Xv design for torsion

Department of Civil Engineering viTüasßanCatiBhubec©keTskm<úCa

XV. karKNnasMrab;kMlaMgrmYl
Design for Torsion

1> esckþIepþIm Introduction
kugRtaMgrmYlekItmanenAkñúgmuxkat;FñwmenAeBlEdlm:Um:g;manGMeBIRsbeTAnwgmuxkat;enaH.
m:Um:g;rmYleFVIeGayGgát;vil nigmansñameRbHenAelIépÞrbs;va CaTUeTAEtgekItmanenAelImuxkat;mUl.
edIm,IbgðajkugRtaMgrmYl eKGnuvtþkMlaMgrmYl T Fñwm cantilever muxkat;mUlEdleFVIBI elastic
homogenous material dUcbgðajkñúgrUbTI 15>1. kMlaMgrmYlnwgeFVIeGayFñwmvil. cMnuc B clt½eTA

cMnuc B' enAxagcugrbs;Fñwm b:uEnþcugmçageTotrbs;FñwmRtUv)anbgáb;. mMu θ RtUv)aneKehAfa mMurmYl
(angle of twist). bøg; AO' OB nwgdUrrageTACarag AO'OB' . edaysnμt;fa Ggát;enArkSaRbEvgrbs;

vadEdl enaH shear strain KW
BB' rθ
γ = =
L L
Edl L CaRbEvgrbs;Fñwm nig r CakaMrbs;muxkat;rgVg;.

enAkñμúgeRKOgbgÁúMebtugGarem: Ggát;nwgrgm:Um:g;rmYlenAeBlGgát;enaHekagenAkñúgbøg;/ RT
cantilever slab/ mannaTICa spandrel beam (end beam)/ b¤CaEpñkrbs;CeNþIrvil.

Design for Torsion 346

T.Chhay NPIC

Ggát;eRKOgbgÁúMGacrgnUvEtkMlaMgrmYlsuT§ b¤enAkñúgkrNICaeRcIn vargCamYyKñakñúgeBlEtmYy
nUvkMlaMgkat;TTwg nigm:Um:g;Bt;. ]TahrN_TI15>1 bgðajBIkMlaMgepSg²EdlGacGnuvtþmkelImuxkat;
epSgKñaénFñwm cantilever.
]TahrN_TI15>1³
KNnakMlaMgEdlmanGMeBIenAmuxkat; !/ @ nig # énFñwm cantilever EdlbgðajenAkñúgrUbTI 15>2. Fñwm
rgnUvkMlaMgbBaÄr P1 = 67kN / kMlaMgedk P2 = 53.5kN EdleFVIGMeBIenAcMnuc C nigbnÞúkedk
P3 = 89kN EdlGnuvtþenAcMnuc B nigEkgeTAnwgTisedArbs;kMlaMg P2 .

dMeNaHRsay³
yk N = kMlaMgEkg (normal force)/ V = kMlaMgkat; (shear force)/ M = m:Um:g;Bt; (bending
moment)/ T =m:Um:g;rmYl (torsional moment). kMlaMgTaMgGs;RtUv)anbgðajenAkñúgtaragxageRkam³

muxkat; N (kN ) M x (kN .m) M y (kN .m) V x (kN ) V y (kN ) T (kN .m)
! 0 − 180.9 144.45 53.5 67 0
@ − 53.5 ¬sgát;¦ 0 144.45 89 67 180.9
# − 53.5 ¬sgát;¦ 241.2 464.85 89 67 180.9

karKNnasMrab;kMlaMgrmYl 347


RbsinebI P1 / P2 nig P3 CabnÞúkemKuN ¬ Pu = 1.2PD + 1.6PL ¦ enaHral;tMélenAkñúgtaragCakMlaMg
KNnaemKuN.
2> m:Um:g;rmYlenAkñúgFñwm Torsional Moments in Beams
dUcbgðajenAkñúgrUbTI 15>1 kMlaMgGacGnuvtþenAelIeRKagsMNg;GKar edayeFVIeGaymanm:Um:g;
rmYl. RbsinebIkMlaMgcMcMnuc P GnuvtþenARtg;cMnuc C enAelIeRKag ABC dUcbgðajenAkñúgrUb 15>3 a
vabegáItm:Um:g;rmYl T = PZ enAkñúgFñwm AB Rtg;cMnuc D . enAeBl D sßitenAkNþalElVgénFñwm AB
enaHm:Um:g;rmYlKNnaenAkñúgkMNat; AD esμInwgm:Um:g;rmYlKNnaenAkñúgkMNat; DB b¤esμInwg 1 T . Rb
2
sinebIkMral cantilever slab RtUv)anRTedayFñwm AB ¬rUbTI 15>3 b¦ enaHkMralxNÐ½begáItm:Um:g;rmYl
BRgayesμI mt tambeNþayFñwm AB . m:Um:g;rmYlBRgayesμIenH KWekItBIbnÞúkenAelIcMerokTTwkmYy
Éktþarbs;kMralxNÐ½. RbsinebI S CaTTwgén cantilever slab nig w CabnÞúkenAelIkMralxNÐ½
¬ kN / m 2 ¦ enaH mt = wS 2 / 2 ¬ kN .m / m ¦énFñwm AB . m:Um:g;rmYlKNnaGtibrmaenAkñúgFñwm AB KW
T = mt L / 2 EdlGnuvtþenARtg;cMnuc A nig B . krNIbnÞúkepSgeTotRtUv)anbgðajenAkñúgtarag

15>1. CaTUeTA düaRkamm:Um:g;rmYlenAkñúgFñwmmanrag nigmantMéldUcKñanwgdüaRkamkMlaMgkat;TTWgsM
rab;FñwmEdlrgnUvkMlaMg mt nig T .


T.Chhay NPIC



3> kugRtaMgrmYl Torsional Moments in Beams

edayBicarNaelIFñwm cantilever Edlmanmuxkat;mUl ¬rUbTI15>1¦ enAeBlEdlm:Um:g;rmYl T
manGMeBIelIFñwm vanwgbegáIteGaymankMlaMgkat;TTwg dV EkgeTAnwgkaMrbs;muxkat;. BIlkçxNÐl<nwg
m:Um:g;rmYlxageRkARtUv)anTb;edaym:Um:g;rmYlxagkñúgEdlmantMél T esμIKñaEtTisedApÞúyKña . Rbsin
ebI dV CakMlaMgkat;TTwgeFVIGMeBIelIépÞ dA ¬rUbTI 15>4¦ enaHGaMgtg;sIueténkMlaMgrmYlKW
T = ∫ rdV

edayyk v CakugRtaMgkMlaMgkat;TTWgenaH
dV = vdA nig T = ∫ rvdA
kMlaMgkat;TTwgeGLasÞicGtibrmaekItmanenAépÞxageRkArbs;muxkat;rgVg;Rtg;kaM r CamYynwgkM
ras; dr enaHkMlaMgrmYlGacRtUv)ankMNt;edayKitm:Um:g;eFobnwgcMnuc 0 sMrab;RkLaépÞkg³
dT = (2πrdr )vr
Edl 2πrdr CaRkLaépÞkg nig v CakugRtaMgkMlaMgkat;TTwgenAkñúgkg. dUcenH
T = ∫ (2πrdr )vr = ∫ 2πr 2 dr
R
0
R
0
¬!%>!¦
sMrab;muxkat;RbehagEdlmankaMxagkñúg R1 /
R
T = ∫ 2πr 2 dr
R1
¬!%>@¦
sMrab;muxkat;tan; edayeRbIsmIkar ¬!%>!¦ nig v = vmax r / R
R ⎛v r⎞ ⎛ 2π ⎞ R 3
T = ∫ 2πr 2 ⎜ max ⎟dr = ⎜ ⎟vmax ∫0 r dr
0 ⎝ R ⎠ ⎝ R ⎠
⎛ 2π ⎞ ⎛π ⎞
4
R
=⎜ ⎟vmax = ⎜ ⎟vmax R 3
⎝ R ⎠ 4 ⎝2⎠


T.Chhay NPIC

vmax =
2T
πR 3
¬!%>#¦
m:Um:g;niclPaBb:UElrénmuxkat;rgVg;KW J = πR 4 / 2 . dUcenH kugRtaMgkMlaMgkat;GacRtUv)an
sresrCaGnuKmn_énm:Um:g;niclPaBb:UElrdUcxageRkam³
vmax =
TR
J
¬!%>$¦
4> m:Um:g;rmYlenAkñúgmuxkat;ctuekaN Torsional Moments in Rectangular Sections
karKNnakugRtaMgenAkñúgGgát;manmuxkat;minmUlEdlrgbnÞúkrmYlminsamBaØdUckarKNnasM
rab;muxkat;mUleT. b:uEnþ lT§plEdlTTYlBIRTwsþIeGLasÞic (theory of elasticity) bgðajfakugRtaMg
kMlaMgkat;TTwgGtibrmasMrab;muxkat;ctuekaNEkgGacRtUv)ankMNt;dUcxageRkam³
vmax = 2
T
αx y
¬!%>%¦
Edl T= kMlaMgrmYlEdlGnuvtþ
x = RCugxøIrbs;muxkat;ctuekaN

y = RCugEvgrbs;muxkat;ctuekaN

α = emKuNEdlGaRs½ynwgpleFobén y / x tMélrbs;vaRtUv)aneGayenAkñúgtarag
xageRkam.
y/x 1 .0 1 .2 1 .5 2 .0 4 .0 10

α 0.208 0.219 0.231 0.246 0.282 0.312
kugRtaMgkMlaMgkat;TTwgGtibrmaekItmanenAtamGkS½énRCugEvg y ¬rUbTI 15>5¦.
sMrab;Ggát;EdlekItBIkarpÁúMénmuxkat;ctuekaNEkg dUcCamuxkat;GkSr L / T nig I tMél α
GacRtUv)ansnμt;faesμInwg 1/ 3 ehIymuxkat;GacRtUv)anEckecjCamuxkat;ctuekaNCaeRcInEdlman
RCugEvg yi nigRCugxøI xi . kugRtaMgkMlaMgkat;TTwgGacRtUv)anKNnaBI
vmax =
3T
¬!%>^¦
∑ i ix2 y

Edl ∑ xi2 y i CatMélEdl)anBIplbUkmuxkat;ctuekaNEkgtUc². enAeBlEdl y / x ≤ 10 eK
GaceRbIsmIkarsMrYlxageRkam³



v max =
3T
¬!%>&¦
⎛ x⎞
∑ x 2 y⎜1 − 0.63
⎜
⎟
y⎟
⎝ ⎠

5> kMlaMgpÁÜbrvagkMlaMgkat; nigkMlaMgrmYl Combined Shear and Torsion

enAkñúgkrNIGnuvtþn_CaeRcIn Ggát;eRKOgbgÁúMGacrgnUvTaMgkMlaMgkat; nigkMlaMgrmYlCamYyKña.
kugRtaMgkMlaMgkat;GacnwgekItmanenAkñúgmuxkat;CamYynwgkugRtaMgkMlaMgkat;mFüm = v1 enAkñúgTis

T.Chhay NPIC

edAénkMlaMgkat; V ¬rUbTI 15>6 a¦. kMlaMgrmYl T begáItkugRtaMgrmYlenAelIRKb;RCugrbs;muxkat;
ctuekaN ABCD ¬ rUbTI 15>6 a¦ CamYynig v3 > v2 . karBRgaykugRtaMgcugeRkayRtUv)anTTYlBI
karbUkbBa©ÚlnUvT§iBlénkugRtaMgkMlaMgkat; nigkugRtaMgrmYl edIm,IbegáIttMélGtibrmaesμI v1 + v3 enA
elIRCug CD b:uEnþRCug AB nwgmankugRtaMgcugeRkayesμI v1 − v3 . RCug AD nig BC nwgrgEtkugRtaMg
rmYl v2 . muxkat;RtUvd)anKNnasMrab;kugRtaMgGtibrma v = (v1 + v3 ) .
6> RTwsþIkarrmYlsMrab;Ggát;ebtug Torsion Theories for Concrete Members
eKmanviFICaeRcInsMrab;viPaKGgát;ebtugBRgwgedayEdkEdlrgkarrmYl b¤rgkarrmYl karBt;
nigkarkat;kñúgeBlEtmYy. CaTUeTAviFIKNnasMGageTAelIRTwsþIeKalBIrKW³ the skew bending theory
nig space truss analogy.
6>1> Skew Bending Theory
viFIeKalrbs; skew bending theory EdlENnaMeday Hsu CaviFIEdlsikSakar)ak;énmuxkat;
ctuekaNedaykarrmYlEdlekItedaykarBt;eFobGkS½RsbeTAnwgépÞénmuxkat; y FMCag nigeRTteday
mMu 45o eTAnwgGkS½EvgénFñwm ¬rUbTI 15>7¦. QrelIviFIsaRsþenH m:Um:g;rmYlGb,brma Tn GacRtUv)an
KNnadUcxageRkam³

⎛ x2 y ⎞
Tn = ⎜
⎜ 3 ⎟ r
⎟f ¬!%>*¦
⎝ ⎠
Edl f r KWm:UDuldac;rbs;ebtug. f r RtUv)ansnμt;esμInwg 5 f 'c / 12 enAkñúgkrNIenH Edl
RtUv)aneRbobeFobCamYy 7.5 f 'c /12 EdlTTYleday ACI Code sMrab;KNnaPaBdabenAkñúgFñwm.



kMlaMgrmYlTb;edayebtugsMEdgdUcxageRkam³
⎛ 1 ⎞ 2
Tc = ⎜ ⎟ x y f 'c ¬!%>(¦
⎝ x⎠
nigkMlaMgTb;karrmYledayEdkTb;karrmYlKW
α1 ( x1 y1 At f y )
Ts =
s
¬!%>!0¦
dUcenH Tn = Tc + Ts Edl Tn lT§PaBTb;m:Um:g;rmYl nominal énmuxkat;.
6>2> Space Truss Analogy

viFIsaRsþén space truss analogy KWQrelIkarsnμt;falT§PaBTb;Tl;karrmYlrbs;ebtugGar
em:muxkat;ctuekaNRtUv)anKitecjEtBIEdknigebtugEdlBT§½CMuvijEdkb:ueNÑaH. kñúgkrNIenH muxkat;
thin-wall RtUv)ansnμt;mannaTICa space truss ¬rUbTI 15>8¦. cMerokebtugvNÐ½eRTtcenøaHsñameRbH

Tb;kMlaMgsgát; b:uEnþEdkbeNþayenARCug nigEdkkgTb;nwgkMlaMgTajEdlekItedaym:Um:g;rmYl.
kareFVIkarrbs;FñwmebtugGarem:EdlrgkarrmYlsuT§GacbgðajedayRkaPicénTMnak;TMngrvagkar
rmYlnigmMurmYl dUcbgðajenAkñúgrUbTI15>9. eyIgemIleXIjfa muxnwgeRbH ebtugTb;nwgkugRtaMgrmYl
nigEdkswgEtKμanrgkugRtaMg. eRkayeBleRbH kareFVIkarrbs;FñwmCalkçN³eGLasÞicminGacGnuvtþ)an
dUcenHmMurmYlekIteLIgPøam² EdlekIneLIgrhUtdl;lT§PaBTb;Tl;m:Um:g;rmYlekItman. karkMNt;Edl
manlkçN³Rbhak;RbEhlénlT§PaBTb;karrmYlsMrab;muxkat;eRbHGacnwgsMEdgdUcxageRkam³
⎛A f ⎞
Tn = 2⎜ t s ⎟ x1 y1
⎝ s ⎠
¬!%>!!¦
Edl At = éneCIgmçagrbs;Edkkg


T.Chhay NPIC

s=KMlatEdkkg
x1 nig y1 = RbEvgxøI nigRbEvgEvg KitBIGkS½eTAGkS½énEdkkgbiTCit b¤BIEdkenARCug.

smIkarmunecalnUvlT§PaBTb;karrmYlrbs;ebtug. Mitchell nig Collins ENnaMnUvsmIkarxag
eRkamedIm,IKNnamMurmYlkñúgmYyÉktþaRbEvg ψ ³
⎛ P ⎞ ⎡⎛ ε ⎞ ⎛ P (ε tan α ) ⎞ 2ε d ⎤
ψ = ⎜ o ⎟ ⎢⎜ l ⎟ + ⎜ h h
⎜ 2 A ⎟ tan α ⎜ ⎟+
⎟ sin α ⎥ ¬!%>!@¦
⎝ o ⎠ ⎣⎝ P ⎠ ⎝ o ⎠ ⎦
Edl εl = bMErbMrYlrageFob (strain) enAkñúgEdkbeNþay (longitudinal reinforcing steel)
ε h = bMErbMrYlrageFobenAkñúgEdkkg (hoop steel)
ε d = bMErbMrYlrageFobebtugGgát;RTUgenARtg;TItaMgénkMlaMgpÁÜbénFarkMlaMgkat;
(shear flow)
Ph = brimaRtrbs;EdkkgKitRtwmGkS½Edk
⎡ ⎛ P ⎞⎤
α = mMuénkMlaMgsgát;Ggát;RTUg = (ε d + ε l ) / ⎢ε d + ε h ⎜ h ⎟⎥
⎜P ⎟
⎣ ⎝ o ⎠⎦
Ao =RkLaépÞEdlBT§½CMuvijedaykMlaMgkat; b¤
= torque / 2q ¬Edl q = FarkMlaMgkat;¦

Po = brimaRténKnøgFarkMlaMgkat; ¬brimaRtrbs; Ao ¦

smIkarmMurmYlxagelImanlkçN³RsedogKñanwgsmIkarmMukMeNagkñúgkarBt; ¬rUbTI 15>10¦
ε + εs
φ = curvature = c
d
¬!%>!#¦



Edl ε c nig ε s CabMErbMrYlrageFobenAkñúgebtug nigEdk erogKña. smIkard¾samBaØRtUv)anbk
Rsayeday Solanki edIm,IkMNt;lT§PaBTb;nwgkarrmYlsuT§rbs;FñwmebtugGarem: edayQrelI space
truss analogy dUcxageRkam³
1
⎡⎛ ∑ As f sy ⎞ ⎛ Ah f hy ⎞⎤ 2
Tu = (2 Ao )⎢⎜
⎜
⎟×⎜
⎟ ⎜
⎟⎥
⎟ ¬!%>!$¦
⎢⎝ Po
⎣ ⎠ ⎝ s ⎠⎥⎦
Edl / nig s RtUv)anBnül;BIxagelI
Ao Po

∑ As f sy = kMlaMg yield énEdkbeNþayTaMgGs;
Ah f hy = kMlaMg yield énEdkkg

ACI Code )anTTYlykRTwsþIenHedIm,IKNnaGgát;eRKOgbgÁúMebtugEdlrgkarrmYl b¤karrmYl

nigkarkat; enAkúñgviFIsaRsþd¾sMrYl.

7> ersIusþg;rmYlénGgát;ebtugsuT§ Torsional Strength of Plain Concrete Memgers
Ggát;ebtugrgkarrmYlCaTUeTARtUv)anBRgwgedayEdkTb;nwgkarrmYlBiess. kñúgkrNIEdlkug
RtaMgrmYlmantMéltUc nigRtUvkarKNnasMrab;Ggát;ebtugsuT§ kugRtaMgkMlaMgkat; vtc GacRtUv)ankMNt;
edayeRbIsmIkar !%>^³
3T f 'c
vtc = ≤
φ∑ x y 2 2


T.Chhay NPIC

nigmMurmYlKW θ = 3TL / x3 yG / Edl T Cam:Um:g;rmYlEdlGnuvtþmkelImuxkat; ¬tUcCagm:Um:g;
rmYlEdleFVIeGayeRbH¦ nig G KWCam:UDulkMlaMgkat; nigGacRtUv)ansnμt;esμInwg 0.45 dgénm:UDuleG-
LasÞicrbs;ebtug Ec Edl G = 2135 f 'c . kMlaMgkat;TTwgeFVIeGayeRbHedaysarkarrmYl
(torsional cracking shear) vc enAkñúgebtugsuT§GacRtUv)ansnμt;esμI 0.5 f 'c . dUcenH sMrab;muxkat;

ctuekaNebtugsuT§
φ 2
Tc =
12
x y f 'c ¬!%>!%¦
nigsMrab;muxkat;EdlpSMeLIgedayctuekaNEkgeRcIn
φ
Tc =
12
f 'c ∑ x 2 y ¬!%>!^¦
8> karrmYlenAkñúgGgát;ebtugBRgwgedayEdk Torsion in Reinforced Concrete
Memebers (ACI Code Procedure)
8>1> sBaØaNTUeTA General
dMeNIrkarKNnasMrab;karrmYlmanlkçN³RsedogKñaeTAnwgkMlaMgkat;TTwgedaykarBt;. enA
eBlEdlm:Um:g;rmYlemKuNGnuvtþenAelImuxkat;FMCaglT§PaBTb;m:Um:g;rmYlkñúgrbs;ebtugGacTb;)an
enaHsñameRbHEdlekItedaykarrmYl (torsional crack) ekIteLIg dUcenHEdkTb;karrmYl (torsional
reinforcement) kñúgTMrg;CaEdkkgbiTCit (closed stirrup or hoop reinforcement) RtUv)andak;.

bEnßmBIelIEdkkgbiTCit EdkbeNþayk¾RtUv)andak;enAtamRCugrbs;Edkkg nigRtUv)anBRgayy:ag
l¥enACMuvijmuxkat;. TaMgEdkkgbiTCit nigEdkbeNþaymansarsMxan;Nas;kñúgkarTb;nwgkMlaMgTaj
Ggát;RTUgEdlbNþaymkBIkMlaMgrmYl EdkEtmYyRbePTnwgKμanRbsiT§PaBeTebIKμanEdkmYyRbePT
eTot. EdkkgRtUvEtbiTCit edaysarkugRtaMgrmYlekItmanenARKb;RCugrbs;muxkat;.
EdkcaM)ac;sMrab;karrmYlRtUv)anbEnßmelIEdkcaM)ac;sMrab;kMlaMgkat;/ sMrab;karBt; nigkMlaMg
tamGkS½. EdkEdkcaM)ac;sMrab;karrmYlRtUv)andak;edIm,IeFVIeGayersIusþg;m:Um:g;rmYlrbs;muxkat; φTn
FMCagb¤esμInwgm:Um:g;rmYlemKuN Tu EdlRtUv)anKNnaBIbnÞúkemKuN.
φTn ≥ Tu ¬!%>!&¦
enAeBleKRtUvkarEdkTb;karrmYl ersIusþg;m:Um:g;rmYl φTn RtUv)anKNnaedaysnμt;kMlaMg
rmYl Tu TaMgGs; RtUv)anTb;edayEdkkg nigEdkbeNþayCamYynwgersIusþg;Tb;karrmYlrbs;ebtug



Tc = 0 . kñúgeBlCamYyKña ersIusþg;kMlaMgkat;EdlTb;edayebtug vc RtUv)ansnμt;enAdEdledayKμan
karERbRbYledaysarvtþmanrbs;ersIusþg;rmYl.
8>2> )a:ra:Em:RtFrNImaRténkarrmYl Torsional Geometric Parameters

enAkñúg ACI Code, Section 11.6 karKNnasMrab;karrmYlKWQrenAelI space truss analogy
dUcbgðajenAkñúgrUbTI 15>8. eRkayeBlEdlsñameRbHedaykarrmYlekIteLIg karrmYlRtUv)anTb;
edayEdkkgbiTCit EdkbeNþay nigersIusþg;kMlaMgsgát;Ggát;RTUgrbs;ebtug. sac;ebtugenAxageRkA
EdkkgkøayeTACaKμanRbsiT§PaB nigRtUv)anecalenAkñúgkarKNna. RkLaépÞBT§½CMuvijedayGkS½énEdk


T.Chhay NPIC

kgbiTCitxageRkAbMput RtUv)ankMNt;eday Aoh ¬épÞqUtenAkñúgrUbTI 15>11¦. edaysarGgÁdéTeTot
RtUv)aneRbIenAkñúgsmIkarKNna vak¾RtUv)anENnaMCadMbUgenATIenHedIm,ICYyeGaykaryl;nUvsmIkarman
lkçN³gayRsYl. BIrUbTI 15>11 GgÁEdleGayRtUv)ankMNt;dUcxageRkam³
Acp = RkLaépÞmuxkat;ebtugEdlBT§½CMuvijedaybrimaRtxageRkAénmuxkat;ebtug

Pcp = brimaRténmuxkat;ebtugTaMgmUl Acp

Aoh = RkLaépÞEdlBT§½CMuvijedayGkS½énEdkrgkarrmYlTTwgbiTCitxageRkAbMput ¬épÞqUtkñúgrUbTI

15>11¦
Ao = RkLaépÞEdlBT§½CMuvijedayKnøgFarkMlaMgkat;TTwg nigGacykesμInwg 0.85 Aoh

Ph = brimaRtebtugrbs;EdkrgkarrmYlTTwgbiTCitxageRkAbMput

θ = mMuénkMlaMgsgát;Ggát;RTUgcenøaH 30 o eTA 60 o ¬b¤GacykesμInwg 45o sMrab;Ggát;ebtugGarem:¦
sMrab;muxkat;GkSr T nig L TTwgRbsiT§PaBénsøabmçag²RtUv)ankMNt;esμInwgkMBs;FñwmEdl
sßitenABIelI b¤BIeRkamkMralxNÐ½ edayykmYyNaEdlFMCag b:uEnþminRtUvFMCag 4 dgkMras;kMralxNÐ½eT
¬ACI Code, Sections 11.6.1 and 13.2.4¦.
8>3> m:Um:g;rmYleFVIeGayeRbH Tcr Cracking Torsional Moment Tcr
m:Um:g;eFVIeGayeRbHeRkamm:Um:g;rmYlsuT§ Tcr GacRtUv)anTajecjedayCMnYsmuxkat;BitR)akd
munnwgeRbH CamYynwg thin-walled tube smmUl t = 0.75 Acp / Pcp / CamYynwgRkLaépÞEdlBT§½CMuvij
edayGkS½CBa¢aMg A0 = 2 Acp / 3 . enAeBlEdl kugRtaMgTajGtibrma ¬kugRtaMgem¦ mantMélesμI
f 'c / 3 sñameRbHnwgekItman ehIyCaTUeTAm:Um:g;rmYl T esμInwg

T = 2 Aoτt ¬!%>!*¦
Edl τ = kugRtaMgkMlaMgkat;edaykarrmYl = f 'c / 3 sMrab;sñameRbHedaykarrmYl.
CMnYs τ eday f 'c / 3
f 'c ⎛ Acp ⎞
2
Tcr =
3
⎜
⎜P ⎟ n
⎟ =T nig Tu = φTcr ¬!%>!(¦
⎝ cp ⎠
edaysnμt;fam:Um:g;rmYltUcCagb¤esμInwg Tcr / 4
nwgmineFVIeGaymankarkat;bnßyersIusþg;Tb;karBt; b¤Tb;kMlaMgkat;enAkñúgGgát;énrcnasm<n§½ ACI
Code, Section 11.6.1
GnuBaØateGayecalnUvT§iBlm:Um:g;rmYlenAkñúgGgát;ebtugGarem:enAeBlEdlm:Um:g;rmYlemKuN
Tu ≤ φTcr / 4 b¤



f 'c ⎛ Acp ⎞
2
Tu ≤ φ
12
⎜ ⎟=T
⎜ Pcp ⎟ a ¬!%>@0¦
⎝ ⎠
enAeBlEdl Tu FMCagtMélenAkñúgsmIkar !%>@0 Tu TaMgGs;RtUv)anTb;edayEdkkgbiTCit
nigEdkbeNþay. m:Um:g;rmYl Tu RtUv)anKNnaBImuxkat;EdlmanTItaMgRtg;cMgay d BIépÞénTMr nig
Tu = φTn Edl φ = 0.75 .

]TahrN_TI15>1³
sMrab;muxkat;bIEdlbgðajenAkñúgrUbTI 15>12 nigQrelIkarkMNt; ACI Code cUrkMNt;
a. m:Um:g;eFVIeGayeRbH φTcr

b. m:Um:g;rmYlemKuNGtibrma φTn EdlGacGnuvtþelImuxkat;nImYy²edaymineRbIEdkRTnugTb;kar

rmYl.
snμt; f 'c = 28MPa / f y = 400MPa / kMras;ebtugkarBarEdk 40mm nigeRbIEdkkg DB12 .

dMeNaHRsay³
1> muxkat; !
a. mU:m:g;eFVIeGayeRbH φTcr GacRtUv)anKNnaBIsmIkar !%>!(
f 'c ⎛ Acp ⎞
2
φTcr = φ ⎜ ⎟
3 ⎜ Pcp ⎟
⎝ ⎠


T.Chhay NPIC

sMrab;muxkat;enH Acp = xo yo RkLaépÞmuxkat;TaMgmUl Edl xo = 400mm nig
yo = 610mm
Acp = 400 × 610 = 244000mm 2

Pcp = brimaRténmuxkat;ebtugTaMgmUl
= 2( xo + yo ) = 2(400 + 610 ) = 2020mm
28 ⎛ 244000 2 ⎞
⎜ ⎟ = 39kN .m
φTcr = 0.75
3 ⎜ 2020 ⎟
⎝ ⎠
b. φTn GnuBaØatEdlGacGnuvtþedaymineRbIEdkTb;karrmYlRtUv)anKNnaBIsmIkar !%>@0
φTcr 39
Ta = = = 9.75kN .m
4 4
2> muxkat; @
a. dMbUgKNna Acp nig Pcp sMrab;muxkat;enH nigGnuvtþsmIkarTI !%>!( edIm,IKNna φTcr .
edaysnμt;søabRtUv)andak;CamYyEdkkgbiTCit enaHsøabRbsiT§PaBEdlRtUveRbIenA
RCugmçag²énRTnugesμInwg $dgkMras;søab b¤ 4(100) = 400mm = hw = 400mm
Acp = web area + area of effective flanges

Acp = 500 × 350 + 2 ×100 × 400 = 255000mm 2

Pcp = 2(b + h ) = 2(350 + 2 × 400 + 500) = 3300mm
28 ⎛ 255000 2 ⎞
⎜ ⎟ = 26kN .m
φTcr = 0.75
3 ⎜ 3300 ⎟
⎝ ⎠
cMNaM³ RbsinebIsøabRtUv)anecal ehIyEdkTb;karrmYlRtUv)andak;EtenAkñúgRTnug
enaH
Acp = 350 × 500 = 175000mm 2

Pcp = 2(350 + 500) = 1700mm

φTcr = 23.8kN .m
b. φTn GnuBaØatEdlGacGnuvtþedaymineRbIEdkTb;karrmYl
φTcr 26
Ta = = = 6.5kN .m
4 4
3> muxkat; 3



a. snμt;søabRtUv)andak;EdkkgbiTCit RbEcgRbsiT§PaBesμInwg
hw = 370mm < 4 ×150 = 600mm
Acp = 350 × 520 + 370 ×150 = 237500mm 2

Pcp = 2(b + h) = 2(350 + 370 + 520) = 2480mm
28 ⎛ 237500 2 ⎞
⎜ ⎟ = 30kN .m
φTcr = 0.75
3 ⎜ 2480 ⎟
⎝ ⎠
cMNaM³ RbsinebIsøabRtUv)anecal enaH
Acp = 350 × 520 = 182000mm 2

Pcp = 2(350 + 520) = 1740mm

φTcr = 25.2kN .m
b. φTn GnuBaØat φTn = φTcr = 30 = 7.5kN.m
4 4
8>4> m:Um:g;rmYllMnwg nwgm:Um:g;rmYlRtUvKña Equilibrium Torsion and Compatibility Torsion
kñúgkarviPaKeRKOgbgÁáúMGgát;ebtug kMlaMgepSg²EdlGnuvtþrYmman kMlaMgEkg (normal force)/
m:Um:g;Bt; (bending moment)/ kMlaMgkat; (shear force) nigm:Um:g;rmYl Edl)anBnül;enAkñúg]TahrN_
TI 15>1. karKNnaGgát;ebtugGarem:KWQrelIkar)ak;rbs;Ggát;GMeBIrbs;bnÞúkemKuN. sMrab;Ggát;sþa
TicminkMNt; (statically indeterminate member) karEbgEckm:Um:g;mþgeTot (redistribution of
moments) ekItmanmuneBl)ak; dUcenHm:Um:g;KNnaGacnwgRtUv)ankat;bnßy b:uEnþ sMrab;Ggát;sþaTickM

Nt; (statically determinate member) dUcCaFñwmsamBaØ (simple beam) b¤Fñwm cantilever Kμankar
EbgEckm:Um:g;mþgeTotekIteLIgeT.
enAkñúgkarKNnaGgát;Edlrgm:Um:g;rmYl eKmanBIrkrNIEdlGacGnuvtþbnÞab;BIkareRbH.
!> krNIm:Um:g;rmYllMnwg (equilibrium torsion case) ekItmanenAeBlm:Um:g;rmYlEdlRtUvkar
sMrab;eRKOgbgÁúMsßitkñúgsßanPaBlMnwg ehIy Tu minGacRtUv)ankat;bnßyedaykarEbg
EckeLIgvijrbs;m:Um:g;eT dUckrNIFñwmTMrsamBaØ. kñúgkrNIenHEdkTb;rmYlRtUv)andak;
edIm,ITb;RKb; Tu . rUbTI 15>13 FñwmEdlenAEKmRTkMralxNÐ½ cantilever EdlKμankar
EbgEckm:Um:g;mþgeTotekItman.
@> krNIm:Um:g;rmYlRtUvKña (compatibility torsion case) ekItmanenAeBlm:Um:g;rmYl Tu Gac
RtUv)ankat;bnßyedaykarEbgEckkMlaMgkñúgmþgeTotbnÞab;BIeRbH enAeBlEdlPaBRtUvKña


T.Chhay NPIC

énkMhUcRTg;RTay (compatibility of deformation) RtUv)anrkSa enAkñúgGgát;eRKOgbgÁúM.
rUbTI 15>14 bgðajBI]TahrN_sMrab;krNIenH EdlFñwmTTwgBIrmanGMeBIelIFñwmEKmbegáIt
m:Um:g;rmYl. mMurmYlFMekItman enAeBlsñameRbHedaykarrmYlelcecj Edlpþl;nUvkar
bgEckbnÞúkd¾FMenAkñúgeRKOgbgÁúM. vanwgeTAdl;m:Um:g;rmYlEdleFVIeGayeRbH Tcr eRkamGM
eBIénbnSM karBt; karkat; nigkarrmYl enAeBlEdlkugRtaMgem (principle stress) mantM
élRbEhl f 'c / 3 . enAeBlEdl Tu > Tcr m:Um:g;rmYlesμInwg Tcr ¬smIkar !%>!(¦
EdlGacsnμt;ekItmanenARtg;muxkat;eRKaHfñak;enACitépÞénTMr.
ACI Code kMNt;m:Um:g;rmYlKNnaesμInwgtMéltUcCageKén Tu Edl)anBIbnÞúkemKuN b¤ φTcr

BIsmIkar !%>!(.

8>5> karkMNt;énersIusþg;m:Um:g;rmYl Limitation of Tortional Moment Strength
ACI Code,Section 11.6.3 kMNt;TMhMmuxkat;edaysmIkarxageRkamBIr³

!> sMrab;muxkat;tan;



2 2
⎛ Vu ⎞ ⎛ Tu Ph ⎞ ⎡⎛ ⎞ ⎤
⎜ ⎟ +⎜ ⎟ ≤ φ ⎢⎜ Vc ⎟ + 2
⎜ b d ⎟ ⎜ 1 .7 A 2 ⎟ ⎜ ⎟ f 'c ⎥ ¬!%>@!¦
⎝ w ⎠ ⎝ oh ⎠ ⎣⎝ bw d ⎠ 3 ⎦
@> sMrab;muxkat;Rbehag
⎛ Vu ⎞ ⎛ Tu Ph ⎞ ⎡⎛ ⎞ ⎤
⎜ ⎟+⎜
⎜ b d ⎟ ⎜ 1.7 A2 ⎟
⎟ ≤ φ ⎢⎜ Vc ⎟ + 2
⎜b d ⎟ 3 f 'c ⎥ ¬!%>@@¦
⎝ w ⎠ ⎝ oh ⎠ ⎣⎝ w ⎠ ⎦
Edl Vc = f 'c bwd / 6 = ersIusþg;kMlaMgkat;sMrab;ebtugTMgn;Fmμta. GgÁdéTeTotRtUv)ankM
Nt;enAkñúgEpñk 8>2.
karkMNt;enHKWQrelIPaBCak;EsþgEdlfaplbUkénkugRtaMgEdlbNþalBIkMlaMgkat; nigm:Um:g;
rmYl ¬GgÁxageqVg¦ minRtUvFMCagkugRtaMgEdleFVIeGayeRbHbUknwg 2 f 'c / 3 . krNIdUcKñaRtUv)an
GnuvtþedIm,IKNnakMlaMgkat;edayKμanm:Um:g;rmYlenAkñúgemeronTI 8. eKRtUvkarkarkMNt; (limitation)
edIm,Ikat;bnßysñameRbH nigedIm,IkarBarEbképÞebtugEdlbNþalmkBIkugRtaMgkMlaMgkat;TTwgeRTt
nigm:Um:g;rmYleRTt.
8>6> muxkat;Rbehag Hollow Section

bnSMénkugRtaMgkMlaMgkat; nigkugRtaMgm:Um:g;rmYlenAkñúgmuxkat;RbehagRtUv)anbgðajenAkñúgrUb
15>6 EdlkMras;CBa¢aMgRtUvOansnμt;faefr. enAkñúgmuxkat;RbehagxøH kMras;CBa¢aMgGacERbRbYlCMuvij
brimaRt. sMrab;krNIenH smIkar !%>@@ RtUv)ankMNt;enATItaMgEdlGgÁxageqVgmantMélGtibrma. cM
NaMfa enAnwgsøabxagelI nigsøabxageRkam CaTUeTAkugRtaMgkMlaMgkat;RtUv)anecal. CaTUeTA Rbsin
ebIkMras;CBa¢aMgénmuxkat;Rbehag t tUcCag Aoh / Ph enaHsmIkar !%>@@ køayCa
⎛ Vu ⎞ ⎛ Tu Ph ⎞ ⎡⎛ V ⎞ 2 ⎤
⎜ ⎟+⎜⎜
⎜ b d ⎟ 1.7 A t ⎟ ⎟ ≤ φ ⎢⎜ c ⎟ +
⎜b d ⎟ 3 f 'c ⎥ ¬!%>@#¦
⎝ w ⎠ ⎝ oh ⎠ ⎣⎝ w ⎠ ⎦
(ACI Code, Section 11.6.3) .
8>7> EdkRTnug Web Reinforcement
dUcEdl)anBnül;rYcehIy viFI ACI Code sMrab;KNnaGgát;Edlrgm:Um:g;rmYlKWQrelI space
truss analogy enAkñúgrUbTI 15>8. bnÞab;BIkareRbHedaykarrmYl eKRtUvkarEdkBIrRbePTedIm,ITb;nwg

m:Um:g;rmYlEdlGnuvtþ Tu KW EdkTTwg (transverse reinforcement) At enAkñúgTMrg;CaEdkkgbiTCit nig
EdkbeNþay (longitudinal reinforcement) Al . ACI Code )anbgðajnUvsmIkarxageRkamedIm,I
KNna At nig Al ³
!> EdkkgbiTCit At EdlGacKNnadUcxageRkam³


T.Chhay NPIC

2 Ao At f yt cot θ
Tn =
s
¬!%>@$¦
Edl Tn = Tφu nig φ = 0.75
At = RkLaépÞéneCIgmYyrbs;EdkkgbiTCit

f yt = ersIusþg;yal (yield strength) rbs; At At ≤ 400MPa

s = KMlatEdkkg

Ao nig θ RtUv)ankMNt;enAkñúgEpñk 8>2. smIkar !%>@$ GacRtUv)ansresrdUcxageRkam
At
=
Tn
s 2 A f cot θ
¬!%>@%¦
o yt

RbsinebI θ = 45o enaH cot θ = 1.0 nigRbsinebI enaHsmIkar !%>@% køayCa
f yt = 400MPa
At
=
s 800 Ao
Tn
¬!%>@^¦
Edl Tn KitCa N .mm . KMlatEdkkg s minRtUvFMCagéntMéltUcCageKkñúgcMeNam Ph / 8 nig
300mm . sMrab;muxkat; RbehagrgkarrmYl

cMgayEdlvas;BIGkS½énEdkkgeTAépÞxagkñúgrbs;CBa¢aMgminRtUvtUcCag 0.5 Aoh / Ph .
@> EdkbeNþaybEnßm Al EdlcaM)ac;sMrab;karrmYlminKYrtUcCagtMélxageRkam³
⎛ A ⎞ ⎛ f yt ⎞ 2
Al = ⎜ t ⎟ Ph ⎜
⎜ f ⎟
⎟ cot θ ¬!%>@&¦
s ⎝ ⎠ ⎝ y ⎠
Rbsin θ = 45 nig o
f yt = f y = 400MPa sMrab;TaMgEdkkg nigEdkbeNþay
enaHsmIkar !%>@& køayCa
⎛A ⎞ ⎛A ⎞
Al = ⎜ t ⎟ Ph = 2⎜ t ⎟( x1 + y1 ) ¬!%>@*¦
⎝ s ⎠ ⎝ s ⎠
Ph RtUv)ankMNt;enAkñúgEpñk 8>2. cMNaMfa EdkEdlcaM)ac;sMrab;karrmYlKYrRtUv)anbEnßmBIelI

EdlEdlcaM)ac;sMrab;kMlaMgkat; m:Um:g;Bt; nigkMlaMgtamGkS½EdleFVIGMeBIrYmKñaCamYykMlaMgrmYl. kar
kMNt;epSgeTotsMrab;EdkbeNþay Al mandUcxageRkam³
a. Ggát;p©itEdktUcbMputsMrab;EdkbeNþayKW DB10 b¤KMlatEdkkgelI 24 b¤ s / 24
edayykmYyNaEdlmantMéltUcCageK.
b. EdkbeNþayKYrRtUv)anBRgayCMuvijbrimaRtrrbs;EdkkgCamYyKMlatGtibrma 300mm .



c. EdkbeNþayKYrEtdak;enAkñúgEdkkg y:agehacNas;k¾dak;EdkenARKb;mMurbs;Edkkg.
EdkEdldak;enAnwgmMurbs;EdkkgRtUv)aneKrkeXIjfamanRbsiT§PaBkñúgkarbegáItersIu
sþg;m:Um:g;rmYl nigkñúgkarkarBarsñameRbH.
d. EdkTb;m:Um:g;rmYlRtUvdak;enAcMgay (bt + d ) BIcMnucEdlRTwsþIRtUvkar Edl bt CaTTwgén
Epñkrbs;muxkat;EdlmanEdkkgTb;kMlaMgrmYl.
8>8> EdkTb;karrmYlGb,brma Minimum Torsional Reinforcement

enAkEnøgNaEdlEdkTb;karrmYlGb,brmaRtUvkar EdkTb;karrmYlGb,brmaRtUv)ankMNt;dUc
xageRkam (ACI Code, Section 11.6.5) ³
!> EdkkgbiTCitGb,brmasMrab;bnSMénkMlaMgkat;TTwg nigkarrmYl ¬emIlEpñk 8>6¦³
Av + 2 At ≥
0.35bw s
f
¬sMrab; f 'c < 31MPa ¦
yt

⎛b s⎞
Av + 2 At ≥ 0.063 f 'c ⎜ w ⎟
⎜ f yt ⎟
¬sMrab; f 'c ≤ 31MPa ¦ ¬!%>@(¦
⎝ ⎠
Edl Av = RkLaépÞeCIgTaMgBIrrbs;EdkkgEdlkMNt;)anBIkMlaMgkat;
At = RkLaépÞeCIgEtmYyrbs;EdkkgEdlkMNt;BIm:Um:g;rmYl

s = KMlatEdkkg

f yt = ersIusþg;yal (yield strength) rbs;Edkkg ≤ 400 MPa

KMlatEdkkg s minKYrFMCagtMéltUcCagkñúgcMeNam Ph / 8 nig 300mm . KMlatenHRtUvkar
edIm,IRKb;RKgsñameRbH.
@> RkLaépÞEdksrubGb,brmarbs;EdkbeNþayTb;karrmYl³
5 f 'c Acp ⎛ At ⎞ ⎛ f yt ⎞
Al min = − ⎜ ⎟ Ph ⎜
⎜ f ⎟
⎟ ¬!%>#0¦
yf ⎝ s ⎠ ⎝ y ⎠
Edl At / s minRtUvyktUcCag 173bw / f yt .
Al Gb,brmaenAkñúgsmIkar !%>#0 RtUv)ankMNt;edIm,Ipþl;nUvGRtaGb,brmaénmaDEdkTb;kM

laMgrmYlelImaDebtug mantMélRbEhl 1% sMrab;ebtugGarem:EdlrgkMlaMgrmYlsuT§.
9> segçbviFIsaRsþKNnaeday ACI Code Summary of ACI Code Procedures
viFIsaRsþKNnasMrab;bnSMkMlaMgkat;TTwg nigkMlaMgrmYlGacRtUv)ansegçbdUcxageRkam³


T.Chhay NPIC

!> KNnakMlaMgkat;TTwgemKuN Vu nigm:Um:g;rmYlemKuN Tu BIkMlaMgEdlGnuvtþmkelIeRKOg
bgÁúM. tMéleRKaHfñak;sMrab;kMlaMgkat;TTwg nigkMlaMgrmYlKWsßitenARtg;muxkat;EdlmancM
gay d BIépÞrbs;TMr.
@> a. eKRtUvkarEdkkMlaMgkat;TTwgenAeBl Vu > φVc / 2 Edl Vc = f 'c bwd / 6 .
b. EdkTb;karrmYlRtUvkarenAeBlEdl
f 'c ⎛ Acp ⎞
2
Tu > φ
12
⎜
⎜P ⎟
⎟ ¬!%>@0¦
⎝ cp ⎠
RbsinebIEdkRTnugRtUvkar GnuvtþviFIsaRsþxageRkam.
#> KNnasMrab;kMlaMgkat;TTwg
a. KNnaersIuisþg;kMlaMgkat; nominal Edlpþl;edayebtug Vc . kMNt;kMlaMgkat;TTwg

EdlTb;edayEdkRTnug³
V − φVc
Vu = φVc + Vs b¤ Vs = u
φ
b. eRbobeFob Vs Edl)anKNnaCamYynwgtMélGnuBaØatGtibrma 2 f 'c bw d / 3 eyag

tam ACI Code. RbsinebI Vs tUcbnþkarKNna EtpÞúymkvijtMeLIgTMhMmuxkat;rbs;
ebtug.
c. EdkRTnugkMlaMgkat;TTwgRtUv)anKNnadUcxageRkam³
Vs s
Av =
f yt d

Edl Av =RkLaépÞéneCIgTaMgBIrrbs;Edkkg
s = KMlatEdkkg

EdkkMlaMgkat;TTwgkñúgmYyÉktþaRbEvgKW
Av V
= s
s f yt d

d. RtYtBinitü Av / s Edl)anKNnaCamYynwg Av / s Gb,brma³
Av ⎛b ⎞ ⎛ ⎞
(min) = 0.063 f 'c ⎜ w ⎟ ≥ 0.35⎜ bw ⎟
s ⎜ f yt ⎟ ⎜ f yt ⎟
⎝ ⎠ ⎝ ⎠
Av Gb,brma RtUv)ankMNt;edaybTdæaneRkambnSMénGMeBIrbs;kMlaMgkat;TTwg nigkM
laMgrmYlRtUv)aneGayenAkñúgCMhanTI5



$> KNnasMrab;karrmYl³
a. RtYtBinitüfaetIm:Um:g;rmYlemKuN Tu begáItm:Um:g;rmYllMnwg (equilibrium torsion)

b¤m:Um:g;rmYlRtUvKña (compatibility torsion). sMrab; equilibrium torsion eRbI Tu .
sMrab; compatibility torsion m:Um:g;rmYlKNnaKWtMéltUcCageKén Tu BIbnÞúkemKuN
nig
f 'c ⎛ Acp ⎞
2
Tu 2 = φ
3
⎜
⎜P ⎟
⎟ ¬!%>!(¦
⎝ cp ⎠
b. RtYtBinitüfaetITMhMénmuxkat;RKb;RKan;b¤Gt;. vaTTYl)anedayeRbIsmIkar !%>@! sMrab;
muxkat;tan; b¤smIkar !%>@@ sMrab;muxkat;Rbehag. RbsinebItMélenAGgÁxageqVgFM
Cag φ (Vc / bwd + 2 f 'c / 3) enaHbegáInmuxkat; pÞúymkvijKNnabnþ. sMrab;muxkat;
Rbehag RtYtBinitüfaetIkMras;CBa¢aMg t tUcCag Aoh / Ph b¤Gt;. RbsinebIvatUcCageRbI
smIkar !%>@# pÞúymkvijeRbIsmIkar !%>@@.
c. kMNt;EdkkgbiTCitcaM)ac;BIsmIkar !%>@%
At
=
Tn
s 2 A f cot θ
¬!%>@%¦
o yt

At / s minRtUvtUcCag 173bw / f yt . ehIy mMu θ Gacsnμt;esμI 45o / Tn = Tu / φ nig
φ = 0.75 .
snμt; Ao = 0.85 Aoh = 0.85(x1 y1 )
Edl x1 nig y1 CaTTwg nigkMBs;rbs;muxkat;KitBIGkS½eTAGkS½Edkkg ¬emIlrUb
TI !%>!!¦. sMrab; θ = 45o nig f y = 400MPa
At
=
s 800 Ao
Tn
¬!%>@^¦
KMlatGnuBaØatGtibrma s KWtMéltUcCageKén 300mm b¤ Ph / 8 .
d. kMNt;EdkbeNþaybEnßmBIsmIkar !%>@&³
⎛ A ⎞ ⎛ f yt ⎞ 2
Al = ⎜ t ⎟ Ph ⎜
⎜ f ⎟
⎟ cot θ ¬!%>@& a ¦
⎝ s ⎠ ⎝ y ⎠
EtminRtUvtUcCag
⎛ 5 f 'c Acp ⎞ ⎛ A ⎞ ⎛ f yt ⎞
Al min = ⎜
⎜ 12 f y
⎟ − ⎜ t ⎟P ⎜
⎟ ⎝ s ⎠ h⎜ fy ⎟
⎟ ¬!%>@& b ¦
⎝ ⎠ ⎝ ⎠


T.Chhay NPIC

sMrab; θ = 45o nig f yt = 400MPa enaH Al = ( At / s )Ph ¬!%>@*¦
EdkbeNþayTb;karrmYlKYrmanGgát;p©ity:agticesμIKMlatEdkkgelI 24 b¤ s / 24 b:uEnþ
minRtUvtUcCag DB10 . EdkbeNþayRtUvdak;enAkñúgEdkkgbiTCitCamYyKMlatGtibrma
esμI 300mm . y:agehaceKRtUvdak;EdkmYyedImenARKb;mMurbs;Edkkg. CaTUeTAmYy
PaKbIén Edk Al RtUv)anbEnßmeTAelIEdkTaj mYyPaKbIenABak;kNþalkMBs;rbs; mux
kat; nigmYyPaKbIeTotenAEpñksgát;.
%> kMNt;RkLaépÞsrubénEdkkgbiTCitEdlbNþalBI Vu nig Tu .
Avt = ( Av + 2 At ) ≥
0.35bw s
f
¬!%>@(¦
yt

eRCIserIsEdkkgbiTCitsmrmüCamYyKMlat s EdlmantMéltUcCageKkñúgcMeNam 300mm
nig Ph / 8 .
EdkkgKYrRtUv)andak;enAcMgay (bt + d ) eRkaycMnucEdlRTwsþIRtUvkar Edl bt = TTwgén
muxkat;EdlTb;nwgkMlaMgrmYl.
]TahrN_15>3³ (Equilibrium Torsion)
kMNt;brimaNEdkRTnugcaM)ac;sMrab;muxkat;ctuekaNEkgdUcbgðajenAkñúgrUbTI 15>15. muxkat;rgnUvkM
laMgkat;emKuN Vu = 213.5kN nigkMlaMgrmYllMnwg (equilibrium torsion) Tu = 41kN .m enATItaMg
EdlmancMgay d BIépÞénTMr. eKeGay f 'c = 28MPa nig f y = 400MPa .



dMeNaHRsay³
CMhanxageRkambgðajBIviFIsaRsþkñúgkarKNna
1> kMlaMgKNnaKW Vu = 213.5kN nig Tu = 41kN .m
2> a. EdkTb;kMlaMgkat;RtUvkarenAeBl Vu > φVc / 2 .
φ
28 (400 )(520) ⋅10 − 3 = 137.6kN
0.75
φVc = f 'c bd =
6 6
φV
Vu = 213.5kN > c = 68.8kN
2
eKRtUvkarEdkTb;kMlaMgkat;.
b. eKRtUvkarEdkTb;karrmYlenAeBl
f 'c ⎛ Acp ⎞
2
Tu > φ ⎜ ⎟ =T
12 ⎜ Pcp ⎟ a
⎝ ⎠
Acp = xo yo = 400 × 580 = 232000mm 2

Pcp = 2( xo yo ) = 2(400 + 520) = 1840mm
0.75 28 (232000 )2 − 6
Ta = 10 = 9.7 kN .m
12 × 1840
Tu = 41kN .m > 9.7kN .m
EdkTb;kMlaMgrmYlRtUvkarcaM)ac;. cMNaMfa RbsinebI Tu tUcCag 9.7kN.m enaHEdkTb;kar
rmYlnwgminRtUvkar b:uEnþEdkTb;kMlaMgkat;RtUvkar.
3> KNnakMlaMgkat;TTwg³
a. Vu = φVc + φVs / Vs = 101.2kN

28 (400)(520 ) = 733.8kN > Vs
2 2
b. Vs (max) = f 'c bd =
3 3
101.2 ⋅10 3
c.
Av
s
V
= s =
f y d 400 × 520
= 0.5mm 2 / m ¬eCIgBIr¦
Av
2s
= 0.25mm 2 / m ¬eCIgmYy¦
4> KNnasMrab;karrmYl
a. kMlaMgrmYlKNna Tu = 41kN .m . KNnalkçN³muxkat; edaysnμt;kMras;ebtugkarBar

Edk 40mm nigeRbIEdkkg DB12 ³
x1 = 400 − 2(40 + 6 ) = 308mm


T.Chhay NPIC

y1 = 580 − 2(40 + 6 ) = 488mm
CakarGnuvtþn_ eKGacsnμt; x1 = b − 90mm nig y1 = h − 90mm
Aoh = x1 y1 = 308 × 488 = 150304mm 2

Ao = 0.85 Aoh = 127758.4mm 2

Ph = 2(x1 + y1 ) = 2(308 + 488) = 1592mm
θ = 45o nig cot θ = 1.0
b. RtYtBinitüPaBRKb;RKan;rbs;muxkat;edayeRbIsmIkar !%>@!³
2 2
⎛ Vu ⎞ ⎛ Tu Ph ⎞ ⎡⎛ ⎞ ⎤
⎜ ⎜ ⎟ ≤ φ ⎢⎜ Vc ⎟ + 2
⎜b d ⎟ +⎜
⎟ 2 ⎟ ⎜b d ⎟ 3 f 'c ⎥
⎝ w ⎠ ⎝ 1.7 Aoh ⎠ ⎣⎝ w ⎠ ⎦
φVc = 137.6kN nig Vc = 183.5kN
2 2
⎛ 137600 ⎞ ⎛ 41000000 × 1592 ⎞
Left − hand side = ⎜ ⎟ +⎜ ⎟ = 1.82 MPa
⎝ 400 × 520 ⎠ ⎝ 1.7 × 150304 2 ⎠
⎛ 183500 2 ⎞
Right − hand side = 0.75⎜ + 28 ⎟ = 3.3MPa > 1.82MPa
⎝ 400 × 520 3 ⎠
muxkat;RKb;RKan;
c. kMNt;EdkkgbiTCitcaM)ac;EdlbNþalBIkarrmYlBIsmIkar !%>@%³
At Tn
=
s 2 Ao f yt cot θ
Tu 41
Tn = = = 54.7 kN .m cot θ = 1.0 Ao = 127758.4mm 2
φ 0.75
54.7 ⋅10 6
At
=
s 2 × 127758.4 × 400
d. kMNt;EdkbeNþaybEnßmBIsmIkarTI !%>@&³
⎛A ⎞ ⎛ f yt ⎞ 2
Al = ⎜ t ⎟ Ph ⎜ ⎟ cot θ
⎝ s ⎠ ⎜ fy
⎝
⎟
⎠
At
= 0.535 Ph = 1592mm f yt = f y = 400MPa cot θ = 1.0
s
Al = 0.535 × 1592 = 851.72mm 2
5 f 'c Acp ⎛ A ⎞ ⎛ f yt ⎞
Al (min) = − ⎜ t ⎟ Ph ⎜ ⎟
12 f y ⎝ s ⎠ ⎜ fy ⎟
⎝ ⎠
At
Acp = 232000mm 2 = 0.535 f yt = f y = 400MPa
s



5 28 (232000)
Al (min) = − (0.535)(1592) = 427mm 2
12 × 400
Al = 851.72mm 2 lb;
5> kMNt;RkLaépÞEdkkgsrub
a. sMrab;eCIgmYyrbs;Edkkg
Avt At Av
= +
s s 2s
EdkkgEdlcaM)ac; Avt = 0.535 + 0.25 = 0.785mm 2 / m ¬eCIgmYy¦
eRbIEdk DB12 RkLaépÞmuxkat;rbs;EdkkgsMrab;eCIgmYyKW 113mm 2
= 144mm yk 140mm
113
spacing of stirrups =
0.785
b. KMlatGtibrma s = h = = 199mm b¤ 300mm mYyNaEdltUcCag.
P 1592
8 8
KMlatEdleRbIKW 140mm < 199mm
0.35bw 0.35 × 400
c. Avt / s Gb,brma = = = 0.35mm 2 / m < 0.785mm 2 / m
f yt 400

6> edIm,IrkkarBRgayEdkbeNþay cMNaMfa Al srub = 851.72mm 2 . eRbImYyPaKbIenAEpñkxag
elI b¤ 851.72 / 3 = 283.9mm 2 edIm,IbEnßmenAkñúgEdkrgkarsgát; A's . eRbImYyPaKbIdak;enA
EpñkxageRkam edIm,IbEnßmBIelIEdkrgkarTaj nigEdkmYyPaKbIeTotdak;enAkMBs;Bak;kNþal.
a. RkLaépÞEdksrubenAEpñkxagelIesμI 226 + 283.9 = 509.9mm 2 . eRbI 3DB16

¬ As = 603mm 2 ¦
b. RkLaépÞEdksrubenAEpñkxageRkamesμI 3078.8 + 283.9 = 3362.7 mm 2 . eRbI 3DB 28

nig 2DB32 enARCugmMu ¬ As srub = 3455.8mm 2 ¦
Al srubEdleRbI = (603 − 226 ) + (3455.8 − 3078.8) = 754mm 2

c. enAkMBs;Bak;kNþal eRbIEdk 2DB12 ¬ As = 226mm 2 ¦

bøg;srésEdklMGitRtUv)anbgðajenAkñúgrUbTI 15>15. KMlatEdkbeNþayesμInwg
230mm EdltUcCagKMlatEdkGtibrmaEdlRtUvkar 300mm 2 . Ggát;p©itEdkkg DB12 Edl

eRbIFMCagGgát;p©itGb,brma DB10 b¤KMlatEdkkgelI 24 ¬ s / 24 = 5.8mm ¦.
]TahrN_15>4³ (Compatibility Torsion)
edaHRsay]TahrN_TI 15>3 eLIgvij RbsinebIkMlaMgrmYlemKuNCa compatibility torsion.
dMeNaHRsay³


T.Chhay NPIC

eyagtamdMeNaHRsaykñúg]TahrNITI 15>3
!> kMlaMgKNnaKW V u = 213.5kN nig compatibility torsion Tu = 41kN .m
@> CMhan (a) nig (b) dUcKñaenAkñúg]TahrN_TI 15>3. eKRtUvkarEdkRTnug.
#> CMhan (c) KWdUcKña.
$> KNnasMrab;kMlaMgrmYl³
edaysar compatibility torsion Tu = 41kN .m enaH Tu KNnaRtUvtUvCag 41kN .m b¤ φTcr
RtUv)aneGayenAkñúgsmIkar !%>!(
f 'c ⎛ Acp ⎞ 0.75 28 ⎛ 232000 2 ⎞ − 6
2
φTcr = φ ⎜ ⎟= ⎜ ⎟ ⋅10 = 38.7 kN .m
3 ⎜ Pcp ⎟ 3 ⎜ 1840 ⎟
⎝ ⎠ ⎝ ⎠
edaysarEt φTcr < Tu / eRbI Tu = 38.7kN .m . GnuvtþeLIgvijRKb;CMhanenAkñúg]TahrN_TI
15>3 edayeRbI Tu = 38.7kN .m edIm,IkMNt;famuxkat;RKb;RKan;.
At
s
Al = 0.5 × 1592 = 796mm 2
eRbI Al = 852mm 2 > Al (min)
%> Avt caM)ac; = 025 + 0.5 = 0.75mm 2 / m ¬eCIgmYy¦
.

113
s= = 150.6mm
0.75
eRbI 150mm . eRCIserIsEdkbeNþay nigEdkkgdUckñúg]TahrN_TI 15>3.
]TahrN_15>5³ (L-section with Equilibrium Torsion)
FñwmxagénRbBn§½kMralxNÐ½rbs;GKardUcbgðajenAkñúgrUbTI 15>16. muxkat;enAcMgay d BIépÞénTMrrg
nUv Vu = 235kN nig equilibrium torque Tu = 27kN .m . KNnaEdkRTnugcaM)ac;edayeRbI
f 'c = 28MPa nig f y = 400MPa sMrab;RKb;EdkEdleRbIenAkñúgFñwm.

dMeNaHRsay³
1> kMlaMgKNnaKW Vu = 235kN nig Tu = 27kN .m
2> a. EdkTb;kMlaMgkat;RtUvkarenAeBl Vu > φVc / 2
φ f 'c 0.75 28
φVc = bw d = 350 × 455 ⋅10 − 3 = 105.3kN
6 6
φVc
Vu > = 52.65kN
2



eKRtUvkarEdkkMlaMgkat;TTwg
b. RtYtBinitüfaetIEdkTb;karrmYlRtUvkarb¤Gt;. snμt;fasøabcUlrYmkñúgkarTb;karrmYl RbEvg

søabRbsiT§PaBKW hw = 380mm < 150 × 4 = 600mm .
xo = 350mm nig yo = 530mm
Acp = (350 × 530) + (150 × 380 ) = 242500mm 2

Pcp = 2(730 + 530) = 2520mm
⎛ 242500 2 ⎞ − 6
BIsmIkar !%>@0 Ta =
0.75
12
28 ⎜
⎜ 2520 ⎟
⎟ ⋅10 = 7.7kN .m
⎝ ⎠
Tu > Ta
muxkat;RtUvkarEdkTb;karrmYl.
3> KNnaEdkTb;kMlaMgkat;TTwg³
a. Vu = φVc + φVs

235 = 105.3 + 0.75Vs

Vs = 173kN
2
b. Vs (max) = f 'c bw d = 561.8kN > Vs
3
c.
Av
s
V
= s =
173000
f y d 400 × 455
= 0.95mm 2 / m ¬eCIgBIr¦
Av 0.95
= = 0.475mm 2 / m
2s 2
4> KNnaEdkTb;karrmYl³ Tu = 27kN .m
a. KNnalkçN³muxkat;edaysnμt; kMras;ebtugkarBarEdk 40mm nigEdkkg DB12 .

RTnug x1 = 350 − (2 × 40) − 12 = 258mm y1 = 530 − (2 × 40) − 12 = 438mm
søab x1 = 380mm ¬EdkkghYscUleTAkñúgRTnug¦
y1 = 150 − 92 = 58mm
Aoh = (58 × 380) + (258 × 438) = 135044mm 2

Ao = 0.85 Aoh = 114787.4mm 2

Ph = 2(58 + 380) + 2(258 + 438) = 2268mm

θ = 45o cot θ = 1.0


T.Chhay NPIC

b. RtYtBinitüPaBRKb;RKan;rbs;muxkat;edayeRbIsmIkarTI !%>@!³ Vu = 235kN /
φVc = 105.3kN / Vc = 140.4kN / Tu = 27kN .m

⎛ 235000 ⎞ ⎛ 27 ⋅10 × 2268 ⎞
2 6
Left − hand side = ⎜ ⎟ +⎜ ⎟ = 2.47 MPa
⎝ 350 × 455 ⎠ ⎜ 1.7 × 135044 2 ⎟
⎝ ⎠
⎛ 140400 2 ⎞
Right − hand side = 0.75⎜ + 28 ⎟ = 3.3MPa
⎝ 350 × 455 3 ⎠
muxkat;manlkçN³RKb;RKan;
c. kMNt;EdkkgbiTCitedIm,ITb;karrmYl At / s BIsmIkar !%>@%³
27 ⋅ 10 6
At
s
=
Tn
=
2 Ao f yt cot θ 0.75 × 2 × 114787.4 × 400
d. KNnaEdkbeNþaybEnßmBIsmIkar !%>@* ¬sMrab; f 'c = 400 MPa nig cot θ = 1.0 ¦
⎛A ⎞
Al = ⎜ t ⎟ Ph = 0.392 × 2268 = 889mm 2
⎝ s ⎠
Al min ¬BIsmIkar !%>#0¦ KW
5 28 × 242500
Al min = − 889 = 447.7mm 2
12 × 400
karcUlrYmrbs;søabRtUv)anecaledaysarTTYl)anlT§plxusKñatictYc nigtMélBlkmμ
ticCag.
5> kMNt;RkLaépÞmuxkat;EdkkgbiTCit
a. sMrab;eCIgmYy vt = t + v
A A A
s s 2s
muxkat;cM)ac; Avt = 0.392 + 0.475 = 0.867mm 2 / m ¬eCIgmYy¦
eRCIserIsEdk DB12 ¬ As = 113mm 2 ¦
KMlatEdkkg = 0113 = 130mm eRbI 125mm
.867
b. KMlatEdkGtibrma s max = h = = 283.5mm . eRbI s = 125mm dUckarKNna.
P 2268
8 8
Avt 0.35bw 0.35 × 350
c. = = = 0.31mm 2 / m < 0.867mm 2 / m dUcenHeRbI
s f yt 400

DB12 @125
6> kMNt;karBRgayrbs;EdkbeNþay. Al srubKW 889mm 2 . eRbImYyPaKbI b¤
889 / 3 = 296.3mm 2 enAEpñkxagelI EpñkkNþal nigEpñkxageRkam.



a. brimaNEdksrubenAEpñkxagelI = 628.3 + 296.3 = 924.6mm 2 eRbI 3DB20
¬ As = 942.5mm 2 ¦
b. brimaNEdksrubenAEpñkxagelI = 2463 + 296.3 = 2759.3mm 2 eRbI 5DB28
¬ As = 3078.8mm 2 ¦
Al srubEdleRbI = (942.5 − 628.3) + (3078.8 − 2463) = 930mm 2

c. eRbIEdk 2DB12 enABak;kNþalkMBs; ¬ As = 226mm 2 ¦. bøg;EdklMGitRtUv)anbgðajenA
kñúgrUbTI 15>16. KMlatEdkbeNþayKW 190mm < 300mm . Ggát;p©itrbs;EdkkgKW
12mm EdlFMCagGgát;p©itEdk DB10 b¤KMlatEdkkgelI 24 ¬ s / 24 = 5.2mm ¦.

bEnßmEdkbeNþay DB12 enARKb;mMurbs;EdkkgenAkñúgRTnugFñwm nigsøabFñwm.


T.Chhay NPIC


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