This document discusses shear and torsion strength design of beams. It introduces the concepts of shear stress and torsion stress, and how they are related to the internal forces in a beam. The document explains homogeneous and non-homogeneous beam behavior under shear and torsion loading based on classical beam mechanics. It provides equations to calculate maximum shear stresses and strains in homogeneous and non-homogeneous beams. Failure modes such as flexural failure, diagonal tension failure, and shear compression failure are also discussed for beams without diagonal tension reinforcement.

1. T.Chhay
V. karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl
Shear and Torsion Strength Design
1> esckþIepþIm Introduction
CMBUkenHnwgBN’naBIdMeNIrkarsikSaKNnamuxkat;ebtugeRbkugRtaMgedIm,ITb;Tl;kMlaMgkat; nigkM
laMgrmYlEdlekItBIkMlaMgGnuvtþn_xageRkA. edaysarersIusþg;Tajrbs;ebtugexSayCagersIusþg;sgát;
rbs;va karsikSaKNnasMrab;kMlaMgkat; nigkMlaMgrmYlkøayCaerOgsMxan;sMrab;RKb;RbePTeRKOgbgÁúM
ebtugTaMgGs;.
karEbgEckrvagkar)ak;rbs;FñwmebtugeRbkugRtaMgeRkamGMeBIkMlaMgkat; b¤bnSMkMlaMgkat; nigkM
laMgrmYl KWxusBIkar)ak;eRkamGMeBIkMlaMgBt;begáag. va)ak;y:agelOnedaymin)anRbkasGsnñCamun
RKb;RKan; ehIysñameRbHGgát;RTUgEdlekItmanmanTMhMFMCagsñameRbHedaysarGMeBIkMlaMgBt;begáag.
TaMgkMlaMgkat; nigkMlaMgrmYlbegáItCakugRtaMgkat;. kugRtaMgenHGacegáItCakugRtaMgTajem (principal
tensile stress) enARtg;muxkat;eRKaHfñak;EdlGacnwgmantMélFMCagersIusþg;Tajrbs;ebtug. cMNaMfa
bøg;ekagrgkar)ak;edaykarrmYlKWbNþalmkBIm:Um:g;rmYlEdl)ak;bEnßmBIelI. kugRtaMgenAkñúgFñwmFmμta
minRtwmEtekIteLIgedaysarkMlaMgkat;edaypÞal; (direct shear) b¤kMlaMgrmYlsuT§ (pure torsion)
b:ueNÑaHeT b:uEnþvak¾ekIteLIgedaysarbnSMénkMlaMg nigm:Um:g;xageRkA. kMlaMgTaMgenaHbegáIteGayman
kugRtaMgTajGgát;RTUg (diagonal tension stress) b¤kugRtaMgTajedaysarkarbegáag (flexural shear
stress) enAkñúgGgát;. kugRtaMgkMlaMgkat;edaypÞal; b¤kugRtaMgrmYlsuT§ekItmanEtenAkñúgRbB½n§eRKOg
bgÁúMxøHb:ueNÑaH dUcCakrNI corbel b¤ bracket EdlBak;B½n§nwgkMlaMgkat;edaypÞal; b¤ cantilever
balcony EdlTak;TgCaBiessnwgkMlaMgrmYlsuT§enAelIFñwmTMr.
2> kareFVIkarrbs;Fñwm homogeneous eRkamGMeBIkMlaMgkat;
Behavior of Homogeneous Beams in Shear
BicarNaFatuGnnþtUcBIr A1 nig A2 rbs;FñwmctuekaNenAkñúgrUbTI 5>1 (a) EdlplitBIsMPar³
EdlmanlkçN³sac;mYy (homogeneous), lkçN³esμIsac; (isotropic) nig linearly elastic. rUbTI
5>2 (b) bgðajBIkarBRgaykugRtaMgBt; nigkarBRgaykugRtaMgkat;elIkMBs;rbs;muxkat;. kugRtaMg
Shear and Torsion Strength Design 214

2. NPIC
TajEkg (tensile normal stress) ft nigkugRtaMgkat; v CatMélenAkñúgFatu A1 enAelIkat;bøg; a − a
Rtg;cMgay y BIG½kSNWt.
BIeKalkarN_ classical emkanic eKGacsresrkugRtaMgEkg (normal stress) f nigkugRtaMg
kat; v sMrab;Fatu A1 dUcxageRkam³
My
f = (5.1)
I
nig v=
VA y VQ
Ib
=
Ib
(5.2)
Edl M nig V = m:Um:g;Bt; nigkMlaMgkat;enARtg;muxkat; a − a
A = RkLaépÞrbs;muxkat;enARtg;bøg;Edlkat;tamTIRbCMuTMgn;rbs;Fatu A1
y = cMgayBIFatuGnnþtUceTAG½kSNWt
y = cMgayBITIRbCMuTMgn;rbs; A eTAG½kSNWt
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 215

3. T.Chhay
m:Um:g;niclPaBrbs;muxkat;
I=
Q = m:Um:g;sþaTicrbs;RkLaépÞmuxkat;EdlenABIxagelI b¤BIxageRkamG½kSNWt
b = TTwgrbs;Fñwm
rUbTI 5>2 bgðajBIkugRtaMgxagkñúgEdlmanGMeBIelIFatuGnnþtUc A1 nig A2 . edayeRbIrgVg;m:
(Mohr’s cicle) enAkñúgrUbTI 5>2(b) kugRtaMgemsMrab;Fatu A1 enAkñúgtMbn;TajxageRkamG½kSNWtkøayCa
2
⎛f ⎞
f t (max)
f
= t + ⎜ t ⎟ + v2
2 ⎝2⎠
kugRtaMgTajem (5.3a)
2
⎛f ⎞
f c (max ) =
ft
2
− ⎜ t ⎟ + v2
⎝2⎠
kugRtaMgsgát;em (5.3b)
nig tan 2θ max =
v
ft / 2
Shear and Torsion Strength Design 216

4. NPIC
3> kareFVIkarrbs;FñwmebtugGarem:CalkçN³minEmnsac;mYy
Behavior of Concrete Beams as Nonhomogeneous Sections
kareFVIkarrbs;FñwmebtugGarem: nigFñwmebtugeRbkugRtaMgxusBIFñwmEdk EdlersIusþg;Tajrbs;
ebtugmantMélRbEhlmYyPaKdb;énersIusþg;sgát;rbs;ebtug. ersIusþg;sgát; fc enAkñúgFatu A2 énrUbTI
5>2(b) EdlenAxagelIG½kSNWtkarBareRbH edaysarkugRtaMgemGtibrmaenAkñúgFatuGnnþtUcCakugRtaMg
sgát;. sMrab;Fatu A1 EdlenAxageRkamG½kSNWt kugRtaMgemGtibrmaCakugRtaMgTaj dUcenHnwgekItman
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 217

5. T.Chhay
sñameRbH. edaysarm:Um:g;Bt; dUcKñanwgkugRtaMgTajfycuH ehIykugRtaMgkat;ekIneLIgkñúgTisedAeTArk
TMr enaHkugRtaMgTajem ft (max ) nwgeFVIGMeBIelIbøg;RbEhl 45o Rtg;muxkat;Ek,rTMr dUceXIjenAkñúgrUbTI
5>3. edaysarersIusþg;Tajrbs;ebtugtUc sñameRbHGgát;RTUgekItmantambøg;Ekgnwgbøg;énkugRtaMg
Tajem ehIysñameRbHenHeKeGayeQμaHfa sñameRbHGgát;RTUgTaj (diagonal tension crack). edIm,I
karBarsñameRbHEbbenHBITIkEnøgcMh (opening) eKRtUvdak;EdkTajGgát;RTUgBiess.
RbsinebIeKsnμt; ft EdlenAEk,rTMrénrUbTI 5>3 esμIsUnü enaHFatuGnnþtUcnwgesÞIrkøayeTACa
sßanPaBénkugRtaMgkat;suT§ ehIykugRtaMgTajemEdleRbIsmIkar 5.3a nwgesμInwgkugRtaMgkMlaMgkat; v
enAelIbøg; 45o . vaCakugRtaMgTajGgát;RTUgEdlbgáeGaymansñameRbHeRTt.
karyl;y:agc,as;las;BI correct shear mechanism enAkñúgebtugGarem:enAminTan;RKb;RKan;
enAeLIgeT. b:uEnþ viFIén ACI-ASCE Joint Committee 426 pþl;nUveKalkarN_mUldæanEdl)anBI
lT§plénkarBiesaFy:ageRcInsn§wksn§ab;.
4> FñwmebtugEdlKμanEdkTajGgát;RTUg
Concrete Beams without Diagonal Tension Reinforcement
enAkñúgtMbn;énm:Um:g;Bt;FM sñameRbHekItmanesÞIrEtEkgnwgG½kSrbs;Fñwm. sñameRbHTaMgenHman
eQμaHfa sñameRbHBt;begáag (flexural crack). enAkñúgtMbn;kMlaMgkat;FMedaysarkMlaMgTajGgát;RTUg
sñameRbHeRTtekItmanbnþBI flexural crack ehIyRtUv)aneKehAfasñameRbHkMlaMgkat;begáag (flexural
shear crack). rUbTI 5>4 bgðajBIRbePTsñameRbHEdlrMBwgnwgekItmanenAkñúgebtugGarem:edayman b¤
KμanEdkTajGgát;RTUgRKb;RKan;.
Shear and Torsion Strength Design 218

6. NPIC
enAkñúgFñwmeRbkugRtaMg muxkat;esÞIrEtrgkugRtaMgsgát;TaMgGs;eRkamGMeBIbnÞúkeFVIkar (service
load). BIrUbTI 5>2 (c) nig (d) kugRtaMgemsMrab;Fatu A2 KW
f t (max ) = − c + ( f c / 2)2 + v 2
f
2
kugRtaMgTajem (5.4a)
f c (max ) = − c − ( f c / 2)2 + v 2
f
2
kugRtaMgsgát;em (5.4b)
nig tan 2θ max =
v
f /2 c
k> KMrU)ak;rbs;FñwmEdlKμanEdkTajGgát;RTUg
Modes of Failure of Beams without Diagonal Tension Reinforcement
pleFobElVgkat;elIkMBs; (slenderness ratio) rbs;FñwmkMNt;nUvKMrU)ak;rbs;Fñwm. rUbTI 5>5
bgðajBIkar)ak;sMrab;EdnkMNt;én slenderness ratio epSg². RbEvgElVgkMlaMgkat; (shear span) a
sMrab;bnÞúkcMcMnucCacMgayrvagcMnucénkarGnuvtþbnÞúk nigépÞénTMr. sMrab;bnÞúkBRgay shear span lc Ca
clear beam span. CaeKalkarN_ KMrUénkar)ak;manbIEbbKW kar)ak;edaykarbegáag (flexural failure),
kar)ak;edaykMlaMgTajGgát;RTUg (diagonal tension failure) nigkar)ak;edaybnSMkarkat; nigkarsgát;
(shear compression failure or web shear). Fñwmkan;EtRsav kareFVIkarrbs;vakan;EtxiteTArklkçN³
begáagdUckarerobrab;xageRkam.
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 219

7. T.Chhay
x> kar)ak;edaykarBt;begáag Flexural Failure [F]
sMrab; flexural failure sñameRbHeRcInEtQrelIRbEvgmYyPaKbIEpñkkNþalrbs;Fñwm ehIyEkg
eTAnwgExSénkugRtaMgem. sñameRbHenHekItBIkugRtaMgkMlaMgkat; v tUc nigkugRtaMgBt; f FM EdlbegáIt
nUvkugRtaMgem ft (max ) esÞIrEtedk. enAkñúgKMrU)ak;EbbenHsñameRbHbBaÄrtUcb:unsréssk;cab;epþImekIt
manenAtMbn;kNþalElVgeRkamGMeBIrbs;bnÞúkRtwm 50% én failure load. edaysarbnÞúkxageRkAekIn
eLIg sñameRbHbEnßmekItmanenAtMbn;kNþalrbs;ElVg ehIysñameRbHdMbUgrIkFM ehIymanTisedAeq<aH
eTArkG½kSNWt CamYynwgkarekIneLIgPaBdabrbs;FñwmKYreGaykt;sMKal;. RbsinebIFñwmenHCa under-
reinforced member kar)ak;rbs;vaekIteLIgkñúglkçN³ ductile eday longitudinal flexural rein-
forcement eFVIkareTAdl; yield. kareFVIkarRbePTenHpþl;nUvkarRbkasGasnñénkarrlMrbs;Fñwm.
pleFobElVgkat;elIkMBs;sMrab;kareFVIkarenHmantMélFMCag 5.5 sMrab;krNIbnÞúkcMcMnuc nigFMCag 16
sMrab;bnÞúkBRgay.
K> kar)ak;edaykMlaMgTajGgát;RTUg
Diagonal Tension Failure [Flexure shear, FS]
kar)ak;edaykMlaMgTajGgát;RTUgekItmanRbsinebIersIusþg;kMlaMgTajGgát;RTUgrbs;FñwmtUcCag
ersIusþg;Bt;begáagrbs;va. pleFobElVgkat;elIkMBs;mantMélkNþalEdlERbRbYlcenøaH 2.5 eTA 5.5
sMrab;krNIbnÞúkcMcMnuc. eKKitFñwmEbbenHCa intermidate slenderness. sñameRbHcab;epþImCamYykar
ekIteLIgén flexural crack bBaÄresþIg²enAkNþalElVg nigbnþedaykar)at;bg;PaBs¥itrvagEdkCamYy
nwgebtugEdlB½T§CMuvijvaenARtg;TMr. bnÞab;mkeTot edayminmankarRbkasGsnñRKb;RKan;BIkarekItman
kar)ak; sñameRbHGgát;RTUgBIr b¤bIekItmanenARtg;RbEhl 1.5d eTA 2d éncMgayBIépÞénTMrkñúgkrNIFñwm
ebtugGarem: nigCaTUeTAekItmanRtg;RbEhlmYyPaKbYnénElVgkñúgkrNIFñwmeRbkugRtaMg. enAeBlEdl
vaenAzitezr sñameRbHGgát;RTUgmYykñúgcMeNamenaHrIkhMkøayCasñameRbHTajGgát;RTUgem ehIylat
sn§wgeTArksréssgát;xagelIrbs;Fñwm dUceXIjenAkñúgrUbTI 5>5(b) nig 5>5(c). cMNaMfa flexural
crack minlatrIkraleTArkeTArkG½kSNWtenAkúñgkrNIKrU)ak;RsYyEbbenHeT. KMrU)ak;enHmanPaBdab
M
tUcCagKMrU)ak;elIkmunenAeBl)ak;.
eTaHbICakMlaMgkat;xageRkAGtibrmaenARtg;TMrkþI TItaMgeRKaHfñak;rbs;kugRtaMgTajemGtibrma
minsßitenARtg;TMreT. vaRtUv)ankat;bnßyenARtg;mxkat;enaH edaysarkMlaMgsgát;FMrbs;EdkeRbkug
u
RtaMgbEnßmBIelIkMlaMgsgát;bBaÄrénRbtikmμrbs;FñwmenARtg;TMr. vaCamUlehtuEdlsñameRbHGgát;RTUg
Shear and Torsion Strength Design 220

8. NPIC
EdlenAzitezrmanTItaMgq¶aycUlmkkñúgElVgRbEhlmYyPaKbYnénElVgsMrab;FñwmeRbkugRtaMgEdlman
søab ehIyTItaMgenHGaRs½ynwgTMhMénkMlaMgeRbkugRtaMg nigPaBERbRbYlrbs;cMNakp©it. CarYm
diagonal tension failure CalT§plénbnSMrbs;kugRtaMgBt; nigkugRtaMgkat; edayKitnUvkarcUlrYmrbs;
bgÁúMbBaÄrrbs;kMlaMgeRbkugRtaMg nigRtUv)ankt;sMKal;edaysñameRbHbegáag nigsñameRbHGgát;RTUg.
CakarRbesIr eKehAvafa flexure shear sMrab;FñwmeRbkugRtaMg ehIyvamanlkçN³gayRsYlBnül;Ca
web shear Edlnwgerobrab;enAeBlbnÞab;.
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 221

9. T.Chhay
X> kar)ak;edaybnSMkarkat; nigkarsgát;
Shear Compression Failure [Web Shear, WS]
FñwmEdlRbQm shear compression failure EtgmanpleFobElVgkat;elIkMBs; 2.5 sMrab;
krNIbnÞúkcMcMnuc nigtUcCag 5.0 sMrab;bnÞúkBRgay. dUckñúgkrNIkar)ak;edaykMlaMgTajGgát;RTUg
sñameRbHedaysarkarBt;esþIg²mYycMnYncab;epþImekItmanenAkNþalElVg ehIybBaÄb;karrIkral
edaysarkar)at;bg; PaBs¥itrvagEdkbeNþayCamYynwgebtugEdlB½T§CMuvijvaenARtg;TMr. bnÞab;mk
sñameRbHeRTtEdlmanlkçN³ecaTCagsñameRbHkñμúgkrNIkar)ak;edaykMlaMgTajGgát;RTUgekIteLIg
Pøam² nigbnþrIkraleq<aHeTArkG½kSNWt. GRtaénkarrIkralrbs;vaRtUv)ankat;bnßyedaysarkarEbk
rbs;ebtugenAelIsréssgát;xagelI nigkarEbgEckkugRtaMgeLIgvijenAkñúgtMbn;xagelI. kar)ak;Pøam²
ekIteLIgedaysarsñameRbHGgát;RTUgemrYmpSMngtMbn;ebtugEbk dUcbgðajenAkñúgrUbTI 5>5(c). eKKit
w
fakar)ak;RbePTenHmanlkçN³minsUvRsYydUckar)ak;edaysar diagonal tension failure edaysar
karBRgaykugRtaMgeLIgvij. CakarBit vaCaRbePTénkar)ak;EdlmanlkçN³RsYyedaymankarRbkas
GasnñEdlmanEdnkMNt; ehIyeKRtUveCosvagkarsikSaKNnaEbbenHdac;xat.
FñwmebtugminmanlkçN³sac;mYy (homogeneous) ehIyCaTUeTAkarBRgayersIusþg;rbs;ebtug
enAelIElVgTaMgmUlmanlkçN³minesμIKñaeT. dUcenH eKminGacrMBwgBIkarekIteLIgénsñameRbHGgát;RTUg
enAelIcugTaMgsgçagrbs;Fñwm)aneT. ehIyedaysarlkçN³TaMgenH karbnSMCan;Kñaénkar)ak;edaykar
Bt; nigkar)ak;edaykMlaMgTajGgát;RTUg CamYynwgkar)ak;edaysarkMlaMgTajGgát;RTUg nigkar)ak;
edaybnSMkMlaMgkat; nigkMlaMgsgát;GacekItmanenAeBlpleFobElVgkat;elIkMBs;Can;Kña. RbsinebI
eKdak;EdkkMlaMgkat; (shear reinforcement) smrmü eKGackat;bnßykar)ak;edaylkçN³RsYyrbs;
Ggát;edkCamYynwgfvikarbEnßmtictYc.
kar)ak;PaKeRcInEtgEt)ak;edaykMlaMgTajGgát;RTUg EdlCabnSMénT§iBlrbs;karBt;begáag
nigkarkat;. RbePTénkar)ak;edaybnSMkMlaMgkat; nigkMlaMgsgát; ¬CalT§plénkarEbkénépÞsgát;xag
elIrbs;ebtug nigkar)at;bg;lT§PaBTb;Tl;kMlaMgBt;¦ naMdl;karEbgEcksøabrgkarTajecjBIRTnug
sMrab;muxkat;mansøabedaysarsñameRbHeRTtlatsn§wgeTArkTMr. karEbkénFñwmrbs;muxkat;eFVIeGay
FñwmeFVIkardUcnwgFñÚ (tied arch). CakarRbesIr eKKYrehARbePTénkar)ak;enHCa kar)ak;kMlaMgRTnug
(web-shear failure) sMrab;FñwmeRbkugRtaMg eKcaM)ac;kMNt;lT§PaBénbnSMkMlaMgBt; nigkMlaMgkat; nig
lT§PaBkMlaMgkat;RTnugrbs;muxkat;Rtg;TItaMgeRKaHfñak;edIm,IkMNt;ersIusþg;kMlaMgkat;rbs;muxkat;eb-
tug.
Shear and Torsion Strength Design 222

10. NPIC
karEbgEckkugRtaMgkat;tamTisedkGtibrmaenAkñúgmuxkat;eRbHrbs;muxkat;mansøabRtUv)an
bgðajenAkñúgrUbTI 5>6. edaysarkarpøas;bþÚrTTwgmuxkat;y:agrh½senARtg;RCug A dUcenHeKcaM)ac;
RtYtBinitülT§PaBrbs;muxkat;enARtg;TItaMgeRKaHfñak;tambeNþayElVg CaBiesssMrab;kar)ak;edaysar
kMlaMgkat;RTnug (web-shear failure).
5> kugRtaMgkat; nigkugRtaMgemenAkñúgFñwmeRbkugRtaMg
Shear and Principal Stresses in Prestressed Beams
dUcEdl)anerobrab;enAkñúgEpñk 4/ flexure shear enAkñúgFñwmebtugeRbkugRtaMgrYmbBa©ÚlTaMgT§i-
BlénkMlaMgeRbkugRtaMgsgát;EdlGnuvtþBIxageRkA. bgÁúMkMlaMgbBaÄrrbs;kMlaMgEdkeRbkugRtaMgkat;
bnßykugRtaMgbBaÄrEdlekIteLIgedaybnÞúkTTwgG½kSxageRkA (external transverse load) ehIy net
transverse load EdlFñwmRtUvTTYlmantMéltUcKYreGaykt;sMKal;sMrab;FñwmeRbkugRtaMgCagsMrab;Fñwm
ebtugGarem:.
elIsBIenH kMlaMgsgát;rbs;EdkeRbkugRtaMg ¬eTaHbIenAkñúgkrNI straight tendon¦ kat;bnßy
T§iBlrbs; tensile flexural stresses y:ageRcIn dUcenHkarrIkraldalén flecural cracking nigTMhMén
flexural cracking enAkñúgFñwmeRbkugRtaMgRtUv)ankat;bnßy. CalT§pl kMlaMgkat; nigkugRtaMgemenA
kñúgFñwmeRbkugRtaMgmantMéltUcCagkMlaMgkat; nigkugRtaMgemenAkñúgFñwmebtugGarem:xøaMgNas;. dUcenH
smIkarmUldæanEdlbegáIteLIgsMrab;kMlaMgkat;enAkñúgebtugeRbkugRtaMgmanlkçN³dUcKñanwgsmIkarmUl
dæanEdlbegáIteLIgsMrab;ebtugeRbkugRtaMg. rUbTI 5>7 bgðajBIkarcUlrYmrbs;bgÁúMbBaÄrrbs;kMlaMgeRb
kugRtaMgenAkñúgEpñkEdlminmanlMnwg b¤kMlaMgkat;bBaÄr V PaKeRcInEdlekItedaybnÞúkTTwgG½kSxag
eRkA. kMlaMgkat;suT§ (net shearing force) Vc EdlTb;Tl;edayebtugKW
Vc = V − V p (5.5)
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 223

11. T.Chhay
BIsmIkar 5.2 kugRtaMgkat;ÉktþasuT§ v enARtg;kMBs;Nak¾edayrbs;muxkat;KW
Vc Q
vc = (5.6)
Ib
karEbgEckkugRtaMgsréssgát; fc EdlbNþalBIm:Um:g;Bt;xageRkAKW
Pe Pe ec M T c
fc = − ± m (5.7)
Ac Ic Ic
ehIyBIsmIkar 5.4a kugRtaMgTajemKW
f 't = ( f c / 2)2 + vc2 − fc
(5.8)
2
k> Flexural-Shear Strength
edIm,IsikSaKNnasMrab;kMlaMgkat; eKcaM)ac;kMNt;faetI flexural shear b¤ web shear lubedIm,I
eFVIkareRCIserIsersIusþg;kat; Vc rbs;ebtug. sñameRbHeRTtEdlmanlMnwg (inclined stabilized crack)
enAcMgay d / 2 BI flexural crack EdlekItmanenAeBlrg first cracking load enAkñúgkrNI flexure
shear RtUv)anbgðajenAkñúgrUbTI 5>8. RbsinebI kMBs;RbsiT§PaBCa d p ¬kMBs;BIsréssgát;eTATIRbCMu
TMgn;rbs;EdkeRbkugRtaMgbeNþay¦ bMErbMrYlm:Um:g;rvagmuxkat; @ nig # KW
Vd p
M − M cr ≅ (5.9a)
2
b¤ V=
M cr
M /V − d p / 2
(5.9b)
Shear and Torsion Strength Design 224

12. NPIC
Edl V CakMlaMgkat;enARtg;muxkat;EdlBicarNa. lT§plénkarBiesFCaeRcInbgðajfaeKRtUvkar
kMlaMgkat;bBaÄrbEnßmEdlmanTMhM 0.6bwd p f 'c sMrab;xñat US nig bwd p f 'c / 20 sMrab;xñat SI
edIm,IeFVIeGaymansñameRbHeRTtenAkñúgrUbTI 5>8 eBjelj. dUcenH kMlaMgkat;bBaÄrsrubEdleFVIGMeBI
enARtg;bøg;elx @ rbs;rUbTI 5>8 KW
Vci =
M cr
M /V − d / 2
+ 0.6bw d p f 'c + Vd ¬xñat US¦ (5.10)
p
Vci =
M cr
M /V − d p / 2
+
bw d p f 'c
20
+ Vd ¬xñat SI¦
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 225

13. T.Chhay
Edl Vd CakMlaMgkat;bBaÄrEdlbNþalBITMgn;pÞal;. bgÁúMbBaÄr V p rbs;kMlaMgeRbkugRtaMgminRtUv)an
KitbBa©ÚlenAkñúgsmIkar 5.10 eT edaysarvamanTMhMtUctambeNþaymuxkat;ElVgEdlEdkeRbkugRtaMg
minecatxøaMg.
tMélrbs; V enAkñúgsmIkar 5.10 KWCakMlaMgkat;emKuN Vi enARtg;muxkat;EdlBicarNaEdl
bNþalBIbnÞúkxageRkAEdlekIteLIgtMNalKñaCamYynwgm:Um:g;Gtibrma M max EdlekIteLIgenARtg;mux
kat;enaH
Vci = 0.6λ f 'c bw d p − Vd +
Vi
(M cr ) ≥ 1.7λ f 'c bwd p ¬xñat US¦ (5.11)
M max
λ f 'c bw d p λ f 'c bw d p
Vci =
20
− Vd +
Vi
M max
(M cr ) ≥
7
¬xñat SI¦
≤ 5.0λ f 'c bw d p ¬xñat US¦
≤ 0.42λ f 'c bw d p
Edl sMrab;ebtugTMgn;Fmμta
λ = 1 .0
= 0.85 sMrab; sand-lightweight concrete
= 0.70 sMrab; all-lightweight concrete
Vd = kMlaMgkat;Rtg;muxkat;EdlbNþalBIbnÞúkefrEdlKμanemKuN
Vci = ersIusþg;kMlaMgkat;Fmμta (nominal shear strength) Edlpþl;eGayedayebtugenAeBl
EdlsñameRbHkMlaMgTajekItBIbnSMénkMlaMgkat;bBaÄr nigm:U:m:g;
Vi = kMlaMgkat;emKuNRtg;muxkat;EdlbnÞúkxageRkAekIteLIgenAeBlCamYyKñanwg M max
sMrab;ebtugTMgn;Rsal λ = f ct / 6.7 f 'c sMrab;xñat US nig λ = fct / 0.556 f 'c sMrab;xñat
SI RbsinebIeKsÁal; tensile splitting strength f ct . cMNaMfatMélrbs; f 'c minKYrFMCag 100 psi
(0.69MPa ) .
smIkarsMrab; M cr ¬m:Um:g;EdlbegáIt flexural cracking EdlbNþalBIbnÞúkxageRkA¦ RtUv)an
eGayeday
M cr = c (6 f 'c + f ce − f d ) ¬xñat US¦
I
(5.12)
y t
M cr
I
(
= c 0.5 f 'c + f ce − f d
yt
) ¬xñat IS¦
enAkñúg ACI Code eKeRbI f pe CMnYseGay f ce
Shear and Torsion Strength Design 226

14. NPIC
Edl ersIusþg;sgát;rbs;ebtugEdlbNþalBIeRbkugRtaMgRbsiT§PaBeRkaykMhatbg;enARtg;
f ce =
srésxageRkAbMputrbs;muxkat;EdlkugRtaMgTajRtUv)anbgáeLIgedaybnÞúkxageRkA.
enARtg;TIRbCMuTMgn; fce = f c .
f d = kugRtaMgEdlbNþalBIbnÞúkGefrKμanemKuNenARtg;srésxageRkAbMputrbs;muxkat;Edl
bNþalEtBIbnÞúkpÞal;EdlkugRtaMgTajRtUv)anbgáedaybnÞúkxageRkA.
yt = cMgayBIG½kSTIRbCMuTMgn;eTAsrésrgkarTajxageRkA
ehIy M cr = EpñkxøHrbs;m:Um:g;énbnÞúkxageRkAEdlbgáeGaymansñameRbH. CakarsMrYl eKGacCMnYs
Sb CMnYseGay I c / yt .
rUbTI 5>9 bgðajBIdüaRkamrbs;smIkar 5.10 CamYynwgTinñn½yénkarBiesaF.
cMNaMfa eKeRbIkarsikSaKNnadUcKñaEdlGnuvtþsMrab;muxkat;cak;Rsab; sMrab;karsikSaKNna
kMlaMgkat;énmuxkat;smas. BIeRBaHkarsikSaKNnasMrab;kMlaMgkat;KWQrelIsßanPaBkMNt;edA
eBl)ak;eRkamGMeBIbnÞúkemKuN. eTaHbICa muxkat;TaMgmUlrbs;muxkat;smasTb;Tl;nwgkMlaMgkat;
smasdUcmuxkat;Edlcak;kñúgeBlEtmYy (monolithic section) k¾eday k¾karKNnaersIusþg;kMlaMg
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 227

15. T.Chhay
kat; Vc KYrQrelIlkçN³rbs;muxkat;cak;Rsab; edaysarersIusþg;kMlaMgkat;PaKeRcInpþl;eGayeday
RTnugrbs;muxkat;cak;Rsab;. dUcenH fce nig f d enAkñúgsmIkar 5.12 RtUv)anKNnaedayeRbIragFrNI
maRtrbs;muxkat;cak;Rsab;.
x> ersIusþg;kMlaMgkat;RTnug Web-Shear Strength
sñameRbHkMlaMgkat;RTnug (web-shear crack) enAkñúgFñwmeRbkugRtaMgekIteLIgedaysarkugRtaMg
EdlminGackMNt;)an (indeterminate stress) EdlCakarRbesIreKKYrKNnavaedaykugRtaMgTajemenA
Rtg;bøg;eRKaHfñak;BIsmIkar 5.8. eKGaccat;TukkugRtaMgkat; vc CakugRtaMgkat;RTnug vcw nigmantMél
GtibrmaenAEk,rTIRbCMuTMgn; cgc énmuxkat;EdlsñameRbHGgát;RTUgCak;EsþgekItman dUckarBiesaFeTA
dl;kar)ak;CaeRcIn)anbgðaj. RbsinebIeKCMnYs vc sMrab; vcw nig fc sMrab; f c ¬EdlCakugRtaMgeb-
tug fc EdlbNþalBIeRbkugRtaMgRbsiT§PaBenARtg;nIv:U cgc¦ enAkñúgsmIkar smIkarEdleGaykugRtaMg
TajemenAkñúgebtugesμInwgersIusþg;TajpÞal; (direct tensile strength) køayCa
( f c / 2) + vcw − f2c
f 't = 2
(5.13)
Edl vcw = Vcw / (bwd p ) CakugRtaMgkat;enAkñúgebtugEdlbNþalBIbnÞúkTaMgGs;EdleFVIeGayman
ersIusþg;kMlaMgkat;bBaÄrFmμta Vcw enAkñúgRTnug. edaHRsayrk vcw enAkñúgsmIkar 5.13
vcw = f 't 1 + f c / f 't (5.14a)
edayeRbI f 't = 3.5 f 'c psi(0.3 f 'c MPa) CatMéld¾smrmüsMrab;kugRtaMgTajedayQrelIlT§pl
énkarBiesaFCaeRcIn smIkar 5.14(a) køayCa
vcw = 3.5 f 'c ⎛ 1 + f c / 3.5 f 'c ⎞
⎜
⎝
⎟
⎠
¬xñat US¦ (5.14b)
vcw = 0.3 f 'c ⎛ 1 + f c / 0.3 f 'c ⎞
⎜
⎝
⎟
⎠
¬xñat SI¦
EdleyIgGacsMrYl)andUcxageRkam
vcw = 3.5 f 'c + 0.3 f c ¬xñat US¦ (5.14c)
vcw = 0.3( f 'c + f c ) ¬xñat SI¦
enAkñúg ACI Code eKeRbI f pc CMnYseGay f c . nimitþsBaØaEdleRbIenATIenHKWcg;bBa¢ak;favaCakugRtaMg
enAkñúgebtugminEmnenAkñúgEdkeRbkugRtaMgeT. ersIusþg;kMlaMgkat;Fmμta Vcw EdleGayedayebtugenA
eBlsñameRbHGgát;RTUgekItBIkugRtaMgTajemd¾FMenAkñúgRTnugkøayCa
Vcw = (3.5λ f 'c + 0.3 f c )bw d p + V p ¬xñat US¦ (5.15)
Shear and Torsion Strength Design 228

16. NPIC
( )
Vcw = 0.3 λ f 'c + f c bw d p + V p ¬xñat SI¦
Edl V p = bgÁúMbBaÄrénkMlaMgeRbkugRtaMgRbsiT§PaBenARtg;muxkat;BiessEdlcUlrYmnwgersIusþg;
FmμtabEnßm
λ = 1.0 sMrab;ebtugTMgn;Fmμta nigmantMéltUcCagenHsMrab;ebtugTMgn;Rsal
d p = cMgayBIsréssgát;xageRkAeTATMRbCMuTMng;rbs;EdkeRbkugRtaMg b¤ 0.8h edayykmYy
NaEdlFMCag
ACI Code yktMél f c CakugRtaMgsgát;pÁÜbrbs;ebtugenARt;gTIRbCMuTMgn;rbs;muxkat; b¤Rtg;
kEnøgEdlkat;KñarvagRTnug nigsøabenAeBlEdlTIRbCMuTMgn;sßitenAkñúgsøab. enAkñúgkrNImuxkat;smas
eKKNna f c edayQrelIkugRtaMgEdlekIteLIgedaykMlaMgeRbkugRtaMg nigm:Um:g;EdlTb;Tl;eday
Ggát;cak;Rsab;EdleFVIkarEtÉg. düaRkaménTMnak;TMngrvagkugRtaMgkat;RTnugFmμta (nominal web
shear stress) vcw nigkugRtaMgsgát;rbs;ebtugRtg;TIRbCMuTMgn;RtUv)aneGayenAkñúgrUbTI 5>10. cMNaMfa
PaBdUcKñarvagExSekagénsmIkar 5.14b nig c bgðajfasmIkar 5.14c RtUv)anEktMrUvBIsmIkar 5.14b
edIm,IeGaymanlkçN³bnÞat;. Code GnuBaØateGayeRbIemKuN 1.0 CMnYseGayemKuN 0.3 sMrab;tYTIBIr
énsmIkar 5.15.
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 229

17. T.Chhay
K> karRtYtBinitütMélrbs;V nigV sMrab;KNnaersIusþg;ebtugRTnugV
ci cw c
Controlling Values of Vci and Vcw for the Determination of the
Web Concrete Strength Vc
ACI Code mansmIkarbEnßmsMrab;kMNt;Vci nig Vcw edIm,IeRCIserIstMél Vc EdlRtUvkarenA
kñúgkarKNna³
(a) enAkñúgGgát;eRbkugRtaMgEdlmuxkat;enAcMgay h / 2 BIépÞénTMrenAEk,rcugénGgát;CagRbEvgepÞr
rbs;EdkeRbkugRtaMg enaHeKRtUvBicarNatMéleRbkugRtaMgkat;bnßyenAeBlKNna Vcw . tMél
Vcw enHRtUv)anKitCaEdlGtibrmarbs; Vc enAkñúgsmIkar
⎛ Vu d p ⎞
Vcw = ⎜ 0.6λ f 'c + 700
⎜ ⎟bw d p ≥ 2λ f 'c bw d p
⎝ Mu ⎟ ⎠
≤ 5λ f 'c bw d p ¬xñat US¦ (5.16)
⎛ Vu d p ⎞
Vcw = ⎜ 0.05λ f 'c + 5
⎜ ⎟bw d p ≥ 0.2λ f 'c bw d p
⎝ Mu ⎟ ⎠
≤ 0.4λ f 'c bw d p ¬xñat SI¦
tMél Vu d p / M u minGacFMCag 1.0 eT.
(b) enAkñúgGgát;eRbkugRtaMgEdl bonding rbs; tendon xøHmin)anBnøÚtdl;cugrbs;Ggát; enaHeKRtUv
KiteRbkugRtaMgkat;bnßyenAeBlkMNt; Vc edayeRbIsmIkar 5.16 b¤eRbItMéltUcCageKkñúg
cMeNamtMél Vc EdlTTYl)anBIsmIkar 5.11 nigBIsmIkar 5.15. dUcKña tMélrbs; Vcw Edl
KNnaedayeRbIeRbkugRtaMgkat;bnßyRtUvyktMélGtibrmaénsmIkar 5.16.
(c) eKGaceRbIsmIkar 5.16 kñúgkarkMNt; Vc sMrab;Ggát;EdlkMlaMgeRbkugRtaMgRbsiT§PaBmintUc
Cag 40% énersIusþg;Tajrbs;EdkrgkarBt; (flexural reinforcement) ebImindUcenaHeT luH
RtaEteKGnuvtþkarviPaKlMGitedayeRbIsmIkar 5.11 sMrab; Vci nigsmIkar 5.15 sMrab; Vcw
ehIyedayeRCIserIsyktMéltUcCageKéntMélTaMgBIrCatMélkMNt; Vc edIm,IeRbICaersIusþg;
rbs;RTnugkñúgkarKNnaEdkRTnug.
(d) bøg;dMbUgsMrab;ersuIsþg;kat;FmμtaEdlRtUvkarsrub (total required nominal shear strength)
Vn = Vu / φ EdlRtUv)aneRbIsMrab;KNnaEdkRTnugk¾sßitenARtg;cMgay h / 2 BIépÞrbs;TMr.
Shear and Torsion Strength Design 230

18. NPIC
6> EdkkMlaMgkat;RTnug Web-Shear Reinforcement
k> Web Steel Planar Truss Analogy
edIm,IkarBarsñameRbHGgát;RTUgenAkñúgGgát;eRbkugRtaMg eTaHbIekIteLIgedaysar flexural-
shear b¤ web-shear action k¾eday eKRtUvdak;EdkBRgwgtamTMrg;énExSditEdlBN’naBIKnøgrbs;kug
RtaMgTajenAkñúgrUbTI 5>3. b:uEnþdMeNaHRsayEbbenHRtUv)anRcanecal ehIyTMrg;epSgeTotrbs;Edk
RtUv)anécñRbDiteLIgedIm,ITb;Tl;nwgkugRtaMgTajenARtg;bøg;)ak;edaysarkMlaMgkat;eRKaHfñak; (critical
shear failure plane).
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 231

19. T.Chhay
Shear and Torsion Strength Design 232

20. NPIC
KMrUénkar)ak;edaykMlaMgkat;eFVIeGayFñwmman stimulated arched section rgkarsgát;enAEpñk
xagelI ehIyRtUv)ancgP¢ab;KñaenAEpñkxageRkamedayEdkTaj dUceXIjenAkñúgrUbTI 5>11(a). Rbsin
ebIeKBinitüEtFatusgát;EdlbgðajenAkñúgrUbTI 5>11 (b) eKGacKitvaCaGgát;sgát; enAkñúg truss RtIekaN
dUcbgðajenAkñúgrUbTI 5>11 (c) EdlmanBhuekaNénkMlaMg Cc / Tb nig Ts EdltMNageGaykMlaMg
EdlmanGMeBIeTAelIGgát;rbs; truss dUcenHeKGacehAvafa truss analogy. kMlaMg Cc kMlaMgsgát;enA
kñúg simulated concrete strut, kMlaMg Tb CakMeNInkMlaMgTajénEdkTaj beNþayem ehIy Ts CakMlaMg
enAkñúgEdkBt; (bent bar). rUbTI 5>12 (a) bgðajBI analogy truss sMrab; krNIénkareRbIEdkkgbBaÄr
(vertical stirrup) CMnYseGayEdkeRTt CamYynwgBhuekaNkMlaMgEdlmankMlaMgTajbBaÄr Ts CMnYs
eGaykMlaMgTajeRTtenAkñúgrUbTI 5>11 (c).
EdkkMlaMgkat;mantYnaTIsMxan;bYn³
- vaTb;Tl;EpñkxøHénkMlaMgkat;emKuNxageRkA Vu
- vaRKb;RKgkarrIkFMénsñameRbHGgát;RTUg
- vaeFVIeGayEdkbeNþayemsßitcMTItaMg dUcenHeKGacdak;Edk dowel EdlcaM)ac;edIm,IRT
flexural load
- vapþl;nUv confinement xøHdl;ebtugenAkñúgtMbn;sgát; RbsinebIeKeRbIEdkkgkñúgTMrg;biTCit
x> Web Steel Planar Resistance
RbsinebI Vc ¬ersIusþg;kMlaMgkat;FmμtaénRTnugebtugsuT§¦ mantMéltUcCagkMlaMgbBaÄrsrub
Fmμta Vu / φ = Vn eKRtUvdak;EdkRTnugedIm,ITb;Tl;nUvPaBxusKñaénkMlaMgTaMgBIr. dUcenH
Vs = Vn − Vc (5.17)
enATIenH Vc CatMéltUcCageKén Vci nig Vcw . eKGackMNt; Vc BIsmIkar 5.11 b¤ 5.15 ehIyeKkMNt;
Vs BIsmIkarlMnwgenAkñúg analogous triangular truss. BIsmIkar 5.11(c)
Vs = Ts sin α = Cc sin β (5.18a)
Edl Ts CakMlaMgpÁÜbénEdkkgRTnugTaMgGs;EdlEkgnwgbøg;sñameRbHGgát;RTUg ehIy n CacMnYnénKM
lat s . RbsinebI s1 = ns enAkñúgGgát;rgkarTajxageRkamrbs; analogous truss enaH
s1 = jd (cot α + cot β ) (5.18b)
snμt;fa édXñas; jd ≅ d / kMlaMgEdkkgkñúgmYyÉktþaRbEvgBIsmIkar 5.18a nig b Edlman s1 = ns
nwgkøayCa
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 233

21. T.Chhay
Ts Ts V 1
= = s (5.18c)
s1 ns sin α d (cos α + cos β )
RbsinebIeKmanEdkkgeRTtcMnYn n Edlman analoguos truss chord RbEvg s1 ehIyRbsin
ebI Av CaRkLaépÞénEdkkgeRTtmYy enaH
Ts = nAv f y (5.19a)
dUcenH nAv =
Vs ns
d sin α (cot β + cot α ) f y
(5.19b)
b:uEnþGacsnμt;fa enAkñúgkrNI)ak;edaykMlaMgTajGgát;RTUg (diagonal tension failure) Ggát;RTUgrgkar
sgát;manmMu β = 45o dUcenHsmIkar 5.19b nwgkøayCa
Av f y d
Vs = [sin α (1 + cot α )]
s
Av f y d
b¤ Vs =
s
(sin α + cos α ) (5.20a)
edaHRsayrk s edayeRbI Vs = Vu − Vc
Av f y d
s= (sin α + cos α ) (5.20b)
Vu − Vc
RbsinebIEdkRTnugeRTtpÁúMeLIgedayEdkeTal b¤RkuménEdkeTalEdlEdkTaMgenARtUv)anBt; nigRtUv)an
dak;enAcMgaydUcKñaBIépÞénTMr enaH
Vs = Av f y sin α ≤ 3.0 f 'c bw d ¬xñat US¦
Vs = Av f y sin α ≤ 0.25 f 'c bw d ¬xñat SI¦
RbsinebIeKeRbIEdkkgbBaÄr mMu α nwgesμInwg 90o enaHeK)an
Av f y d
Vs = (5.21a)
s
Av f y dAvφf y d
b¤ s= =
(Vu / φ ) − Vc Vu − φVc
(5.21b)
enAkñúgsmIkar 5.21a nig b/ d p CacMgayBIsréssgát;xageRkAbMputeTAkan;TIRbCMuTMgn;rbs;EdkeRbkug
RtaMg ehIy d CacMgayBIsréssgát;xageRkAbMputeTATIRbCMuTMgn;rbs;EdkFmμta. tMélrbs; d p min
RtUvtUcCag 0.80h eT.
Shear and Torsion Strength Design 234

22. NPIC
K> EdlkMNt;énTMhM nigKMlatrbs;Edkkg
Limitation on Size and Spacing of Stirrups
smIkar 5.20 nig 5.21 eGaynUvTMnak;TMngRcasKñarvagKMlatEdkkg nigkMlaMgkat; b¤kugRtaMg
kat;EdlvaRtUvTb;Tl;. enAeBlEdl s fycuH (Vu − Vc ) nwgekIneLIg. edIm,IeGayEdkkgbBaÄrTb;Tl;
sñameRbHGgát;RTUg dUcbgðajenAkñúgrUbTI 5>11 (c) eKRtUvGnuvtþEdnkMNt;KMlatGtibrmasMrab;Edkkg
bBaÄrdUcxageRkam³
(a) smax ≤ 3 h ≤ 24in.(60cm) Edl h CakMBs;srubrbs;muxkat;
4
(b) RbsinebI Vs > 4λ f 'c bw d p ¬xñat US¦ Vs > λ f 'c bw d p / 3 ¬xñat SI¦ eKRtUvkat;
bnßyKMlatGtibrmarbs; (a) Bak;kNþal ¬ smax ≤ 83 h ≤ 12in.(30cm)
(c) RbsinebI Vs > 8λ f 'c bw d p ¬xñat US¦ Vs > 2λ f 'c bw d p / 3 ¬xñat SI¦
eKRtUvBRgIkmuxkat;.
(d) RbsinebI Vu = φVn > φVc / 2 / eKRtUvdak;EdkkMlaMgkat;Gb,brma. eKKNnaRkLaépÞ
EdkGb,brmaenHedaysmIkar
Av = 0.75 f 'c w
b s
f
b¤ Av = 50fbws edayykmYyNaEdlFMCag
y y
RbsinebIkMlaMgeRbkugRtaMgRbsiT§PaB Pe FMCag b¤esμInwg 40% énersIusþg;Tajrbs;EdkBt;
(flexural reinforcement) enaH
A ps f pu s dp
Av = (5.22b)
80 f y d bw
Edlvapþl;nUv Av Gb,brmaRtUvkartUcCag ehIyEdleKGaceRbIvaCMnYs)an.
(e) edIm,IRbsiT§PaB EdkRTnugRtUvEtmanRbEvgbgáb; (development lengt) RtUvkareBjelj.
enHmann½yfaEdkkgRtUvBnøÚtcUleTAkñúgEpñkrgkarsgát; nigEpñkrgkarTajrbs;muxkat;/
RtUvkarkMras;ebtugkarBarEdk (clear concrete cover) tUc nigeKGaceRbITMBk; 90o b¤
135o enAkñúgtMbn;sgát;.
rUbTI 5>13 bgðajBIdüaRkaménkardak;EdkkgRTnugeTAtamtMbn;énRbEvgElVgrbs;FñwmeRbkug
RtaMgEdlrgGMeBIénbnÞúkBRgayesμI. épÞqUtCakMlaMgkat;elIs Vs EdlRtUvkarEdkRTnug.
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 235

23. T.Chhay
7> ersIusþg;kMlaMgkat;edkenAkñúgeRKOgbgÁúMsmas
Horizontal Shear Strength in Composite Construction
snμt;fakMlaMgkat;tamTisedkepÞreBjeljRtg;épÞb:HénkEnøgEdlCYbKña.
k> eRkamGMeBIbnÞúkeFVIkar Service-Load Level
eKGackMNt;kugRtaMgkat;tamTisedkGtibrma vh BIeKalkarN_mUldæanrbs;emkanic
VQ
vh = (5.23)
I c bv
Edl V= kugRtaMgkat;KNnaKμanemKuN (unfactored design vertical shear) EdleFVIGMeBIelImux
kat;smas
Q = m:Um:g;RkLaépÞeFob cgc énkMNt;muxkat;EdlenABIxagelI b¤BIxageRkam cgc
I c = m:Um:g;niclPaBénmuxkat;smasTaMgmUl
bv = TTwgRtg;kEnøgb:Hrbs;muxkat;RTnugénGgát;cak;Rsab; b¤TTwgénmuxkat;EdleKKNna
kMlaMgkat;edk
Shear and Torsion Strength Design 236

24. NPIC
eKGacsMrYlsmIkar 5.23 dUcxageRkam
V
vh = (5.24)
bv d pc
Edl d pc CakMBs;RbsiT§PaBBIsréssgát;xageRkAénmuxkat;smaseTATIRbCMuTMgn; cgc rbs;EdkeRbkug
RtaMg.
x> Ultimate-Load Level
Direct Method: sMrab;kar)ak;enAkñúgsßanPaBkMNt; eKGacEkERbsmIkar 5.24 edayCMnYs V
eday Vu dUcenHeyIgTTYl)an
Vu
vuh = (5.25a)
bv d pc
b¤ sMrab;ersIusþg;kMlaMgkat;bBaÄrFmμta Vn
Vu / φ V
vnh = = n (5.25b)
bv d pc bv d pc
Edl φ = 0.75 / RbsinebI Vnh CaersIusþg;kMlaMgkat;edkFmμta enaH Vu ≤ Vnh ehIyersIusþg;kMlaMgkat;
FmμtasrubKW
Vnh = vnhbv d pc (5.25c)
ACI Code kMNt; vnh Rtwm 80 psi(0.55MPa ) RbsinebIeKmineRbIEdkEdkrgcaM (dowel) b¤EdkkgbBaÄr
ehIyépÞ b:HmanlkçN³eRKIm b¤RbsinebIeKeRbIEdkkgbBaÄrGb,brma b:uEnþépÞb:HminmanlkçN³eRKIm.
vnh GaceTAdl; 500 psi (3.45MPa ) EteKRtUveRbI friction theory CamYynwgkarsnμt;xageRkam
(a) enAeBlEdlminmanEdkkgbBaÄr b:uEnþépÞb:HénGgát;cak;Rsab;manlkçN³eRKIm
enaHeKeRbI
Vnh ≤ 80 Ac ≤ 80bv d pc (5.26a)
Edl Ac CaRkLaépÞrbs;ebtugEdlTb;Tl;kMlaMgkat; = bv d pc
(b) enAeBlEdleKeRbIEdkkgGb,brma Edl Ac = 50(bws ) / f y b:uEnþépÞb:Hrbs;Ggát;cak;
Rsab;minmanlkçN³eRKIm
vnh ≤ 80bv d pc
(c) RbsinebIépÞb:Hrbs;Ggát;cak;Rsab;manlkçN³eRKImEdlmankMBs; 1 / 4in.(6mm) ehIy
EdkbBaÄrGb,brmaenAkñúg (b) RtUv)andak; enaHeKeRbI
Vnh ≤ 500bv d pc (5.26b)
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 237

25. T.Chhay
(d) RbsinebIkMlaMgkat;emKuN Vu > φ (500bv d pc )/ eKGaceRbI shear friction theory edIm,I
KNnaEdk dowel. enAkñúgkrNIenH kMlaMgkat;edkTaMgGs;RtUv)anKitedaybøg;EkgEdl
Vnh = μAvf f y (5.27)
Edl Avf = RkLaépÞén shear-friction reinforcement/ in.2
f y = design yield strength/ minRtUvelIs 60,000 psi (414 MPa )
μ = emKuNkkit
= 1.0λ sMrab;ebtugEdlcak;elIépÞebtugEdlmaneRKIm
= 0.6λ sMrab;ebtugEdlcak;elIépÞebtugEdlminmaneRKIm
λ = emKuNsMrab;RbePTebtug
enAkñúgkrNITaMgGs; ersIusþg;kat;Fmμta Vn ≤ 0.20 f 'c Acc ≤ 800 Acc Edl Acc CaRkLaépÞb:Hrbs;
ebtugEdlTb;Tl;nwgkMlaMgkat;epÞr (shear transfer). cMNaMfa enAkñúgkrNICaeRcIn kugRtaMgkat; vuh
EdlTTYl)anBIkMlaMgkat;emKuNGt;FMCag 500 psi(3.45MPa ) eT. dUcenH eKmincaM)ac;RtUvkareRbIEt
shear friction theory kñúgkarKNnaEdkrgcaM (dowel) sMrab;skmμPaBsmas (composite action)
enaHeT.
KMlatGnuBaØatGtibrmaénEdkrgcaM (dowel) b¤ tie sMrab;kMlaMgkat;edkKWtMéltUcCageKkñúg
cMeNam 4 dgénTMhMtUcCageKénmuxkat;TMr nig 24in.(60cm) .
Basic Method: ACI Code GnuBaØateGayeRbIviFIepSgeTotEdlkMlaMgkat;edkRtUv)anGegát
edayKNnabMErbMrYlCak;EsþgénkMlaMgsgát; b¤kMlaMgTajenAkñúgbøg;NamYy nigedayepþrkMlaMgkat;edk
enaHeTAGgát;EdlCaTMr. eKCMnYsRkLaépÞb:H Acc sMrab; bv d pc enAkñúgsmIkar 5.25b nig c enaHeK
TTYl)an
Vnh = vnh Acc (5.28)
Edl Vnh ≥ Fh / kMlaMgkat;edk ehIyy:agehacNas;vaRtUvesμInwgkMlaMgsgát; C b¤kMlaMgTaj T enA
kñúgrUbTI 5>14. ¬emIlsmIkar 5.30 sMrab;tMélrbs; Fh ¦
eKGackMNt;RkLaépÞb:H Acc dUcxageRkam
Acc = bv lvh (5.29)
Edl lvh CaRbEvgkMlaMgkat;edk (horizontal shear length) EdlkMNt;enAkñúgrUbTI 5.15(a) nig (b)
sMrab;Ggát;TMrsamBaØ nigsMrab;Ggát;TMrCab; erogKña.
Shear and Torsion Strength Design 238

26. NPIC
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 239

27. T.Chhay
K> karKNnaEdkrgcaMskmμsmas
Design of Composite-Action Dowel Reinforcement
Edk tie sMrab;kMlaMgkat;edkGacpSMeLIgBIr)arEdkeTal (single bars or wires)/ BIEdkkgeCIg
eRcIn (multiple leg stirrup) b¤BI vertical legs of welded wire fabric. KMlatrbs;vaminGacFMCagbYn
dgénTMhMEdltUcCageKénGgát;TMr b¤ 24in.(60cm) edayykmYyNaEdltUcCageK. RbsinebI μ Ca
emKuNkMlaMgkkit enaHeKGackMNt;kMlaMgkat;edkFmμta Fh enAkñúgrUbTI 5>14 dUcxageRkam
Fh = μAvf f y ≤ Vnh (5.30)
tMél ACI rbs; μ KWQrelIersIusþg;kkit-kMlaMgkat;kMNt; (limit shear-friction strength) 800 psi
(5.5MPa ) ¬vaCatMélEdlmanlkçN³suvtßiPaBbnþicEdlbgðajedaykarBiesaF¦. viTüasßanebtugeRb
kugRtaMg (Prestressed Concrete Institute) ENnaM μe = 2.9 CMnYseGay μ = 1.0λ sMrab;ebtug
Edlcak;elIépÞebtugeRKIm ehIykMlaMgkat;KNnaGtibrma (maximum design shear force)
Vu ≤ 0.25λ2 f 'c Ac ≤ 1,000λ2 Acc (5.31a)
CamYynwgRkLaépÞcM)ac;rbs;Edkkkit-kMlaMgkat; (shear-friction steel)
Vuh
Avf = (5.31b)
φf y μ e
b¤ Avh =
Vnh F
= h
μe f y μe f y
(5.31c)
edayeRbItMél PCI EdlminsUvsuvtßiPaB smIkar 5.31c køayCa
Fh ≤ μ e Avf f y ≤ Vnh (5.32)
λ 2
CamYynwg μe = 1,000F bv I vh ≤ 2.9
h
Edl bvlvh = Acc / EdkGb,brmaKW
50bv s 50bv lvh
Av = = (5.33)
fy fy
8> CMhanKNnaEdkRTnugsMrab;kMlaMgkat;
Web Reinforcement Design Procedure for Shear
xageRkamCakarsegçbBICMhanénkarKNnaEdkRTnugsMrab;kMlaMgkat;³
!> kMNt;tMélersIusþg;kMlaMgkat;FmμtaEdlRtUvkar Vn = Vu / φ enARtg;cMgay h / 2 BIépÞénTMr Edl
φ = 0.75 .
Shear and Torsion Strength Design 240

28. NPIC
@> KNnaersIusþg;kMlaMgkat;Fmμta (nominal shear strength) Vc EdlRTnugmanedayeRbIviFImYy
kñúgcMeNamviFIBIrxageRkam³
(a) ACI conservative method RbsinebI f pe > 0.40 f pu
⎛ 700Vu d p ⎞
Vc = ⎜ 0.60λ f 'c +
⎜ ⎟bw d p ¬xñat US¦
⎟
⎝ M u ⎠
⎛ λ f 'c V d⎞
Vc = ⎜
⎜ 20
+ 5 u ⎟bw d p
Mu ⎟
¬xñat SI¦
⎝ ⎠
Edl 2λ f 'c bw d p ≤ Vc ≤ 5λ f 'c bw d p sMrab;xñat US
λ f 'c bw d p / 5 ≤ Vc ≤ 0.4λ f 'c bw d p sMrab;xñat SI
Vu d p
≤ 1.0
Mu
ehIyeKKNna Vu enARtg;muxkat;dUcKñasMrab;karKNna M u .
RbsinebIersIusþg;eRcokTajmFüm (average tensile splitting strength) fct sMrab;
ebtugTMgn;Rsal enaH λ = fct / 6.7 f 'c sMrab;xñat US b¤ λ = fct / 0.556 f 'c
sMrab;xñat SI CamYynwg f 'c Gt;FMCag 100 psi(0.67MPa ) .
(b) Detailed analysis Edl Vc CatMéltUcCageKkñúgcMeNam Vci nig Vcw
¬xñat US¦ 1.7λ f 'c bw d p ≤ Vci = 0.60λ f 'c bw d p + Vd +
Vi
M max
(M cr ) ≤ 5.0λ f 'c bwd p
λ f 'c bw d p λ f 'c bw d p
¬xñat SI¦ 7
≤ Vci =
20
+ Vd +
Vi
M max
(M cr ) ≤ 0.4λ f 'c bwd p
¬xñat US¦ ( )
Vcw = 3.5λ f 'c + 0.3 f c bw d p + V p
¬xñat SI¦ ( )
Vcw = 0.3 λ f 'c + f c bw d p + V p
edayeRbItMélNaEdlFMCageKkñúgcMeNam d p nig 0.8h
Edl M cr = (I c / yt )(6λ f 'c + fce − f d ) ¬xñat US¦
M cr = (I c / yt )(0.5λ f 'c + f ce − f d ) ¬xñat SI¦
b¤ M cr = Sb (6λ f 'c + fce − f d ) ¬xñat US¦
M cr = Sb (0.5λ f 'c + f ce − f d ) ¬xñat SI¦
Vi = kMlaMgkat;emKuNEdlbNþalBIbnÞúkGnuvtþn_BIxageRkAEdlekItmankñúg
eBldMNalKñaCamYynwg M max
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 241

29. T.Chhay
kugRtaMgsgát;enAkñúgebtugeRkayekItmankMhatbg;TaMgGs;enARtg;
f ce =
srésxageRkArbs;muxkat;EdlbnÞúkxageRkAbgáeGaymankugRtaMgTaj.
f ce køayCa f c sMrab;kugRtaMgenARtg;TIRbCMuTMgn;rbs;muxkat;.
#> RbsinebI Vu / φ ≤ Vc / 2 vaminRtUvkarEdkRTnugeT. RbsinebI Vc / 2 < Vu / φ < Vc vaRtUvkar
EdkGb,brma. RbsinebI Vu / φ > Vc nig
RbsinebI Vs = Vu / φ − Vc ≤ 8λ f 'c bw d p ¬xñat US¦
Vs = Vu / φ − Vc ≤ 2λ f 'c bw d p / 3 ¬xñat SI¦
eKRtUvKNnaEdkRTnug.
RbsinebI Vs = Vu / φ − Vc > 8λ f 'c bw d p b¤ Vs > φ (Vc + 8λ f 'c bw d p ) ¬xñat US¦
Vs = Vu / φ − Vc > 2λ f 'c bw d p / 3 ¬xñat SI¦
eKRtUvtMeLIgmuxkat;.
$> KNnaEdkRTnugGb,brmaEdlRtUvkar. KMlatKW s ≤ 0.75h b¤ 24in.(60cm) edayykmYyNa
EdltUcCageK.
Av min = 0.75 f 'c w b¤ Av min = ¬US¦ edayykmYyNaEdlFMCag
b s 50bw s
f y f y
Av min = f 'c
bw s
16 f y
b¤ Av min = 50fbws ¬SI¦ edayykmYyNaEdlFMCag
y
RbsinebI /
f pe ≥ 0.40 f pu Av min EdlmanlkçN³suvtßiPaBticCagCatMéltUcCageKkñúg
cMeNam
A ps f pu s dp
Av =
80 f y d p bw
Edl d p ≥ 0.80h
CamYynwg Av min = 0.75 f 'c
bw s
fy
b¤ Av min = 50fbws ¬US¦
y
Av min = f 'c
bw s
16 f y
b¤ Av min = 50fbws ¬SI¦
y
%> KNnaTMhM nigKMlatEdkRTnugEdlRtUvkar. RbsinebI
¬xñat US¦
Vs = (Vu / φ − Vc ) ≤ 4λ f 'c bw d p
Vs = Vu / φ − Vc ≤ λ f 'c bw d p / 3 ¬xñat SI¦
enaHKMlatEdkkg s EdlRtUvkarKWesμInwgtMélEdlKNnaedaysmIkarEdleGayenAkñúgCMhan ^.
Shear and Torsion Strength Design 242

30. NPIC
EtRbsinebI
¬xñat US¦
Vs = (Vu / φ − Vc ) > 4λ f 'c bw d p
Vs = Vu / φ − Vc > λ f 'c bw d p / 3 ¬xñat SI¦
enaHKMlatEdkkg s EdlRtUvkarKWesμInwgBak;kNþaléntMélEdlKNnaedaysmIkarEdleGay
enAkñúgCMhan ^.
^> s = (VAvφf)y− V = VAv −yφdVp ≤ 0.75h ≤ 24in.(60cm) ≥ s Gb,brmaEdl)anBICMhan $
dp f
u c u c
&> sg; shear envelope enAelIElVgFñwm nigKUsbBa¢ak;tMbn;EdlRtUvkarEdkRTnug
*> KUrBRgayEdkRTnugtambeNþayElVgedayeRbIEdkkgTMhM #3 b¤ #4 tamEdlcUlcitþ b:uEnþEdk
kgminRtUvmanTMhMFMCag #6 eT.
(> KNnaEdkrgcaM (dowel reinforcement) bBaÄrkñúgkrNImuxkat;smas
(a) Vnh ≤ 80bv d pc sMrab;TaMgépÞb:HeRKImedayKμanEdkrgcaM b¤Edk tie bBaÄr nigsMrab;TaMgépÞb:H
EdlmineRKImb:uEnþeRbIEdk tie bBaÄrGb,brma. eRbI
50bw s 50bv I vh
Av = =
fy fy
(b) Vnh ≤ 500bw d pcsMrab;épÞeRKImEdlmanecjk,alfμkMBs; 1 / 4in.(6mm)
(c) sMrab;krNIEdl Vnh > 500bw d pc / KNnaEdk tie bBaÄrsMrab; Vnh = Avf f y μ
Edl Avf = RkLaépÞrbs;EdkrgcaMEdlmanlkçN³kkit (frictional steel dowel)
μ = emKuNkkit = 1.0λ sMrab;épÞEdlmanlkçN³eRKIm Edl λ = 1.0 sMrab;
ebtugTMgn;Fmμta. sMrab;RKb;krNITaMgGs; Vn ≤ Vnh ≤ 0.2 f 'c Acc
≤ 800 Acc Edl Acc = bv lvh .
viFIepSgeToténkarKNnaRkLaépÞEdkrgcaM Avf KWedayKNnakMlaMgedk Fh enARtg;épÞkkit
rbs;ebtugEdl
Fh ≤ μ e Avf f y ≤ Vnh
1,000λ2bv lvh
Edl μe =
Fh
≤ 2.9
rUbTI 5>16 bgðajBICMhanKNnaEdl)anerobrab;xagelICaTMrg; flowchart.
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 243

31. T.Chhay
Shear and Torsion Strength Design 244

32. NPIC
9> kugRtaMgTajemenAkñúgmuxkat;mansøab nigKNnaEdkrgcaMbBaÄrenA
kñúgmuxkat;smas
Principal Tensile Stresses in Flanged Sections and Design of
Dowel-Action Vertical Steel in Composite Sections
]TahrN_ 5>1³ FñwmeRbkugRtaMgmuxkat;GkSr T mankarBRgayénkugRtaMgeFVIkarsgát;dUcbgðajenAkñúg
rUbTI 5>17. kMlaMgkat;bBaÄrxageRkAKNnaEdlKμanemKuN V = 120,000lb(554kN ) ehIykMlaMg
kat;emKuN Vu = 190,000lb(845kN ) .
(a) KNnakugRtaMgTajemenARtg;G½kSTIRbCMuTMgn; cgc nigenARtg;cMnucRCugEkg A énkEnøgkat;Kña
rvagsøab nigRTnug nigKNnakugRtaMgkMlaMgkat;edkGtibrmaeRkamGMeBIbnÞúkeFVIkarsMrab;TI
taMgTaMgenH.
(b) KNnaersIusþg;kMlaMgkat;edkFmμtaEdlRtUvkarenARtg;épÞGnþrkmμ A − A rvagRTnugEdlcak;
Rsab; nigsøabEdlcak;enAnwgkEnøg nigKNnaEdkrgcaM b¤Edk tie bBaÄrcaM)ac;edIm,IkarBarkar
rGildac;enARtg; A − A EdlFanaskmμPaBsmaseBjelj. eRbI ACI direct method nig
snμt;faépÞb:HmanlkçN³eRKIm. smμtikmμmandUcxageRkam³
f 'c sMrab;RTnug = 6,000 psi ebtugTMgn;Fmμta (normal-weigth concrete)
f 'c sMrab;søab = 3,000 psi ebtugTMgn;Fmμta (normal-weigth concrete)
TTwgsøabRbsiT§PaB bm = 60in.(152.4cm)
Edl bm CaTTwgEktMrUvEdl)anKitbBa©ÚlPaBxusKñaénm:UDuleGLasÞicrbs;ebtugépñkEdlcak;
Rsab; nigEpñkxagelIEdlcak;enAnwgkEnøg.
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 245

33. T.Chhay
dMeNaHRsay³
kugRtaMgkMlaMgkat;eFVIkaredk
kugRtaMgkMlaMgkat;edkGtibrma
VQ
vh =
Ibv
Edl (
Q A = 60 × 12(19.32 − 6 ) = 9,590in.3 157,172cm3 )
12 × (7.32 )2
Qcgc = 60 × 12(19.32 − 6 ) +
2
(
= 9,912in.3 162,429cm.3 )
dUcenHkugRtaMgkMlaMgkat;edkeRkamGMeBIbnÞúkeFVIkarKW
enARtg;cMnuc A / vh = 120,000 × 9,12 = 235 psi(1.6MPa)
408,240 ×
590
enARtg; cgc / vh = 120,000 × 9,12 = 243 psi(1.7MPa)
408,240 ×
912
BIsmIkar 5.13 kugRtaMgTajemEdlRtUvKñaKW
2
⎛f ⎞
enARtg; A / ⎝ 2 ⎠
2 f
f 't = ⎜ cA ⎟ + vh − cA
2
2
⎛ 2,160 ⎞
⎟ + (235) − = 25 psi (111Pa )
2,160
= ⎜ 2
⎝ 2 ⎠ 2
2
nigenARtg; cgc / f 't = ⎛ 1,831 ⎞ + (235)2 − 1,831 = 32 psi(221Pa )
⎜
⎝ 2 ⎠
⎟
2
dUcenH kugRtaMgTajemmantMéltUc nigminbgáeGaymansñameRbHeRkamGMeBIbnÞúkeFVIkareT.
eKRtUvRtYtBinitükugRtaMgkMlaMgkat;edk vh = 235 psi enARtg;épÞb:H A − A edIm,IepÞóg
pÞat;favasßitenAkñúgEdnkMNt;EdlGacTTYlyk)an. GnuelameTAtam AASTHO kugRtaMg
GnuBaØatGtibrmaKW 160 psi(1.1MPa ) < 235 psi(1.6MPa ) dUcenHkarpþl;eGayCaBiesssMrab;
EdkrgcaM b¤Edk tie bBaÄrbEnßmRtUv)aneFVIRbsinebIeKGnuvtþtamtMrUvkarrbs; AASTHO.
KNnaEdkrgcaM (Dowel Reinforcement Design)
Vu = 190,000lb
Vnh EdlRtUvkar = Vφu = 190.,75 = 253,333lb(1126kN )
0
000
bv = 12in.(30.5cm ) d pc = 57in.(145cm )
BIsmIkar 5.26b
Shear and Torsion Strength Design 246

34. NPIC
EdlGacman = 500bv d pc = 500 × 12 × 57 = 342,000lb(1520kN )253,333lb
Vnh
BIsmIkar 5.22 a/ 0.75 f 'c = 0.75 6,000 = 58 > 50 dUcenHeRbI 58 enAkñúgsmIkarenaH
Av
l
ÉktþaGb,brma = 58bv = 60,× 12 = 0.0116in.2 / in tambeNþayElVg
f
58
000
vh y
dUcenHeRbIEdkkgbBaÄr #3 Edlman Av = 2 × 0.11 = 0.22in.2 ehIy s = 0.22 / 0.0116
= 18.9in.(48cm ) EdlKitBIG½kSeTAG½kS < 24in. dUcenHKWTTYlyk)an (O.K.). EdkRTnugbBaÄrsMrab;
kMlaMgkat;enAkñúgRTnugKYrRtUvkarKMlattUcCagenH. dUcenH BnøÚtEdkkgRTnugTaMgGs;eTAkñúgkMralxNÐ
xagelIEdlcak;enAnwgkEnøg.
10> KNnaEdkrgcaMsMrab;skmμPaBsmas
Dowel Steel Design for Composite Action
]TahrN_ 5>2³ edayeRbI (a) emKuNkkit ACI nig (b) emKuNkkit PCI cUrKNnaEdkrgcaM (dowel
reinforcement) én]TahrN_ 5>1 sMrab;skmμPaBsmaseBjelj (full composite action) edayviFI
epSgeTot (alternative method). edaysnμt;ElVgRbsiT§PaBrbs;FñwmTMrsamBaØesμInwg 65 ft.(19.8m) .
dMeNaHRsay³ BIrUbTI 5>14 nig 5>17
Atop = 60 × 12 = 720in.2 (4,645cm 2 )
Cc = 0.85 f 'c Atop = 0.85 × 3,000 × 720 = 1,836,000lb(8,167kN )
snμt; Aps f ps > Cc enAeBlEdleKmineGaykMlaMgeRbkugRtaMg. enaH
Fh = 1,836,000lb
65 × 12
lvh = = 390in.
2
bv = 12in.
80bv lvh = 80 × 12 × 390 = 374,400lb(1,665kN ) < 1,836,000lb
dUcenH eKRtUvkarEdk tie bBaÄr.
sMrab;épÞeRKImEdlmanecjk,alfμkMBs; 1 / 4in.(6mm) nigmanEdkGb,brma
Vnh = 500bv d = 50 × 12 × 57 = 342,000
Vu
φ
EdlRtUvkar
=
190,000
0.75
= 253,333lb < Vnh = 342,000
< Fh EdlGacman = 1,836,000lb
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 247

35. T.Chhay
eRbI Fh = 253,333lb sMrab;kMNt;EdkskmμPaBsmasEdlRtUvkar (required composite
action reinforcement).
(a) edayeRbItMél μ rbs; ACI
BIsmIkar 5.27 nig μ = 1.0 CamYynwg lvh = 390in.
Avf srub = = 4.2in.2 (26.3cm 2 )
253,333
1.0 × 60,000
Avf Gb,brma = 50bfvlvh = 50 × 12000390 = 3.90in.2 (25.1cm2 )
60,
×
y
BI]TahrN_ 5>1 Avf Gb,brma = 0.0116in.2 / in. = 0.139in.2 / 12in. / lub. dUcenHsakl,g
Edkkg #3 eyIgTTYl)an Av = 2 × 0.11 = 0.22in.2 enaH
390 × 0.22
= 20.42 BIG½kSeTAG½kS < 24in. < tMélGnuBaØatGtibrma 4 × 12 = 48in.
l A
s = vh v =
A vf 4.2
dUcenHeRbIEdk tie GkSr U #3 EdlmanKMlat 20in. KitBIG½kSeTAG½kS.
(b) edayeRbItMél μ e rbs; PCI
λ = 1 .0
1,000λ2bv lvh
μe = ≤ 2.9
Fh
1,000 × 1 × 12 × 390
= = 2.55 < 2.9
1,836,000
dUcenHeRbI μe = 2.55 . bnÞab;mk BIsmIkar 5.32
Avf tMrUvkar = = 1.66in.2 < Avf Gb,brma = 3.90in.2 lub
253,333
2.55 × 60,000
dUcenHeRbIEdk tie GkSr U #3 EdlmanKMlat 20in. EdlKitBIG½kSeTAG½kS ¬Ggát;p©it 9.5mm
KMlat 55cm ¦
11> Dowel Steel Design for Composite Action in an Inverted T
Beam
]TahrN_ 5>3³ FñwmGkSr T bRBa©asEdlRTedayTMrsamBaØmanElVgRbsiT§PaB 24 ft (7.23m) . mux
kat;rbs;FñwmenHRtUv)anbgðajenAkñúgrUbTI 5>18 edaymankMralxagelIEdlcak;enAnwgkEnøgkMras; 2in.
(5.1cm) enAelIépÞGt;eRKIm. KNnaEdkkgrgcaMcaM)ac;edIm,IbegáItskmμPaBsmaseBjelj (full
Shear and Torsion Strength Design 248

36. NPIC
composite behavior) edaysnμt;fakMlaMgkat;emKuN Vu EdlFñwmRtUvrgenARtg;muxkat;eRKaHfñak;KW
160,000lb(712kN ) . smμtikmμmandUcxageRkam³
f 'c ¬cak;Rsab;¦ = 6,000 psi (41.4 MPa ) / ebtugTMgn;Fmμta
f 'c ¬kMralxagelI¦ = 3,000 psi (20.7 MPa ) / ebtugTMgn;Fmμta
EdkeRbkugRtaMg³ tendon 270k Ggát;p©it 1 / 2in.(12.7mm) cMnYn 12
f pu = 270,000 psi (1,862MPa )
f ps = 242,000 psi (1,669 MPa )
rbs;Edk tie = 60,000 psi(414MPa )
fy
edayeRbITaMg ACI direct method nig alternative method CamYynwg μe RbsiT§PaBsMrab;karKNna.
dMeNaHRsay³ d p = 2 + 2 + 10 + 12 − 3 = 23in.
Aps = 12 × 0.153 = 1.836in.2
Tn = Aps f ps = 1.836 × 242,000 = 444,312lb(1,976kN )
bv = 12in.
24 × 12
lvh = = 144in.
2
Atop = 2 × 48 + 2 × 12 = 120in.2
Cc = 0.85 f 'cc Atop = 0.85 × 3,000 × 120 = 306,000lb(1,316kN )
< Tn = 444,312lb
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 249

37. T.Chhay
dUcenH eRbI Fh = 306,000lb(1,361kN ) . bnÞab;mk sMrab;épÞmineRKIm
Vnh EdlGacman = 80bv lvh = 80 × 12 × 144 = 138,240lb(615kN )
< Cc = 306,000lb
dUcenH Edk tie EdlRtUvkarcaM)ac;sMrab;begáItskmμPaBsmaseBjeljedayeRbI λ = 1.0 .
(a) ACI Direct Method
tMrUvkar = Vφu = 160.,75 = 213,333lb(949kN )
Vnh
0
000
eRbI μ = 1.0 . bnÞab;mk BIsmIkar 5.26a CamYynwgEdkrgcaM (dowel reinforcement)
eyIgman
Vnh EdlGacman = 80bv d pc = 80 × 12 × 23 = 22,000lb << Vnh tMrUvkar
BIsmIkar 5.27 sMrab;épÞEdlmineRKIm μ = 0.6λ = 0.60 . bnÞab;mk
Avf srubtMrUvkar = n =
V 213,333
= 5.93in.2
μf 0.60 × 60,000
y
BIsmIkar 5.22(a)/ 0.75 f 'c = 0.75 6,000 = 58 > 50 dUcenHeRbI 58 enAkñúgsmIkar
58bv lvh 58 × 12 × 144
Avf Gb,brmatMrUvkar = = = 1.67in.2 < 5.93in.2
f y 60,000
dUcenHeRbI Avf = 5.23in.2 (33.7cm2 ) ehIysakl,gEdk tie GkSr U páab; #3 . bnÞab;mk
Avf = 2 × 0.11 = 0.22in.2 ( .4cm 2 ) nigmanKMlat
1
lvh Av 144 × 0.22
s= = = 5.34in.(13.7cm )
Avf 5.93
KMlatGnuBaØatGtibrmaKW s = 4(2 + 2) = 16in. b¤ 0.75h = 0.75 × 26 = 19.5in. < 24in. .
dUcenHeRbIEdk tie GkSr U páab; #3 KMlat 5in.(13cm) KitBIG½kSelIG½kSelIElVgTaMgmUl.
(b) Alternativ Method edayeRbI μe
Fh = 306,000lb
1,000λ2bv lvh 1,000 × 1.0 × 12 × 144
μe = = = 5.65 > 2.9
Fh 306,000
dUcenHeRbI μe = 2.9 bnÞab;mk BIsmIkar 5.31c eyIgTTYl)an
Avf tMrUvkar = h =
F 306,000
= 1.76in.2
μ f 2.9 × 60,000
e y
Shear and Torsion Strength Design 250

38. NPIC
Gb,brmatMrUvkarEdl)anBI (a) = 1.67in.2 < 1.76in.2
Avf
dUcenHeRbI Avf = 1.76in.2 enaHKMlatKW
l A 144 × 0.22
s = vh v = = 18in. BIG½kSeTAG½kS
A vf 1.76
ehIyKMlatGnuBaØatGtibrmaKW
s = 4(2 + 2 ) = 16in. < 24in.
dUcenH eRbIEdk tie GkSr U páab; #3 manKMlat 16in. KitBIG½kSeTAG½kSelIElVgTaMgmUl.
12> Shear Strength and Web-Shear Steel Design in a Prestressed
Beam
]TahrN_ 5>4³ KNna bonded beam én]TahrN_ 4>2 edIm,IeGaymansuvtßiPaBRbqaMgnwgkar)ak;
edaykMlaMgkat; ehIyKNnaEdkRTnugtMrUvkar.
dMeNaHRsay³
Tinñn½y nigkarkMNt;ersIusþg;kMlaMgkat;Fmμta (data and nominal shear strength determination)
f pu = 270,000 psi (1,862MPa )
f y = 60,000 psi (414MPa )
f pe = 155,000 psi (1,069MPa )
f 'c = 5,000 psiebtugTMgn;Fmμta
Aps = tendon Ggát;p©it 1 / 2in.(12.7 mm ) Edlman wire 7 cMnYn 13
= 1.99in.2 = (12.8cm 2 )
As = 4#6 = 1.76in.2 (11.4cm 2 )
RbEvgElVg = 65 ft (19.8m)
bnÞúkeFVIkar WL = 1,100 plf (16.1kN / m)
bnÞúkeFVIkar WSD = 100 plf (1.46kN / m)
bnÞúkeFVIkar WD = 393 plf (5.7kN / m)
h = 40in.(101.6cm )
d p = 36.16in(91.8cm )
d = 37.6in(95.5cm )
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 251

39. T.Chhay
bw = 6in.(15cm )
ec = 15in.(38cm )
ee = 12.5in.(32cm )
(
I c = 70,700in.4 18.09 × 106 cm 4 )
(
Ac = 377in.2 2,432cm 2 )
(
r 2 = 187.5in.2 1,210cm 2 )
cb = 18.84in.(48cm )
ct = 21.16in.(54cm )
Pe = 308,255lb(1.371kN )
bnÞúkemKuN Wu = 1.2D + 1.6L
= 1.2(100 + 393) + 1.6 × 1,100 = 2,352 plf
kMlaMgkat;enARtg;épÞTMr Vu = Wu L / 2
= (2,352 × 65) / 2 = 76,440lb
Vn tMrUvkar = Vu / φ = 76,440 / 0.75 = 101,920lb enARtg;TMr
bøg;enARtg; 12 d p BIépÞénTMr
!> ersIusþg;kMlaMgkat;Fmμta (nominal shear strength) Vc rbs;RTnug ¬CMhanTI2 nigTI3¦
1 36.16
dp = ≅ 1.5 ft
2 2 × 12
Vn = 101,920 ×
[(65 / 2) − 1.5] = 97,216lb
65 / 2
Vu enARtg;1
2
d p = 0.75 × 97,216 = 72,912lb
f pe = 155,000 psi
0.40 f pu = 0.40 × 270,000 = 108,000 psi (745MPa )
< f pe = 155,000 psi (1,069 MPa )
eRbI ACI alternate method
edaysar d p > 0.8h / eRbI d p = 36.16in. edaysnμt;faEdkeRbkugRtaMgxøHRtg;rhUtdl;TMr.
BIsmIkar 5.16
Shear and Torsion Strength Design 252

40. NPIC
⎛ Vu d p ⎞
Vc = ⎜ 0.60λ f 'c + 700
⎜ ⎟bw d p ≥ 2λ f 'c bw d p ≤ 5λ f 'c bw d p
⎝ Mu ⎟ ⎠
λ = 1 .0sMrab;ebtugTMgn;Rsal
Wu (1.5)2
M u enARtg; d / 2 BIépÞ = Rbtikmμ × 1.5 −
2
2,352(1.5)2
= 76,440 × 1.5 − = 112,014 ft − lb
2
= 1,344,168in. − lb
Vu d p 72,912 × 36.16
= = 1.96 > 1.0
Mu 1,344,168
dUcenHeRbI Vu d p / M u = 1.0 enaH
Vc Gb,brma = 2λ f 'c bw d p = 2 × 1.0 5,000 × 6 × 36.16 = 30,683lb
Vc Gtibrma = 5λ f 'c bw d p = 76,707lb(341kN )
Vc = (0.60 × 1.0 5,000 + 700 × 1.0 )6 × 36.16
= 161,077lb > Vc Gtibrma = 76,707lb
bnÞab;mk Vc = 76,707lb / lub. dUcKña Vu / φ > Vc / 2 dUcenH eKRtUvkarEdkRTnug.
Vu
Vs = − Vc = 97,216 − 76,707 = 20,509lb
φ
8λ f 'c bw d p = 8 × 1.0 5,000 × 6 × 36.16 = 122,713lb(546kN )
> Vs = 20,509lb
dUcenHkMBs;rbs;muxkat;RKb;RKan;.
@> EdkRTnugGb,brma ¬CMhanTI4¦
BIsmIkar 5.22b
Av
s
Gb,brma = 80psf fdpu d p
A
b y p w
1.99 × 270,000 36.16
= = 0.0076in.2 / in.
80 × 60,000 × 36.16 6
#> EdkRTnugtMrUvkar ¬CMhanTI5 nigTI6¦
BIsmIkar 5.21b
Av f y d p
s= ≤ 0.75h ≤ 24in.
Vu / φ − Vc
b¤ Av
s
V
= s =
20,509
f y d p 60,000 × 36.16
= 0.0095in.2 / in.
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 253

41. T.Chhay
¬kMlaMgeRbkugRtaMgKWRtUvFMCag 0.4 × ersIusþg;Taj¦
EdkRTnugkMlaMgkat; (web-shear steel) EdlRtUvkarGb,brma Av / s = 0.0095in.2 / in. .
dUcenHsakl,gEdkkgGkSr U #3 / Av = 2 × 0.11 = 0.22in.2 . enaHKMlatGtibrma
= 23.2in.(59cm )
0.22
s=
0.0095
nig 4λ f 'c bwd p = 4 × 1.0 5,000 × 6 × 36.16 = 61,366lb > Vs
dUcenH eyIgminRtUvkareRbI 12 s . LÚv
0.75h = 0.75 × 40.0 = 30.0in.
dUcenH eRbI web-shear reinforcement #3 KMlat 22in. KitBIG½kSeTAG½kS ¬EdkkgGgát;p©it
9.5mm KMlat 62cm KitBIG½kSeTAG½kS¦
bøg;EdlminRtUvkarEdkkg
snμt;fabøg;enHsßitenAcMgay x BITMr. tamRtIekaNdUc
1 76,707 65 / 2 − x
Vc = = 101,920 ×
2 2 65 / 2
b¤ 65
2
−x=
76,707 65
101,920 4
×
x = 20.3 ft (6.11m ) ≈ 244in.
dUcenHeRbIEdkkgGkSr U #3 KMlat 22in KitBIG½kSeTAG½kSelIRbEvgRbEhl 244in. edayEdk
kgTImYycab;epþImenARtg; 18in. BIépÞénTMr. BRgayEdkkgeTAdl;kNþalElVgRbsinebIeKRtUvkarskmμ-
PaBsmas.
13> Web-Shear Steel Design by Detailed Procedures
]TahrN_ 5>5³ edaHRsay]TahrN_ 5>4 edaydMeNIrlMGitEdlkMNt;tMélrbs; V CatMéltUcCageK
c
én flexure shear Vci nig web shear Vcw . snμt;fa tendon RtUv)an harp enAkNþalElVg. ehIy
snμt;fa f 'c = 6,000 psi .
dMeNaHRsay³ Profile rbs;EdkeRbkugRtaMgRtuv)anbgðajenAkñúgrUbTI 5>19.
bøgenARtg; d / 2 BIépÞrbs;TMr
BI]TahrN_ 5>4/ Vn = 97,216lb
!> Flexure-shear cracking, Vci ¬CMhanTI2¦
Shear and Torsion Strength Design 254

42. NPIC
BIsmIkar 5.11
Vci = 0.60λ f 'c bw d p + Vd +
Vi
(M cr ) ≥ 1.7λ f 'c bwd p
M max
BIsmIkar 5.12
M cr =
Ic
yt
(
6λ f 'c + f ce − f d )
Edl I c / yt = Sb edaysar yt CacMgayBITMRbCMuTMgn;eTAsrésTajxageRkA.
eyIgman I c = 70,700in.4
cb = 18.84in.
Pe = 308,255lb
Sb = 3,753in.3
r 2 = 187.5in.2
dUcenHBIsmIkar 4.3b kugRtaMgebtugenARtg;srésxageRkambMputEdlbNþalmkEtBIeRbkug
RtaMgKW
Pe ⎛ ecb ⎞
f ce = − ⎜1 + 2 ⎟
Ac ⎝ r ⎠
ehIycMNakp©itEdkeRbkugRtaMgenARtg; d p / 2 ≅ 1.5 ft BIépÞrbs;TMrKW
e = 12.5 + (15 − 12.5)
1.5
= 12.62in.
65 / 2
308,255 ⎛ 12.62 × 18.84 ⎞
dUcenH f ce =−
377 ⎝
⎜1 +
187.5
⎟ ≅ −1,855 psi (12.8MPa )
⎠
BI]TahrN_ 4>2/ bnÞúkefrKμanemKuNEdlbNþalBITMgn;pÞal; WD = 393 plf (5.7kN / m) KW
WD x(l − x ) 393 × 1.5(65 − 1.5) × 12
Md /2 = = = 224,600in. − lb(25.4kN .m )
2 2
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 255

43. T.Chhay
ehIykugRtaMgEdlbNþalBIbnÞúkGefrKμanemKuNenARtg;srésebtugxageRkAbMputEdlkugRtaMg
TajRtUv)anbegáItedaybnÞúkxageRkAKW
M d / 2cb 224,600 × 18.84
fd = = = 60 psi
Ic 70,700
ehIy (
M cr = 3,753 6 × 1.0 × 6,000 + 1,855 − 60 )
= 8,480,872in. − lb(958kN .m )
⎛l ⎞ ⎛ 65 ⎞
Vd = WD ⎜ − x ⎟ = 393⎜ − 1.5 ⎟ = 12,183lb(54.2kN )
⎝2 ⎠ ⎝ 2 ⎠
WSD = 100 plf
WL = 1,100 plf
WU = 1.2 × 100 + 1.6 × 1,100 = 1,880 plf
kMlaMgkat;emKuNenARtg;muxkat;EdlbNþalBIbnÞúkGnuvtþn_xageRkAEdlekIteLIgtMNalKñaCa
mYynwg M max KW
⎛l ⎞ ⎛ 65 ⎞
Vi = WU ⎜ − x ⎟ = 1,880⎜ − 1.5 ⎟ = 58,280lb(259kN )
⎝2 ⎠ ⎝ 2 ⎠
1,880 × 1.5(65 − 1.5)
nig M max = U (l − x ) =
W x
2 2
× 12 = 1,074,420in. − lb(122kN .m )
dUcenH Vci = 0.6 × 1.0 6,000 × 6 × 36.16 + 12,183 +
58,280
1,074,420
(8,480,872)
= 482,296lb(54.5kN .m )
1.7λ f 'c bw d p = 1.7 × 1.0 6,000 × 6 × 36.16 = 28,569lb(127kN )
< Vci = 482,296lb
dUcenH Vci = 482,296lb(214.5kN )
@> Web-shear cracking, Vcw ¬CMhanTI2¦
BIsmIkar 5.15
( )
Vcw = 3.5 f 'c + 0.3 f c bw d p + V p
f c = kugRtaMgsgát;enAkñúgebtugRtg; cgc
≅ 818 psi (5.6 MPa )
Pe 308,255
= =
Ac 377
Vp = bgÁúMbBaÄrrbs;eRbkugRtaMgRbsiT§PaBenARtg;muxkat;
= Pe tan θ
Edl θ CamMurvag tendon eRTtCamYynwgbøg;edk. dUcenH
Shear and Torsion Strength Design 256

44. NPIC
V p = 308,255
(15 − 12.5) = 1,976lb(8.8kN )
65 / 2 × 12
dUcenH Vcw = (3.5 6,000 + 0.3 × 818)× 6 × 36.16 + 1,976 = 114,038lb(507kN )
enAkñúgkrNIenH web-shear cracking manlkçN³lub ¬Edl Vc = Vcw = 114,038lb(507kN )
RtUv)aneRbIsMrab;KNnaEdkRTnug¦. eRbobeFobtMélenHCamYynwg Vc = 76,707lb(341kN )
EdlTTYl)anBIsmIkar 5.4 eday alternative method EdlmanlkçN³suvtßiPaBCag.
BIsmIkar 5>4
− Vc = (97,216 − 114,038)lb
Vu
Vs =
φ
dUcenHeKminRtUvkarEdkkgeT elIkElgEt Vu / φ > 12 Vc . dUcenH eyIgKNnatYxageRkaydUc
xageRkam
= 57,019lb(254kN ) < 97,216lb(432kN )
1 114,038
Vc =
2 2
edaysar Vu / φ > 12 Vc b:uEnþ < Vc dUcenHeRbIEdkkgGb,brmaenAkñúgkrNIenH
#> EdkkgGb,brma ¬CMhanTI4¦
BI]TahrN_ 5>4
Av
s
tMrUvkar = 0.0077in.2 / in.
dUcenH sakl,gEdkkg #3 /
eyIgTTYl)an Av = 2 × 0.11 = 0.22in.2 ehIy
= 28.94in.(73cm )
0.22
s=
0.0077
bnÞab;mkeyIgRtYtBinitü Av Gb,brmaEdlCatMéltUcCageKkñúgcMeNamtMélTaMgBIrxageRkam
50bw s
Av =
fy
nig Av =
A ps f pu s
80 f y d p
dp
bw
dUcenHKMlatGnuBaØatGtibrma ≤ 0.75h ≤ 24in. . eRbIEdkkgGkSr U #3 EdlmanKMlat 22in.
KitBIG½kSeTAG½kSelIRbEvg 84in. BIépÞrbs;TMrdUcenAkñúg]TahrN_ 5>4.
muxkat;sMrab;karerobEdk ¬CMhanTI8¦ RtUv)anbgðajenAkñúgrUbTI 5>20.
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 257

45. T.Chhay
14> Design of Web Reinforcement for a PCI Double T-Beam
]TahrN_ 5>6³ Fñwm T DubTMrsamBaØ PCI 12 DT 34 manElVg 70 ft (21.3m) . varg service dead
edayrYbbBa©ÚlTaMgkMralBIelIbEnßm nig service live load WL = 720 plf .
load 200 plf (29.kN / m )
KNnaEdkkgEdlRtUvkaredIm,IkarBar shear cracking enARtg;muxkat;mYyPaKbYnénElVg 17 ft 6in.
(5.3m ) BITMr edayKNna nominal web-shear strength Vc tam detailed design method. ehIy
KNna dowel reinforcement RbsinebIcaM)ac; edaysnμt;faépÞxagelIrbs;Fñwm T cak;Rsab;Gt;eRKIm.
lkçN³muxkat;RtUv)anbgðajenAkñúgrUbTI 5>21 CamYynwgTinñn½yxageRkam³
Shear and Torsion Strength Design 258

46. NPIC
Tinñn½yepSgeTot³
f 'c ¬ebtugcak;Rsab;¦ = 5,000 psi (34.5MPa ) ebtugTMgn;Fmμta
f 'cc ¬ebtugsMrab;cak;kMralxagelI¦ = 3,000 psi (20.7 MPa ) ebtugTMgn;Fmμta
f 'ci = 4,000 psi (27.6MPa )
/
f pu = 270,000 psi (1,862MPa ) low-relaxation steel
f ps = 240,000 psi (1,655MPa )
f pe = 148,000 psi (1,020MPa )
ee = 11.38in.(28.3cm )
ec = 21.77in.(57.2cm )
Aps = strandGgát;p©it 1 / 2in.(12.7mm) cMnYn 18
f yv sMrab;Edkkg = 60,000 psi (414 MPa )
eRbItMélsMrab;RbEvgRbsiT§PaB d p sMrab;muxkat;kNþalElVgk¾dUcCamuxkat;epSgeTot. cMNaMfa bw
sMrab;RTnugTaMgBIr = 2(4.75 + 7.75) / 2 = 12.5in.(32cm) .
dMeNaHRsay³
Wu = 1.2(200 + 1,091) + 1.6 × 720 ≅ 2,615 plf (38.15kN / m )
2,615 × 70
Vu enARtg;épÞrbs;TMr
=
2
= 91,525lb
enARtg; BIépÞrbs;TMr
=
1 ⎛ (35 − 17.5) ⎞
Vn 17 ft 6in. ⎜ 91,525 × ⎟
0.75 ⎝ 35 ⎠
= 61,017lb(271kN )
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 259

47. T.Chhay
!> Flexure-shear cracking Vci ¬CMhanTI2¦
d p = 34 − 25.77 + 21.77 = 30.0in.(76cm )
Pe = 18 × 0.153 × 148,000 = 407,592lb(18,176kN )
enARtg;
e BITMr
17 ft 6in. = 11.38 + (21.77 − 11.38)
17.5
35
= 16.58in.
eRbIlkçN³muxkat;cak;Rsab;sMrab;KNna f ce nig f d dUcerobrab;enAkñúgEpñkTI 5
Pe ⎛ ecb ⎞ 407,592 ⎛ 16.58 × 25.77 ⎞
f ce = − ⎜1 + 2 ⎟ = − ⎜1 + ⎟
Ac ⎝ r ⎠ 978 ⎝ 88.0 ⎠
= 2,440 psi (16.8MPa )
eRbIkugRtaMgsgát;srésxageRkAGnuBaØatdUcxageRkam³
(a) eRbkugRtaMg + bnÞúkGcié®nþy_ ³ fc = 0.45 f 'c
(b) eRbkugRtaMg + bnÞúksrub ¬edayGnuBaØatedayekIneLIg 33% EdlbNþalBI transient
load³ f c = 0.60 f 'c ¦.
cMNaMfaeTaHbICa fce = 0.45 f 'c k¾eday k¾vaminCHT§iBldl;ersIusþg;kMlaMgkat;Edr edaysar
f ce bNþalmkBIEtkMlaMgeRbkugRtaMgb:ueNÑaH ehIyedaysarkarrYmbBa©ÚlTaMgbnÞúkpÞal;)an
kat;bnßyvaeGaytUcCag 0.45 f 'c . dUcenHeyIgman
bnÞúkpÞal; WD = 1,019 plf
WD x(l − x ) 1,019 × 17.5(70 − 17.5)
M 17.5 = = × 12
2 2
= 5,617,238in. − lb(634kN .m )
Mcb 5,617,238 × 25.77
fd = = = 1,682 psi (11.6MPa )
Ic 86,072
(
M cr = Sb 6.0λ f 'c + f ce − f d )
(
= 3,340 6.0 × 1.0 5,000 + 2,440 − 1,682 )
= 3,948,762in. − lb(445kN .m )
eKKYrcMNaMfaemKuN 6.0 enAkñúgsmIkar cracking moment mantMéltUc edaysareKyk
modulus of rupture 7.5 . RbsinebIeKeRbI 7.5 enAkñúgsmIkarxagelI enaH cracking
moment nwgmantMél 4,303,022in. − lb GBa©wgvanwgkat;bnßycMnYnrbs;EdkkgenAkñúgkar
KNnaenH.
kMlaMgKμanemKuNEdlbNþalBIbnÞúlpÞal;KW
Shear and Torsion Strength Design 260

48. NPIC
⎛l ⎞ ⎛ 70 ⎞
Vd = WD ⎜ − x ⎟ = 1,019⎜ − 17.5 ⎟ = 17,833lb
⎝2 ⎠ ⎝ 2 ⎠
WSD = 200 plf
WL = 720 plf
GaMgtg;sIuetbnÞúkxageRkAemKuNKW
WU = 1.2 × 200 + 1.6 × 720 = 1,392 plf (20.4kN / m )
⎛l ⎞ ⎛ 70 ⎞
Vi = WU ⎜ − x ⎟ = 1,392⎜ − 17.5 ⎟ = 24,360lb(108kN )
⎝2 ⎠ ⎝ 2 ⎠
⎛ l − x ⎞ 1,392 × 17.5(70 − 17.5)
M max = WU x⎜ ⎟= × 12
⎝ 2 ⎠ 2
= 7,673,400in. − lb(867 kN .m )
× (M cr ) ≥ 1.7λ f 'c bw d p
Vi
Vci = 0.6λ f 'c bw d p + V p +
M max
= 0.6 × 1.0 × 5,000 × 12.5 × 30.0 + 17,833 +
24,360
(3,948,762)
7,673,400
= 46,279lb(201kN )
1.7λ f 'c bw d p = 1.7 × 1.0 5,000 × 12.5 × 30.0 = 45,078lb < 46,279lb
dUcenH Vci = 46,279lb lub
@> Web-shear cracking, Vcw ¬CMhanTI2¦
= 417 psi (2.9MPa )
Pe 407,592
fc = =
Ac 978
sMrab;bgÁúMbBaÄrrbs;kMlaMgeRbkugRtaMg
VP = Pe tan θ = 407,592
(21.77 − 11.38) = 10,083lb(44.0kN )
70 / 2 × 12
( )
Vcw = 3.5λ f 'c + 0.3 f c bw d p + V p
= (3.5 )
5,000 + 0.3 × 417 12.5 × 30.0 + 10,083
nig Vci = 46,279lb
= 149,803lb
Vc CatMéltUcCageKén Vci nig Vcw dUcenH
Vc = Vci = 46,279lb
#> KNnaEdkkg ¬CMhanTI3-8¦
eyIgman Vc = 46,279lb
dUcenH 1
2
Vc = 23,140lb
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 261

49. T.Chhay
enARtg;muxkat; 17.5 ft BITMr = 61,017lb > Vc > 12 Vc dUcenHeKcaM)ac;KNnaEdkkg.
Vu / φ
RbsinebI Vu / φ < Vc > 12 Vc eKRtUvkarRtwmEtEdkkgGb,brma.
Av
s
tMrUvkar = Vnf − Vc = (Vu f/ φd) − Vc = 61,017 −×46,279
d 60,000 30.0
y p y p
= 0.0082in.2 / in.
edayeRbI d ≅ d p = 30.0in. nig bw = 12.5in.
⎛ Av ⎞ A ps f pu dp 18 × 0.153 270,000 30.0
⎜ ⎟ = = × = 0.0080
⎝ s ⎠ min 80 f y d p bw 80 60,000 × 30.0 12.5
0.75 f 'c = 0.75 5,000 = 53
⎛ Av ⎞ 53bw 53 × 12.5
b¤ ⎜ ⎟ = = = 0.011in.2 / in.
⎝ s ⎠ min fy 60,000
dUcenH tMéltUcCageKéntMélGb,brmaTaMgBIrKW
⎛ Av ⎞
⎜ ⎟ = 0.0080in.2 / in. = 0.010in.2 / ft sMrab;RTnugTaMgBIr b¤ 0.005in.2 / ft
⎝ s ⎠ min
sMrab;RTnugmYy
sakl,g D5 deformed welded wire fabric mYyCYredayKMlat 10in. KitBIG½kReTAG½kS.
KMlatGnuBaØatGtibrmaKW 0.75h ≤ 24in. dUcenHeyIgman
0.75h = 0.75 × 34 = 25.5in.
dUcenH TTYlykEdkkg D5 WWF mYyCYrkñúgmYyRsTab;edaymanKMlat 10in. KitBIG½kSeTA
G½kSenARtg;muxkat;énmYyPaKbYnElVg.
cMNaMfa edayeRbobeFobdMeNaHRsaysMrab; Vci nig Vcw enAkñúg]TahrN_ 5>6 tMélx<s;Cag
eKrbs; Vci sßitenAEk,rTMr ehIyfycuHy:agelOneTArkkNþalElVg enAeBlEdlbMErbMrYlrbs; Vcw
mantMéltUcCag dUcEdleXIjenAkñúgrUbTI 5>13. eKcaM)ac;KNna flexure shear Vci nig web shear
Vcw enAeRcInmuxkat;tambeNþayElVgedIm,IkMNt;karrayEdkkgRbkbedayRbsiT§PaBbMput. kmμviFIkMu
BüÚT½rsMrYlkarKNnatMélTaMgenHelIcenøaHefrNamYy ¬dUcCa 10 énElVg¦ ehIyeKGacsg;düaRkam
1
RsedogKñanwgdüaRkamenAkñúgrUbTI 5>13 EdlbgðajBIbMErbMrYlénersIusþg;kMlaMgkat;rbs;RTnugtam
beNþayElVg.
$> KNna dowel steel sMrab; full composite section Edlman topping bEnßm 2in. ¬CMhanTI9¦
RbsinebIeKbEnßm topping EbbenHeTAelI pretopped section enAeBleRkay.
Shear and Torsion Strength Design 262

50. NPIC
muxkat;enARtg; 12 d p BIépÞrbs;TMr
d p = 30.0 + 2.0 = 32.0in.
Vu enARtg;TMr = 91,525(408kN )
= 1.33 ft (40cm )
1 32.0
dp =
2 2 × 12
h / 2 = 17in. = 1.33 ft
⎛ 35 − 1.33 ⎞
Vu = 91,525 × ⎜ ⎟ = 88,047lb(393kN )
⎝ 35 ⎠
Vnh tMrUvkar V
= u =
φ
88,047
0.75
= 117,393lb(522kN )
bv = 12 ft
htopping = 2in.
BIrUbTI 5>14
Cc = 0.85 f 'cc Atop = 0.85 × 3,000 × 12 × 12 × 2 = 734,400lb(3,267kN )
Ts = Aps f ps = 18 × 0.153 × 240,000 = 660,960lb(2,940kN )
< Cc = 734,400lb
dUcenH Fh = 660,960lb(2,178kN )
70 × 12
lvh = = 420in.(1,067cm )
2
bv = 144in.(366cm )
80bv lvh = 80 × 144 × 420 = 4,838,400lb(21,520kN ) >> 660,960lb
eKminRtUvkar dowel reinforcement edIm,ITak;enAkñúg topping EdlRtUvbEnßmenAeBleRkay
2in. edIm,I)an full composite action eT. eKTTYlykmuxkat;enH enAeBlvamanlkçN³RKb;RKan;
sMrab; flexurl, PaBdab nigtMrUvkarsMrab;karRKb;RKgsñameRbH.
15> Brackets and Corbels
Bracket nig corbel Ca short-haunched cantilever EdllyecjBIépÞxagrbs;ssr b¤
CBa¢aMgebtugedIm,IRTbnÞúkcMcMnucEdlF¶n; b¤Rbtikmμrbs;Fñwm. vaCaFatuy:agsMxan;rbs;eRKOgbgÁúMsMrab;RT
Fñwmcak;Rsab;/ gantry girder nigTMrg;epSgeToténRbB½n§eRKOgbgÁúMcak;Rsab;. ebtugeRbkugRtaMg nigeb
tugcak;Rsab;RtUv)aneRbIy:ageRcIn nigekIneLIgy:agrh½s ehIyElVgkan;EtEvgRtUv)ansagsg; Edl
karKNnaersIusþg;kMlaMgkat; nigersIusþg;kMlaMgrmYl 263