Analysis of Statically Determinate Structures

mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa

@> karviPaKrcnasm<½n§kMNt;edaysþaTic
Analysis of Statically Determinate Structures
enAkñúgCMBUkenH eyIgnwgBiPakSaBITMrg;Fmμtarbs;eRKOgbgÁú<MEdlvisVkrRtUveFVIkarsikSaviPaK
ehIyvak¾CaTMrg;EdlsßitenAkñúgbøg; ehIymüa:geTotRbB½n§kMlaMgGnuvtþn_k¾sßitenAelIbøg;eRKOgbgÁúMEdr.
eyIgcab;epþImBiPakSaCamYynwgPaBcaM)ac;kñúgkareRCIserIsm:UEDlrbs;eRKOgbgÁúMd¾smRsbedIm,IeyIgGac
kMNt;kMlaMgenAkñúgeRKOgbgÁúMeGaymantMélsuRkitEdlGacTTYlyk)an. bnÞab;mkeTot eyIgnwg
BiPakSaBIlkçxNÐcaM)ac;sMrab;sßirPaBrbs;eRKOgbgÁúM. ehIycugeRkay mankarbgðajBIkarviPaKeRKOg
bgÁúMsßitkñúgbøg;kMNt;edaysþaTicEdltP¢ab;edaytMN pin.

@>!> m:UEDleRKOgbgÁúM (Idealized structure)
Cak;Esþg eKminGacviPaKeRKOgbgÁúM)ansuRkiteT edaysareKRtUveFVIkar)a:n;RbmaNbnÞúk
nigPaBFn;rbs;sMPar³. elIsBIenH eKRtUveFVIkarsnμt;BITItaMgEdlbnÞúkRtUveFVIGMeBIr. dUcenH va
CakarsMxan;Nas;EdlvisVkrRtUvGPivDÆsmtßPaBkñúgkarbegáItm:UEDleRKOgbgÁúM. enAkñúgkfaxNÐ
enHeyIgnwgENnaMBIbec©keTsCamUldæanEdlcaM)ac;sMrab;kargarenH.
tMNTMr (Support Connection): Ggát;eRKOgbgÁúMRtUv)antamP¢ab;KñatamviFIeRcIn edayGaRs½yeTA
elIectnarbs;visVkr. tMNsamBaØbIRbePTEdleKsÁal;PaKeRcInKW tMNsnøak; (pin connection)/ TMr
rGil (roller support) nigtMNbgáb; (fixed joint). Pin-connection joint nig roller support GnuBaØat
eGayGgát;vil)an Et fixed joint minGnuBaØateGayGgát;vileT ehIyCaTUeTAkareFVI fixed joint TamTar
tMélx<s;. ]TahrN_éntMNTaMgenH ¬sMrab;Edk nigsMrab;ebtug¦RtUv)anbgðajenAkñúgrUbTI2-1 nig2-2
erogKña. tMNenAkñúgeRKOgbgÁúMeFVIGMeBIeQIPaKeRcInRtUv)aneKsnμt;faCa pin connection edaysarkar
P¢ab;edayb‘ULúg nigedayEdkeKalminGacpþl;nUvkarTb;karvil (restraint) )anRKb;RKan;eT.

Analysis of Statically Determinate Structures T.Chhay -24

Department of Civil Engineering NPIC

m:UEDleRKOgbgÁúMEdleRbIenAkñúgkarviPaKeRKOgbgÁúMEdltMNageGayTMrsnøak; (pinned support)
nigTMrbgáb; (fixed support); nigtMNsnøak; (pin-connected joint) nigtMNbgáb; (fixed-connected
joist) RtUv)anbgðajenAkñúgrUbTI 2-3 a nigrUbTI 2-3 b erogKña. EtkarBit RKb;tMNTaMgGs;manPaB

rwgRkajxøHkñúgkarTb;Tl;nwgkarvil. sMrab;krNIEbbenH m:UEDlEdlsaksmCagsMrab;TMr b¤tMNRtUv
)anbgðajenAkñúgrUbTI 2-3 c. RbsinebIefrkMrajTb;karrmYl (torsional spring constant) k = 0
eKKitfatMNCasnøak; ehIyRbsinebI k → ∞ eKKitfatMNCatMNbgáb;.

enAeBleRCIserIsm:UEDlsMrab;TMr b¤tMNnImYy²rYcehIy visVkrRtUvdwgfakarsnμt;nwgCHT§iBl
dl;kareFVIkarCak;Esþgrbs;Ggát; ehIyRtUvdwgfakarsnμt;enaHsmrmüsMrab;karsikSaKNnaeRKOgbgÁúM
b¤Gt;. ]TarhN_ eKmanFñwmdUcbgðajenAkñúgrUbTI 2-4 a edayeKeRbIvaeGayRTbnÞúkcMcMnuc P . tMN
EkgenARtg;cMnuc A dUckarbgðajenAkñúgrUbTI 2-1 a dUcenHeKGaceFVIm:UEDlTMrenHCaRbePTTMrsnøak; (pin
support). elIsBIenH TMrRtg;cMnuc B RKan;EtTl;Ggát;kMueGayFøak; dUcenHeKeFVIm:UEDlvaCaTMrkl; (roller).

kMras;rbs;FñwmRtUv)anecaledaysarvamanTMhMtUceFobnwgRbEvgrbs;Fñwm dUcenHm:UEDlrbs;FñwmRtUv)an
karviPaKrcnasm<½n§kMNt;edaysþaTic T.Chhay -25


bgðajenAkñúgrUbTI 2-4 b. kardak;bnÞúkenAelIm:UEDlFñwmmanlkçN³RsedogKñanwgkardak;bnÞúkenAelI
FñwmCak;Esþg. edIm,Ibgðajfam:UEDlenHmanlkçN³smRsb eyIgnwgBicarNakrNICak;lak;rbs;FñwmEdl
eFVIBIEdkEdlmanRbEvg 6.1m nigrgbnÞúk P = 35.6m ehIysnμt;TMrenARtg;cMnuc A Ca TMrsnøak;. kar
sikSaKNnaFñwmedayeRbI standard code procedures bgðajfa W 11× 22 mansmtßPaBRKb;RKan;kñúg
karRTbnÞúkenH. edayeRbIviFIkMNt;PaBdabmYyenAkñúgCMBUk8 mMurgVil (rotation) enARtg;TMrsnøak;KW
θ = 0.007rad = 0.40 o . BIrUbTI 2-4 c mMurgVilEbbenHeFVIeGaysøabxagelI nigsøabxageRkam pøas;TI
)ancMgay Δ = θr = (0.0070rad )(127mm) = 0.889mm . tMNEdlmanbgðajenAkñúgrUbTI 2-1 a BitCa
nwgpþl;nUvbrimaNd¾tUcEbbenHEmn dUcenHTMrsnøak;BitCasaksmsMrab;m:UEDlenH.

RbePTtMNepSgeTotEdlEtgEtCYbRbTHenAelIeRKOgbgÁúMkñúgbøg;RtUv)anbgðajenAkñúgtarag 2-
1. vaBitCamansarsMxan;Nas;kñúgkarsÁal;nimitþsBaØa nigRbePTRbtikmμrbs;tMN b¤TMr. TMrGacbegáIt
kMlaMgenAelIGgát;RbsinebIvaTb;Tl;nwgkarcl½trbs;Ggát; ehIyvanwgbegáItm:Um:g;RbsinebIvaTb;Tl;kar
vilrbs;Ggát;. Ca]TahrN_ sMrab;Ggát;Edlb:HnwgépÞrelag (3) RtUv)ankarBarnwgkarcl½tEtmYyTis
EdlEkgnwgépÞ. dUcenH épÞGnuvtþEdlkMlaMgEkg F eTAelIGgát;enAkñúgTisenaH. TMhMrbs;kMlaMgenH
tMNageGaymYyGBaØat (unknown). ]TahrN_epSgeTot TMrbgáb; (fixed support) (7) karBarTaMgkar
cl½t TaMgkarvilrbs;Ggát;enARtg;tMN. dUcenH TMrRbePTenHGnuvtþbgÁúMkMlaMgBIr nigm:Um:g;mYyeTAelI
Ggát;. dUcenH sMrab;TMrbgáb;manGBaØatTaMgGs;cMnYnbI.



enAkñúglkçN³Cak;Esþg TMrTaMgGs;GnuvtþkMlaMgBRgayelIépÞ (distributed surface load) eTAelI
Ggát;Edlb:HnwgTMr. kMlaMgcMcMnuc (concentrated load) nigm:Um:g; (moment) EdlbgðajenAkñúgtarag
2-1 tMNageGaykMlaMgpÁÜbénkarBRgaybnÞúkTaMgenH.
m:UEDleRKOgbgÁúM (Idealized structure): eday)anerobrab;BIviFIkñúgkarKitm:UEDléntMNenAelI
eRKOgbgÁúM eyIgGacBiPakSaBIbec©keTskñúgkarbegáItm:UEDleRKOgbgÁúMsMrab;RbB½n§eRKOgbgÁúMepSg².
dUc]TahrN_TImYy BicarNadgsÞÚc jib crane nig trolley enAkñúgrUbTI 2-5 a. sMrab;karviPaK
eRKOgbgÁúMeyIgGacecalkMras;rbs;Ggát;emTaMgBIr ehIyeyIgsnμt;fatMNenARtg;cMnuc B CatMNrwg
(rigid). elIsBIenH eKGaceFIVm:UEDltMNTMrenARtg;cMnuc A CaTMrbgáb; ehIykarlMGitBI trolley minRtUv

)anKit. dUcenH Ggát;rbs;m:UEDleRKOgbgÁúMRtUv)ansMEdgedaybnÞat;BIrtP¢ab;Kña ehIybnÞúkenAelI hook
RtUv)ansMEdgedaybnÞúkcMcMnuceTal F ¬rUbTI 2-5 b¦.

CaerOy² eKeRbIFñwm nigr:tedIm,IRTkMralrbs;GKar. r:t (girder) CaGgát;emEdlRTbnÞúkrbs;kM-
ral ÉGgát;muxkat;tUcCagEdlmanRbEvgElVgxøICag nigRtUv)antP¢ab;eTAnwgr:tRtUv)aneKeGayeQμaHfa
Fñwm (beam). bnÞúkEdlmanGMeBImkelIFñwm b¤r:tCabnÞúkEdl)anBIkMralEdlFñwm b¤r:tRT. Ca]TahrN_
eKmaneRKagEdleRbIedIm,IRTkMralkMral ¬rUbTI 2-6 a¦. kMralenHRtUv)anRTeday floor joist ehIy
Edl floor joist enHRtUv)anRTedayr:tBIr AB nig BC EdlenAsgxag. sMrab;karviPaKeKsnμt;fatMN
CatMNsnøak; b¤tMNrGilEdltP¢ab;eTAnwgr:t ehIyr:tRtUv)anRTedayssrEdleKsnμt;faCaTMrsnøak;
nig / b¤ TMrkl;. KMehIjBIxagelIrbs;RbB½n§eRKOgbgÁúMenHRtUv)anbgðajenAkñúgrUbTI 2-6 b. enAkñúgrUb
enHbnÞat; EdltageGay joist minb:Hnwgr:t ehIybnÞat;EdltMNageGayr:tminb:Hnwgssr. karminb:H


enHtMNageGaytMNTMrsnøak; nig / b¤tMNTMrkl;. müa:gvijeTot RbsinebIeKcg;bgðajbøg;eRKagenHCa
tMNbgáb; ¬eRbItMNpSarCMnYseGaytMNsamBaØedayb‘ULúg¦ enaHbnÞat;sMrab;Fñwm b¤r:tRtUvb:HnwgssrdUc
enAkñúgrUbTI 2-7. dUcKña eKRtUvbgðaj Fixed-connected overhanging beam kñúgKMehIjBIelIdUcbgðaj
enAkñúgrUbTI 2-8. RbsinebIkarsagsg;eRbIebtugGarem: Fñwm nigr:tRtUvtMNagedaybnÞat;Dub. CaTUeTA
RbB½n§enH Ggát;TaMgGs;RtUvP¢ab;edaytMNbgáb; dUcenHeKKUrP¢ab;Ggát;eTAnwgTMr. ]TahrN_ rUbTI 2-9 a
bgðajBIRkaPiceRKOgbgÁúMsMrab;RbB½n§ebtugGarem:Edlcak;enAnwgkEnøg (cast-in-place) ehIyrUbTI 2-9 b
bgðajBIbøg;emIlBIelI. bnÞat;sMrab;FñwmCabnÞat;dac;edaysarvasßitenAeRkamkMral.



RkaPiceRKOgbgÁúM nigm:UEDleRKOgbgÁúMsMrab;rcnasm<½n§GMBIeQImanlkçN³RsedogKñanwgrcnasm<½n§
GMBIEdk. RbB½n§eRKOgbgÁúMEdlbgðajenAkñúgrUbTI 2-10 a bgðajBIsMNg; beam-wall ÉkMraldMbUl (roof
deck) RtUv)anRTedayrnUteQI (wood joist) EdlbBa¢ÚnbnÞúkeTACBa¢aMgdæ. eKGacsnμt;farnUtRtUv)an

RTedayTMrsamBaØenAelICBa¢aMg dUcenHeKGaceFVIm:UEDleRKOgbgÁúMdUcbgðajenAkñúgrUbTI 2-10 b.

karepÞrbnÞúkBIépÞrgsMBaF (tributary loadings): enAeBlEdlépÞerobesμIdUcCaCBa¢aMg kMral b¤
dMbUlRtUv)anRTedayeRKag eKcaM)ac;RtUvkMNt;BIrebobénkarepÞrbnÞúkEdlmanGMeBIelIépÞenaHeTAGgát;
eRKOgbgÁúMEdlRTva. CaTUeTA eKmanviFIBIrkñúgkarkMNt;va. kareRCIserIsviFIGaRs½ynwgragFrNImaRt
rbs;RbB½n§eRKOgbgÁúM sMPar³EdlvaeFVI nigviFIkñúgkarsagsg;va.
RbB½n§mYyTis (One-way system): karRTkMralEdlbBa¢ÚnbnÞúkeTAGgát;TMrtamskmμPaBmYyTis
RtUv)aneKeGayeQμaHfa kMralxNÐmYyTis (one-way slab). edIm,IbgðajBIviFIénkarbBa¢ÚnbnÞúk cUrBi-
carNaRbB½n§eRKagdUcbgðajenAkñúgrUbTI 2-11 a EdlFñwm AB / CD nig EF sßitenABIelIr:t AE nig
BF . RbsinebIeKdak;bnÞúkrayesμI 4.8kN / m 2 enAelIkMral enaHeKsnμt;faFñwm CD RTbnÞúkEdlmanGMeBI

enAelIépÞrgsMBaF (tributary area) AB EdlbgðajedayBN’ditenAelIbøg;eRKagbgÁúMenAkñúgrUbTI 2-11b.
dUcenH Ggát; CD rgnUvkarBRgaybnÞúkefr (4.8kN / m 2 )(1.5m) = 7.2kN / m dUcbgðajenAelIm:UEDlFñwm
enAkñúgrUbTI 2-11c. bnÞab;RbtikmμenAelIFñwmenH (10.8kN ) GnuvtþeTAelIr:t AE ¬nig BF ¦ cMkNþal
dUcbgðajenAkñúgrUbTI 2-11d. edayeRbIviFIKMnitkñúgkarbBa¢ÚnbnÞúkEbbenH etIGñkmaneXIjcMENkrbs;
kMralEdlenAsl;RtUv)anepÞreTAcugrbs;r:tKW 5.4kN b¤eT?



sMrab;RbB½n§kMralxøH eKtP¢ab;Fñwm nigr:teTAnwgssrenARtg;nIv:UdUcKña dUcbgðajenAkñúgrUbTI 2-12
a. RbsinebIeKCYbkrNIEbbenH eBlxøHeKk¾GacKitfakMralCakMralxNÐmYyTisEdr. ]TahrN_ Rbsin

ebIkMralxNÐebtugGarem:EdlmanEdkBRgwgEtkñúgTismYy b¤eKcak;ebtugenAelIkMragEdlpñt; (corrugated
metal deck) enaHeKsnμt;fabnÞúkRtUv)anbBa¢ÚntamskmμPaBmYyTis. müa:gvijeTot RbsinebIkMralCa

kMralerobesμITaMgEpñkxagelI nigEpñkxageRkam ehIyRtUv)anBRgwgedayEdkkñúgTisTaMgBIr enaHeKKitfa
bnÞúkRtUv)anbBa¢ÚneTAGgát;TMrtammYy Tis b¤tamBIrTis. ]TahrN_ eKmankMral nigbøg;eRKagdUcenAkñúg
rUbTI 2-12b tamlkçxNÐTUeTA RbsinebI L2 ≥ L1 ehIyRbsinebIpleFobElVg (L2 / L1 ) ≥ 2 kMralxNÐ
nwgeFVIkarCakMralxNÐmYyTis. edaysarEt L1 kan;EttUc Fñwm AB / CD nig EF pþl;nUvPaBrwgRkaj
kan;EtFMkñúgkarRTbnÞúk.



RbB½n§BIrTis (Two-way system): RbsinebIpleFobElVgenAkñúgrUbTI 2-12 b KW (L 2 / L1 ) < 2

eKsnμt;fabnÞúkRtUv)anbBa¢ÚneTAFñwm nigr:ttamskmμPaBBIrTis. sMrab;kMralEdlCYbkrNIEbbenH RtUv
)aneKeGayeQμaHfa kMralxNÐBIrTis (two-way slab). edIm,IbgðajBIviFIkñúgkaredaHRsaykrNIenH cUr
BicarNakMralebtugGarem:ragkaerenAkñúgrUbTI 2-13 a EdlRTedayFñwmxagcMnUnbYn AB / BD / DC
nig DC EdlmanRbEvg 3m . enATIenH L2 / L1 = 1 . edaysarkMralxNÐrgskmμPaBBIrTis épÞrgsMBaF
sMrab;Fñwm AB RtUv)anbgðajedayBN’ditenAkñúgrUbTI 2-13b. épÞenHRtUv)ankMNt;edaybnÞat;Ggát;
RTUg 45o dUcbgðaj. dUcenH RbsinebIbnÞúkrayesμI 4.8kN / m 2 GnuvtþenAelIkMral GaMgtg;sIuetrbs;bnÞúk
EdlGnuvtþenARtg;cMnuckNþalFñwmKW (4.8kN / m 2 )(1.5m) = 7.2kN / m ehIybnÞúkEdlBRgayenAelIFñwm
enHmanragRtIekaNdUcbgðajenAkñúgrUbTI 2-13c. sMrab;]ThrN_enAkñúgkrNI L2 / L1 = 1.5 eKcaM)ac;
sg;bnÞat; 45o dUcbgðajenAkñúgrUbTI 2-14a. bnÞúk 4.8kN / m 2 EdlGnuvtþenAelIkMralnwgbegáItbnÞúk
BRgayragctuekaNBñay nigragRtIekaNenAelIGgát; AB nig AC dUcbgðajenAkñúgrUbTI 2-14b nig2-
14c erogKña.



]TahrN_2-1³ kMralrbs;fñak;eronmYyRtUv)anRTedayrnUtr)ar (bar joist) dUcbgðajenAkñúgrUbTI 2-
15 a. rnUtnImYy²manRbEvg 4.5m ehIyvamanKMlatBIKña 0.75m KitBIG½kS. kMralenHRtUv)ansagsg;BI
ebtugTMgn;RsalEdlmankMras;100mm . edayminKitTMgn;pÞal;rbs;rnUt nigkMralEdkpñt; (corrugated
metal deck) cUrkMNt;bnÞúkEdlmanGMeBIenAelIrnUtnImYy².

dMeNaHRsay³
bnÞúkefrenAelIkMralKWbNþalBITMgn;rbs;kMralebtug. BItarag 1-3 TMgn;ebtugTMgn;RsalkMras; 100mm
KW (100)(0.015kN / m 2 ) = 1.50kN / m 2 . BItarag1-4 bnÞúkGefrsMrab;fñak;eronKW1.92kN / m 2 . dUcenH
bnÞúkkMralsMrubKW1.50klN / m 2 + 1.92kN / m 2 = 3.42kN / m 2 . sMrab;RbB½n§kMral L1 = 0.75m nig
L2 = 4.5 . edaysar L2 / L1 > 2 eKKitkMralxNÐCakMralxNÐmYyTis. épÞrgsMBaFsMrab;rnUtnImYy²

RtUv)anbgðajenAkñúgrUbTI 2-15 b. dUcenH bnÞúkrayesμIenArnUtKW
w = 3.42kN / m 2 (0.75m ) = 2.57 kN / m
bnÞúk nigRbtikmμenAelIGgát;rnUtnImYy²RtUv)anbgðajenAkñúgrUbTI 2-15c.

]TahrN_2-2³ kMraldMbUlrabesμIénGKareRKagEdkenAkñúgrUbTI 2-16a RtUv)aneKeRbIedIm,IRTbnÞúk
srub 2kN / m 2 EdlmanGMeBIelIépÞkMral. RbsinebIFñwm AD nig BC manRbEvg 5m ehIymanKMlat
3m ¬Fñwm AB nig DC ¦. cUrkMNt;bnÞúkdMbUl (roof load) enAkñúgtMbn; ABCD EdlRtUv)anbBa¢ÚneTAFñwm

BC .



dMeNaHRsay³
enAkñúgkrNIenH L1 = 5m ehIy L2 = 3m . edaysar L2 / L1 < 2 enaHkMralxNÐeFVIkartamBIrTis.
bnÞúkenAelIépÞrgsMBaFRtUv)anbgðajenAkñúgrUbTI 2-16b EdlépÞctuekaNBñayénbnÞúkEdlmanBN’dit
RtUv)anbBa¢ÚneTAGgát; BC . GaMgtg;sIuetrbs;bnÞúkenARtg;kMBUlKW (2kN / m 2 )(1.5m) = 3kN / m . Ca
lT§pl karBRgaybnÞúkenAelIGgát; BC RtUv)anbgðajenAkñúgrUbTI 2-16b. eKk¾KYreFVIkarKNnakar
bBa¢únbnÞúkenAelIépÞrgsMBaFsMrab;tMbn;kaerBIrEdlenAxagsþaMrbs;Ggát; BC enAkñúgrUbTI 2-16a ehIy
bnÞúkEdl)anKNnabEnßmenHk¾RtUv)andak;enAelIGgát; BC Edr.

@>@> eKalkarN_tMrYtpl (Principle of superposition)
eKalkarN_tMrYtplbegáItnUvmUldæansMrab;RTwsþIviPaKeRKOgbgÁúM (theory of structural) CaeRcIn.
eKalkarN_BN’nafa³ eKGackMNt;bMlas;TIsrub b¤kMlaMgxagkñúg (internal loading or force) srubenA
Rtg;cMnucNamYyenAelIeRKOgbgÁúMEdlrgbnÞúkxageRkACaeRcInedaykarbUkbMlas;TI b¤kMlaMgxagkñúgEdl
bNþalBIbnÞúkxageRkAnImYy²bBa©ÚlKña. edIm,IGnuvtþeKalkarN_tMrYtplenH)an bnÞúk kugRtaMg nigbM-
las;TIRtUvEtCalIenEGr (linear).
edIm,IGnuvtþeKalkarN_pl eKRtUvbMeBjtMrUvkarcMnYnBIrxaeRkam³
sMPar³RtUvEteFVIkarenAkñúgTMrg;lIenEGreGLasÞic (linear-elastic) dUcenHmann½yfaeKGac
eRbIc,ab;h‘Uk (Hook’s law) )an ehIybnÞúksmamaRteTAnwgbMlas;TI.



ragFrNImaRtrbs;eRKOgbgÁúMminRtUvrgkarERbRbYlFMeRkamGMeBIrbs;bnÞúkeT mann½yfaeK
GnuvtþRTwsþIbMlas;TItUc (small displacement theory). bMlas;TIFMnwgpøas;bþÚrTItaMg nig
Tisrbs;bnÞúky:agxøaMg. ]TahrN_ r)arEdkkugs‘ulesþIgEdlrgkMlaMgenAxagcugrbs;va.
enAkñúgesovePAenH BN’naEtBIsMPar³lIenEGreGLasÞic ehIybMlas;TIEdlekItedaysabnÞúk
minpøas;bþÚrTisedArbs;bnÞúkGnuvtþn_FM ehIyk¾minpøas;bþÚrTMhMEdleRbIsMrab;KNnam:Um:g;énkMlaMgFMEdr
dUcenHtMrUvkarTaMgxagelIRtUv)anbMeBj.

@>#> smIkarlMnwg (Equation of equilibrium)
enAkñúgsþaTic)anEcgfa eRKOgbgÁúM b¤Ggát;mYyéneRKOgbgÁúMmanlMnwg enAeBlNaEdlvaGacrkSa
lMnwgénkMlaMg niglMnwgénm:Um:g;. CaTUeTA eKRtUvbMeBjnUvlMnwgénsmIkarkMlaMg nigsmIkarm:Um:g;tam
G½kSTaMgbI mann½yfa³
∑ Fx = 0 ∑ Fy = 0 ∑ Fz = 0
∑Mx = 0 ∑M y = 0 ∑Mz = 0 (2-1)

b:uEnþ PaKeRcIneRKOgbgÁúMeRcInRTcMENkbnÞúkEdlsßitenAkñúgbøg;eTal ehIyedaysarEtbnÞúkk¾sßitenAkñúg
bøg;enaH smIkartMrUvkarxagelIkat;bnßymkRtwm
∑ Fx = 0 ∑ Fy = 0 ∑ MO = 0 (2-2)

enATIenH ∑ Fx nig ∑ Fy CaplbUkBiCKNiténbgÁúMkMlaMgtamG½kS x nigtamG½kS y erogKña EdlmanGMeBI
mkelIeRKOgbgÁúM b¤Ggát;mYyrbs;eRKOgbgÁúM ehIy ∑ M O CaplbUkBiCKNiténm:Um:g;énkMlaMgCMuvijG½kS
EdlEkgnwgbøg; x − y ¬G½kS z ¦ ehIykat;tamcMnuc O .
enARKb;eBlEdleKGnuvtþsmIkarTaMgenH dMbUgeKcaM)ac;RtUvKUrdüaRkamGgÁesrI (free body
diagram) rbs;eRKOgbgÁúM b¤rbs;Ggát;. edIm,IKUrdüaRkamGgÁesrI eKRtUvpþac;Ggát;ecjBIKña nigecjBITMr.

kMlaMgxageRkA kMlaMgkñúg b¤kMlaMgRbtikmμRtUvbgðajenAelIGgát;Edl)anpþac;ecj. tamry³enH eKGac
kMNt;kMlaMgRbtikmμenARtg;TMredayeRbItarag 2-1. ehIyeyIgRtUvcaMpgEdrfa enAelIdüaRkamGgÁesrI
kMlaMgkñúgénGgát;BIrEdlP¢ab;KñamanTMhMdUcKña EtmanTispÞúyKña.
RbsinebIeKcg;kMNt;kMlaMgkñúgenARtg;cMnucCak;lak;enAkñúgGgát; eKRtUveRbIviFImuxkat; (method
of section). viFIenHtMrUveGaykat;Ggát;tamTisEkgeTAnwgG½kSrbs;Ggát;Rtg;cMnucEdleKRtUvkarkMNt;

kMlaMgkñúg. eKRtUvKUrdüaRkamGgÁesrIénkMNat;Ggát;nImYy²Edl)ankat; rYckMNt;kMlaMgkñúgedayGnuvtþ
smIkarlMnwgeTAelIkMNat;Ggát;enaH. CaTUeTA kMlaMgkñúgEdlmanGMeBIenAelImuxkat;rbs;Ggát;Edlkat;
rYmmankMlaMgEkg (normal force) N / kMlaMgkat; (shear force) V nigm:Um:g;Bt; (bending moment)


M dUcbgðajenAkñúgrUbTI 2-17.
eyIgnwgBiBN’naeKalkarN_sþaTicEdleRbIedIm,IkMNt;RbtikmμxageRkAenAelIeRKOgbgÁúMenAkñúg
kfaxNÐ 2.5. eyIgnwgBiPakSaBIkMlaMgkñúgenAkñgGgát;eRKOgbgÁúMenAkñúgCMBUkTI 4.
ú

@>$> PaBkMNt; nigsßirPaB (Determinacy and stability)
muxnwgcab;epþImviPaKkMlaMgrbs;eRKOgbgÁúM eKcaM)ac;kMNt;nUvPaBkMNt; nigsßirPaBrbs;eRKOgbgÁúM.
PaBkMNt; (determinacy): smIkarlMnwgpþl;nUvlkçxNÐcaM)ac; nigRKb;RKan;sMrab;lMnwg. enAeBl
EdlRKb;kMlaMgTaMgGs;enAkñúgeRKOgbgÁúMGackMNt;edayeRbIsmIkarlMnwgxagelI enaHeKehAeRKOgbgÁúM
enaHCa eRKOgbgÁúMkMNt;edaysþaTic (statically determinate structure). sMrab;eRKOgbgÁúMEdlmancMnYn
kMlaMgCaGBaØateRcInCagcMnYnsmIkarEdlGacman eKeGayeQμaHvafa eRKOgbgÁúMminkMNt;edaysþaTic
(statically indeterminate structure). tamc,ab;TUeTA eKGackMNt;faeRKOgbgÁúMmYyCaeRKOgbgÁúMkMNt;

edaysþaTic b¤k¾eRKOgbgÁúMminkMNt;edaysþaTicedayKUrdüaRkamGgÁesrIénGgát;TaMgGs;rbs;va b¤épñkén
Ggát;EdleRCIserIs ehIybnÞab;mkeRbobeFobcMnYnGBaØatsrubEdlCakMlaMg nigm:Um:g;CamYynwgcMnYn
srubénsmIkarlMnwgEdlGacman*. sMrab;eRKOgbgÁúMkñúgbøg; eKmansmIkarlMnwgbIsMrab;Ggát;nImYy²
dUcenHRb sinebIeKmanGgát;srubcMnYn n ehIykMlaMgRbtikmμ nigm:Um:g;RbtikmμcMnYn r enaHeyIg)an
r = 3n / kMNt;edaysþaTic

r > 3n / minkMNt;edaysþaTic (2-3)

CaebIvaCaeRKOgbgÁúMminkMNt;edaysþaTic
enaHeKRtUvkarsmIkarbEnßmedIm,IedaHRsayrkRbtikmμEdlCaGBaØatedayP¢ab;TMnak;TMngrvagbnÞúkGnuvtþ
n_ nigRbtikmμeTAnwgbMlas;TI b¤mMurgVilenARtg;cMnucepSgKñaenAelIRKOgbgÁúM. smIkarTaMgenHCasmIkarRtUv
Kña (compatibility equation) ehIyvaRtUvmancMnYnesμInwgdWeRkénPaBminkMNt;rbs;eRKOgbgÁúM. smIkar
compatibility Tak;TgnwglkçN³FrNImaRt niglkçN³rUbrbs;eRKOgbgÁúM ehIyeyIgnwgelIkykvamk

*
karKUrdüaRkamGgÁesrImincaM)ac;CaTUeTAeT edaysareKGacrab;cMnUnGBaØat ehIyeFVIkareRbobeFobCamYynwgcMnUnsmIkar
lMnwg)an.


BN’naenAkñúgCMBUkTI 10.
]TahrN_ 2>3³ cat;cMNat;fñak;rbs;FñwmnImYy²EdlbgðajenAkñúgrUbTI 2-18 a rhUtdl; 2-18 d Ca
eRKOgbgÁúMkMNt;edaysþaTic b¤eRKOgbgÁúMminkMNt;edaysþaTic. eKsnμt;faeKsÁal;kMlaMgEdlGnuvtþmin
elIFñwm ehIyvaGacmanGMeBIenARKb;TItaMgTaMgGs;rbs;Fñwm.
dMeNaHRsay³
eKRtUvpþac;FñwmpSM (compound beam) enAkñúgrUbTI 2-18c nig 2-18d EdlpSMeLIgGgát;EdltP¢ab;Kña
edaytMNsnøak;ecjBIKña. cMNaMfa enAkñúgkrNITaMgenH kMlaMgRbtikmμEdlCaGBaØatEdlmanGMeBIenA
cenøaH Ggát;nImYy²RtUv)anbgðajCaKU EtmanTispÞúyKña. düRkamGgÁesrIRtUv)anbgðaj. edayGnuvtþ
r = 3n b¤ r > 3n lT§plénkarcat;cMNat;fñak;RtUv)anbgðaj.

]TahrN_ 2>4³ cat;cMNat;fñak;rbs;eRKOgbgÁúMEdltP¢ab;edaytMNsnøak;EdlbgðajenAkñúgrUbTI 2-19
a rhUtdl; 2-19 d CaeRKOgbgÁúMkMNt;edaysþaTic b¤CaeRKOgbgÁúMminkMNt;edaysþaTic. RbsinebIvaCa
eRKOgbgÁúMminkMNt;edaysþaTic cUrR)ab;BIcMnYnéndWeRkénPaBminkMNt;. eRKOgbgÁúMrgnUvbnÞúkxageRkAEdl


eKsnμt;sÁal; ehIyvamanGMeBIenARKb;TItaMgTaMgGs;rbs;eRKOgbgÁúM.
dMeNaHRsay³
karcat;cMNat;fñak;én;eRKOgbgÁúMEdltP¢ab;edaytMNsnøak;manlkçN³RsedogKñanwgFñwmEdr. düRkamGgÁ
esrIRtUv)anbgðaj. edayGnuvtþ r = 3n b¤ r > 3n lT§plénkarcat;cMNat;fñak;RtUv)anbgðaj.

]TahrN_ 2>5³ cat;cMNat;fñak;rbs;eRKagEdlbgðajenAkñúgrUbTI 2-20a nig2-20d CaeRKOgbgÁúM
kMNt;edaysþaTic b¤CaeRKOgbgÁúMminkMNt;edaysþaTic. RbsinebIvaCa eRKOgbgÁúMminkMNt;edaysþaTic
cUrR)ab;BIcMnYnéndWeRkénPaBminkMNt;. eRKOgbgÁúMrgnUvbnÞúkxageRkAEdl eKsnμt;sÁal; ehIyvaman
GMeBIenARKb;TItaMgTaMgGs;rbs;eRKOgbgÁúM.
dMeNaHRsay³
karkMNt;sßirPaB nigPaBkMNt;rbs;eRKOgbgÁúMeRKagmindUcKñanwgkarkMNt;sMrab;Fñwm nigeRKOgbgÁúMEdlt


P¢ab;edaytMNsnøak;eT eRBaHeRKagpSMeLIgedayGgát;EdltP¢ab;Kñaedaysnøak;rwg. eBlxøHGgát;begáIt
tMbn;biTCitxagkñúg dUcenAkñúgrUbTI 2-20a. enATIenH ABCD begáIttMbn;biTCit. edIm,Icat;cMNat;fñak;
rbs;eRKOgbgÁúMTaMgenH eKcaM)ac;RtUveRbIviFImuxkat; ehIykat;EpñkEdlbiTCit. eRkayeBleKKUrdüaRkam
GgÁesrIrbs;EpñkEdlkat;rYcehIy enaHeKGaccat;cMNat;fñak;rbs;eRKag)an. cMNaMfa eKRtUvkarEtEpñk
mYyrbs;muxkat;Edlkat;tamtMbn;biTCit. enAeBlEdleKkMNt;GBaØatenARtg;muxkat;rYcehIy eKGac
kMNt;kMlaMgkñúgenARKb;cMnucTaMgGs;enAkñúgGgát;edayeRbIviFImuxkat; nigsmIkarlMnwg. eRKagenAkñúgrUbTI
2-20b manmintMbn;biTCit dUcenHedIm,IrkPaBkMBt;vaminRtUvkarkarkat;cenøaHTMreT. lT§plénkarcat;
cMNat;fñak;RtUv)anbgðajenAkñúgrUbTI 2-20a nig 2-20b.

sßirPaB (stability): edIm,IFananUvsßanPaBlMnwgrbs;eRKOgbgÁúM nigGgát;rbs;va eKminRtwmEtbMeBj
smIkarlMnwgb:uENÑaHeT b:uEnþGgát;RtUvEtTb; (constraint) y:agl¥ nigRtwmRtUvedayTMrrbs;va. sßanPaB
BIrnwgekItmaneLIgenAeBlNaEdlTMrmin)anTb;RtwmRtUv.
karTb;mineBjelj (partial constraints): enAkñúgkrNIxøH eRKOgbgÁúM b¤Ggát;rbs;vaGacman
kMlaMgRbtikmμticCagsmIkarlMnwgEdlRtUvbMeBj eBlenaHeRKOgbgÁúMnwgkøayCaeRKOgbgÁúMEdlTb;mineBj
elj (partially constrained structure). ]TahrN_ eKmanGgát;EdlbgðajenAkñúgrUbTI 2-21 CamYy
nwgdüaRkamGgÁesrIrbs;va. enATIenHsmIkar ∑ Fx = 0 minRtUv)anbMeBjsMrab;lkçxNÐdak;bnÞúk dUcenH
Ggát;KμansßirPaB.


karTb;minRtwmRtUv (improper constraints): enAkñúgkrNIxøHEdlcMnYnénkMlaMgEdlCaGBaØat nig
cMnYnénsmIkarlMnwgesμIKña b:uEnþvaenAEtGaceFIVeGayeRKOgbgÁúMcl½t)an b¤KμansßirPaB edaysarEtkarTb;
minRtwmRtUvrbs;TMr. krNIEbbenHekItmanenAeBlEdlExSskmμénkMlaMgRbtikmμrbs;TMrkat;KñaRtg;cMnuc
mYy. ]TahrN_sMrab;krNIRtUv)anbgðajenAkñúgrUbTI 2-22. BIdüaRkamGgÁesrIrbs;Fñwm eyIgeXIjfa
plbUkm:Um:g;Rtg;cMnuc O nwgminesμIsUnüeT ¬ Pd ≠ 0 ¦ dUcenHvanwgekItmankarvilCuMvijcMnuc O .



viFImü:ageTotEdlkarTb;minRtwmRtUvrbs;TMrnaMeGaymaneRKOgbgÁúMKμansßirPaBekItmanenAeBl
EdlkMlaMgRbtikmμTaMgGs;manTisRsbKña. ]TahrN_sMrab;krNIRtUv)anbgðajenAkñúgrUbTI 2-23. enA
eBlEdleKGnuvtþkMlaMgeRTt P plbUkkMlaMgtamTisedknwgminesμIsUnüeT.
CaTUeTA eRKOgbgÁúMnwgKμansßirPaBedaylkçN³FrNImaRt ¬rMkiltictYc b¤dYlrlM¦ RbsinebIva
mankMlaMgRbtikmμticCagsmIkarlMnwg b¤RsinebIvamankMlaMgRbtikmμRKb;RKan; EtGsßirPaBrbs;nwgekIt
manRbsinebIExSskmμénkMlaMgRbtikmμkat;KñaRtg;cMnucmYy b¤k¾RsbKña. RbsinebIeRKOgbgÁúMpSMeLIgeday
Ggát;eRcIn eKGackMNt;GsßirPaBrbs;Ggát;mYy b¤rbs;Ggát;eRcInedaykarGegát. RbsinebIGgát;begáIt
collapsible mechanism )an eRKOgbgÁúMnwgminmansßirPaB. eyIgGackMNt;sßirPaBtamrUbmnþ)ansMrab;

eRKOgbgÁúMkñúgbøg;Edlman n nigGBaØatEdlCakMlaMgRbtikmμ r ehIyedaysarGgát;nImYy²RtUvman
smIkarlMnwgcMnYnbI enaH
r < 3n KμansßirPaB
r ≥ 3n KμansßirPaBenAeBlEdlkMlaMgRbtikmμrbs;Ggát;
kat;KñaRtg;cMnucmYy b¤RsbKña b¤Ggát;xøHbegáIt
collapse mechanism (2-4)
eKRtUveCosvaRbePTeRKOgbgÁúMEdlKμansßirPaBeTaHbICavaCaeRKOgbgÁúMkMNt;edaysþaTic b¤eRKOg
bgÁúMminkMNt;edaysþaTick¾eday.

]TahrN_ 2>6³ cUrcat;cMNat;fñak;rbs;eRKOgbgÁúMnImYy²enAkñúgrUbTI 2-24a rhUtdl;2-24e CaeRKOg
bgÁúMmansßirPaB b¤KμansßirPaB. eRKOgbgÁúMrgnUvbnÞúkxageRkANamYyEdleyIgsnμt;vasÁal;.
dMeNaHRsay³ eRKOgbgÁúMRtUv)ancat;cMNat;fñak;dUcbgðajkñúgrUb.



@>%> karGnuvtþénsmIkarlMnwg (Application of the equation of equilibrium)
eBlxøH Ggát;rbs;eRKOgbgÁúMRtUv)antP¢ab;KñaedaytMNEdleKsnμt;faCatMNsnøak;. eRKagGKar
nig truss CaRbePTeRKOgbgÁúMEdlCaerOy²sagsg;edayeRbItMNEbbenH. kñúgkrNIeRKOgbgÁúMkñúgbøg;
EdltP¢ab;edaysnøak;mankarTb;RtwmRtUv nigminmanTMrelIs b¤Ggát;elIsedIm,IkarBarkardYlrlM enaHeK
GackMlaMgEdlmanGMeBIenARtg;tMN nigenARtg;TMredayGnuvtþsmIkarlMnwgTaMgbI ¬ ∑ Fx = 0 / ∑ Fy = 0
∑ M O = 0 ¦ eTAelIGgát;nImYy². enAeBlEdleKTTYl)ankMlaMgenARtg;tMN eKGacsikSaKNaTMhM

rbs;Ggát; kartP¢ab; nigTMredayEp¥kelI design code specification.

edIm,IbgðajBIviFIénkarviPaKkMlaMg BicarNaeRKagEdlmanGgát;bIdUcbgðajenAkñúgrUbTI 2-25a
EdlrgnUvbnÞúk P1 nig P2 . düaRkamGgÁesrIrbs;Ggát;nImYy²RtUv)anbgðajenAkñúgrUbTI 2-25b. Ca
srub eKmanGBaØatTaMgGs; 9 ehIyeKmansmIkarlMnwgTaMgGs; 9 EdlGgát;nImYy²mansmIkarlMnwg 3
dUcenHeRKOgbgÁúMenHCaeRKOgbgÁúMkMNt;edaysþaTic. sMrab;dMeNaHRsayCak;Esþg ¬eBlxøHmanlkçN³
gayRsYl¦ eKGacBicarNaeRKagCaEpñk b¤k¾eRKagTaMgmUl enAeBlEdlGnuvtþsmIkarxøHénsmIkarTaMg
9 enH. Ca]TahrN_ düaRkamGgÁesrIéneRKagTaMgmUlRtUv)anbgðajenAkñúgrUbTI 2-25c. eKGackMNt;

kMlaMgRbtikmμTaMgbI Ax / Ay nig C x enAelIRbB½n§eRKOgbgÁúMEdltP¢ab;edaytMNsnøak;rwgraMenH bnÞab;mk
viPaKGgát;BIrkñúgcMeNamGgát;TaMgGs;enAkñúgrUbTI 2-25b edIm,ITTYl)anGBaØat 6 epSgeTot. elIsBI
enH eKGacRtYtBinitücMeLIyCaEpñkedayGnuvtþsmIkarlMnwgTaMgbIeTAelIGgát;TIbIEdlenAsl;. Casegçb
eKGacedaHRsaybBaðaenHedaysresrsmIkarlMnwgy:ageRcInNas;cMnYn 9 edayeRbIdüaRkamGgÁesrIén
Ggát;nImYy² nig / b¤bnSMénGgát;EdltP¢ab;. smIkareRkABIsmIkarTaMg 9 xagelIRtUv)aneRbIedIm,IRtYt
BinitülT§pl.


LÚvBicarNaeRKagEdlmanGgát;BIrdUcbgðajenAkñúgrUbTI 2-26a . enATIenHdüaRkamGgÁesrI
rbs;Ggát;bgðajGBaØatcMnYn 6 ¬rUbTI 2-26b¦ ehIyeyIgGacsresrsmIkarlMnwgcMnYn 6 EdlGgát;
nImYy²mansmIkarlMnwgcMnYn 3 dUcenHvaCaeRKOgbgÁúMkMNt;edaysþaTic. dUckñúgkrNIelIkmun eKk¾Gac
eRbIdüaRkamGgÁesrIrbs;eRKOgbgÁúMTaMgmUl ¬rUbTI 2-26c¦ edIm,IviPaKpgEdr. dUcbgðajenAkñúgrUb
eRKagnwgdYlrlMRbsinebIKμanTMr edayvanwgvilCMuvijsnøak; B EtkrNIenHnwgminekItmaneT edaysar
RbB½n§kMlaMgEdlmanGMeBIenAelIeRKagTaMgGs;eFVIeGayeRKagmanlMnwg. dUcenH
eKGacKNnaGBaØatTaMg 6 enHedayGnuvtþsmIkarlMnwgTaMgbIeTAelIeRKagTaMgmUl ¬rUbTI 2-26c¦
ehIydUcKñaGnuvtþsmIkarTaMg bIeTAelIGgát;mYykñúgcMeNamGgát;TaMgBIr.
]TahrN_TaMgBIrxagelIbgðajfa RbsinebIeRKagRtUv)anRTedayTMrd¾RtwmRtUv nigminmanTMr b¤
Ggát;caM)ac;eRcInCagenHedIm,IkarBarkardYlrlM enaHeRKagCaeRKagkMNt;edaysþaTic ehIyeKGackMNt;
GBaØatEdlCakMlaMgenARtg;TMr nigtMNBIsmIkarlMnwgedayGnuvtþeTAelIGgát;nImYy². ehIyRbsinebI
eRKOgbgÁúMenAEtrwgmaM ¬minGacdYlrlM¦ enAeBleKdkTMrecj ¬rUbTI 2-25c¦ enaHeKGackMNt;Rbtikmμ
TMrTaMgbIedayGnuvtþsmIkarlMnwgeTAelIeRKOgbgÁúMTaMgmUl. b:uEnþRbsinebIeRKOgbgÁúMbgðajPaBminrwgmaM
¬GacdYlrlM¦ bnÞab;BIeKdkTMrecj ¬rUbTI 2-26c¦ enaHeKRtUvpþac;Ggát;TaMgenaHecjBIKña ehIyeKRtUv
BicarNalMnwgrbs;Ggát;nImYy²edIm,ITTYl)ansmIkarRKb;RKan;edIm,IkMNt;RbtikmμTMrTaMgGs;.



dMeNIrkarkñúgkarviPaK (Procedure for analysis)
CMhanxageRkampþl;nUvviFIsMrab;kMNt;kMlaMgRbtikmμsMrab;eRKOgbgÁúMEdlpSMedayGgát;EdltP¢ab;
edaytMNsnøak;.
düaRkamGgÁesrI (Free-body diagrams)
pþac;rcnasm½<n§ nigKUrdüaRkamGgÁesrIénGgát;nImYy². ehIy vanwgkan;EtgayRsYl Rbsin
ebIeyIgKUrdüaRkamGgÁesrIrbs;eRKOgbgÁúMTaMgmUlbEnßmBIelIdüaRkamGgÁesrIrbs;Ggát;nI-
mYy². eKGackMNt;RbtikmμTMrxøH b¤TaMgGs;edayeRbIdüaRkamenH.
eyIgRtUvcgcaMfakMlaMgRbtikmμrbs;Ggát;BIreFVIGMeBICamYynwgTMhMesμIKña b:uEnþmanTisedApÞúy
KñaenAelIdüaRkamGgÁesrIrbs;Ggát;.
enAkñúgkrNIxøH eyIgGacdwgBITisedArbs;kMlaMgEdlCaGBaØat b:uEnþRbsinebIeyIgBi)akkñúg
karkMNt;TisedArbs;va eyIgGacsnμt;)an.
smIkarlMnwg (Equation of equilibrium)
rab;cMnYnGBaØatsrubedIm,IeGayc,as;faeKGacsresrcMnYnsmIkarlMnwgsmmUlsMrab;kar
edaHRsay. CaTUeTAeKGacsresrsmIkarlMnwgbIsMrab;Ggát;nImYy² elIkElgGgát;Edl
mankMlaMgBIr.
dMeNaHRsaysMrab;GBaØatnwgminsμúKsμaj RbsinebIeKGnuvtþsmIkarm:Um:g; ∑ M O = 0 kan;
EteRcIn.
enAeBlGnuvtþsmIkarkMlaMg ∑ Fx = 0 nig ∑ Fy = 0 eKRtUvbMEbkkMlaMgpÁÜbeGayeTACa
bgÁúMkMlaMgtamG½kS x nigG½kS y .
RbsinebIdMeNaHRsayénsmIkarlMnwgeGayCaTMhMGviC¢mansMrab;kMlaMg b¤m:Um:g;KUEdlCa
GBaØat vabgðajfaTisedArbs;k,alRBYjpÞúyBIGVIEdleyIg)ansnμt;enAelIdüaRkamGgÁesrI.
]TahrN_ 2>7³ kMNt;kMlaMgRbtikmμenAelIFñwmEdlbgðajenAkñúgrUbTI 2-27a.



dMeNaHRsay³
düaRkamGgÁesrI³ dUcbgðajenAkñúgrUbTI 2-27b kMlaMg 270kN RtUv)anbMEbkCabgÁúMkMlaMgtamG½kS x nig
y . elIsBIenH eKminRtUvkarrgVas;RbEvg 2.1m eT edaysarm:Um:g;KUKμanviucT½r (free vector) ehIyeKGac

GnuvtþvaenARKb;TItaMgTaMgGs;rbs;FñwmsMrab;KNnakMlaMgRbtikmμ.
smIkarlMnwg³ edayGnuvtþsmIkar 2-2 CabnþbnÞab; edayeRbIlT§plEdlKNnaBIxagelI eyIg)an³
+
→ ∑ Fx = 0 Ax − 270 cos 60 o = 0 Ax = 135kN

/
+ ∑ M A = 0 − 270 sin 60 o (3) + 270 cos 60 o (0.3) + B y (4.2) − 67.5 = 0 B y = 173.4kN

+ ↑ ∑ Fy = 0 − 270 sin 60o + 173.4 + Ay = 0 Ay = 60.4kN

]TahrN_ 2>8³ kMNt;kMlaMgRbtikmμenAelIFñwmenAkñúgrUbTI 2-28 a.

dMeNaHRsay³
düaRkamGgÁesrI³ dUcbgðajenAkñúgrUbTI 2-28b kMlaMgBRgayragctuekaNBñayRtUv)anbMElgCakMlaMg
BRgayragRtIekaN nigBRgayesμI. kMlaMgBRgayragRtIekaN nigBRgayesμIRtUv)ansMEdgCakMlaMgpÁÜb.
kMlaMgpÁÜbTaMgenHmanGMeBIkat;tamTIRbCMuTMgn;rbs;RkLaépÞkMlaMgEdlRtUvKña.
smIkarlMnwg³ edayGnuvtþsmIkar 2-2 CabnþbnÞab; edayeRbIlT§plEdlKNnaBIxagelI eyIg)an³
+
→ ∑ Fx = 0 Ax = 0
+ ↑ ∑ Fy = 0 Ay − 60 − 60 = 0 Ay = 120kN

/
+ ∑ M A = 0 − 60(4 ) − 60(6 ) + M A = 0 M A = 600kN .m

c]TahrN_ 2>9³ kMNt;kMlaMgRbtikmμenAelIFñwmenAkñúgrUbTI 2-29 a. snμt;fa A CaTMrsnøak; ehIy
B CaTMrkl; ¬épÞrelag¦.



dMeNaHRsay³
düaRkamGgÁesrI³ dUcbgðajenAkñúgrUbTI 2-29b TMrkl;enARtg;cMnuc B GnuvtþkMlaMgEkgeTAelIFñwmenA
Rtg;cMnucEdlvab:H. ExSskmμénkMlaMgTaMgenHRtUv)ankMNt;edayRtIekaN 3-4-5.
smIkarlMnwg³ bMEbkkMlaMg N B eGayeTACabgÁúMkMlaMgtamTis x nigtamTis y ehIyedayeFVIpl
bUkm:Um:g;Rtg;cMnuc A eyIgnwgTTYl)ankMlaMg N B edaypÞal;. edayeRbIlT§plenH eyIgGacTTYl)an
Ax nig Ay .

⎛3⎞ ⎛4⎞
+∑MA = 0 − 15.75(10.5) + N B ⎜ ⎟(3) + N B ⎜ ⎟(1.2) = 0 N B = 6kN
⎝5⎠ ⎝5⎠
+
Ax − (6) = 0
4
→ ∑ Fx = 0 Ax = 4.8kN
5
Ay − 15.75 + (6 ) = 0
3
+ ↑ ∑ Fy = 0 Ay = 12.15kN
5

]TahrN_ 2>10³ FñwmpSMenAkñúgrUbTI 2-30 a RtUv)anbgáb;enARtg;cMnuc A . kMNt;kMlaMgRbtikmμenA
Rtg; A / B nig C . snμt;fa B CatMNsnøak; ehIy C CaTMrkl;.



dMeNaHRsay³
düaRkamGgÁesrI³ düaRkamGgÁesrIénkMNat;Ggát;nImYy²RtUv)anbgðajenAkñúgrUbTI 2-30b. ehtuGVIk¾
lMhat;enHCaeRKOgbgÁúMkMNt;edaysþaTic?
smIkarlMnwg³ eKmanGBaØatTaMgGs;6. edayGnuvtþsmIkarlMnwgTaMg6 eyIg)an
sMrab;kMNat; BC ³
+ ∑MC = 0 − 8 + B y (4.5) = 0 B y = 1.78kN
+
→ ∑ Fx = 0 Bx = 0
+ ↑ ∑ Fy = 0 − 1.78 + C y = 0 C y = 1.78kN

sMrab;kMNat; AB ³
+∑MA = 0 M A − 36(3) + 1.78(6 ) = 0 M A = 97.3kN .m
+
→ ∑ Fx = 0 Ax − 0 = 0
+ ↑ ∑ Fy = 0 Ay − 36 + 1.78 = 0 Ay = 34.2kN

]TahrN_ 2>11³ eRKOgbgÁúMEdlbgðajenAkñúgrUb
RTkMral nigBüÜrTUk. m:UEDleRKOgbgÁúMEdlbgðaj
enAkñúgrUbenHRtUv)anbgðajenAkñúgrUbTI 2-31a
EdleyIgsnμt;fa A CaTMrkl; ehIy B CaTMrsnøak;.
edayeRbIkUdkñúgtMbn; (local code) bnÞúkkMralEdl
snμt;RtUv)aneTAelIr:tKW 6kN / m . xül;Gnuvtþ
kMlaMgpÁÜbtamTisedk 4kN dUcbgðaj ehIym:as;rbs;TUkEdlBüÜreTAnwgr:tKW 23Mg . TIRbCMuTMgn;rbs;
m:as;TUksßitenARtg; G . kMNt;kMlaMgRbtikmμenARtg;TMr.



dMeNaHRsay³
düaRkamGgÁesrI³ enATIenH eyIgnwgBicarNaTUk nigr:tCaRbB½n§EtmYy ¬rUbTI 2-31b¦. dUcbgðaj eK
RtUvdak;bnÞúkpÁÜbCMnYseGaybnÞúkBRgay.
smIkarlMnwg³ edayGnuvtþsmIkar 2-2 CabnþbnÞab;. eyIg)an
+
→ ∑ Fx = 0 4 − Bx = 0

+ ∑MB = 0 22.8(1.90 ) − Ay (2) + 225.6(5.40) − 4(0.30) = 0 Ay = 630.2kN = 630kN

+ ↑ ∑ Fy = 0 630.2 − B y − 22.8 − 225.6 = 0 B y = 382kN

cMMNaMfa³ RbsinebIeKBicarNar:tEtmYysMrab;karviPaK enaHkMlaMgEkgenARtg;cMnuc C nig D RtUv)anKit
dMbUgedayeRbIdüaRkamGgÁesrIénTUk. kMlaMgTaMgenHnwgmanRbsinebIkabTajTUkeGayb:Hp¥wbnwgcMnuc
TaMgBIr. kMlaMgEkgEdlmanTMhMesμIKña EtTisenApÞúyKñaenARtg;cMnuc C nwgeFVIGMeBIeTAelIr:t enAeBleK
EdleKBicarNadüaRkamGgÁesrIrbs;va. eKnwgTTYl)anlT§pldUcKña. b:uEnþ edayBicarNaRbB½n§TUk
nigr:t kMlaMgEkg nigkMlaMgkabkøayeTACakMlaMgkñúg ehIyminRtUv)aneKykmkKiteT.
]TahrN_ 2>12³ kMNt;bgÁúMkMlaMgRbtikmμedk nigQrenARtg;snøak; A / B nig C rbs;eRKagEdlman
GgÁt;BIrdUcbgðajenAkñúgrUbTI 2-32a.



dMeNaHRsay³
düaRkamGgÁesrI³ düaRkamGgÁesrIénGgát;nImYy²RtUv)anbgðajenAkñúgrUbTI 2-32b.
smIkarlMnwg³ edayGnuvtþnUvsmIkarlMnwgcMnYn6 enAkñúgCMhanbnþbnÞab;GnuBaØateGayeyIgTTYl)an
dMeNaHRsaysMrab;GBaØatTaMg6edaypÞal;.
sMrab;Ggát; BC ³
+ ∑MC = 0 − B y (2) + 6(1) = 0 B y = 3kN

sMrab;Ggát; AB ³
+∑MA = 0 − 8(2 ) − 3(2 ) + Bx (1.5) = 0 Bx = 14.7kN
+
→ ∑ Fx = 0 Ax +
3
(8) − 14.7 = 0 Ax = 9.87kN
5
Ay − (8) − 3 = 0
4
+ ↑ ∑ Fy = 0 Ay = 9.40kN
5
sMrab;Ggát; BC ³
+
→ ∑ Fx = 0 14.7 − C x = 0 C x = 14.7kN
+ ↑ ∑ Fy = 0 3 − 6 + Cy = 0 C y = 3kN

]TahrN_ 2>13³ RCugrbs;GKarenAkñúgrUbTI 2-33a rgnUvbnÞúkxül;EdlbegáItCasMBaFEkgBRgay
esμI 15kPa enAelICBa¢aMgEdlRbQmnwgTisxül;bk; (windward side) nigrgsMBaFbWt (suction
pressure) 5kPa enAelICBa¢aMgEdlminRbQmnwgTisxül;bk; (leeward side). kMNt;bgÁúMkMlaMgRbtikmμ

edk nigQrenARtg;snøak; A / B nig C énTMrrbs; gable arch.



dMeNaHRsay³
edaysarbnÞúkCabnÞúkBRgayelIépÞ enaHeKRtUvbMElgbnÞúkenHeGayeTACabnÞúkBRgayelIbnÞat; mann½y
fa gable arch kNþalRTnUvbnÞúkEdleFVIGMeBIenAelICBa¢aMg nigdMbUlrbs;épÞrgsMBaFEdlmanBN’dit.
bnÞúkBRgayelIenAelIRCugEdlRtUvnwgTisxül;bk;KW (15kN / m 2 )(4m) = 60kN / m ehIyenAelIRCug
mYyeTotKW (5kN / m 2 )(4m) = 20kN / m ¬rUbTI 2-33b¦.

düaRkamGgÁesrI³ edaysMrYlKitbnÞúkBRgayCabnÞúkpÁÜbcMcMnuc düaRkamGgÁesrIrbs;eRKagTaMgmUl
nigrbs;Ggát;nImYy²RtUv)anbgðajenAkñúgrUbTI 2-33c.
smIkarlMnwg³ eKRtUveCosvagdMeNaHRsaykñúgeBlEtmYy
edayGnuvtþsmIkarlMnwgenAkñúgCMhanbnþbnÞab;dUcbgðajenAkñúg]TahrN_xagelI . *

eRKagTaMgmUl³
+∑MA = 0 − (180 + 60 )(1.5) − (254.6 + 84.9 )cos 45o (4.5) − (254.6 sin 45o )(1.5)
+ (84.9 sin 45o )(4.5) + C y (6) = 0 C y = 240.0kN

+ ↑ ∑ Fy = 0 − Ay − 254.6 sin 45o + 84.9 sin 45o + 240.0 = 0 Ay = 120.0kN

sMrab;Ggát; AB ³
+ ∑MB = 0 − Ax (6) + 120.0(3) + 180(4.5) + 254.6(2.12) = 0 Ax = 285.0kN
+
→ ∑ Fx = 0 − 285.0 + 180 + 254.6 cos 45o − Bx = 0 Bx = 75.0kN
+ ↑ ∑ Fy = 0 − 120.0 − 254.6 sin 45o + B y = 0 B y = 300.0kN

sMrab;Ggát; BC ³
+
→ ∑ Fx = 0 − C x + 60 + 84.9 cos 45o + 75.0 = 0 C x = 195.0kN

*
eKk¾GacedaHRsay]TahrN_enHedayGnuvtþsmIkarlMnwgEtcMnYn6eTAelIGgát;TaMgBI. RbsinvaGaceTArYc vaCakarRbesIrEdl
dMbUleyIgRtUv eFVIplbUkm:Um:gRtg;cMnuc A enAelIGgát; AB bnÞab;mkRtg;cMnuc C enAelIGgát; CB. edayeFVIEbbenH eyIgnwg
TTYl)anBIrsmIkarsMrab;edaHRsayrk B nig B kñúgeBlCamYyKña.
x y



cMeNaT
2>1> eKeRbIeRKagmYyeGayRTkMraleQI ¬min BE nig FD . yk b = 3m / a = 2.4m . CMnYy³
bgðaj¦ EdlvargbnÞúkBRgayesμI 6.5kN / m 2 . emIltarag 1-3.
KUrbgðajBIbnÞúkEdlmanGMeBIenAelIGgát; BG nig
ABCD . yk b = 3m / a = 1.5m .

2>4> edaHRsaycMeNaT 2-3/ CamYynwg b =
3 .6 m / a = 1 .2 m .

2>2> kMraldMbUlénGKarEdlmankMBs;mYyCan; 2>5> eKeRbIeRKageQIedIm,IRTkMraleQI. bgðaj
rgnUvbnÞúkefr nigbnÞúkGefr 6kN / m 2 . Rbsin bnÞúkEdlmanGMeBIelIGgát; BG nig ABCD . yk
b = 3m / a = 1.5m . CMnYy³ emIltarag 1-4.
ebIrnUtmanKMlatBIKña1.2m ehIyr:tmanKMlat
7.5m BIKña cUrkMNt;bnÞúkBRgayEdleFVIGMeBIelI

rnUt DF nigbnÞúkEdleFVIGMeBIelIr:tenARtg;cMnuc
A / B / C / D nig E .

2 .4 m / a = 2 .4 m .

4.5m / a = 3m .

2>8> cat;cMNat;fñak;éneRKOgbgÁúMnImYy²CaeRKOg
bgÁúMkMNt;edaysþaTic eRKOgbgÁúMminkMNt;edaysþa
2>3> eKeRbIeRKagEdkedIm,IRTkMralebtugGarem: Tic eRKOgbgÁúMmansßirPaB b¤eRKOgbgÁúMKμanesßr PaB.
TMgn;RsalkMras; 100mm EdlRTbnÞúkBRgayesμI RbsinebICaeRKOgbgÁúMminkMNt;edaysþaTic
25kN / m 2 . bgðajbnÞúkEdlmanGMeBIelIGgát; cUrkMNt;BIdWeRkénPaBminkMNt;.
Problems T.Chhay -52


2>10> cat;cMNat;fñak;éneRKOgbgÁúMnImYy²Ca
eRKOgbgÁúMkMNt;edaysþaTic eRKOgbgÁúMminkMNt;
edaysþaTic b¤eRKOgbgÁúMKμansßirPaB. RbsinebICa
eRKOg bgÁúMminkMNt;edaysþaTic cUrkMNt;BIdWeRkén
PaBminkMNt;.

2>9> cat;cMNat;fñak;éneRKOgbgÁúMnImYy²CaeRKOg
bgÁúMkMNt;edaysþaTic eRKOgbgÁúMminkMNt;edaysþa
Tic b¤eRKOgbgÁúMKμansßirPaB. RbsinebICaeRKOg
bgÁúMminkMNt;edaysþaTic cUrkMNt;BIdWeRkénPaB
minkMNt;.

eRKOgbgÁúMminkMNt;edaysþaTic cUrkMNt;BIdWeRkén
PaBminkMNt;.

cMeNaT T.Chhay -53


PaBminkMNt;. TMr nigtMNTaMgGs;RtUv)ansnμt;
dUcbgðaj.

PaBminkMNt;. TMr nigsnøak;RtUv)ansnμt;dUc
bgðaj.

PaBminkMNt;. tMNTaMgGs;CatMNrwg b¤tMNbgáb;.



PaBminkMNt;. TMr nigtMNTaMgGs;RtUv)ansnμt;
2>15> cUrkMNt;BIdWeRkénPaBminkMNt;rbs; dUcbgðaj.
eRKOgbgÁúMminkMNt;edaysþaTic. tMNTaMgGs;
CatMNrwg b¤tMNbgáb;.

cMeNaT T.Chhay -55


2>20> kMNt;kMlaMgRbtikmμenAelIFñwm.


2>17> kMNt;kMlaMgRbtikmμenAelIFñwm. eKsnμt;
TMrenARtg; B CaTMrkl; nigminKitBIkMras;rbs;Fñwm. 2>22> kMNt;kMlaMgRbtikmμenARtg;TMr A nig B .
kMral CD / DE / EF nig FG bBa¢ÚnbnÞúkrbs;va
eTAelIr:ttamTMrkl;. snμt; A CaTMrkl; nig B
CaTMrsnøak;.

2>18> kMNt;kMlaMgRbtikmμenARtg;TMr A nig B .
snμt; A CaTMrkl; nig B CaTMrsnøak;.

2>23> KNnakMlaMgRbtikmμenARtg;TMr A nig B
énFñwmpSM. enARtg; C CatMNsnøak;.



2>24> eKeRbIRKwHeCIgtagedIm,IRTbnÞú 50kN . kM-
Nt;GaMgtg;sIuet w1 nig w2 énbnÞúkBRgayEdl
manGMeBIenAelI)atrbs;RKWHedIm,IeGaymanlMnwg.

2>28> eKeRbIeCIgtagkugs‘ul (cantilever
footing) edIm,IRTCBa¢aMgenAEk,rcugEKmrbs;RKwH

A dUcenH vaeFVIeGaymansMBaFdIrayesμIenABI

eRkamRKwH. kMNt;bnÞúkBRgayesμIcaM)ac; wA nig
wB EdlKitCa kN / m enARtg;RKwH A nig B edIm,I
2>25> kMNt;kMlaMgRbtikmμenAelIFñwm. RTbnÞúkCBa¢aMg 40kN nig100kN .

2>26> KNnakMlaMgRbtikmμenARtg;TMrrelag C
nigenARtg;TMrsnøak; A . snμt; B CatMNbgáb; . 2>29> kMNt;kMlaMgRbtikmμenARtg;TMr A nig B
rbs; truss. bnÞúkBRgayRtUv)anbgáeLIgeday
sarxül;.

2>27> KNnakMlaMgRbtikmμenARtg;TMrrelag A
nigenARtg;TMrsnøak; B . tMN C CatMNbgáb; .

cMeNaT T.Chhay -57


2>30> dgsÞÚc jib crane mantMNsnøak;enARtg;
A nigmanTMr smooth collar enARtg; B . kMNt;

bMlas;TI x énbnÞúk 22.5kN dUcenHvaeFVIeGayman
kMlaMgRbtikmμGtibrma nigGb,brma. KNnakM-
laMgRbtikmμenAkñúgkrNInImYy². snμt;eKecal
TMgn;rbs;dgsÞúc. eKtMrUveGay1.2m ≤ x ≤ 3m .
2>33> kMNt;bgÁúMkMlaMgRbtikmμedk nigQrenA
Rtg;TMr A nig B . tMNenARtg; C nig D CatMN
bgáb; .

2>31> kMNt;kMlaMgRbtikmμenARtg;TMr A nig B 2>34> kMNt;kMlaMgRbtikmμenARtg;TMr A / B nig
éneRKag. snμt;fa A CaTMrkl;. E . snμt; B CaTMrkl;.

2>35> kMNt;kMlaMgRbtikmμenARtg;TMr A / B /
C nig D .

2>32> kMNt;kMlaMgRbtikmμenARtg;TMr A nig B
rbs; truss. bnÞúkBRgayRtUv)anbgáeLIgeday
sarxül;.


2>36> kMNt;kMlaMgRbtikmμenRtg;TMrsMrab;FñwmpSM.
eKmansnøak;enARtg; A / E nig F .

2>37> sMNg;s<anRtUv)anbgðajenAkñúgrUb. enATI
enHeyIgeXIjfa truss kNþal CD RtUv)anRT 2>40> kMNt;kMlaMgRbtikmμenARtg; A / B . snμt;
eday truss ly ABC nig DEF . C nig D B CaTMrkl; ¬TMrrGil¦ C CatMNrwg.

CatMNsnøak;. kMNt;RbtikmμbBaÄrenARtg;TMr A /
B / E nig F RbsinebIbnÞúk 60kN manGMeBIenA

elI truss kNþal.

2>38> kMNt;kMlaMgRbtikmμenARtg;TMr A / C nig
D. 2>41> kMNt;kMlaMgRbtikmμenARtg;TMr A nig B .

2>42> eRKags<anmYypSMeLIgedaybIkMNat; Edl
2>39> kMNt;kMlaMgRbtikmμenARtg; A nig C . eKKitfa A / D nig E Casnøak; ehIy C nig F
snμt;fa A CaTMrkl; B CatMNbgáb; nig C Ca CaTMr rocker É B CaTMrkl;. kMNt;bgÁúMkMlaMgRbti
tMNsnøak;. kmμtamTisedk nigTisQrenARKb;TMrTaMgGs;
EdlbNþalBIbnÞúkEdl)anbgðaj.
cMeNaT T.Chhay -59


Rtg; A / B nig C . snμt;eKP¢ab;eRKagedaytMN
snøak;enARtg;cMnucTaMgenH ehIy D nig E CatMN 2>46> kMNt;kMlaMgRbtikmμenARtg;TMr A nig D .
rwg. snμt;fa A CaTMrbgáb; ehIy B / C nig D Casnøak;.

2>44> kMNt;kMlaMgRbtikmμenARtg;TMr A nig B .
tMN C nig D CatMNrwg.
2>47> kMNt;kMlaMgRbtikmμenARtg;TMr A nig C .
eRKagRtUv)antP¢ab;edaytMNsnøak;enARtg; A / B
nig C ehIytMNTaMgBIrCaTMrrwg.

Rtg;TMr A nig B .



2>48> kMNt;kMlaMgRbtikmμenARtg;tMN A / B
nig C . snμt;tMNnImYy²CatMNsnøak; ehIytMN
B CatMNrwg.

KMerag³ pøÚvrfePøIgRtUv)anRTedayeRKagebtug
Garem:. snμt;r:tTMrsmBaØBIrRTEdk rail nigkMral
pøÚvrfePøIgEdlmanTMgn; 7.5kN / m nigRTbnÞúk
rbs;rfePøIg 100kN / m ¬rUbxageRkam¦. r:t
nImYy²manRbEvg 100kN / m . GnuvtþbnÞúkTaMg
2>49> kMNt;bgÁúMkMlaMgRbtikmμenARtg;tMN A nig enHeTAelIeRKags<anTaMgmUl ehIykMNt;kMlaMg
C én gable frame. snμt;fa A / B nig C Ca
sgát;enAkñúgssrrbs;va. sMrab;karviPaKsnμt;fa
tMNsnøak;. eKdak;bnÞúkenARtg;cMnuc D / E / F tMNTaMgGs;CatMNsnøak; ehIyecalTMrg;rbs;
nig G EkgeTAnwgG½kSrbs;r:tnImYy². eRKag. etIkarsnμt;tMNTaMgGs;CatMNsnøak;Gac
TTYlyk)aneTsMrab;PaBCak;Esþg?

Rtg;TMr A / B nig C . snμt;faeRKagRtUv)antP¢ab;
edaytMNsnøak;enARtg; A / B / D / E nig F
É C CaTMrbgáb;.

cMeNaT T.Chhay -61

Analysis of Statically Determinate Structures

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