNþalBIkMlaMgrmYlsMrab;Ggát;Edlmanmuxkat;minEmnmUl
edayeRbIkarviPaKEdlmanlkçN³hμt;ct;CagGVIEdl)anerobrab;enATIenH.
kMlaMgrmYl³ enAkñúgkfaxNÐ 9-4 eyIg)anBicarBaelIT§iBlénbMErbMrYlsItuNðPaBesμI dT enAelIGgát;
truss ehIy)anbgðajfaGgát;nwglUt b¤rYjxøIedaybrimaN dL = αΔTL . b:uEnþ enAkñúgkrNIxøH rcna-

sm<½n§ GacrgsItuNðPaBxusKñaelImuxkat;TaMgmUl dUckñúgkrNIénFñwmEdl)anbgðajenAkñúgrUbTI 9-21a.
kñúgkrNIEbbenH eKGackMNt;bMlas;TIrbs;cMNuctambeNþayExSekageGLasÞicrbs;FñwmedayeRbI
eKalkarN_kmμnþnimitþ. edIm,IGnuvtþeKalkarN_kmμnþnimitþ dMbUgeyIgRtUvKNnabrimaNmMurgVilrbs;Fatu
GnnþtUc dx rbs;FñwmEdlbNþalmkBIbMErbMrYlkMedA (thermal gradient) EdlmanGMeBIelImuxkat;Fñwm.
eyIgnwgeRCIserIskrNIsamBaØbMputrbs;FñwmEdlmanG½kSNWtsßitenABak;kNþalkMBs; (c ) rbs;muxkat;
Fñwm. RbsinebIeyIgKUrTMrg;énbMErbMrYlsItuNðPaBenAkñúgmuxkat; ¬rUbTI 9-21b¦ eyIgeXIjfasItuNð-
PaBmFümKW Tm = (T1 + T2 ) / 2 . RbsinebI T1 > T2 enaHbMErbMrYlsItuNðPaBenATItaMgxagelIbMputeFVI
eGayFatuGnnþtUcmanbMErbMrYlrageFoblUt ehIybMErbMrYlrageFobrbs;FatuGnnþtUcenATItaMgxageRkam
bMputrYj. enAkñúgkrNITaMgBIr bMErbMrYlsItuNðPaBKW ΔTm = T1 − Tm = Tm − T2 . edaysarbMErbMrYlkMedA
rbs;RbEvgenAxagelIbMput nigenAxageRkambMputKW δx = αΔTm dx ¬rUbTI 9-21c¦ enaHmMurgVilrbs;Fatu
GnnþtUcKW
αΔTm dx
dθ =
c
RbsinebIeyIgGnuvtþkMlaMgnimitþÉktþaenARtg;cMNucmYyenAelIFñwmEdleKRtUvkarkMNt;bMlas;TI b¤Gnuvtþ
m:Um:g;nimitþÉktþaenARtg;cMNucEdlRtUvkarkMNt;bMlas;TIEdlKitCamMurgVilrbs;bnÞat;b:H enaHkMlaMgTaMg


enHnwgbegáItm:Um:g;nimitþ m enAkñúgFñwmRtg;cMNucEdlFatuGnnþtUc dx sßitenA. enAeBlEdlbMErbMrYl
sItuNðPaBmanGMeBI enaHfamBlbMErbMrYlrageFobnimitþenAkñúgFñwmKW
L mαΔTm dx
U temp = ∫ (9-23)
0 c

Edl m= m:Um:g;nimitþxagkñúgEdlsmIkarrbs;vaRtUv)ansresrCaGnuKmn_eTAnwg x ehIyEdlbNþal
BIbnÞúknimitþÉktþaxageRkA b¤m:Um:g;nimitþÉktþa.
α = emKuNlUtkMedA.
ΔTm = bMErbMrYlsItuNðPaBrvagsItuNðPaBmFüm nigsItuNðPaBenAxagelI b¤enAxageRkambMput

rbs;Fñwm.
c = kMBs;Bak;kNþalrbs;muxkat;Fñwm.

elIkElgEtmankarbBa¢ak; ebImindUcenaHeTcMeNaTenAkñúgesovePAenHnwgBicarNaEtPaBdabrbs;Fñwm
nigPaBdabrbs;eRKagEdlbNþalBIm:Um:g;Bt;Etb:ueNÑaH. b:uEnþ CaTUeTAGgát;Fñwm nigeRKagGacrgnUvbnÞúk
epSg²CaeRcIndUcEdl)anerobrab;enAkñúgkfaxNÐenH. b:uEnþ dUcEdl)anerobrab;rYcenAkñúgkfaxNÐmun²
PaBdabbEnßmEdlbNþalBIkMlaMgkat; kMlaMgtamG½kS eFIVeGayPaBdabrbs;FñwmERbRbYlRtwmEtBIr b¤bI
PaKryb:ueNÑaH. edaysarmUlehtuenH CaTUeTAeKecalvakñúgkarviPaK. RbsinebIeKKitT§iBlEdl)an
erobrab;enAkñúgkfaxNÐenH eKGacbUkbBa©ÚlfamBlbMErbMrYlrageFobnimitþEdl)ankMNt;enAkñúgsmIkar
9-20 dl; 9-23 eTAkñúgsmIkarkmμnþnimitþEdl)ankMNt;edaysmIkar 9-18 b¤ 9-19. ]TahrN_xag

eRkambgðajBIkarGnuvtþénsmIkarTaMgenH.

]TahrN_ 9-10³ kMNt;bMlas;TItamTisedkrbs;cMNuc C enAelIeRKagEdlbgðajenAkñúgrUbTI 9-
22a edayKitbBa©ÚlfamBlbMErbMrYlrageFobxagkñúgEdlbNþalBIkMlaMgtamG½kS nigkMlaMgkat;. yk
E = 200GPa / G = 80GPa / I = 235(10 6 )mm 4 ehIy A = 50(103 )mm 2 sMrab;Ggát;TaMgBIr. muxkat;


rbs;Ggát;manragctuekaN.

dMeNaHRsay³
enATIenH eyIgRtUvEtGnuvtþkMlaMgÉktþaedkRtg;cMNuc C . düaRkamGgÁesrIEdlcaM)ac;sMrab;bnÞúkBit nig
bnÞúknimitþRtUv)anbgðajenAkñúgrUbTI 9-22b nig 9-22c.
m:Um:g;Bt;³ famBlbMErbMrYlrageFobnimitþEdlbNþalBIm:Um:g;Bt;RtUv)anKNnaenAkñúg]TahrN_ 9-8.
vaRtUv)anbgðajdUcxageRkam³


L mMdx 1660.5kN 2 .m 3 1660.5kN 2 .m 3
Ub = ∫
0 EI
=
EI
=
[ ( ) ][ ( )( ) ]
200 10 6 kN / m 2 235 10 6 10 −12 m 4

= 0.0353kN .m = 35.3kN .mm

kMlaMgtamG½kS³ BITinñn½yenAkñúgrUbTI 9-22b nig 9-22c eyIgman
nNL
Ua = ∑
AE
1.25kN (112.5kN )(3m ) 1kN (0 )(2.4m )
=
[50(10 )(10 )m ][80(10 )kN / m ] [50(10 )(10 )m ][80(10 )kN / m ]
3 −6 2 6 2
+ 3 −6 2 6 2

= 0.000105kN .m = 0.105kN .mm

kMlaMgkat;³ edayGnuvtþsmIkar 9-21 CamYynwg K = 1.2 sMrab;muxkat;ctuekaN nigedayeRbIGnuKmn_
kMlaMgkat;EdlbgðajenAkñúgrUbTI 9-22b nig 9-22c eyIg)an
L ⎛ vV ⎞
Us = ∫ K⎜ ⎟dx
0 ⎝ GA ⎠
31.2(1)(180 − 60 x1 )dx1 2.4 1.2(− 1.25)(− 112.5)dx2
=∫ +∫
0 GA 0 GA
2
729kN .m
=
[ ( ) ][ ( )( ) ]
80 10 6 kN / m 2 50 103 10 − 6 m 2
= 0.000182kN .mm = 0.182kN .mm

edayGnuvtþsmIkarkmμnþnimitþ eyIg)an
1kN ⋅ Δ C h = 35.3kN .mm + 0.105kN .mm + 0.182kN .mm

Δ C h = 35.59mm

T§iBlénkMlaMgkat; nigkMlaMgtamG½kSeFVIeGaybMlas;TIekIneLIgRtwmEt 0.8% b:ueNÑaH ebIeRbobeFob
nwgbMlas;TIEdlbNþalBIm:Um:g;Bt;;.

]TahrN_ 9-11³ FñwmEdlbgðajenAkñúgrUbTI 9-23a RtUv)aneRbIenAkñúgGKarEdlrgnUvkMedAxusKñaBIr.
RbsinebIsItuNðPaBenAépÞxagelIrbs;FñwmKW 25o C ehIysItuNðPaBenAépÞxageRkamKW 70o C cUrkMNt;
PaBdabtamTisbBaÄrrbs;FñwmenARtg;cMNuckNþalFñwmEdlbNþalBIbMErbMrYlsItuNðPaB. yk α =
11.7( −6 )/ o C .
10

dMeNaHRsay³
edaysareKRtUvkarkMNt;PaBdabenAkNþalElVgrbs;Fñwm eKRtUvdak;bnÞúknimitþÉktþaenATIenaH ehIyeK
RtUvKNnam:Um:g;Bt;nimitþenAkñúgFñwm ¬rUbTI 9-23b¦.


sItuNðPaBmFümenAcMNuckNþalrbs;muxkat;FñwmKW (70o C + 25o C )/ 2 = 47.5o C . sMrab;kar
GnuvtþsmIkar 9-23, ΔTm = 47.5o C − 25o C = 22.5o C . ehIy C = 250mm / 2 = 125mm . eday
GnuvtþeKalkarN_kmμnþnimitþ eyIg)an
L mαΔTm dx
1N ⋅ Δ C v = ∫
0 c
⎛1 ⎞
1500 mm ⎝ 2 ⎠
( ) −6 o
( )
⎜ x ⎟11.7 10 / C ⋅ 22.5 C
o

= 2∫ dx
0 125mm
Δ C v = 2.37 mm

lT§plbgðajfavaCaPaBdabEdlGacecal)an.

(>&> RTwsþIbTrbs;kasÞIKøIy:aNU (Castigliano’s theorem)
enAqñaM 1879 Gal;Eb‘tU kasÞIKøIy:aNU visVkrpøÚvrfePøIgsBaØatiGIutalI)ane)aHBum<pSayesovePA
mYyk,alEdlENnaMBIviFIsMrab;kMNt;PaBdab b¤mMurgVilenARtg;cMNucmYyenAkñúgrcnasm<½n§EdlGacCa
truss, Fñwm b¤k¾eRKag. viFIenH ¬EdleKeGayeQμaHfaRTwsþITIBIrrbs;kasÞIKøIy:aNU b¤viFIkmμnþGb,brma¦

eRbI)ancMeBaHEtrcnasm<½n§EdlmansItuNðPaBefr TMrEdlminyal (unyielding support) nigeFVIBI
sMPar³eGLasÞiclIenEG‘r. RbsinebIeKRtUvkarkMNt;bMlas;TI RTwsþIeBalfabMlas;TIesμInwgedrIev eday
EpñkelIkTImYyénfamBlbMErbMrYlrageFobenAkñúgeRKOgbgÁúMtamkMlaMgEdlmanGMeBIenARtg;cMNucenaH
nigkñúgTisrbs;bMlas;TI. tamrebobdUcKña mMurgVilenARtg;cMNucmYyenAkñúgeRKOgbgÁúMesμInwgedrIev eday
EpñkelIkTImYyénfamBlbMErbMrYlrageFobenAkñúgeRKOgbgÁúMtamm:Um:g;EdlmanGMeBIenARtg;cMNucenaH nig
kñúgTisrbs;mMurgVil.


edIm,ITTYl)anRTwsþITIBIrrbs;kasÞIKøIy:aNU eKRtUvBicarNaGgÁFatu ¬rcnasm<½n§¦EdlmanragTUeTA
NamYyEdlrg n es‘rIénkMlaMg P1 / P2 ..., Pn . edaysarkmμnþxageRkAeFVIedaykMlaMgTaMgenHesμInwg
famBlbMErbMrYlragxagkñúgEdlrkSaTukenAkñúgGgÁFatu eyIgGacsresr
Ui = Ue
kmμnþxageRkACaGnuKmn_énbnÞúkxageRkA ¬U e = ∑ ∫ Pdx ¦. dUcenH
U i = U e = f (P , P2 ,..., Pn )
1

LÚv RbsinebIkMlaMg Pi mYyénkMlaMgxagelIekIneLIgedaybrimaNDIegr:g;Esül dPi kmμnþxagkñúgk¾
ekIneLIg ehIyfamBlbMErbMrYlrageFobfμIkøayCa
∂U i
U i + dU i = U i + dPi (9-24)
∂Pi
b:uEnþ tMélenHminGaRs½yeTAnwglMdab;énkarGnuvtþ n kMlaMgeTAelIGgÁFatueT. ]TahrN_ RbsinebIeyIg
GnuvtþCadMbUgnUv dPi eTAelIGgÁFatu bnÞab;mkvanwgeFVIeGayGgÁFatumanbMlas;TIedaybrimaNDIepr:g;-
Esül dΔi eTAtamTisedArbs; dPi . tamsmIkar 9-3 ¬U c = PΔ / 2 ¦ ehIykMeNInrbs;famBlbMEr
bMrYlrageFobKW 12 dPi dΔ i . b:uEnþ brimaNenHCaDIepr:g;EsüldWeRkTIBIr dUcenHeyIgGacecalva)an. kar
GnuvtþbnþénbnÞúk P1, P2 , ..., Pn EdleFVIeGayGgÁFatupøas;TI Δ1, Δ 2 , ..., Δ n begáIt)anCafamBl
bMErbMrYlrageFob.
U i + dU i = U i + dPi Δ i (9-25)

enATIenH EdldUcBIelIkmun U i CafamBlbMErbMrYlrageFobxagkñúgGgÁFatuEdlbgáeLIgedaybnÞúk P1,
P2 , ..., Pn nig dU i = dPi Δ i CafamBlbMErbMrYlrageFobbEnßmEdlbgáeday dPi ¬smIkar 9-4/ U e =

PΔ' ¦.

Casegçb smIkar 9-24 CafamBlbMErbMrYlrageFobenAkñúgGgÁFatuEdlkMNt;CadMbUgedayGnuvtþ
P , P2 , ..., Pn bnÞab;mk dPi ehIysmIkar 9-25 CafamBlbMErbMrYlrageFobEdlkMNt;CadMbUgeday
1

Gnuvtþ dPi bnÞab;mk P1, P2 , ..., Pn . edaysarsmIkarTaMgBIrenHRtUvEtesμIKña eyIgtMrUveGay
∂U i
Δi = (9-26)
∂Pi
bMlas;TI Δ i eTAtamTIrbs; Pi esμInwgedrIevedayEpñkelIkTImYyénfamBlbMErbMrYlrageFobtam Pi . *

*
RTwsþITImYyrbs;køasÞIKøIy:aNUmanlkçN³RsedogeTAnwgRTwsþITIBIrrbs;Kat;Edr EtP¢ab;TMnak;TMngrvagkMlaMg P eTAnwgedrIev
i

edayEpñkénfamBlbMErbMrYlrageFobtambMlas;TIEdlRtUvKña eBalKW P = ∂U / ∂Δ . karedaHRsaymanlkçN³RsedogKña
i i i



eKRtUvcMNaMfasmIkar 9-26 CaBMenalGMBIPaBRtUvKñaénrcnasm<½n§ (structure’s compatibility).
ehIykareFVIedrIevxagelITamTareGayBicarNaEtkarrkSakMlaMg (conservative force) sMrab;karviPaK.
kMlaMgTaMgenHeFVIkmμnþedayminGaRs½yeTAnwgKnøg dUcenHvamin)aneFVIeGay)at;bg;famBleT. eday
sareKeGaykMlaMgmanGMeBIEtkñúglkçxNÐeGLasÞiclIenEG‘r dUcenHRTwsþIenHGnuvtþcMeBaHEtsMPar³Edl
eFVIkarkñúgtMbn;eGLasÞiclIenEG‘rb:ueNÑaH. vamindUcnwgviFIkMlaMgnimitþEdl)anerobrab;enAkñúgkfaxNÐ
elIkmun EdlGacGnuvtþeTAelIkareFVIkarCalkçN³eGLasÞic nigmineGLasÞic (inelastic).

(>*> RTwsþIbTrbs;kasÞIKøIy:aNUsMrab; trusses (Castigliano’s theorem for trusses)
famBlbMErbMrYlrageFobsMrab;Ggát;rbs; truss RtUv)aneGayedaysmIkar 9-9/U i = N 2 L / 2 AE
edayCMnYssmIkarenHeTAkñúgsmIkar 9-26 niglubecalsnÞsS i eyIg)an
∂ N 2L
Δ= ∑
∂P 2 AE
CaTUeTA vamanlkçN³gayRsYlCagkñúgkareFVIDIepr:g;EsülmunkareFVIplbUk. enAkñúgkrNITUeTA L / A
nig E CacMnYnefrsMrab;Ggát;EdleGay dUcenHeyIgGacsresr
⎛ ∂N ⎞ L
Δ = ∑ N⎜ ⎟ (9-27)
⎝ ∂P ⎠ AE
Edl bMlas;TIéntMNrbs; truss.
Δ=

P = kMlaMgxageRkAEdlGnuvtþeTAelItMN truss tamTisedArbs; Δ .

N = kMlaMgxagkñúgrbs;Ggát;EdlbNþalBIkMlaMg P nigbnÞúkenAelI truss.




smIkarenHRsedogeTAnwgsmIkarEdleRbIsMrab;viFIkmμnþnimitþ ¬smIkar 9-15, 1⋅ Δ = ∑ nNL / AE ¦
Edl n RtUv)anCMnYseday ∂N / ∂P . cMNaMfa edIm,IkMNt;edrIevedayEpñk eKcaM)ac;RtUvcat;Tuk P dUcCa
Gefr ehIyelIsBIenH eKRtUvsresrkMlaMgkñúgGgát; N nImYy²CaGnuKmn_eTAnwgkMlaMg P . CalT§pl
karKNna ∂N / ∂P TamTarnUvkarKNnaeRcInCagkarKNnakMlaMg n nImYy²edaypÞal;. tYTaMgenHman

eTAnwgGVIEdleGaykñúgkfaxNÐ 9-7 nigdUceTAnwgviFIbMlas;TInimitþ. RTwsþIkasÞIKøIy:aNUGnuvtþ)ancMeBaHsMPar³eGLasÞic nigmin
eGLasÞic. mü:ageTot viFIenHbgðajBItMrUvkarlMnwgsMrab;eRKOgbgÁúM ehIyedaysarvamankMritkñúgkareRbIR)as;kñúgkarviPaKeRKOg
bgÁúM vaminRtUv)anelIkykmkniyayenAkñúgesovePAenHeT.


lkçN³dUcKña edaysar n b¤ ∂N / ∂P CabMErbMrYlénkMlaMgkñúgGgát;tambnÞúk P b¤bMErbMrYlenAkñúgkMlaMg
Ggát;kñúgelIbnÞúkÉktþa.

eKGaceRbICMhanxageRkamedIm,IkMNt;bMlas;TIrbs;tMN truss edayeRbIRTwsþIkasÞIKøIy:aNU.
kMlaMgxageRkA P
dak;kMlaMg P enAelI truss enARtg;tMNEdleKcg;kMNt;bMlas;TI. eKsnμt;eGaykMlaMgenHman
TMhMERbRbYl ehIymanTistamExSskmμrbs;bMlas;TI.
kMlaMgkñúg N
kMNt;kMlaMg N enAkñúgGgát;nImYy²EdlbNþalBIbnÞúkBit nigkMlaMgERbRbYl P . snμt;kMlaMg
TajviC¢man nigkMlaMgsgát;GviC¢man.
KNnaedrIevedayEpñkén ∂N / ∂P sMrab;Ggát;nImYy².
eRkayeBlkMNt; N nig ∂N / ∂P rYcehIy dak;tMéleGaykMlaMg P CatMélénkMlaMgBit Rbsin
ebIvaCMnYskMlaMgBitEdlmanGMeBIenAelI truss. ebImindUcenaHeT yk P esμInwgsUnü.
RTwsþIkasÞIKøIy:aNU
GnuvtþRTwsþIkasÞIKøIy:aNUedIm,IkMNt;bMlas;TI Δ Edlcg;)an. eKcaM)ac;RtUvrkSasBaØaBiCKNit
sMrab;tMélRtUvKñaén N nig ∂N / ∂P enAeBlCMnYstYTaMgenHeTAkñúgsmIkar.
RbsinebI ∑ N (∂N / ∂P )L / AE viC¢man Δ manTisedAdUcnwg P . RbsinebIlT§plmantMél
GviC¢man Δ manTispÞúyeTAnwgkMlaMg P .

]TahrN_ 9-12³ kMNt;bMlas;TItamTisQrrbs;tMN C rbs; truss EdlbgðajenAkñúgrUbTI 9-24a.
RkLaépÞmuxkat;rbs;Ggát;nImYy²KW A = 400mm2 nig E = 200GPa .
dMeNaHRsay³
kMlaMgxageRkA P³ eKdak;kMlaMgbBaÄr P eTAelI truss enARtg;tMN C edaysarvaCacMNucEdleyIgRtUv
karkMNt;bMlas;TIbBaÄr ¬rUbTI 9-24b¦.



kMlaMgkñúg N³ kMNt;kMlaMgRbtikmμenARtg;TMr truss A nig B ehIylT§plrbs;vaRtUv)anbgðajenAkñúg
rUbTI 9-24b. eKGackMNt;kMlaMg N edayeRbIviFItMN ¬rUbTI 9-24c¦*. sMrab;PaBgayRsYl
lT§plTaMgenH CamYynwgedrIevedayEpñkén ∂N / ∂P RtUv)anerobCataragdUcxageRkam³

edaysareKminman P dUckMlaMgBitenAelI truss eyIgyk P = 0 enAkñúgtaragxagelI.
RTwsþIkasÞIKøIy:aNU³ edayGnuvtþsmIkar 9-27 eyIg)an
⎛ ∂N ⎞ L 10.67kN .m
Δ Cv = ∑ N ⎜ ⎟ =
⎝ ∂P ⎠ AE AE
edayCMnYs A = 400mm2 = 400(10−6 )m 2 / E = 200GPa = 200(109 )Pa nwgbMEbgxñatrbs; N BI kN
eTA N enaHeyIg)an
10.67( 9 )N .m
10
400(10 )m (200( 9 )N / m 2 )
ΔC = v −6 2
= 0.000133m = 0.133mm
10
dMeNaHRsayenHdUcKñanwgdMeNaHRsayedayviFIkmμnþnimitþén]TahrN_ 9-2.

*
vamanlkçN³gayRsYlCagkñúgkarviPaK truss EtCamYynwgkMlaMg 4kN EdlmanGMeBIelIva bnÞab;mkviPaK truss CamYynwg
kMlaMg P elIva. eKGacbUkbBa¢ÚllT§plTaMgBIredIm,ITTYl)ankMlaMg N.


]TahrN_ 9-13³ kMNt;bMlas;TItamTisedkrbs;tMN D rbs; truss EdlbgðajenAkñúgrUbTI 9-25a.
yk E = 200GPa . RkLaépÞmuxkat;rbs;Ggát;nImYy²RtUv)anbgðajenAkñúgrUb.

dMeNaHRsay³
kMlaMgxageRkA P³ edaysareKRtUvkarkMNt;bMlas;TItamTisedkrbs; D eKRtUvdak;kMlaMgGefr P enA
Rtg;tMN D ¬rUbTI 9-25b¦.
kMlaMgkñúg N³ kMNt;kMlaMg N enAkñúgGgát; edayeRbIviFItMN*. lT§plRtUvbgðajenAkñúgrUbTI 9-25b.
edayerobcMTinñn½yenAkñúgtarag eyIg)an³

⎛ ∂N ⎞ L
Δ Cv = ∑ N ⎜ ⎟
⎝ ∂P ⎠ AE
468.75kN .m 202.5kN .m
= 0+0+0+
( ) [ ( )
385 10 − 6 m 2 200 10 6 kN / m 2 ] +
( ) [ ( )
577.5 10 − 6 m 2 200 10 6 kN / m 2 ]
= 0.00784m = 7.84mm

*
dUcenAkñúg]TahrN_elIkmun eKGacGnuvtþkarviPaK truss CamYynwgkMlaMg 45kN nigkMlaMg P dac;edayELkBIKña ehIyeFVIkar
bUklT§plrbs;vabBa©ÚlKña.


]TahrN_ 9-14³ kMNt;bMlas;TItamTisQrrbs;tMN C rbs; truss EdlbgðajenAkñúgrUbTI 9-26a.
snμt;fa A = 300mm2 / E = 200GPa .

dMeNaHRsay³
kMlaMgxageRkA P³ kMlaMg 20kN enARtg;cMNuc C RtUv)anCMnYsedaykMlaMgGefr P enARtg;tMN C
¬rUbTI 9-26b¦.
kMlaMgkñúg N³ kMNt;kMlaMg N enAkñúgGgát; edayeRbIviFItMN. lT§plRtUv)ansegçbenAkñúgrUbTI 9-26b.
Tinñn½yEdlRtUvkarRtUv)anerobcMenAkñúgtarag eyIg)an³

RTwsþIkasÞIKøIy:aNU³ edayCMnYsTinñn½yeTAkñúgsmIkar 9-27 eyIg)an
⎛ ∂N ⎞ L 369.7kN .m
Δ Cv = ∑ N ⎜ ⎟ =
⎝ ∂P ⎠ AE EA
edaybMEbkxñatrbs;RbEvgGgát;eTACamIlIEm:Rt nigedayCMnYstMélCaelxsMrab; AE eyIg)an
369.7 kN .m(1000mm / m)
Δ Cv =
[300(10 )m ][200(10 )kN / m ] = 6.16mm
−6 2 6 2

eyIgKYrcMNaMfa lT§plenHmanlkçN³RsedogKñanwglT§plenAkñúg]TahrN_ 9-1 EdleRbIviFIkmμnþ
nimitþ.



(>(> RTwsþIbTrbs;kasÞIKøIy:aNUsMrab;Fñwm nigeRKag
(Castigliano’s theorem for beams and frames)
famBlbMErbMrYlrageFobsMrab;Fñwm b¤eRKagEdlekItBIm:Um:g;Bt;RtUv)aneGayedaysmIkar 9-11
¬U i = ∫ M 2 dx / 2EI ¦. edayCMnYssmIkarenHeTAkñúgsmIkar 9-26 ¬ Δ i = ∂U i / ∂Pi ¦ nigedaylub
ecalsnÞsS i eyIg)an
∂ L M 2 dx
∂P ∫0 2 EI
Δ=

kareFVIDIepr:g;EsülmunkareFVIGaMgetRkalmanlkçN³gayRsYlCagkareFVIGaMgetRkaleTAelIkaerén
m:Um:g;Bt; M rYceTIbeFVIedrIevedayEpñktameRkay. kñúgkrNIEdl E nig i CatMélefr eyIg)an
L ⎛ ∂M ⎞ dx
Δ = ∫ M⎜ ⎟ (9-28)
0 ⎝ ∂P ⎠ EI

Edl Δ= bMlas;TIéncMNucEdlbNþaledaybnÞúkBitEdlmanGMeBIelIFñwm b¤eRKag.
P = kMlaMgxageRkAEdlGnuvtþeTAelIFñwm b¤eRKagtamTisedArbs; Δ .

M = m:Um:g;Bt;enAkñúgFñwm b¤eRKag EdlRtUv)ansresrCaGnuKmn_eTAnwg x nigedaysarkMlaMg P

nigkMlaMgBitenAelIFñwm.
E = mU:DuleGLasÞicrbs;Ggát;.

I = m:Um:g;niclPaBénRkLaépÞmuxkat;EdlKNnaRtg;G½kSNWt.

RbsinebIeKRtUvkarkMNt;mMurgVilenARtg;cMNucmYy eyIgRtUvrkedrIevedayEpñkénm:Um:g;Bt; M tam
m:Um:g;xageRkA M ' EdlmanGMeBIenARtg;cMNucenaH eBalKW
L ⎛ ∂M ⎞ dx
θ = ∫ M⎜ ⎟ (9-29)
0 ⎝ ∂M ' ⎠ EI

smIkarxagelImanlkçN³RsedogKñanwgsmIkarEdleRbIsMrab;viFIkmμnþnimitþ ¬smIkar 9-18 nig
9-19¦ elIkElgRtg;eKCMnYs m nig mθ eday ∂M / ∂P nig ∂M / ∂M ' erogKña. dUcenAkñúgkrNI truss,

CaTUeTAeKRtUvkarKNnaeRcInCagkareRbIviFIkmμnþbnþicedaytMrUveGayKNnaedrIevedayEpñk ehIyGnuvtþ
RTwsþIkasÞIKøIy:aNU. ehIyeKKYrcgcaMfa RTwsþIenHGnuvtþEteTAelIsMPar³EdlmanlkçN³eGLasÞic
lIenEG‘r. RbsinebIeKRtUvkarKitfamBlbMErbMrYlrageFobenAkñúgeRKOgbgÁúMEdlekItkMlaMgkñúgTaMgGs;
eKRtUvKitbBa©ÚlfamBlbMErbMrYlrageFobEdlbNþalBIkMlaMgkat; kMlaMgtamG½kS nigkMlaMgrmYl.


karRsayrUbmnþsMrab;kMlaMgkat; nigkMlaMgrmYlmanlkçN³RsedognwgkarRsayrUbmnþénsmIkar 9-21
nig 9-22. famBlbMErbMrYlrageFob nigedrIevrbs;vaKW
L V 2 dx ∂U s L V ⎛ ∂V ⎞
Us = K∫ =∫ ⎜ ⎟dx
0 2 AG ∂P 0 AG ⎝ ∂P ⎠

LT 2
dx ∂U t L T ⎛ ∂T ⎞
Ut = ∫ =∫ ⎜ ⎟dx
0 2 JG ∂P 0 JG ⎝ ∂P ⎠

b:uEnþT§iBlenHminRtUv)aneKrab;bBa©ÚleTAkñgkarviPaKenAkñúgemeronenHeT edaysarPaBdab
ú
rbs;Fñwm nigeRKagekIteLIgCacMbgedaysarfamBlbMErbMrYlragedaym:Um:g;Bt;. eKeRcInEtviPaKeRKag
EdlmanElVgFM b¤eRKagEdlmanragFrNImaRtminFmμtaedayeRbIkMuBüÚT½rCaCMnYy EdlT§iBlTaMgenH
RtUv)anKitbBa©ÚleTAkñúgkarviPaKCaeRsc.

eKGaceRbICMhanxageRkamedIm,IkMNt;bMlas;TI nig/b¤mMurgVilenARtg;cMNucmYyenAkñúgFñwm b¤
eRKagedayeRbIRTwsþIkasÞIKøIy:aNU.
kMlaMgxageRkA P b¤m:Um:g;xageRkA M’
dak;kMlaMg P enAelIFñwm b¤eRKagenARtg;cMNuc nigtamTisedAénbMlas;TIEdlcg;)an.
RbsinebIeKRtUvkarkMNt;mMurgVil eKRtUvdak;m:Um:g; M ' enARtg;cMNucenaH.
eKRtUvsnμt;fa P nig M ' manTMhMERbRbYl.
m:Um:g;Bt; M
begáItkUGredaen x smRsbmann½yenAkñúgtMbn;énFñwm b¤eRKagEdlminmanPaBminCab;énkMlaMg
bnÞúkBRgay b¤m:Um:g;.
KNnam:Um:g;Bt; M CaGnuKmn_én P b¤ M ' nigkUGredaen x nImYy². ehIy KNnaedrIeveday
Epñkén ∂M / ∂P b¤ ∂M / ∂M ' sMrab;kUGredaen x nImYy².
eRkayeBlkMNt; M nig ∂M / ∂P b¤ ∂M / ∂M ' rYcehIy dak;tMéleGaykMlaMg P b¤m:Um:g; M '
CatMélénkMlaMgBit RbsinebIvaCMnYskMlaMg b¤m:Um:g;Bit. ebImindUcenaHeT yk P b¤ M ' esμInwg
sUnü.
RTwsþIkasÞIKøIy:aNU
GnuvtþsmIkar 9-28 b¤ 9-29 edIm,IkMNt;bMlas;TI Δ b¤mMurgVil θ Edlcg;)an. eKcaM)ac;RtUvrkSa
sBaØaBiCKNitsMrab;tMélRtUvKñaén M nig ∂M / ∂P b¤ ∂M / ∂M ' .


RbsinebIplbUkénGaMgetRkalkMNt;TaMgGs;mantMélviC¢man enaH Δ b¤ θ manTisedAdUcnwg P
b¤ M ' .

]TahrN_ 9-15³ kMNt;bMlas;TIrbs;cMNuc B énFñwmEdlbgðajenAkñúgrUbTI 9-27a. yk E =
/
200GPa I = 500 106 mm 4 ( ) .

dMeNaHRsay³
kMlaMgxageRkA P³ dak;kMlaMgbBaÄr P enAelIFñwmRtg;cMNuc B dUcbgðajenAkñúgrUbTI 9-27b.
m:Um:g;Bt; M³ sMrab;karedaHRsay]TahrN_enH eKRtUvkarkUGredaenH x EtmYy edaysarminmanPaB
minCab;énbnÞúkenAcenøaH A nig B . edayeRbIviFImuxkat; ¬rUbTI 9-27¦ eyIg)an³
⎛ x⎞
+∑M = 0 − M − (12 x )⎜ ⎟ − Px = 0
⎝2⎠
∂M
M = −6 x 2 − Px = −x
∂P
edayyk P = 0 eyIg)an
∂M
M = −6x 2 = −x
∂P
L ⎛ ∂M ⎞ dx 10 (− 6 x 2 )(− x )dx
ΔB = M ⎜ ∫0 ⎟
⎝ ∂P ⎠ EI
= ∫0 EI

=
( )
15 103 kN .m 3
EI

b¤ ( ) 15 10 3 kN .m 3
200(10 )kN / m [500( )mm ](10
ΔB = 6
102 6 4 −12
m 4 / mm 4 )
= 0.150m = 150mm



eyIgKYrcMNaMfa lT§plenHmanlkçN³RsedogKñanwglT§plenAkñúg]TahrN_ 9-4 EdleRbIviFIkmμnþ
nimitþ.

]TahrN_ 9-16³ kMNt;mMurgVilenARtg;cMNuc B énFñwmEdlbgðajenAkñúgrUbTI 9-28a. yk E =
/
200GPa I = 60 10 6 mm 4 ( ) .

dMeNaHRsay³
m:Um:g;xageRkA M’³ edaysareKRtUvkarkMNt;mMurgVilRtg;cMNuc B dUcenHeKRtUvdak;m:Um:g; M ' enAelIFñwm
enARtg;cMNucenaH ¬rUbTI 9-28b¦.
m:Um:g;Bt; M³ eKRtUveRbIkUGredaen x1 nig x2 edIm,IkMNt;m:Um:g;Bt;enAkñúgFñwmedaysarvamanPaBdab; M '
enARtg;cMNuc B . dUcbgðajenAkñúgrUbTI 9-28a/ x1 latsn§wgBI A eTA B nig x2 latsn§wgBI B eTA C .
edayeRbIviFImuxkat; ¬rUbTI 9-28c¦ eK)anm:Um:g;Bt; nigedrIevedayEpñkrbs;vadUcxageRkam.
sMrab; x1 ³
+∑M = 0 M 1 + 3 x1 = 0

M 1 = −3x1
∂M 1
=0
∂M '
sMrab; x2 ³
+∑M = 0 M 2 − M '+3(5 + x2 ) = 0

M 2 = M '−3(5 + x2 )
∂M 2
=1
∂M '
RTwsþIkasÞIKøIy:aNU³ edayeGay M ' = 0 nigedayGnuvtþsmIkar 9-29 eyIg)an


L⎛ ∂M ⎞ dx 5 (− 3 x1 )(0 )dx1 5 − 3(5 + x2 )(1)dx2
θB = ∫ M ⎜ ⎟ =∫ +∫
0 ⎝ ∂M ' ⎠ EI 0 EI 0 EI
112.5kN .m 2
=−
EI
− 112.5kN .m 2
b¤ θB =
( ) [ ( ) ](
200 10 6 kN / m 2 60 10 6 mm 4 10 −12 m 4 / mm 4 )
= −0.00938rad
sBaØadkbgðajfa θ B manTisedApÞuúyeTAnwgTisedArbs;m:Um:g; M ' . eyIgKYrcMNaMfa lT§plenHman
lkçN³RsedogKñanwglT§plenAkñúg]TahrN_ 9-6 EdleRbIviFIkmμnþnimitþ.

]TahrN_ 9-17³ kMNt;bMlas;TIbBaÄréncMNuc C rbs;FñwmEdlbgðajenAkñúgrUbTI 9-29a. yk E =
/
200GPa I = 150 106 mm 4 ( ) .

dMeNaHRsay³
kMlaMgxageRkA P³ dak;kMlaMgbBaÄr P enAelIFñwmRtg;cMNuc C ¬rUbTI 9-29b¦. eyIgnwgeGaykMlaMg
P enHesμInwgkMlaMgxageRkA 20kN enAeBleRkay.

m:Um:g;Bt; M³ enAkñúgkrNIenH eKRtUvkarkUGredaen x cMnYnBIrsMrab;kareFVIGaMgetRkal ¬rUbTI 9-29b¦
edaysarbnÞúkmanlkçN³minCab;enARtg;cMNuc C . edayeRbIviFImuxkat; ¬rUbTI 9-29c¦ eyIg)an³
sMrab; x1
⎛x ⎞
+∑M = 0 − (24 + 0.5 P )x1 + 8 x1 ⎜ 1 ⎟ + M 1 = 0
⎝2⎠
∂M 1
M 1 = (24 + 0.5 P )x1 − 4 x1
2
= 0.5 x1
∂P
sMrab; x2


+∑M = 0 − M 2 + (8 + 0.5 P )x2 = 0
∂M 2
M 2 = (8 + 0.5 P )x2 = 0.5 x2
∂P
RTwsþIkasÞIKøIy:aNU³ edayyk P = 20kN nigedayGnuvtþsmIkar 9-28 eyIg)an
L ⎛ ∂M ⎞ dx 4 (34 x1 − 4 x1 )(0.5 x1 )dx1
2 4(18 x2 )(0.5 x2 )dx2
ΔC = M ⎜
v ∫0 ⎟ =
⎝ ∂P ⎠ EI
∫0 EI ∫0+
EI
234.7kN .m 3 192kN .m 3 426.7kN .m 3
= + =
EI EI EI
3
b¤ Δ Cv =
( )
426.7 kN .m
[ ( ) ](
200 10 kN / m 150 10 6 mm 4 10 −12 m 4 / mm 4
6 2
)
= 0.0142m = 14.2mm

]TahrN_ 9-18³ kMNt;mMurgVilenARtg;cMNuc C rbs;eRKagEdlmanGgát;BIrEdlbgðajenAkñúgrUbTI 9-
30a. TMr A CaTMrbgáb;. yk E = 200GPa / I = 235(106 )mm4 .

dMeNaHRsay³
m:Um:g;xageRkA M’³ eKdak;m:Um:g;Gefr M ' eTAelIeRKagenARtg;cMNuc C edaysarRtUvkarkMNt;mMurgVil
Rtg;cMNucenH ¬rUbTI 9-30b¦. eyIgnwgeGaym:Um:g; M ' enHesμIsUnüenAeBleRkay.
m:Um:g;Bt; M³ edaysarPaBminCab;rbs;bnÞúkxagkñúgenARtg;cMNuc B eKRtUveRbIkUGredaen x1 nig x2 dUc
bgðajenAkñúgrUbTI 9-30b. edayeRbIviFImuxkat; ¬rUbTI 9-30c¦ eyIg)an


sMrab; x1 ³
⎛x ⎞
+∑M = 0 − M 1 − 30 x1 ⎜ 1 ⎟ − M ' = 0
⎝2⎠
(
M 1 = − 15 x1 + M '
2
)
∂M 1
= −1
∂M '
sMrab; x2 ³
+∑M = 0 ( )
− M 2 − 108 x2 cos 60 o + 1.8 − M ' = 0

M2 = −108(x 2 cos 60
o
+ 1 .8 ) − M '
∂M 2
= −1
∂M '
RTwsþIkasÞIKøIy:aNU³ edayeGay M ' = 0 nigedayGnuvtþsmIkar 9-29 eyIg)an
L ⎛ ∂M ⎞ dx 3.6 (− 15 x1 )(− 1)dx1
2 3 − 108(x2 cos 60 o + 1.8)(− 1)dx2
θC = M ⎜ ∫0 ⎟
⎝ ∂M ' ⎠ EI
=
0 ∫EI 0
+ ∫ EI
233.28kN .m 2 826.2kN .m 2 1059.48kN .m 2
= + =
EI EI EI
2
b¤ θC =
( )
1059.48kN .m
[ ( ) ](
200 10 kN / m 235 10 6 mm 4 10 −12 m 4 / mm 4
6 2
=
)
= 0.00225rad



cMeNaT
(>!> eRbIviFIkmμnþnimitþedIm,IkMNt;bMlas;TIbBaÄr (>^> edaHRsaycMeNaT (>%> edayeRbIRTwsþIkasÞI
éntMN B rbs; truss. Ggát;EdknImYy²manRk- KøIy:aNU.
LaépÞmuxkat; 300mm2 / E = 200GPa . (>&> kMNt;bMlas;TIbBaÄrrbs;tMN C én truss
kñúgcMeNaT (>%>. Ggát;nImYy²manRkLaépÞmux
kat; 400mm2 / E = 200GPa . eRbIviFIkmμnþ
nimitþ.
(>*> edaHRsaycMeNaT (>&> edayeRbIRTwsþIkasÞI
KøIy:aNU.
(>(> kMNt;bMlas;TIbBaÄrrbs; truss Rtg;tMN
F . snμt;RKb;Ggát;TaMgGs;tP¢ab;KñaedaytMN
(>@> edaHRsaycMeNaT (>!> edayeRbIRTwsþIkasÞIKøI
y:aNU. snøak;. yk A = 300mm2 / E = 200GPa
(>#> eRbIviFIkmμnþnimitþedIm,IkMNt;bMlas;TItamTis sMrab;Ggát;nImYy². eRbIviFIkmμnþ nimitþ.
edkrbs;tMN B én truss kñúgcMeNaT (>!>. Ggát;
EdknImYy²manRkLaépÞmuxkat; 300mm2 / E =
200GPa .

(>$> edaHRsaycMeNaT (>#> edayeRbIRTwsþIkasÞI
KøIy:aNU.
(>%> kMNt;bMlas;TItamTisedkrbs;tMN B én
truss. Ggát;nImYy²manRkLaépÞmuxkat; (>!0> edaHRsaycMeNaT (>(> edayeRbIRTwsþIkasÞI
400mm 2 / E = 200GPa . eRbIviFIkmμnþnimitþ. KøIy:aNU.
(>!!> kMNt;bMlas;TIbBaÄrrbs; truss enAkñúg
cMeNaT (>!0> Rtg;tMN B . snμt;RKb;Ggát;TaMg
Gs; tP¢ab;KñaedaytMNsnøak;. yk
A = 300mm 2 / E = 200GPa sMrab;Ggát;

nImYy². eRbIviFIkmμnþnimitþ.

cMeNaT T.Chhay -326


(>!@> edaHRsaycMeNaT (>!!> edayeRbIRTwsþIka (>!^. Ggát;Edk A − 36 nImYy²manRkLaépÞ
sÞIKøIy:aNU. muxkat; 400mm2 / E = 200GPa .
(>!#> kMNt;bMlas;TIbBaÄrrbs;éntMN A . snμt; (>!*> kMNt;bMlas;TItamTisQrrbs;tMN B .
RKb;Ggát;TaMgGs;tP¢ab;KñaedaytMNsnøak;. yk RkLaépÞmuxkat;rbs;Ggát;nImYy²RtUv)ankMNt;
A = 100mm 2 / E = 200GPa sMrab;Ggát; enAkñúgrUb. snμt;Ggát;nImYy²RtUv)antP¢ab;eday
nImYy². eRbIviFIkmμnþnimitþ. tMNsnøak;. yk E = 200GPa . eRbIviFIkmμnþ
nimitþ.

(>!$> edaHRsaycMeNaT (>!#> edayeRbIRTwsþIka
sÞIKøIy:aNU. (>!(> edaHRsaycMeNaT (>!*> edayeRbIRTwsþIka
(>!%> eRbIviFIkmμnþnimitþedIm,IkMNt;bMlas;TItam sÞIKøIy:aNU.
Tisedkrbs;tMN C . Ggát;EdknImYy²man (>@0> kMNt;bMlas;TItamTisQrrbs;tMN E .
RkLaépÞmuxkat; 400mm2 / E = 200GPa . Ggát;nImYy²manRkLaépÞmuxkat; 2700mm2 /
E = 200GPa . eRbIviFIkmμnþnimitþ.

(>@!> edaHRsaycMeNaT (>@0> edayeRbIRTwsþIka
(>!^> edaHRsaycMeNaT (>!%> edayeRbIRTwsþIka sÞIKøIy:aNU.
sÞIKøIy:aNU. (>@@> eRbIviFIkmμnþnimitþedIm,IkMNt;bMlas;TItam
(>!&> eRbIviFIkmμnþnimitþedIm,IkMNt;bMlas;TItam TisQrrbs;tMN A . Ggát;EdknImYy²man
TisQrrbs;tMN D én truss enAkñúgcMeNaT RkLaépÞmuxkat; 1800mm2 / E = 200GPa .

Problems T.Chhay -327


(>@#> edaHRsaycMeNaT (>@@> edayeRbIRTwsþIka (>@(> dkbnÞúkBI truss enAkñúgcMeNaT(>@^ ehIy
sÞIKøIy:aNU. kMNt;bMlas;TItamTisQrrbs;tMN A RbsinebI
(>@$> eRbIviFIkmμnþnimitþedIm,IkMNt;bMlas;TItam eKplitGgát; AE manRbEvgxøICagGgát;edIm
TisQrrbs;tMN B én truss enAkñúgcMeNaT (>@@. 12mm .

Ggát;EdknImYy²manRkLaépÞmuxkat; 1800mm2 / (>#0> eRbIviFIkmμnþnimitþedIm,IkMNt;bMlas;TItam
E = 200GPa . TisQrrbs;tMN C . yk E = 200GPa . Ggát;
(>@%> edaHRsaycMeNaT (>@$> edayeRbIRTwsþIka EdknImYy²manRkLaépÞmuxkat; 1500mm2 .
sÞIKøIy:aNU.
(>@^> kMNt;bMlas;TItamTisQrrbs;tMN A .
snμt;faGgát;nImYy²RtUv)antP¢ab;Kñaedaysnøak;.
yk A = 1200mm2 nig E = 200GPa sMrab;
Ggát;nImYy². eRbIviFIkmμnþnimitþ. (>#!> edaHRsaycMeNaT (>#0> edayeRbIRTwsþIka
sÞIKøIy:aNU.
(>#@> kMNt;bMlas;TItamTisQrrbs;tMN B .
sMrab;Ggát;nImYy² A = 900mm2 / E = 200GPa .
eRbIviFIkmμnþnimitþ.

(>@&> edaHRsaycMeNaT (>@^> edayeRbIRTwsþIka
sÞIKøIy:aNU.
(>@*> dkbnÞúkBI truss enAkñúgcMeNaT (>@^
ehIykMNt;bMlas;TItamTisQrrbs;tMN A .
RbsinebIGgát; AB nig BC rgnUvkMeNIn (>##> edaHRsaycMeNaT (>#@> edayeRbIRTwsþIka
sItuNðPaB ΔT = 110o C . yk A = 1200mm2 sÞIKøIy:aNU.
nig E = 200GPa . ehIy α = 12.0(10−6 )/ o C .
cMeNaT T.Chhay -328


(>#$> kMNt;bMlas;TItamTisQrrbs;tMN E én (>#(> edaHRsaycMeNaT (>#*> edayeRbIRTwsþIka
truss enAkñúgcMeNaT (>#@. sMrab;Ggát; sÞIKøIy:aNU.
nImYy² A = 900mm2 / E = 200GPa . eRbIviFI (>$0> kMNt;bMlas;TItamTisedkrbs;tMN B én
kmμnþnimitþ. truss kñúgcMeNaT (>#*. Ggát;nImYy²manmux-

(>#%> edaHRsaycMeNaT (>#$> edayeRbIRTwsþIka kat; A = 1800mm2 / E = 200GPa . eRbIviFI
sÞIKøIy:aNU. kmμnþnimitþ.
(>#^> kMNt;bMlas;TItamTisQrrbs;tMN D én (>$!> edaHRsaycMeNaT (>$0> edayeRbIRTwsþIka
truss. Ggát;nImYy²manmuxkat; A = 300mm 2 / sÞIKøIy:aNU.
E = 200GPa . eRbIviFIkmμnþnimitþ. (>$@> kMNt;bMlas;TItamTisedkrbs;tMN C .
eRbIviFIkmμnþnimitþ snμt;Ggát;TaMgGs;tP¢ab;Kña
edaytMNsnøak;. Ggát;nImYy²manmuxkat; AE
mantMélefr.

(>#&> edaHRsaycMeNaT (>#^> edayeRbIRTwsþIka
sÞIKøIy:aNU.
(>#*> kMNt;bMlas;TItamTisedkrbs;tMN C én
truss. Ggát;nImYy²manmuxkat; A = 1800mm 2 /

E = 200GPa . eRbIviFIkmμnþnimitþ.

(>$#> dkbnÞúkBI truss enAkñúgcMeNaT (>$@
ehIykMNt;bMlas;TItamTisedkrbs;cMNuc C
RbsinebIGgát; AB nig BC rgnUvkMeNIn
sItuNðPaB ΔT = 110o C . yk A = 1200mm2
nig E = 200GPa . ehIy α = 1.80(10−6 )/ o C .



(>$$> dkbnÞúkBI truss enAkñúgcMeNaT (>$@ (>%!> kMNt;bMlas;TIrbs;cMNuc C énFñwmEdlman
ehIykMNt;bMlas;TItamTisedkrbs;cMNuc C m:Um:g;niclPaB I = 20(106 )mm 4 yk E =
RbsinebIGgát; CD RtUv)aneKplitedayman 200GPa .

RbEvgxøICag12mm .
(>$%>eRbIviFIkmμnþnimitþedIm,IkMNt;bMlas;TItam
TisQrrbs;tMN A . Ggát;nImYy²manmuxkat;
A = 400mm 2 / E = 200GPa .
(>%@> kMNt;bMlas;TIrbs;cMNuc C énFñwmenAkñúg
cMeNaT (>%! EdlFñwmenHmanm:Um:g;niclPaB
I = 20(106 )mm 4 yk E = 200GPa .

(>%#>eRbIviFIkmμnþnimitþedIm,IkMNt;bMlas;TIrbs;cM
Nuc C énFñwmEdleFVIBIEdk. E = 200GPa / I =
100( 6 )mm 4 .
10

(>$^> edaHRsaycMeNaT (>$%> edayeRbIRTwsþIka
sÞIKøIy:aNU.
(>$&>eRbIviFIkmμnþnimitþedIm,IkMNt;bMlas;TIrbs;cM
Nuc C . EI CatMélefr. (>%$>eRbIviFIkmμnþnimitþedIm,IkMNt;mMurgVilrbs;cM-
Nuc A énFñwmEdleFVIBIEdkkñúgcMeNaT (>%#.
E = 200GPa / I = 100( 6 )mm 4 .
10

(>%%> edaHRsaycMeNaT (>%$> edayeRbIRTwsþIka
(>$*> edaHRsaycMeNaT (>$&> edayeRbIRTwsþIka sÞIKøIy:aNU.
sÞIKøIy:aNU. (>%^> kMNt;mMurgVilenARtg;cMNuc A . EI man
(>$(>eRbIviFIkmμnþnimitþedIm,IkMNt;mMurgVilrbs;cM tMélrfr.
Nuc C éncMeNaT (>$&. EI CatMélefr.
(>%0>eRbIviFIkmμnþnimitþedIm,IkMNt;mMurgVilrbs;cM
Nuc A éncMeNaT (>$&. EI CatMélefr.
(>%&> edaHRsaycMeNaT (>%^> edayeRbIRTwsþIka
sÞIKøIy:aNU.
cMeNaT T.Chhay -330


(>%*> eRbIviFIkmμnþnimitþedIm,IkMNt;bMlas;TItam bNþalBIbMErbMrYlsItuNðPaB. FñwmenHmanmux-
Tisedkrbs;cMNuc A EdlbNþalBIkMlaMgcMNuc kat;ctuekaNEdlmanCMerA 350mm .
P . TMrRtg;cMNuc C CaTMrbgáb;. EI CatMélefr.

KitEtT§iBlénm:Um:g;Bt;b:ueNÑaH.

(>^#> EpñkxagelIrbs;FñwmrgnUvsItuNðPaB Tt =
200o C xN³sItuNðPaBénEpñkxageRkamrbs;

FñwmKW Tb = 30o C . RbsinebI α = 12(10−6 )/ o C
cUrkMNt;bMlas;TItamTisQrrbs;cMNuc C Edl
(>%(>edaHRsaycMeNaT (>%*> edayeRbIRTwsþIka bNþalBIbMErbMrYlsItuNðPaB. FñwmenHmanmux-
sÞIKøIy:aNU. kat;ctuekaNEdlmanCMerA 350mm .
(>^0> r)ar ABC manmuxkat;ctuekaN 300mm
× 100mm . EdkBüÜr BD manGgát;p©it 20mm .

kMNt;bMlas;TIrbs;cMNuc C EdlbNþalBIbnÞúk.
BicarNaEtT§iBlm:Um:g;Bt;enAkñúgr)ar ABC nig (>^$> kMNt;bMlas;TItamTisedkrbs;cMNuc C .
EI CatMélefr. eKmanTMrbgáb;enARtg;cMnuc A .
kMlaMgtamG½kSenAkñúg DB . E = 200GPa .
BicarNaEtT§iBlm:Um:g;Bt;.

(>^!> kMNt;mMurgViléncMNuc A éncMeNaT (>^0.
(>^@> )atrbs;FñwmrgnUvsItuNðPaB Tb = 120o C
xN³sItuNðPaBénEpñkxagelIrbs;FñwmKW Tt = (>^%> eRbIviFIkmμnþnimitþedIm,IkMNt;bMlas;TItam
10o C . RbsinebI α = 11.7(10 −6 )/ o C cUr
TisedkenARtg;cMNuc C . RkLaépÞmuxkat;rbs;
kMNt;bMlas;TItamTisQrrbs;cMNuc B Edl Ggát; nImYy²RtUv)anbgðajenAkñúgrUb. snμt;Ggát;



TaMgGs;RtUv)antP¢ab;edaytMNsnøak;. E =
200GPa .

(>&0> Edkbt;man E = 200GPa / G = 75GPa
nigmankaM 30mm . eRbIviFIkmμnþnimitþedIm,IkMNt;
bMlas;TItam TisQrenARtg;cMNuc C . Kit
(>^^> edaHRsaycMeNaT (>^%> edayeRbIRTwsþIka bBa©ÚlT§iBlfamBlbMErbMrYlrageFobEdlekItBI
sÞIKøIy:aNU. m:Um:g;Bt; kMlaMgkat; nigkMlaMgrmYl.
(>^&> eRbIviFIkmμnþnimitþedIm,IkMNt;bMlas;TItam
TisQrenARtg;TMr D . EI mantMélefr.

(>&!> edaHRsaycMeNaT (>&0> edayeRbIRTwsþIka
sÞIKøIy:aNU.
(>^*> edaHRsaycMeNaT (>^&> edayeRbIRTwsþIka (>&@> eRKagxageRkamrgnUvbnÞúk 20kN . kMNt;
sÞIKøIy:aNU. bMlas;TItamTisQrenARtg;cMNuc C . snμt;va
(>^(> eKmankgmYyenAelIépÞrwg ehIykgenHrg Ggát;tP¢ab;edaytMNsnøak;enARtg;cMNuc A / C
bnÞúkbBaÄr P . kMNt;bMlas;TItamTisQrenA nig E ehIytMN B nigtMN D CatMNbgáb;. EI
Rtg;cMNuc B . EI mantMélefr. mantMélefr. eRbIviFIkmμnþnimitþ.

cMeNaT T.Chhay -332

9. deflection using energy method

9. deflection using energy method

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