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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
Department of Civil Engineering                                                             NPIC




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
RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                  555
T.Chhay                                                                        viTüasßanCatiBhubec©keTskm<úCa

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
Two-Way Prestressed Concrete Floor Systems                                                       556
Department of Civil Engineering                                                              NPIC




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



RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                   557
T.Chhay                                                               viTüasßanCatiBhubec©keTskm<úCa

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.




Two-Way Prestressed Concrete Floor Systems                                              558
Department of Civil Engineering                                                              NPIC




       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.

RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                   559
T.Chhay                                                                 viTüasßanCatiBhubec©keTskm<úCa

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;.




Two-Way Prestressed Concrete Floor Systems                                                560
Department of Civil Engineering                                                             NPIC




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:

RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                  561
T.Chhay                                      viTüasßanCatiBhubec©keTskm<úCa




Two-Way Prestressed Concrete Floor Systems                     562
Department of Civil Engineering                                                             NPIC




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

RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                  563
T.Chhay                                                                    viTüasßanCatiBhubec©keTskm<úCa

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 ¬kMralxNЦCab; 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¦.

Two-Way Prestressed Concrete Floor Systems                                                   564
Department of Civil Engineering              NPIC




RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis   565
T.Chhay                                                                viTüasßanCatiBhubec©keTskm<úCa

          ∑ 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




Two-Way Prestressed Concrete Floor Systems                                               566
Department of Civil Engineering                                                               NPIC




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.




RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                    567
T.Chhay                                                                  viTüasßanCatiBhubec©keTskm<úCa

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.




Two-Way Prestressed Concrete Floor Systems                                                 568
Department of Civil Engineering                                                              NPIC




        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.



RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                 569
T.Chhay                                                                  viTüasßanCatiBhubec©keTskm<úCa

        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

Two-Way Prestressed Concrete Floor Systems                                                 570
Department of Civil Engineering                                                                  NPIC




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;.


RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                       571
T.Chhay                                      viTüasßanCatiBhubec©keTskm<úCa




Two-Way Prestressed Concrete Floor Systems                     572
Department of Civil Engineering              NPIC




RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis   573
T.Chhay                                                                 viTüasßanCatiBhubec©keTskm<úCa

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.




Two-Way Prestressed Concrete Floor Systems                                                574
Department of Civil Engineering                                                          NPIC




        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 ) .




RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                               575
T.Chhay                                                                     viTüasßanCatiBhubec©keTskm<úCa

       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




Two-Way Prestressed Concrete Floor Systems                                                    576
Department of Civil Engineering                                                                    NPIC




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



RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                         577
T.Chhay                                                                 viTüasßanCatiBhubec©keTskm<úCa

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.



Two-Way Prestressed Concrete Floor Systems                                                578
Department of Civil Engineering                                                            NPIC




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.




RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                 579
T.Chhay                                                                    viTüasßanCatiBhubec©keTskm<úCa

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

Two-Way Prestressed Concrete Floor Systems                                                   580
Department of Civil Engineering                                                                      NPIC




                      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


RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                           581
T.Chhay                                                                     viTüasßanCatiBhubec©keTskm<úCa

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



Two-Way Prestressed Concrete Floor Systems                                                    582
Department of Civil Engineering                                                                   NPIC




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 )


RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                        583
T.Chhay                                                                           viTüasßanCatiBhubec©keTskm<úCa

                 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.

Two-Way Prestressed Concrete Floor Systems                                                          584
Department of Civil Engineering                                                                                   NPIC



                             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


RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                                        585
T.Chhay                                                               viTüasßanCatiBhubec©keTskm<úCa

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.


Two-Way Prestressed Concrete Floor Systems                                              586
Department of Civil Engineering                                                                  NPIC




      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
RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                      587
T.Chhay                                      viTüasßanCatiBhubec©keTskm<úCa




Two-Way Prestressed Concrete Floor Systems                     588
Department of Civil Engineering                                                             NPIC




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

RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                589
T.Chhay                                                                         viTüasßanCatiBhubec©keTskm<úCa

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
Two-Way Prestressed Concrete Floor Systems                                                        590
Department of Civil Engineering                                                           NPIC




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


RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                591
T.Chhay                                                                     viTüasßanCatiBhubec©keTskm<úCa

             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.



Two-Way Prestressed Concrete Floor Systems                                                    592
Department of Civil Engineering                                                              NPIC




          (> 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.


RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                   593
T.Chhay                                                                  viTüasßanCatiBhubec©keTskm<úCa

                   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




Two-Way Prestressed Concrete Floor Systems                                                 594
Department of Civil Engineering              NPIC




RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis   595
T.Chhay                                      viTüasßanCatiBhubec©keTskm<úCa




Two-Way Prestressed Concrete Floor Systems                     596
Department of Civil Engineering                                                            NPIC




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 .

RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                 597
T.Chhay                                      viTüasßanCatiBhubec©keTskm<úCa




Two-Way Prestressed Concrete Floor Systems                     598
Department of Civil Engineering                                                                  NPIC




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




RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                       599
T.Chhay                                                                  viTüasßanCatiBhubec©keTskm<úCa
                           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


Two-Way Prestressed Concrete Floor Systems                                                 600
Department of Civil Engineering                                                                               NPIC



                                      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
                     Ks =                    = 110in. − lb / rad / Ecs
                          12 × 17.5 − 20 / 2
       ehIy sMrab;kMralenAxagsþaMssrxagkñúg B
RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                                    601
T.Chhay                                                                       viTüasßanCatiBhubec©keTskm<úCa

                            4 × 1× 20(6.5)3
                    Ks =                     = 79in. − lb / rad / Ecs
                            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;.




Two-Way Prestressed Concrete Floor Systems                                                      602
Department of Civil Engineering                                                                             NPIC




    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.

RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                                  603
T.Chhay                                                                           viTüasßanCatiBhubec©keTskm<úCa




@> 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 bal = 34,460in. − lb / ft      ¬BItarag 9>3¦
                    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 bal = 39,320in. − lb / ft      ¬BItarag 9>3¦

Two-Way Prestressed Concrete Floor Systems                                                          604
Department of Civil Engineering                                                                NPIC




                     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
                                                       .




RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis                                                     605
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems
Ix. two way prestressed concrete floor systems

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12. displacement method of analysis moment distribution
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Ix. two way prestressed concrete floor systems

  • 1. 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
  • 2. T.Chhay viTüasßanCatiBhubec©keTskm<úCa Two-Way Prestressed Concrete Floor Systems 554
  • 3. Department of Civil Engineering NPIC 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 RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 555
  • 4. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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 Two-Way Prestressed Concrete Floor Systems 556
  • 5. Department of Civil Engineering NPIC 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 RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 557
  • 6. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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. Two-Way Prestressed Concrete Floor Systems 558
  • 7. Department of Civil Engineering NPIC 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. RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 559
  • 8. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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;. Two-Way Prestressed Concrete Floor Systems 560
  • 9. Department of Civil Engineering NPIC 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: RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 561
  • 10. T.Chhay viTüasßanCatiBhubec©keTskm<úCa Two-Way Prestressed Concrete Floor Systems 562
  • 11. Department of Civil Engineering NPIC 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 RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 563
  • 12. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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 ¬kMralxNЦCab; 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¦. Two-Way Prestressed Concrete Floor Systems 564
  • 13. Department of Civil Engineering NPIC RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 565
  • 14. T.Chhay viTüasßanCatiBhubec©keTskm<úCa ∑ 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 Two-Way Prestressed Concrete Floor Systems 566
  • 15. Department of Civil Engineering NPIC 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. RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 567
  • 16. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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. Two-Way Prestressed Concrete Floor Systems 568
  • 17. Department of Civil Engineering NPIC 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. RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 569
  • 18. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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 Two-Way Prestressed Concrete Floor Systems 570
  • 19. Department of Civil Engineering NPIC 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;. RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 571
  • 20. T.Chhay viTüasßanCatiBhubec©keTskm<úCa Two-Way Prestressed Concrete Floor Systems 572
  • 21. Department of Civil Engineering NPIC RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 573
  • 22. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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. Two-Way Prestressed Concrete Floor Systems 574
  • 23. Department of Civil Engineering NPIC 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 ) . RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 575
  • 24. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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 Two-Way Prestressed Concrete Floor Systems 576
  • 25. Department of Civil Engineering NPIC 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 RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 577
  • 26. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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. Two-Way Prestressed Concrete Floor Systems 578
  • 27. Department of Civil Engineering NPIC 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. RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 579
  • 28. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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 Two-Way Prestressed Concrete Floor Systems 580
  • 29. Department of Civil Engineering NPIC 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 RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 581
  • 30. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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 Two-Way Prestressed Concrete Floor Systems 582
  • 31. Department of Civil Engineering NPIC 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 ) RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 583
  • 32. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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. Two-Way Prestressed Concrete Floor Systems 584
  • 33. Department of Civil Engineering NPIC 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 RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 585
  • 34. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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. Two-Way Prestressed Concrete Floor Systems 586
  • 35. Department of Civil Engineering NPIC 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 RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 587
  • 36. T.Chhay viTüasßanCatiBhubec©keTskm<úCa Two-Way Prestressed Concrete Floor Systems 588
  • 37. Department of Civil Engineering NPIC 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 RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 589
  • 38. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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 Two-Way Prestressed Concrete Floor Systems 590
  • 39. Department of Civil Engineering NPIC 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 RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 591
  • 40. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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. Two-Way Prestressed Concrete Floor Systems 592
  • 41. Department of Civil Engineering NPIC (> 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. RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 593
  • 42. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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 Two-Way Prestressed Concrete Floor Systems 594
  • 43. Department of Civil Engineering NPIC RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 595
  • 44. T.Chhay viTüasßanCatiBhubec©keTskm<úCa Two-Way Prestressed Concrete Floor Systems 596
  • 45. Department of Civil Engineering NPIC 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 . RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 597
  • 46. T.Chhay viTüasßanCatiBhubec©keTskm<úCa Two-Way Prestressed Concrete Floor Systems 598
  • 47. Department of Civil Engineering NPIC 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 RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 599
  • 48. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 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 Two-Way Prestressed Concrete Floor Systems 600
  • 49. Department of Civil Engineering NPIC 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 Ks = = 110in. − lb / rad / Ecs 12 × 17.5 − 20 / 2 ehIy sMrab;kMralenAxagsþaMssrxagkñúg B RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 601
  • 50. T.Chhay viTüasßanCatiBhubec©keTskm<úCa 4 × 1× 20(6.5)3 Ks = = 79in. − lb / rad / Ecs 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;. Two-Way Prestressed Concrete Floor Systems 602
  • 51. Department of Civil Engineering NPIC 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. RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 603
  • 52. T.Chhay viTüasßanCatiBhubec©keTskm<úCa @> 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 bal = 34,460in. − lb / ft ¬BItarag 9>3¦ 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 bal = 39,320in. − lb / ft ¬BItarag 9>3¦ Two-Way Prestressed Concrete Floor Systems 604
  • 53. Department of Civil Engineering NPIC 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 . RbB½n§kMralxNÐebtugeRbkugRtaMgBIrTis 605