X. connections for prestressed concrete element

Department of Civil Engineering NPIC

X. tMNsMrab;Ggát;ebtugeRbkugRtaMg
Connections for Prestressed Concrete Elements

10.1. esckþIepþIm Introduction
tYnaTIrbs;tMNKWkarepÞrbnÞúk nigkugRtaMgBIEpñkmYyrbs;rcnasm<½n§eTAEpñkEdlenAEk,rRbkb
edaylkçN³esdækic© ehIypþl;nUvesßrPaBdl;RbB½n§eRKOgbgÁúM. bnÞúkEdleFVIGMeBIRtg;tMNminRtwmEt
ekItecjEtBIbnÞúkTMnajb:ueNÑaHeT vak¾GacekItBIbnÞúkxül; T§iBlrBa¢ÜydI karpøas;bþÚrmaDEdlekIteLIg
edaysar long-term creep nig shrinkage/ differential movement rbs;kMral nigT§iBlrbs;
sItuNðPaB.
edaysartMNCacMnuctP¢ab;EdlmanlkçN³exSayCageKenAkñúgRbB½n§eRKOgbgÁúMTaMgmUl dUcenH
vaRtUvman nominal design strength FMCag nominal design strength rbs;Ggát;EdlvaRtUvtP¢ab;. em
KuNbnÞúkbEnßmy:agehacNas; 1.3 RtUv)aneKeRbIenAkñúgkarsikSaKNnakartP¢ab; EtelIkElgkrNI
insensitive connection dUcCa pad sMrab; column base. eKsikSaKNnatMNTaMgGs;sMrab;kMlaMgTaj

tamTisedkGb,brma 0.2 dgénbnÞúkefrbBaÄr elIkElgEteKRtUveRbI bearing pad Edl)ansikSa
KNnad¾RtwmRtUv.
krNIEdlRtUv)aneKKitBicaNasMrab;ersIusþg;enAkñúgkarsikSaKNnatMNmandUcxageRkam³
!> Load transfer mechanism
@> emKuNbnÞúk (load factors)
#> karpøas;bþÚrmaD (volumetric changes)
$> PaBsVit (ductility)
%> PaBrwgmaM (Durability)
^> karTb;Tl;nwgGKÁIP½y (fire resistance)
&> kMritGt;eGan nigRbeLaHtMrUvkar (required tolerance and clearance)
*> karBicarNaEdlTak;TgnwgkargartMeLIg (erection-related consideration)
(> karBicarNaEdlTak;TgnwgGakasFatuekþA nigGakasFatuRtCak;
!0> esdækic©énkarlMGittMN (economics of the details of the connection)

tMNsMrab;Ggát;ebtugeRbkugRtaMg 639

T.Chhay viTüasßanCatiBhubec©keTskm<úCa

10.2. kMritGt;eGan Tolerance

eKRtUvkMNt;)a:n;RbmaNRbeLaHrvagGgát;tamPaBCak;Esþg. eKRtUvkMNt;TItaMgEdlmankMrit
Gt;eGanx<s;RtUv)anGnuBaØat nigkEnøgEdlkMritlMeGanminRtUv)anGnuBaØat ehIyeKk¾KitbBa©ÚlRbeLaH
sMrab;ktþaTaMgenH. xageRkamCakMritlMeGanEdlENnaMsMrab;TMhMlMgakenAkñúgFñwm ssr nig spandrel
panel³

!> karERbRbYlkñúgbøg;BITItaMgEdl)ankMNt;enAkñúgbøg;³ ± 0.5in. sMrab;ssr b¤Fñwm
@> kargakecjenAkñúgbøg;BIbnÞat;Rtg;EdlRsbeTAnigG½kSGaKar³ 1 / 40in. kñúg 1 ft RKb;FñwmEdl
xøICag 20 ft b¤KMlatrbs;ssrBIrEdlenAEk,rKñatUcCag 20 ft / 0.5in. sMrab;KMlatssr
EdlXøatq¶ayBIKña 20 ft .
Connections for Prestressed Concrete Elements 640


#> PaBxusKñaén relative position rbs;ssrEdlenAEk,rBI relative position Edl)ankMNt;³
0.5in. enARtg;nIv:UkMral (deck level).

$> lMgakBIkUnRbeyal (plumb)³ ± 0.25in. sMrab;ral;kMBs; 10 ft / GtibrmaRtwm 1in. sMrab;
kMBs;TaMgmUl.
%> PaBERbRbYlénkMritkMBs;rbs; bearing surface BIkMritkMBs;Edl)ankMNt;³ ± 0.5in. sMrab;
ssr nigFñwmTaMgGs; nigsMrab;RKb;TItaMg.
^> lMgakEpñkxagelIrbs; spandrel BIkMritkMBs;Edl)ankMNt;³ 0.5in. / sMrab;RKb; spandrel
&> lMgakénkMritkMBs;rbs; bearing surface BIExSRsbeTAExSrnIv:UEdl)ankMNt;³ 1 / 40in. kñúg
1 ft sMrab;RKb;FñwmxøICag 20 ft b¤ssrEk,rBIrEdlXøatBIKñaticCag 20 ft / GtibrmaRtwm

0.5in. sMrab;RKb;FñwmEvgCag b¤esμI 20 ft b¤ssrEk,rBIrXøatBIKñaeRcInCag b¤esμI 20 ft .

*> bMErbMrYlBI bearing length Edl)ankMNt;enAelITMr³ 3 / 4in. .
(> bMErbMrYlBI bearing width enAelITMr³ ± 0.5in. .
!0> PaBrt;Rtg;rbs;RCugEKm³ 0.25in.
tarag 10>1 eGaynUvkMritlMeGogEdlGacGnuvtþsMrab;tMN.

10.3. Ggát;smas Composite Members
dUcEdl)anerobrab;lMGitenAkñúgCMBUkTI5 BIEpñkEpñk5>7 eTAEpñk5>11/ eKRtUvFanakarepÞr
kMlaMgkat;tamTisedkenARtg;épÞb:HrvagGgát;cak;Rsab; nig situ-cast-topping. ]TahrN_ 5>14
bgðajBIkMlaMgGnþrGMeBI (interaction forces) nig flowchart énEpñk 5>8>2 eGay operational step-
by-step design procedure nigsmIkarKNnaEdlGacGnuvtþ)an (applicable design equation). rUbTI

5>18 én]TahrN_ 5>3 ehIykarsikSaKNnapþl;nUvTMhM nigKMlatrbs; dowel EdlmanT§iBldl;kar
epÞreBj eljénkMlaMgkat;tamTisedkrvagGgát;EdlP¢ab;Kña.

10.4. RTnab;TMrebtugGarem:enAkñúgGgát;smas
Reinforced Concrete Bearing in Composite Members
rUbTI 10>1 bgðajBI composite-action dowel reinforcement. edIm,IkarBarebtugEdlb:H
bearing edaypÞal;kuMeGaypÞúHEbkedaysarkMlaMgsgát;FelIslub eKRtUvGnuvtþkMlaMgxageRkAeTAelI
M
bearing EdlmanTMhMFMRKb;RKan;. kareFVIEbbenHkugRtaMgEdlTTYl)anBIsßanPaBkMNt;nwgminFMelIs



ersIusþg;sgát;rbs;ebtugeT. eKGackMNt; nominal bearing strength rbs;ebtugsuT§tamsmIkarxag
eRkam

Vn = C r (0.85 f ' c A1 ) A2 / A1 ≤ 1.2 f ' c A1 (10.1)
Edl C r = 1 .0enAeBleKdak;EdkBRgwgenAkñúgTisrbs;kMlaMgb:HtamTisedk (horizontal frictional
force) N u dUceXIjenAkñúgrUbTI 10>2 b¤enAeBlEdleKyk N u = 0 . eKGackMNt; C r

= (S × W / 200) Nu / Vu EdlRkLaépÞ S × W minRtUvFMCag 9.0in.2 ehIyvaRtUv)anbgðaj

enAkñúgrUbTI 10>3 .
A1 = RkLaépÞ direct bearing

A2 = RkLaépÞGtibrmarbs;cMENkénépÞTMrEdlmanragFrNImaRtRsedogKñanwgRkLaépÞrg

bnÞúk dUcbgðajkñúgrUbTI 10>3.
Design bearing strength KW

Vu = φVn
Edl φ = 0.70 . edIm,IeCosvagsñameRbH nig spalling EdlekIteLIgedayécdnüenAxagcugrbs; thin-
stemmed member, eKENnaMeGayeRbIEdkGb,brmaEdlesμInwg N u / φf y b:uEnþminRtUvtUcCag 1#3

¬Ggát;p©it 9.52mm ¦ enAeBlEdl bearing area tUcCag 2in.2 (12.9cm 2 ).
RbsinebIbnÞúkemKuN Vu FMCag design bearing strength Vu = φVn dUcEdl)anKNnaBI
smIkar 10>1/ enaHeKRtUvkarEdkBRgwgenAkñúg bearing area. eKGacsikSaKNnaEdkenHedayRTwsþI



-
shear friction Edlerobrab;enAkñúgCMBUk 5. eKRtUvsikSaKNna reinforced bearing sMrab;Ggát;cak;
Rsab;TaMgGs; elIkElgEtkMraltan; nig hollow-core slab edIm,IkarBarsñameRbHtamTisedk nigsñam
eRbHtamTisbBaÄrenARtg;EdkxageRkAbMputrbs;FñwmRtg;TMr. eKGacsnμt;PaBeRTtrbs;sñameRbHxag
cugedaysuvtßiPaBRbhak;RbEhlnwg 20o dUceXIjenAkñúgrUbTI 10>2. RbsinebI Vu esμInwgkMlaMgkat;
emKuN ¬EdlRsbeTAnwgbøg;sñameRbHsnμt;¦ eKKYrkMNt;tMélrbs;kMlaMgkat;dUcbgðajenAkñúgtarag
10>2 sMrab;emKuN shear-friction RbsiT§PaBGtibrma μe .

eKGacrkRkLaépÞEdkEdlEkgeTAnwgbøg;sñameRbHsnμt;BIsmIkarxageRkam³
Vup
Avf = (10.2)
φμe f y
Edl Vu / φ = nominal strength Vn

f y = yield strength rbs; Avf


Vup = kMlaMgkat;emKuNGnuvtþn_ EdlkMNt;edaytMélEdleGayenAkñúgtarag 10>2 ehIy
1,000λAcr μ
μe =
Vup

Edl λ = 1 .0sMrab;ebtugTMgn;Rsal/ 0.85 sMrab; sand-lightweight nig 0.75 sMrab; all-lightweight
concrete.

Acr = RkLaépÞrbs;épÞb:Hbøg;sñameRbH EdleKGacykvaesμInwg l d b Edl l d Ca development

length rbs; Avf ehIy b CaTTwgmFümrbs;Ggát;.

tarag 10>3 eGay development length ld sMrab;TMhMEdkepSg². eKGackMNt;EdkbBaÄr
Ash Edlkat;tamsñameRbHtamTisedkdUcxageRkam



Ash =
(Avf + An ) f y (10.4)
μ 'e f ys
1,000λAcr μ
Edl μ 'e =
(
Avf + An f y) (10.5)

ehIy rbs; Ash
f ys = yield strength

An = RkLaépÞrbs;EdkedIm,ITb;Tl;kMlaMgTajtamG½kS N u enAkñúgrUbTI 10>2 Edl

( )
An = N u / φf y (10.6)

Edl Nu = kMlaMgTajtamTisedkGnuvtþn_emKuNEdlEkgeTAnwgbøg;sñameRbHsnμt;
φ = emKuNkat;bnßyersIusþg; = 0.75



cMNaMfa eKRtUvf<k;EdkBRgwgTaMgGs;enAelIRCugNak¾edayrbs;bøg;sñameRbHsnμt;eGay)an
l¥eday development length b¤edaykarpSareTAnwgEdkEkg (angles)/ EdkbnÞH b¤EdkTMBk; (hooks)
edIm,IbegáItkMlaMgTb;Tl;Edl)anKNna.

KNnaRTnab;TMrebtugGarem:
10.4.1. Reinforced Bearing Design
]TahrN_ 10>1³ FñwmebtugeRbkugRtaMgragctuekaN PCI standard 16RB28 rgkMlaMgkat;emKuNbBaÄr
Vu = 90,000lb(400kN ) nigkMlaMgTajtamTisedk N u = 21,000lb(93.4kN ) . FñwmRtUv)anRTenAelI
Teflon pad TMhM 4in. × 4in.(10cm × 10cm ) . KNna end reinforcement enAkñúgFñwmEdlGackarBarkar

ekItman bearing crack tamTisedk b¤tamTisQr. eKeGayTinñn½yxageRkam³
f 'c = 5,000 psi (34.47 MPa ) ebtugTMgn;Fmμta

f y = 60,000 psi sMrab;EdkFmμtaTaMgGs; (413.7 MPa )

θ = 20 o
dMeNaHRsay³
EdktamTisedk (Avf + An )
sMrab;karkMNt;EdkBRgwgtamTisedk/ sakl,gEdk #6
kMBs;Fñwm h = 28in. b = 16in.
BItarag 10>3/ ld = 29in.
Acr = l d b = 29 × 16 = 464in.2
BItarag 10>2/ μ = 1.4 nigBIsmIkar 10.3
1,000λAcr μ 1,000 × 1.0 × 464 × 1.4
μe =
Vup
=
90,000
= 10.61 > μ e GnuBaØat = 3.4
dUcenHeRbI μe = 3.4
BIsmIkar 10.2
Avf =
Vup
φf y μ e
=
90,000
0.75 × 60,000 × 3.4
(
= 0.59in.2 3.4cm 2 )
N u = 21,000lb
N u 21,000
=
Vu 90,000
= 0.23 > tMélGb,brma 0.20
dUcenH yk N u = 21,000lb .



BIsmIkar 10.6/ An = N u / φf y = 21,000 /(0.75 × 60,000) = 0.47in.2 (2.94cm 2 )
Edksrub
As = Avf + An = 0.59 + 0.47 = 1.06in.2 (6.63cm 2 )

dUcenH eRbI 3#6 = 1.32in.2 (8.52cm 2 )
EdkbBaÄr ¬ Ash ¦
BItarag 10>3/ ld = development length rbs;Edk #6 = 29in.(74cm) nig Acr = ld b =
29 × 16 = 464in 2 (3,159cm 2 ). BIsmIkar 10.5
1,000λAcr μ 1,000 × 1.0 × 464 × 1.4
μ 'e =
(A + A ) f = 0.93 × 60,000 = 11.64 > μe GnuBaØat = 3.4
vf n y

dUcenH eRbI μ 'e = 3.4 . BIsmIkar 10.4
Ash =
(Avf + An ) f y = 0.93 × 60,000 = 0.27in.2 (1.74cm 2 )
μ 'e f ys 3.4 × 60,000

dUcenH eRbIEdkkg (stirrup) (
= 0.66in.2 4.26cm 2 )

10.5. Dapped-End Beam Connections

-
Dapped end beam CaGgát;eRKOgbgÁúMEdlmankarbnßykMBs;FñwmPøam²enAxagcugrbs;vaedIm,I
nUv seating b¤ bearing caM)ac;enAelI corbel b¤ bracket edayKμankar)at;bg; clear height rvagkMral.


rUbTI 10>4 bgðajBIFñwmebtugeRbkugRtaMgEdlman dapped end KMrUenAxagcug. sñameRbHBIrRbePTGac
ekItman³ sñameRbHelx @ CasñameRbHkMlaMgkat;edaypÞal; (direct shear crack) cMENkÉsñameRbH
elx #/ elx $ nigelx % CasñameRbHkMlaMgTajGgát;RTUgEdlbgáedaykMlaMgTajtamG½kS nigkMlaMg
begáagenAkñúgkMBs;FñwmEdlkat;bnßy ehIykugRtaMgRbmUlpþúMenAmþúMkac;RCug. dUcenH eKRtUvdak;RbePT
EdkBRgwgxageRkam dUcbgðajenAkñúgrUb³
!> EdkrgkarBt; (flexural reinforcement) A f bUknwgEdkrgkarTajtamG½kS An Edl
As = A f + An edIm,IkarBar cantilever bending stresses.

@> Shear-friction reinforcement A f + An bUknwgEdkrgkarTajtamG½kS An edIm,IkarBar
kMlaMgkat;bBaÄredaypÞal; (direct vertical shear force) enARtg;RbsBVénEpñk dapped nig
EpñkEdlmin dapped rbs;FñwmEdlbgáeGaymansñameRbHelx @.
#> EdkrgkMlaMgkat; (shear reinforcement) Ash edIm,ITb;Tl;nwgkugRtaMgTajGgát;RTUgEdl
ekItmanenARtg;cMnuckac;RCugEdlbgáeGaymansñameRbHelx #.
$> EdkrgkMlaMgTajGgát;RTUg (diagonal tension reinforcement) Ah + Av edIm,IkarBarsñam
eRbHelx$ EdlekItBIkugRtaMgTajGgát;RTUgenAkñúgEpñk papped rbs;Fñwm.
%> Development length As = A f + Ah edIm,IkarBarsñameRbHelx% EdlbNþalBIkugRtaMg
TajGgát;RTUg enAkñúgEpñkmin dapped rbs;Fñwm.

10.5.1. karkMNt;EdkBRgwgedIm,ITb;Tl;kar)ak;
Determination of Reinforcement to Resist Failure
10.5.1.1. EdkrgkarBt; nigEdkrgkarTajtamG½kS Flexure and Axial Tension
sMrab;lMnwgm:Um:g;enAkñúgrUbTI 10>4/ m:Um:g;emKuNsrubEdleFVIGMeBIenAelIEpñk cantilever dapped
enARtg;bøg;rbs; As KW
M u = Vu a + N u (h + d ) (10.7a)
Edl h= kMBs;rbs;Ggát;BIelI dap
d = kMBs;RbsiT§PaBrbs; dap eTATIRbCMuTMgn;rbs;EdkBRgwg As

a = ElVgkMlaMgkat; (shear span)

M u RtUvTb;Tl;edayersIusþg;m:Um:g; nominal M n = M u / φ / b¤



Vu a + N u (h − d )
Mn = (10.7b)
φ
edaysnμt;faédXñas;m:Um:g; jd ≅ 0.9d
Vu a + N u (h − d )
Fn = (10.8)
0.9φd
Edl φ = 0.90 sMrab;karBt;begáag. edaysar 0.9φ = 0.81 edIm,ICakarsMrYleKeRbItMél φ = 0.85 enA
kñúgsmIkar 10.8 edIm,ITTYl)an
Vu a + N u (h − d )
Fn = (10.9a)
φd
V ⎛a⎞ N ⎛h−d ⎞
b¤ Fn = u ⎜ ⎟ + u ⎜
φ ⎝d ⎠ φ ⎝ d ⎠
⎟ (10.9b)

enaHEdkrgkarBt;begáagKW
Fn Vu a + N u (h − d )
As = = (10.10)
fy φf y d
ehIyEdkrgkarTajedaypÞal;EdlbNþalBIkMlaMgTaj Nu KW
Nu
An = (10.11)
φf y
BIsmIkar 10.10 nig 10.11/ RkLaépÞsrubrbs;EdkrgkarBt;begáag nigEdkrgkarTajedaypÞal;køayCa
1 ⎡ ⎛a⎞ ⎛ h ⎞⎤
As = A f + An = ⎢Vu ⎜ d ⎟ + N u ⎜ d ⎟⎥ (10.12)
φf y ⎣ ⎝ ⎠ ⎝ ⎠⎦
Edl tMélEksMrYlrbs; φ = 0.85 .

10.5.1.2. EdkrgkMlaMgkat;bBaÄredaypÞal; Direct Vertical Shear
sñameRbHelx@ EdlekItBIkMlaMgkat;edaypÞal;RtUv)anTb;edaybnSMénEdk As nig Ah enAkñúg
rUbTI 10>4. eKGackMNt;EdkBRgwgtamTisedk Ah EdlRtUvkaredIm,ITb;Tl;nwgkMlaMgkat;edaypÞal;
tamsmIkarxageRkam
Ah = 0.5( As − An ) (10.13)

Edl As =
2Vu
3φf y μ e
+ An (10.14a)

Nu
An = (10.14b)
φFy
1,000λbhμ
μe =
Vu
CamYynwg φ = 0.85 nig μe dUcenAkñúgsmIkar 10.3. dUcenH


1 ⎛ 2Vu ⎞
As = ⎜ 3μ + N u ⎟
⎜ ⎟ (10.15)
φf y ⎝ e ⎠
tMélrbs; As EdleRbIenAkñúgsmIkar 10.13 KYrFMCagtMélTaMgBIrEdlTTYl)anBIsmIkar 10.12
nig 10.15.
eKRtUvbgðÚtEdk As edaytMélGb,brma 1.7ld kat;tamcMnuccugénEpñk dap b¤ ld kat;sñameRbH
elx% ehIyf<k;enAxagcugrbs;FñwmedaypSarP¢ab;eTAnwg cross bar/ angle b¤ plates. ekRtUvbgðÚtEdk
tamTisedk Ah dUcKña ehIyEdkbBaÄr Ash nigEdkbBaÄr b¤EdkeRTt Av k¾RtUv)anf<k;edayTMBk;
(hook) tamkarTamTarrbs; ACI Code.

ersIusþg;kMlaMgkat; nominal rbs; dap end RtUv)ankMNt;Rtwm
Vn ≤ 0.30 f 'c bd ≤ 1,000bd (10.16a)
sMrab;ebtugTMgn;Fmμta/
⎛ 0.07a ⎞
Vn ≤ ⎜ 0.20 − ⎟ f 'c bd (10.16b)
⎝ d ⎠
⎛ 280a ⎞
b¤ Vn ≤ ⎜ 800 −
⎝ d ⎠
⎟bd (10.16c)

sMrab; sand-lightweight b¤ all-lightweight concrete, edayykmYyNaEdltUcCag Edl a Ca shear
span nig d CakMBs;RbsiT§PaBrbs;Fñwm.

10.5.1.3. EdkrgkMlaMgTajGgát;RTUgRtg;kac;RCug
Diagonal Tension at Reentrant Corner
eKTTYl)anEdkBRgwgEdlRtUvkaredIm,ITb;Tl;nwgsñameRbHedaysarkMlaMgTajGgát;RTUgeRTtE
dlralBIp©iténkugRtaMgRbmUlpþúMenARtg;kac;RCugeTAkan;EpñkEdlmin dapped BIsmIkarxageRkam
Vu
Ash = (10.17)
φf y
Edl φ = 0.85 ehIy f y Ca yield strength rbs;EdkBRgwg Ash .

10.5.1.4. EdkrgkMlaMgTajGgát;RTUgenAkñúg Dapped end
Diagonal Tension in the Dapped end
edIm,IkarBarsñameRbHGgát;RTUgelx$ enAkñúg dapped end/ eKRtUvdak;EdkbEnßm As y:agNa
edIm,IeGayersIusþg;kMlaMgkat; nominal srub Vn bMeBjsmIkar



Vu
Vn = = Av f y + Ah f y + 2λbd f 'c (10.18)
φ
y:agehacNas;k¾eKRtUvdak;EdkBak;kNþalénEdkBRgwgenHbBaÄr dUcenHsmIkar 10.18 eGay
1 ⎛ Vu ⎞
Av, min = ⎜ − 2λbd f 'c ⎟
⎜φ ⎟ (10.19)
2 fy ⎝ ⎠
cMNaMfa karKitBIkareFVIkarTamTardUcxageRkam³
!> kMBs;rbs; dapped end y:agehack¾esμIBak;kNþalénkMBs;Fñwm elIkElgkMBs;FñwmFMCag
tMrUvkar.
@> RbsinebIkugRtaMgBt;begáagEdlKNnasMrab;kMBs;eBjelj (full depth) rbs;muxkat;eday
eRbIbnÞúkemKuN nig gross section propertied FMCag 6 f 'c Pøam²BIeRkay dap/ eKKYrdak;
EdkBRgwgbeNþaybEnßmenAkñúgFñwmedIm,IbegáItersIusþg;Bt;begáagtMrUvkar.
#> eKRtUvdak;EdkrgkarTajGgát;RTUg Ash eGaykan;EtEk,rkac;RCug. EdkBRgwgenHCaEdk
bEnßmeTAelIEdkrgkMlaMgkat;KNna (design shear reinforcement) EdlRtUvkarsMrab;mux
kat;FñwmEdlmankMBs;eBj.

10.5.2. KNnatMNrbs; Dapped end Beam
Dapped-End Beam Connection Design
]TahrN_ 10>2³ FñwmebtugeRbkugRtaMg PCI standard 16RB28 Edl dapped enAxagcugsMrab;
bearing enAelI column corbel/ rgnUvkMlaMgkat;TMnajemKuNenAxagcug Vu = 110,000lb(489kN )
nigkMlaMgTajtamG½kStamTisedk Nu = 20,000lb(97.9kN ) . KNnaEdkrgkarBt;begáag Edkrg
kMlaMgkat;edaypÞal; nigEdkrgkarTajGgát;RTUg As / Ash / Ah nig Av EdlRtUvkarsMrab;karBarsñam
eRbH EdlbNþalBI dapping énFñwmxagcug. Tinñn½yEdleKeGayman f 'c = 5,000 psi(34.5MPa)
ebtugTMgn;Rsal ehIy f y = 60,000 psi(414Mpa) .
dMeNaHRsay³
snμt;fa shear span a = 6in.(152mm) / kMBs;RbsiT§PaB dapped-end d = 16in.(406mm)
nig h = 18in.(457mm) .
EdkrgkarBt; nigEdkrgkarTajtamG½kS As
Nu 20,000
= = 0.18 < 0.20
Vu 110,000
dUcenH N u = 0.20 ×110,000 = 22,000lb(97.9kN )


1 ⎡ ⎛a⎞ ⎛ h ⎞⎤
As = ⎢Vn ⎜ d ⎟ + N u ⎜ d ⎟⎥
φf y ⎣ ⎝ ⎠ ⎝ ⎠⎦
1 ⎡ 6 18 ⎤
= ⎢110,00 × 16 + 22,000 × 16 ⎥ = 1.46in.
2
0.75 × 60,000 ⎣ ⎦
EdkrgkMlaMgkat;edaypÞal; As nig Ah
BItarag 10>2 / μ = 1.4λ Edl λ = 1.0 . bnÞab;mk BIsmIkar 10.14c Edl b sMrab;muxkat; 16RB28
esμInwg 16in.
1,000λbhμ 1,000 × 1.0 × 16 × 18 × 1.4
μe = = = 3.67 > μ e GnuBaØatGtibrma = 3.4
Vu 110,000
dUcenH eRbI μe = 3.4 . bnÞab;mk BIsmIkar 10.5
1 ⎛ 2Vu ⎞ 1 ⎛ 2 × 110,000 ⎞
As = ⎜
⎜ 3μ + N u ⎟ = 0.75 × 60,000 ⎜ 3 × 3.4 + 22,000 ⎟ = 0.96in.
⎟
2
φf y ⎝ e ⎠ ⎝ ⎠
)anmkBIelImun
< As = 1.46in.2

dUcenH eRbI As = 1.46in.2 (9.1cm2 ) . enaHEdk 3#7 = 1.80in.2 EdlRKb;RKan;.
BIsmIkar 10.4b
= 0.49in.2 (3.0cm 2 )
N 22,000
An = u =
φf 0.75 × 60,000
y

BIsmIkar 10.13/ EdkrgkMlaMgkat;tamTisedkkat;tamkMBs;rbs;FñwmKW Ah = 0.5( As − An ) =
0.5(1.29 − 0.43) = 0.43in.2 (2.77cm 2 ) . dUcenH sakl,gEdk 2#3 = 2(2 × 0.11) = 0.44in.2

(2.84cm 2 ) EdleKRtUvepÞógpÞat;vaCabnþbnÞab;. tamkarRtYtBinitüBIsmIkar 10.16a ersIusþg;kMlaMgkat;
nominal KW

Vn EdlGacekItman = 800bd = 800 ×16 ×16 = 204,800lb

Vn tMrUvkar = u =
V 110,000
= 146,667lb < 204,800lb O.K.
φ 0.75
EdkBRgwgbBaÄrrgkarTajGgát;RTUgenAkac;RCug
BIsmIkar 10.19
= 2.45in.2 (15.3cm 2 )
V 110,000
Ash = u =
φf 0.75 × 60,000
y

dUcenH sakl,gEdkkgbiTCit #4 / As = 2 × 0.20 = 0.40in.2 . cMnYnrbs;Edkkg 2.16 / 0.4 = 5.4
dUcenHeRbI 6#4 EdlRtUv)anRbmUlpþúMenAmþúMkac;RCug.
EdkBRgwgkarTajGgát;RTUg Av enAkñúg Dapped End
BIsmIkar 10.19


1 ⎛ Vu ⎞
Av = ⎜ − 2λbd f 'c ⎟
2 fv ⎜ φ
⎝
⎟
⎠
ersIusþg;kMlaMgkat; nominal rbs;ebtugsuT§KW
2λbd f 'c = 2 × 1.0 × 16 × 16 5,000 = 36,204lb
⎛ 110,000 ⎞
bnÞab;mk Av =
1
⎜
2 × 60,000 ⎝ 0.75
− 36,204 ⎟ = 0.92in.2
⎠
sakl,gEdkkgGkSr U 4#4 = 4(2 × 0.20) = 1.60in.2 . BIelIkmun/ Ah = 0.44in.2 . dUcenH
BIsmIkar 10.18 ersIusþg; kMlaMgkat; nominal srubrbs;muxkat;KW
Vn EdlGacekItman = Av f y + Ah f y + 2λbd f 'c

= 1.60 × 60,000 + 0.44 × 60,000 + 36,034
V
= 158,434 > u = 146,667lb O.K.
φ
RtYtBinitütMrUvkar Development Length sMrab;karf<k;
EdkBRgwg As KW 3#7 . BItarag 10>3/ sMrab;Edk #7 / f 'c = 5,000 psi nig ld = 42in. .
kMBs;FñwmEdlmin dapped = 2 ft 4in. = 28in. ehIy development length srub = 28 − d + ld =
28 − 16 + 42 = 54in. . edaysar development length Gb,brma l d = 42in. eRbI l d = 54in.

= 4 ft 6in.(108cm ) .



EdkBRgwg Ah KWEdkGkSr U #4 . dUcenHBItarag 10>3/ 1.7ld = 32in.(81cm) BIeRkayFñwm
dap. rUbTI 10>5 bgðajBIkarlMGitEdksMrab;tMNFñwm dapped.

10.6. Brackets nig Corbel ebtugGarem:
Reinforced Concrete Brackets and Corbels
Corbel CaFñwm cantilever xøI EdlpleFob shear span elIkMBs; a / d minRtUvFMCag 1.0 . varg
kMlaMgkat;edaypÞal; Vu nigkMlaMgTajtamTisedk N u . Epñk 5.14 enAkñúgCMBUk5/ design flowchart
enAkñúgEpñk 5.14.4 nig]TahrN_ 5>7 bgðajBIkMlaMgsgát; nigkarGnuvtþénRTwsþI shear-friction enAkñúg
karsikSaKNna corbel. karlMGitsrésEdkrbs;tMNCakargard¾sMxan;mYyedIm,ITTYleCaKC½ykñúgkar
sikSaKNna corbel edayKitBIlT§PaBrbs;vaedIm,ITb;Tl;nwgGnuvtþkMlaMgGnuvtþn_. srésEdklMGit
rbs; corbel KMrURtUv)anbgðajenAkñúgrUbTI 10>6.



10.7. Epñklyecjrbs;FñwmebtugGarem:
Concrete Beam Ledges
eKeRbI beam ledge edIm,IRTbnÞúkcMcMnuccugFñwmebtugeRbkugRtaMgcak;Rsab;tamTisTTwg ehIyva
eFVIkarkñúgTMrg;RsedogKñanwg corbel Edr. kM;laMgedaypÞal;EdleFVIGMeBIelI ledge GacbgáeGaymansñam
bBaÄrdUcbgðajenAkñúgrUbTI 10>7. RbsinebIbnÞúkCabnÞúkminCab; ehIymkBIRCugmçag/ ledge beam kñúg
TMrg;GkSr L eFVIGMeBIdUc spandrel beam nigrgm:Um:g;rmYlbEnßmBIelIkMlaMgkat;edaypÞal;. karKNna
ledge beam KWGnuvtþtamkarsikSaKNna nig]TahrN_enAkñúgCMBUk5. enAkñúgemeronenHbgðajBIkar

sikSaKNnaEdkrgkMlaMgkat;sMrab; cantilevering ledge EdlCaTUeTAmanpleFob shear span elI
kMBs; l p / d tUcCag b¤esμInwg 0.5 .

eKRtUvkMNt;esIusþg;kMlaMgkat; nominal rbs; ledge Rtg;kac;RCugedaytMéltUcCageKkñúg
cMeNamtMélEdlTTYlBIsmIkar nigeRkamlkçxNÐEdleKeGaydUcxageRkam
!> s > b + h



(
Vn = 3hλ f 'c 2l p + b + h ) (10.20a)
(
Vn = hλ f 'c 2l p + b + h + 2d e ) (10.20b)

@> s < b + h nigbnÞúkcMcMnucesμIKña
(
Vn = 1.5hλ f 'c 2l p + b + h + s ) (10.21a)
⎛ b+h ⎞
Vn = hλ f 'c ⎜ l p + + de + s ⎟ (10.21b)
⎝ 2 ⎠
Edl RbEvglyecjrbs; ledge
lp =

b = TTwgrbs; bearing area

h = kMBs;rbs; ledge

s = KMlaténbnÞúkcMcMnuc

d e = cMgayBIG½kSénbnÞúkeTAcugFñwm

RbsinebI ledge RTbnÞúkCab; b¤bnÞúkcMcMnucEdlmanKMlatEk,rKña eKkMNt;ersIusþg;kMlaMgkat;
nominal rbs;muxkat; ledge BI
Vn = 24hλ f 'c (10.22)

Edl Vn CakMlaMgkat;kñúgmYyÉktþaRbEvg. y:agehacNas; eKRtUveGayersIusþg;KNna Vu esμInwgkM
laMgemKuN Vu = φVn sMrab; φ = 0.85 . RbsinebIbnÞúkemKuNGnuvtþn_ Vu FMCagersIusþg;KNna dUc
EdlkMNt;BIsmIkar 10.20, 10.21 b¤ 10.22/ eKRtUvdak;EdkBiessEdlKNnaRsedogKñaeTAnwgEdk
EdlRtUvkarenAkñúgcug dapped beam dUcEdl)anerobrab;enAkñúgEpñk 10.5. enAkñúgkrNIenH eKRtUv
kMNt;EdkrgkarBt; As BIsmIkar 10.12/ EdkBRgwgrgkarTajGgát;RTUgbBaÄr (hanger) Ash BI
smIkar 10.17 ehIyEdkBRgwgbEnßm At Edldak;enAsrésxagelI nigsrésxageRkamrbs; ledge BI
200l p d
At = (10.23)
fy

Edl At CaRkLaépÞrbs;EdkbeNþayenAkñúg ledge. eKdak;EdkBRgwg Ash edayKMlatesμIKñaelITTwg
6h énRCugnImYy²rbs; bearing b:uEnþminRtUvFMCagBak;kNþalcMgayeTAkan;bnÞúkbnÞab;. KMlatEdkmin

RtUvFMCagkMBs; ledge h b¤ 18in. ehIy Ash RtUv)ansikSaKNnasMrab; ledge EdlminRtUvbEnßmeTAelI
EdkrgkMlaMgkat; nigEdkrgkMlaMgrmYlrbs; ledge beam srub.

karKNnatMNFñwmlyecj
10.7.1. Design of Ledge Beam Connection
]TahrN_ 10>3³ eRKOgbgÁúMkMralGaKarcMNtrfynþRtUv)anpSMeLIgBI 10 ft -wide double-T RtUv)an



RTenAmuxkat;FñwmGkSr L sþg;dar. eKRtUvdak;eCIgrbs; double-T y:agNaenAelIRKb;cMnucTaMgGs;enA
elI ledge. kMlaMgkat;emKuNbBaÄrxagcug Vu = 24,000lb(107kN ) kñúgeCIgmYy nigkMlaMgTajtamTis
edk Nu = 5,000lb(22.4kN ) kñúgeCIgmYy. KNnaersIusþg;kMlaMgkat; nominal rbs; ledge nigsikSa
KNnaEdkRbsinebIcaM)ac;. eKeGay
b = 4in.

h = 12in.

d = 10.5in.
l p = 6in.(15cm )

s = 48in.(122cm )
f 'c = 5,000 psi (34.5MPa ) ebtugTMgn;Rsal
f y = 60,000 psi (414MPa )

dMeNaHRsay³
Vu = 24,000lb

N u = 5,000lb

s = 48in.

b + h = 4 + 12 = 16in.
tMélGb,brmarbs; d e = 1 b = 2in.
2
2l p + b + h = 2 × 6 + 4 + 12 = 28in.

edaysar s > b + h nig d e < 2l p + b + h ehIyGnuvtþsmIkr 10.20b ehIyersIusþg;kMlaMgkat;Edl
GacekItman Vn = hλ f 'c (2l p + b + h + 2de ) = 12 ×1.0 5,000 (2 × 6 + 4 + 12 + 2 × 2) = 27,153lb
(120.8kN ) . dUcenHersIusþg;kMlaMgkat;KNna Vu = φVu = 0.75 × 27,153 = 20,365lb < kMlaMgkat;em
KuN Vu = 24,000lb ehIyeyIgRtUveRbIEdkBRgwgEdlKNnadUcEdkBRgwgsMrab;muxkat; dapped Edr.
EdkrgkarBt;begáag As
Shear span a ≅ 3l p / 4 + 1.5 = 3 × 6 / 4 + 1.5 = 6in.(15cm )

edaysar Nu / Vu = 5,000 / 24,000 = 0.21 > 20% dUcenHeKeRbI Nu = 5,000lb .
BIsmIkar 10.12
1 ⎡ ⎛a⎞ ⎛ h ⎞⎤
As = ⎢Vu ⎜ d ⎟ + N u ⎜ d ⎟⎥
φf y ⎣ ⎝ ⎠ ⎝ ⎠⎦



=
1 ⎡ 6 12 ⎤
( )
⎢24,000 10.5 + 5,000 10.5 ⎥ = 0.38in. 2.45cm
0.85 × 60,000 ⎣ ⎦
2 2

edaysar 6h = 6 ×12 > s / 2 = 24in. / EbgEckEdkBRgwg s / 2 = 24in. enARCugnImYy²énbnÞúk.
TTwgrbs; band sMrab;kardak;EdkrgkarBt;begáag A = 2 × 24 = 48in. ehIyKMlatEdkGti-
s

brma h = 12in. . dUcenHeRbIEdk 4#3 enAkñúg band width 48in. nImYy² = 0.44in. > EdktMrUvkar
2

0.38in. . dUcenH dak;EdkbEnßmBIrenARtg;cugFñwmedIm,Ipþl;EdksmmUlsMrab;eCIgFñwmEdldak;Ek,r
2

xagcug.
EdkbBaÄrrgkarTajGgát;RTUg A sh

BIsmIkar 10.17
= 0.53in. (3.42cm )
V 24,000
A = = u 2 2
φf
sh
0.75 × 60,000
y

elI hand width 48in. . dUcenH Ash / ft = 0.47 / 4 = 0.12in.2 / ft b¤ #3@11in. . Cavi)akeRbIEdkkg
biTCit 5#3 enAkñúg bad width 48in. = 0.55in.2 > muxkat;EdktMrUvkar 0.53in.2 . bnÞab;mk sMrab;kargar
Gnuvtþn_ eRbIcMnYn nigKMlatdUcKñasMrab;Edk As nig Ash ¬EdkkgbiTCit 5#3 ¦. cMNaMfa manEteCIgmçag
rbs;Edkkg Ash RtUv)anKitbBa©ÚleTAkñúgmuxkat;én 5#3 edIm,Ipþl;karRbmUlpþúMtMrUvkarénEdkEk,rkac;
RCug.
EdkbeNþay Al
BIsmIkar 10.23
200l p d 200 × 6 ×10.5
Al = = = 0.21in.2
fy 60,000

sMrab;kargarGnuvtþn_ eRbIEdk #4 mYyenARtg;kac;RCugrbs; ledge edayeGay 4#4 = 0.80in.2 ¬Ggát;
p©it 12.7mm ¦ > 0.21in.2 / O.K.
CakarBit karKNnaEdlmanlkçN³eBjeljTamTarkarviPaKkMlaMgkat; nigkMlaMgrmYlrbs;
muxkat;srubedIm,ITb;Tl;nwgkMlaMgkat;srubEdlbBa¢ÚnedayeCIgrbs; double-T TaMgGs; nigm:Um:g;rmYl
EdlekIteLIgedaykarGnuvtþrbs;kMlaMgcakp©itBIeCIgrbs; double-T. karerobrab;BIRkLaépÞEdkBRgwg
rbs; ledge Edl)anKNnaenAkñúg]TahrN_enHCaRkLaépÞEdkbEnßmeTAelIelIEdkrgkMlaMgTaj nig
Edkrg kMlaMgrmYlEdlTamTarsMrab;Fñwmsrub.
rUbTI 10>8 bgðajBIkarlMGitrbs;srésEdksMrab;tMN ledge, b:uEnþmin)anrab;bBa©ÚlEdkrgkM-
laMgTaj nigEdkrgkMlaMgrmYlEdlRtUvsikSasMrab;FñwmGkS L TaMgmUl.



10.8. lMGittMNEdl)aneRCIserIs Selected Connection Details
dUcEdl)aneerobrab;enAkñúgEpñk 10.1 tMNCaeRKOgP¢ab;cMbgenAkñúgRbB½n§eRKOgbgÁúMTaMgmUl Edl
kareFVIkarrbs;vakMNt;faeRKOgbgÁúMmansuvtßiPaB nigmanesßrPaB. dUcenH design engineer RtUvmankar
Rby½tñy:agxøaMgkñúgkarsikSaKNna nigeRCIserIsnUvmuxkat;EdlsmRsbsMrab;mUlehtu suvtßiPaB nig
esdækic©. BIrUbTI 10>9 dl;rUbTI 10>16 manbgðajBIkarlMGiténRbePTtMNEdl)aneRCIserIsCaeRcIn.


X. connections for prestressed concrete element

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