Ix. two way prestressed concrete floor systems

Department of Civil Engineering NPIC

IX. RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis
Two-Way Prestressed Concrete Floor Systems

9.1. esckþIepþIm³ rMlwkBIviFIsaRsþ Introduction: Review of Method
CaTUeTA RbB½n§kMralEdlmanTMr (supported floor system) RtUv)aneKsg;BIebtugGarem:Edl
cak;enAnwgkEnøg. kMralxNÐBIrTisCakMralxNÐEdlmanpleFobbeNþayelITTwgtUcCagBIr. karsikSa
viPaK nigkarsikSaKNnaRbB½n§kMralxNÐEdlbgðajenAkñúgrUbTI 9>1 rYmbBa©ÚlTMrg;kMralxNÐeRcInCag
mYyRbePT. enAkñúgemeronenH nwgbgðajBIrebobkMNt; (1) moment capacity, (2) slab-column shear
capacity nig (3) serviceability behavior EdlmandUcCakarRKb;RKgPaBdab nigsñameRbH. cMNaMfa

flat plate CakMralxNÐEdlRTEdlQrelIssredaypÞal;edayminmanFñwm dUcEdlbgðajenAkñúgrUbRtg;

cMnuc (a) cMENkkMralxNÐQrelIFñwmRtUv)anbgðajenAkñúgcMnuc (b) ÉcMnuc (c) bgðajBI waffle slab floor.
eKeRbIeKalkarN_dUcKñakñúgkarsikSaviPaKRbB½n§ flat plate ebtugGarem: edIm,IsikSaviPaK flat
plate ebtugeRbkugRtaMgCab;BIrTis. b:uEnþ bec©keTskñúgkarsagsg;manlkçN³xusKña . CaerOy²

lkçN³esdækic©EtmYyminTMngGacbgðajBIlkçN³smehtuplkñúgkareRbIRbePTRbB½n§kMralxNÐBIrTisE
dlbgðajenAkñúgrUbTI 9.1(b) nig (c) eT. CaTUeTA eKniymeRbIRbB½n§ post-tensioned sMrab;RbB½n§ two-
way plate Edlcak;ehIy. eBlxøH eKeRbIkMralxNÐBIrTisEdlcak;Rsab;enAkardæan EdleKehAfa lift

slabs CaRbB½n§eRKOgbgÁúMdac;edayELkBIKñaEdleFVIeGaykarsagsg;manel,ÓnelOn nigmanlkçN³

esdækic©CagkMralxNÐBIrTiseRbkugRtaMgcak;enAnwgkEnøg. b:uEnþ karxVHbec©keTskñúgkarsagsg; lift
slab nigGvitþmanénGñkCMnajÉkeTsxagkargarsagsg;EbbenHGacbegáItnUvlkçxNÐeRKaHfñak;Edl

GaceFVIeGay)at;bg;nUvesßrPaB nigeFVIeGayeRKOgbgÁúMdYlrlM.
bec©keTssMrab;plit lift slab rYmmankarcak; ground-level slab EdlmantYnaTIBIrKW casting
bed EdlenABIelIvaeKGaccak;kMralxNÐepSg²TaMgGs; nigdak;KelIKña EdlEckdac;BIKñaeday mem-

brane b¤ sprayed parting agent. ssr EdlGacCaEdk b¤ebtugRtUv)ansg;rhUtdl;kMBs;rbs;GKar

munnwgcak; basic bottom slab. kMralxNÐdéTeTotTaMgGs;RtUv)aneKcak;CMuvijssr edayman steel
collar pþl;nUvKMlatRKb;RKan;edIm,IGnuBaØat listing (jacking) kMralxNÐdl;kMritnIv:Usmrmürbs;kMral

xNÐ dUcbgðajenAkñúgrUbTI 9>1 (d). eKsMerc)ankargar lifting tamry³kareRbI jack Edldak;enAelI
kMBUlrbs;ssr nigP¢ab;eTAnwgr)arEdkEdlmaneFμj (threaded rod) EdlbgðÚtcuHeRkamelIépÞrbs;ssr

RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 553

T.Chhay viTüasßanCatiBhubec©keTskm<úCa

Two-Way Prestressed Concrete Floor Systems 554


eTAnwg lifting collar Edlbgáb;enAkñúgkMral. skmμPaBénkar jack TaMgGs;RtUvrkSaPaBedkrbs;kMral
edIm,IeCosvagkar)at;bg;lMnwg.
karsikSaviPaKkareFVIkarrbs;kMralxNÐeRkamkarBt;begáagrhUtdl;TsvtSr× 1940 nigedImTs-
vtSr× 1950 eFVIeLIgedayGnuvtþtamRTwsþIeGLasÞic (classical theory of elasticity). RTwsþIPaBdabtUc
rbs; plate (small-deflection of plates) Edlsnμt;sMPar³manlkçN³sac;mYy (homogeneous) nig
esμIsac; (isotropic) EdlbegáItCaeKalkarN_én ACI Code recommendation CamYynwgtaragemKuN
m:Um:g;.
enAqñaM 1943, Johansen )anbgðajRTwsþI yield-line sMrab;kMNt;lT§PaBTb;Tl;kardYlrlMrbs;
kMralxNÐ. cab;taMgBIeBlenaHmk kargarsikSaRsavRCavCaeRcInRtUv)aneFVIeLIg EdlkargarTaMgenHTak;
TgnwgkareFVIkarrbs;kMralebtugGarem:eRkamGMeBI ultimate. karsikSaedayGñkRsavRCavCaeRcIndUcCa


Ockleston, Mansfield, Rzhanitsyn, Powell, Wood, Sawcczuk, Gamble-Sozwn-Siess nig Park
)ancUlrYmd¾mhimasMrab;karyl;dwgBI limit-state behavior rbs;kMralxNÐ nig plate enAeBl)ak; k¾dUc
enAeBlrgbnÞúkesvakmμ.
eKmanviFICaeRcInEdlRutUv)aneRbIsMrab;sikSaviPaK nigsikSaKNnakMralxNÐ nig plate BIrTis
RtUv)ansegçbdUcxageRkam³

9.1.1. viFI ACI Code Bak;kNþaleGLasÞic The Semielastic ACI Code Approach

viFI ACI pþl;nUvCMerIsBIrsMrab;sikSaviPaK nigsikSaKNnaRbB½n§ plate b¤kMralxNÐBIrTisKW³ viFI
sikSaKNnaedaypÞal; (direct design method) nigviFIeRKagsmmUl (equivalent frame method).
viFITaMgBIrnwgRtUv)anykmkBiPakSaenAkñúgcMnuc 9.3.. eKeRbI equivalent frame method enAkñúgkar
sikSaKNna nigsikSaviPaK plate nigkMralxNÐeRbkugRtaMg.

9.1.2. The Yield-Line Method
eKGnuvtþ semielastic code approach sMrab;krNI nigragsþg;dar ehIyvamanemKuNsuvtßiPaB
sMrab;lT§PaBRTRTg;FMNas; cMENkÉ yield-line theory CaRTwsþI)aøsÞicEdlgayRsYlGnuvtþeTAelIlkç-
xNÐRBMEdn nigrUbragminRbRktI. RbsinebIeKGnuvtþ serviceability constraints, yield-line theory
rbs; Johansen bgðajnUvkareFVIkarBItR)akdrbs; plate nigkMralxNÐebtug EdlGnuBaØatdl;karkMNt;
m:Um:g;begáagBI collapse mechanism Edl)ansnμt;EdlGnuKn_eTAnwgRbePTénbnÞúkxageRkA nigrUbrag
rbs; floor panel. eyIgnwgBiPakSaRTwsþIenHkan;EtlMGitenAkñúgcMnuc 9.14..

9.1.3. The Limit Theory of Plates
cMNab;GarmμN_kñúgkarbegáIt limit solution køayCacaM)ac;edaysarlT§PaBkñúgkarkMNt;
collapse field CaeRcIn EdlGaceGayeKkMNt;)annUv lower failure load. dUcenH upper bound

solution EdlTamTarnUv valid mechanism enAeBlEdkrksmIkarkmμnþ (work equation) k¾dUc lower

bound solution EdlTamTareGayEdnkugRtaMg (stress field) bMeBjlkçxNÐsmIkarlMnwgDIepr:g;Esül

RKb;TIkEnøg (differential equation of equilibrium) Edl
∂2M x ∂ 2 M xy ∂2M y
−2 + = −w (9.1)
∂x 2 ∂x∂y ∂y 2
Edl M x / M y nig M xy Cam:Um:g;Bt; nig w CaGaMgtg;sIueténbnÞúkÉktþa. brimaNEdkEdlERbRbYl


GnuBaØateGay lower bound solution enAEtmann½y. Wood, Park nigGñkRsavRCavdéTeTot)anpþl;
nUv semiexact prediction EdlsuRkitrbs; collaps load.
sMrab; limit-state solution eKsnμt;fakMralxNÐmanlkçN³rwgdac;xatrhUtdl;eBldYlrlM.
eRkaymk Nawy )anbBa©ÚlT§iBlénPaBdabeRkamGMeBIbnÞúkFM k¾dUcT§iBlkMlaMg membrane rgkar
sgát;eTAkñúgkar)a:n;sμan collapse load.

9.1.4. viFIcMerok The Stripe Method
viFIenHRtUv)anesñIeLIgeday Hillerborg kñúgkic©RbwgERbgedIm,ItMerobEdkenAkñúgEdncMerok
(stripe field). edaysarkarKitkñúgkarGnuvtþenAkardæanCak;EsþgTamTarnUvkardak;EdkkñúgTisEkgKña

(orthogonal direction), Hillerborge kMNt;eGaym:Um:g;rmYl (twisting moment) esμIsUnü ehIybMElg

kMralxNÐeGayeTACacMerokFñwmEdlkat;Kña (intersection beam stripe) dUcenHeTIbeKeGayeQμaHfa
stripe method.

elIkElg yield-line theory rbs; Johansen ecj dMeNaHRsayPaKeRcInCa lower bound.
Upper-bound solution rbs; Johansen Gacpþl;nUv collaps load FMbMput kñúgkrNIeKeRbI valid

failure mechanism kñúgkar)a:n;RbmaN collapse load.

9.1.5. esckþIsegçb Summary
viFIeRKagsmmUlCaviFIcMbgEdlnwgRtUvBiPakSa edaysareRbIR)as;én direct design method
enAkñúgkarGnuvtþrbs;vasMrab;RbB½n§kMralebtugeRbkugRtaMgBIrTismanEdnkMNt; nigtMrUvkarnUvkarkMNt;
stiffness enARtg;tMNrvagssr nigkMralxNÐenAkñúgdMeNIrkarénkarsikSaKNna. RTwsþI yield-line

sMrab;sikSakMralxNÐ nig plate eRkamsßanPaBkMNt;enAeBl)ak;k¾RtUv)anbgðajy:agsegçb.

9.2. kareFVIkarrbs;kMralxNÐBIrTiseRkamkarBt;begáag
Flexural Behavior of Two-Way Slabs and Plates
9.2.1. GMeBIBIrTis Two-Way Action
eKmankMralctuekaNeTalEdlRCugTaMgbYnrbs;vaRtUv)anRTeday unyielding support dUcCa
shear wall b¤ stiff beam. eyIgBinitükareFVIkarrbs;kMraleRkamGMeBI gravity load. kMralnwgdabkñúg



TMrg;EdleKmincg;)aneRkamGMeBIbnÞúkxageRkA ehIykac;RCugrbs;vanwgehIbeLIgRbsinebIvaminRtUv)an
cak;ebtugCamYynwgTMrkñúgeBlEtmYyeTenaH. ExSvNÐ (contour) EdleXIjenAkñúgrUbTI 9>2 (a) bgðaj
faExSekagm:Um:g;enARtg;cMnuckNþal C maneRKaHfñak;enAelIRCugxøItamTis y CagenAelIRCugEvgtamTis
x.

karkMNt;m:Um:g;tamTis x nigtamTis y BitCamanlkçN³sμúKsμaj edaysarvaeFVIkarCaeRKOg
bgÁúMsþaTicminkMNt;dWeRkx<s;. eKbriyaykMralenAkñúgEpñk (a) énrUbTI 9>2 CakrNIsamBaØedayykcM-
erok AB nig DE EdlenAkNþalElVg ¬dUckñúgEpñk (b)¦ EdlPaBdabrbs;cMerokTaMgBIrenAcMnuckNþal
C mantMélesμIKña.



FñwmTMrsamBaØEdlrgbnÞúkBRgayesμImanPaBdab 5wl 4 / 384EI b¤ Δ = kwl 4 Edl k cMnYnefr.
RbsinebIkMras;rbs;cMerokTaMgBIrdUcKña enaHPaBdabrbs;cMerok AB KW kwAB L4 ehIyPaBdabrbs;cMerok
DE KW kwDE S 4 Edl w AB nig wDE CacMENkénGaMgtg;sIuetbnÞúksrub w EdlepÞreTAcMerok AB nig

cMerok DE erogKña Edl w = wAB + wDE . dak;eGayPaBdabRtg;cMnuckNþal C éncMerokTaMgBIresμI
Kña enaHeyIg)an
wS 4
w AB = (9.2a)
L4 + S 4
wL4
nig wDE = 4
L + S4
(9.2b)

BIsmIkarTaMgBIrxagelIenH eyIgeXIjfaElVgEdlxøI S rbs;cMerok DE RTnUvcMENkénbnÞúkFMCag. dUc
enH ElVgxøIrbs;kMralEdlenAelI unyielding support KWrgnUvm:Um:g;FMCag EdlRTnUvExSekagEdlman
lkçN³ecatenAkñúgrUbTI 9>2 (a).

9.2.2. Relative Stiffness Effects
eKmankMralxNÐmYyEdlRTeday flexible support dUcCaFñwm nigssr b¤ flat plate EdlRT
edayssr. enAkñúgkrNImYyNak¾eday karEbgEckm:Um:g;enAkñúgTisxøI nigTisEvgmanlkçN³sμúKsμaj
Nas;. PaBsμúKsμajenHekIteLIgBIdWeRkén stiffness rbs; yielding support EdlkMNt;PaBecatén
ExS contour enAkñúgrUbTI 9>2 (a) enAkñúgTis x nigTis y nigkMNt;nUvkarEbgEckm:Um:g;eLIgvij.
pleFob stiffness rbs;FñwmTMrelI stiffness rbs;kMralxNÐGaceFVIeGayExSekag nigm:Um:g; enA
elITisEvgFMCagExSekag nigm:Um:g;enAelITisxøI edaysarkMralsrubeFVIkardUc orthotropic plate Edl
QrelIssrEdlKμanFñwm. RbsinebIElVgEvg L enAkñúgRbB½n§kMralenHFMCagElVgxøI S xøaMg
enaHm:Um:g;Gti-
brmaenAcMnuckNþalrbs;kMralnwgmantMélRbhak;RbEhlnwgm:Um:g;Rtg;kNþalElVgrbs;cMerokElVg L
EdlrgbnÞúkBRgayesμI ehIyEdlcugrbs;vaRtUv)anTb;mineGayvil.
Casegçb edaysarkMralmanlkçN³rlas; (flexible) nigmanbrimaNEdkticEmnETn enaHkar
EbgEckm:Um:g;eLIgvijenAkñúgTisTaMgBIrGaRs½ynwg relative stiffness rbs;TMr nigrbs;kMral. kugRtaMg
EdlFMRCulenAkñúgtMbn;mYyRtUv)ankat;bnßyedaykarEbgEckm:Um:g;eLIgvijenHeTAkan;tMbn;Edlmankug
RtaMgtUc.



9.3.viFIeRKagsmmUl The Equivalent Frame Method

9.3.1. esckþIepþIm Introduction
xageRkamCakarerobrab;BIviFI equivalent frame sMrab;karviPaKkMralxNÐBIrTisEdlsegçb
ACI Code approach edIm,IkMNt; nigEbgEckm:Um:g;srubenAkñúgkMralBIrTis. kUdsnμt;kat;tambøg;GKar

ragctuekaNeRcInCan;tamTisbBaÄrtambeNþayExS AB nig CD enAkñúgrUbTI 9>3 cenøaHkNþalElVg.
eKTTYl)an rigid frame enAkñúgTis x . dUcKña bøg;kat;bBaÄr EF nig HG pþl;nUv rigid frame enAkñúg
Tis y . Idealized frame EdlpSMeLIgedayFñwmedk b¤kMralxNÐsmmUl nigssrEdlCaTMrGaceGay
eKKNnakMralxNÐedaycat;TukvadUcCaFñwm. dUcenH viFI equivalent frame cat;Tuk idealized frame
eFVIkarRsedogKñanwgeRKagBitR)akd ehIyvamanEdnkMNt; nigpþl;nUvlT§plsuRkitCag direct design
method. viFIenHRtUvkarEbgEckm:Um:g;eLIgvijeRcIndg cMENkÉ direct design method RtUvkarkarEbg

Eckm:Um:g;eLIgvijEtmþgKt;.



9.3.2. EdnkMNt;rbs;viFIKNnaedaypÞal; Limitations of the Direct Design Method

xageRkamCaEdnkMNt;rbs; direct design method:
!> enAkñúgTisnImYy²RtUvmanbIElVgCab;Kñay:agtic.
@> pleFobElVgEvg elIElVgxøIminRtUvFMCag 2.0.
#> ElVgkñúgTisnImYy²minRtUvxusKñaedaytMélFMCagmYyPaKbIénElVgEdlEvg.
$> ssrGacsßitenAxusBIG½kSedaytMélGtibrma 10% énElVgenAkñúgTisEdlvasßitenA.
%> bnÞúkTaMgGs;KYrEtCabnÞúkTMnaj nigCabnÞúkBRgayesμIelIkMralTaMgmUl. bnÞúkGefrminRtUvFM
CagbnÞúkefrbIdgeT.
^> RbsinebI kMralRtUv)anRTedayFñwmRKb;Tis/ relative stiffness rbs;FñwmkñúgTisBIEkgKñamin
RtUvtUcCag 0.2 b¤FMCag 5.0.
edayeKeGayEdnkMNt;TaMgenHsMrab;karviPaKkMralxNÐebtugeRbkugRtaMg eKcaM)ac;eRbI
equiva-lent frame method RbesIrCag.

9.3.3. karkMNt;m:Um:g;sþaTic M o Determination of the Statical Moment M o
eKmanCMhankñúgkarKNnakMralxNÐsMxan;cMnUn 4dUcxageRkam³
!> kMNt;m:Um:g;sþaTicsrubenAkñúgTisnImYy².
@> EbgEckm:Um:g;srubsMrab;karKNnamuxkat;sMrab;m:Um:g;viC¢man nigGviC¢man.
#> EbgEckm:Um:g;GviC¢man nigm:Um:g;viC¢maneTAcMerokelIssr nigcMerokkNþalElVg nigeTAFñwmRb
sinebIvaman. cMerokelIssr (column strip) CacMerokEdlmanTTwgesμInwg 25%énTTwgeRKag
smmUlenAelIRCugnImYy²énG½kSssr ehIyTTwgrbs;cMerokkNþalElVgCaRbEvgEdlenA
sl;.
$> kMNt;TMhM nigkarEbgEckEdkenAkñúgTisnImYy².
tamGVIEdleGaydUcxagelI eKRtUveFVIkarEksMrYltMélrbs;m:Um:g;Edl)anEbgEckehIy. eKman
kMralxNÐxagkñúgEdlmanTMhM l1 KitBIG½kSenAkñúgTisénm:Um:g;EdlRtUvBicarNa nigTMhM l2 enAkñúgTis
Ekgnwg l1 dUceXIjenAkñúgrUbTI 9>4. Clear span ln CaTMhMEdlKitBIépÞeTAépÞrbs;ssr b¤ capital
b¤CBa¢aMg. tMélrbs;vaminRtUvtUcCag 0.65l1 ehIysMrab;TMrEdlmanmuxkat;mUl
eKRtUvKitCamuxkat;kaer:



EdlmanRkLaépÞmuxkat;esμIKña. m:Um:g;sþaTicsrubrbs;FñwmTMrsamBaØrgbnÞúkBRgayesμIEdlCa one-
dimensional member KW M o = wl 2 / 8 . enAkñúgkMralxNÐBIrTisEdlCa two-dimensional member,

eKbMElgeRKOgbgÁúMeGayeTACaeRKagsmmUlEdlGaceGayeKKNna M o mþgenAkñúgTis x nigmþgenA
kñúgTis y . RbsinebIeKKitkMralxagkñúgenHCadüaRkamGgÁesrIdUcbgðajenAkñúgrUbTI 9>5 (a) PaBsIuemRTI
kat;bnßykMlaMgkat; nigm:Um:g;rmYl (twisting moment) rhUtdl;sUnütambeNþayRCugénkMNat;Edlrg
karkat;. RbsinebIenAxagcugcMnuc A nigcMnuc B minmankarTb;nwgkarvil (restraint) enaHeKKitkMral
CakMralTMrsamBaØenAkñúgTisénElVg ln . RbsinebIeKkat;enARtg;kNþalElVg dUcenAkñúgrUbTI 9>5 (b)
ehIyKitkMralBak;kNþalCadüaRkamGgÁesrI enaHm:Um:g; M o enAkNþalElVgKW
wl 2 l n1 l n1 wl 2 l n1 l n1
Mo = −
2 2 2 4
wl (l ) 2
b¤ M o = 2 n1
8
(9.3)

edaysarman restraint enARtg;TMr/ eKEbgEck M o enAkñúgTis x eTATMr nigeTAkNþalElVgdUcxag
eRkam
Mo = MC +
1
(M A + M B ) (9.4a)
2
karEbgEckenHGaRs½ynwgdWeRkénPaBrwgRkajrbs;TMr. enAkñúgrebobdUcKña M o enAkñúgTis y CaplbUk



m:Um:g;enAkNþalElVg nigtMélmFüménm:Um:g;elITMrenAkñúgTisenaH.
enAkñúgTisEdlEkg smIkar 9.4a køayCa
M ' o = M 'C +
1
(M A + M B ) (9.4b)
2
Edl M 'o / M ' A / M 'B nig M 'C Cam:Um:g;enAelIG½kSEdlEkgKñanwg M o / M A / M B nig M C erogKña.
dUcKña enAkñúgrebobRsedogKñanwgsmIkar 9.3 eyIg)an
wl1 (l n 2 )2
M 'o = (9.5)
8
GaMgtg;sIuetbnÞúk w eRkambnÞúkesvakmμenAkñúgkMralebtugeRbkugRtaMgGacCa Ww kñúgmYyÉktþaépÞ.

9.3.4. viPaKeRKagsmmUl Equivalent Frame Analysis
eRKOgbgÁúM ¬EdlEckecjCaeRKagCab;dUcbgðajenAkñúgrUbTI 9>6 sMrab;eRKagkñúgTisTaMgBIrEkg
Kña¦ manssrmYyCUr nigFñwm ¬kMralxNÐ¦Cab; ABCDE sMrab;bnÞúkTMnaj. kMralnImYy²RtUv)anviPaK
dac;edayELkBIKña EdlssrRtUv)ansnμt;fa fixed enARtg;kMralxagelI nigxageRkam. edIm,IbMeBj
lkçxNÐsþaTic niglkçxNÐlMnwg eRKagsmmUlnImYy²RtUvRTbnÞúkGnuvtþn_srub. kardak;bnÞúkelIElVg
qøas;RtUv)aneRbIsMrab;rklkçxNÐbnÞúkGefrGaRkk;bMput.
eKcM)ac;KitBIersIusþg;Tb;mMurgVil (rotational resistance) rbs;ssrenARtg;tMN enAeBlKitBI
moment relaxation b¤karEbgEckm:Um:g; elIkElgenAeBlssrmanlkçN³EvgEdleFVIeGayvaman

PaBrwg (rigidity) tUceFobeTAnwg rigidity rbs;kMralenARtg;tMN. enAkñúgkarsagsg; lift slab eKcaM
)ac;KitEtFñwmCab;Etb:ueNÑaH. rUbTI 9>7 bgðajBIbNþaGgát;éneRKagsmmUl. cMerokkMralRtUv)ansnμt;
RTedaykMralTTwg (transverse slab). ssrpþl;nUversIusþg;Tb;m:Um:g;rmYl M T EdlsmmUleTAnwgGaMg
tg;sIuetm:Um:g;rmYlGnuvtþn_ mt . muxkat;cugxageRkArbs;cMerokkMralvilFMCagmuxkat;Rtg;kNþaleday
sarkMhUcRTg;RTayedaykarrmYl. edIm,IKitBImMurgVil nigkMhUcRTg;RTayenH eKRtUvCMnYsssrCak;Esþg
nig transverse slab strip edayssrsmmUlEdl flexibility rbs;ssrsmmUlesμInwgplbUk flexibi-
lity énssrCak;Esþg nigcMerokkMral. karsnμt;enHsMEdgedaysmIkarxageRkam³
1 1 1
= + (9.6)
K ec ∑ K c K t
Edl K ec = PaBrwgRkajTb;karBt; (flexural stiffness) rbs;ssrsmmUl ¬m:Um:g;kñúgmYyÉktþa
mMurgVil¦.



∑ Kc = plbUk flexural stiffness rbs;ssrxagelI nigxageRkamenARtg;tMN
K t = flexural stiffness rbs;FñwmEdlrmYl

müa:gvijeTot eKGacsresrsmIkar 9>6 CasmIkar stiffness
∑ Kc
K ec = (9.7)
∑ Kc
1+
Kt
ehIyeKGackMNt;PaBrwgRkajrbs;ssrsMrab;eRKagsmmUlCa
EI ⎡ ⎛L⎞ ⎤
2
Kc = ⎢1 + 3⎜ ⎟ ⎥ (9.8)
l' ⎢
⎣ ⎝ L' ⎠ ⎥⎦
Edl I Cam:Um:g;niclPaBrbs;ssr/ L CaRbEvgElVgKitBIG½kS/ L' CaRbEvgElVgrbs;FñwmsmmUlEdl
KitBIépÞssr. eKykemKuN carryover RbEhlnwg − 12 (1 + 3h / L ) . eKGacKNnaemKuN carry-
over )anCak;Elkeday column-analogy method edayeRbIkMralxNÐCa analogous column.

smIkarsMrYlsMrab; K c eGaylT§plxusBItMélEdl)anBIsmIkar 9.8 RbEhl 5%.
4 EI
Kc = (9.9)
Ln − 2h
Edl h CakMras;kMralxNÐ. PaBrwgRkajkñúgkarrmYl (torsional stiffness) rbs;kMralxNÐenAkñúgCYr
ssr
9 Ecs C
Kt = ∑ 3
(9.10a)
⎛ c ⎞
L2 ⎜1 − 2 ⎟
⎜ L ⎟
⎝ 2⎠

Edl L2 = TTwg band
Ln = ElVg

c2 = TMhMrbs;ssrkñúgTisRsbnwgFñwgrgkarrmYl ehIyefrrmYl (torsional constant) KW
⎛ x⎞
⎜1 − 0.63 ⎟ x 3 y
⎜ y⎟
C =∑⎝ ⎠ (9.10b)
3
Edl x= TMhMxøIénEpñkctuekaNrbs;muxkat;enARtg;TIRbsBVrbs;ssr ¬dUcCakMBs;kMral¦
y = TMhMEvgénEpñkctuekaNrbs;muxkat;enARtg;TIRbsBVrbs;ssr ¬dUcCaTTwgssr¦

PaBrwgRkajrbs;kMralxNÐRtUv)aneGayedaysmIkar
4 Ecs I s
Ks = (9.11)
Ln − c1 / 2



enAeBlEdleKkMNt;PaBrwgRkajRbsiT§PaB (effective stiffness) K ec rbs;ssr nigPaBrwgRkajrbs;
kMralxNÐ K s eKGacviPaKeRKagsmmUledayviFINamYyk¾)an dUcCa relaxation method b¤ moment
distribution method.

emKuNEbgEck (distribution) sMrab;m:Um:g;bgáb;cug (fixed-end moment) x = KW
Ks
DF = (9.12)
∑K
Edl ∑ K = K ec + K s(left ) + K s(right ) . eKGaceRbIemKuN carryover COF ≅ 1/ 2 edayminman)at;
bg;PaBsuRkit edaysar nonprismatic section bgáT§iBlticNas;eTAelI fixed-end moment nigeTA
elIemKuN carryover. m:Um:g;bgáb;cug FEM sMrab;m:Um:g;BRgayesμIKW wl2 (ln )2 /12 enARtg;TMr EdlenA
eRkayeBlkarEbgEckm:Um:g;eLIgvij plbUkm:Um:g;EdlEbgEckGviC¢manenARtg;TMr nigm:Um:g;enAkNþal
ElVgEtgEtesμInwgm:Um:g;sþaTic M o = wl2 (ln )2 / 8 .

9.3.5. KMrUénkardak;bnÞúkenAelIElVg Pattern Loading of Spans
eKmincaM)ac;eFVIkarBRgaybnÞúkelIElVgTaMgGs;kúgeBlEtmYyeT eRBaHvamin)anbegáItkugRtaMg
ñ
Bt;begáagGviC¢man nigviC¢manGtibrmaeT. dUcenH eKENnaMeGayviPaKeRKageRcInCan;edayeRbIKMrUénkar
BRgaybnÞúkqøas;sMrab;bnÞúkGefr. sMrab;eRKagbIElVg KMrUénkardak;bnÞúkEdlesñIeLIgsMrab;bnÞúkGefr
RtUv)anbgðajenAkñúgrUbTI 9>8.



9.4. bnÞúklMnwgBIrTis Two-Directional Load Balancing
dUcEdl)anerobrab;enAkñúgCMBUkTI 1 bnÞúklMnwg (load balancing) CakMlaMgRbqaMgnwgbnÞúkTMnaj
xageRkA. bnÞúkenHRtUv)anbegáIteLIgedaybgÁúMTTwg (transverse component) énkMlaMgeRbkugRtaMgtam
beNþayenAkñúg parabolic b¤ harped tendon. bnÞúk w enAkñúgsmIkar 9.3 eTAdl; 9.5 CabnÞúkTTwgG½kS
xageRkAEdlmanTiscuHeRkam (downward external transverse load) EdlGacCabnÞúkeFVIkar ww b¤Ca
bnÞúkemKuN wu . bnÞúkEdlmanTiseLIgelI (upward load) enAkñúgkMralxNÐEdlbNþalBI transver
component énkMlaMgeRbkugRtaMg ¬Edlmanerobrab;enAkñúgCMBUk 1¦ kat;bnßyT§iBlrbs; ww ehIy

eKGaceRCIserIsvaCabnÞúklMnwgBitR)akdEdlmanTisedAcuHeRkam. eRkamlkçxNÐEbbenH kMralxNÐBIr
Tisminrgm:Um:g;begáag ehIyk¾minrgm:Um:g;rmYl ehIykarviPaKRtUv)ansMrYly:ageRcIn.
bnÞúklMnwgBIrTisenAkñúgkMralxNÐBIrTisxusBIbnÞúklMnwgmYyTisenAkñúgFñwm. bnÞúklMnwgEdl
begáIteday tendon enAkñúgTismYyGacbegáIn b¤kat;bnßybnÞúklMnwgEdlbegáIteday tendon kñúgTis
Ekg. dUcenH kMlaMgeRbkugRtaMg nig tendon profile enAkñúgTisBIrEkgKñaKWmanTMnak;TMngKñaeTAvijeTA
mk edayrkSanUveKalkarN_sþaTic. plRbeyaCn_d¾FMCageKrbs;bnÞúklMnwgKWkarKNnakMraleRbkug
RtaMgEdlbgÁúMeLIgelIrbs;kMlaMgeRbkugRtaMgpþl;nUvkarBRgaybnÞúkenAkñúgTisnImYy²smmUleTAnwgbnÞú
kxageRkAEdlmanTisedAcuHeRkam. karsikSaKNnaEbbenHRtUv)aneKeGayeQμaHfa pure balanced
design. eKRtUvviPaKral;kargakecjBIlkçxNÐlMnwg (balanced condition) dUcbnÞúkEdlmanGMeBIelI

kMralEdlminrgT§iBlBI bgÁúMeLIgelIrbs;kMlaMgeRbkugRtaMg.



RbsinebIkMralxNÐBIrTisEdlmanTMrrwgdUcCaCBa¢aMgrgeRbkugRtaMgBIrTisEkgKñaEdlmanElVg
LS nig LL kñúgTisxøI nigTisEvg erogKña dUcbgðajenAkñgrUbTI 9>9 enaHeKTTYlbnÞúklMnwgEdlmanTis
ú
eLIgelIEdlRtUvkaredIm,IbegáIt balanced design load Edl)anBIsmIkar 1.15a edaysmIkar
8PS eS
Wbal (S ) = (9.13a)
L2S

nig Wbal (L ) =
8 PL eL
L2
(9.13b)
L

Edl PS nig PL CakMlaMgeRbkugRtaMgRbsiT§PaBeRkaykMhatbg;kñúgmYyÉktþaTTwgrbs;kMralxNÐenA
kñúgTisxøI LS nigTisEvg LL erogKña/ eS nig eL CacMNakp©itGtibrmarbs;EdkeRbkugRtaMg. bnÞúklMnwg
srubkñúgmYyÉktþaTTwgnwgkøayCa
8PS eS 8PL eL
Wbal = Wbal (S ) + Wbal (L ) = + (9.14)
L2
S L2
L

GñksikSaKNnaKYreRCIserIs Wbal ehIykMNt;tMélrbs;kMlaMgeRbkugRtaMg PS nig PL . bnSMén
PS nig PL GacbMeBjsmIkarsßaTic 9.14. RbsinebIkMralxNÐQrelIFñwm b¤RbsinebIkMralxNÐsamBaØ

QrelICBa¢aMg enaHkarKNnaEdlmanlkçN³esdækic©CageKGacRTbnÞúk W EtkñúgTisxøI b¤RT W / 2 kñúg
TisnImYy²sMrab;krNIkMralxNÐragkaer:. kMralxNÐEdlrgedaybnÞúk Wbal nigrgkugRtaMgedaykMlaMg
eRbkugRtaMg PS nig PL nwgRbQmnwgkarBRgaykugRtaMgesμI PS / h nig PL / h enAkñúgTisnImYy² Edl
kñúgenaH h CakMras;kMralxNÐ. kMralxNÐRtUvEtrabesμI edayminmanPaBdab nig camber. KMlatén
bnÞúkGnuvtþn_BI Wbal nwgTamTarnUvkareRbIRTwsþIeGLasÞicFmμtasMrab;viPaK two-way plate.
CaTUeTA edaysarkMralxN§ÐBIrTisebtugeRbkugRtaMgrgkarTajCaeRkay (prestressed post-
tensioned two-way slab) Ca flat plate EdlRTedayssredaypÞal; enaHbnÞúkTaMgGs;RtUv)anRTkñúg

TisTaMgBIredayeRbIEdkeRbkugRtaMgBRgayesμI b¤ banded tendon CamYynwgkarRbmUlpþúMEdkeRbkugRtaMg
enAmþúMcMerokssrrbs;kMralBIrTis.
karEbgEckkugRtaMgesμI nigPaBdab b¤ camber sUnüminmansar³sMxan;sMrab;karkMNt;sma-
maRtmuxkat;RbB½n§kMraleT. RbsinebImindUecñaHeT bnÞúklMnwgminEmnCaviFIEdlmanlkçN³esdækic©
CageKkñúgkarkMNt;kMlaMgeRbkugRtaMgeT. müa:gvijeTot CaerOy²GñksikSaKNnaEtgeRbIbnÞúklMnwg
edayEpñk (partial balancing load) Wbal < WD + WL sMrab;RbB½n§kMraleRcIn dUcbgðajenAkñúgrUbTI
9>2. RbsinebIbnÞúk Ww = WD + WL FMCagbnÞúklMnwg Wbal Edl)anBIsmIkar 9.14 enaHm:Um:g;Éktþa
M S nig M L nigekItmanenAkñúgTis S nigTis L erogKña.



kugRtaMgÉktþaenAkñúgebtugenAkñúgTisxøI nigTisEvgEdlbNþalBIbnÞúkKμanlMnwg (unbalanced
loading) RtUv)anTTYledaykardak;bEnßmkMlaMgsgát;esμIEdlbNþalBIbnÞúklMnwgeTAelIkugRtaMgbegáag

enAkñúgebtugEdlbgáeLIgedaym:Um:g;Bt;begáag M S nig M L EdlekItBI unbalanced load Ww − Wbal .
kugRtaMgebtugenAsrésxagelI nigsrésxageRkamkñúgTisnImYy²RtUv)aneGaydUcxageRkam³
TisxøI
PS M S c
ft =− − (9.15a)
bh IL
P M c
fb = − S + S (9.15b)
bh IL
TisEvg
PL M L c
ft =− − (9.16a)
bh IL
P M c
fb = − L + L (9.16b)
bh IL
enAkñúgsmIkarTaMgenH GkSr t tMNageGaysrésxagelIbMputrbs;kMral ehIyGkSr b tMNageGay
srésxageRkambMputrbs;ebtug/ c = h / 2 / TTwg b = 12in. ehIy
PS total
PS =
L
ehIy P total
PL = L
S
CakMlaMgeRbkugRtaMgÉktþa. emKuNm:Um:g;bnÞúkesvakmμ (service-load moment coefficient) sMrab;kM
Nt; M S nig M L GacTTYl)anBI chart enAkñúgrUbTI 9>10 sMrab;lkçxNÐRBMEdnTaMgGs;. eKmanem
KuNm:Um:g;Bt;sMrab;m:Um:g;Bt;viC¢man nigGviC¢manGtibrma Edl βx2 nig βx'2 GnuvtþeTAelI + M nig − M
enAelIElVgxøI Lx erogKña. dUcKña βy2 nig βy'2 GnuvtþeTAelIm:Um:g;Bt;viC¢man nigGviC¢manGtibrmaenA
elIElVgEvg Ly erogKña. tamrebobdUcKña chart enAkñúgrUbTI 9>11 pþl;nUvviFIy:asrh½skñúgkarkMNt;
ultimate bending moment coefficient enAkñúg plate ebtugeRbkugRtaMgBIrTisCab;.

9.5.ersIusþg;begáagrbs;kMraleRbkugRtaMg Flexural Strength of Prestressed Plates
9.5.1. m:Um:g;KNna M uDesign Moments M u

eKkMNt; design moment sMrab; bonded member eRbkugRtaMgsþaTicminkMNt;edaybnSMrvag
m:Um:g; M u EdlbNþalBIbnÞúkemKuNefrbUknwgbnÞúkemKuNGefr CamYynwg secondary moment M s



EdlekItmanenAkñúgeRKagedaysar tendon. m:Um:g;em (primary moment) M1 nig secondary moment
M s k¾manerobrab;enAkñúgviFIbnÞúklMnwgEdr. dUcenH sMrab;tMélbnÞúkesvakmμ eKcaM)ac;BicarNa Etm:Um:g;

net load M net enAkñúgkarKNnam:Um:g;bgáb;cugemKuN xN³EdleKRtUvKit Wbal sMrab;karviPaKersIusþg;

begáag.

m:Um:g;bgáb;cug M sMrab;karEbgEckm:Um:g;
u

RbsinebI M1 = Pee = Fe Ca primary moment/ M net Cam:Um:g;lMnwgEdlbNþalmkBI Wbal /
M S = M bal − M 1 Ca secondary moment/ ehIy M u Cam:Um:g;bgáb;cugemKuNEdlbNþalBIbnÞúkemKuN

Wu enaHy:agehacNas;k¾ design ultimate moment
Mu = M u − Ms (9.17)
ehIyersIusþg;m:Um:g;EdlGacekItmanKW
Mu
Mn = (9.18)
φ
eKGnuvtþkarEbgEckm:Um:g;eLIgvijCa enlastic EdlbNþalBIPaBCab;eTAelIersIusþg;m:Um:g;EdlGacekIt
man M n enARtg;TMreTAkan;ersIusþg;m:Um:g;tMrUvkar M n enAkNþalElVg.
enAeBlEdleKdak; bonded reinforcement enARtg;TMr ehIyEdkminrgeRbkugRtaMgGb,brma
RtUv)andak;edayGnuelamtamsmIkar 9.19 nig 9.20 enaHm:Um:g;GviC¢manEdlKNnaedayRTwsþIeGLasÞic
sMrab;kardak;bnÞúksnμt;GacnwgekIneLIg b¤fycuHedayPaKryEdlminFMCagPaKryEdleGayedayem
KuNEbgEckm:Um:g;eLIgvij inelastic Edlerobrab;enAkñúgCMBUk 4 nigCMBUk 6.
eKKYreRbIm:Um:g;GviC¢manEkERb (modified negative moment) sMrab;KNnam:Um:g;vIC¢manenARtg;
muxkat;enAkñúgElVg sMrab;kardak;bnÞúkdUcKña. eKGaceFVIkarEbgEckm:Um:g;eLIgvij inelastic sMrab;m:Um:g;
GviC¢manEtenAeBlNaEdlm:Um:g;enARtg;muxkat;enaHRtUv)ankat;bnßy ehIyvaRtUv)aneKsikSaKNna
edaymineGay ω p b¤ ω p + (d / d p )(ω / ω ') FMCag 0.24β1 eT ehIymüa:gvijeTotemKuNEbgEckm:Um:g;
eLIgvijminRtUvFMCag 1000ε t eT.
]TahrN_ 9>2 bgðajy:aglMGitBIviFIsaRsþviPaKeRKagsmmUlTaMgsMrab;lkçxNÐ service load
nig ultimate load nigkarEbgEckm:Um:g;eLIgvij enealsic EdlbNþalBIPaBCab;EdlRtUv)aneRbIenAkñúg
karviPaKersIusþg;.



9.6. Banding of Prestressing Tendons and Limiting Concrete Stresses
9.6.1. karBRgayEdkeRbkugRtaMg Distribution of Prestressing Tendons
eKsnμt;fakMral plate nImYy²manTMrCab;tambeNþayTTwgG½kSssr. karsnμt;RtUv)aneFVIeLIg
dUckarerobrab;BIelIkmuxfakMralxNÐeFVIkardUckMralFñwmBIrEkgKñaEdlTTwgrbs;vaesμInwgTTwgrbs;kMral
EdlRtUv)anRTtambeNþayG½kSssr. dUcenH eKKitfabnÞúk 100% EdlRtUveFVIeGaymanlMnwgRtUv)an
RTedaykMralFñwmkñúgTisedABIrEkgKña.
eKk¾dwgfakarEbgEckm:Um:g;minmanlkçN³esμItamTTwgrbs;kMral b:uEnþvaeRcInEtRbmUlpþúMenAelI
cMerokelIssr. Cavi)ak eKminmanehtuplkñúgkarRbmUlpþúMPaKryy:ageRcInén tendon enAkñúgcMerok
elIssreT dUckarkMNt;enAkñúgrUbTI 9>4 ehIyeKRtUvBRgay tendon EdlenAsl;enAkñúgcMerokkNþal
ElVg. sMrab;ElVgCab; m:Um:g;BI65 eTA75% kñúgTisnImYy²RtUv)anRTedaycMerokssr xN³EdleKRtUv
rkSaRkLaépÞsrub nigcMnYnrbs; tendon EdlTamTaredaym:Um:g;Gnuvtþn_srub.



TTwgrbs;knøHcMerokelIssresμInwgmYyPaKbYnénTMhMEdltUcCageKrbs;kMralxNÐ. cMerok
kNþalElVgCa slab band EdlenAGmedaycMerokelIssrBIr. dUcenH karEbgEck b¤ banding rbs;
EdkeRbkugRtaMgRtUveFVIeLIgeTAtamPaKryénkarEbgEckm:Um:g;rvagcMerokelIssr nigcMerokkNþal
ElVg. Cavi)ak RbsinebI 70%énEdkeRbkugRtaMgRtUv)anRbmUlpþúMenAkñúgcMerokelIssr enaHeKrMBwgfa
cMerokelIssrnwgRT 70%énm:Um:g;srub ehIycMerokkNþalElVgRTnUv 30%énm:Um:g;srubEdlenAsl;.
rUbTI 9>12 bgðajBIkarEbgEckEdkeRbkugRtaMgenAkñúgTisBIrEkgKña. tameKalkarN_ENnaM
TUeTA EdkeRbkugRtaMgkñúgcMerolelIssrRtUvmanKMlatesμInwg3 eTA4dgénkMras;kMralxNÐ ehIyKMlat
Gtibrmarbs;kabeRbkugRtaMgenAkñúgcMerokkNþalElVgminRtUvFMCag 6dgénkMras;kMralxNÐeT. kug
RtaMgsgát;mFümenAkúñgebtugkñúgTisnImYy²KYrmantMély:agticbMputesμInwg 125 psi(0.90MPa ) .



karGegát)anepÞógpÞat;tamry³karBiesaF plate eRbkugRtaMgbYndUcbgðajenAkñúgrUbTI 9>13
bgðajfakarERbRbYlénkarBRgay tendon eRbkugRtaMgEdlmanbrimaNdUcKñaminmanT§iBleTAelIPaB
dabeT. Banding rbs;kabeRbkugRtaMgdUcbgðajenAkñúgEpñk (b) énrUb EdlmanEdkeRbkugRtaMg 65%
eTA75% enAkñúgcMerokelIssr hak;manRbsiT§PaBCageK CaBiessbegáInlT§PaBepÞr shear-moment
enARtg;muxkat;TMrssrrbs;kMralxNÐBIrTis.

9.6.2. kugRtaMgTajkMNt;rbs;ebtugeRkamlkçxNÐbnÞúkesvakmμ
Limiting Concrete Tensile Stresses at Service Load
9.6.2.1. karBt;begáag Flexure
ACI 318 Code kMNt;kugRtaMgTajkñúgebtugsMrab;Ggát;eRbkugRtaMgedIm,IRKb;RKgkarekItman
sñamedaykarBt;begáag (flexural crack). xageRkamCatMélkugRtaMgGnuBaØatGtibrmaenAkñúgGgát;eRb
kugRtaMgsMrab;tMbn;m:Um:g;epSg²³
!> RkLaépÞm:Um:g;GviC¢manCamYynwgkarbEnßmEdkminrgeRbkugRtaMg 6 f 'c psi(0.5 f 'c MPa)
@> RkLaépÞm:Um:g;GviC¢manEdlKμankarbEnßmEdlminrgeRbkugRtaMg 0

#> RkLaépÞm:Um:g;viC¢manCamYynwgkarbEnßmEdkminrgeRbkugRtaMg 2 f 'c psi(0.166 f 'c MPa)
$> RkLaépÞm:Um;g;viC¢manEdlKμankarbEnßmEdkminrgeRbkugRtaMg 0

%> kugRtaMgsgát;enAkñúgebtug ¬eRkamlkçxNÐCak;Elk/ 0.60 f 'c ¦ f c = 0.45 f 'c

9.6.2.2. EdkBRgwg Reinforcement
RkLaépÞGb,brmarbs; bonded reinforcement edayelIkElgGIVEdlTamTaredaysmIkar
9.20 xageRkam KW
As = 0.004 A (9.19a)

Edl A CaRkLaépÞrbs;Epñkénmuxkat;cenøaHépÞrgkarTajedaykarBt;begáagCamYynwgTIRbCMuTMgn;rbs;
gross section. sMrab;RkLaépÞm:Um:g;viC¢manEdlkugRtaMgTajenAkñúgebtugeRkamlkçxNÐbnÞúkesvakmμFM

Cag 2 f 'c psi(0.166 f 'c MPa) enaHeKKNnaRkLaépÞGb,brmarbs; bonded reinforcement BI
Nc
As = (9.19b)
0.5 f y



Edl N c CakugRtaMgTajenAkñúgebtugEdlbNþalBIplbUkbnÞúkefr nigbnÞúkGefrKμanemKuN ehIy
f y = 60,000 psi (414MPa ) . sMrab;RkLaépÞm:Um:g;GviC¢manenARtg;ssrTMr RkLaépÞGb,brmarbs;

bonded reinforcement enAkñúgTisnImYy²RtUv)ankMNt;BI
As = 0.00075hL (9.20)
Edl RbEvgElVgenAkñúgTisRsbeTAnwgEdkBRgwgEdlRtUv)ankMNt;
L=

h = kMras;kMralxNÐ

eKRtUvBRgayEdkBRgwgEdlTTYl)anBIsmIkar 9.20 kñúg slab band width cenøaHExSEdlmanRbEvg
1.5h BIxageRkAépÞQmrbs;ssr. y:agehacNas;eKRtUvdak; bar b¤ wire 4 kñúgTisTaMgBIr.

RbEvgGb,brmarbs; bonded reinforcement enAkñúgRkLaépÞviC¢manKYresμInwgmYyPaKbIén
clear span ehIyvaRtUv)aneKdak;enARtg;kNþalRkLaépÞm:Um:g;viC¢man. RbEvgGviC¢manrbs; bonded

reinforcement enAkñúgRkLaépÞGviC¢manKYrRtUv)andak;bgðÚt 1/6 én clear span enAelIRCugnImYy²rbs;

TMr ehIyeKdak;vaenAsrésxagelI. kugRtaMg f ps enAkñúgEdkeRbkugRtaMgeRkam nominal strength
EdlTamTareday ACI 318 Code RtUv)anpþl;eGaydUcxageRkam
sMrab; Bonded Tendon
⎛ γp ⎡ f pu ⎤⎞
f ps = f pu ⎜1 −
⎜ β1
⎢ρ p +
d
(ω − ω ')⎥ ⎟ (9.21)
⎝ ⎢
⎣ f 'c d p ⎥⎟
⎦⎠
Edl ω ' = ρ ' = f y / f 'c
nig γ p = 0.40 sMrab; f py / f pu ≥ 0.85
= 0.28 sMrab; f py / f pu ≥ 0.90
RbsinebIeKKitEdkrgkarsgát; enaHtY [ρ p f pu / f 'c +(d / d p )(ω − ω ')] enAkñúgsmIkar 9.21 RtUv)aneK
ykmineGaytUcCag 0.17/ ehIy d ' minRtUvFMCag 0.15d p .
sMrab; Unbonded Tendon EdlmanpleFobElVgelIkMras;kMralxNÐ ≤ 35
f 'c
f ps = f pe + 10,000 + (9.22)
100 ρ p

Edl f ps ≤ f py ≤ f pe + 60,000

sMrab; Unbonded Tendon EdlmanpleFobElVgelIkMras;kMralxNÐ > 35
f 'c
f ps = f pe + 10,000 + (9.23)
300 ρ p

Edl f ps ≤ f py ≤ f pe + 30,000



9.6.2.3. kMlaMgkat; Shear
muxkat;TMrssrenAkñúg flat plate: nominal shear strength Edlpþl;edayebtugenARtg;TIRbsBVrbs;
ssrénkMralebtugeRbkugRtaMgRtUv)aneGayeday
Vc = (β ρ f 'c + 0.3 f c )bo d + V p (9.24a)

b¤ nominal unit shearing strength KW
Vp
vc = β ρ f 'c + 0.3 f c + (9.24b)
bo d
Edl bo = brimaRtrbs;muxkat;rgkMlaMgkat;eRKaHfñak;enAcMgay d / 2 BIépÞrbs;TMr
f c = tMélmFümrbs;kugRtaMgrgkarsgát;RbsiT§PaBenAkñúgebtugEdlbNþalBIbnÞúkGnuvtþn_xag

eRkAsMrab;TisBIrEkgKñaEdlKNnaenARtg;TIRbCMuTMgn;rbs;muxkat;eRkaykMhatbg;eRbkug
RtaMg ¬enAkñúg ACI Code eKeRbI f pc ¦
V p = bgÁúMbBaÄrénkMlaMgeRbkugRtaMgRbsiT§PaBTaMgGs;Edlkat;tammuxkat;eRKaHfñak;

β ρ = tMéltUcCageKkñúgcMeNam 3.5 nig (α s d / bo + 1.5) Edl α s esμInwg 40 sMrab;ssrxag
kñúg nig 30 sMrab;ssrxag nig 20 sMrab;ssrkac;RCug.
enAkñúgkMralxNÐEdlmankarBRgaykabeRbkugRtaMg eKminKittY V p eT ebImindUecñaHeTvakøayCacMa)ac;
kñúgkareRbIragFrNImaRtkMeNagkabeRbkugRtaMgbRBa©asenAkñúgkarKNnaedIm,I)a:n;RbmaNkMlaMgkat;
EdlRTeday tendon Edlkat;tammuxkat;eRKaHfñak;. eyagtam ACI 318 Code/ KμancMENkNarbs;
muxkat;ssrKYrenAEk,rcugEdlminCab;FMCagbYndgkMras;kMralxNÐ/ f 'c enAkñúgsmIkar 9.24 minKYrFM
Cag 5,000 psi ehIy f c enAkñúgTisnImYy²minRtUvtUcCag 125 psi b¤FMCag 500 psi eT.
RbsinebIeKminGacbMeBjlkçxNÐTaMgenHeT eKKYryktMél
Vc CatMélEdltUcCageKkñúgcMeNam smIkarxageRkam³
⎛ 4 ⎞
(i) Vc = ⎜ 2 +
⎜ ⎟ f 'c bo d (9.25a)
⎝ βc ⎟
⎠
⎛α d ⎞
(ii) Vc = ⎜ s + 2 ⎟ f 'c bo d
⎜ b ⎟ (9.25b)
⎝ o ⎠
(iii) Vc = 4 f 'c bo d (6.25c)

Edl β c = pleFobRCugEvgelIRCugxøIrbs;ssr b¤RkLaépÞbnÞúkRbmUlpþúM.



smIkar 9.25(a) nig (b) CalT§plrbs;karBiesaFEdlbgðajfa enAeBlpleFob bo / d ekIneLIg enaH
nominal shear strength Vc EdlGacekItmanfycuH dUcenHenAkñúgsßanPaBEbbenH smIkar 9.25(c)

nwgminlubedaysarvakøayCaKμansuvtßiPaB.
TMrcugCab; (Continuous Edge Support): sMrab;bnÞúkBRgay nigTMrcugCab;dUcCaFñwm nigCBa¢aMg/ Rbsin
ebIkMlaMgeRbkugRtaMgRbsiT§PaBmintUcCag 40%énkugRtaMgTajrbs;EdkBRgwg enaHkugRtaMgkat;GnuBaØat
GtibrmaKW
⎡ V d⎤
Vc = ⎢0.60 f 'c + 700 u ⎥bw d p ≥ 2 f 'c bw d
⎣ Mu ⎦

< 5 f 'c bw d ¬xñat US¦ (9.26)
⎡ f 'c V d⎤
Vc = ⎢ + 5 u ⎥bw d p ≥ 0.166 f 'c bw d
⎢ 20
⎣ Mu ⎥
⎦
< 0.415 f 'c bw d ¬xñat SI¦
Edl bw RtUv)anykCaTTwgcMerok ehIy Vu d / M u enAcMgay d p / 2 BIépÞrbs;TMr/ d p ≥ 0.80h .
tMél
f 'c enAkñúgRKb;smIkarxagelITaMgGs;RtUvKuNnwgemKuN λ = 1.0 sMrab;ebtugTMgn;Fmμta/

λ = 0.85 sMrab; sand-lightweight concrete nig λ = 0.75 sMrab; all-lightweight concrete.
emKuNkMlaMgkat; (Shear Force Coefficients): eKGackMNt;tMélRbhak;RbEhlrbs;kMlaMgkat;
GtibrmaenARtg;cugrbs;kMralxNÐBIrTisEdlRTbnÞúkBRgayesμI ehIyvaRtUv)anRTtambeNþayRbEvg
brimaRtrbs;vadUcxageRkam³
1
V = wLS
3
¬RCugxøI¦ (9.27a)

V = kwLS / (2k + 1) ¬RCugEvg¦ (9.27b)

Edl k CapleFobElVgEvg LL elIElVgxøI LS . eKGaceRbItMéldUcKñasMrab;kMralEdlRtUv)anbgáb; b¤
Cab;tambeNþayRCugTaMgbYn. sMrab;krNIdéTeTot eKRtUvEktMrUvkarEbgEckkMlaMgkat; nigkarEbgEck
kugRtaMgEdlbNþalBIGVIEdlsUveRKaHfñak;edayQrelIeKalkarN_kMlaMgkat;enAelIRCugCab;FMCagkMlaMg
kat;enAelIRCugsamBaØbnþicbnþÜc.
ACI Code GnuBaØateGaybegáInkMlaMgkat; 15%enARtg;TMrCab;xagkñúgTImYysMrab; one-way

action.



9.7. Load-Balancing Design of a Single-Panel Two-way Floor Slab
]TahrN_ 9>1³
Two-way single-panel prestressed warehouse lift slab 20 ft × 24 ft (6.10m × 7.32m ) manbøg;dUc
bgðajenAkñúgrUbTI 9>14. vaRtUv)anRTenAelICBa¢aMgdæTaMgbYnRCug edayminKit rotational restraint enA
Rtg;RBMEdnTaMgenH b:uEnþkac;RCugRtUv)anTb;nwgkarrmYl (torsional restraint) . kMralxNÐRtUvRTnUv
superimposed service dead load 15 psf (0.72kPa ) bEnßmBIelIbnÞúkpÞal;rbs;va nigRTnUv service live

load 75 psf (3.59kPa ) . eKminGnuBaØaeGaymanPaBdabeRkamGMeBI full dead load.

sikSaKNnakMralxNÐCa post-tensioned nonbonded prestressed two-way floor edayeRbI
kabeRbkugRtaMg 7-wire 270-K Ggát;p©it 1 / 2in.(12.7mm) . eKeGayTinñn½ydUcxageRkam³
f 'c = 5,000 psi (34.5MPa ) ebtugTMgn;Fmμta
f 'ci = 3,750 psi(25.9MPa )
fc GtibrmakñúgTis E-W EdlbNþalBI net prestress eRkaykMhatbg; = 200 psi



fc GtibrmakñúgTis N-S EdlbNþalBI net prestress eRkaykMhatbg;minRtUvFMCag
350 psi ¬ACI GnuBaØatrhUtdl; 500 psi ¦

f c GtibrmaEdlbNþalBIkugRtaMgpÁÜb (combined stress) = 0.45 f 'c

Ec = 57,000 f 'c = 4.03 ⋅ 106 psi (27.8 ⋅ 106 MPa )

f ps ≤ 0.70 f pu = 189,000 psi(1,303MPa ) dUckarTamTareday ACI Code

f py = 240,000 psi(1,655MPa )

f pe = 159,000 psi(1,096MPa )

(
E ps = 29 ⋅ 10 6 psi 200 ⋅ 103 MPa )
f y = 60,000 psi (414MPa )

(
E s = 29 ⋅ 10 6 psi 200 ⋅ 103 MPa )
dMeNaHRsay³
− 1 = 2.0in.(51mm )
6
eS = e L =
2
eRCIserIskMras;kMralxNÐsakl,gedayQrelIpleFobElVgelIkMras; (span-to-depth ratio) ≅ 45
h=
(20 + 24) × 12 × 1
= 5.87in.
2 45
dUcenH sakl,gkMras;kMralxNÐ 6in.(153mm) edaysnμt;Ggát;p©itbMBg; (duct) ≅ 0.5in. ehIykMBs;
RbsiT§PaB d p = 6.0 − (0.5 / 2 + 3 4 ) = 5.0in.(127mm) .
bnÞúklMnwg (Balancing Load)
× 150 = 90 psf (4.31kPa )
6
WD = 15 psf +
12
edaysar balancing load RtUv)antMrUvsMrab;PaBdab b¤ camber EdlbNþalBIbnÞúkefresμIsUnü enaH
snμt; Wbal = WD = 90 psf (4.31kPa) . ehIyedaysar f c EdlbNþalBIkMlaMgeRbkugRtaMg = 200 psi
¬smμtikmμ¦ snμt;vaCakugRtaMgenAkñúgTis E-W. bnÞab;mkkMlaMgeRbkugRtaMgRbsiT§PaBenAkñúgTis E-W
KW PL = 200 × 6 × 12 = 14,400lb kñúgmYycMerok nigBIsmIkar 9.13b eyIg)an
8 ×14,400 × 2
≅ 33 psf (1.58kPa )
8 PL eL
Wbal (L ) = =
L2
L (24)2 ×12
Uplift EdlRtUvpþl;eday tendon enAkñúgTisxøI ¬bnÞúkTMngRtUvRTedayElVgenAelITisxøI¦ køayCa
Wbal (S ) = WD − Wbal (L ) = 90 − 33 = 57 psf (2.73kPa ) . BIsmIkar 9.13a
Wbal (S ) L2 57 × (20)2 × 12
= 17,100lb / ft (249.7kN / m ) bnÞab;BIkMhatbg;
S
PS = =
8e S 8× 2



kugRtaMgsgát;enAkñúgebtugeRkaykMhatbg;enAkñúgkabeRbkugRtaMgkñúgTis N-S KW
PS 17,100
fc = = = 238 psi < 350 psi
bh 12 × 6
EdlvabMeBjlkçxNÐ. dUcenH eRbIkabeRbkugRtaMg 7-wire 270-K Ggát;p©it 1/ 2in. EdlmankMlaMgeRb
kugRtaMgRbsiT§PaB Pe = 159,000 × 0.153 = 24,327lb(108.2kN ) .
KMlattMrUvkarenAkñúgTis N-S KW
= 1.42 ft = 17in.(432mm )
24,327
sS =
17,100
KMlattMrUvkarenAkñúgTis E-W KW
= 1.69 ft ≅ 20in.(508mm )
24,327
sL =
14,400
cMNaMfa KMlatTaMgBIrRtUvKñanwgKMlatEdl)anENnaM ¬3eTA 5dgénkMras;kMral¦.
edIm,IkarBarkarEbkebtugenARtg;tMbn; anchorage enAmþúMCBa¢aMg bEnßmEdkFmμtaminrgeRbkugRtaMg 2#4
¬12.7mm cMnYnBIr¦ tambeNþay anchorage line enAelIbrimaRtkMral.
kugRtaMgbnÞúkesvakmμ (Service-load Stresses)
bnÞúkGefresvakmμ
WL = 75 psf (3.59kPa )
aspect ratio
L 24
k= L = = 1.20
LS 20
BIrUbTI 9>10/ emKuNm:Um:g;sMrab;m:Um:g;kNþalElVgGtibrmaenAkñúgTisxøI nigTisEvgKW α N − S = 0.062
nig α E −W = 0.035 erogKña edaysnμt;fakac;RCugrbs;kMralxNÐBIrTisRtUv)anTb;nwgkarrmYl
(torsionally restrained).

eyIgsnμt;faRbEvgRbsiT§PaBtamTisxøI nigTisEvg
LS = 19.5 ft nig LL = 23.5 ft

m:Um:g;bnÞúkGefr (Live-load Moment) KW
M S = 0.062 × 75 × (19.5)2 × 12 = 21,218in. − lb / ft
nig M L = 0.035 × 75 × (23.5)2 × 12 = 17,396in. − lb / ft

m:Um:g;niclPaBKW
12(6 )3
Is = = 216in.3
12



kugRtaMgebtugEdlbNþalBIbnÞúkGefr³
kugRtaMgebtugEdlbNþalBIbnÞúkGefrenAkñúgTisxøUIKW
M S c 21,218 × 3
f = = = 295 psi (2.03MPa )
Is 216
kugRtaMgebtugEdlbNþalBIbnÞúkGefrenAkñúgTisEvgKW
M L c 17,396 × 3
f = = = 242 psi (1.67 MPa )
Is 216
kugRtaMgtamG½kSpÁÜb (combined axial stresses) EdlbNþalBIbnÞúklMnwg nigkugRtaMgBt;begáagpÁÜb
(combined flexural stresses) EdlbNþalBIbnÞúkGefr ¬BIsmIkar 9.15 nig 9.16¦ enAkñúgTisxøI (N-S)

køayCa
= −238 − 295 = −533 psi (C )(3.68MPa )
PS M S c
ft =− −
bh Is
nig f b = −238 + 295 = +57 psi (T ) ¬edayvamantMéltUc eKGacecal)an¦
ehIyenAkñúgTisEvg (E-W)
f t = −200 − 242 = −442 psi (C )(3.05MPa )
nig f b = −200 + 242 = +42 psi (T ) ¬Gacecal)an¦
kugRtaMgsgát;GnuBaØat ACI KW f c = 0.45 × 5,000 = 2,250 psi EdlvamantMélFMCagkugRtaMgCak;Esþg
dUcenH vabMeBjlkçxNÐ. CamYynwgkugRtaMgtUcTaMgenH eKGacEksMrYlkMras;kMralxNÐeGayesþIgCag
6in. kñúgkrNIEdlPaBdabedaysarbnÞúkGefrGacTTYl)an. cMNaMfa kMralxNÐbegáItPaBdab nig

camber eRkamGMeBIbnÞúkGefrsUnüenAkñúg]TahrN_enH bNþalmkBIbnÞúklMnwg.

RtYtBinitüPaBdab (Deflection Check)³
eyIgRtYtBinitüEtPaBdabedaysarbnÞúkGefrEtb:ueNÑaH. BIeKalkarN_emkanic eyIgman
5 ML2
Δ=
48 Ec I s

I s = 216in.4

Ec = 4.03 ⋅10 6 psi
5 17,396(24 × 12)2
Δ E −W = = 0.17in.
48 4.03 ⋅10 6 × 216
5 21,218(20 × 12)2
Δ N −S = = 0.15in.
48 4.03 ⋅10 6 × 216
0.17 + 0.15
PaBdabkNþalElVgmFüm Δ=
2
= 0.16in.(4.1mm )



PaBdabEdlGacTTYlyk)an = 360 = 20 ×12 = 0.67in.(17mm) >> 0.16in.
LS
360
dUcenH eyIgGaceFVIkarKNnaCMuTIBIrCamYynwgkMras;kMralxNÐ 5.5in. kñúgkrNIEdlersIusþg;m:Um:g;
nominal rbs;kMralmanlkçN³RKb;RKan;edIm,IRTbnÞúk. kñúgkrNIenH h = 5.5in. minmanlkçN³RKb;

RKan;sMrab;ersIusþg;m:Um:g; nominal dUckarbgðajxageRkam.
ersIusþg;m:Um:g; nominal
Wu = 1.2 × 90 + 1.6 × 75 = 228 psf (11.0kPa )
dUcKña ElVgRbsiT§PaBtamTisxøI LS = 19.5 ft

ElVgRbsiT§PaBtamTisEvg LL = 23.5 ft
BIrUbTI 9>11 emKuNm:Um:g;sMrab;m:Um:g;emKuNGtibrmaKW
α N − S = 0.072
nig α E −W = 0.038
enAkñúgTis N-S eyIgman
M u = 0.072 × 228(19.5)2 × 12 = 74,906in. − lb / ft
Mu 74,906
Mn = = = 83,229in. − lb / ft
φ 0.9
cMNaMfa kMlaMgeRbkugRtaMgenAkñúgeRKOgbgÁúMenHmin)anbegáIt secondary moment M s edaysarvamin
manPaBCab;enARtg;RBMEdnkMralxNÐ. eyIgman Aps = 0.153in.2 enAelI 1.42 ft BIG½kSeTAG½kS ¬Edl
)anBIelIkmun¦ nig Aps / f = 0.153 /1.42 = 0.11in.2 / ftt . dUcKña kMlaMgeRbkugRtaMgRbsiT§PaB f pe
= 159,000 psi . dUcenH kñúgkrNIEdl A ps EdleRbIFMCag Pe / initial A ps eKRtUvkat;bnßy f pe ;

eyIgman
0.11
ρN −S = = 0.0018
12 × 5
20 × 12
pleFobElVgelIkMras;kMral =
6
= 40

BIsmIkar 9.23b
f 'c
f ps = f pe + 10,000 + ≤ f py ≤ f pe + 30,000
300 ρ p
5,000
f ps = 159,000 + 10,000 + = 178,259 psi
300 + 0.0018
< f py = 240,000 psi < f pe + 30,000 = 189,000 psi

< f ps lImIt = 189,000 psi O.K.



A ps f ps 0.11× 178,259
a= = = 0.38in.
0.85 f 'c b 0.85 × 5,000 × 12

m:Um:g; nominal EdlGacman M n = Aps f ps ⎛ d − a ⎞ = 0.11×178,259⎛ 5 − 0.2 ⎞
⎜
⎝ 2⎠
⎟ ⎜
⎝
38
⎟
⎠
= 94,316in. − lb / ft > M n tMrUvkar = 83,229in. − lb O.K.

enAkñúgTis E-W eyIgman
M u = 0.038 × 228(23.5)2 × 12 = 57,417in. − lb / ft
Mu 57,417
Mn = = = 63,797in. − lb / ft
φ 0.9
A ps = 0.153in.2 kñúg 1.69 ft. EdlKitBIG½kSeTAG½kS ¬BIelIkmun¦
0.153
A ps / ft = = 0.09in.2 / ft
1.69
0.09
ρ E −W = = 0.0015
12 × 5
5,000
f ps = 159,000 + 10,000 + = 180,111 psi O.K.
300 × 0.0015
0.09 × 180,111
a= = 0.32in.
0.85 × 5,000 × 12
⎛ 0.32 ⎞
m:Um:g; nominal EdlGacman = 0.09 × 180,111⎜ 5 −
⎝ 2 ⎠
⎟

= 78,456in. − lb / ft > M n tMrUvkar = 63,797in. − lb / ft O.K.

(29.1kN .m / m > 23.6kN .m / m )
ersIusþg;kMlaMgkat;
BIelIkmun/ aspect ratio k = 1.2 nigBIsmIkar 9.27
1 1
Vu = wu LS = × 228 × 19.5 = 1482lb / ft (N-S)
3 3
kw L
Vu = u S (E-W)
2k + 1
= 1,569lb / ft (22.9kN / m )
19.5
= 1.2 × 228 ×
2 × 1.2 + 1
BIsmIkar 9.26
⎛ V d⎞
2 f 'c bw d p ≤ Vc = ⎜ 0.6 f 'c + 700 u ⎟bw d p ≤ 5 f 'c bw d p
⎜
⎝ Mu ⎟
⎠
eKman 700(Vu d ) / M u = 0 enARtg;RBMEdnén single-panel wall-support slab enAkñúg]TahrN_enH.
enAkñúgkrNIEbbenH
Vc = 0.6 5,000 × 12 × 5 = 2,546lb / ft (37.2kN / m ) >> 1,569lb / ft



EdlvabMeBjlkçxNÐ. dUcenH TTYlykkarsikSaKNnaenH
h = 6in.(152mm )
d p = 5in.(127mm )

eRbIkabeRbkugRtaMg 7-wire 270-K Ggát;p©it 1/ 2in. EdlmanKMlat 17in.(432mm) KitBIG½kSeTAG½kS
kñúgTis N-S nig 20in.(508mm) G½kSeTAG½kS kñúgTis E-W. dUcKña eRbIEdk 2#4 ¬Ggát;p©it 12.7mm ¦
tambeNþay anchorage zone EdlB½T§CMuvijbrimaRtkMralTaMgGs;.

9.8. RbB½n§kMralmYyTis One-Way Slab Systems
kMralxNÐebtugeRbkugRtaMgmYyTiseFVIkarRsedogKñaeTAnwgFñwmedayminKitfavaCakMralTMr
samBaØ b¤kMralTMrCab;KñaeRcIneT. dUcenH eKsikSaKNnakMralmYyTisCaFñwmEdlmanTTwg 12in. . eK
dak;kabeRbkugRtaMgemenAkñúgTisénbeNþayrbs;kMral Edltamsn§wgtamElVgCab;. pleFobElVgelI
TTwgrbs; slab band RtUvman tMélFMCag 2 edIm,IeGayeKKitkMralxNÐCakMralxNÐmYyTis.
eyIgGaceRbIdMeNIrkarsikSaKNna nigkarsikSaviPaK nigeRbI]TahrN_enAkñúgCMBUkTI 6 sMrab;
karsikSaviPaK nigkarsikSaKNnaRbB½n§kMralxNÐeRbkugRtaMgmYyTisCab;.

9.9. karepÞr Shear-Moment eTAssrEdlRT Flat Plate
Shear-Moment Transfer to Column Supporting Flat Plates
9.9.1. ersIusþg;kMlaMgkat; Shear Strength
kMlaMgkat;rbs; plate nigkMralxNÐBIrTisKWCa three-dimensional stress problem. bøg;Edl
)ak;edaykMlaMgkat;eRKaHfñak;RbRBwtþtambrimaRtrbs;RkLaépÞrgbnÞúk nigmanTItaMgenARtg;cMgayEdl
pþl;eGayedaybrimaRtkMlaMgkat;Gb,brma bo . edayEp¥kelIkarepÞógpÞat;tamkarBiesaF nigkarviPaK
CaeRcIn bøg;kMlaMgkat;minKYrenAEk,rcMgay d / 2 BIRkLaépÞRbtikmμ b¤RkLaépÞbnÞúkRbmUlpþúM.
RbsinebIeKmindak;EdkBRgwgkMlaMgkat;Biess ersIusþg;kMlaMgkat; nominal Vc dUckarTamTar
eday ACI RtUv)ankMNt;enAkñúgsmIkar 9.24, 9.25 nig 9.26. eKGaceRbIsmIkar 9.27 sMrab;kMNt;tMél
Rbhak;RbEhlrbs;emKuNsMrab;KNnakMlaMgkat;emKuNxageRkA Vu enAkñúgkMralxNÐCab;BIrTisEdl
brimaRtrbs;manTMrB½T§CMuvij.



9.9.2. Shear-Moment Transfer
m:Um:g;KμanlMnwg (unbalanced moment) enARtg;épÞssrEdlCaTMrrbs;kMralminmanFñwmCakrNI
mYyénkarsikSaKNnaEdleRKaHfñak;CageKenAkñúgkarkMNt;smamaRtmuxkat; flat plate b¤ flat slab.
edIm,IFanaPaBRKb;RKan;rbs;ersIusþg;kMlaMgkat; eKTamTareGaymanepÞrm:Um:g;eTAssredaykarBt;begáag
Edlkat;tambrimaRtrbs;ssr nigedaykugRtaMgkat;cakp©it EdlRbEhl 60%RtUv)anepÞrdaykarBt;
begáag nig 40%RtUv)anepÞredaykMlaMgkat;.
cMENk γν énm:Um:g;EdlepÞredaycMNakp©iténkugRtaMgkMlaMgkat;fycuH enAeBlEdlTTwgrbs;
épÞénmuxkat;eRKaHfñak;EdlTb;Tl;m:Um:g;ekIneLIg
1
γv = 1− (9.28)
2 b1
1+
3 b2

Edl b2 = c2 + d CaTTwgénépÞrbs;muxkat;eRKaHfñak;EdlTb;Tl;m:Um:g; ehIy b1 = c1 + d CaTTwgénépÞ
EdlEkgeTAnwg b2 .
cMENkEdlenAsl; γ f énm:Um:g;KμanlMnwgEdlepÞredaykarBt;begáag nigEdlGMeBIelITTwgkMral
xNÐRbsiT§PaBcenøaHExSEdlesμInwg 1.5 dgénkMras;kMralsrub h enAelIRCugTaMgBIrrbs;ssr.
1
γf = = 1− γv (9.29)
2 b1
1+
3 b2

sMrab;ssrxageRkA b1 = c1 + d / 2 . tMélrbs; γ f GacekIneLIgrhUtdl; 1.0 RbsinebI Vu tUcCag
0.75φVc . enARtg;TMrxagkñúg eKGacbegáIn γ f 25% RbsinebI Vu ≤ 0.4φVc nig ρ ≤ 0.375 ρ b .

karEbgEckkugRtaMgkMlaMgkat;CMviujEKmssrRtUv)anbgðajenAkñúgrUbTI 9>15. vaERbRbYlCa
lkçN³bnÞat;CMuvijTIRbCMuTMgn;rbs;muxkat;eRKaHfñak;. kMlaMgkat;emKuN Vu nigm:Um:g;emKuNKμanlMnwg
M u EdlRtUv)aneKsnμt;favamanGMeBIenARtg;épÞssrRtuv)anepÞreTAkat;G½kSTIRbCMuTMgn; c − c rbs;mux

kat;eRKaHfñak;. dUcenH G½kSRtUv)ankMNt;TItaMgedayTTYl)anBIédXñas;kMlaMgkat; g ¬cMgayBIépÞssr
eTAbøg;G½kSTIRbCMuTMgn;¦ énmuxkat;eRKaHfñak; c − c sMrab;karepÞr shear-moment.
sMrab;karkMNt;kugRtaMgkMlaMgkat;GtibrmaEdlRtUvRTRTg;eday plate enAkñúgtMbn;RCugssr/ ACI
Code TamTarkareRbIR)as; full nominal moment strength M n RtUv)anpþl;eGayedaycMerok

ssrenAkñúgsmIkar 9.30 edIm,IeFVItamdUcCam:Um:g;KμanlMnwgEdlRtUv)anKuNedayemKuNcMENkepÞr (tran-
sfer fraction factor) γ v . m:Um:g;KμanlMnwg M n ≥ M ue / φ RtUv)anpÁúMeLIgedayBIrEpñk³ m:Um:g;cugkMral

GviC¢man (negative end panel moment) M ne = M e / φ enARtg;épÞrbs;ssr nigm:Um:g; (Vu / φ )g Edl


bNþalBIkMlaMgkat;brimaRtemKuNcakp©it (eccentric factored perimetric shear factor) Vu . tMél
kMNt;rbs;kugRtaMgkat;RtUv)ankMNt;edaysmIkarxageRkam³
vu ( AB ) Vu γ v M ue c AB
= + (9.30a)
φ φAc φJ c
vu (CD ) Vu γ v M ue cCD
= − (9.30b)
φ φAc φJ c
EdlersIusþg;kMlaMgkat; nominal KW
vu
Vn = (9.30c)
φ
Edl RkLaépÞrbs;ebtugénmuxkat;eRKaHfñak;snμt;
Ac =

= 2d (c1 + c2 + 2d ) sMrab;ssrxagkñúg

J c = lkçN³énmuxkat;eRKaHfñak;snμt;EdlRsedogKñanwgm:Um:g;niclPaBb:UElrénmuxkat;

tMél J c sMrab;ssrxagkúñgKW
Jc = 1
(c + d / 2)(d )3 + 2(d ) (c 3 + c 3 )+ (c + d )dc 2
AB CD 2 AB
6 3
BIeKalkarN_eKalénemkanicsMPar³ ersIusþg;kMlaMgkat;KW
Vu Mc
vu = +γv
Ac J
EdltYTIBIrenAGgÁxagsþaMCakugRtaMgkMlaMgkat;EdlekItBIm:Um:g;rmYlenARtg;épÞssr.
RbsinebIersIusþg;m:Um:g; nominal M n éntMbn;epÞrm:Um:g;-kMlaMgkat;eRkayBIkarKNnaénEdk
BRgwgmantMélFMCag M ue / φ enaHeKKYreRbI M n enAkñúgsmIkar 9.30a nig b CMnYseGay M ue / φ . enA
eBlEdlersIusþg;m:Um:g; M n = M ne + (Vu / φ )g mankarekIneLIgedaysarkareRbIEdkrgkarBt;begáag
eRcInCagtMrUvkarsMrab;Tb;Tl;nwg M ue / φ enaHPaBrwgRkajrbs;kMralmankarekIneLIg dUcenHkarekIn
eLIgkugRtaMgkMlaMgkat;EdlepÞr vu EdlKNnaBIsmIkar 9.30a nig b sMrab;begáIt full moment transfer.
dUcenH eKENnaM eGayrkSa design moment M ue EdlmantMélEk,rnwgtMélm:Um:g;emKuN M ue Rbsin
ebIeKcg;eCosvagkarekIneLIgkugRtaMgkMlaMgkat;EdlbNþalBIkarepÞrm:Um:g;bEnßm nigkarBarkarekIneLIg
bEnßmeTotén kMras;kMralxNÐ.
]TahrN_ 9>2 bgðajBIviFIsaRsþsMrab;KNnakugRtaMgkMlaMgkat;brimaRtkMNt;enAkñúg plate Rtg;
tMbn;EKmssr.
kñúgkrNIssrxagkñúg kugRtaMgkMlaMgkat;brimaRt vu GacmantMélFMCag kugRtaMgkMlaMgkat;Edl



KNnaedaysmIkar 9.30a nig b enAeBlEdlElVgEdlenAEk,rminmanTMhMesμIKña b¤minrgbnÞúkesμIKña. sM
rab;muxkat;kMralxNÐEdlCab;Tak;Tgnwgm:Um:g;emKuNenAkñúgssr nigCBa¢aMg/ ACI Code kMNt;faGgát;
EdlCaTMrdUcCassr b¤CBa¢aMgRtUvTb;Tl;nwgm:Um:g;KμanlMnwg
[
M ' = 0.07 (wnd + 0.5wnl )l2ln 2 − w'nd l '2 (l 'n )2 ] (9.31)
Edl w'nd / l '2 nig l'n sMedAdl;ElVgxøI. dUcenH eKRtUvEfmtYbEnßmeTAkñúgsmIkar 9.30a b¤ b
Vu γ v M u c AB γ v M ' c
vu = + + (9.32)
Ac Jc J 'c
Edl J 'c Cam:Um:g;niclPaBb:UElrEdlmanRkLaépÞm:Um:g;EdlRtUv)anykkñúgTisedAEkgnwgTisEdleRbI
sMrab; J c .

9.9.3. tMrUvkarPaBdabsMrab;kMras;Gb,brma³ viFIminpÞal;
Deflection Requirements for Minimum Thickness: An Indirect Approach
sMrab;kar)a:n;RbmaNkMras;kMralxNÐBIrTisdMbUg eKcaM)ac;eRbInUveKalkarN_ENnaMsMrab;tMélRb
hak;RbEhledIm,IeRCIserIskMras;sakl,gelOn nigmanRbsiT§PaB. eKrMBwgfapleFobElVgelIkMras;
kMralxNÐsMrab;kMralxNÐebtugeRbkugRtaMgnwgmantMélFMCagpleFobElVgelIkrM as;kMralxNÐsMrab;kM
ralxNÐebtugGarem: RbsinebIminmankar)at;bg;KuNsm,tþ×énGgát;eRbkugRtaMg.
eKniymeRbI service live load CaplbUksrubénbnÞúkefr nigbnÞúkGefredIm,IkMNt;PaBdab.
eKeRbIbnÞúklMnwgEdl)anBIbgÁúMTTwgrbs;kMlaMgeRbkugRtaMgedIm,IeFVIeGayPaBdabEdlekItBI dead load
NWt b¤begáIteGayman camber RbsinebIbnÞúkGefrmantMélFMEmnETn. eKeRbIeKalkarN_ENnaMsMrab;
tMélRbhak;RbEhlénpleFobElVgelIkMras;kMralxNÐ 16 eTA 25 sMrab; solid cantilever slabs nig
40 eTA 50 sMrab;kMralxNÐCab;BIrTis. sMrab; waffle slab eKENnaMeGayeRbItMél 35 eTA 40. sMrab;

ElVgTMrsamBaØ nigsMrab; single-T nig double-T eRbI 90%éntMélTaMgenHsMrab;karsakl,gelIkTImYy.
ACI tMrUvfapleFobElVgelIPaBdabGb,brmaRtUv)ankMNt;y:agtwgrwgEdlGaRs½ynwgRbePT

énkardak;bnÞúk niglkçxNÐénkareRbIR)as;. karkMNt;enHminRtUv)aneRbIsMrab;karkarBarsñameRbHénkar-
garbegðIyenAelIBIdan nigsñameRbHelI partition nigkardk;TwkenAelIdMbUl. taragTI 9>1 eGaynUvtMél
ENnaMénpleFobElVgelIPaBdabsMrab;karRKb;RKgPaBdab.
karkMNt;PaBdab b¤ camber rbs; plate nigkMralxNÐBIrTisebtugeRbkugRtaMg nigebtugeRbkug
RtaMgEdlmanlkçN³suRkitCagRtUv)anbgðajenAkñúgcMnuc 9.12. viFIenHeRbIPaBrwgRkajénGgát;EdlRb-
sBVKñaedayeRbIviFIeRKagsmmUlkñúgkarsikSaviPaKPaBdab. viFIenHmanlkçN³gayRsYl nigsmehtu


pledaysaremKuNPaBrwgRkajrbs;Ggát;epSg²RtUv)anKNnarYcehIyenAkñúgkarviPaKkarBt;begáag
(flexural analysis) éneRKagCab;smmUl.

9.10. Step-By-Step Trial-and-Adjustment Procedure for the Design of a
Two-Way Prestressed Slab and Plate System
xageRkamenHCaCMhanbnþbnÞab;EdlRtUv)anesñIreLIgsMrab;kargarsikSaKNna nigsMrab;kargar
viPaKkMralxNÐebtugeRbkugRtaMgBIrTis³
!> kMNt;faetIragFrNImaRtrbs;kMralxNÐ nigkardak;bnÞúktMrUvrviPaKtamlkçN³BIrTiseday
viFIeRKagsmmUlb¤Gt;.
@> eRCIserIskMras;kMralxNÐsakl,gsMrab;beNþayGtibrma h = L / 45 b¤TTwgGtibrma h =
L / 45 . KNnabnÞúkefresvakmμsrub bnÞúkGefresvakmμsrub nigbnÞúkemKuN.

#> snμt; tendon profile kat;tamElVgCab;kñúgTis E-W nigTis N-S ehIykMNt;kMlaMgeRbkug
RtaMg F / kugRtaMgebtug f c = F / Ac / nigcMnYn strand kñúgmYyElVg. KNna balancing
load intensity Wbal = 8Fa / L2 nigKNna net load Wnet ↓ = Ww↓ − Wbal ↓ .

$> kMNt;lkçN³eRKagsmmUl (equivalent frame characteristics) tamviFIeRKagsmmUl
nigkMNt;PaBrwgRkajTb;karBt; nigPaBrwgRkajTb;karrmYlrbs;kMralEdleGayeday
4 EI
Kc ≅
Ln − 2h



nig Kt = ∑
9 Ecs C
3
⎛ c ⎞
L2 ⎜1 − 2 ⎟
⎜
⎝ L2 ⎟
⎠
Edl C = ∑(1 − 0.63x / y )x3 y / 3 . nigbnÞab;mkKNna
−1
⎛ 1 1 ⎞
K ec =⎜
⎜K + K ⎟⎟
⎝ c t ⎠

sMrab;ssrxageRkA nigxagkñúg ehIyPaBrwgRkajrbs;kMralxNÐ
4 EI
Ks ≅
L1 − c1 / 2
Edl L1 CaElVgEdlKitBIG½kS nig c1 CakMras;ssrsMrab;tMNrvagssr nigkMralxNÐnImYy².
%> KNnaemKuNEbgEckm:Um:g;sMrab;kMralBItMél K ec nig K s EdlTTYl)anenARtg;tMNnImYy²
Ks
DF =
∑K
Edl ∑ K = K ec + K S (left ) + K S (rkght ) . bnÞab;mkKNna fixed-end moment FEM enA
Rtg;tMNsMrab; net load EdleGayeday FEM = WL2 / 12 sMrab;bnÞúkBRgay.
^> GnuvtþkarEbgEckm:Um:g;sMrab; net load moment M net nigEktMrUvm:Um:g;EdlEbgEckeLIgvij
edIm,ITTYl)antMél net moment enARtg;épÞrbs;TMr. smIkarKW M n = M n(centerline) − Vc / 3 .
bnÞab;mkepÞógpÞat;fakugRtaMgebtug
P M net
ft = − +
A S
EdlTTYl)anm:Um:g;TaMgBIrtUcCagkugRtaMgGnuBaØatGtibrma ft = 6 f 'c psi(0.5 f 'c MPa)
sMrab;muxkat;TMr nig ft = 2 f 'c psi(0.166 f 'c MPa ) sMrab;muxkat;kNþalElVg.
&> KNna balanced service-load fixed-end moment
Wbal L2
FEM bal =
12
nigGnuvtþkarEbgEckm:Um:g;énm:Um:g;bnÞúklMnwg M bal . bnÞab;mkkMNt; primary moment
M 1 = Pe e nig secondary moment M s = (M bal − M 1 ) .

*> KNna fixed-end factored load moment FEM u− = (Wu L2 )/ 12 ehIyGnuvtþkarEbgEck
m:Um:g;én factored moment. bnÞab;mkKNna required design moment M u = M u− − M s
sMrab;kMralxNÐenARtg;RKb;tMN nigenARtg;m:Um:g;viC¢manGtibrma M u tambeNþayElVg.



(> kMNt; required nominal moment strength M n = M u / φ sMrab;m:Um:g;TMrGviC¢man − M u
nigm:Um:g;ElVgviC¢man + M u . bnÞab;mkRtYtBinitüemIlfa − M n nig + M n EdlGacman
sMrab;kMralxNÐ nigsMrab;EdkeRbkugRtaMgRKb;RKan;b¤Gt;. bnÞab;mkeTot kMNt; inelastic
moment redistribution ΔM R BIdMeNIrkarEdlmanerobrab;enAkñúgcMnuc 4.12.4 nig 6.7.2.

Edl ΔM R = ρ D (support M u ) . bEnßmEdkFmμtaenARtg;TMr nigkNþalElVgRbsinebI
caM)ac; edayrMlwkfaEdkminrgeRbkugRtaMgGb,brma As = 0.00075hL .
!0> RtYtBinitü nominal shear strength rbs;kMralxNÐenARtg;TMrxageRkA nigTMrxagkñúg rYcKNna
karepÞr shear-moment nigkarepÞr flexure-moment eTAssr. emKuNkMlaMgkat;m:Um:g;
(moment shear factor) KW
1
γ v = 1−
2
1+ b1 / b2
3
ehIyemKuNkarBt;begáagm:Um:g; (moment flexure factor) KW
1
γf =
2
1+ b1 / b2
3
Edl b1 = c1 + d / 2 sMrab;ssrxageRkA
b1 = c1 + d sMrab;ssrxagkñúg
b2 = c2 + d
eKGacbegáIntMél γ f 25%enARtg;TMrxagkñúg nigbegáInrhUtdl;esμInwg 1.0 enARtg;TMrepSg
eTotdUcbgðajenAkñúgsmIkar 9.29. bnÞab;mkKNna c AB nig cCD sMrab;ssrxageRkA k¾dUc
total nominal unbalanced moment strength M n = M ue + Ve g .

!!> KNna shear ultimate stress EdlbNþalBIkMlaMgkat;brimaRt nigT§iBlrbs; γν M n ³
γ c M
vn = u + ν AB n ≤ vc GnuBaØatGtibrma
V
φ A J
v c c
V
Edl kugRtaMgGnuBaØatGtibrma vc = β p f 'c + 0.3 f c + b pd
o

β p = tMélEdltUcCageKkñúgcMeNam 3.5 nig (α s d / bo + 1.5)
φ = 0.75 sMrab;kugRtaMgkat; nigkugRtaMgrmYl
Edl α s = 40 sMrab;ssrxagkñúg/ 30 sMrab;ssrxag nig 20 sMrab;ssrkac;RCug.



ssrRtUvmanmuxkat;y:agticbMput 4in. BIépÞénRCugGt;Cab; ehIy f 'c minKYrFMCag
5,000 psi nig f RtUvmantMélGb,brmaesμInwg 125 psi nigGtibrmaesμInwg 500 psi ebImin

dUecñaHeT eKKYrKNna vc BItMélEdltUcCageKEdlTTYl)anBIsmIkarxageRkam
⎛ 4 ⎞ ⎛α d ⎞
vc = ⎜ 2 +
⎜ ⎟ f 'c b¤ vc = ⎜ s + 2 ⎟ f ' c b¤ vc = 4 f 'c
⎝ β ⎟ c ⎠ ⎝
⎜ b
o ⎠
⎟

!@> KNnatMélm:Um:g;emKuN γ f M n nigRtYtBinitüersIusþg;EdlGacekItman M n énmuxkat;
EdlRbmUlpþúMEdkenAkñúg column band [c + 2(1.5h)] .
!#> RtYtBinitüPaBdab nig camber rbs;kMralxNÐ
!$> TTYlykkarsikSaKNnaRbsinebIvabMeBjRKb;lkçxNÐEdl)anerobrab;xagelI. bnÞab;mk
GnuvtþkarKNnasMrab;Tis E-W nigTis N-S rbs;RbB½n§kMralxNÐ.
rUbTI 9>16 bgðajBI flowchart sMrab;karsikSaKNna nigkarsikSaviPaK plate nigkMralxNÐeb
tugeRbkugRtaMgBIrTis



9.11. sikSaKNnaRbB½n§kMral Flat-Plate ebtugeRbkugRtaMgTajCaeRkay
Design of Prestressed Post-Tensioned Flat-Plate Floor System
]TahrN_ 9>2³ RbB½n§kMral post-tensioned prestressed nonbonded flat-plate sMrab;GKarsñak;enA
RtUv)anbgðajenAkñúgrUbTI 9>17. kMralxagmanTMhM 17 ft 6in. × 20 ft (5.33m × 6.10m) EdlKitBIG½kS
eTAG½kS ehIykMralxagkñúgmanTMhM 24 ft × 20 ft (7.32m × 6.10m) . kMBs; lu rbs;Can;KW 8 ft 9in.
(2.67m ) . sikSaKNnakMralxNÐenHedIm,IRTnUv working live load WL = 40 psf (1.92kPa ) nig
superimposed dead load WSD = 20 psf (0.96kPa ) EdlbNþalBI partition nig flooring. snμt;enA

kñúgdMeNaHRsayenHfaRKb;kMralTaMgGs;TTYlbnÞúkGefrkñúgeBlCamYyKña nigepÞógpÞat;lT§PaBkarepÞr
kMlaMgkat;-m:Um:g; (shear-moment transfer capacity) rbs;kMralenARtg;ssr. eRbIkabeRbkugRtaMg 7-
wire 270-K Ggát;p©it 1 / 2in. ehIyeRbIviFIeRKagsmmUl (equivalent frame method) kñúgkarsikSa

KNnaenH. xageRkamenHCaTinñn½yEdleKeGay³
f 'c = 4,000 psi (27.6MPa ) ebtugTMgn;Fmμta
f 'ct = 3,000 psi (20.7 MPa )
enARtg;TMr f t = 6 f 'c = 380 psi(2.62MPa)
enARtg;kNþalElVg f t = 2 f 'c = 127 psi(0.88MPa)
kugRtaMgkMlaMgkat;rbs;ebtugGtibrma vc RtUv)anTamTareday ACI Code
f pu = 270,000 psi (1,862MPa )

f ps minRtUvFMCag 185,000 psi(1,276MPa )
f py = 243,000 psi (1,675MPa )

f pe = 159,000 psi (1,096MPa )

(
E ps = 29 ⋅10 6 psi 200 ⋅10 3 MPa )
f y = 60,000 psi (414MPa )

dMeNaHRsay
Tis N-S
I. Service Load analysis
!> bnÞúk
edIm,IRKb;RKgPaBdab snμt;fakMras;kMralxNÐ h ≅ L / 45 . TisbeNþay 20 ×12 / 45 = 5.33in.
ehIy h = 24 ×12 / 45 = 6.40in. . dUcenHsakl,g h = 6 12 in.(165mm) TMgn;pÞal;rbs;kMral = 81 psf .



bnÞúkGefrEdldak;bEnßmBIelI = 20 psf dUcenHeyIg)an
bnÞúkGefrsrub WD = 101 psf
WL = 40 psf
bnÞúkesvakmμ Ww = WD + L = 141 psf (6.75kPa )

Wu = 1.2WD + 1.6WL = 1.2 × 101 + 1.6 × 40 ≅ 186 psf (8.9kPa )
sMrab;EpñkéndMeNaHRsayenH)
Ln = bay span (N-S

L2 = band width (Tis E-W)

@> bnÞúklMnwg nig tendon profile
edIm,IeFVIkar)a:n;RbmaNelIkdMbUgsMrab;bnÞúklMnwg snμt;tMélkugRtaMgsgát;enAelIebtugmFüm
EdlbNþalBIbnÞúklMnwgKW f c = 170 psi(1.17MPa) . kMlaMgÉktþa F = 170 × 6.5 ×12 = 13,260lb / ft
(193.6kN / m ) . dUcenH sakl,gEdkeRbkugRtaMg 7-wire 270-K Ggát;p©it 1 / 2in. . eyIgeXIjfakM-
laMgRbsiT§PaB Pe kñúgkabeRbkugRtaMgmYy = Aps f pe = 0.153 ×159,000 = 24,327lb . sMrab; L =
20 ft tamTisbeNþayrbs;eRKOgbgÁúM kMlaMgsrubKW Fe = FL = 13,260 × 20 = 265,200lb(1,180kN ) .

cMnYnrbs; strand kñúgmYy bay KW Fe / Pe = 265,200 / 24,327 ≅ 11 ehIykMlaMgeRbkugRtaMgRb
siT§PaBsrub Pe = Fe = 24,327 ×11 = 267,597lb . kMlaMgÉktþaCak;Esþg F = 267,597 / 20 =
13,380lb / ft (195.3kN / m ) ehIykugRtaMgsgát;enAkñúgebtugCak;Esþg f c = F / A = 13,380 / (6.5 × 12 )

≅ 172 psi ≅ 170 psi KWbMeBjlkçxNÐ. dUcenH yk f c = 172 psi EdlbNþalBIbnÞúklMnwg ehIysnμt;

parabolic tendon profile dUcbgðajenAkñúgrUbTI 9>18.

ElVgxageRkA AB b¤ CD enARtg;kNþalElVg
3.25 + 5.50
a1 = a3 = − 1.75 = 2.625in.
2
BIsmIkar 1.16 sMrab; parabolic tendon


8Fa
W=
L2n
8 × 13,380 × 2.625 / 12
Wbal = ≅ 72 psf
(18)2
Net load EdlbegáItm:Um:g;Bt;KW
Wnet = Ww − Wbal = 141 − 72 = 69 psf (3.30 KPa )
ElVgxagkñúg BC
a 2 = 6.5 − 1 − 1 = 4.5in.
8Fa 8 × 13,380 × 4.5 / 12
Wbal = 2 = ≅ 70 psf
Ln (24)2
Wnet = 141 − 70 = 71 psf (3.40kPa )
#> lkçN³rbs;eRKagsmmUl (Equivalent Frame Characteristics)
ykeRKagsmmUlenAkñúgTis N-S Edlbøg;RtUv)anbgðajedaykarqUtenAkñúgrUbTI 9>17.
PaBrwg RkajTb;nwgkarBt;Rbhak;RbEhlrbs;ssrxagelI nigxageRkamtMNkMralxNÐ
¬m:Um:g;kñúgmYyÉktþa mMurgVil¦ nigBIsmIkar 9>9 KW
4 Ec I c
Kc =
Ln − 2h
Edl Ln = Lu = 8 ft 9in. = 105in.
(a) PaBrwgRkajssrxageRkA (14in.×12in. )
sMrab;ssrxageRkA b = 14in. dUcenH I c = 14(12)3 /12 = 2,016in.4 . snμt;fa Ecol / Eslab =
Ecc / Ecs = 1.0 nigeRbI Ecc = Ecs = 1.0 enAkñúgkarKNna eday Ecs minRtUv)anKitenAkñúgsmIkar

sMrab; K c . bnÞab;mk eyIgTTYl)an
4 × 1× 2,016
K c srub = × 2 ¬sMrab;cug nigKl;ssr¦
105 − (2 × 6.5)
= 175.3in. − lb / rad / Ecc
BIsmIkar 9.10b efrkMlaMgrmYlKW
⎛ x ⎞ x3 y
C = ∑⎜1 − 0.63 ⎟
⎜
⎝ y⎟ 3
⎠
⎛ 6.5 ⎞ 3 12
= ⎜1 − 0.63 × ⎟6.5 × = 724
⎝ 12 ⎠ 3
PaBrwgRkajTb;karrmYlrbs;kMralenARtg;G½kSssrKW



9 Ecs C
Kt = ∑ 3
⎛ c ⎞
L2 ⎜1 − 2 ⎟
⎜ L ⎟
⎝ 2⎠
9 × 1 × 724 9 × 1 × 724
= 3
+ 3
⎛ 14 ⎞ ⎛ 14 ⎞
20 × 12⎜1 − ⎟ 20 × 12⎜1 ⎟
⎝ 12 × 20 ⎠ ⎝ 12 × 20 ⎠
= 65.0in. − lb / rad / Ecs
BIsmIkar 9.7/ PaBrwgRkajsmmUlrbs;ssrKW
−1 −1
⎛ 1 1 ⎞ ⎛ 1 1 ⎞
K ec =⎜
⎜K + ⎟ =⎜ + ⎟ = 47in. − lb / rad / Ecc
⎝ c Kt ⎟
⎠ ⎝ 175.3 65 ⎠
(b) PaBrwgRkajssrxagkñúg (14in.× 20in. )
sMrab;ssrxagkñúg b = 14in. / dUcenH I = 14(20)3 /12 = 9,333in.4 . dUcenH eyIgman
4 × 1× 9,333
K c srub = × 2 = 812in. − lb / rad / Ecc
105 − 2 × 6.5
⎛ 6.5 ⎞
⎟ × (6.5) ×
3 20
C = ⎜1 − 0.63 × = 1,456
⎝ 20 ⎠ 3
9 × 1,456 9 × 1,456
Kt = 3
+ 3
= 131in. − lb / rad / Ecs
⎛ 14 ⎞ ⎛ 14 ⎞
20 × 12⎜1 − ⎟ 20 × 12⎜1 − ⎟
⎝ 12 × 20 ⎠ ⎝ 12 × 20 ⎠
−1
⎛ 1 1 ⎞
K ec = ⎜ + ⎟ = 113in. − lb / rad / Ecc
⎝ 812 131 ⎠
(c) PaBrwgRkajrbs;kMralxNÐ
BIsmIkar 9.9
4 Ecs I s
Ks =
c
Ln − 1
2
Edl Ln CaRbEvgElVgEdlKitBIG½kSeTAG½kS nig c1 CakMras;ssr. TTwg slab band enAkñúgTis
E-W KW 20 / 2 + 20 / 2 = 20 ft . dUcenH I s = 20 × 12(6.5)3 / 12 = 5,493in.4 ehIysMrab;kMralenA

xagsþaMssrxageRkA A
4 × 1× 20(6.5)3
Ks = = 108in. − lb / rad / Ecs
12 × 17.5 − 12 / 2
sMrab;kMralxNÐenAxageqVgssrxagkñúg B
4 × 1× 20(6.5)3
12 × 17.5 − 20 / 2
ehIy sMrab;kMralenAxagsþaMssrxagkñúg B


4 × 1× 20(6.5)3
12 × 24 − 20 / 2
BIsmIkar 9.12/ emKuNEbgEckm:Um:g;enAelIkMralxNÐRtg;tMNKW DF = K s / ∑ K Edl ∑ K =
K ec + K s (left ) + K s (right ) . dUcenHsMrab;tMNkMralxNÐxageRkA A / DF = 108 / (47 + 108) =

0.697 sMrab;tMNkMralxNÐxageqVg B / DF = 110 / (113 + 110 + 79 ) = 0.364 nigsMrab;tMN

kMralxNÐxagsþaM B / DF = 79 /(113 + 110 + 79) = 0.262 .
$> Design Service-Load Moment and Stresses
Design net load moment
sMrab;ElVgxageRkA AB nig CD / Wnet = 69 psf . dUcenHm:Um:gbgáb;cug (fixed-end moment) KW
WL2 69 × (17.5)2
FEM = n
= × 12 = 21.1 ⋅10 3 in. − lb
12 12
dUcKña sMrab;ElVgxagkñúg BC / Wnet = 71 psf . dUcenHm:Um:g;bgáb;cugKW
71(24 )2
FEM = × 12 = 40.9 ⋅10 3 in. − lb
12
edayGnuvtþkarviPaKkarEbgEckm:Um:g;dUcbgðajkñúgtarag 9>2/ eKGaceRbIemKuN carryover COF
= 1 / 2 sMrab;RKb;ElVgTaMgGs;. eKRtUveFVIkarEksMrYlkarsnμt;EbbenH edaysareKecalT§iBlén

nonprismatic section eTAelI fixed-end moment nigemKuN carryover. enAkñúgeRKageRcInElVg

eKGacsnμt;faeRKagenARtg;tMNénElVgBIrEdlKitBIxageqVgtMN C RtUv)anKitfaCaTMrbgáb;kñúg
karEbgEckm:Um:g;.



kugRtaMgTajrbs;ebtugkMralenARtg;TMr
Net moment enARtg;épÞxagkñúgrbs;ssr B Caplsgénm:Um:g;Rtg; centerline CamYynwg
Vc / 3 Edl
20 ⎛ 71× 24 ⎞
M net , max = 39.56 ⋅10 3 − ⎜ ⎟ = 33,880in. − lb / ft
3 ⎝ 2 ⎠
m:UDulmuxkat;rbs;kMralxNÐ S = bh 2 / 6 = 12(6.5)2 / 6 = 84.5in.3 ehIyeyIgmankugRtaMg
ebtugsMrab;TMr
= +229 psi (1.63MPa )(T ) ]
P M 33,880
ft = − + = −172 +
A S 84.5
dUcenH kugRtaMgGnuBaØat f t = 6 f 'c = 380 psi > 229 psi RKb;RKan;.
kugRtaMgTajrbs;ebtugkMralenARtg;kNþalElVg
Net moment GtibrmakNþalElVgKW WL2 / 8 − 39.56 ⋅103 b¤
71(24 )2
M net , max = × 12 − 39.56 ⋅ 10 3 = 21,784in. − lb / ft (7.85kN / m )
8
ehIy f t Rtg;kNþalElVg =− +
P M
A S
= −172 +
21,784
84.5
= +86 psi (0.545MPa )(T )

dUcenH kugRtaMgGnuBaØati ft = 2 f 'c = 127 psi > 86 psi RKb;RKan;.
RbsinebI f t > kugRtaMgGnuBaØat f t / kMlaMgTajTaMgmUlRtUv)anykedayEdkBRgwgFmμtaCamYy
kugRtaMg f s = f y / 2 .
Ultimate Flexural Strength Analysis
II. Design Moment M u
!> Balanced moments M bal
Secondary moment RtUv)aneGayeday M s = M bal − M 1 / Edl M bal Ca balanced

moment nig M 1 Ca primary moment = Pe e = Fe . sMrab;ElVg AB b¤ CD
72(17.5)2
FEM bal = × 12 = 22,050in. − lb / ft
12
nigsMrab;kMral BC
70(24 )2
FEM bal = × 12 = 40,320in. − lb / ft
12
karGnuvtþkarEbgEckm:Um:g;dUcenAkñúgtarag 9>3 nwgkMNt;m:Um:g; M bal GtibrmasMrab;tMNssr
xageRkA.



@> Secondary moments M s nigm:Um:g;bnÞúkemKuN M u
ElVg AB
BI tendon profile énrUbTI 9>18/ e = 0 . dUcenHeyIgman³
Primary moment M 1 / ft enARtg;TMr A = Pe e = 0

M bal = 5,670in. − lb / ft ¬BItarag 9>3¦
M s = M bal − M 1 = 5,670 − 0 = 5.67 ⋅10 3 in. − lb / ft
Wu l 2 186(17.5)2
m:Um:g;bgáb;cugbnÞúkemKuN FEM u =
12
=
12
× 12 = 56,963in. − lb / ft

ElVg BA
BI tendon profile enAkñúgrUbTI 9>18/ e = 6.5 / 2 − 1 = 2.25in. . dUcenHeyIgman³
M 1 = 13,380 × 2.25 = 30,105in.lb / ft (11.16kN .m )
M s = 34,460 − 30,105 = 4,355in. − lb / ft (1.61kN .m / m )
m:Um:g;bgáb;cugbnÞúkemKuN FEM u = 56,963in. − lb / ft (21.1kN .m / m)
ElVg BC
e = 2.25in.
M 1 = 30,105in. − lb / ft



M s = 39,320 − 30,105 = 9,215in. − lb / ft (3.4kN .m / m )

m:Um:g;bgáb;cugbnÞúkemKuN FEM u = 18612 ) ×12 = 107,136in. − lb / ft (39.7kN .m / m)
(24 2

GnuvtþkarEbgEckm:Um:g;sMrab;m:Um:g;emKuNdUcenAkñúgtarag 9>4. sikSaviPaKKMrUénkardak;bnÞúkelI
ElVgqøas;edIm,ITTYllkçxNÐGaRkk;bMputsMrab;m:Um:g;esvakmμ nigm:Um:g;bnÞúkemKuN.

#> Design moments M u
m:Um:g;KNna (design moment) M u Caplsgénm:Um:g;bnÞúkemKuN M u− nig secondary
moment M s b¤ M u = M u − M s ¬BIsmIkar 9.17¦.
−

m:Um:g; − M u Rtg;tMN A ¬ElVg AB ¦
sMrab;m:Um:g;Rtg;tMN A ¬ElVg AB ¦/ M s = 5,670in. − lb / ft ¬)anBIelIkmun¦ ehIym:Um:g;Rtg;
centerline M u = 12,310 − 5,670 = 6,640in. − lb / ft . karkat;bnßym:Um:g;BIssr A = Vc / 3 . dUcenH
− −
Wu L M u @ B − M u @ A 186 × 17.5 103 (89.88 − 12.31)
V AB = − = −
2 Ln 2 17.5 × 12

= 1627.5 − 369.4 = 1231.1lb / ft

c = 12in.
m:Um:g;Rtg; centerline M u = 12,310 − 5,670 = 6,640in. − lb / ft
m:Um:g;Rtg;épÞssrtMrUvkar M u = 6,640 − 123131×12
.



= 6,640 − 4,924 = 1,716in. − lb / ft (0.64kN .m / m )

m:Um:g;tMrUvkar M
− Mn = u =
φ
1,716
0.9
= 1,987in. − lb / ft (0.4kN .m / m )

m:Um:g; − M Rtg;tMN B ¬ElVg BA ¦
u

sMrab;m:Um:g;Rtg;tMN B ¬ElVg BA ¦/ M s = 4,355in. − lb / ft ¬)anBIelIkmun¦ ehIym:Um:g;Rtg;
centerline M u = 89,880 − 4,355 = 85,525in. − lb / ft . dUcenH

V AB = 1627.5 + 369.4 = 1,996.9lb / ft

c = 20in.
m:Um:g;Rtg;épÞssrtMrUvkar M u = 85,525 − 1,996.39 × 20
= 85,525 − 13,313 = 72,212in. − lb / ft (26.7 kN .m / m )

m:Um:g;tMrUvkar M
− Mn = u =
φ
72,212
0.9
= 80,236in. − lb / ft (31.7 kN .m / m )

m:Um:g; − M Rtg;tMN B ¬ElVg BC ¦
u

sMrab;m:Um:g;Rtg;tMN B ¬ElVg BA ¦/ M s = 9,215in. − lb / ft ehIym:Um:g;Rtg; centerline
M u = 93,440 − 9,215 = 84,225in. − lb / ft . dUcenH
186 × 24
VBC = = 2,232lb / ft
2
m:Um:g;Rtg;épÞssrtMrUvkar M u = 84,225 − 22323× 20
= 84,225 − 14,880 = 69,345in. − lb / ft (25.7 kN .m / m )

m:Um:g;tMrUvkar M
− Mn = u =
φ
69,345
0.9
= 77,050in. − lb / ft (28.56kN .m / m )

m:Um:g; + M GtibrmakNþalElVg AB
u

snμt;facMnuckMlaMgkat;sUnü nigm:Um:g;GtibrmaKW x BIépÞ A . enaH
x = V AB / Wu = 1231.1 / 186 = 6.62 ft . dUcKña BItarag 9>4 m:Um:g;cug M u Rtg;
−

A = 12,310in. − lb / ft nigBIelIkmun M s = 1 (5,670 + 4,355) = 5,013in. − lb / ft . dUcenH eyIgman
2
2
m:Um:g;Gtibrma + M u = V AB x − Wu2x − M u− + M s
186(6.62 )2
= 1231.1× 6.62 × 12 − × 12 − 12,310 + 5,013
2
= 97,799 − 48,908 − 12,310 + 5,013
= 41,594in. − lb / ft (15.4kN .m / m ) enARtg; 6.62 ft BI A



m:Um:g;tMrUvkar + M n = M u = 410,594 = 46,216in. − lb / ft (17.2kN .m / m)
φ .9
m:Um:g; + M GtibrmakNþalElVg BC
u

BIelIkmun VBC = 2,232lb / ft nig x = Ln / 2 = 24 / 2 = 12 ft . m:Um:g;kNþalElVgrbs;ElVg
samBaØKW
Ln ⎛ L ⎞ (L )
M u = V AB × − ⎜Wu × ⎟
2 ⎝ 2⎠ 4
24 186(24 )2
= 2,232 × − = 13,392 ft − lb / ft = 160,704in. − lb / ft
2 8
müa:gvijeTot/ m:Um:g;kNþalElVgrbs;ElVgsamBaØKW
Wu L2 186(24 )2
M= = × 12 = 160,704in. − lb / ft
8 8
−
+ Mu = M − Mu + M s
BItarag 9>4 eyIgman M u− = −93,440in. − lb / ft nig M s = 9,215in. − lb / ft .
dUcenHersIusþg;m:Um:g;GtibrmatMrUvkarenAkNþalElVgKW
+ M u = 160,704 − 93,440 + 9,215 = 76,479in. − lb / ft (27.13kN .m / m )
ehIyersIusþg;m:Um:g; nominal tMrUvkarKW
Mu
= 84,977in. − lb / ft (30.14kN .m / m )
79,479
+ Mn = =
φ 0.9
rUbTI 9>19 eGaynUvdüaRkaménersIusþg; design moment tMrUvkar M u tambeNþayElVg nigtMélx<s;
bMputrbs;m:Um:g;.
III. Flexural Strength M n (Nominal Moment Strength)
ACI Code RtUvkarbrimaNEdkGb,brmarbs;EdkFmμtaminrgeRbkugRtaM. BIsmIkar 9.20
As = 0.00075hLn
!> muxkat;TMrxagkñúg B
sMrab;muxkat;TMrxagkñúgRtg;cMnuc B / m:Um:g; nominal tMrUvkarEdllubKW M n = 77,050in. − lb / ft
dUcenH/ RkLaépÞGb,brmarbs;EdkBRgwgminrgeRbkugRtaMgkñúgTisnImYy²EdlsßitenAkñúgRkLaépÞm:Um:g;
GviC¢manrbs;kMralRtg;ssrRtUv)anKNnadUckñúgsmIkar 4.55(b) KW As = 0.00075hl Edl h = kMras;
kMralxNÐsrub nig l = RbEvgElVgkñúgTisRsbeTAnwgTisrbs;EdkEdlRtUvkMNt;.
⎟ × 12 = 1.23in. (7.93cm )
⎛ 18 + 24 ⎞
As = 0.00075 × 6.5⎜ 2 2
⎝ 2 ⎠



dUcenH sakl,g 6#4 EdlmanRbEvg 11 ft ehIyKMlatGtibrmarbs;EdkKW 6in.(152mm) edayKitBI
G½kSeTAG½kS EdlvaRtUv)anRbmUlpþúMenAmþMússrelI band width EdlesμIeTAnwgTTwgssrbUkCamYynwg
1.5 dgénkMras;kMralelIRCugnImYy²rbs;ssr. enaH

As = 6 × 0.20 = 1.20in.2 ≅ 1.23in.2 EdlRtUvkar O.K.

TTwgkMral = 20 ft
1.2
As / ft = = 0.06in.2
20
BIsmIkar 9.23b/ design stress enAkñúg tendon KW
f 'c
f ps = f pe + + 10,000 psi
300 ρ p
A ps 11× 0.153
ehIy ρp =
bd
=
(20 × 12)5.5
= 0.0013

f pe = 159,000 psi

+ 10,000 = 179,256 pis (1,236 MPa )
4,000
f ps = 159,000 +
300 × 0.0013



179,256 × 0.153 × 11
F ps = = 15,084lb / ft
20
Fs = 60,000 × As / ft = 60,000 × 0.06 = 3,600lb / ft
kMlaMgsrub F / ft = Fps + Fs = 15,084 + 3,600 = 18,684lb / ft ehIyeyIgk¾man
f +
kMBs;bøúksgát; a = As 0.y85 fA'psbf ps
c

= 0.46in.(11.7mm )
18,684
=
0.84 × 4,000 × 12
eKRtUvdak; tendon nig bar elInIv:UdUcKña d = 6.5 − 1 = 5.5in. . ehIy M n = (As f y + Aps f ps )(d − a / 2)
m:Um:g;EdlGacekItman − M n = 18,684 × (5.5 − 0.46 / 2) = 98,465in. − lb / ft (36.5MPa) ehIym:Um:g;
tMrUvkar M n = 77,050in. − lb / ft < 98,465in. − lb / ft . dUcenH vaminRtUvkarersIusþg;m:Um:g;bEnßmeT.
kñúgkrNIenHeKminRtUvkarkarEbgEckm:Um:g;GviC¢man inelastic eLIgvijeT edaysarEdkm:Um:g;
viC¢manEdlGacekItmanRKb;RKan;. RbsinebIeKcg;GnuvtþkarEbgEckm:Um:g;eLIgvij/ BIrUbTI 4>46³
⎛ dt ⎞ ⎛ 5.5 ⎞
ε t = 0.003⎜ − 1⎟ = 0.003⎜ − 1⎟ = 0.027in. / in. > 0.0075in. / in. O.K.
⎝ c ⎠ ⎝ 0.46 / 0.85 ⎠
eKGnuvtþkarEbgEckm:Um:g;eLIgvijBITMreTAkNþalElVg tMélkarEbgEckm:Um:g;eLIgvijGtibrma
= 1000ε t ≤ 20% .

emKuNEbgEckm:Um:g;eLIgvijCak;Esþg = 1000 × 0.027 = 27% FMCagtMélGnuBaØatGtibrma.
GnuvtþemKuNEbgEckm:Um:g;eLIgvij 15% eTAelIm:Um:g;kNþalElVgviC¢man.
dUcenH + M n = 1.15 × 84,977 = 97,724in. − lb / ft
eKFanakarkat;bnßyRkLaépÞrbs;EdkBRgwgFmμtasMrab;m:Um:g;GviC¢man kñúgkrNIEdlbrimaN
EdkGb,brmaRKb;RKan;.
@> muxkat;kNþalElVgenARtg;ElVg BC
BIelIkmun Fps = Aps f ps = 15,084lb / ft ehIy
A ps f ps 15,084
a= = = 0.37in.
0.85 f 'c b 0.85 × 4,000 × 12
dUcenH m:Um:g;EdlGacman − M n = Aps f ps (d − a / 2) = 15,084(5.5 − 0.37 / 2) = 80,171in. − lb / ft
ehIym:Um:g;tMrUvkar M n = 97,724in. − lb / ft > 80,171in. − lb / ft dUcenHvaminRKb;RKan;. dUcenH bEnßm
6#4 Rtg;kNþalElVgelITTwg 20 ft edIm,ITTYl)an

As = 6 × 0.20 = 1.20in.2


1.20 × 60,000
As f y = = 3,600lb / ft
20
a=
(15,084 + 3,600) = 0.46in.
0.85 × 4,000 ×12

ersIusþg;m:Um:g;EdlGacman + M n = (As f y + Aps f ps )⎛ d − a ⎞
⎜
⎝ 2⎠
⎟

⎛ 0.46 ⎞
= (15,084 + 3,600)⎜ 5.5 − ⎟ = 98,465in. − lb / ft
⎝ 2 ⎠
> +M n tMrUvkar = 97,724in. − lb / ft O.K.

karsegçbsrésEdk
eRkaykarEbgEckm:Um:g;eLIgvij eRbIEdkFmμtaminrgeRbkugRtaMg #4 ¬Ggát;p©it 12.7mm ¦ cMnYn 6 edIm
enAsrésxageRkamRtg;kNþalElVgbEnßmBIelI tendon eRbkugRtaMgCab;elIkMNat; 20 ft . ehIyeRbI
EdkFmμtaminrgeRbkugRtaMg 6#4 enAsrésxagelIenARtg;TMr edaydak;ecjBIG½kSrbs;ssrCamYynwgKM
lat 6in. edayKitBIG½kSeTAG½kS ¬EdkGgát;p©it 12.7mm cMnYn 6 edImCamYynwgKMlat 152mm BIG½kS
eTAG½kS¦ CabrimaNEdkGb,brmaEdlRtUvepÞógpÞat;sMrab;karepÞr shear-moment.
muxkat;kNþalElVgrbs;ElVg AB nig CD KYrmantMélFMCagersIusþg;m:Um:g; nominal viC¢man
edIm,ITb;Tl;nwgm:Um:g;emKuNviC¢man. ersIusþg;m:Um:g; nominal GviC¢manrbs;muxkat;Rtg;TMrxageRkA A nig
D RtUv)anRKb;RKgeday moment-shear transfer stress.

#> Banding the reinforcement enARtg;tMbn;ssr
kabeRbkugRtaMgGgát;p©it 12 in. cMnYn 11 ehIyTTwgrbs;cMerokelIssr = 2(14 × 20 ×12) = 120in. .
snμt;fa EdkeRbkugRtaMg 70% RtUv)anRbmUlpþúMenARtg;cMerokelIssr. enaHcMnYnkabeRbkugRtaMg =
0.7 × 11 = 7.7 . dUcenH kabeRbkugRtaMgcMnYn 7 RtUv)andak;enAelIcMerokssr EdlkñúgenaHkabeRbkug

RtaMg 3 RtUvkat;tammuxkat;ssr.
enAelIcMerokkNþalElVgmankabeRbkugRtaMg 11 − 4 = 7 . eKGacsnμt;PaKryEbgEckm:Um:g;
cenøaHcMerokelIssr nigcMerokkNþaledaytMélRbhak;RbEhldUcxageRkam³
emKuNm:Um:g;cMerokelIssr = 7 /11 = 0.64
emKuNm:Um:g;cMerokkNþal = 0.36
m:Um:g;srubGtibrma + M Rtg;épÞssr B = 33,880in. − lb / ft ¬emIltarag 9>2¦
m:Um:g;srubGtibrma + M Rtg;kNþalElVg 21,784in. − lb / ft
dUcenH EbgEck tendon eRbkugRtaMgcenøaHcMerokelIssr nigcMerokkNþalElVgdUckarbgðajcxageRkam³



IV. Nominal Shear Strength
!> ssrxageRkA A nig D
(a) rUbragFrNImaRt nigbnÞúkxageRkA

BIelIkmun eyIgman VAB = 1231.1lb / ft ehIykMlaMgkat;TTwgsrubKW VB = 1231.1 × 20 =
24,622lb . snμt;CBa¢aMgxageRkA nigkBa©k;manTMgn;mFüm 500 plf ³

kMlaMgkat;EdlekItBIbnÞúkCBa¢aMg Vu = 1.2 × 500 × 20 = 12,000lb
kMlaMgkat;EdlekItBIkMralxNÐ Vu = 24,622lb
kMlaMgkat;emKuNsrub VuA = 36,622lb(162.9kN )
muxkat;kMlaMgkat;eRKaHfñak;RtUv)anKitenARtg; d / 2 BIépÞrbs;ssr dUcbgðajenAkñúgrUbTI 9>20.

eyIgman d = 6.5 − 1.0 = 5.5in.

tMélGtibrma d p = dv = 0.8h = 0.8 × 6.5 = 5.2in.(132mm)
c1 = 12in.

c2 = 14in.
d 5.2
b1 = c1 + = 12 + = 14.6in.
2 2
b2 = c2 + d = 14 + 5.2 = 19.2in.
Ac = bo d = 5.2(2 ×14.6 + 19.2) = 252in.2


2
⎛ d⎞
BIrUbeyIg)an d (2c1 + c2 + 2d )x = d ⎜ c1 + ⎟
⎝ 2⎠
b¤ 5.2(2 × 12 + 14 + 2 × 5.2 )x = 5.2(14.6 )2

x = c AB = 4.40in.
d 5.2
g = x − = 4.4 − = 1.8in.
2 2
b12 d (14.6 )2 × 5.2
müa:gvijeTot c AB =
Ac
=
252
= 4 .4

cCD = b1 − c AB = 14.6 − 4.4 = 10.2in.
BIlkçN³FrNImaRtrbs;ssrxageRkAEdlbgðajkñúgrUbTI 9>20 nigBIsmIkar 9.28 nig 9.29
1 1
γ v = 1− = 1−
b1 14.6
1+ 2
3
1+ 2
3
b2 19.2

= 1 − 0.63 = 0.37
edayeRbI d v sMrab; d / enaHm:Um:g;niclPaBb:UElrKW
⎛ d⎞ 3
⎜ c1 + ⎟d
Jc = ⎝
6
2⎠
+
2d 3
3
(3
)
c AB + cCD + (c2 + d )(d )(c AB )2

14.6(5.2)3 2 × 5.2
=
6
+
3
( )
4.4 3 + 10.2 3 − 19.2 × 5.2(4.4 )2

= 342 + 3,974 + 1,933 = 6,249in.4
BIelIkmun m:Um:g;Éktþa − M u = 6,640in. − lb / ft enARtg; column centerline. dUcenH bay moment
srubenARtg; column centerline KW − M c = 6,648 × 20 = 132,960in. − lb . snμt;fakMlaMgkat;pÁÜb
Vu manGMeBIRtg;épÞrbs;ssrsMrab; shear-moment transfer. kMlaMgkat;-m:Um:g;EdlepÞredaycMNak

p©itKW Vu g = −24,644 ×1.8 = 44,320in. − lb / m:Um:g;xageRkAemKuN M ue = 132,960 + 44,320 =
177,280in. − lb / ft ehIyersIusþg;m:Um:g;KμanlMnwgtMrUvkarsrub M n = M ue / φ = 177,280 / 0.9 =

196,978in. − lb .

(b) Shear-moment transfer
cMENkénersIusþg;m:Um:g; nominal EdlRtUvepÞredaykMlaMgkat;KW γ v M n = 0.37 ×196,978 =
72,882in. − lb . BIsmIkar 9.30a/ kugRtaMgkMlaMgkat;EdlbNþalBIkMlaMgkat;brimaRt (perimeter

shear)/ T§iBlrbs; γ v M n nigTMgn;rbs;CBa¢aMgKW
Vu γ v c AB M n
vn = +
φAc Jc



36,622 0.37 × 4.4 × 196,978
= +
0.75 × 252 6,249

= 193.8 + 51.2 = 245 psi
BIEpñkbnÞúklMnwgéndMeNaHRsay kugRtaMgsgát;mFümenAkñúgebtugRtg;TIRbCMuTMgn;muxkat;Edl
bNþalBIkMlaMgGnuvtþxageRkA Pe KW f c = Pe / Ac = 172 psi .
BIsmIkar 9.24 nig 9.25 edayminKitBIT§iBlrbs;bgÁúMkMlaMgeRbkugRtaMgbBaÄr V p ersIusþg;
kMlaMgkat;GnuBaØatGtibrmakøayCa
vc = β p f 'c + 0.3 f c

Edl emKuN β p CatMéltUcCageKén (α s d / bo + 1.5) nig 3.5 ehIy α s = 3.0 sMrab;cugssr. BIrUbTI
9>20/ bo = 2 × 14.6 + 19.2 = 48.2in. ehIy
αsd 30 × 5.5
+ 1.5 = + 1.5 = 4.92 > 3.5
bo 48.2
dUcenH eRbI β p = 3.5
kugRtaMgkat;GnuBaØatGtibrma vc = 3.5 4,000 + 0.3 × 172

Cak;Esþg = 245 psi O.K.
= 221 + 52 = 273 psi > vu

RbsinebIeKKit V p bBa©ÚlkñúgkarKNna enaH vc GnuBaØatGtibrmaRtUvmantMélFMCag 273 psi .
(c) Flexure moment transfer
cMENkénersIusþg;m:Um:g; nominal EdlRtUvepÞreday flexure KW M n = 0.63 ×196,978 =
124,096in. − lb . BIsmIkar 9.20 RkLaépÞEdkGb,brma As , min = 0.00075hl =

0.00075 × 6.5 × 17.5 × 12 = 1.02in.2 . dUcenH eRbI 6#4 × 6 ft EdlrYmbBa©ÚlTaMg standard

hook/ RkLaépÞEdkEdleFVIkardl; yielding As = 6 × 0.2 = 1.2in.2 . kugRtaMgenAkñúgEdkeRb

kugRtaMgRtUv)ankMNt;edaysmIkar 9.23 edaysnμt;eGayEdkeRbkugRtaMgbIedImkat;tammux
kat;ssrenARtg;TMrkNþalRtg; e = 0 . eyIgman d p = 6.5 / 2 = 3.25in. ehIyTTwgRbsiT§PaB
b = c2 + 2(1.5 × h ) = 14 + 2(1.5 × 6.5) = 33.5in. . ehIy
A ps 3 × 0.153
ρp = = = 0.0042
bd p 33.5 × 3.25
f 'c
f ps = f pe + 10,000 +
300 ρ p
4,000
= 159,000 + 10,000 +
300 × 0.0042
= 172,174 psi



A ps = 3 × 0.153 = 0.459in.
As f y + A ps f ps 1.20 × 60,000 + 0.459 × 172,174
a= =
0.85 f 'c b 0.85 × 4,000 × 33.5

= 1.33in.
ersIusþg;m:Um:g;EdlGacmanenAtMbn;ssr M n = As f y ⎛ d − a ⎞ + Aps f ps ⎛ d p − a ⎞
⎜
⎝ 2⎠
⎟ ⎜
⎝ 2⎠
⎟

⎛ 1.33 ⎞ ⎛ 1.33 ⎞
= 1.2 × 60,000⎜ 5.5 − ⎟ + 0.459 × 172,101⎜ 3.25 − ⎟
⎝ 2 ⎠ ⎝ 2 ⎠

= 347,400 + 203,410 = 550,810in. − lb
>> γ f M n = 124,096in. − lb



dUcenH ersIusþg;m:Um:g; nominal EdlGacekItmanmantMélFMCagm:Um:g;EdlnwgRtUvepÞreday
flexure. rUbTI 9>21 bgðajBIKMrUmYysMrab; banding kñúgkardak;EdkeRbkugRtaMg nigEdkFmμtasMrab;

shear-moment transfer Rtg;tMbn;ssrxageRkA.

@> ssrxagkñúg B nig C
(a) ragFrNImaRt nigbnÞúkxageRkA

BIelIkmun/ VBA + VBC = 1996.9 + 2232 ≅ 4229 plf . kMlaMgkat;srubKW VuB = 4229 × 20 =
84,578lb(376kN ) ehIy c1 = 20in. / c2 = 14in. nig d = 6.5 − 1 = 5.5in.

snμt;fa d v = 0.8h ≅ 5.2in. / KNna g = 12 c1 = 20 / 2 = 10in.
b1 = c1 + d = 20 + 5.2 = 25.2in.

b2 = c2 + d = 14 + 5.2 = 19.2in.
Ac = bo d = 2(25.2 × 5.2 + 19.2 × 5.2) = 462in.2
edayeRbI d v sMrab; d / m:Um:g;niclPaBb:UElrKW
d (c1 + d )3 d 3 (c1 + d ) d (c2 + d )(c1 + d )2
Jc = + +
6 6 2
5.2(25.2 )3 (5.2 )3 (25.2 ) 5.2(19.2 )(25.2 )2
= + +
6 6 2
= 46,161in.4



rUbTI 9>22 bgðajlkçN³FrNImaRtrbs;ssrxagkñúg³
1
γ v = 1− = 0.433
1+ 2
3
25.2 / 19.2

γ f = 1 − 0.433 = 0.567
m:Um:g; M ue = M e sMrab;ssrxagkñúgnImYy² ehIy net unit moment M e = 80,236 − 77,050 =
3,186in. − lb . m:Um:gEdl)anBIkMlaMgkat;KμanlMnwg (unbalanced shear momnet) KWesμInwg Vu × g =

10(2,232 − 1,996.9 ) = 2,351in. − lb . cugeRkay m:Um:g;srubKW

M ue = 3186 × 20 + 2351 = 66,071in. − lb
ehIyersIusþg;m:Um:g;KμanlMnwgtMrUvkarKW
M n = M ue / φ = 66,071 / 0.9 = 73,412in. − lb

(b) Shear-moment transfer
cMENkersIusþg;m:Um:g; nominal EdlRtUvepÞredaykMlaMgkat;KW
γ v M n = 0.433 × 73,412 = 31,787in. − lb
ehIy c AB = 12 (c1 + d ) = 12 b1 = 25.2 / 2 = 12.6in.
BIsmIkar 9.30a/ kugRtaMgkMlaMgkat;EdlbNþalBIkMlaMgkat;brimaRt (perimeter shear)
nigT§iBlrbs; M n KW
Vu γ v c AB M n
vn = +
φAc Jc
0.433 × 73,412 × 12.6
= 244.0 + 8.68 = 253 psi (173MPa )
84,578
= +
0.75 × 462 46,161
< vc GnuBaØat = 273 psi / O.K.

(c) Flexure moment transfer
cMENkénersIusþg;m:Um:g; nominal EdlRtUvepÞreday flexure KW γ f M n = 0.567 × 73,412 =
41,625in. − lb ehIy b = c2 + 2(1.5 × h ) = 14 + 2(1.5 × 6.5) = 33in. dUcKñasMrab;ssrxageRkA A .

dUckrNIssrxageRkA snμt;fakabeRbkugRtaMgbIkat;tamssrxagkñúg B nig C . eyIgman
d p = 6.5 − 1 = 5.5
A ps 3 × 0.153
ρp = = = 0.0025
bd p 33.5 × 5.5
f 'c
f ps = f pe + 10,000 +
300 ρ p



4,000
= 159,000 + 10,000 + = 174,333 psi
300 × 0.0025
EdlmantMélEk,rnwgtMél f ps sMrab;ssr A . dUcenH eRbI 6#4 ×12 ft CabrimaNEdkGb,-
brmadUcssrxageRkA a ≅ 1.48in. ehIyersIusþg;m:Um:g;EdlGacekItman (available moment capacity)
enAkñúgssrKW
M n = 1.2 × 60,000(5.5 − 1.33 / 2) + 0.459 × 174,333(5.5 − 1.33 / 2)

= 348,120 + 386,891 = 735,011in. − lb
>> M ntMrUvkar = 73,412in. − lb dUcenHvaRKb;RKan;
rUbTI 9>23 bgðajBI schematic layout rbs;EdkBRgwgenAkñúg flat plate Cab;. kabeRbkug
RtaMgGgát;p©it 1/ 2in. cMnYnbIkñúgTisnImYy²RtUvkat;tambrievNkMlaMgkat;eRKaHfñak;rbs;ssr. Cakar
BiteKRtUvRtYtBinitütMrUvkar serviceablity sMrab;PaBdab dUcenAkñúgEpñk 9>13.



BIkarsikSaviPaK eyIgTTYlykkarsikSaKNna ehIyeRbIKMrUénkardak;EdkBRgwgdUcKñasMrab;Tis
TaMgBIr N-S nig E-W rbs;RbB½n§kMral edaysarTMhMrbs;kMralxNÐTaMgBIrTisesÞIresμIKña.
eKcaM)ac;RtUvcMNaMfakarcUlrYmrbs;kabeRbkugRtaMgeTAkñúgersIusþg;m:Um:g;EdlGacekItmanrbs;kM
ralRtg;tMbn;ssrmann½yEtkñúgkrNIeKdak;EdkeRbkugRtaMgRtg;kat;tamssr dUcbgðajkñúgrUbTI 9>21.
kñúgkarsikSaKNnaCaeRcIn EdkeRbkugRtaMgRtUg)aneKdak;eGaygakecjBIG½kSrbs;ssredayCMraltic
tYc dUcenHmanEtEdkBRgwgFmμtab:ueNÑaHEdlbegáItersIusþg;m:Um:g;edIm,ITb;Tl;nwgm:Um:g;KμanlMnwg. eTaHbI
kñúgkrNIEbbenHk¾edayk¾ersIusþg;m:Um:g;EdlGacekItmanEdl)anBIEdkBRgwgFmμtaEdlRbmUlpþúMtamry³
muxkat;ssrenAEtmantMélFMCagm:Um:g;tMrUvkar dUckarKNnaEdlbgðajkñúg]TahrN_enHRsab;.

9.12. viFIedaypÞal;kñúgkarKNnaPaBdab
Derect Method of Deflection Evaluation
9.12.1. viFIeRKagsmmUl
The Equivalent Frame Approach
dUcKñanwgviFIeRKagsmmUlsMrab; flwcural analysis Edlerobrab;;y:aglMGitenAkñúgEpñkelIkmun
eKEckeRKOgbgÁúMCaeRKagCab;EdlsßittamG½kSrbs;ssrkñúgTisEkgTaMgBIr. eRKagnImYy²pÁúMeLIgeday
ssrmYyCYr nig band rbs;kMrald¾FMEdlenAcenøaH panel centerline rYmCamYynwgFñwmtambeNþay
G½kSssr.
tamtMrUvkarrbs;sþaTic eKRtUvKitbBa©ÚlbnÞúkGnuvtþenAkñúgTisEkgnImYy². edIm,IKitbBa©Úl
torsional deformation rbs;Fñwm eKRtUveRbIssrsmmUlEdl flexibility rbs;vaCaplbUkén

flexibility rbs;ssrCak;EsþgCamYynwg torsional flexibility rbs;FñwmTTwg (transverse beam)

b¤cMerokkMralxNÐ (stiffness CacMras;én flexibility).
1 1 1
= + (9.33)
K ec ∑ K c K t
Edl rbs;ssrsmmUl/ m:Um:g;Bt;kñúgmYyÉktþamMulrgVil.
K ec = flexural stiffness

∑ K c = plbUkPaBrwgRkajTb;karBt; (flexural stiffness) rbs;ssrxagelI nigssrxag

eRkam/ m:Um:g;Bt;kñúgmYyÉktþamMurgVil.
K t = PaBrwgRkajTb;karrmYl (torsional stiffness) rbs;FñwmTTwg (transverse beam)

b¤cMerokkMralxNÐ/ m:Um:g;rmYlkñúgmYyÉktþamMurgVil.
dUcenH eKRtUvsÁal;tMélrbs; K ec edIm,IKNnaPaBdabedayviFIsaRsþenH.


eKKitfacMerokkMralxNÐ-FñwmminRtUv)anRTedayssreT b:uEnþcMerokkMralxNÐ-FñwmTTwgsßitenA
elIG½kSssr. rUbTI 9>24 (a) bgðajBIcMnucenH. eKKitkMhUcRTg;RTayrbs;KMrUkMralxNÐkñúgmYyTismþg.
dUcenH eKeFVIplbUkPaBdabtamTisnImYy² ¬Tis x nigTis y ¦ edIm,ITTYl)anPaBdabsrubRKb;cMnucTaMg
Gs;enAelIkMralxNÐ b¤ plate.



dMbUg eKRtUvKNnaPaBdabEdlbNþalBIkarBt;begáagkñúgTis x ¬rUbTI 9>24 (b)¦. bnÞab;mk
eKRtUvrkPaBdabEdlbNþalBIkarBt;begáagkñúgTis y . eKGacTTYl)anPaBdabkNþalkMralxNÐ
(midpanel) CaplbUkénPaBdabkNþalElVg (senter-span) rbs;cMerokelIssrkñúgTismYyCamYynwg

PaBdabkNþalElVgrbs;cMerokkNþalElVgkñúgTisEdk ¬rUbTI 9>24 (c)¦.

9.12.2. PaBdabcMerokelIssr nigPaBdabcMerokkNþalElVg
The Equivalent Frame Approach
eKGacKitPaBdabrbs;kMralnImYy²BIplbUkénFatupSMbIKW³
!> PaBdabkNþalElVgeKalrbs;kMralxNÐ edaysnμt;cugTaMgBIrbgáb;KW
wl 4
δ '= (9.34)
384 Ec I frame

PaBdabenHRtUvsmamaRteTAnwgPaBdab δ c rbs;cMerokelIssr nigPaBdab δ s rbs;cMerok
kNþalElVg Edl
M col , strip Ec I cs
δc = δ ' (9.35a)
M frame Ec I c



M slab, strip Ec I cs
nig δs = δ '
M frame Ec I s
(9.35b)

Edl I cs Cam:Um:g;niclPaBrbs;eRKagsrub/ I c Cam:Um:g;niclPaBrbs;cMerokelIssr nig I s Ca
m:Um:g;niclPaBrbs;cMerokkNþalElVg.
@> PaBdabRtg;kNþal (center deflection) δ "θL = 1 θL bNþalBImMurgVilRtg;cugxageqVg xN³
8

EdleKKitfacugxagsþaMRtUv)anbgáb; (fixed) Edl θL Ca M net / K ec xageqVg ehIy K ec Ca
flexural stiffness rbs;ssrsmmUl ¬m:Um:g;kñúgmYyÉktþamMurgVil¦.

#> PaBdabRtg;kNþal (center deflection) δ "θR = 1 θL bNþalBImMurgVilRtg;cugxagsþaM xN³
8

EdleKKitfacugxageqVgRtUv)anbgáb; (fixed) Edl θL Ca M net / K ec xagsþaM. dUcenH
δ cx b¤ δ cy = δ c + δ "θL +δ "θR (9.36a)

δ sx b¤ δ sy = δ s + δ "θL +δ "θR (9.36a)

kñúgsmIkar 9.36a nig 9.36b eRbItMél δ c / δ "θL nig δ "θR EdlRtUvKñanwgTisrbs;ElVg. BIrUbTI
9>24 (b) nig (c) PaBdabsrubKW
Δ = δ sx + δ cy = δ sy + δ cx (9.37)

9.13. KNnaPaBdabrbs;kMralxNÐebtugeRbkugRtaMgBIrTis
Deflection Evaluation of Two-Way Prestressed Concrete Floor Slabs
]TahrN_ 9>3³ KNna central deflection rbs;kMralxageRkArbs;kMralebtugeRbkugRtaMgBIrTisEdl
karTajCaeRkayEdl)ansikSaKNnaenAkñúg]TahrN_ 9>2 sMrab;kardak;bnÞúkry³eBlxøI nigkardak;
bnÞúkry³eBlyUr. snμt;fa PaBdabGnuBaØat GtibrmaKW l / 480 énElVg.
dMeNaHRsay³
Tinñn½yrbs;eRKOgbgÁúM³ BI]TahrN_ 9>2/ eyIgmanTinñn½ydUcxageRkam
kMras; plate h = 6.5in.(165mm)
bnÞúk Wd = 101 psf (4.84kPa)
WL = 40 psf (1.92kPa )
bnÞúklMnwgElVg AB / Wbal = 72 psf (3.45kPa)
Wnet = WD + WL − Wbal = 101 + 40 − 72 = 69 psf (3.3kPa )
bnÞúklMnwgElVg BC / Wbal = 70 psf (3.35kPa)



Wnet = 141 − 70 = 71 psf (3.4kPa )
bøg;kMralRtUv)anbgðajenAkñúgrUbTI 9>25/ ehIybøg;lMGitTaMgGs; nigmuxkat;bBaÄrrbs;sMNg;
RtUv)anbgðajenAkñúgrUbTI 9>17. eKykm:Um:g;begáagEdl)anEbgEckenAkñúgTis N-S BI flexural
analysis sMrab; Wnet enAkñúgtarag 9>2 EdlbgðajenAkñúgrUbTI 9>26.

Stiffness Fatores nig strip moment Tis N-S ¬ElVg 18 ft ¦
tMélemKuNPaBrwgRkajrbs;ssrsmmUl K ec RtUv)anKNnaenAkñúg]TahrN_ 9>2 CamYynwg
lT§pldUcxageRkam³
ssrxageRkA A ³ K ec = 47 Ec in. − lb / rad
ssrxagkñúg B ³ K ec = 113Ec in. − lb / rad
Net frame moment M A = 5.30 × 10 3 in. − lb / ft
Net frame moment M B = (39.56 − 33.94 )10 3 = 5.62 × 10 3 in. − lb / ft



dUcbgðajenAkñúg]TahrN_ 9>2/ cMerokelIssrTTYlykm:Um:g; 64% ehIycMerokkNþalElVgTTYlyk
m:Um:g; 36% . m:Um:g;niclPaBrbs;eRKagsrub I cs = bh3 /12 = 20 ×12(6.5)3 /12 = 5,493in.4 xN³Edl
m:Um:g;niclPaBrbs;cMerokelIssr I c = m:Um:g;niclPaBcMerokkNþalElVg I c = 5,493 / 2 = 2,747in.4 .

BIsmIkar 9.34 PaBdabkNþalElVgeKalkñúgTis N-S Rtg;cMNuckNþal O kñúgrUbTI 9>27
eday`snμt;cugTaMgsgxagrbs;kMralxNÐbgáb; (fixed) KW
WL4 69 × 20(18)4 (12)3
δ '= = = 0.029in.
384 Ec I cs 384 × 4.03 ⋅10 6 × 5,493

PaBdabenHRtUvsmamaRteTAnwgPaBdab δ c rbs;cMerokelIssr nig δ s rbs;cMerokkNþalElVg³
M col , strip Ec I cs
δc = δ '
M frame Ec I c

BI]TahrNI 9>2/ M col, strip / M frame = 0.64 dUcenH PaBdabtamTis N-S δ c = 0.029 × 0.64 × 2 =
0.037in. nig δ s = 0.029 × 0.36 × 2 = 0.021in. ehIymMurgVilenARtg;cMnuc A KW
M A 5.30 ⋅10 3 × 20
θA = = = 5.6 × 10 − 4 rad
K ec 47 × 4.03 ⋅10 6
MB 5.62 ⋅10 3 × 10
θB = = = 2.5 × 10 − 4 rad
K ec 113 × 4.03 ⋅10 6

δ "=
θl
=
(5.6 + 2.5)10 −4 (18 × 12) = 0.022in.
8 8
dUcenH/ N-S net δ cy = 0.037 + 0.022 = 0.059in. ehIy N-S net δ sy = 0.021 + 0.022 = 0.043in.



Stiffness Fatores nig strip moment Tis E-W ¬ElVg 18 ft ¦
sMrab;Tis E-W/ TTwg b rbs;eRKagsmmUl = 12 (18 + 24) = 21.0 ft .
bh 3 21× 12(6.5)3
m:Um:g;niclPaBeRKagsrubKW I cs = 12 = 12 = 5,767in.4
m:Um:g;niclPaBcMerokelIssr I c = m:Um:g;niclPaBcMerolkNþalElVg I s = 5,767 / 2 = 2,884in.4
BIsmIkar 9.24/ central deflection Rtg;cMnuc O Edlmancugbgáb;KW
WL4 69 × 21(20 )4 (12)3
δ '= = = 0.045in.
384 Ec I cs 384 × 4.03 ⋅10 6 × 5,767

sMrab;krNIEdlkMralTaMgGs;rgbnÞúkenAkñúg]TahrN_enH/ net moment enARtg;ssrnImYy²
EdlbNþalBIplsgénm:Um:g;GviC¢manBIkMraleTAxagssrxageqVg nigm:Um:g;GviC¢manBIkMraleTAssr
xagsþaMesμIsUnü. dUcenH eKman net rotation θ = 0 nigeRbI E-W net δ cx = 0.058in. nig E-W net
δ sx = 0.032in. .



rUbTI 9>27 eGayPaBdabcMerolelIssr nigPaBdabcMerokkNþalElVgkñúgTisTaMgBIr N-S nig
E-W .
Central Deflection srub (Total Immediate Central Deflection)
Central deflection srub
Δ = δ sx + δ cy = δ sy + δ cx

dUcenH Δ N − S = δ sy + δ cx = 0.043 + 0.058 = 0.101in.
ehIy Δ E −W = δ sx + δ cy = 0.032 + 0.059 = 0.091in.
dUcenH PaBdabPøam²mFümEdlbNþalBI net load KW Δ net = 12 (Δ N − S + Δ E −W )
=
1
(0.101 + 0.091) = 0.096in.(2.44mm)
2
PaBdabry³eBlEvg (Long-term Deflection)
sMrab;PaBdabry³eBlEvg/ bnÞúk Wnet = 69 psf nigbnÞúkGefr WL = 40 psf edaysnμt;fa
kMralxNÐRtUvRTbnÞúkGefr 65% dUcenHbnÞúkGcié®nþsrubEdlkMralxNÐRtUvRTKW
Wsusl = (69 − 40) + 40 × 0.65 = 55 psf
edaysnμt;emKuN creep srubesμInwg 2 / eyIg)an
PaBdabry³eBlEvg = 55 × 0.096 × 2 = 0.153in.(4.09mm)
69
PaBdabsrub = 0.096 + 0.153 = 0.249in.(6.33mm)
PaBdabGnuBaØatGtibrmaenAkñúgeRKOgbgÁúMKW
20 × 12
= 0.50in.(12.7 mm) > Δ Cak;Esþg = 0.249in.
L
Δ allow = = O.K.
480 480

9.14. RTwsþI Yield-Line sMrab; Plates BIrTis
Yield-Line Theory for Two-Way-Action Plates
karsikSaBI hinge-field mechanism enAkMralxNÐ b¤ plate eRkamGMeBIrbs;bnÞúkrhUtdl;Cit
)ak;CYydl;nisitSvisVkrkñúgkarbegáItGarmμN_sMrab;kareFVIkarCalkçN³BIrTisrbs; plates. Hing field
CadMeNIrEdlmant²Kñaén hing band EdlRtUv)aneKKitfamanragCabnÞat; dUcenHvaRtUv)anGñkR)aCJ
K.W.Johansen eGayeQμaHfa yield-line theory.

edIm,IbgðajBIPaBl¥RbesIrrbs;RTwsþIenH eKRtUvkarBN’nalMGitBIvaelIeRcInCMBUk. eKalbMNgenA
eBlenHKWRKanEtENnaMGñksikSaBIeKalkarN_dMbUgrbs;RTwsþI yield-line nigkarGnuvtþrbs;vaEtb:ueNÑaH.



RTwsþI yeild-line CadMNaHRsay upper-bound kñúgkarKNna plate. enHmann½yfaersIusþg;m:Um:g;
EdlTsSn_Tayrbs;kMralxNÐmantMélFMNas;ebIeRbobeFobnwglT§plBiesaFn_. elIsBIenH RTwsþIenH
snμt;kMralxNÐkMralxNÐenAEtmansPaBrabesμIenAeBl)ak; (totally rigid-plastic behavior). dUcenH
PaBdabmikKitbBa©ÚlkMlaMgsgát;EdleFVIGMeBienAkñúgbøg;rbs;kMral b¤ plate EdlBicarNaeT. eKsnμt;fa
kMrallCakMralxNÐEdlmanbrimaNEdkticEmnETn (considerably underreinforced)/ kñúgTMrg;Edl
PaKryEdkGtibrma ρ minFMCag 0.5% énmuxkat; bd .
edaysardMeNaHRsaymanklçN³ upper bound, kMras;kMralxNÐEdlTTYl)anBIviFIenHnwg
esþIgCagkMras;kMralxNÐEdlTTYl)anBI lower bound solution dUcCaviFIeRKagsmmUl. dUcenH eKcaM
)ac;RtUvKNna serviceablilty requirment sMrab;PaNdab nigsñameRbH.
KuNsm,tþid¾sMxan;mYyrbs;RTwsþIenHKWeKGacsikSaKNnakMralxNÐRKb;ragFrNImaRtTaMgGs;
EdlviFIPaKeRcInGnuvtþ)anEtsMrab;ragctuekaN. visVkrGacrkersIusþg;m:Um:g;sMrab;ragRtIekaN ctuekaN
Bñay ctuekaN rgVg; b¤ragepSgeTot RbsinebIeKsÁal; b¤GacTsSn_TayBI failure mechanism. enAeBl
EdleKGackMNt;KMrUénkar)ak;)an eKnwgTTYl)annUvdMeNaHRsayrbs;va.

9.14.1. eKalKMnitén Hinge-Field Failure Mechanism kñúgkarBt;begáag
Fundamental Concepts of Hinge-Field Failure Mechanisms in Flexure

eRkamskmμPaBrbs;m:Um:g;Bt;begáagBIrTis/ yielding rbs; rigid-plastic plate ekItmanenA
eBlm:Um:g;embMeBjlkçNvinicä½y Johansen’s square yield dUcbgðajkñúgrUbTI 9>28. eyagtamlkçN


vinicä½yenH eKKitfa yielding ekItmanenAeBlm:Um:g;emEdlmantMélFMCageKxiteTArktMélrbs; ± M
enARtg; yield line crack. TisedArbs; principal curvature rates RtUv)anKitfaRtYtKñaCamYynwg
kMeNagrbs;m:Um:g;em. TMnak;TMngrvagm:Um:g; nigkMeNagRtUv)anbgðajCaExSDitkñúgrUbTI 9>29. eKKitfa
ExS OA esÞIrEtQrRtg;enARtg;cMnuc O ehIy strain hardening RtUv)anecal.

RbsinebIeKBicarNakrNIsamBaØbMputrbs;kMralxNÐkaer:EdlmanTMr CamYynwgdWeRkénkarbgáb;
(fixity) i EdlERbRbYlBI i = 0 sMrab;TMrsamBaØeTA i = 1.0 sMrab;TMrEdlTb;nwgkarvileBjeljenAelI

RCugTaMgbYn/ failure mechanism rbs;vanwgmanlkçN³dUcbgðajkñúgrUbTI 9>30 enAeBleKGnuvtþbnÞúk
BRgayesμI.

eKmankrNIkMralxNÐTMrsamBaØ (a). m:Um:g; yield line tambeNþay yield line Cam:Um:g;em. dUcenH m:U
m:g;rmYlesμIsUnükñúg yield line ehIyenAkñúgkrNICaeRcInkMlaMgkat;k¾esμIsUnüEdl. dUcenHmanEtm:Um:g;


M kñúgmYyÉktþaRbEvgrbs; yield line eFVIGMeBItamExS AD nig BE kñúgrUbTI 9>31. eKtagm:Um:g;srub
edayviucT½renAkñúgTisrbs; yield line EdltMélrbs;vaCaplKuNrbs; M nigRbEvgrbs; yield line
Edl M (a / 2)cosθ kñúgrUbTI 9>31 (c). Virtual work rbs; yield moment rbs;kMNt;RtIekaNEdl
qUt ABO plKuNsáaElrrbs;viucT½rm:Um:g;TaMgBIr Ma / 2 cosθ enAelIExS)ak; AO nig BO nigmMurgVil
θ . müa:gvijeTot kmμnþxagkñúgKW

EI = ∑ M θ
RbsinebIkMhUcRTg;RTayrbs;kMNat;EdlqUtenARtg;TIRbCMuTMgn; c KW δ enaHkmμnþxageRkAKW
E E = force × displacement = ∑ ∫∫ wu dxdyδ

Edl wu CaGaMgtg;sIuetrbs;bnÞúkxageRkAkñúgmYyÉktþaépÞ. b:uEnþ EI = EE . dUcenH
∑ M θ = ∑ ∫∫ wu d x d y δ (9.38)



edayGnuvtþsmIkar 9.38 sMrab;krNIEdlkMBugBicarNa eyIgTTYl)an
Δ
M θ = Ma
a/2
edaysarmMu θ enAkñúgrUbTI 9>31 (b) tUc [θ = Δ /(a / 2)] .
kmμnþkñúgmYykMNat;RtIekaNKW
E I = M θ = 2MΔ
wu a 2 Δ
EE = ×
4 3
EdlPaBdabenARtg;TIRbCMuTMgn;rbs;RtIekaNKW Δ / 3 . dUcenH
⎛ w a2 ⎞
4(2MΔ ) = 4⎜ u Δ ⎟
⎜ 12 ⎟
⎝ ⎠
2
ehIym:Um:g;Éktþa M= u
w a
24
(9.39)

RbsinebIRCugTaMgbYnrbs;kMralxNÐkaer:RtUv)anbgáb;eBjelj/ EI = 4(4MΔ ) edaysarExSrdac;
(fracture line) ekItmanCMuvijGgÁt;RTUg k¾dUcRCugTaMgbYnrbs;va dUcbgðajenAkñúgrUbTI 9>30(c). dUcenH

sMrab;kMralkaer:bgáb;eBjelj (fully fixed squared slab)
wu a 2
m:Um:g;Éktþa M = 48 (9.40)

eyIgsegáteXIjfa lower bound solution EdlesñIeLIgeday Mansfield’s fialure pattern kñúgrUbTI
9>30 (c) eGaytMél M = wu a 2 / 42.88 . dUcenH sMrab;kMralkaer:EdlrgbnÞúkrayesμICamYynwgGaMg
tg;sIuetbnÞúk wu kñúgmYyÉktþaépÞ nigdWeRkénkarbgáb;TMr (support fixity) i enARKb;RCugTaMgGs;
wu a 2 = M [24(1 + i )] (9.41)
smIkarTUeTAsMrab;ersIusþg;m:Um:g; yield line rbs;kMralctuekaNenAelIFñwmEdlmanTMhM a × b dUcbgðaj
kñúgrUbTI 9>32 EdlRCug a CaRCugxøIKW
2
w d2 ⎡ ⎛a ⎞ ar ⎤
m:Um:g;Éktþa M = u r ⎢ 3+⎜ r
24 ⎢ ⎜b ⎟− ⎥
⎟ b (9.42)
⎣ ⎝ r ⎠ r ⎥
⎦
Edl ar =
2a
1 + i2 + 1 + i 4
2b
br =
1 + i1 + 1 + i3

i= dWeRkénkarTb;nwgkarvil (restraint) EdlGaRs½ynwgpleFobPaBrwgRkaj (stiffness ratio)
Edlmanerobrab;enAkñúgEpñk 9>2.



cMNaMfa smIkar 9.42 nwgkøayeTACaTMrg;samBaØénsmIkar 9.40 b¤ 9.41 sMrab;krNIkMralxNÐ
kaer:Edl restrained RCugTaMgbYn ¬ i = 1.0 ¦.
Affine Slabs: kMralxNÐEdlBRgwgedayEdkkñúgTisEkgTaMgBIrxusKñaRtUv)aneKeGayeQμaHfa

orthotropic slab ¬b¤ plate¦. m:Um:g;kñúgTis x esμI M ehIyenAkñúgTis y esμI μM Edl μ CargVas;

éndWeRk orthotrophy b¤CapleFobén
My ( As ) y
=
Mx ( As )x
edIm,IsMrYlkarsikSaviPaK eKbMElgkMralxNÐeGayeTACa affine (isotropic) slab EdlersIusþg; nig
RkLaépÞEdkkñúgTisTaMgBIr x nig y mantMéldUcKña. karbMElgenHRtUvGnuvtþdUcxageRkam³
!> EckTMhMkñúgTis M eday μ sMrab;kMralxNÐEdlRtUvBRgwgedayEdksMrab;m:Um:g; M kñúgTis
TaMgBIredayeRbIGaMgtg;sIuetbnÞúk wu kñúgmYyÉktþaépÞdUcKña.
@> bnÞúkk¾RtUvEcknwg μ kñúgkrNIbnÞúkcMcMnuc b¤bnÞúksrub.
#> kñúgkrNIbnÞúkragbnÞat; eKRtUvEckbnÞúknwg μ cos 2 θ + μ sin 2 θ / Edl θ CamMurvagbnÞúkrag
bnÞat;CamYynwgTis M .
RbsinebIeKsikSaviPaKkMralCa affine slab CamYynwgm:Um:g; μM kñúgTisedATaMgBIr eKRtUvKuN
TMhMkñúgTis μM CamYynwg μ . kñúgkrNINak¾eday lT§plEdlTTYl)anBitCadUcKña.



9.14.2. Failure Mechanism and Moment Capacities of Slabs of Various Shapes
Subjected to Distributed or Concentrated Loads
karENnaMy:agsegçbxagelIBI virtual work method kñúgkarkMNt; yield line moment sMrYl
dl;karyl;dwgy:agl¥BIdMeNIrkarKNitviTüaénkMralxNÐragctuekaNEdlrgbnÞúkBRgay. kMralxNÐ
EdlmanragsμúKsμaj nigkardak;bnÞúkRbePTsIuemRTI b¤minsIuemRTITamTarnUvcMeNHdwgBImuxviC¢aenHeRcIn.
ehIy muxkat;)ak;snμt; nigkareFVIeGayfamBl principle mantMélGb,brmaGaceGaytMélsMrab;
krNIBiessxusKñaBIlT§plBiesaFn_bnþicbnþÜc edayGaRs½yeTAnwgkarsnμt;CalkçN³KNitviTüanUvrUb
rag)ak;.
karsegçbxageRkamBIKMrUénkar)ak; nigersIusþg;m:Um:g;EdlKitCabnÞúk.
!> bnÞúkcMcMnucenARtg;kac;RCugrbs; cantilever plate ctuekaN

@> kMralkaer:rgbnÞúkcMkNþal nigmanTMrsamBaØRbqaMgnwgclnaeLIgcuH

#> kMral n RCugEdlmanTMrsamBaØ nigrgbnÞúkcMkNþal ¬ n > 4 ¦



$> kMralkaer:rgbnÞúkcMkNþal nigmanTMrsamBaØRbqaMgniwgclnacuHeRkam EtGnuBaØatclnaeLIg
elI

%> kMralxNÐmUlrgbnÞúkcMkNþalnwgmanTMrsamBaØtambeNþayRCugEKm

^> kMralmUlrgbnÞúkcMkNþal P CamYynwgEKmbgáb;

&> bnÞúkcMcMnuc P GnuvtþRtg;cMnucNak¾edayelIkMralragTUeTAEdlmanTMrbgáb;tambeNþayRCug



*> kMralRtIekaNsamBaØCamYynwgTMrsamBaØ ehIyrgbnÞúkcMkNþaledaybnÞúkcMcMnuc P

(> kMralRtIekaNsamBaØelITMrsamBaØEdlrgbnÞúkcMcMnucRtg;p©itrgVg;carikkñúgRtIekaN

!0> kMralRtIekaNEdlmanmMuTalmanTMrsamBaØ ehIybnÞúk P manGMeBIRtg;p©itrgVg;carikkñúgrgVg;

!!> kMralxNÐRTEvgEdlmanTMrsamBaØ ehIyrgbnÞúkcMcMnuc P Rtg;cMnuckNþal

!@> kMralRTEvgTMrsamBaØCamYynwgbnÞúk P esμIKñasßitenAcenøaHRCugEKm



!#> kMralRTEvgTMrsamBaØrgbnÞúkminesμIKña P nig kP Rtg;cenøaHRCugEKm Edl k < 1.0 ehIy
bnÞúkmancMgayq¶ayBIKñaRKb;RKan;

!$> kMralkaer:rgbnÞúkBRgayCamYynwgdWeRkbgáb; i EdlERbRbYlcenøaHBIsUnü eTAmYy



!%> kMralRtIekaNsm½gS ¬ λ = 60o ¦ rgbnÞúkBRgayesμI



!^> kMralctuekaNrgbnÞúkrayesμIEdlmanGaMgtg;sIuetbnÞúkÉktþa wu ehIyTMrTaMgbYnRCugrbs;
vamandWeRk restraint i EdlERbRbYlBIsUnü eTAmYy ¬cMNaMfa eKRtUvbg;elxerogbnþeTA
elIRCugrbs;kMral¦³

eKRtUvcMNaMCaTUeTAfa sMrab;smIkarBImunEdlP¢ab;TMnak;TMngbnÞúk P eTAnwgm:Um:g; M eKsnμt;
bnÞúk P eFVIGMeBIRtg;cMcMnuc. edIm,IEktMrUvsMrab;karBitEdlfa P eFVIGMeBIelIépÞkMNt;mYy eKRtUvsnμt;fava
eFVIGMeBIelIRkLaépÞrgVg;EdlmankaM ρ . sMrab;kMralxNÐEdlRBMEdnRtUv)anTb;eBjelj/ eKRtUvP¢ab;
hinge field edayrgVg;Edlb:HeTAnwgRBMEdnkMral ¬kaMrgVg; = r ¦. kñúgkrNIenH
ρ ⎛ 2ρ ⎞
M + M '= ⎜1 − ⎟ (9.43)
2π ⎝ 3r ⎠
Edl M Cam:Um:g;ÉktþaviC¢man nig M ' Cam:Um:g;ÉktþaGviC¢man.
eKGacKitRbtikmμrbs;ssrEdlRT flat plate RsedogKñanwgkarviPaK flexural local capacity
rbs; plate enAkñúgtMbn;ssr. sMrab;TMrctuekaN eKGacKittMélRbhak;RbEhlCaTMrrgVg;smmUleday
eRbIsmIkar 9.43.

9.15. ersIusþg;m:Um:g; Yield Line rbs;kMralebtugeRbkugRtaMgBIrTis
Yield-Line Moment Strength of a Two-Way Prestressed Concrete Plate
]TahrN_ 9>4³ kMNt;ersIusþg;m:Um:g; nominal rbs;kMralebtugeRbkugRtaMgBIrTisenAkñúg]TahrN_ 9>2
edaysnμt;fakabeRbkugRtaMgs¥itCab;CamYyebtug ¬prestressing strands are bonded¦.
dMeNaHRsay³
bnÞúk³
BI]TahrN_ 9>2/ GaMgtg;sIuetbnÞúksrubenAsßanPaBkMNt;énkar)ak;KW
Wu = 1.2WD + 1.6WL = 186 psf



edaysnμt;fa RbtikmμssrCabnÞúkcMcMnucbRBa©asenAkñúgEdnkMralCab; (continuous plate
field) eKGackMNt;ersIusþg;m:Um:g;tMrUvkar M n BIkrNITI & rbs;Epñk 9.4.12 dUcxageRkam³

PA = 4πM n
+
bnÞúkemKuN Pu = 186 × 20⎛ 24 2 18 ⎞ = 78,120lb(34.8kN )
⎜
⎝
⎟
⎠
¬TMgn;ssrRtUv)anecal¦
bnÞúktMrUvkar Pn = Pu = 780,120 = 86,800lb(38.6)
φ .9

m:Um:g;ÉktþatMrUvkar M n sMrab;bnÞúkcMcMnuc = 4π = 486,3.14 = 6910lb(30.7kN )
Pn
×
800

kaMsmmUl ρ = 20π×214 = 28.4in.
snμt; r ≅ 17.5 ft = 210in. ehIy M = M ' . enaH
Pn ⎛ 2ρ ⎞ ⎛ 2 × 28.4 ⎞
M 'n = ⎜1 − ⎟ = 6910⎜1 − ⎟ = 6287lb(28kN )
4π ⎝ 3r ⎠ ⎝ 3 × 210 ⎠
ersIusþg;m:Um:g;EdlGacman M n ³
EdkkMralxNÐtMbn;ssrEdlGacmanRtUv)ankMNt;dUcxageRkam
EdkeRbkugRtaMg
Aps = kabeRbkugRtaMg 270-K Ggát;p©it 0.5in. cMnYnbI = 3 × 0.153 = 0.459in.2
f py = 243,000 psi (1,675MPa )

f ps = 179,256 psi enARtg;ssrxagkñúg
f 'c = 4,000 psi (27.58MPa )
dUcenH eRbI f py enAsßanPaBkMNt;énkar)ak;.
EdkminrgeRbkugRtaMg
As = 6#4 = 6 × 0.2 = 1.2in.2
f y = 60,000 psi

ersIusþg;m:Um:g; (moment strength) M n

d p = d = 6.5 − 1 = 5.5in.

b = 33.5in. ¬BI]TahrN_ 9>2¦
As f y + A ps f py 1.2 × 60,000 + 0.459 × 243,000
a= = = 1.61in.(40.9mm )
0.85 f 'c b 0.85 × 4,000 × 33.5

ersIusþg;m:Um:g;EdlGacman M n = As f y ⎛ d − a ⎞ + Aps f py ⎛ d p − a ⎞
⎜
⎝ 2⎠
⎟ ⎜
⎝ 2⎠
⎟



⎛ 1.61 ⎞ ⎛ 1.61 ⎞
= 1.2 × 60,000⎜ 5.5 − ⎟ + 0.459 × 243,000⎜ 5.5 − ⎟
⎝ 2 ⎠ ⎝ 2 ⎠

= 338,040 + 523,666 = 861,706in. − lb
m:Um:g;Éktþa M n = 861,706 = 25,723in. − lb / in. = 25,723lb
33.5
RtYtBinitü M sMrab;TTwgkMralTaMgmUl
n

TTwg slab band Tis N-S = 18 + 24 = 21 ft (6.4m)
2
TTwg slab band Tis E-W = 20 ft
dUcenH eRbI b = 21 ft = 252in. . ehIykabeRbkugRtaMgsrub Aps = 270-K Ggát;p©it 0.5in.
cMnYn 11. edaysarEdkrgeRbkugRtaMgxagelIsßitenAEtkñúgtMbn;ssr EdlminKitfavasßitenAelIRCug
suvtßiPaB. enaHeyIgman
kabeRbkugRtaMgÉktþa Aps = 11×252153 = 0.0067in.2 / in.
0.

0.0067 × 243,000
a= = 0.48in.
0.86 × 4,000 × 1
⎛ 0.48 ⎞
m:Um:g;Éktþa M n = 0.0067 × 243,000⎜ 5.5 −
⎝ 2 ⎠
⎟ = 8.564in. − lb / in.

= 8,564ib(38.09kN )
m:Um:g;tMrUvkar M n = 6287lb. < M n EdlGacman = 8,564lb O.K.
eyIgeXIjy:agc,as;fa BI limit theoru solution/ eyIgGaceFVIkarviPaKkMraleRbkugRtaMg)an
y:agelOn. b:uEnþ karviPaKEbbenHRtUvKitbBa©ÚlkarkMNt; ersIusþg; yield-line shear enARtg;TMr nigRtYt
Binitü serviceability sMrab;karRKb;RKgsñameRbH nigPaBdab. GñksikSaKNnaGaceRCIserIstMélm:U
m:g;y:agRsYlsMrab; failur mechanism Edl)anbgðajenAkñúgEpñk 9.14.2. eKGaceFVIkarRtYtBinitü ser-
viceability sMrab;karRKb;RKgsñameRbH)any:agRsYldUcEdlnwgerobrab;kñúgEpñk 11.9 enAesovePAenH

sþIBIkarRKb;RKgsñameRbHenAelICBa¢aMgGagebtugeRbkugRtaMg.


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