1. This document discusses continuous beams and frames, which are structural elements made of concrete slabs, beams, columns, and footings that are monolithically connected. 2. It describes how to calculate the maximum moment in continuous beams using basic elastic analysis and considering the loading application and moment redistribution. The maximum positive moments within a span and maximum negative moments at supports are also addressed. 3. Formulas are provided to calculate the maximum and minimum positive moments based on the beam's properties and span between supports. The analysis considers both statically determinate and indeterminate continuous beams.

1. T.Chhay NPIC
XVI. FñwmCab; nigeRKag
Continuous Beams and Frames
1> esckþIepþIm Introduction
GKarebtugBRgwgedayEdkpÁúMeLIgedayGgát;eRKOgbgÁúMCaeRcInRbePTdUcCa kMralxNн (slab) Fñwm
(beam) ssr (column) nigeCIgtag (footing) . Ggát;eRKOgbgÁúMTaMgenHGacRtUv)ancak;dac;eday
ELkBIKña dUcCakMralebtugcak;eRsc (precast concrete slab) Fñwmcak;eRsc nigssrcak;eRsc eday
manbnSl;TuksrésEdkedIm,IP¢ab;KñaBIGgát;eRKOgbgÁúMmYyeTAGgát;eRKOgbgÁúMmYyeTot edim,IeFVIeGayva
køayCarcnasm<n§½EtmYy (monolithic structure). Ggát;Edlcak;eRscRtUv)anKNnadUcGgát;eRKOgbgÁúM
enAelITMrsamBaØ eTaHbICaRbePTénPaBCab;xøHRtUv)anpþl;eGayenAEpñkcugrbs;va. sMrab;Ggát;Edlman
lkçN³EtmYy (monolithic member) PaBCab;énGgát;nImYy²RtUv)anpþl;eGay EdleFVIeGayGgát;
eRKOgbgÁúMRtUv)anviPaKdUcrcnasm<n§½sþaTicminkMNt; (statically indeterminate structure) .
karviPaK nigkarKNnakMralxNнmYyTisCab;RtUv)anbgðajenAkñúgemeronTI 9 ehIyemKuNKNna
(design coefficient) nigsrésEdklMGitRtUv)anbgðajenAkñúgrUbTI 9>8 nig 9>9. enAkñúgRbBn§½kMral
xNнmYyTis bnÞúkkMralxNнRtUv)anbBa¢ÚnmkFñwmTMr dUcbgðajenAkñúgrUb 16>1 a . RbsinebIbnÞúkemKuN
enAelIkMralxNнKW wu enaHbnÞúkBRgayesμIenAelIFñwm AB nig BC kñúgmYyÉktþaRbEvgKW wu s bUknwg
TMgn;pÞal;rbs;Fñwm. bnÞúkBRgayesμIenAelIFñwm DE nig EF KW wu s / 2 bUknwgTMgn;pÞal;rbs;Fñwm.
bnÞúkenAelIssr B esμInwg wu Ls b:uEnþbnÞúkenAelIssr E / A nig D KW ws Ls / 2 / ws Ls / 2 nig
ws Ls / 4 erogKña.
enAkñúgkMralxNнBIrTisEdlRTedayFñwmenARCugTaMgbYn bnÞúkkMralxNнRtUv)anbBa¢ÚneTAFñwmBIRkLa
épÞcMNuHEdlBT§½edaybnÞat; 45o dUcbgðajenAkñúgrUbTI 16>1 b. EpñkxøHénbnÞúkkMralxNнRtUv)anbBa¢Ún
eTAFñwmEvg AB / BC / DE nig EF BIRkLaépÞctuekaNBñay b:uEnþbnÞúkkMralxNнEdlenAsl;RtUv)an
bBa¢ÚneTAFñwmxøI AD / BE nig CF BIRkLaépÞRtIekaN. sMrab;kMralxNнkaer bnÞúkRtUv)anbBa¢ÚneTAFñwm
EdlBT§½CMuvijBIRkLaépÞRtIekaN. FñwmenAxagkñúgTTYlbnÞúkBIRCugTaMgBIr b:uEnþFñwmxagrgbnÞúkEtmYyxag
b:ueNÑaH. edaysarFñwmTaMgBIrTisRtUv)ancak;ebtugkñúgeBlCamYyKñanwgkMralxNн enaHvaRtUv)anviPaKCa
FñwmCab;sþaTicminkMNt; (statically indeterminate continuous beam). FñwmbBa¢ÚnbnÞúkbnþeTAssr.
bnÞúkenAelIssr B esμInwg wu Ls enAeBlEdl bnÞúkenAelIssr E / A nig D KW ws Ls / 2 /
FñwmCab; nigeRKag 378

2. T.Chhay NPIC
ws Ls / 2nig ws Ls / 4 erogKña. RkLaépÞcMNuHsMrab;ssrnImYy²RtUv)anKitBIGkS½rbs;ElVgEdlenA
Ek,rRKb;Tis.
2> m:Um:g;GtibrmaenAkñúgFñwmCab; Maximum Moment in Continuous Beams
2> 1> eKalkarN_viPaK Basic Analysis
CaTUeTA karKNnam:Um:g;Bt; nigkMlaMgkat;TTwgenAkñúgFñwmCab;ebtugGarem:KWQrelIRTwsþIeGLasÞic
(elastic theory). enAeBlmuxkat;ebtugGarem:RtUv)anKNnaedayeRbIviFIKNnaersIusþg; (strength
design method) lT§plEdlTTYl)annwgmineqøIytbCamYynwgkarviPaKeGLasÞic (elastic analysis)
eT. b:uEnþ ACI Code min)anbBa¢ÚllkçxNÐsMrab;karKNna)øasÞic (provision for a plastic design) b¤
karKNnasßanPaBkMNt; (limit state design) éneRKOgbgÁúMCab;ebtugGarem: RKan;EtGnuBaØatkarEbg
Eckm:Um:g;eLIgvij (moment redistribution) dUcEdlnwgBnül;enAkñúgemeronenH.
2> 2> karGnuvtþn_kardak;bnÞúk Loading Application
m:Um:g;Bt;enARtg;cMnucNamYyenAkñgFñwmCab; vaminGaRs½yEtnwgbnÞúkEdlmanTItaMgenAelIFñwmdUcKña
ú
enaHeT b:uEnþvak¾GaRs½ynwgbnÞúkEdlsßitenAelIFñwmd¾éTeTotEdr. kñúgkrNIbnÞúkefr RKb;FñwmTaMgGs;RtUv
Etdak;bnÞúkkñúgeBldMNalKña edaysarEtbnÞúkefrmanTItaMgnigtMélkMNt;. kñúgkrNIbnÞúkgl½t b¤
bnÞúkGefrEdlekIteLIgmþgmáal EbbbTénkardak;bnÞúkRtUv)anBicarNay:gNaedIm,IkMNt;m:Um:g;Gti -
FñwmCab; nigeRKag 379

3. T.Chhay NPIC
brmaenATItaMgeRKaHfñak;. ExST§iBlGacRtUv)aneRbIedIm,IkMNt;TItaMgrbs;bnÞúkGefredIm,IKNnam:Um:g;
Gtibrma nigm:Um:g;Gb,brma. b:uEnþenAkñúgemeronenH k,Ünd¾samBaØEdlQrelIExSekagbnÞúk-PaBdab
(load-deflection curve) RtUv)aneRbIedIm,IkMNt;EbbbTénkardak;bnÞúkEdlbegáItm:Um:g;Gtibrma.
2> 3> m:Um:g;viC¢manGtibrma nigGb,brmaenAkñúgElVg Maximum and Minimum Positive
Moments within a Span
m:Um:g;Bt;viC¢manGtibrmaenAkñúgFñwmTMrsamBaØEdlrgbnÞúkBRgayesμI w KWsßitenAkNþalElVg CamYy
wl 2
nwg M = 8 . RbsinebIcugrbs;vamYy b¤TaMgBIrRtUv)anbnþ karTb;enAcugEdlCab;nwgbegáItm:Um:g;GviC¢
manenAelITMr ehIyTItaMgénbnÞúkviC¢manGtibrmaGacpøas;bþÚrbnþicBIkNþalElVg. ragdabénFñwmCab;sM
rab;kardak;bnÞúkEtelIElVgEtmYy RtUv)anbgðajenAkñúgrUbTI 16>2 a . PaBdabcuHeRkambgðajfam:Um:g;
FñwmCab; nigeRKag 380

4. T.Chhay NPIC
viC¢man ehIyPaBdabeLIgelIbgðajm:Um:g;GviC¢man. RbsinebIRKb;ElVgEdldabcuHeRkamRtUv)andak;
bnÞúk enaHbnÞúknImYy²nwgbegáInm:Um:g;viC¢manenAkñúgElVg AB ¬rUbTI 16>2 d ¦. dUcenHedIm,IKNnam:U
m:g;viC¢manGtibrmaenAkñúgElVgNamYy bnÞúkGefrRtUv)andak;enAelIElVgenaH nigenAelIElVgqøas;TaMg
sgçag. m:Um:g;bnÞúkGefremKuNEdlKNnadUcBnül;BImunRtUv)anbEnßmBIelIm:Um:g;bnÞúkGefremKuNenA
elImuxkat;dUcKñaedIm,ITTYl)anm:Um:g;viC¢manGtibrma.
düaRkamm:Um:g;Bt;Edl)anBIbnÞúkBRgayesμIenAelIElVg AB RtUv)anbgðajenAkñúgrUbTI 16>2 b.
PaBdab nigm:Um:g;Bt;fycuHy:agelOnCamYynwgcMgayBIElVg AB EdlrgbnÞúk. dUcenH edIm,IsMrYlkar
viPaKFñwmCab; m:Um:g;enAkñúgElVgNamYyGacRtUv)anKNnaedayBicarNaEtFñwmEdlrgbnÞúk AB nigFñwm
BIreTotEdlenAsgçagva nigsnμt;TMrbgáb;enAcugEdlq¶ayTaMgsgçag ¬rUbTI 16>2 c ¦.
RbsinebIElVgEdlCab;nwgElVg AB RtUv)andak;bnÞúk ExSekagdabRtUv)anbgðajenAkñúgrUb 16>2 e .
PabdabenAkñúgElVg AB nwgeLIgelI ehIym:Um:g;GviC¢mannwgRtUv)anbegáItenAkñúgFñwm AB . m:Um:g;GviC¢ -
manenHnwgRtUv)anbEnßmeTAelIm:Um:g;viC¢manEdl)anBIbnÞúkefredIm,ITTYl)anm:Um:g;Bt;cugeRkay. dUcenH
edIm,IKNnam:Um:g;viC¢manGb,brma ¬b¤m:Um:g;GviC¢manGtibrma¦ enAkñúgElVg AB bnÞúkGefrRtUv)andak;
enAElVgEk,rnwgElVg AB nigRKb;ElVgqøas;nwgElVgEdlrgbnÞúk ¬rUbTI 16>2 e ¦.
2> 4> m:Um:g;GviC¢manGtibrmaenAelITMr Maximum Negative Moments at Supports
kñúgkrNIenH eKcaM)ac;kMNt;m:Um:g;GviC¢manGtibrmaenAelITMrNamYy dUcCaTMr A ¬rUbTI 16>3¦.
enAeBlEdlElVg AB RtUv)andak;bnÞúk m:Um:g;GviC¢manRtUv)anbegáItenAelITMr A . dUcKña bnÞúkénElVg
AF k¾nwgbegáItm:Um:g;GviC¢manenAelI A Edr. dUcenH edIm,IKNnam:Um:g;GviC¢manGtibrmaenAelITMrNa
mYy bnÞúkGefrRtUv)andak;enAelIElVgEk,rTaMgBIr nigenAelIRKb;ElVgqøas;TaMgsgçag ¬rUbTI 16>3¦.
enAkñúgkarviPaKeRKOgbgÁúM (structural analysis) énFñwmCab; RbEvgElVgRtUv)anKitBIGkS½eTAGkS½TMr
EdlRTedayTMrcugkaMbiT (knife-edge support). enAkñúgkarGnuvtþn_ TMrRtUv)aneFVIeLIgedaymanTMhM
RKb;RKan;edIm,ITTYlbnÞúkEdlbBa¢ÚnedayFñwm CaTUeTAm:Um:g;eFVIGMeBIenAnwgépÞénTMr. edIm,IKNnam:Um:g;enA
nwgépÞénTMr eKGacdkedaym:Um:g;EdlesIμnwg Vu c / 3 BIm:Um:g;emKuNenAGkS½rbs;TMr Edl Vu KWCakMlaMg
kat;TTwgemKuN nig c CaTTwgssr.
FñwmCab; nigeRKag 381

5. T.Chhay NPIC
2> 5> m:Um:g;enAkñúgFñwmCab; Moments in Continuous Beams
FñwmCab; nigeRKagCab;GacRtUv)anviPaKedayeRbIviFIRbhak;RbEhl (approximate method) b¤kmμ -
viFIkMuBüÚT½r. viFIepSgeTotdUcCa viFIbnÞúk nigbMlas;TI (displacement and force method) énkarviPaK
EdlQrelIkarKNnaénm:aRTIsénPaBrwgRkaj nigPaBrlas; (stiffness and flexibility matrices)
GacRtUvykmkeRbI. viFI slope deflection nigviFI moment-distibution k¾GacRtUv)aneRbI. viFITaMgenH
manBnül;enAkñúgesovePAEdlniyayBIkarviPaKeRKOgbgÁúM (structural analysis) énFñwm nigeRKag.
ACI Code, Section 8.3 pþl;nUvemKuNRbhak;RbEhlsMrab;KNnam:Um:g;Bt; nigkMlaMgkat;TTwgenAkñúg
Fñwm nigkMralxNнCab;. emKuNTaMgenHRtUv)aneGayenAkñúgemeronTI 9. m:Um:g;EdlTTYl)anBIemKuN
ACI nwgmantMélFMCagm:Um:g;EdlTTYl)anBIkarKNnaviPaKbnþic. eKRtUveKarBtamkarkMNt;Edlman
EcgenAkñúgkareRbIR)as;emKuNTaMgenaH.
]TahrN_TI 16>1³
RbBn§½kMralxNн-Fñwm Edl)anbgðajenAkñúgrUbTI 16>4 RTnUvbnÞúkGefrBRgayesμI 6.2kN / m 2
nigbnÞúkefrEdlrYmmanbnÞúkpÞal;rbs;kMralxNнbUknwg 3.8kN / m 2 . edayeRbIemKuNm:Um:g; ACI
KNnaFñwmCab;enAxagkñúg nigKUrnUvmuxkat;lMGit. eKeGay f 'c = 28MPa / f y = 400MPa / TTWgFñwm
b = 300mm / muxkat;ssr 300 × 300mm nigkMras;kMralxNн 125mm .
FñwmCab; nigeRKag 382

6. T.Chhay NPIC
dMeNaHRsay³
1> KNnakMralxNн³ kMralxNнmannaTICakMralxNнmyTis edaysarpleFobRCugEvgelIRCugxøI
Y
FMCag 2 . karKNnaénRbePTkMralxNнCab;RtUv)anBnül;enAkñúg]TahrN_TI 9>4.
2> bnÞúkenAelIkMralxNн³
bnÞúkefr Dead load = 0.125 × 25 + 3.8 = 6.925kN / m 2
bnÞúkGefr Live load = 6.2kN / m 2
bnÞúkemKuN Ultimate load (wu ) = 1.2 × 6.925 + 1.6 × 6.2 = 18.23kN / m 2
bnÞúkenAelIFñwm³ Fñwmxagkñúg ABC RTbnÞúkkMralxNнTaMgsgçag CamYynwgTTwgkMralxNнsrub
3 .6 m
bnÞúkemKuNenAelIFñwm
Factored load on beam = 3.6 × 18.23 + 1.2(self - weight of beam web )
kMBs;rbs;FñwmGacRtUv)ankMNt;edayeRbIemKuNénkMras;FñwmGb,brmadUcEdl)anbgðajenAkñúg
tarag]bsm<n§½ B.6. sMrab; f y = 420MPa kMBs;FñwmGb,brmarbs;FñwmTI1 AB KW L /18.5
= 7.2 / 18 = 0.389m . snμt;kMBs;Fñwmsrub 550mm kMBs;RTnugFñwmKW 550 − 125 = 425mm .
dUcenHbnÞúkemKuNenAelIFñwm ABCD KW
wu = 3.6 × 18.23 + 1.2(0.425 × 0.3) × 25 = 69.45kN / m
yk wu = 69.5kN / m
3> m:Um:g;enAkñúgFñwm ABC ³ emKuNm:Um:g;RtUv)anbgðajenAkñúgrUbTI 9>8. FñwmCab;Kñaman 5 ElVg
ehIysIuemRTIKñaeFobGkS½kNþalRtg;cMnuc D . dUcenH eKcaM)ac;KNnaEtFñwmBak;kNþal
ABCD BIeRBaHFñwmBak;kNþaleTotnwgmanTMhMehIynwgbrimaNEdkdUcKña. edaysarElVg
AB nig BC minesμIKñaEtmanpleFob 7.9 / 7.2 mantMéltUcCag 1.2 enaHemKuNm:Um:g; ACI
GacGnuvtþ)ansMrab;FñwmenH. elIsBIenHRbEvg clear span mFümEdlenAEk,rRtUv)aneRbIedIm,I
KNnam:Um:g;GviC¢manenAelITMr.
m:Um:g;enAmuxkat;eRKaHfñak;RtUv)anKNnadUcxageRkam ¬rUbTI 16>4¦
2
M u = coefficient × wu l n
FñwmCab; nigeRKag 383

7. T.Chhay NPIC
TItaMg ! @ # $ % ^
1 1 1 1 1 1
emKuNm:Um:g; − 16 +
14
−
10
+
16
−
11
+
16
M u ¬ kN .m ¦ − 206.8 236.35 − 365.3 250.9 − 364.9 250.9
4> kMNt;TMhMFñwm nigsrésEdk
a. m:Um:g;GviC¢GtibrmaKW − 365.3kN .m . edayeRbI ρ max = 0.016 enaH Ru = 5MPa / φ = 0.9
Mu 365.3 ⋅10 6
d= = = 493.5mm
Ru b 5 × 300
sMrab;EdkmYyRsTab; kMBs;FñwmsrubKW 493.5 + 60 = 553.5mm yk 600mm
d EdleRbIR)as;BitR)akdKW 600 − 60 = 540mm
As = 0.016 × 300 × 540 = 2592mm 2
eRbI 4DB30 ¬mYyRsTab;¦
cMNaMfa³ kMBs;FñwmEdleRbIenATIenHKW 600mm EdlFMCagkMBs;Fwñmsnμt;EdlesμInwg 550mm
edIm,IkMNt;TMgn;rbs;Fñwm. edaysarbnÞúkbEnßmmantMéltUcGacecal)an eKmincaM)ac;eFVIkar
KNnaeLIgvijeT.
b. muxkat;enATMreFVIGMeBICamuxkat;ctuekaNEkgCamYyEdkTajenAkñúgsøab. EdkcaM)ac;enAkñúgTMr
mandUcxageRkam³
TItaMg ! # %
M u (kN .m) − 206.8 − 365.3 − 364.9
Ru ( MPa) 2.36 4.17 4.17
ρ (%) 0.74 1.39 1.39
As (mm 2 ) 1199 2252 2252
cMnYnEdk DB30 2 4 4
c. sMrab;muxkat;GkSr T enAkNþalElVg M u = 250.9kN .m . sMrab; a = 25mm nigTTwgsøab
b f = 1.8m
Mu 250.9 ⋅10 6
As = = = 1321.2mm 2
⎛ a ⎞ 0.9 × 400(540 − 12.5)
φf y ⎜ d − ⎟
⎝ 2⎠
FñwmCab; nigeRKag 384

8. T.Chhay NPIC
RtYtBinitü a ³ a = 0.85s ff y' b = 0.132128××400 = 12.34mm
A
85 ×
.2
1800
c
CamYy a = 12.34mm eKTTYl)an As = 1305.5mm 2 . dUcenHeRbIEdk 3DB25
¬ As = 1472.6mm 2 ¦ sMrab;RKb;muxkat;kNþalElVgTaMgGs;. bøg;EdklMGitRtUv)anbgðajenA
kñúgrUbTI 16>5.
5> KNnaFñwmsMrab;kMlaMgkat;TTwg dUcEdl)anBnül;enAkñúgemeronTI 8.
6> RtYtBinitüPaBdab nigsñameRbH dUcEdl)anBnül;enAkñúgemeronTI 9.
3> eRKagsMNg;GKar Building Frames
eRKagGKarKWCaRbBn§½eRKOgbgÁúMbITMhM ¬kñúglMh¦ (three-dimensional structural system) EdlpÁúM
eLIgeday Ggát;Rtg;²Edlsagsg;CamYyKña (built monolithically) nigmantMNrwg. eRKagGacman
mYyElVg nigmankMBs;mYyCan; dUcCa portal frame nig gable frame dUcbgðajenAkñúgrUbTI 16>6 a b¤
GacpSM eLIgedayeRcInElVg nigeRcInCan; dUcbgðajenAkñúgrUbTI 16>6 b. RKb;Ggát;rbs;eRKagRtUv)an
cat;Tuk faCab;sMrab;TisTaMgbI ehIyssrcUlrYmCamYynwgFñwmedIm,ITb;bnÞúkxageRkA. eRkABIkarkat;
bnßym:Um:g; edaysarPaBCab; eRKagGKarcg;)ankarBRgaybnÞúkkan;EtesμIkan;Etl¥. T§iBlénbnÞúk
xag dUcCabnÞúkxül; nigbnÞúkrBa¢ÜydI RtUv)anBRgayenAelIeRKagTaMgmUl edIm,IbegáInsuvtßiPaBrbs;va.
FñwmCab; nigeRKag 385

9. T.Chhay NPIC
sMrab;karKNna viFIRbhak;RbEhl (approximate method) GacRtUv)aneRbIedaysnμt;RbBn§½eRKag
BIrTMhM ¬kñúgbøg;¦.
eRKagEdlrgnUvRbBn§½bnÞúkGacRtUv)anviPaKedayviFIeRKagsmmUl (equivalent frame method).
enAkñúgviFIenH karviPaKtamCan;KWRtUveFVIeLIgedaysnμt;facugssrEdlenAq¶ayBIelI nigBIeRkamnIv:UkM-
ralxNнRtUv)anbgáb; ¬rUbTI 16>7¦. CaTUeTA karviPaKRtUv)aneFVIeLIgedayeRbIviFIEbgEckm:Um:g;
(moment-distribution method).
enAkñúgkar TMhMkMral (panel) cMgaycenøaHssr/ cMnYnCan; nigkMBs;énCan;nImYy²RtUv)ansÁal;BI
eRBaHBYkvaRtUv)anpþl;eGayedaykarKNnasßabtükmμ ehIyRbBn§½brikçaRtUv)anKitrYcCaeRsc. TMhM
ssr nigTMhMFñwmRtUv)aneFVIkar)a:n;sμanCamun ehIypleFobPaBrwgRkajrbs;vaEdlQrelImuxkat;eb
tugRtUv)aneRbI. enAeBlm:Um:g;RtUv)ankMNt; eKRtUvRtYtBinitümuxkat;Edl)ansnμt;BImun nigeFVIkaeEktMrUv
RbsinebIcaM)ac;. karviPaKEdlmanlkçN³suRkitCagGaceFVIeLIgedayeRbIkmμviFIkMuBüÚTr½ EdlRtUv)an
FñwmCab; nigeRKag 386

10. T.Chhay NPIC
ENnaMsMrab;karviPaK eRKOgsþaTicminkMNt;EdlmanGBaØtieRcIn. viFIénkarviPaKRtUv)anBN’naenAkñúg
esovePACaeRcInEdlniyayBIkarviPaKeRKOgbgÁúM.
4> Portal Frames
Portal frames pSMeLIgedayFñwmebtugGarem:EdlrwgRtUv)ancak;CamYynigTMrssrrbs;va. tMN
rvagFñwm nigssrRtUv)anBicarNabgáb;rwgCamYynwgplbUkm:Um:g;enARtg;tMNesμIsUnü. Portal frames
RtUv)aneRbIenAkñúgsMNg;salEdlmanElVgFM (large-span halls)/erag (sheds)/ s<an nigs<ankat;RC
lgPñM (viaducts). Ggát;EpñkxagelIrbs;eRKagGacedk (portal frames) b¤eRTt (gable frames) ¬rUb
TI 16>8¦. eCIgeRkamrbs;eRKagGacCaTMrbgáb; b¤TMrsnøak;.
Portal frames sþaTicminkMNt;GacRtUv)anviPaKedayviFIEbgEckm:Um:g; b¤viFIdéTeTotEdleRbI
edIm,IviPaKeRKagsþaTicminkMNt;. Ggát;rbs;eRKagRtUv)anKNnasMrab;m:Um:g; kMlaMgkat;TTwg nigkMlaMg
Ekg b:uEnþeCIgtagRtUv)anKNnaedIm,IRTkMlaMgEdlmanGMeBIenA)atssr.
Fñwm nigssrrbs;eRKagGacmanTMhMesμI b¤ERbRbYl dUcbgðajenAkñúgrUbTI 16>8. kMlaMgenAkñúg
portal frames EdlmanmYyElVgehIymanmuxkat;esμIGacRtUv)anKNnadUcxageRkam³
4> 1> cugsnøak;BIr Two Hinged Ends
kMlaMgenAkñúgGgát;rbs; portal frames Edlmancugsnøak;BIrGacRtUv)anKNnaedayeRbIsmIkar
xageRkam ¬rUbTI 16>9¦³
sMrab;krNIbnÞúkBRgayesμIenAelIGgát; BC / yk
⎛I h⎞
K = 3 + 2⎜ 2 × ⎟
⎜I ⎟
⎝ 1 L⎠
Edl I1 nig I 2 = m:Um:g;niclPaBrbs;ssr nigFñwm
h nig L = kMBs; nigRbEvgElVgrbs;eRKag
m:Um:g;Bt;enAtMN B nig C KW
wL2
M B = MC =−
4K
2
m:Um:g;viC¢manGtibrmaenAkNþalElVg BC = wL + M B
8
kMlaMgRbtikmμtamTisedkenARtg;cMnuc A KW H A = M B / h = H D
kMlaMgRbtikmμtamTisQrenARtg;cMnuc A KW V A = wL / 2 = VD
FñwmCab; nigeRKag 387

11. T.Chhay NPIC
sMrab;bnÞúkBRgayesμIenAelIEtBak;kNþalFñwm BC ¬rUbTI 16>9 b¦
wL2
M B = MC = −
8K
MB
H A = HD =
h
V A = 3wL / 8 nig VD = wL / 8
4> 2> cugbgáb;BIr Two Fixed Ends
kMlaMgenAkñúgGgát;rbs; portal frames Edlmancugbgáb;BIrGacRtUv)anKNnaedayeRbIsmIkar
xageRkam ¬rUbTI 16>10¦³
sMrab;krNIbnÞúkBRgayesμIenAelIGgát; BC / yk
⎛I h⎞
K1 = 2 + ⎜ 2 × ⎟
⎜I ⎟
⎝ 1 L⎠
wL2
M B = MC =−
4K
FñwmCab; nigeRKag 388

12. T.Chhay NPIC
MB
MA = MD =
2
wL2
M (midspan) = + MB
8
3M A
HA = HD =
2
nig VA = VD = wL
2
sMrab;bnÞúkBRgayesμIenAelIEtBak;kNþalFñwm BC
⎛I h⎞
K 2 = 1 + 6⎜ 2 × ⎟
⎜I ⎟
⎝ 1 L⎠
wL2 ⎛ 1 1 ⎞ wL2 ⎛ 2 1 ⎞
MA = ⎜ − ⎟ MB = ⎜ + ⎟
8 ⎜ 3 K1 8 K 2 ⎟
⎝ ⎠ 8 ⎜ 3 K1 8 K 2 ⎟
⎝ ⎠
wL2 ⎛ 2 1 ⎞ wL2 ⎛ 1 1 ⎞
MC = ⎜ 3K − 8 K ⎟
⎜ ⎟ MD = ⎜ 3K − 8 K ⎟
⎜ ⎟
8 ⎝ 1 2⎠ 8 ⎝ 1 2⎠
wL2 1
H A = HD = ×
8 K1 h
wL ⎛ 1 ⎞
VA =
wL
2
− VD nig VD = ⎜ 4K ⎟
⎜1 −
8 ⎝ ⎟
2⎠
FñwmCab; nigeRKag 389

13. T.Chhay NPIC
5> eRKagTUeTA General Frames
FñwmCab; nigeRKag 390

14. T.Chhay NPIC
rUbragcMbgrbs;eRKagKWtMNrwg EdlP¢ab;FñwmdMbUledk b¤eRTteTAnwgGgát;eRKOgbgÁúMEdlRTva.
PaBCab;rvagGgát;nwgEbgEckm:Um:g;Bt;EdlmanCab;CamYynwgRbBn§½bnÞúkeTAGGát;eRKOgbgÁúMepSgeTot
eTAtampleFobénPaBrwgRkajrbs;va. eRKagGacRtUv)ancat;cMNat;fñak;dUcxageRkam³
- eRKagsþaTickMNt; (statically determinate frame) ¬rUbTI 16>11 a¦
- eRKagsþaTicminkMNt; (statically indeterminate frame) ¬rUbTI 16>12¦
- eRKagsþaTicminkMNt; CamYynwgcMNg (statically indeterminate frame with tie) ¬rUbTI
16>13¦
FñwmCab; nigeRKag 391

15. T.Chhay NPIC
FñwmCab; nigeRKag 392

16. T.Chhay NPIC
6> karKNnasnøak;rbs;eRKag Design of Frame Hinges
RbePTd¾sMxan;énsnøak;EdleRbIenAkñúgeRKagebtugKW Mesnager hinges, Considère hinges
nig lead hinges.
6> 1> Mesnager hinges
CaTUeTA kMlaMgEdlmanGMeBIenAelIsnøak;KWkMlaMgedk H nigkMlaMgbBaÄr P . kMlaMgpÁÜbén
kMlaMgTaMgBIr R RtUv)anbBa¢ÚneTAeCIgtagtamry³EdkExVg A nig B dUcEdl)anbgðajenAkñúgrUbTI
16>14. PaBeRTtrbs;r)ar A nig B eTAnwgGkS½edkERbRbYlBI 30o eTA 60o CamYynwgcMgay
Gb,brma a ¬Edlvas;BIEpñkeRkameKrbs;eRKagssr¦ esμInwg 8D Edl D CaGgát;p©itrbs;EdkeRTt.
cenøaH y rvageRKagssr nigEpñkxagelIrbs;eCIgtagERbRbYlcenøaHBI 25mm eTA 1.3h' Edl h' Ca
TTwgrbs;muxkat;ebtugenAnIv:Usnøak;. kMBs;cenøaHsMrab;karGnuvtþn_ERbRbYlBI 50mm eTA 100mm .
karvilrbs;cugeRKagRtUv)aneFVIeLIgedaysnøak; ehIyCaTUeTAcenøaHRtUv)anbMeBjeday bituminous
cork nigsMPar³EdlmanlkçN³bt;EbnRsedogKña. Bitumen karBar corkkMueGayxUcedaysarkarb:H
CamYydI. EdkExVg A nig B rgnUvkugRtaMgsgát;EdlminRtUvFMCag 1/ 3 énersIusþg;yal (yield
strength) rbs;Edk f y eRkambnÞúkeFVIkar (service load) b¤ 0.55 f y eRkambnÞúkemKuN (factored
load). kugRtaMgtUcRtUv)ansnμt;edaysarkarvilenAsnøak;cg;Bt;Edk nigegáItkugRtaMgBt;TIBIr
FñwmCab; nigeRKag 393

17. T.Chhay NPIC
. CaTUeTA eKcaM)ac;RtUvrkSakugRtaMgsgát;eGaytUclμmRbesIrCagKNna
(secondary flexural stresses)
kugRtaMgTIBIr. RkLaépÞEdk A nig B RtUv)anKNnadUcxageRkam³
RkLaépÞmuxkat;Edk A ³ As1 = 0.551 f
R
¬!^>!¦
y
RkLaépÞmuxkat;Edk B ³ As 2 = 0.552f
R
¬!^>@¦
y
Edl R1 nig R2 CabgÁúMénkMlaMgpÁÜb R enAkñúgTisedAénEdkeRTt A nig B edayeRbIbnÞúkem
KuN. CaTUeTA bgÁúMkMlaMg R1 nig R2 RtUv)anTTYledaysmIkarsþaTicxageRkam³
H + R2 sin θ = R1 sin θ nig R2 = R1 − sin θ
H
¬!^># a¦
dUcKña (R1 + R2 )cosθ = Pu
dUcenH R1 = cosuθ − R2 = cosuθ − ⎡ R1 − sin θ ⎤
P P
⎢
⎣
H
⎥
⎦
1 ⎡ Pu H ⎤
R1 =
2⎣⎢ cosθ + sin θ ⎥ ¬!^># b¦
⎦
Edksnøak;eRTtbBa¢ÚnkMlaMgrbs;vatamry³cMNgenAtamRbEvgEdkbgáb;eTAkñúgssreRKag nig
eCIgtag. dUcenH EdkeRTtbeBa©jkMlaMgFak;ecjeRkA EdlRtUvTb;edayEdkcMNg. EdkcMNgRtUv)an
dak;elIcMgay a = 8D ¬Ggát;p©itEdkEdlFMCageKkñúgcMeNam A nig B ¦ TaMgenAkñúgssr nigeCIgtag.
kMlaMgFak;ecj (bursting force) GacRtUv)ankMNt;dUcxageRkam³
P
F = u tan θ +
2
Ha
0.85d
¬!^>$¦
RbsinebIkarcUlrYmrbs;ebtugRtUv)anecal enaHRkLaépÞmuxkat;rbs;EdkcMNg Ast EdlcaM)ac;
Tb;kMlaMg F KW³
Ast =
F
φf
=
F
0.85 f
¬!^>%¦
y y
kugRtaMgenAkñúgEdkcMNgk¾GacRtUv)ankMNt;dUcxageRkam³
Pu Ha
tan θ +
f s ( tie) = 2 0.85d ≤ 0.85 f
0.005ab + Ast ( tie)
y ¬!^>^¦
Edl Ast = RkLépÞmuxkat;EdkkgkñúgcMgay a = 8D
d = kMBs;RbsiT§PaBrbs;muxkat;ssr
b = TTwgrbs;muxkat;ssr
FñwmCab; nigeRKag 394

18. T.Chhay NPIC
snøak;RbePTenHRtUv)aneRbIsMrab;kMlaMglμm nigRtUv)ankMNt;edaycMnYnEdkeRTtGtibrmaEdl
GacRtUv)andak;enAkñúgTTwgssr.
6> 2> Considère hinges
PaBxusKñarvag Considère hinges nig Mesnager hinges KWfakMlaMgEkg Pu RtUv)ansnμt;fa
vaRtUv)anbBa¢ÚneTAeCIgtagedaykUnssrEdkkgvNн (spirally reinforced short columns) mYyb¤eRcIn
Edldak;bBa©ÚleTAkñúgeCIgtag b:uEnþkMlaMgedk H RtUv)ansnμt;Tb;edayEdkeRTt A nig B
¬rUbTI !^>%¦. lT§PaBRTbnÞúkrbs;kUnssrEdkkgvNнGacRtUv)ankMNt;edayeRbIsmIkar !^>&
edayecalemKuN 0.85 sMrab;cMNakp©itGb,brma.
Pu = φPn = 0.70[0.85 f 'c (Ag − Ast ) + Ast f y ] ¬!^>&¦
FñwmCab; nigeRKag 395

19. T.Chhay NPIC
Edl Ag CaRkLaépÞmuxkat;rbs;muxkat;snøak;ebtug b¤ bh' / nig Ast CaRkLaépÞmuxkat;Edk
beNþayenAkñúgEdkkgvNн. EdkcMNgRtUv)andak;enAkñúgssrrhUtdl;cMgayEdlesμInwgRCugEvgrbs;
muxkat;ssr h .
6> 3> Lead hinges
eBlxøH Lead hinges RtUv)aneRbIenAkñúgeRKagssrebtugGarem:. enAkñúgsnøak;RbePTenH CaTU
eTA lead plate EdlmankMras;BI 20mm eTA 25mm RtUv)anebIedIm,IbBa¢ÚnkMlaMgtamGkS½ Pu eTAeCIg
tag. kMlaMgedk H RtUv)anTb;edayEdkbBaÄrEdldak;enARtg;GkS½rbs;ssr nigRtUv)andak;bBa©Úl
FñwmCab; nigeRKag 396

20. T.Chhay NPIC
eTAkñúgssr ¬rUbTI 16>16¦. enARtg;)atrbs;ssr kMlaMgEkg Pu minRtUvFMCagersIusþg;Tb;
φ (0.85 f 'c A1 ) ¬EdlkMNt;eday ACI Code, Section 10.15¦ Edl φ = 0.65 nig A1 = bh' . RkLa
épÞmuxkat;EdkbBaÄrKW As = H / 0.6 f y Edl H = kMlaMgedkemKuN.
]TahrN_16>2
salmYymanTMhM 25.2 ×12m RtUv)anRKbedaykMralebtugGarem:EdlRTeday portal frame Edlman
snøak;enATMr. Portal frame manKMlatBIGkS½eTAGkS½ 3.6m ¬rUbTI 16>17¦. kMBs;eRKagKW 4.5m
ehIyKμanssrenAkñúgsaleT. bnÞúkefrenAelIkMralxNнKW)anBIbnÞúkpÞal;bUknwg 3.6kN / m 2 Edl)an
mkBIkargarbegðIyrbs;dMbUl. bnÞúk efrenAelIkMralxNнKW 4kN / m 2 . KNnaeRKagenAxagkñúgeday
eRbI f 'c = 28MPa nig f y = 400MPa sMrab;eRKag ehIymuxkat;ssrmanTTwg b = 400mm .
dMeNaHRsay³
karKNnaeRKageménGKarrYmmandUcxageRkam³
- KNnakMralxNнmYyTis
- viPaK Portal frame
- KNnaFñwmeRKagEdlbNþalmkBIm:Um:g;
FñwmCab; nigeRKag 397

21. T.Chhay NPIC
- KNnaFñwmeRKagEdlbNþalmkBIkMlaMgkat;
- KNnassr
- KNnasnøak;
- KNnaeCIgtag
1> kMralxNнmYyTis³ kMras;Gb,brmarbs;kMralxNнTI1KW L / 30 edaysarcugmçagCab;
nigcugmçageTotminCab; ¬tarag B.6 enAkñúg]bsm<n§½ B¦.
kMBs;Gb,brma = 3600 = 120mm
30
karKNnakMralxNнGnuvtþtamCMhanEdlmanenAkñúg]TahrN_9>5.
2> viPaK Portal frame enAxagkñúg³
a. bnÞúkenAelIkMralxNнKW
bnÞúkefrenAelIkMralxNн = 3.6 + 0.12 × 25 = 6.6kN / m 2
bnÞúkemKuNenAelIkMralxNн = 1.2 × 6.6 + 1.6 × 4.0 = 14.32kN / m 2
b. kMNt;bnÞúkenAelIeRKag³ eRKagxagkñúgRTbnÞúkBIkMralxNн 3.6m bEnßmBIelIbnÞúkpÞal;rbs;
FñwmeRKag. snμt;kMBs;rbs;FñwmKW L / 24 = 12000 / 24 = 500mm . edayeRbIkMBs;BIeRkam
kMralxNн 400mm enaHkMBs;srubrbs;FñwmKW 520mm .
bnÞúkefrBIbnÞúkpÞal;rbs;Fñwm = 0.4 2 × 25 = 4kN / m
bnÞúkemKuNsrubenAelIeRKag = 14.32 × 3.6 + 1.2 × 4 = 56.352kN / m
dUcenH wu = 56.5kN / m
c. kMNt;m:Um:g;niclPaBénmuxkat;Fñwm nigmuxkat;ssr. FñwmeFVIkarCalkçN³muxkat;GkSr T .
TTwgRbsiT§PaBrbs;kMralxNнEdleFVIskmμPaBCamYynwgFñwmKWCatMéltUcCageKén 1/ 4 én
RbEvgElVg = 12 / 4 = 3m / 16hs + bw = 16 × 0.12 + 0.4 = 2.32m nig 3.6m . yk
b = 2.32m . TIRbCMuTMgn;énmuxkat;BIsrésxagelIeKKW
2320 × 120 × 60 + 400 × 400 × 320
y= = 154.89mm
2320 × 120 + 400 × 400
⎞ ⎛ 400 ⎞
4
⎛ 2320
I b (beam) = ⎜ × 120 3 + 2320 × 120 × 94.89 2 ⎟ + ⎜ + 400 2 × 165.112 ⎟
⎝ 12 ⎠ ⎜ 12
⎝
⎟
⎠
= 9336 ⋅10 6 mm 4
FñwmCab; nigeRKag 398

22. T.Chhay NPIC
vaCakarGnuvtþn_TUeTAedayKitm:Um:gniclPaBRbhak;RbEhlrbs;FñwmGkSr T esμInwg 2 dg
;
énm:Um:g;niclPaBrbs;muxkat;ctuekaNEkgEdlmankMBs;srubCaplbUkrvagkMBs;RTnugFñwm
nigkMras;kMralxNн³
400 × 520 3
I b (beam) = 2 × = 9374 ⋅10 6 mm 4
12
sMrab;Fñwmxag m:Um:g;niclPaBRbhak;RbEhlKW I = 1.5 × bh3 /12 .
snμt;muxkat;ssresμI 400 × 500mm ¬EdlmanTTwgdUcTTwgFñwm¦
400 × 5003
I c (column) = = 4167 ⋅10 6 mm 4
12
d. KNnaemKuN K
⎛I h ⎞ ⎛ 9374 ⋅10 6 4500 ⎞
K = 3 + 2⎜ b ×
⎜L I ⎟ = 3 + 2⎜
⎟ + ⎟ = 4.68
⎜ 12000 4167 ⋅10 6 ⎟
⎝ c ⎠ ⎝ ⎠
tamrUbTI 16>17 nigsMrab;bnÞúkBRgayesμI wu = 56.5kN / m enAelIGgát; BC
wu L2 56.5 × 12 2
M B = MC = − =− = −434.6kN .m
4K 4 × 4.68
m:Um:g;viC¢manGtibrmaenAkNþalElVg BC esμInwg
L2 56.5 × 12 2
wu + MB = − 434.6 = 582.4kN / m
8 8
kMlaMgRbtikmμedkenARtg;cMnuc A KW³
M B 434.6
HA = HD = = = 96.6kN
h 4.5
kMlaMgRbtikmμbBaÄrenARtg; A KW³
wu L
V A = VD = + weight of column
2
56.5 × 12
VA = + 0.4 × 0.5 × 4.5 × 25 = 361.5kN
2
düaRkamm:Um:g;Bt;RtUv)anbgðajenAkñúgrUbTI 16>17.
e. edIm,IBicarNaT§iBl sideway enAelIeRKag bnÞúkGefrRtUv)andak;enAelIEtBak;kNþalFñwm
BC ehIym:Um:g;RtUv)anKNnaenARtg;muxkat;eRKaHfñak;. krNIenHmineRKaHfñak;enAkñúg]Ta-
hrN_enHeT.
FñwmCab; nigeRKag 399

23. T.Chhay NPIC
f. kMlaMgkat;TTWgGtibrmaenAcugTaMgBIrrbs;Fñwm BC ekItmanenAeBlFñwmRtUv)andak;bnÞúkCa
mYynwgbnÞúkemKuN wu b:uEnþkMlaMgkat;TTwgGtibrmaekItmanenAeBlEdlFñwmRtUv)andak;
bnÞúkCamYynwgEtBak;kNþalbnÞúkGefr nigbnÞúkefreBj³
56.5 × 12
Vu (at support ) = = 339kN
2
L 12
Vu (at midspan) = W1 = 1.7 × 4 × 3.6 × = 36.72kN
8 8
g. kMlaMgtamGkS½enAkñúgssrnImYy²KW VA = VD = 361.5kN
h. ykcMnucm:Um:g;sUnüenAkñúgFñwm BC sßitenAcMgay x BI B / enaH
x x2
M B = wu L − wu
2 2
x2
x
434.6 = 56.5 × 12 − 56.5
2 2
b¤ x 2 − 12 x + 15.384 = 0
x = 1.46m BITMr B
3> KNnaFñwm BC
a. KNnamuxkat;eRKaHfñak;enAkNþalElVg. M u = 582.4kN .m / TTwgRTnugKW bw = 400mm
TTwgsøabKW b = 2320mm nig d = 520 − 90 = 430mm ¬snμt;eRbIEdkBIrRsTab;¦.
RtYtBinitüfaetImuxkat;eFVIkarCaragmuxkat;ctuekaNCamYynwgTTwgRbsiT§PaB
b = 2320mm b¤Gt;? snμt; a = 25mm enaH
Mu 582.4 ⋅10 6
As = = = 3874.9mm 2
φf y (d − ) 0.9 × 400⎛ 430 − ⎞
a 25
⎜ ⎟
2 ⎝ 2 ⎠
As f y 3874.9 × 400
a= = = 28mm < 120mm
0.85 f 'c b 0.85 × 28 × 2320
a snμt;mantMélRbhak;RbEhl a KNna. muxkat;eFVIkarCamuxkat;ragctuekaN dUcenH
eRbIEdk 6DB28 . RtYtBinitü bmin ¬edIm,Idak;EdkmYyCYr¦
bmin = 11× 28 + 2 × 10 + 80 = 408mm > 400mm
dak;EdkBIrCYr/ bgðajenAkñúgrUbTI 16>18.
b. KNnamuxkat;eRKaHfñak;enARtg; B ³ M u = 434.6kN .m / b = 400mm nig d = 520 − 60
= 460mm ¬sMrab;EdkmYyRsTab;¦. kMralxNнrgkarTaj dUcenHEdkRtUv)andak;enAkñúgmux
kat;EpñkxagelI.
FñwmCab; nigeRKag 400

24. T.Chhay NPIC
Mu 434.6 ⋅10 6
Ru = = = 5.135MPa
bd 2 400 × 460 2
ρ = 0.0166 < ρ max = 0.018
As = 0.0166 × 400 × 460 = 3054.4mm 2
eRbI 5DB28 kñúgmYyCYr.
4> KNnaFñwm BC edaysarkMlaMgkat;TTwg
a. muxkat;eRKaHfñak;enAcMgay d BIépÞssrCamYycMgayBIGkS½ssr 250 + 460 = 710mm
dUcenH
Vu (at distance d) = 339 - 56.5 × 0.71 = 298.9kN
b. ersIusþg;kMlaMgkat;Edlpþl;edayebtug
φ 0.75
φVc = f 'c bw d = 28 × 400 × 460 ⋅10 − 3 = 121.7kN
6 6
kMlaMgkat;Edlpþl;edayEdkRTnug (web reinforcement) KW
φVs = Vu − φVc = 298.9 − 121.7 = 177.2kN
177.2
Vs = = 236.3kN
0.75
c. eRCIserIsEdkkg DB12 nig Av = 2 ×122 π = 226mm 2
4
dUcenH s = AvVf y d = 226 × 40010460 = 176mm yk 175mm
×
236.3 ⋅ 3
s
d. KMlatGtibrmaénEdkkg DB12 KW
d
s max = = 230mm
2
yk 225mm
b¤ smax = 0A35fby = 0.35× 400 = 646mm
.
v 226
× 400
w
RtYtBinitüsMrab;KMlatGtibrmaén d ³ Vs ≤ 1
2 3
f 'c bw d b¤
28
Vs ≤ 400 × 460 ⋅10 − 3 = 324.5kN
3
tMél Vs = 236.3 < 324.5kN . dUcenHeRbI smax = 225mm .
Av f y d 226 × 400 × 460 − 3
V ' s (for s max = 220mm) = = 10 = 184.8kN
s 225
φV ' s = 0.75 × 184.8 = 138.6kN
cMgayBIépÞssreTATItaMgEdl smax = 225mm RtUv)aneRbIesμInwg 1.56m
FñwmCab; nigeRKag 401

25. T.Chhay NPIC
karBRgayEdkkg
e.
EdkkgTImYymanKMlat s / 2 = 75mm
7 EdkkgmanKMlat 175mm = 1225mm
19 EdkkgmanKMlat 225mm = 4275mm ¬srub 5575mm ¦
cMgayBIépÞssreTAGkS½rbs;FñwmesμInwg 6000mm − 250mm = 5750mm . BRgayEdkkg
dUcKñasMrab;Bak;kNþalFñwmEdlenAsl; nigdak;EdkkgmYyeTotenAkNþalElVg. cMnYnEdk
kgsrubKW 55 kg.
5> KNnamuxkat;ssrenAcMnuc B ³ M u = 434.6kN .m / Pu = 339kN / b = 400mm nig
h = 500mm
a. snμt;eRkamGMeBIbnÞúkEdleGayeRKagnwgminrgnUv sideway enaHT§iBlrbs;PaBrlas;
(slenderness ration) GacRtUv)anecal nigssrGacRtUv)anKNnaCassrxøIenAeBl
12 M 1
KLu
4
< 34 −
M2
¬emIlEpñk 12>5¦
M1 = 0 nig M 2 = 434.6kN.m
yk K = 0.8 ¬rUbTI 12>2¦ Lu = 4.5 − 0.2 = 4.24m nig r = 0.3 × 500 = 150mm enaH
52
KLu 4.24
= 0 .8 = 22.6 < 34
r 0.15
RbsinebI K RtUv)ansnμt;esμInwg 1.0 enaH
KLu
= 28.3 < 34
r
dUcenH/ karKNnaGgát;CassrxøI.
b. dMeNIrkarKNnaRsedogKñanwg]TahrN_TI 11>16 nig 11>3.
cMNakp©it (e) = M u = 434.6 = 1.282m
Pu 339
vaCacMNakp©itFM nigvaRtUv)ansnμt;Camuxkat;enAkñúgtMbn; transition region/ φ < 0.9
d = 500 − 60 = 440mm
c. edaysar e = 1.282m KWFMCag d q¶ayNas;/ kMNt;tMélRbhak;RbEhl As nig A's EtBI
M u nigbnÞab;mkRtYtBinitümuxkat;cugeRkayedaysþaTic Edl)anBnül;enAkñúg]TahrN_
FñwmCab; nigeRKag 402

26. T.Chhay NPIC
11>3. sMrab; M u = 434.6kN .m / b = 400mm / h = 500mm nig d = 440mm /
Ru = M u / bd 2 = 434.6 ⋅10 6 / 400(440)2 = 5.61MPa
ρ = 0.0183 nig As = ρbd = 3220.8mm 2
eRCIserIs 3DB28 nig 2DB32 ¬ As = 3456mm 2 ¦.
eRCIserIs A's = As / 3 = 3220.8 / 3 = 1074mm 2 nig 3DB22 ¬ A's = 1140mm 2 ¦ ¬rUbTI
16>18¦. enAeBlcMNakp©it e mantMélFM/ vaCakarGnuvtþn_Fmμta A's = As / 3 b¤ As / 2
CMnYseGay As = A's .
d. RtYtBinitülT§PaBénmuxkat;cugeRkayedayeRbI As = 3456mm 2 nig A's = 1140mm 2
Rsedognwg]TahrN_ 11>3 GaRs½yeTAtamCMhanxageRkam³
i. Pn = Cc + C s − T
Cc = 0.85 f 'c ab = 0.85 × 28 × 400a = 9520a
C s = A' s ( f ' s −0.85 f 'c ) = 1140(400 − 0.85 × 28) = 428868kN
T = As f y = 3456 × 400 = 1382400kN
Pn = 9520a + 428868 − 1382400 = 9520a − 953532 (I)
ii. Kitm:Um:g;eFobRtg; As ³
1⎡ ⎛ a⎞ ⎤
Pn = ⎢Cc ⎜ d − 2 ⎟ + C s (d − d ')⎥
e' ⎣ ⎝ ⎠ ⎦
Edl d " sßitenAcMgay As eTATIRbCMuTMgn;énmuxkat;. TIRbCMuTMgn;ekItman
e' = e + d "
enAcMgay 276.68mm BIsréssgát;xageRkAbMput nig d " = d − x = 163.32mm
e' = 1282 + 163.32 = 1445.32mm
1 ⎡ ⎛ a⎞ ⎤
Pn = ⎢9520a⎜ 440 − 2 ⎟ + 428868(440 − 60)⎥
1445.32 ⎣ ⎝ ⎠ ⎦
= −3.3a 2 + 2898.2a + 112757 (II)
iii. KNnasmIkar I nig II eKTTYl)an a = 150mm nig Pn = 474.5kN .
epÞógpÞat; f 's = 600(c − d ') / c ≤ f y / c = a / 0.85 = 176.47mm / enaHeyIgTTYl)an
f ' s = 600(176.47 − 60 ) / 176.47 = 396MPa EdlmantMélEk,r 400 MPa dUc
Edl)ansnμt;kñúgkarKNna. eRCIserIsEdkkg DB10 EdlmanKMlat 400mm .
iv. epÞótpÞat; φ ³ dt = 440mm
FñwmCab; nigeRKag 403

27. T.Chhay NPIC
⎛ dt − c ⎞ ⎛ 440 − 176.47 ⎞
εt = ⎜ ⎟0.003 = ⎜ ⎟0.003 = 0.0045
⎝ c ⎠ ⎝ 176.47 ⎠
⎛ 250 ⎞
φ = 0.65 + (ε t − 0.002 )⎜ ⎟ = 0.858
⎝ 3 ⎠
φPn = 0.858 × 474.5 = 407kN > 339kN
muxkat;RKb;RKan;.
FñwmCab; nigeRKag 404

28. T.Chhay NPIC
6> epÞógpÞat;muxkat;ssrenAkMBs; 2.25m ¬Bak;kNþal¦BI A . M u = 434.6 / 2 = 217.3kN .m
Pu = 339 + 11.25 = 350.25kN
eRbIEdk As = 3DB28 nig A's = 3DB22 . tamviFIedaHRsayRsedogKñakñúgCMhan 5>
φPn = 632.8kN > 350.25kN ¬Edk DB32 Gacpþac;)an EtRtUvBnøÚtRbEvgbgáb;BIeRkam
Bak;kNþalkMBs;ssr¦.
7> KNnasnøak;enARtg;cMnuc A ³ M u = 0 / H = 96.6kN / Pu = 361.5kN
a. eRCIserIssnøak; Mesnager hinge . edayeRbIsmIkar 16.3 a nig 16.3b/ R1 = 305.3kN
nig R2 = 112.1kN ¬eyagtamrUbTI 16>19 CamYynwg θ = 30o ¦
R1 305300
As1 = = = 1387.7mm 2
0.55 f y 0.55 × 400
eRCIserIs 3DB25 ¬ As = 1472.6mm 2 ¦
R2 112100
As 2 = = = 509.5mm 2
0.55 f y 0.55 × 400
eRCIserIs 2DB22 ¬ As = 760mm 2 ¦. tMerobEdkExVgedaydak; DB25 nigbnÞab;mkdak;
DB 22 dUcEdlbgðajenAkñúgrUbTI 16>19 ¬b¤eRbI 5DB 25 ¦
b. EdkcMNgTTwgKYrdak;tambeNþaycMgay a = 8D = 8 × 25 = 200mm enAkñúgssr
nigeCIgtag. kMlaMgFak;KW
Pu Ha
F= tan θ +
2 0.85d
/
θ = 30 o d = 440mm nig a = 200mm
361.5 96.6 × 200
F= tan 30 o + = 156kN
2 0.85 × 440
156000
RkLaépÞmuxkat;EdkcMNg =
0.85 × 400
= 458.8mm 2
RbsinebIEdkcMNgbiTCit DB10 ¬eCIgBIr¦ RtUv)anerIs enaHRkLaépÞmuxkat;KW 157mm 2 .
cMnYnEdkcMNgKW 458.8 = 2.9 . eRbIEdkcMNg 3 kgEdlmanKMlat 200 / 2 = 100mm
157
dUcbgðajenAkñúgrUbTI 16>19.
8> KNnaeCIgtag³ RbsinebIkMBs;eCIgtagRtUv)ansnμt;esμI h' enaHkMlaMgEdlmanGMeBIelIeCIgtag
CakMlaMgEkg P nigm:Um:g; M = H / h' . sMBaFdIRtUv)anKNnaBIsmIkar !#>!$ BIemeronTI 13³
P Mc
q=+ ± ≤ allowable soil pressure
A I
FñwmCab; nigeRKag 405

29. T.Chhay NPIC
viFIsaRsþKNnaeCIgtagmanlkçN³RsedogKñaeTAnwg]TahrN_TI 13>7.
7> esckþIepþImBIkarKNnasßanPaBkMNt; Introduction to Limit Design
7>1> lkμN³TUeTA General
karKNnasßanPaBkMNt;éneRKOgbgÁúMmanbICMhandUcxageRkam³
- karkMNt;bnÞúkKNnaemKuN EdlTTYledayKuNbnÞúkefr nigbnÞúkGefrCamYynwgemKuNbnÞúk.
ACI Code TTYlnUvemKuNdUcEdleGayenAkñúgemeronTI3.
- viPaKeKOgbgÁúMeRkamGMeBIbnÞúkemKuN (ultimate or factor load) edIm,IkMNt;m:Um:g;emKuN nigkM
laMgemKuNenAeBlEdleKOgbgÁúMdYl b¤)ak;. viFIénkarviPaKenHRtUv)aneRbIsMrab;karKNnaeRKOg
bgÁúMEdk EtsMrab;eRKOgbgÁúMebtugGarem: ACI Code min)anTTYlykviFIenHTaMgGs;eT edaysar
karxVHPaBsVit (ductility) rbs;Ggát;ebtugGarem:. ACI Code GnuBaØatEtEpñkénkarEbgEck
eLIgvijnUvm:Um:g;edayEpñk (partial redistribution of moment) enAkñúgrcnasm<n§½edayQrelI
PaKryEdl)anmkBIkarBiesaFn_ EdlnwgBnül;kñúgemeronenH.
FñwmCab; nigeRKag 406

30. T.Chhay NPIC
- KNnaGgát;nImYy²éneRKOgbgÁúMedaysnμt;fava)ak;edaym:Um:g;emKuN nigkMlaMgemKuNEdl)an
mkBIkarviPaKeRKOgbgÁúM. viFIenHRtUv)aneRbIR)as;CaTUeTAsMrab;karKNnaebtugGarem: ehIy ACI
Code GnuBaØatnUvkareRbIR)as;viFIKNnalT§PaBRTRTg; (strength design method) dUcEdl)an
Bnül;kñúgemeronknøgmk.
7> 2> KMniténkarKNnasßanPaBkMNt; Limit Design Concept
karKNnasßanPaBkMNt;enAknúgebtugGarem:sMedAelIkarEbgEckm:Um:g;eLIgvijEdlekItman
enAelIeRKOgbgÁúMTaMgmUl dUcEdlEdkBRgwgenAmuxkat;eRKaHfñak;eFVIkardl;ersIusþg;yal (yield
strength) rbs;va. ersIusþg;cugeRkay (ultimate strength) rbs;eRKOgbgÁúMekIneLIgenAeBlmuxkat;rbs;
vaekIneLIg. eTaHbICakaryal (yielding) rbs;EdkeFVIeGaymanPaBdabFM EdlKYrEteCosvageRkam
bnÞúkeFVIkar (service load) eRKOgbgÁúMsþaTicminkMNt;mindYlb¤)ak;enAeBlEdlEdkBRgwgénmuxkat;TI
mYyyal. elIsBIenH ersIusþg;EdlbMrugTuky:ageRcInsßitenAcenøaHkaryaldMbUg nigkar)ak;rbs;eRKOg
bgÁúM.
kñúgkarKNnaeRKOgbgÁúMEdk BakükarKNna)øasÞic (plastic design) RtUv)aneRbIedIm,Ibgðajkar
pøas;bþÚrkñúgkarEbgEckm:Um:g;enAkñúgeRKOgbgÁúMdUcCa steel fiber ¬enARtg;muxkat;eRKaHfñak;¦ rgkugRtaMg
rhUtdl;ersIsþg;yal. karekIneLIgnUvkugRtaMgtamkMBs;rbs;muxkat;EdkeRkamkarekIneLIgrbs;bnÞúk
RtUv)anbgðajenAkñúgrUbTI 16>20. Portal frame RtUv)anBiesaFn_edIm,IGegátlT§PaBRTRTg;karvil
rbs;snøak;)øasÞicrbs;ebtugGarem:. b:uEnþ ACI Code tMrUveGaymankarsikSaRsavRCavCaeRcIneTotmun
nwgTTYlykkarKNnasßanPaBkMNt;.
7> 3> eKalkarN_énsnøak;)aøsÞic Plastic Hinge Concept
ExSkMeNag φ rbs;Ggát;ekIneLIgCamYynwgm:Um:g;Bt; M . sMrab;FñwmebtugEdlmanbrmaNEdk
tic düaRkamm:Um:g;-ExSkMeNag nigExSekagbnÞúk-PaBdab RtUv)anbgðajenAkñúgrUbTI 16>21. FñwmEdl
FñwmCab; nigeRKag 407

31. T.Chhay NPIC
manbrimaNEdk balanced nigbrimaNEdkeRcInminRtUv)anGnuBaØateGayeRbIeday ACI Code eT
edaysarva)ak;edaykarpÞúHEbkebtug nigbgðajnUvRbeLaHExSkMeNagtUcenAeBlm:Um:g;emKuN ¬rUbTI
16>22¦.
FñwmCab; nigeRKag 408

32. T.Chhay NPIC
EpñksMxan;rbs;düaRkamExSrkMeNag-m:Um:g;enAkñúgrUbTI 16>21 KWsßitenAcenøaHcMnuc B nig C
Edlm:Um:g; M u rkSatMélefrsMrab;RbeLaHéntMél φ d¾FM. kñúgkarviPaKsßanPaBkMNt; ExSekagExSkM
eNag-m:Um:g;GacRtUv)ansnμt;manTMrg; idealized form dUcbgðajenAkñúgrUbTI 16>23 EdlExSkMeNag
φ enAcenøaH B nig C RtUv)ansnμt;faefr edIm,IbegáItrUbragsnøak;)aøsÞic. edaysarebtugCasMPar³
RsYy CaTUeTAeK)anKitnUvEdnkMNt;edaleGayGgát;)ak;enARtg;ExSkMeNagGtibrmaRtg; C .
elak Cranston )anbgðajfaeRKagebtugGarem:EdlKNnaCaTUeTA GacmanlT§PaBTb;nwgkar
vil)anFM ehIyExSkMeNagGtibrmaenARtg;cMnuc C nwgminGaceTAdl; Tal;EteRKag)ak;sin. dUcenH
enAeBlGgát;RTm:Um:g;esμInwgm:Um:g;emKuN M u rbs;va ExSkMeNagbnþekIneLIgcenøaHcMnuc B nig C
edayKμankarERbRbYlm:Um:g; edIm,IbegáItsnøak;)aøsÞic. karekIneLIgénExSkMeNagGnuBaØateGayEpñkd¾éT
eToténeRKOgbgÁúMsþaTicminkMNt;edIm,ITTYlbnÞúkbEnßm.
8> emkanicénkar)ak; The Collapse Mechanism
kñúgkarKNnasßanPaBkMNt; Ggát;ebtugGarem:GaceTAdl;ersIusþg;cugeRkay (ultimate
strength) enAeBlvaCitnwgrlM b¤)ak;. Ggát;dYlrlMenAeBlEdlcMnYnsnøak;)aøsÞicRKb;RKan;bMElgva
eGaycUleTACaFñwmemkanicénkar)ak;. cMnYn caM)ac;énsnøak;)aøsÞic n GaRs½yeTAnwgdWeRk
redundancy r rbs;eRKOgbgÁúM. TMnak;TMngrvag n nig r edIm,IbegáItFñwmemkanicénkar)ak;KW
n = 1+ r
]TahrN_ enAkñúgFñwmTMrsamBaØKμanGBaØtielIseT Edl r = 0 . dUcenH FñwmkøayeTACaKμanesßrPaB
ehIy)ak;enAeBlEdlsnøak;)aøsÞicmYyekItmanenARtg;muxkat;énm:Um:g;Gtibrma dUcbgðajenAkñúgrUbTI
16>24 a. karGnuvtþn_eTAelIFñwm nigeRKagk¾RtUv)anbgðajenAkñúgrUbTI 16>24.
9> eKalkarN_énkarKNnasßanPaBkMNt; Principles of Limit Design
eRkambnÞúkeFVIkar (working load) karEbgEckm:Um:g;enAkñúgeRKagsþaTicminkMNt;KWQrelIRTwsþI
eGLasÞic ehIyeRKOgbgÁúMTaMgmUlenAEtrkSaenAkñgEdneGLasÞic. enAkñúgkarKNnasßanPaBkMNt; enA
ú
eBlEdlvaeTAdl;emkanicénkarrlM karEbgEckm:Um:g;enAeBl)ak;edaybnÞúkemKuN xusKñaBIkarEbg
EckedayQrelIRTwsþIeGLasÞic. karxusKñaenHbgðajeGayeXIjnUvkarEbgEckm:Um:g;eLIgvij
(moment redistribution).
FñwmCab; nigeRKag 409

33. T.Chhay NPIC
sMrab;karKNnasßanPaBkMNt;mann½y lkçxNÐ 4 RtUvEteKarBdac;xat
- lkçxNÐemkanicénkar)ak; (Mechanism condition)³ snøak;)aøsÞicRKb;RKan;RtUv)anbegáIteLIg
edIm,IbMElgeRKOg bgÁúMTaMgmUl b¤EpñkxøHrbs;eRKOgbgÁúMeTACaGgát;emkanicénkar)ak;.
- lkçxNÐlMnwg (Equilibrium condition)³ karEbgEckm:Um:g;RtUvEtmanlMnwgCamYynwgbnÞúkEdl
Gnuvtþ.
FñwmCab; nigeRKag 410

34. T.Chhay NPIC
- lkçxNÐyal (Yield condition)³ m:Um:g;emKuNminRtUvelIsersIusþg;m:Um:g;enARKb;TItaMgrbs;
eRKOgbgÁúM.
- lkçxNÐrgVil (Rotation condition)³ snøak;)aøsÞicRtUvEtmanlT§PaBTb;rgVil (rotation
capacity)RKb;RKan; edIm,IGnuBaØatkarekItmanFñwmemkanicénkar)ak;.
manEtlkçxNÐbIdMbUgeTEdlGnuvtþkñúgkarKNna)aøsÞic edaysarlT§PaBTb;rgVilmanenAkñúg
sMPar³sVitdUcCaEdkrYceTAehIy. lkçxNÐTIbYnbegáInEdnkMNt;bEnßmeTAelIkarKNnakMNt;énGgát;eb
tugGarem:edayeRbobeFobCamYykarKNna)aøsÞic (plastic design).
10> Ednx<s; nigEdnTabénemKuNbnÞúk Upper and Lower Bounds of Load
Factors
eRKOgbgÁúMEdlCitdYlrlMRtUvEtmancMnYnsnøak;)aøsÞicRKb;RKan;edIm,IbMElgvaeTACaGgát;emkanic
énkar)ak;. sMrab;TItaMgNamYyénsnøak;)aøsÞicenAelIeRKOgbgÁúM bnÞúkEdleFVIeGayrlMGacRtUv)anKNna
EdlGacFMCag b¤esμIbnÞúkBit. edaysarbnÞúkKNnaminGacFMCagbnÞúkEdleFVIeGayrlMBitsMrab;eRKOg
bgÁúM enaHviFIsaRsþKNnabgðajnUvEdnsIueNm:aTic b¤Edkx<s; (upper or kinematic bound) rbs;bnÞúk
EdleFVIeGayrlMBit. dUcenHRKb;Ggát;emkanicénkarrlMEdlGacekItmanRtUv)aneFVIkarGegát M u
EdltUcCageK nwgekIteLIgedaybnÞúkBit. elak Horne )anBnül;Ednx<s;edaysnμt;eRKOgbgÁúMCa
Ggát;emkanicénkar)ak; nigbnÞab;mkKNnakmμnþxageRkA We EdleFVIedaybnÞúkxageRkA nigkmμnþkñúg
Wi EdleFVIenAsnøak;)aøsÞic. RbsinebI We = Wi enaHbnÞúkEdlGnuvtþFMCag b¤esμIbnÞúkEdleFVIeGay)ak;.
RbsinebIdüaRkamm:Um:g;enARtg;cMnucNamYyRtUv)anbegáItedIm,IbRgÁb;lMnwgsþaTiceRkamGMeBIbnÞúk
GnuvtþenAeBl)ak; enaHbnÞúkEdlGnuvtþtUcCag b¤esμIbnÞúkEdleFVIeGay)ak;Bit. sMrab;düaRkamm:Um:g;
epSgeTot eKGacTTYlbnÞúkemKuNepSgeTot. tMélEdlFMCagénEdntUc b¤EdnsþaTic (lower or static
bound) RtUv)anTTYlenAeBlm:Um:g;enAmuxkat;CaeRcInsMrab;düaRkamm:Um:g;Edlsnμt;xiteTACitm:Um:g;Edl
eFVIeGayrlM. elak Horne)anBnül;EdnTabedaysnμt;karEbgEckm:Um:g;xusKñaedIm,ITTYlnUveRKOgbgÁúM
EdlsßitenAkñúgsßanPaBlMnwgCamYybnÞúkxageRkA nigeKarBlkçxNÐyalenAelIeRKOgbgÁúMTaMgmUl. kñúg
krNIenH bnÞúkxageRkAtUcCag b¤esμIbnÞúkEdleFVIeGayrlM.
FñwmCab; nigeRKag 411

35. T.Chhay NPIC
11> karviPaKsßanPaBkMNt; Limit Analysis
sMrab;karviPaKedaydMeNIrkarKNnasßanPaBkMNt; viFIBIrGacRtUv)aneRbI viFIkmμnþCak;Esþg
(virtual work method) nigviFIlMnwg (equilibrium method). sMrab;viFI virtual work kmμnþRtUv)aneFVI
edaybnÞúkemKuN Pu ¬b¤ wu ¦ edIm,IbegáItPaBdab virtual deflection EdleGay Δ RtUv)andak;
eGayesμIkmμnþEdlRsUbykenAsnøak;)aøsÞic. kmμnþxageRkAEdleFVIedaybnÞúkKW We = ∑ (wu Δ ) b¤
∑ (Pu Δ ) . kmμnþEdlRsUbykedaysnøak;)aøsÞicKWkmμnþ = Wi = ∑ (M uθ ) .
]TahrN_16>3
FñwmEdlbgðajenAkñúgrUbTI 16>25 RTnUvbnÞúkcMcMnucenAkNþalElVg. KNnam:Um:g;EdleFVIeGay)ak;enA
muxkat;eRKaHfñak;.
FñwmCab; nigeRKag 412

36. T.Chhay NPIC
dMeNaHRsay
1> FñwmCaGgát;sþaTicminkMNt; 1 dWeRk
ehIycMnYnsnøak;EdlRtUvkaredIm,IbMElgFñwmeTACaFñwmemkanicénkar)ak;KW n = 1 + 1 = 2
snøak;)aøsÞic enARtg;cMnuc A nig C . snøak;)aøsÞicTImYyekItmanenARtg;cMnuc A
ehIyFñwmeFVIkarCaFñwmTMrsamBaØrhUtdl;vaxiteTArkemkanicénkar)ak;.
2> RbsinebImMurgVil θ ekItmanenAsnøak;)aøsÞicenAcugbgáb; A enaHmMurgVilenARtg;cMnuc
C = 2θ . PaBdabenARtg;cMnuc C eRkambnÞúkKW (L / 2 )θ ¬rUbTI 16>25¦.
⎛ Lθ ⎞
We = external work = ∑ Pu Δ = Pu ⎜ ⎟
⎝ 2 ⎠
Wi = internal work = ∑ M uθ = M u1 (θ ) + M u 2 (2θ )
RbsinebImuxkat;BIrenARtg; A nig C manTMhM nigbrimaNEdkdUcKña enaH
M u1 = M u 2 = M u nig Wi = 3M uθ . edayeGay We = Wi
L
M u1 + 2M u 2 = Pu = 3M u
2
nig M u = Pu6L
]TahrN_16>4
KNnam:Um:g;EdleFVIeGay)ak;enAmuxkat;eRKaHfñak;sMrab;FñwmEdlbgðajenAkñúgrUbTI 16>26 EdlbNþal
edaybnÞúkBRgayesμI wu .
dMeNaHRsay
1> cMnYcsnøak;)aøsÞicKW 2 .
2> sMrab;PaBdabenARtg; C = 1.0 mMurgVilenARtg;cMnuc A θ A = 1 / a nig θ B = 1 / b ehIy
a+b L
θc = θ A + θ B = =
ab ab
3> kmμnþxageRkAKW We = ∑ wu Δ = wu ⎛ 1×2L ⎞ = w2L
⎜
⎝
⎟
⎠
u
kmμnþxagkñúgKW Wi = ∑ M uθ = M u1θ A + M u 2θ C
1 ⎛1 1⎞
= M u1 + M u2 ⎜ + ⎟
a ⎝a b⎠
2⎛M M ⎞
dak;eGay We nig Wi M
wu = ⎜ u1 + u 2 + u 2 ⎟
L⎝ a a L−a⎠
¬!^>(¦
RbsinebIm:Um:g;TaMgsgçagesμIKña enaH
FñwmCab; nigeRKag 413

37. T.Chhay NPIC
2M u ⎡ 2 1 ⎤ 2M u ⎡ 2 L − a ⎤
wu =
L ⎣ ⎢ a + (L − a ) ⎥ = L ⎢ a (L − a ) ⎥ ¬!^>!0¦
⎦ ⎣ ⎦
FñwmCab; nigeRKag 414

38. T.Chhay NPIC
4> edIm,IKNnaTItaMgsnøak;)aøsÞicenARtg; C
EdlbegáIttMélGb,brmaénbnÞúkEdleFIVeGay)ak; wu eFVIedrIevsmIkarTI !^>( eFobnwg a
nigdak;eGayvaesμIsUnü³
δwu ⎛ Mu M M u2 ⎞
=0 − ⎜ 21 + u 2 − ⎟=0
δa ⎜ a a2 (L − a )2 ⎟
⎝ ⎠
2 1
RbsinebI M u1 = M u2 = M u enaH =
a 2
( L − a )2
b¤ a = L(2 − 2 ) = 0.586 L
BIsmIkar !^>!0 bnÞúkEdleFVIeGay)ak;KW wu = 11.66(M u / L2 ) nigm:Um:g;EdleFVIeGay)ak;
KW M u = 0.0858wu L2 . RbtikmμenAcMnuc A KW 0.586wu L nig RbtikmμenARtg;cMnuc B
KW 0.414wu L .
enAkñúgviFIlMnwg lMnwgrbs;Fñwm b¤rbs;Ggát;dac;edayELkrbs;FñwmRtUv)ansikSaeRkambnÞúk
EdlbgðajenAeBl)ak;. edIm,IbgðajkarviPaKedayviFIenH ]TahrN_BIxagmuxTaMgBIr
RtUv)aneFVIeLIgvijenATIenH.
]TahrN_16>5
sMrab;FñwmEdlbgðajenAkñúgrUbTI 16>25 kMNt;m:Um:g;EdleFVIeGay)ak;edayeRbIviFIlMnwg.
dMeNaHRsay
snøak;)aøsÞicBIrRtUv)anbegáItenAcMnuc A nig C . edayeyagtamrUbTI 16>25 e kMlaMgRbtikmμenAcMnuc
A KW (Pu / 2) + (M u1 / L ) nig RbtikmμenAcMnuc B KW (Pu / 2) − (M u1 / L )
KitlMnwgénFñwm BC nigKitm:Um:g;eFobcMnuc C /
⎛ Pu M u1 ⎞⎛ L ⎞
⎜ − ⎟⎜ ⎟ = M u 2
⎝ 2 L ⎠⎝ 2 ⎠
L
M u1 + 2M u 2 = Pu
2
EdlRsedogKñanwgsmIkar !^>#. enAeBl M u1 = M u 2 = M u / enaH
L
3M u = Pu
2
b¤ M u = Pu
L
2
]TahrN_16>6
FñwmCab; nigeRKag 415

39. T.Chhay NPIC
KNnam:Um:g;EdleFVIeGay)ak; (collapse moment) sMrab;FñwmEdlbgðajenAkñúgrUbTI 16>26 edayviFI
lMnwg.
dMeNaHRsay
1> snøak;)aøsÞicBIrRtUv)anbegáItenARtg;cMnuc A nig C . eyagtamrUbTI 16>26 d /
kMlaMgRbtikmμenA Rtg; A = wu (L / 2) + (M u1 / L) nigkMlaMgRbtikmμenARtg;
B = wu (L / 2 ) − (M u1 / L ) . bnÞúkenAelI BC KW wu b EdlmanGMeBIenA b / 2 BI B Edl
b = L − a . edayKitlMnwgénkMNat; BC nigKitm:Um:g; eFobcMnuc C /
⎛ L M ⎞
⎜ wu − u1 ⎟b − (wu b ) = M u 2
b
⎝ 2 L ⎠ 2
RbsinebI M u1 = M u 2 = M u enaH
wu
b
(L − b ) = M u (1 + b ) = M u (2 L − a )
2 L L
2M u (2 L − a )
wu = ×
L a (L − a )
EdlRsedogKñanwgsmIkar !^>$
wu L a(L − a )
Mu = ×
2 (2 L − a )
2> TItaMg a GacRtUv)ankMNt;dUcBImun Edl a = 0.586L / M u = 0.0858wu L2 nig
wu = 11.66(M u / L2 ).
12> mMurgVilrbs;snøak;)aøsÞic Rotation of Plastic Hinges
12>1> RbEvg snøak;)aøsÞic Plastic Hinge Length
karsnμt;rbs;RTwsþIeBlfamMurgVilminEmneGLasÞic (inelastic rotation) énebtugekItmanenA
cMnucm:Um:g;GtibrmaenAeBlEdlEpñkd¾éTeTotrbs;Ggát;eFVIkarCalkçN³eGLasÞic. tamBit mMu
rgVil)aøsÞic (plastic rotation) ekItmanenATaMgsgçagénmuxkat;m:Um:g;GtibrmaelIRbEvgkMNt;mYy.
RbEvgenHRtUv)aneKeGayeQμaHfa RbEvgsnøak;)aøsÞic l p . RbEvgsnøak;)aøsÞic l p CaGnuKmn_eTAnwg
kMBs;RbsiT§PaB d nigcMgayBImuxkat;énm:Um:g;x<s;bMput eTAcMnucm:Um:g;sUnü (contra-flexure) .
eyagtamrUbTI 16>27 a RbEvg l p / 2 tageGayRbEvgsnøak;)aøsÞicenAelIEpñkmçagénGkS½
rbs;TMr. M u nig φu Cam:Um:g;emKuN nigkMeNagemKuNenARtg;muxkat;eRKaHfñak; Edl M y nig φ y Ca
FñwmCab; nigeRKag 416

40. T.Chhay NPIC
m:Um:g; nigkMeNagenARtg;cMnucyaldMbUg (first yield). kMeNag)aøsÞicenARtg;muxkat;eRKaHfñak; φ p esμI
(φu − φ y ) ehIylT§PaBTb;rgVilesμInwg (φ p l p ).
RbEvgsnμt;rbs;snøak;)aøsÞicRtUv)ansnμt;famanRbEvgRbEhlkMBs;RbsiT§PaB.
Corley )anesñInUvsmIkarxageRkamsMrab;RbEvgsmmUlénsnøak;)aøsÞic
⎛z⎞
l p = 0.5d + 0.2 d ⎜ ⎟
⎝d ⎠
¬!^>!!¦
Edl z = cMgayénmuxkat;eRKaHfñak;eTAcMnucm:Um:g;sUnü nig d = kMBs;RbsiT§PaBrbs;muxkat;.
Mattock )anesñInUvTMrg;d¾smBaØ
l p = 0.5d + 0.05 z ¬!^>!@¦
karBiesaFn_CaeRcInenAelIebtugGarem:Cab;bgðajfa l p GacRtUv)ansnμt;esμInwg 1.06d . va
k¾)anbgðajEdrfaRbEvgénsnøak;)aøsÞicenAkñúgFñwmebtugGarem:Cab;EdlmanTMBk;enAxagcugekIneLIgCamY
ynwgnwgkarekIneLIgénbrimaNEdk fibers nigEdkemGaRs½ynwgsmIkarxageRkam³
l p = (1.06 + 0.13ρρ s )d ¬!^>!#¦
FñwmCab; nigeRKag 417

41. T.Chhay NPIC
Edl ρ = PaKryEdkemenAkñúgmuxkat; nig ρ s = PaKryén fibers EdkedaymaD/
0 ≤ ρ s ≤ 1.2 . ]TahrN_ RbsinebI ρ = 1% nig ρ s = 0.8% enaH l p = 1.164d .
12>2> emKuNEbgEckkMeNag Curvature Distribution Factor
emKuNsMxan;mYyeTotEdlTak;TgkñúgkarKNnamMurgVil)aøsÞicKWemKuNEbgEckkMeNag β . kM
eNagtambeNþaysnøak;)aøsÞicERbRbYly:agFM ehIyenAkñúgkarKNnamMurgVilemKuNenHminRtUv)anKit
EdleFVIeGaykarKNnasnøak;)aøsÞicmantMélFM. eyagtamrUbTI 16>27 RkLaépÞqUt ABC tag
eGaymMurgVilminEmneGLasÞicEdlGacekItmanenAsnøak;)aøsÞic b:uEnþRkLaépÞGt;qUt EBF tageGay
karcUlrYménlkçN³eGLasÞicrbs;mMurgVilelIRbEvgrbs;Ggát;. RkLaépÞqUt ABC GacRtUv)ansnμt;
FñwmCab; nigeRKag 418

42. T.Chhay NPIC
esμInwgplKuNrvag β nigRkLaépÞsrub ABCD enAkñúgRbEvgsnøak;)aøsÞic l p / 2 EdlenAelIRCugmçag
rbs;muxkat;eRKaHfñak;. emKuNEbgEckkMeNag β CapleFobénmMurgVil)aøsÞicBit θ pc elI φl p Edl
φ CakMeNag nig l p CaRbEvgénsnøak;)aøsÞic. tMél β sßitenAcenøaH 0.5 nig 0.6 . karBiesaFn_)an
bgðajfa β GacRtUv)ansnμt;esμInwg 0.56 . enAeBlEdlEdk fibers EdlmanTMBk;cugRtUv)aneRbIenA
kñúgFñwmebtug tMél β fycuHGaRs½yeTAnwgsmIkarxageRkam³
β = 0.56 − 0.16 ρ s ¬!^>!$¦
Edl ρ s CaPaKry fibers Edk/ 0 ≤ ρ s ≤ 1.2% . karfycuHénemKuNEbgEckkMeNagénebtugEdl
eRbIEdk fibers min)anbgðajfalT§PaBTb;rgVilfycuHeT. kMeNag)aøsÞicrbs;ebtugEdleRbIEdk fibers
EtgEtFMCagkMeNag)aøsÞicrbs;ebtugEdlmineRbIEdk fibers. rUbTI 16>28 bgðajBIkarEbgEckkM
eNagtambeNþayRbEvgsnøak;)aøsÞic. RkLaépÞ ABC1 tageGaymMurgVil)aøsÞicsMrab;ebtugEdlmin
manEdk fibers/ β = 0.56 b:uEnþRkLaépÞ ABC2 nig ABC3 tageGaymMurgVil)aøsÞicsMrab;ebtugEdl
manEdk fibers 0.8% nig 1.2% erogKña.
12>3> snÞsSn_énPaBsVit Ductilty Index
pleFobénkMeNagemKuNnigkMeNagyaldMbUgRtUv)aneKeGayeQμaHfasnÞsSn_énPaBsVit/
μ = φu / φ y . snÞsSn_énPaBsVitrbs;ebtugGarem:ERbRbYlcenøaH 4 nig 6 . RbsinebIEdk fibers
RtUv)aneRbIenAkñúgFñwmebtugGarem: enaHsnÞsSn_énPaBsVitekIneLIgtamsmIkarxageRkam³
μ ' = (1.0 + 3.8ρ s )μ ¬!^>!%¦
Edl μ = pleFobkMeNagemKuNelIkMeNagyaldMbUg
μ ' = snÞsSn_énPaBsViténebtugEdlmanEdk fibers
ρ s = PaKryénEdk fibers edaymaD/ 0 ≤ ρ s ≤ 1.2%
12>4> mMurgVilEdlRtUvkar Required Rotation
eKRtUvkarmMurgVilénsnøak;)aøsÞicenAeRKOgbgÁúMminkMNt;ebtugGarem:edIm,IGnuBaØateGaysnøak;
)aøsÞicekItman ehIyeRKOgbgÁúMEdlxiteTArkemkanicénkar)ak;GacRtUv)ankMNt;eday slope
deflection BIsmIkarxageRkam. sMrab;kMNat; AB cenøaHsnøak;)aøsÞicNIr mMurgVilenARtg; A KW
θA =
L
6 Ec I
[2(M A − M FA ) + (M B − M FB )] ¬!^>!^¦
Edl M A nig M B = m:Um:g;emKuNenARtg;cMnuc A nig B erogKña
M FA nig M FB = m:Um:g;bgáb;cugeGLasÞicenARtg;cMnuc A nig B
FñwmCab; nigeRKag 419

43. T.Chhay NPIC
EC = m:UDuleGLasÞicrbs;ebtug = 4730 f 'c
I = m:Um:g;niclPaBénmuxkat;eRbH ¬emeronTI 5¦
12>5> lT§PaBTb;mMurgVil Rotation Capacity Provided
snøak;)aøsÞicrgkarTajKMrUenARtg;muxkat;TMr nigmuxkat;kNþalElVgéneRKagRtUv)anbgðajenA
kñúgrUbTI 16>29.lT§PaBTb;mMurgVilGaRsy½CacMbgeTAnwg³
- Ultimate strain capacity rbs;ebtug ε 'c EdlGacRtUv)ansnμt;esμI 0.003 b¤ 0.0035 .
- RbEvg l p EdlenAelIRbEvgenHsnøak;)aøsÞiceFVIkardl;yal.
RbEvgenHRtUv)ansnμt;mantMélRbEhlkMBs;RbsiT§PaBénmuxkat;Edlsnøak;)aøsÞicekItman
¬lp = d ¦
- kMBs;rbs;bøúksgát; c enAkñúgebtugenAeBl)ak;enARtg;muxkat;énsnøak;)aøsÞic. mMurgVil θ
énsnøak;)aøsÞicRtUv)anKNnatamsmIkarxageRkam³
ε pI p
θ=
c
¬!^>!&¦
FñwmCab; nigeRKag 420

44. T.Chhay NPIC
Edl ε p KWCakMeNInén strain enAkñúgebtugEdlvas;BI initial yielding rbs;EdkenAkñúgmux
kat; ¬rUbTI 16>29 c ¦³
ε p = ε 'c −ε c1 = 0.0035 − ε c1
RbsinebI l p = d ehIypleFob c / d esμInwg λ ≤ 0.5 /
(0.0035 − ε c1 )d 0.0035 − ε c1
θ= =
λd λ
BIRtIekaNbMErbMrYlrageFob (strain triangle) rUbTI 16>29/
⎛ c ⎞ f y ⎛ λd ⎞ f y ⎛ λ ⎞
ε c1 = ε y ⎜ ⎟= ⎜ ⎟= ⎜ ⎟
⎝ d − c ⎠ E s ⎝ d − λd ⎠ E s ⎝ 1 − λ ⎠
Edl ersIusþg;yalrbs;Edk nig E s = m:UduleGLasÞicrbs;Edk = 2 ⋅105 MPa . dUcenH
fy =
0.0035 ε c1 0.0035 fy
θ=
λ
−
λ
=
λ
−
E s (1 − λ )
¬!^>!*¦
sMrab;Edk f y = 280MPa nigedayeRbItMélGtibrmaén λ = 0.50 enaH
0.0035 280
θ min = − 5
= 4.2 ⋅ 10 −3 rad
0 .5 2 ⋅ 10 × (1 − 0.5)
sMrab;Edk f y = 400MPa nig λmax = 0.44 enaH
0.0035 280
θ min = − 5
= 4.38 ⋅ 10 −3 rad
0.44 2 ⋅ 10 × (1 − 0.44)
θ min Edl)anKNnaenATIenH)anBIEtmçag ehIymMurgVilGnuBaØatsrubenARtg;snøk;)aøsÞicesμInwg
2θ min b¤ 2θ . λ BitGacRtUv)anKNnadUcxageRkam
eKman a = β1c nig β1 = 0.85 sMrab; f 'c ≤ 28MPa
a As f y
c= =
β1 (0.85)2 f 'c b
As f y ρf y
λ=
c
= =
d 0.72 f 'c bd 0.72 f 'c
≤ 0.5 ¬!^>!(¦
Edl ρ = As / bd . eKnwgTTYl)an λmax enAeBlEdleRbI ρ max .
RbsinebIlT§PaBTb;rgVilminRKb;RKan; eKGacbegáInmuxkat; b¤kat;bnßyPaKryEdkedIm,I
TTYl)an c tUc/ λ tUc nig θ FM. RbsinebIeKeRbIEdkkgvNн bMErbMrYlrageFobemKuNEdleFVIeGay
EbkebtugnwgekIneLIgx<s;dl; 0.012 .
sMrab;snøak;)aøsÞicsgát; ¬dUcenAkñúgssr¦
FñwmCab; nigeRKag 421

45. T.Chhay NPIC
ε pl p
θ=
h
¬!^>@0¦
Edl h = kMBs;srubrbs;muxkat; nig l p = RbEvgEdl yielding ekItman. enAkñúgsnøak;sgát; l p
ErbRbYlcenøaH 0.5h nig h .
enAeBlebtugrgkugRtaMgemKuNdl; f 'c / ε c = 0.002 dUcenH ε p = ε 'c −0.002
= 0.0035 − 0.002 = 0.0015 CamMurgVilGb,brmaenARCugmçag. dUcenH
0.0015 × 0.5h
θ min = = 0.00075rad
h
CamYynwgEdkkgvNн/ θ GacnwgekIneLIgdl;
0.5h
θ max = (0.012 − 0.002 ) × = 0.005rad
h
tMél ε 'c = 0.012 CatMélEdlFMNas; dUcenHtMéltUcCagenHGacRtUv)aneRbICamYyEdkkg
vNн. RbsinebImindUcenHeT eKGaceRbImuxkat;epSgeTot.
enAkñúgFñwmCab;ebtugGarem:EdlmanEdk fibers mMurgVil)aøsÞicGacRtUv)anKNnadUcxageRkam³
⎛ 0.0035 fy ⎞
θ p = λβ ⎜ ⎜ λ − ⎟
E (1 − λ ) ⎟
¬!^>@!¦
⎝ s ⎠
Edl λ = (4.3 + 2.24 ρ s − 0.00625 f y + 4.17 ρρ s ) ¬!^>@@¦
β = 0.56 − 0.16 ρ s ¬!^>!$¦
ersIusþg;yalrbs;Edk
fy =
E s = m:UduleGLasÞicrbs;Edkem
ρ = PaKryEdkem
ρ s = PaKryEdk fibers KitmaD
smIkar !^>@! Bgðajy:agc,as;famMurgVil)aøsÞicénebtugGarem:EdleRbIEdk fibers KWGaRs½yeTA
nwg PaKryEdkem/ PaKryEdk fibers nigersIusþg;yalrbs;va. karbegáInersIusþg;yalkat;bnßymMu
rgVil)aøsÞic. smIkar !^>@! k¾rab;bBa©ÚlnUvT§iBlénRbEvgsnøak;)aøsÞicenAelImMurgVilEdr.
xageRkamCaTMrg;EdlRtUv)amsMrYledIm,IKNnamMurgVil)aøsÞic³
⎛ 0.003 ⎞
θ p = λβ ⎜
⎝ λ ⎠
⎟ ¬!^>@#¦
FñwmCab; nigeRKag 422

46. T.Chhay NPIC
]TahrN_ RbsinebI ρ s = 0 nig f y = 400MPa enaH θ p1 = 0.00289 / λ nigRbsinebI ρ s = 1% /
ρ = 1.5% nig f y = 400MPa enaH θ p 2 = 0.01235 / λ . enHmann½yfa lT§PaBTb;mMurgVilrbs;
FñwmebtugGarem:nwgekIneLIgRbEhlbYndgRbsinebIeKeRbIEdk fibers 1% .
13> segçbviFIsaRsþkñúgkarKNnasßanPaBkMNt; Summary of Limit Design
Procedure
- KNnabnÞúkemKuNedayeRbIemKuNbnÞúkEdleGayenAkñúgemeronTI 3³ wu = 1.2D + 1.6L
- kMNt;Ggát;emkanicénkar)ak;/ snøak;)aøsÞic/ nigm:Um:g;emKuN M u
- KNnamuxkat;eRKaHfñak;edayeRbIviFIKNnaersIusþg; (strength design method)
- kMNt;mMurgVilcaM)ac;énsnøak;)aøsÞic
- KNnalT§PaBTb;mMurgVilenARtg;muxkat;énsnøak;)aøsÞic. lT§PaBTb;mMurgVilRtUvEtFMCagmMurgVil
caM)ac;.
- RtYtBinitüemKuNRbqaMg yielding rbs;Edk nigsñameRbHFM² ¬EdlCapleFobrvag
φM u / elastic moment at service load
- epÞógpÞat;PaBdab nigsñameRbHeRkamGMeBIbnÞúkeFVIkar
- RtYtBinitüemIlfaEdkkMlaMgkat;TTwgRKb;RKan;RtUv)andak;enARKb;muxkat;
]TahrN_16>7
FñwmdUcEdlbgðajenAkñúgrUbTI 16>30 RtUv)anbgáb;enAcugTaMgsgçag nigRTbnÞúkBRgayesμIemKuN
80kN / m nigbnÞúkcMcMnucemKuN 213.5kN . KNnaFñwmedayeRbIdMeNIrkarKNnasßanPaBkMNt;.
eKeGay b = 350mm / f 'c = 21MPa nig f y = 280MPa .
dMeNaHRsay
1> bnÞúkBRgayesμIemKuN wu = 80kN / m . bnÞúkcMcMnucemKuN Pu = 213.5kN / m .
2> snøak;)aøsÞicekItmanenARtg;cMnuc A / B nig C EdlbNþaleGaymanemkanicénkar)ak;dUc
bgðajenAkñúgrUbTI 16>30. edayeRbIviFIkmμnþCak;Esþg (virtual work method) nigsnμt;PaB
dabÉktþaenARtg;cMnuc C enaHkmμnþxageRkAesμInwg
We = 213.5 × 1 + 80 × 3.7 = 509.5kN .m
kmμnþxagkñúgEdlRsUbedaysnøak;)aøsÞicKW
Wi = M uθ (at A) + M uθ (at B) + 2 M uθ (at C)
FñwmCab; nigeRKag 423

47. T.Chhay NPIC
4M u
= 4 M uθ =
3 .7
edayeGay We = Wi enaH M u = 471.3kN .m . tamkarviPaKTUeTAeGay
wu L2 L 7 .4 2 7 .4
Mu = + Pu = 80 + 213.5 = 471.3kN .m
16 8 16 8
3> KNnamuxkat;eRKaHfñak;enARtg;cMnuc A / B nig C sMrab; M u = 471.3kN .m . BItaragenA
kñúg]bsm<n§½ B nigsMrab; f 'c = 21MPa / f y = 280MPa CamYynwgPaKryEdk ρ = 0.013
eK)an Ru = 2.95MPa ¬ ρ max = 2.04% ¦
M u = Ru bd 2
471.3 ⋅ 10 6
d= = 676mm
2.95 × 350
kMBs;srubrbs;FñwmKW h = 676 + 60 = 736mm yk 740mm
As = ρbd = 0.013 × 350 × 676 = 3075.8mm 2
eRbIEdk 5DB28 mYyRsTab; ¬ As = 3079mm 2 ¦/
bmin = 9 × 28 + 2 × 10 + 70 = 342mm < 350mm
As f y 3079 × 280
a= = = 138mm
0.85 f 'c b 0.85 × 21 × 350
a c 162.35
c= = 162.35mm λ= = = 0.24
0.85 d 676
4> mMurgVilcaM)ac;rbs;snøak;)aøsÞicKW
a. θ A =
L
[2(M A − M FA ) + (M B − M FB )]
6 Ec I
E c = 4730 f 'c = 4730 21 = 21676MPa
E s = 200000MPa
b. kMNt;m:Um:g;cugbgáb;enARtg; A nig B edayeRbIbnÞúkemKuN³
wu L2 L 7 .4 2 7 .4
M FA = M FB = + Pu = 80 + 213.5 = 562.55kN .m
12 8 12 8
m:Um:g;)aøsÞic M A = M B = 471.3kN .m
c. m:Um:g;niclPaBénmuxkat;eRbHGacRtUv)anKNnadUcxageRkam
x3
I cr =b + nAs (d − x )2
3
FñwmCab; nigeRKag 424

48. T.Chhay NPIC
Edl x CacMgayBIsrésrgkarsgát;eTAGkS½NWt (kd ) . edIm,IkMNt; x ¬emIlemeronTI
6¦/ x = 257.4mm nig I cr = 68.44 ⋅108 mm 4
d. mMurgVilcaM)ac;Gb,brma³ edaycat;Tukm:Um:g;TaMgGs;enARtg; A nig B GviC¢man/ enaH
7400
θA = 8
[2(− 471.3 + 562.55) + (− 471.3 + 562.55)]⋅ 10 6
6 × 21676 × 68.44 ⋅ 10
= 0.00227 rad
5> lT§PaBTb;mMurgVilKW³
0.0035 fy 0.0035 280
θA = − = −
λ E s (1 − λ ) 0.24 200000(1 − 0.24)
= 0.0127 rad > 0.00227 rad
lT§PaBTb;mMurgVilesμIRbEhl 5.5 dgénmMurgVilcaM)ac; enHbgðajfamuxkat;RKb;RKan;.
6> RtYtBinitüpleFobrvagm:Um:g;emKuNelIm:Um:g;eGLasÞiceRkambnÞúkeFVIkar
wL2 L 50 × 7.4 2 133.4 × 7.4
MA = MB = +P = + = 351.6kN .m
12 8 12 8
⎡ ⎛ a ⎞⎤ ⎡ ⎛ 138 ⎞⎤ 6
φM n = φAs f y ⎢d − ⎜ ⎟⎥ = 0.9 × 3079 × 280⎢676 − ⎜ ⎟⎥ ⋅ 10 = 471kN .m
⎣ ⎝ 2 ⎠⎦ ⎣ ⎝ 2 ⎠⎦
471
pleFobKW 351.6
EdltMNageGayemKuNsuvtßiPaBRbqaMgnwg
= 1.34 rbs;Edk
yielding
enARtg;TMr.
7> RtYtBinitüPaBdabGtibrmaEdlbNþalmkBIbnÞúkeFVIkar ¬enAkNþalElVg¦³ ykbnÞúkeFVIkar
BRgayesμI w = 50kN / m nig P = 133.4kN enaH
wL4
Δ1 =
384 EI
sMrab;bnÞúkcMcMnucenAkNþalElVg
PL3
Δ2 =
192 EI
PaBdabsrubKW
50 × 7400 4 133400 × 7400 3
Δ= + = 4.53mm
384 × 21676 × 68.44 ⋅ 108 192 × 21676 × 68.44 ⋅ 108
Δ 4.53 1
= =
L 7400 1633
EdlCapleFobtUcEmnETn
8> brimaNEdkkMlaMgkat;TTwgRKb;RKan;RtUv)andak;edIm,IeCosvagkar)ak;edaykMlaMgkat;TTwg
EdlGacekItman.
FñwmCab; nigeRKag 425

49. T.Chhay NPIC
14> karEbgEckm:Um:g;eLIgvijénm:Um:g;GviC¢manenAkñúgFñwmCab; Moment
Distribution of Negative Moments in Continuous Beams
karEbgEckm:Um:g;GviC¢maneLIgvijenAkñúgGgát;rgkarBt;Cab;KWQrelIsac;lUteFobsuT§ (net
tensile strain NTS) ε t sMrab;TaMgebtugGarem: nigebtugeRbkugRtaMg. rUbTI 16>31 bgðajlImIt
GnuBaØatkñúgkarEbgEckm:Um:g;eLIgvij. vabgðajfaPaKrym:Um:g;GviC¢manekIneLIg b¤fycuHenAelITMr
rbs;FñwmCab; q' EdlKNnaedayRTwsþIeGLasÞicminRtUvFMCag 1000ε t % CamYytMélGtibrma 20% .
karEbgEckm:Um:g;eLIgvijRtUv)anGnuBaØatEtenAeBlEdl ε t ≥ 0.0075 EdlbgðajfavamanPaBsVit
FñwmCab; nigeRKag 426

50. T.Chhay NPIC
RKb;RKan;enARtg;muxkat;Edlm:Um:g;RtUv)ankat;bnßy. enAeBl ε t < 0.0075 karEbgEckm:Um:g;eLIgvij
minRtUv)anGnuBaØateLIy. m:Um:g;GviC¢manEdl)anEktMrYvRtUv)aneRbIsMrab;KNnam:Um:g;viC¢manEdl)anEk
tMrUv (ACI Code, Section 8.4) . karEbgEckm:Um:g;eLIgvijminRtUv)anGnuvtþcMeBaHGgát;EdlKNna
edayviFIKNnaedaypÞal;sMrab;RbBn§½kMralxNнeT ¬eyagtamemeronTI 17¦.
Casegçb PaBryekIneLIg nigPaKryfycuHénm:Um:g;GviC¢manmandUcxageRkam³
- enAeBlEdl ε t < 0.0075 karEbgEckm:Um:g;eLIgvijminRtUv)anGnuBaØat ¬ ρ / ρ b > 0.476 ¦.
- enAeBlEdl ε t = 0.0075 PaKryénkarEbgEckm:Um:g;eLIgvijKW 7.5% ¬ ρ / ρ b = 0.476 ¦.
- enAeBlEdl ε t > 0.020 PaKryénkarEbgEckm:Um:g;eLIgvijKW 20% ¬ ρ / ρ b = 0.217 ¦.
- enAeBlEdl 0.0075 < ε t < 0.020 PaKryEbgEckm:Um:g;eLIgvijKW
q '= 1000ε t ¬!^>@$¦
]TahrN_ RbsinebI ε t = 0.010 enaHPaKryEbgEckm:Um:g;eLIgvijKW 10% . TMnak;TMngrvag
PaKryEdk ρ enAkñúgmuxkat; nigsac;lUteFobsuT§ ε t KWmandUcxageRkam ¬eyagtamEpñk 3>9¦³
FñwmCab; nigeRKag 427

51. T.Chhay NPIC
⎡⎛ fy ⎞⎤
⎢ ⎜ 0.003 +
⎜
⎟⎥
⎟
⎢⎝
εt = ⎢
Es ⎠ ⎥ − 0.003 ¬#>@$¦
⎛ ρ ⎞ ⎥
⎢ ⎜ ⎥
⎢ ⎜ρ ⎟ ⎟ ⎥
⎣ ⎝ b⎠ ⎦
sMrab;Edk f y = 400MPa nig Es = 200000MPa . edaysnμt; f y / E s = 0.002 enaH
⎡ ⎤
⎢ ⎥
εt = ⎢ 0.005 ⎥ − 0.003 ¬#>@%¦
⎢⎛ ρ ⎞ ⎥
⎢⎜⎜ ⎟⎥
⎟
⎢ ⎝ ρb ⎠ ⎥
⎣ ⎦
sMrab; ε t = 0.0075 / lImIténPaBsVitKW ε t / ε y = 0.0075 / 0.002 = 3.75 . PaKryEbgEckm:Um:g;eLIg
vijERbRbYlGaRs½yeTAnwgkarkMNt;TaMgenH nigsMrab; f y = 400MPa RtUv)aneGayenAkñúgtaragTI
16>1 nig 16>2.
RKb;PaKryEbgEckm:Um:g;eLIgvijEdlRtUv)aneRbI vacaM)ac;RtUvFanafamuxkat;minrgnUvkarxUc
xat b¤mansñameRbHFMedaysarbnÞúkeFVIkar ehIyvaRtUvrkSalT§PaBTb;rgVilenARKb;muxkat;eRKaHfñak;enA
kñúgeRKOgbgÁúM. karEbgEckm:Um:g;eLIgvijenAkñúgeRKOgbgÁúMsþaTicminkMNt;eFVIeGaym:Um:g;GviC¢manenAelI
TMr nigm:Um:g;viC¢manenAkNþalElVgfycuH. karfycuHenHmin)aneFVIeGaysuvtßiPaBrbs;eRKOgbgÁúMfycuH
b¤eFVIeGaymaneRKaHfñak;enaHeT ebIeRbobeFobCamYyeRKOgbgÁúMsþaTickMNt;. CakarBit PaBCab;enAkñúg
rcnasm<n§½pþl;nUversIusþg;/ esßrPaB nigesdækic©bEnßmeTotkñúgkarKNna.
emKuNEbgEckm:Um:g;eLIgvij q EdlQrelI (ACI Code 318-99) RtUv)anKNnadUcxageRkam
⎡ (ρ − ρ ') ⎤
q = 20 ⎢1 −
ρ ⎥ ¬!^>@%¦
⎣ b ⎦
enAkñúgsmIkarTI !^>@% PaKryEdk ρ b¤ ( ρ − ρ ' ) enARtg;muxkat;Edlm:Um:g;RtUv)ankMNt;
RtwmPaKryGtibrma 0.5ρ b . PaKryEdkGb,brmaenAkñúgmuxkat;rgkarBt;RtUv)ankMNt;Rtwm
f 'c /(4 f y ) ≥ 1.4 / f y . edayeRbIkarkMNt;dac;xatTaMgenHPaKryEbgEckm:Um:g;eLIgvijGtibrma
nigGb,brmaRtUv)anbgðajenAkñúgtarag 16>3.
taragTI 16>1 karpøas;bþÚrPaKryEbgEckm:Um:g;eLIgvij (q' ) f y = 400MPa
εt 0.0075 0.010 0.0125 0.0150 0.0175 0.020 0.0225
ρ / ρb 0.476 0.385 0.323 0.278 0.244 0.217 0.196
q ' (%) 7 .5 10 12.5 15 17.5 20 20
FñwmCab; nigeRKag 428

52. T.Chhay NPIC
taragTI 16>2 karpøas;bþÚrPaKryEbgEckm:Um:g;eLIgvij (q' ) sMrab;kareGayPaKry ρ / ρ b
ρ / ρb 0.48 0.45 0.40 0.35 0.30 0.25 0.20
εt 0.0074 0.0081 0.0095 0.0113 0.0137 0.017 0.022
q ' (%) 0 .0 8 .1 9 .5 11.3 13.7 17.0 20.0
taragTI 16>3 karEbgEckm:Um:g;eLIgvijGtibrma nigGb,brma q ¬smIkar !^>@%¦
f 'c ( MPa) f y ( MPa) ρb ρ min q max %(for ρ min ) q min %(for 0.5ρ b )
21 400 0.0215 0.0033 16.9 10
28 400 0.0285 0.0033 17.7 10
35 400 0.0339 0.003 17.9 10
]TahrN_16>8
kMNt;m:Um:g;eGLasÞicGtibrmaenARtg;TMr nigenAkNþalElVgénFñwmCab;EdlmanbYnElVgdUcbgðajenA
kñúgrUbTI 16>32 a. Fñwmmanmuxkat;esμI nigRTnUvbnÞúkefrBRgayesμI 117kN / m nigbnÞúkGefr
87.5kN . snμt;PaKryEbEckm:Um:g;eLIgvijGtibrma 10% nigBicarNanUvkrNIBIrxageRkam³
- enAeBlEdlbnÞúkGefrRtUv)andak;enAelIElVgqøas;Kña KNnam:Um:g;viC¢manGtibrmaenAkñúgElVg
- enAeBlEdlbnÞúkGefrRtUv)andak;enAelIElVgCab;Kña KNnam:Um:g;GviC¢manGtibrmaenAelITMr
dMeNaHRsay
1> Fñwmmanm:Um:g;niclPaB I esμIKña nigman E dUcKña. dUcenH EI efr. smIkarbIm:Um:g;
sMrab;viPaKFñwmnigsMrab; EI efrKW³
w1 L1 w2 L3
3
M A L1 + 2 M b (L1 + L2 ) + M C L2 = − − 2
4 4
edaysarElVgesμIKña
L2
M A + 4M B + M C = −
4
(w1 + w2 ) ¬!^>@^¦
enAkñúg]TahrN_enH M A = M E = 0 . krNIénkardak;bnÞúk6EbbepSgKñaRtUv)an
BicarNadUcbgðajenAkñúgrUbTI 16>32³
FñwmCab; nigeRKag 429

53. T.Chhay NPIC
krNITI1> bnÞúkefrRtUv)andak;enAelIFñwm ABCDE TaMgmUl ¬rUbTI 16>32 b¦.
krNITI2> bnÞúkGefrRtUv)andak;enAelIElVg AB nig CD sMrab;m:Um:g;GtibrmaenAkñúg
ElVg AB nig CD ¬rUbTI 16>32 c¦.
krNITI3> dUckrNITI 2 sMrab;ElVg BC nig DE ¬rUbTI 16>32 d¦.
krNITI4> bnÞúkGefrRtUv)andak;enAelIElVg AB / BC nig DE sMrab;m:Um:g;GviC¢manenA
Rtg; B ¬rUbTI 16>32 e¦.
krNITI5> bnÞúkGefrRtUv)andak;enAelIElVg CD nig DE ¬rUbTI 16>32 f¦.
krNITI6> bnÞúkGefrRtUv)andak;enAelI BC nig CD sMrab;m:Um:g;GviC¢manGtibrmaenARtg;
C ¬rUbTI 16>32 g¦.
2> krNITI1> GnuvtþsmIkar !^>@^ eTAelIFñwmkMNat; ABC / BCD nig CDE erogKña
62
4M B + M C = − (117 + 117) = −2106kN .m
4
M B + 4M C + M D = −2106kN .m
M C + 4M D = −2106kN .m
KNnasmIkarTaMgeyIgTTYl)an
M B = M D = −451.28kN .m nig M C = −300.9kN .m
sMrab;karkat;bnßym:Um:g; 10%
M ' B = M ' D = 0.9 × (− 451.28) = −406.2kN .m
M 'C = 0.9 × (− 300.9) = −270.81kN .m
m:Um:g;kNþalElVgEdlRtUvKñanwgm:Um:g;Edl)ankat;bnßyehIyKW
wD L2 1 117 × 6 2 1
ElVg AB = DE = 8 + 2 M ' B = 8 − 2 406.2 = 323.4kN .m
2
ElVg BC = CD = 1178× 6 − 1 (406.2 + 270.81) = 188kN .m
2
3> krNITI2> GnuvtþsmIkar !^>@@ eTAelIkMNat;Fñwm ABC / BCD nig CDE erogKña³
62
4M B + M C = − (87.5) = −787.5kN .m
4
M B + 4M C + M D = −787.5kN .m
M C + 4M D = −787.5kN .m
FñwmCab; nigeRKag 430

54. T.Chhay NPIC
KNnasmIkarTaMgeyIgTTYl)an
M B = M D = −168.75kN .m
nig M C = −112.5kN .m
m:Um:g;eGLasÞickNþalElVgEdlRtUvKñanwgm:Um:g;xagelIKW
wL L2 1 2
Fñwm AB = 8 + 2 M B = 87.58× 6 − 1 168.75 = 309.4kN .m
2
1
BC = 0 − (168.75 + 112.5) = 140.6kN .m
2
w L2 1 87.5 × 6 2
CD = L − (168.75 + 112.5) = − 140.625 = 253.1kN .m
8 2 8
1
DE = 0 − 168.75 = 84.4kN .m
2
edIm,Ikat;bnßym:Um:g;viC¢mankñúgElVg eKRtUvbegáInm:Um:g;Rtg;TMreday 10% nigKNnam:Um:g;
kñúgElVgEdlRtUvKña. m:Um:g;viC¢manEdlTTYl)anRtUvEtesμIy:agtic 90% énm:Um:g;Edl
KNnadMbUg.
M ' B = M ' D = 1.1(− 168.75) = −185.6kN .m
M 'C = 1.1(−112.5) = −123.75kN .m
m:Um:g;eGLasÞickNþalElVgEdlRtUvKñanwgm:Um:g;xagelIKW
2 2
Fñwm AB = wL8L + 1 M ' B = 87.58× 6 − 1 185.6 = 300.95kN .m
2 2
1
BC = 0 − (185.6 + 123.75) = 154.7kN .m
2
wL L2 1 87.5 × 6 2
CD = − (123.75 + 185.6) = − 154.7 = 239.05kN .m
8 2 8
1
DE = 0 − 185.6 = 92.8kN .m
2
4> krNITI3> RsedogKñanwgkrNITI2> ehIym:Um:g;RtUv)anbgðajenAkñúgrUbTI 16>32 d.
5> krNITI4> BicarNaElVg AB / BC nig DE EdlRTbnÞúkGefredIm,IKNnam:Um:g;GviC¢man
GtibrmaenARtg;TM B ³
62
4M B + M C = − (87.5) = −1575kN .m
2
M B + 4M C + M D = −787.5kN .m
M C + 4M D = −787.5kN .m
FñwmCab; nigeRKag 431

55. T.Chhay NPIC
KNnasmIkarTaMgeyIgTTYl)an
M B = −379.7 kN .m / M C = −56.3kN .m
nig M D = −182.8kN .m
sMrab;karkat;bnßym:Um:g; 10% Rtg; B
M ' B = 0.9(− 379.7 ) = −341.7 kN .m
m:Um:g;kNþalElVgEdlRtUvKñaKW
2 2
Fñwm AB = wL8L + 1 M B = 87.58× 6 − 1 341.7 = 222.9kN .m
2 2
wL L2 1 87.5 × 6 2
BC = − (341.7 + 56.3) = − 199 = 194.75kN .m
8 2 8
1
CD = 0 − (56.3 + 182.8) = −119.55kN .m
2
w L2 1 87.5 × 6 2
DE = L − 182.8 = − 91.4 = 302.35kN .m
8 2 8
6> krNITI5> RsedogKñanwgkrNITI4> elIkElgEtElVgxagcugminRtUv)andak;bnÞúkedIm,I
TTYl)anm:Um:g;viC¢manGtibrmarnARtg;TMr B ¬b¤TMr D sMrab;kardak;bnÞúkRsedogKña¦.
düaRkamm:Um:g;Bt;RtUv)anbgðajenAkñúgrUbTI 16>32 f.
7> krNITI6> BicarNaElVg BC nig CD EdlRTbnÞúkGefredIm,IKNnam:Um:g;GviC¢man
GtibrmaenARtg;TM B ³
62
4M B + M C = − (87.5) = −787.5kN .m
4
w L2
M B + 4 M C + M D = − L = −1575kN .m
2
2
w L
M C + 4 M D = − L = −787.5kN .m
4
KNnasmIkarTaMgeyIgTTYl)an
M B = M D = −112.5kN .m
nig M C = −337.5kN .m
sMrab;karkat;bnßym:Um:g; 10% Rtg; C
M 'C = 0.9(− 337.5) = −303.75kN .m
M ' B = M ' D = 0.9(− 112.5) = −101.25kN .m
FñwmCab; nigeRKag 432

56. T.Chhay NPIC
m:Um:g;kNþalElVgEdlRtUvKñaKW
Fñwm AB = DE = − 1 101.2 = −50.6kN .m
2
wL L2 1 87.5 × 6 2
BC = CD = − (101.2 + 303.95) = − 202.6 = 191.15kN .m
8 2 8
8> m:Um:g;Gtibrma nigGb,brmaeRkaykarEbgEckm:Um:g;eLIgvijRtUv)anbgðajenAkñúgtarag
16>4. Moment envelop RtUv)anbgðajenAkñúgrUbTI 16>32 h.
9> enAkñúg]TahrN_renH muxkat;kNþalElVgRtUv)aneRbIsMrab;PaBgayRsYl. m:Um:g;kNþal
ElVgmincaM)ac;Cam:Um:g;viC¢manGtibrmaeT. kñúgkrNIElVgcug AB nig DE m:Um:g;Gtibrma
eRkayeBlBIkarEbgEckbnÞúkeLIgvij 10% KWwesμInwg wD L2 / 12.2 nigekItmanenAcMgay
0.4 L BI BC nig D .
taragTI 16>4 m:Um:g;cugeRkayén]TahrN_TI 16>8 eRkayBIkarEbgEckm:Um:g;eLIgvij
krNI 1 2 3 4 ¬1 + 2¦ 5 ¬1 + 3¦
TItaMg m:Um:;g m:Um:;gGviC¢man m:Um:;gviC¢man m:Um:;gGviC¢man m:Um:;gviC¢man
muxkat; DL Gtibrma LL Gtibrma LL Gtibrma DL + LL Gtibrma DL + LL
TMr
A 0 0 0 0 0
B − 406.2 − 341.7 + 28.1 − 747.9 * − 378.1
C − 300.9 − 303.75 - − 604.65 * − 300.9
D − 406.2 − 341.7 + 28.1 − 747.9 * − 378.1
E 0 0 0 0 0
kNþalElVg
AB + 323.4 − 92.8 + 300.95 + 230.6 + 624.35 *
BC + 188 − 154.7 + 239.05 + 33.3 + 427.05 *
CD + 188 − 154.7 + 239.05 + 33.3 + 427.05 *
DE + 323.4 − 92.8 + 300.95 + 230.6 + 624.35 *
(*) m:Um:g;KNnaGtibrma nigGb,brmacugeRkay.
FñwmCab; nigeRKag 433

57. T.Chhay NPIC
FñwmCab; nigeRKag 434

58. T.Chhay NPIC
FñwmCab; nigeRKag 435

59. T.Chhay NPIC
]TahrN_16>9
kMNt;karEbgEckm:Um:g;GviC¢maneLIgvijGnuBaØatenAelITMr B / C / D nig E énFñwmCab; ABCDE
EdlbgðajenAkñúgrUbTI 1633. FñwmEdlmanmuxkat;ctuekaNEkg b = 300mm / h = 550mm nig
d = 490mm nigvaRtUv)anBRgwgdUcbgðajenAkñúgtarag ¬ f 'c = 28MPa nig f y = 420MPa ¦
- edayeRbI]bsm<n§½ B .
- edayeRbIkarkMNt; ACI Code.
dMeNaHRsay
1> sMrab; f 'c = 28MPa nig f y = 420MPa / ρ b = 0.0285 . emKuNEbgEckm:Um:g;eLIgvij
RtUv)aneGaydUcxageRkam³
⎡ ρ − ρ'⎤
q = 20 ⎢1 −
ρ ⎥
¬!^>@%¦
⎣ b ⎦
2> emKuNEbgEckm:Um:g;eLIgvijrbs; ACI Code CaGnuKmn_énsac;lUtEdksuT§ ε t
nigERbRbYlcenøaHBI 7.5% nig 20% dUcbgðajenAkñúgrUbTI 16>31
q '= 1000ε t ¬!^>@$¦
fy
0.003 +
Es
εt = − 0.003
ρ
ρb
FñwmCab; nigeRKag 436

60. T.Chhay NPIC
nig ε t = 0.005 − 0.003
ρ
sMrab; f y = 420MPa
ρs
taragxageRkambgðajBItMél q nig q' EdlminRtUvKña.
EdkTaj Edksgát; ρ − ρ'
TMr ρ ρ' ρb q% εt q '%
As A' s
B 3DB 28 0.0126 0 0 0.44 11.2 0.008 8
C 3DB32 0.0164 0 0 0.58 8 .4 0.0056 0
D 3DB 20 0.0064 0 0 0.225 15.5 0.0192 19.2
E 4 DB 25 0.0134 3DB 20 0.0064 0.246 15.1 0.0173 17.3
FñwmCab; nigeRKag 437