Xii.lrfd and stan dard aastho design of concrete bridge

Department of Civil Engineering NPIC

XII. karsikSaKNnas<anebtugsþg;dar AASTHO nig LRFD
LRFD and Standard AASTHO Design of Concrete Bridges

12.1. esckþIepþIm³ suvtßiPaB nigTMnukcitþ
Introduction: Safety and Reliability
dUc)anerobrab;enAkñúgEpñk 4.10.1/ Load-resistance factor design method (LRFD) CaviFI
mYyEdlQrelITMnukcitþ sMrab;vaytMélkarsikSaKNnaedayeRbIemKuNEdlQrelIRbU)ab‘IlIet. viFI
enHmanbMNgsMrab;kMNt;smamaRtmuxkat;rbs;eRKOgbgÁMúedayQrelIRbePTbnÞúk edayeFVIy:agNa
eGayersIusþg;Tb;Tl;FMCagbnÞúkemKuN b¤m:Um:g;emKuN.
rUbTI 12>1 (a) nig (b) k¾dUcenAkñúgrUbTI 4>36 énCMBUk 4 bgðajBIdüaRkaménkarEbgEck
frequency dac;edayELkBIKñaénbnÞúkCak;Esþg W nigersIusþg; R CamYynwgtMélmFüm (mean) R .

rUbTI 12>1 (c) bgðajBIkarbUkcUlKñaénkarEbgEckTaMgBIr ehIyExSekagTaMgBIrkat;KñaRtg;cMnuc C .
eKGacrMBwgfaeRKOgbgÁúMman suvtßiPaB nigGacTukcitþ)an RbsinebIbnÞúk W sßitenAxageqVgcMnuc
RbsBV C enAelIExSekagersIusþg;. pÞúymkvij eKrMBwgfaeRKOgbgÁúMnwg)ak; RbsinebIbnÞúkenAelIExSekag
ersIusþg;sßitenAkñúgépÞqUt enAkñúgrUbTI 12>1 (c). RbsinebI β CasnÞsSn_suvtßiPaB enaH
R −W
β= (12.1)
σR
2
+ σW
2

Edl σ R nig σ W Ca standard deviation rbs;ersIusþg; nigbnÞúk erogKña.
karbnSMbnÞúkepSgKñaenAkñúgsmIkar 4.29 ¬CMBUk 4¦ KWQrelIPaBxusKñad¾smrmürvag R nig
W EdlBicarNanUvlkçN³esdækic©.

dUcenH TMnukcitþénkareFVIkarrbs;eRKOgbgÁúMEdlRbkbedaysuvtßiPaBRtUv)anRKb;RKgedaykarBi-
carNarvagTMnak;TMngbnÞúk nigersIusþg;edayeRbIemKuNbnÞúk nigemKuNersIusþg;enAkñúgkarsikSaKNna.
viFI LRFD rbs; AASTHO manbMNgGPivDÄTMnak;TMngrvagbnÞúk nigersIusþg; nUvsmIkarsMrab;
kMhUcRTg;RTay nigbnÞúk nigemKuNersIusþg;EktMrUv φ BIemKuNersIusþg;EdleRbIeday ACI 318. emKuN
φ rbs; LRFD RtUv)anbgðajenAkñúgtarag 12>1 (a).
emeronenHnwgbgðajBIkareRbIR)as;smIkarrbs; LRFD EdlmanlkçN³xusKñaBIsmIkarrbs;
standard AASTHO nig ACI 318. ehIysmIkarEdl)aneRbIenAkñúgCMBUk 3/ 4 nig5 nigeKalkarN_

rbs;vak¾RtUv)anykmkGnuvtþ. dUcenHnisiSt nigvisVkrnwgyl;BIeRbIR)as;smIkaTaMgenHy:agRsYl.

karsikSaKNnas<anebtugsþg;dar AASTHO nig LRFD 745

T.Chhay viTüasßanCatiBhubec©keTskm<úCa

LRFD and Standard AASTHO Design of Concrete Bridges 746


12.2. lkçN³bec©keTsénbnÞúkrfynþrbs; LRFD nig AASTHO Standard (LFD)
AASTHO Standard (LFD) and LRFD Truck Load Specification
karsikSaKNnaBIGgát;ebtugeRbkugRtaMgrbs;s<anRtUv)aneFVIeLIgedayeKarBeTAtam AASTHO
(American Association of Highway and Transportation Officials). kñúgkarsikSaKNnaeRKOgbgÁúM

EpñkxagelI (superstructure) rbs;s<an eKRtuveRCIserIs nigdak;pøÚvcracrN_ (traffic lane) nigbnÞúky:ag
NaEdleFVIeGaymankugRtaMgGtibrmaenAkñúgGgát;eRKOgbgÁúM.
kardak;bnÞúkGefrenAelIs<anRtUvman standard truck b¤ lane load EdlsmmUleTAnwg truck
trains. sMrab;s<anrfePøIg (railway bridge), tMrUvkarRtUv)ankMNt;eday AREA (American Railway

Engineering Association). CaTUeTA tMrUvkarsMrab;kMNt;smamaRtmuxkat;eRKOgbgÁúMrbs;Ggát;TMreRcIn

GnuvtþtambTdæan ACI nig PCI.

12.2.1. bnÞúk Load
kardak;bnÞúk highway mancMNat;fñak;bTdæan 4KW³ H20, H15, Hs20 nig HS15. kardak;bnÞúk
HS15 KWesμInwg 75%énkardak;bnÞúk HS20. RbsinebImankardak;bnÞúkepSgeRkABIenH eKRtUveFVIkarEk



tMrUvTMgn;sMrab; standard truck nig land load EdlRtUvKñatamlkçN³smamaRt. eKsikSaKNna s<an
Highway sMrab;kardak;bnÞúk HS20-44 b¤kardak;bnÞúkEdlmanePøArfynþ (axles) BIrEdlmanKMlatBIKña

14 ft ehIy axle nImYy²manTMgn; 24,000lb edayeKRtUvdak;bnÞúky:agNaedIm,ITTYl)ankugRtaMgFM.

rUbTI 12>2 bgðajBIbnÞúk standard H Truck ehIyrUbTI 12>3 bgðajBIbnÞúk standard HS
truck CamYynwgKMlatkg;rfynþ nigkarEbgEckbnÞúk. rUbTI 12>4 bgðajBI lane load smmUlsMrab; H

20-44, HS 20-44, H 15-44 nig HS 15-44. rUbTI 12>5 bgðajBIRbB½n§kMrals<anepSg²EdleKeRbICa

TUeTA.
(i) T§iBlTgáic (impact): bnÞúkcl½t (movable load) TamTarnUvkarGnuBaØatPaBTgáic EdlCaEpñk

mYyrbs;bnÞúkGefr. eKGackMNt;va)antam standard AASTHO (LFD)
50
I= ≤ 30% (12.2)
L + 125
Edl I= PaKryT§iBlTgáic (impact fraction)
L = RbEvgKitCa feet rbs;cMENkénElVgEdlkardak;bnÞúkeFVIeGaymankugRtaMgGti-

brmaenAkñúgGgát;enaH.



RbEvgdak;bnÞúk L sMrab;Ggát;TTwg (transverse member) dUcCaFñwmkMral (floor beam) CaRb-
Evgrbs;Ggát;EdlKitBIG½kSeTAG½kS.
(ii) kMlaMgtamTisbeNþay (Longitudinal Forces): eKRtUvdak;nUvkMlaMgtambeNþayesμInwg 5%

énbnÞúkGefrenARKb; lane TaMgGs;EdlRTcracrN_kñúgTisedAdUcKña. épÞbnÞúk EdlKμanT§iBl
Tgáic RtUvmanlkçN³dUcxageRkam³
land load + bnÞúkcMcMnuc RtUv)andak;enAelIElVgy:agNaedIm,IbegáItkugRtaMgGtibrma.

bnÞúkcM cMnuc nigbnÞúkBRgayRtUv)anBicarNaBRgayesμIelITTwg 10 ft enAelIExSEdl
EkgeTAnwgG½kSrbs; land. eKsnμt;TIRbCMuTMgn;rbs; longitudinal force manTItaMg
enARtg; 6 ft BIelIkMral.
eKGnuvtþemKuNkat;bnßyenAeBl traffic lane CaeRcInRtUv)andak;bnÞúkkñúgeBlCamYyKña dUcenA
kñúgEpñk (iv) xageRkam.
(iii) kMlaMgcakp©itedk (Centrifugal Horizontal Force): kMlaMgenHekItBIclnarbs;rfynþenA

elIExSekag. eKkMNt;vaCaPaKryrbs;bnÞúkGefr edayKμanT§iBlTgáic dUcxageRkam³
6.68S 2
C = 0.00117 S 2 D = (12.3)
R

Edl C=kMlaMgcakp©itKitCaPaKryrbs;bnÞúkGefrEdlKμamT§iBlTgáic
S = el,ÓnKNna (design speed) KitCa miles kñúgmYyema:g

D = dWeRkrbs;kMeNag

R = kaMrbs;ExSekag KitCa ft

(iv) karkat;bnßyGaMgtg;sIuetbnÞúk (Reduction in Load Intensity): enAeBlEdlkugRtaMgGti-

brmaRtUv)anbegáIteLIgedaykardak;bnÞúkenAelI traffic lanes CaeRcInkñúgeBlCamYyKña eKRtUv
eFVIkarkat;bnßyGaMgtg;sIuetbnÞúkdUcxageRkam³
PaKry
mYy b¤BIr lane 100

bI lane 90

bYn lane b¤eRcInCagenH 75



12.2.2. karEbgEckbnÞúkkg;enAelIkMrals<an³ lkçN³bec©keTssþg;dar AASTHO
Wheel Load Distribution on Bridge Decks: Standard AASHTO
Specifications (LFD)
(i) kMlaMgkat; (Shear): minmankarEbgEckbnÞúkkg; (wheel load) tamTisbeNþaysMrab; wheel
load b¤ axle load enAxagcug enAeBlKNnakMlaMgkat;xagcug nigRbtikmμenAkñúgFñwmTTwg

(transverse beam) b¤FñwmbeNþay (longitudinal beam).

(ii) m:Um:g;Bt;begáag³ FñwmbeNþay (Bending moment: longitudinal beam)

kñúgkarKNnam:Um:g;begáagenAkñúg longitudinal beam b¤ stringer eKminGnuBaØateGaymankar
BRgaybnÞúkkg;tamTisbeNþayeT. sMrab;Ggát;ebtugeRbkugRtaMg kñúgkrNI stringer xagkñúg
m:Um:g;Bt;bnÞúkGefrsMrab; stringer nImYy²RtUv)ankMNt;edayGnuvtþcMENkén wheel load
eTAelI stringer dUcxageRkam

(iii) Side by Side Precast Beams in Multi-Beam Decks
s<anBhuFñwm (multi-beam bridge) RtUv)ansagsg;eLIgCamYynwgFñwmebtugGarem: b¤Fñwmebtug
eRbkugRtaMgcak;Rsab;Edldak;Ek,rKñaelITMr. GnþrGMeBI (interaction) rvagFñwmRtUv)anbegáIteLIg
eday continuous longitudinal shear key EdleRbIrYmpSMCamYynwg transverse tie EdlGacrg
b¤minrgeRbkugRtaMg dUcCab‘ULúg r)arEdk b¤kabeRbkugRtaMg b¤k¾meFüa)ayemkanicepSgeTot.
eKRtUvkar Full-depth rigid end diaphragm edIm,IFanakarBRgaybnÞúkeGayl¥sMrab;FñwmrUbrag
channel, single-stemmed tee beam nig multi-stemmed tee beam.

kñúgkarkMNt;m:Um:g;Bt;begáagenAkñúgs<anebtugBhuFñwmcak;Rsab; ¬ebtugGarem: b¤eRbkug
RtaMg¦ eKsnμt;faminmankarBRgaybnÞúkkg;tamTisbeNþayeT. m:Um:g;Bt;begáagbnÞúkGefr
sMrab;muxkat;nImYy²RtUv)ankMNt;edayGnuvtþcMENkén wheel load ¬TaMgmux nigeRkay¦ Edl
kMNt;edaysmIkarxageRkameTAelIFñwm


S
Load Frcation =
D
Edl S= TTwgrbs;Ggát;cak;Rsab;
D = (5.75 − 0.5 N L ) + 0.7 N L (1 − 0.2C )2 enAeBlEdl C ≤ 5
D = (5.75 − 0.5 N L ) enAeBlEdl C > 5
NL = cMnYn traffic lane
C = K (W / L )
Edl W = TTwgTaMgmUlrbs;s<anEdlvas;EkgeTAnwgr:tbeNþay
L = RbEvgElVgEdlvas;RsbeTAnwgr:tbeNþay. sMrab;r:tEdlman end diaphragm

cak;enAnwgkEnøg ykRbEvgcenøaH end diaphragms
K = [(1 + μ )I / J ]1 / 2
RbsinebItMélrbs; I / J FMCag 5.0 karBRgaybnÞúkRtUv)ankMNt;edayeRbIviFIEdlsuRkitCag
dUcCa Articulated Plate Theory b¤ Grillage Analysis.
Edl I = mU:m:g;niclPaB
J = efrrmYl Saint-Venant

μ = pleFobB½rs‘ugrbs;r:t
sMrab;viFIEdlkat;EtsuRkit eKkMNt;tMélrbs; J edayeRbIsmIkarxageRkam
sMrab;FñwmctuekaNEdlKμanRbehag/ FñwmGkSr C/ Fñwm Tee
J = ∑{1 / 3)bt (1 − 0.630t / b )}
( 3

Edl RbEvgrbs;FatubgÁúMctuekaNnImYy²EdlmanenAkñúgmuxkat;
b=

t = kMras;rbs;FatubgÁúMctuekaNnImYy²EdlmanenAkñúgmuxkat;

søab nigeCIgrbs;muxkat;EdlmaneCIg b¤muxkat;GkSr C RtUv)anKitCaFatubgÁúMctuekaNdac;
edayELkEdltMélrbs;vaRtUv)anbUkbBa©ÚlKñaedIm,IKNna J . cMNaMfa sMrab;FñwmctuekaN
EdlmanRbehagmUl tMélrbs; J GacKitCatMélRbhak;RbEhledayeRbIsmIkarxagelIsMrab;
muxkat;ctuekaN nigedayminKitRbehag.
sMrab;Fñwmmuxkat;RbGb;³
J=
(
2tt f (b − t )2 d − t f )2
bt + dt f − t 2 − t 2
f

Edl b= TTwgsrubrbs;RbGb;


d= kMBs;srubrbs;RbGb;
t = kMras;rbs;RTnugTaMgsgçag

t f = kMras;rbs;søabTaMgBIr

rUbmnþsnμt;fasøabTaMgBIrmankMras;esμIKña ehIyeKeRbIkMras;rbs;søabEtmYyeT. dUcKñasMrab;
RTnug.
sMrab;karsikSaKNnabzm eKGaceRbItMélrbs; K xageRkam³

(iv) kugRtaMgenAkñúgebtug (Stresses in Concrete)
krNI I³ bnÞúkTaMgGs;edayrYmbBa©ÚlTaMgeRbkugRtaMg ¬ D + L + P / S ¦
f c = 0.6 f 'c
f t = 6 f 'c

krNI II³ eRbkugRtaMg + bnÞúkefrTaMgGs; ¬D+ P/S ¦
f c = 0.4 f 'c
f t = 6 f 'c

krNI III³ 12 ¬eRbkugRtaMg + bnÞúkefr¦ + bnÞúkGefr [0.5(D + P / S ) + L]
f c = 0.4 f 'c
f t = 6 f 'c

12.2.3. m:Um:g;Bt;begáagenAkñúgkMrals<an³ lkçN³bec©keTssþg;dar AASTHO (LFD)
Bending Moments in Bridge Decks Slabs: Standard AASHTO
Specifications (LFD)
karKNnam:Um:g;Bt;manBIrRbePT³ RbePT A nigRbePT B sMrab;EdkBRgwgEdlEkg nigRsb
eTAnwgcracrN_ erogKña.
S = RbEvgElVgRbsiT§PaB



TTwgrbs;kMralEdleKEbgEck wheel load BIelIva
E=

P = bnÞúkenAelIkg;xageRkaymYyrbs; truck ¬ P15 b¤ P20 ¦

P15 = 12,000lb sMrab;kardak;;bnÞúk HS 15

P20 = 16,000lb sMrab;kardak;bnÞúk HS 20

(a) krNI A- EdkemEkgeTAnwgcracrN_ ¬ElVgEdlmanRbEvgBI 2 eTA 24 ft ¦

eKkMNt;m:Um:g;bnÞúkGefrsMrab;ElVgsamBaØtamsmIkarxageRkam
sMrab;kardak;bnÞúk H 20
⎛S + 2⎞
ML = ⎜ ⎟ P20 (12.4a)
⎝ 32 ⎠
⎛ S + 2⎞
ML = ⎜ ⎟ P15 (12.4b)
⎝ 32 ⎠
Edl M L KitCa ft − lb / ft énTTwgkMral
sMrab;kMralEdlCab;elITMrbI b¤eRcInCagenH eKRtUvGnuvtþemKuNénPaBCab; 0.80 eTAelI
smIkar 12.4 (a) nig 12.4 (b).
(b) krNI B- EdkemRsbeTAnwgcracrN_

sMrab; wheel load TTwgEbgEck E KYresμInwg = 4 + 0.06S ≤ 7.0 ft . Lane loads RtUv)an
BRgayelITTwg 2E dUcxageRkam³
S ≤ 50 ft ³ M L = 900 S (12.4c)

S = 50 − 100 ft ³ M L = 1000 S (12.4d)
Edl M L KitCa ft − lb
sMrab;kardak;bnÞúk H 15 kat;bnßy 25% BItMélenAkñúgsmIkar 12.4 (c) nig 12.4(d).

12.2.4. bnÞúkxül; Wind Loads
edayKitbBa©ÚlbnÞúkxül; épÞRbQmesμInwgplbUkénépÞrbs;Ggát;TaMgGs;edayrYmbBa©ÚlTaMg
RbB½n§kMral nigbgáan; dUceXIjenAkñúgkMritnIv:U 90o eTAnwgG½kSbeNþayrbs;eRKOgbgÁúM. karsikSaKNna
KUrEp¥kelIel,Ónxül; V = 100mph(160km / h). RkLaépÞrgsMBaFRtUv)ankat;dUcerobrab;enAkñúg
AASTHO.



12.2.5. kMlaMgrBa¢ÜydI Seismic Forces
eKGaceRbITaMgviFIkMlaMgsþaTicsmmUl (equivalent static force method) nig response
spectrum method sMrab;karsikSaKNnaeRKOgbgÁúMEdlGgát;TMrmanPaBrwgRkajRbhak;RbEhlKña.

karlMGitmanenAkñúg AASTHO. karerobrab;bEnßmsMrab; earthquake response, fundamental period
énrMj½r nig International Building Code (IBC 2000) RtUv)anerobrab;enAkñúg Nawy. E.G.,
Reinforced Concrete.

12.2.6. karbnSMbnÞúk AASTHO LFD
AASTHO LFD Load Combination
karsikSaKNnaKYrKitRkuménkarbnSMbnÞúkEbbNaedIm,IeGay)anlkçxNÐkugRtaMgGtibrmaenA
kñúgGgát;EdlBicarNa. eKmanRkuménkardak;bnÞúkeRkamlkçxNÐbnÞúkesvakmμ³
Rkum I³ D + (L + I ) + CF + E + B + SF

Rkum II³ D + E + B + SF + W

Rkum III³ D + (L + I ) + CF + E + B + SF + W + WL + LF

Rkum IV³ D + (L + I ) + CF + E + SF + (R + S + T )

Rkum V³ D + E + B + SF + W + (R + S + T )

Rkum VI³ D + (L + I ) + CF + E + B + SF

Rkum VII³ D + E + B + SF + EQ
Rkum VIII³ D + (L + I ) + CF + E + B + SF + ICE
Rkum IX³ D + E + B + SF + W + ICE

Rkum X³ D + (L + I ) + E

Edl D = bnÞúkefr
L = bnÞúkGefr

I = bnÞúkGefrTgáic

E = sMBaFdI

B = PaBGacGENþt (buoyancy)

W = bgÁúMxül;enAelIGaKar

WL = bnÞúkxül;enAelIbnÞúkGefr − 100lb / ft



bnÞúkbeNþayEdl)anBIbnÞúkGefr
LF =

CF = kMlaMgcakp©it

R = karrYjxøIrbs; rib

S = karrYmmaD

T = sItuNðPaB

EQ = kMlaMgrBa¢ÜydI

SF = sMBaFrMhUrrbs;Twk

ICE = sMBaFTwkkk

sMrab; load factor design eKRtUvKuN)a:ra:Em:RtelIkmunCamYynwgemKuNbnÞúkenAkñúgtarag
12>1(b).
sMrab;bnÞúkemKuN tMélRKumKW
elxRkum (N ) = γ [β D D + β L (L + I ) + β C CF + β E E + β B B + β S SF
+ βW W + βWLWL + β L LF + β R (R + S + T )
+ β EQ EQ + β ICE ICE ] (12.5)

emKuNbnÞúkEdlRtUvGnuvtþeTAelIbnSMbnÞúkKWdUcxageRkam³
β E = 0.7 sMrab;bnÞúkbBaÄrenAelIRbGb;ebtugGarem:
= 1.0 sMrab;bnÞúkxag (lateral load) enAelIRbGb;ebtugGarem:

= 1.0 sMrab;bnÞúkbBaÄr nigbnÞúkxagenAelIlU (culvert) déTeTot

= 1.0 nig 0.5 sMrab;bnÞúkxagenAelIeRKagrwg ¬RtYtBinitükardak;bnÞúkEdllub

sMrab;RkumCak;lak;¦
β E = 1.3 sMrab;sMBaFxagrbs;dIenAeBlRtYtBinitüm:Um:g;viC¢manenAkñúgeRKagrwg (rigid
frame) lU b¤lURbGb;ebtugGarem:

β D = 0.75 enAeBlRtYtBinitüGgát;sMrab;bnÞúktamG½kSGb,brma nigm:Um:g;Gtibrma
sMrab;cMNakp©itGtibrma sMrab;karsikSaKNnassr.
= 1.0 enAeBlRtYtBinitüsMRab;bnÞúktamG½kSGtibrma nigm:Um:g;Gb,brma

= 1.0 sMrab;Ggát;rgkarTaj nigkarBt;begáag.

tarag 12>1 (b) eGaytMélénemKuN β sMrab;)a:ra:Em:RtbnÞúkepSg²enAkñúgsmIkar 12.5
sMrab;lkçN³bec©keTs standard AASTHO.



12.2.7. karbnSMbnÞúk LRFD
LRFD Load Combinations
karbnSMbnÞúkedayeRbIlkçN³bec©keTs LRFD xusKñaBIlkçN³bec©keTs standard AASTHO.
taragxageRkam ³ 12>2 eTAdl; 12>3 bgðajBIbnSMbnÞúktMrUvkar ehIytarag 12>4 dl; 12>7 bgðaj
BIsmIkarkMlaMgkat; nigsmIkarm:Um:g;EdlRtUveRbIenAkñúgkarsikSaKNna. Epñk 12.1.1 bgðajBIemKuNer
sIusþg; LRFD φ EdlxusKñaBIemKuNkat;bnßyersIusþg; standard AASTHO φ . eKRtUvcMNaMfa enAkñúg



lkçN³bec©keTs standard eKeRbI lane load b¤ truck load kñúgkarKNnabnÞúkGefr. lkçN³bec©keTs
LRFD TamTareGayeRbInUvbnSMrvag lane load nig truck load kñúgkarkMNt;bnÞúkGefr.



tMélemKuNersIusþg; LRFD φ RtUv)aneGayenAkñúgtarag 12>1 (a).
eKGaceRbIsmIkarenAkñúgtarag 12>4 nig 12>5 edIm,IKNnam:Um:g;Bt;begáagGtibrma nigkMlaMg
kat;GtibrmaRKb;cMnucTaMgGs;kñúgmYy lane sMrab; HS 20 truck CamYynwgkarkMNt;EdlbgðajenAkñúg
tarag. eKyktMélEdlKNnaRtwmBak;kNþaledIm,ITTYl)ankMlaMgkat; nigm:Um:g;kñúgExSkg; (line of
wheel).

smIkarenAkñúgtaragRtUv)ankMNt;RtwmElVgTMrsamBaØ nigminrYmbBa©ÚlT§iBlTgáiceT.
eKGackMNt;m:Um:g;Bt;Gtibrma nigkMlaMgkat;GtibrmakñúgmYy lane enARtg;cMnucNamYyenAelI
ElVgsMrab; lane load 0.64kip / ft BIsmIkarEdlsMrYlehIydUcxageRkam³
kMlaMgkat;Gtibrma VLL = 02.64 (L − x)2
L
(12.6a)

0.64( x )(L − x )
m:Um:g;Gtibrma M LL =
2
(12.6b)

Edl x = cMgayBITMrxageqVg
L = ElVgFñwm
LL = lane load
TamTarT§iBlTgáic (impact factor) FMCag standard specification. ehIy
LEFD Specification

vak¾TamTarnUvkarBicarNaBIsßanPaBkMNt; fatigue. sMrab; fatigue eKRtUvBicarNa truck load Biess.



vapSMeLIgBI single design truck EdlmanTMgn; axle dUcKñanwgGIVEdleRbIenAkñúgsßanPaBkMNt;déTepSg
eTot b:uEnþvamanKMlatefr 30 ft cenøaH axle 32kips . tarag 12>6 bgðajBI impact factor IM sMrab;
RbePTsßanPaBkMNt;epSg².
tarag 12>7 bgðajBIsmIkarsMrab;KNnam:Um:g;Bt;GtibrmakñúgmYy lane EdlbNþalBIkardak;
bnÞúk HL-93 fatigue truck. eKRtUvKuNtMélEdlTTYl)anBItaragedayemKuN 0.5 edIm,ITTYl)antMél
kñúgmYyExSkg;rfynþ (line of wheel).



Ca HL-93 truck configuration EdlrYmmanbnSM³
LRFD design live load

(a) Design truck b¤ design tandem CamYynwg dynamic allowance. Design truck KWdUcKñanwg

HS20 design truck EdlmankMNt;enAkñúg Standard AASTHO specification. Design

tandem pSMeLIgeday axle 25kips mYyKUr EdlmanKMlatBIKña 4 ft .

(b) Design lane load 0.64kip / ft edayminman dynamic allowance.



12.3. karBicarNaBIkarsikSaKNnakarBt;begáag Flexural Design Considerations
12.3.1. bMErbMrYlénbMErbMrYlrageFob ε nigemKuN φ ³ viFIbMErbMrYlrageFobkMNt;
Strain ε and Factor φ Variations: The Strain Limits Approach
sMrab;kareFVIkarCalkçN³sVit (ductile behavior) rbs;muxkat; PaKryEdkBRgwgRtUvmantMél
tUcCagPaKrybMErbMrYlrageFobkMNt;lMnwg (balanced limit strain) EdlrgkarBt;begáag dUcbgðajenA
kñúgEpñk 4.12.3. eKminRtUvkareRbIEdnx<s;bMputénbrimaNEdkenAkñúgFñwmeT RbsinebIbMErbMrYlrageFob
minFMCagbMErbMrYlrageFobkMNt; ehIyeKeRbIemKuN φ smrmü. bMErbMrYlrageFobrgkarTajEdnx<s;bM
put ε t = 0.005in. / in. edaysarbMErbMrYlrageFobkMNt;mantMélRbhak;RbEhlnwg 75%énPaKry
Edk balanced enAkñúg code elIkmun ehIyvaCaeKalkarN_rbs;viFIenH ¬rUbTI 12>6¦. eKKitBicarNa
bMErbMrYlrageFobkMNt;enARtg;nIv:UEdkrgkarTajxageRkAbMput mann½yfa enARtg;TIRbCMuTMgn;énRsTab;
EdkEdlenAEk,répÞrgkarTajrbs;muxkat;CageK. kan;EtCak;ElkCagenH ε t = 0.0041 EdlRtUvnwg
f y ≅ 230,000 psi enAkñúgEdkeRbkugRtaMg.

sMrab;viFI AASTHO LRFD tMélkMNt;rbs;pleFobkMBs;G½kSNWt c elIkMBs;RbsiT§PaB
rbs;Fñwm d t RtUv)aneKykesμInwg 0.42 enAkñúgviFIbMErbMrYlragkMNt; (strain limits approach) EdleK
GacehAmüa:geTotfa unified approach. viFIenHCaviFIEdlFana strain-compatibility edayeRbIbMEr
bMrYlFmμta nigsmIkarkugRtaMgedayminKitfaGgát;enHBRgwgedayEdk b¤edayeRbkugRtaMg b¤edayeRb
kugRtaMgedayEpñk. kMBs; d t enAkñúgpleFob c / d t køayCa d p RbsinebIeKmineRbIEdkBRgwgFmμtaeT
enaH. tarag 12>9 énEpñk 12.7 bgðajBIkareRbobeFobTUeTArvagviFIsaRsþ ACI nig LRFD sMrab;
kMNt;EdktMrUvkarenAkñúgGgát;rgkarBt;begáag.
eKRtUveRbItMélbMErbMrYlrageFob ε t FMCag 0.005in. / in. dUcCag 0.007 eTA 0.009in. / in. .
sMrab;muxkat; beam-column bMErbMrYlrageFobénmux limit compression-controlled KW ε t = 0.002 .


eKeRbI ε t = 0.002 CaeKalsMrab; yield strain dMbUg ε y = f y / Es = 0.002 eTaHbICatMélenHGacERb
RbYledayGaRs½yeTAnwgRbePTEdkEdleRbIk¾eday. rUbTI 12>7 bgðajBIEdnkMNt;énbMErbMrYlrag
eFobsMrab;muxkat;ebtug tension-controlled nig compression-controlled sMrab;RKb;krNITaMgGs;
¬BRgwgedayEdk nigrgeRbkugRtaMg¦ Edl ε t = 0.003(dt / c − 1) .



enAeBl net tensile strain enAkñúgEdkrgkarTajxageRkAbMputFMRKb;RKan; ¬FMCag b¤esμInwg
0.005 ¦ muxkat;RtUv)ankMNt;Ca tensioned-controlled section EdlvamankarRbkasGasnñBIkar)ak;

RKb;RKan;CamYynwgPaBdabFM nigekItmansñameRbH. enAeBlEdl net tensile strain enAkñúgEdkrgkar
TajxageRkAbMputtUc ¬tUcCag b¤esμInwg compression-controlled strain limit¦ eKrMBwgvanwgekItman
lkçxNÐ)ak;edaylkçN³RsYy CamYynwgkarRbkasGasnñBIkar)ak;d¾tictYcbMput.
lkçxNÐ balanced strain ekItmanenARtg;muxkat;EdlbMErbMrYlrageFobGtibrmaenAsrésrgkar
sgát;xageRkAbMputmantMélesμInwg 0.003 kñúgeBldMNalKñaCamYynwg yield strain dMbUg ε y = f y / Es
enAkñúgEdkrgkarTajEdlRtUvnwg net tensile strain enAkñúgEdkrgkarTajEdlkMNt;enAkñúgviFIenH enA
eBlEdltMélrbs; ε t = 0.002in. / in. .
eKminGaceRbIviFIenHenAkñúgkarKNnaFñwmEdlrgkarBt;begáagEdlminrgkarsgát;eT. enAkñúg
Ggát;EbbenH bMErbMrYlrageFob ε t enAkñúgEdkrgkarTajxageRkAbMputminRtUvFMCag 0.0075 sMrab;kar
Gnuvtþ.

12.3.2. ersIusþg;m:Um:g;Bt;begáagemKuN Factored Flexural Resistance
m:Um:g;Tb;Tl;karBt;begáagemKuN
M t = φM n (12.7)
EdlemKuNersIusþg; φ = 1.0 .
enAkñúgkarviPaK strain compatibility eKENnaMeGaykat;bnßyemKuN φ BItMél 1.0 sMrab; net
tensile strain 0.005 eTA 0.7 sMrab; net tensile strain 0.002 enAkñúgEdkrgkarTajxageRkAbMput man

n½yfa
⎡d ⎤
0.7 ≤ φ = 0.50 + 0.30 ⎢ ext − 1⎥ ≤ 1.0 (12.8)
⎣ c ⎦
Edl dext Ca dt énRsTab;xageRkAbMputrbs;EdkBRgwg KWRsTab;EdkEdlenAEk,rsrésrgkarTajxag
eRkAbMputrbs;muxkat;ebtugeRbkugRtaMg.

12.3.3. )a:r:aEm:RtKNnakarBt;begáag Flexural Design Parameters
smIkarsMrab;KNnaersIusþg;m:Um:g; nominal rbs;muxkat;eRbkugRtaMgeday LRFD method KW
Rsedonwg standard AASTHO ehIyk¾RsedogKñanwgviFIsaRsþKNnaersIusþg;rbs; ACI 318 EdlENnaM



enAkñúgEpñk 4.11 énCMBUk 4. ersIusþg;KNnacugeRkay (ultimate design strength) f ps énEdkBRgwg
EdlGacKNnatamviFI strain-compatibility dUckñúg]TahrN_ 4>19 b¤edayviFIRbEhl (approximate
method) EdleRbIsmIkarxageRkam³
⎛ c ⎞
f ps = f pu ⎜1 − k ⎟ (12.9a)
⎜ dp ⎟
⎝ ⎠
⎛ f py ⎞
Edl k = 2⎜1.04 −
⎜
⎟
f pu ⎟
(12.9b)
⎝ ⎠
= 0.28 sMrab; low relaxation steel
sMrab; unbonded tendons/
⎛ dp −c⎞
f ps = f pe + 900⎜
⎜ l
⎟
⎟ (12.9c)
⎝ e ⎠
Edl le = 2li / (2 + N s )

RbEvg embedment/ li = RbEvgkabeRbkugRtaMgcenøaH anchorage/ N s = cMnYn tendon
le =

enAkñúg standard AASTHO specification eKGac)a:n;RbmaNkugRtaMgmFümdMbUgenAkñúgEdkeRbkug
RtaMgBIsmIkarxageRkam³
⎛ γ f pu ⎞
f ps = f pu ⎜1 − ρ
⎜ β
⎟ (12.9c)
⎝ 1 f 'c ⎟
⎠
eKTTYl)ankMBs;rbs;G½kSNWt c BIsmIkarxageRkam³
(a) muxkat;EdkDub (Doubly reinforced section)³

A ps f pu + As f y − A' s f ' y
c= (12.10)
f pu
0.85 f 'c β1 + kA ps
dp

Edl f ' y = yield strength rbs;Edkrgkarsgát;
(b) muxkat;mansøab (Flanged section)³

Aps f pu + As f y − A' s f ' s −0.85 f 'c β1 (b − bw )h f
c= (12.11)
f pu
0.85 f 'c β1bw + kAps
dp

Edl bw = TTwgRTnug
d p = cMgayBIsrésrgkarsgát;xageRkAbMputeTATIRbCMuTMgn;rbs;EdkeRbkugRtaMg



12.3.4. EdnkMNt;rbs;EdkBRgwg Reinforcement Limits
(a) EdnkMNt;EdkBRgwgGtibrma (Maximum reinforcement limits)³
brimaNGtibrmarbs;EdkrgeRbkugRtaMg b¤EdkminrgeRbkugRtaMgRtUvmantMély:agNaedIm,I
c
≤ 0.42 (12.12a)
de
A ps f ps d p + As f y d s
Edl de =
A ps f ps + As f y
(12.12b)

(b) EdnkMNt;EdkBRgwgGb,brma (Minimum reinforcement)³
enARKb;muxkat;TaMgGs; brimaNEdkrgeRbkugRtaMg nigEdkminrgeRbkugRtaMgRtUvRKb;RKanedIm,I
begáItersIusþg;Tb;Tl;karBt;begáagemKuN M t y:agehacNas;RtUvesμInwg 1.2M cr EdlkMNt;
edayEp¥kelIeKalkarN_viPaKeGLasÞic b¤esIμnwg 1.33dgénm:Um:g;emKuNEdlTamTaredaybnSM
bnÞúk.
⎡S ⎤
M cr = ( f r + f ce )S b − M dnc ⎢ bc − 1⎥ (12.13)
⎣ Sb ⎦
Edl M dnc = m:Um:g;EdlbNþalBI non-composite dead load
Sb = m:UDulmuxkat; non-composite

S bc = m:UDulmuxkat;smas

f r = m:UDuldac; (modulus of rupture) = 7.5 f 'c psi = 0.24 f 'c ksi

f ce = kugRtaMgsgát;enAkñúgebtugedaysarEteRbkugRtaMgRbsiT§PaB ¬eRkaykMhatbg;¦

enARtg;srésrgkarTajxageRkAbMputrbs;muxkat;EdlkugRtaMgTajekItBIbnÞúk
xageRkA.

12.4. karBicarNaBIkarsikSaKNnakMlaMgkat; Shear Design Considerations
12.4.1. The Modified Compression Field Theory
RTwsþI compression field sMrab;kMlaMgkat; nigsMrab;kMlaMgkat;EdlpSMCamYynwgkarrmYlRtUv)an
erobrab;enAkñúgEpñk 5.17.3 énCMBUk 5. enAeBlEdlmankarrmYl eKsnμt;faebtugminTb;Tl;karTajeT
bnÞab;BImansñameRbH ehIyEdn (field) én compressive strut Tb;Tl;kMlaMgkat;Ggát;RTUg. mMu θ rbs;
strut enHERbRbYlGaRs½ynwgbMErbMrYlrageFobbeNþay (longitudinal strain), bMErbMrYlrageFobxag

(transverse strain) nigbMErbMrYlrageFobem (principal strain) enAkñúgRTnug³


ε x − ε2
tan 2 θ = (12.14)
εt − ε2
Edl εx =bMErbMrYlrageFobbeNþayrbs;RTnug ¬rgkarTaj viC¢man¦
ε t = bMErbMrYlrageFobTTWg ¬rgkarTaj viC¢man¦
ε 2 = bMErbMrYlrageFobrgkarsgát;em GviC¢man
rUbTI 12>8 bgðajEdnkugRtaMg (stress field) enAkñúgRTnugrbs;FñwmEdlminrgeRbkugRtaMgmun nigeRkay
eBlmansñameRbH. muneBlFñwmeRbH kMlaMgkat;RtUv)anTb;Tl;edaykugRtaMgTajGgát;RTUg nigkugRtaMg
sgát;Ggát;RTUgesμIKña edayeFVIGMeBItammMu 45o ¬rUbTI 12>8(a)¦. eRkayeBlmansñameRbH sñameRbH
Ggát;RTUgEdlekItBIkugRtaMgTajenAkñgebtugRtUv)ankat;bnßyy:ageRcIn.
ú

enAkñúgRTwsþI compression field eKsnμt;fakugRtaMgTajem f1 = 0 dUcenAkñúgrUbTI 12>8 (b)
eRkayeBlebtugeRbH. RTwsþI modified compression field KitbBa©ÚlkarcUlrYménkugRtaMgTajenA
kñúgebtugcenøaHsñameRbH dUcenAkñúgrUbTI 12>8 (c). BIrgVg;kugRtaMgm: (Mohr’s stress circle) enAkñúgrUb
TI 5>2 (b) enAkñúgCMBUk 5 CamYynwgrUbTI 12>8 (c) eKGacTTYl)ansmIkarxageRkam
f 2 = (tan θ + cot θ )v − f1 (12.15a)
EdlkugRtaMgkMlaMgkat;Gnuvtþn_KW
v=
V
=
(
Vu − θV p ) (12.15b)
bw jd φbw d v
(
dv = d p − a / 2 ) nig bw = TTwgRTnugRbsiT§PaB. eKGackMNt;EdkRTnugrgkarTaj (tension web
reinforcement) Av EdlTamTarsMrab;eFVIeGaykugRtaMgsgát;manlMnwgdUcxageRkam
Av f v = ( f 2 sin 2 θ − f1 cos 2 θ )bw s (12.16)



Edl Av fv CabgÁúMbBaÄrén balancing tensile force edIm,IbiTP¢ab;sñameRbHeRTtEdlmanmMu θ nig f v
CakugRtaMgmFümenAkñúgEdkkgbBaÄr. edayCMnYs f 2 enAkñúgsmIkar 12.15(a) eTAkñúgsmIkar 12.16
eyIg)an³
Av f v
V = f1bw d v cot θ + d v cot θ (12.17)
s
Edl V tMNageGayVn ehIyvaesμInwg (Vc + Vs ) / Edl Vs CakMlaMgkat;EdlTb;Tl;edayEdkkg
bBaÄr.

12.4.2. smIkarKNna Design Expressions
tamkarsnμt;EdlmanlkçN³sMrYl eKGacerobcMsmIkareKalén modified compression field
theory eLIgvij dUcenHeKGackMNt;ersIusþg;kMlaMgkat; nominal Vn enAkñúgebtugeRbkugRtaMg

Vn = Vc + Vs + V p (12.18)

Edl Vc = ersIusþg;kMlaMgkat; nominal EdleGayedaykugRtaMgTajenAkñúgebtug
Vs = ersIusþg;kMlaMgkat; nominal EdleGayedaykugRtaMgTajenAkñúgEdkRTnug

V p = ersIusþg;kMlaMgkat; nominal EdleGayedaybgÁúMkugRtaMgbBaÄrénEdkeRbkugRtaMg

harped b¤ draped tamTisbeNþay.

12.4.2.1. AASTHO Standard Specification (LFD)
karpþl;eGayrbs; AASTHO standard nig ACI 318 KWmanlkçN³RsedogKña edayeKyk Vc
nUv tMélEdltUcCagkñúgcMeNamsmIkarTaMgBIrxageRkamEdlbgðaj nigerobrab;y:aglMGitenAkñúgEpñk
5.5.1 nig 5.5.2 énCMBUk 5³

(a) kMlaMgkat;Bt;begáag (Flexural shear)³

Vi M cr
Vci = 0.6 f 'c bw d + (12.19)
M max
(b) kMlaMgkat;RTnug (Web shear)³
Vcw = [3.5 f 'c + 0.3 f c ]bw d + V p (12.20)

sMrab; AASTHO smIkarm:Um:g;eRbH (cracking moment) KW
(
M cr = St 6 f 'c + f pe − f d )



12.4.2.2. LRFD Specification
karpþl;eGayrbs; LRFD AASTHO TTYlsÁal;viFIBIr³
(a) Strut-and-tie model EdlGacGnuvtþ)anRKb;ragGrNImaRtrbs;muxkat;TaMgGs;Edlman

lkçN³Fmμta nigminCab;.
(b) Modified compression field EdlEp¥lelIm:UEdl truss EdlmanmMuERbRbYl EdlenAkñúgenaHeK

GnuBaØateGayERbRbYlCMralrbs; diagonal compression field. vaxusBIviFI LFD Rtg;kar
snμt;mMu θ eGayesμInwg 45o ¬EdlkñúgenHebtugsuT§k¾cUlrYmeFVIkarEdr¦ ehIy Vc RtUv)anGnuvtþ
karTajEdlqøgkat; compression diagonal dUcerobrab;enAkñúgEpñk 12.4.1.
eKykersIusþg; nominal nUvtMélEdltUcCageKkñúgcMeNam³
Vn = Vc + Vs + V p (12.21)

b¤ Vn = 0.25 f 'c bv d v (12.22)

Edl TTwgRTnugRbsiT§PaB
bv =

d v = kMBs;kMlaMgkat;RbsiT§PaB ≈ (d p − a / 2 )

a = kMBs;rbs;bøúksgát;

muxkat;eRKaHfñaksMrab;kMlaMgkat;sßitenAcMgay dv b¤ (0.5dv cot θ ) edayykmYyNaEdlFM
Cag. eKyktMél dv BIkarKNnaersIusþg;Bt;begáagkNþalElVg.
ersIusþg;kMlaMgkat; nominal rbs;ebtugsuT§ Vc EdlKitCa psi KW
Vc = β f 'c bv d v (12.23)

nigKitCa ksi
Vc = 0.0361β f 'c bv d v (12.24)

emKuN 0.0361 KW 1 / 1000 EdlCaemKuNsMrab;bMElgBI psi eTA ksi .
eKykersIusþg;kMlaMgkat;énkarcUlrYmrbs;EdkRTnugbBaÄrKW
Av f y d v cot θ
Vs = (12.25)
s
eKRtUvdak;Edkkg (transverse shear reinforcement) enAeBlEdlkMlaMgkat;emKuN Vu FMCaglT§PaB
Tb;kMlaMgkat;rbs;ebtugsuT§ b¤
(
Vu > 0.5φ Vc + V p ) (12.26)

EdlemKuNkat;bnßyersIusþg; φ RtUv)anykBItarag 12>1(a).



elIsBIenH enAeBlEdlkMlaMgRbtikmμrbs;FñwmbBa¢Únkarsgát;eTAcugrbs;Ggát;dUcEdlekItman
enAkñúgkrNICaeRcIn eKykmuxkat;eRKaHfñak;enARtg;TItaMgq¶ayCag 0.5dv cot θ b¤ dv EdlvaBIépÞrbs;
TMr.
edIm,IkMNt;ersIusþg;kMlaMgkat; nominal rbs;Ggát;eRbkugRtaMg visVkrKNnaeRKOgbgÁúMRtUvkMNt;
tMélrbs; β nig θ EdlRtUvkatsMrab;kMNt;Vc nig Vs enAkñúgsmIkar 12.21 nig 12.22. sMrab;muxkat;
ebtugminrgeRbkugRtaMg eKyk β = 2.0 nigθ = 45o . sMrab;muxkat;ebtugeRbkugRtaMg/ trial-and-
adjustment eRbItMél β tUc. tarag 12>8 bgðajBItMélrbs; β nig θ sMrab;tMélepSg²rbs; ε x .

eKTTYl)anbMErbMrYlrageFob ε x enAkñúgEdkrgkarTajBIsmIkarxageRkam RbsinebImuxkat;y:ag
ehacNas;manEdkTTwg (transverse reinforcement) Gb,brma
⎡ Mu ⎤
⎢ d + 0.5 N u + 0.5Vu cot θ − Aps f po ⎥
εx = ⎢ u ⎥ ≤ 0.002
⎢ (
2 Es As + E ps A ) ⎥
(12.27)
⎢
⎣ ⎥
⎦
Edl .
f po ≅ 0.70 f pu

eKRtUvKuNtMélenAkñúgsmIkar 12.27 nwgBIr RbsinebImuxkat;manEdkkgticCagEdkkgGb,-
brma.
kugRtaMg f po CakugRtaMgenAkñúgkabeRbkugRtaMgenAeBlGnuvtþeRbkugRtaMg (jacking) sMrab;Ggát;


rgeRbkugRtaMgCamun (pretensioned member) ehIysMrab;suvtßiPaB vaCakugRtaMgmFümsMrab;EdkeRb
kugRtaMgenAkñúgGgát;rgeRbkugRtaMgCaeRkay (post-tensioned member). sMrab;kabeRbkugRtaMgFmμta
eKyk f po = 0.75 f pu sMrab;TaMgGgát; pretensioned nigTaMgGgát; post-tensioned.
f ce = kugRtaMgsgát;rbs;ebtugenARtg;TIRbCMuTMgn;rbs;muxkat;smasEdlTb;Tl;nwgbnÞúkGefr

b¤enARtg;TIRbsBVrvagRTnug nigsøab RbsinebIvasßitenAkñúgsøabEdlbNþalBIeRbkugRtaMg
nigm:Um:g;Bt;EdlTb;Tl;edaymuxkat;cak;Rsab;EdleFVIkarEtxøÜnÉg mann½yfamunnwgeFVI
karCamuxkat;smas.
f pe = kugRtaMgRbsiT§PaBenAkñúgEdkeRbkugRtaMgeRkayeBlxatbg;/ CalkçN³suvtßiPaB eKyk

f po CaeRbkugRtaMgRbsiT§PaB f pe .

RbsinebIbMErbMrYlrageFobenAkñúgEdkrgkarTajGviC¢man eKRtUvKuN ε x CamYynwgemKuN Fε
EdlmansmIkardUcxageRkam
Es As + E ps Aps
Fε = (12.28)
Ec Ac + Es As + E ps Aps

Edl Ac = RkLaépÞrbs;muxkat;ebtugénEpñkrgkarTajedaykarBt;begáagrbs;Ggát; dUcbgðajenA
kñúgrUbTI 12>9.

EdkbeNþayRtUvmansmamaRty:agNaedIm,IeGaymuxkat;FñwmnImYy²bMeBjsmIkarxageRkam³
Mu N ⎛V ⎞
As f s + Aps f ps ≥ + 0.5 u + ⎜ u + 0.5Vs + V p ⎟ cot θ
⎜φ ⎟ (12.29)
d vφ φ ⎝ ⎠
BIsmIkar AASTHO elIkmun GBaØat β CaemKuNy:agsMxan;kñúgkarkMNt;ersIusþg;kMlaMgkat;
Fmμta (nominal) Vc dUcenAkñúgsmIkar 12.21. karbgðajtMélrbs; β EdlEp¥kelI compression



field theory enAkñúgtarag 12>8 bgðajfatMélTaMgenHhak;minGaceRbI)ansMrab;pleFob (v / f 'c )
FMCag 0.125 enAeBlEdlbMErbMrYlrageFobtUcCag 0.005 . Hsu )anerobrab;nUvPaBBi)akenH EdlekIt
eLIgenAeBlkMNt;tMélkugRtaMgkMlaMgkat;sñameRbH (crack shear stress) vci mann½yfa lT§PaBén
crack interface edIm,IepÞrkugRtaMgkMlaMgkat;EdlGaRs½yelITMhMsñameRbH w enAkñúgsmIkarxageRkam
2.16 f 'c 0.18 f 'c
vci ≤ psi, w(in.) vci ≤ MPa, w(mm )
24 w 24 w
0 .3 + 0 .3 +
a + 0.63 a + 16
HsuesñIeGayeRbI vci = 0 edIm,IrkSalMnwg nigPaBRtUvKña (compatibility). ehIy mMusñameRbH
θ enAkñúgtY Vs énsmIkar 12.25 KWCamMucenøaHkugRtaMg¬bMErbMrYlrageFob¦EdkbeNþay nigkugRtaMg ¬bMEr
bMrYlrageFob¦sgát;emrbs;ebtug. edaysarEbbenH kugRtaMgkMlaMgsgát;tambeNþayG½kSemesμInwg
sUnü. ehIy karerobrabenHGnuvtþenAkñúgkarpþl;eGayrbs; LRFD sMrab;krNIénbnSMkMlaMgkat; nigkar
rmYl.

12.4.2.3. KMlatGtibrmarbs;EdkRTnug
Maximum Spacing of Web Reinforcement
KMlatGnuBaØatGtibrma s rbs;EdkRTnugKWtMélEdltUcCageKkñúgcMeNam
s ≤ 0.75h b¤ 24in.
RbsinebI Vs > 4 f 'c bwd / KMlatGnuBaØatGtibrmaRtUv)ankat;bnßy 50% .

12.5. Horizontal Interface Shear
eKalkarN_én horizontal interface shear eRkamGMeBI service load nig ultimate load RtUv)an
erobrab;eBjeljenAkñúgCMBUk 5 Epñk 5.7 EdlrYmman]TahrN_bgðajEdlGnuelamtamtMrUvkar ACI
318 nig PCI. AASTHO standard specification tMrUveGayersIusþg;kMlaMgkat;tamTisedk nominal

Vnh dUcKñanwgersIusþg;kMlaMgkat;rbs; ACI enAeBlEdleKmineRbI dowel reinforcement mann½yfakug

RtaMgGnuBaØatGtibrmaKW 80 psi . vaxusKña enAeBleKeRbI dowel reinforcement Gb,brma Edlkñúg
enaHkugRtaMgkMlaMgkat;edkGnuBaØatGtibrmaKW 350 psi EdlCMnYseGay 500 psi EdlGnuBaØateday
ACI.

tamkarGegát nigBiesaFn_d¾sIuCMerArbs; Nawy )anbgðajeGayeXIjfakugRtaMgGnuBaØatBitCa
tUcEmnETn. karBiesaFbgðajfa sUm,IEtersIusþg;dMbUgeRkamlkçxNÐsItuNðPaB sub-freezing/ vaGac



TTYlersIusþg;eRkamGMeBI ultimate load FMCag 1200 psi(8.3MPa) edayeRbI dowel reinforcement
bBaÄr.
Standard AASTHO TamTardUcxageRkam³

(a) enAeBleKmindak; vertical ties
Vnh = 80bv d (12.30a)
(b) enAeBleKmindak; vertical ties Gb,brma
Vnh = 500bv d (12.30b)
(c) RkLaépÞtMrUvkarrbs; Avh FMCag vertical ties Gb,brma
dp
Vnh = 500bv d + 0.40 Avh f y (12.30b)
s
Edl kMlaMgkat;bBaÄremKuN Vu = φVnh
Vnh = ersIusþg;kMlaMgkat;edk nominal

φ = 0.90
Gb,brma = 50bv s / f y
Avh

bv = TTwgrbs;muxkat;enARtg;épÞb:HEdlRtUvviPaKkMlaMgkat;edk

b p = cMgayBIsrésrgkarsgát;xageRkAbMputeTATIRbCMuTMgn;rbs;EdkeRbkugRtaMg b:uEnþ

minRtUvyktUcCag 0.80h
s = kMlatGtibrmarbs; dowel b:uEnþminRtUvFMCagbYndgénTTwgRTnugEdltUcCageK

rbs;Ggát;TMr b¤ 24in. .
LRFD specification min)anpþl;nUveKalkarN_ENnaMsMrab;KNnakMlaMgkat;edk Vnh eT.

dUcenH eKGaceRbIsmIkarxageRkam
Vu
vuh = (12.31)
bv d v
Edl kugRtaMgkMlaMgkat;edkemKuN
vuh =

Vu = kMlaMgkat;bBaÄremKuN

d v = cMgaycenøaHkMlaMgpÁÜbénkMlaMgTaj nigénkMlaMgsgát; = (d − a / 2 )

bv = TTwg interface

LRFD kMNt;eGayKNnaersIusþg;kMlaMgkat; nominal rbs; interface surface Vn edayeRbI

smIkarxageRkam³
[
Vn = cAcv + μ Av f y + Pc ] (12.32)



nig vuh Acv ≤ φVn (12.33)

Edl c=emKuNs¥it (cohesion factor)
μ = emKuNkkit
Acv = RkLaépÞ interface rbs;ebtugEdlBak;B½n§nwgkarepÞrkMlaMgkat;

Avf = RkLaépÞrbs;EdkkMlaMgkat;Edlkat;tambøg;kMlaMgkat;enAkñúgRkLaépÞ Ac

Pc = kMlaMg net compressive Gcié®nþy_EdlEkgeTAnwgbøg;kMlaMgkat; ¬Gaclubecal eday

lkçxNÐsuvtßiPaB¦
f y = yield strength rbs; dowel reinforcement

CaTUeTA eKEtgEteFVIeGayépÞxagelIbMputrbs;Ggát;cak;Rsab;manlkçN³eRKIm ¬CMerARbEhl
0.24in. ¦ edIm,Icak;ebtugTMgn;FmμtabEnßm dUcerobrab;enAkñúgEpñk 5.7. LRFD ENnaMnUvsmIkarsMrYl

12.32 nig 12.33 dUcxageRkamEdlmanxñatCa ksi ³
⎛ Avf ⎞
vuh ≤ φ ⎜ 0.1 +
⎜
⎟ (12.34)
⎝ Acv ⎟
⎠
EdlRkLaépÞGb,brma
0.05bv s
Avf = (12.35)
fy

ehIy eKRtUvykersIusþg;kMlaMgkat; nominal nUvtMéltUcCageKkñúgcMeNam
Vn ≤ 0.20 f 'c Acv (12.36a)
b¤ Vn = 0.80 Acv (12.36b)

emKuNs¥it c nigemKuNkkit μ enAkñúgsmIkar 12.32 mantMéldUcxageRkamsMrab;lkçxNÐCak;lak;
rbs; interacting surface³
(a) ebtugEdlcak;kñúgeBlCamYyKña

c = 145 psi μ = 1.4λ
(b) ebtugEdlcak;elIebtugrwg nigs¥atehIyépÞrbs;vaeRKIm
c = 100 psi μ = 1.0λ
(c) ebtugEdlcak;elIebtugrwg s¥at nigminmansarFatuehIyépÞrbs;vamineRKIm
c = 75 psi μ = 0.6λ
(d) ebtugRtUv)ancak;f<k;eTAnwg as-rolled structural steel eday headed stud b¤eday reinforcing
bars EdlRKb;EdkTaMgGs;P¢ab;CamYynwgebtugKWs¥atminmanlabfñaM



c = 25 psi μ = 0.7λ
Edl sMrab;ebtugdg;sIuetFmμta
λ = 1 .0
= 0.85 sMrab; sand-low-density concrete

= 0.75 sMrab;ebtugEdlmandg;sIuettUcdéTTaMgGs;

xN³Edl LRFD AASTHO specification TamTareGaydak;EdkGb,brmaedayminKittMél
rbs;kugRtaMgenARtg; interface, GñkKNnaGaceRCIserIsedIm,IkMNt;EdkenHeTAnwgkrNIEdlVuh / φ FM
Cag 100 psi(0.7MPa ) . kareFVIEbbenH edIm,IeGayRsbeTAnwg ACI 318 Code specification nig
standard AASTHO specification.

12.5.1. KMlat Dowel Reinforcement Gtibrma
KMlatGnuBØatGtibrmarbs; dowel KW
(i) RbsinebI Vu < 0.1 f 'c bv d v KMlatGtibrma s ≤ 0.8d v ≤ 24in.
(ii) RbsinebI Vu > 0.1 f 'c bv d v KMlatGtibrma s ≤ 0.4d v ≤ 12in.

12.6. bnSMkMlaMgkat; nigkarrmYl Combined Shear and Torsion
kareobrab;enAkñúgEpñk 12.4.1 GMBI compression field theory CamYynwgEpñk 5.17.3 bgðajBI
karbMErbMrYlrageFob kMlaMgkat; nigGgát;rgkarsgát;Ggát;RTUg. rUbTI 5>38/ 5>39 nig 5>40 bgðajBI
rUbragxUcRTg;RTayrbs;muxkat;eRKaHfñak;enAeBlrgm:Um:g;rmYl (torsional moment). eKsnμt;eGay
bEnßmkugRtaMgkat;EdlbNþalBIkarmYl nigkMlaMgkat;enAelIRCugmçagrbs;muxkat; nigdak;eGayRbqaMg
enAelIRCugQm. Edk transverse closed tie RtUv)ansikSaKNnasMrab;RCugEdlenAkñúgenaHeKRtUv
bEnßmbnSMénT§iBlkMlaMgkat; nigkMlaMgrmYl.
bnÞúkxageRkAEdlbgáeGaymanm:Um:g;rmYlFMbMputmindUcKñanwgbnÞúkEdlbgáeGaymankMlaMgkat;
GtibrmaenARtg;muxkat;eRKaHfñak;eT. eKalbMNgrbs;GñksikSaKNnaKWbUkbBa©ÚltMélx<s;bMputénkar
rmYl nigtMélx<s;bMputrbs;kMlaMgkat;kñúgkarKNnaEdkRTnug. enHBitCamansuvtßiPaB. eKGaceRbIPaB
xusKñaénkMlaMgTaMgBIrenH edIm,IKNna transverse reinforcement sMrab;kMlaMgrmYlx<s;bMput CamYynwg
kMlaMgkat;EdlekIteLIgkñúgeBlCamYyKña b¤k¾KNnasMrab;kMlaMgkat;x<s;bMput CamYynwgkMlaMgrmYlEdl
ekIteLIgkñúgeBlCamYyKña edayykmYyNaEdlnaMeGaymanersIusþg;x<s;bMput. LRFD eRbIm:Um:g;Edl



Tb;Tl;karrmYl nominal dUcKñanwg ACI³
2 Ao At f y cot θ
Tn = (12.37)
s
Edl RkLaépÞmuxkat;EdlB½T§CMuvijeday shear flow path EdlrYmbBa©ÚlTaMgRbehag
Ao =

At = RkLaépÞrbs;eCIgmYyénEdkrgkarTajbiTCit

θ = mMuénsñameRbHEdl)aneRCIserIseday trial-and-adjustment edayeRbItarag 12>8
edIm,IkMNt;tMélrbs; θ / eKTTYl)anbMErbMrYlrageFob ε x enAkñúgEdkrgkarTajBIsmIkar 12.27
elIkElgsMrab;karCMnYs Vu eday
2
⎛PT ⎞
Vu = Vu2 +⎜ h u
⎜ 2A ⎟
⎟ (12.38)
⎝ o ⎠
eKTTYlbrimaN transverse reinforcement tMrUvkarsMrab;kMlaMgkat;BIsmIkar 12.21(a) CamYy
nwgsmIkar 12.23(a) nig 12.25 mann½yfa
Av f y d v cot θ
Vn = β f 'c bv d v + +Vp (12.39)
s
dUcenH sMrab;kMlaMgkat;KitCa lb nigkugRtaMgKitCa psi
Av Vn − (β f 'c bv d v + V p )
= (12.40a)
s f y d v cot θ

RbsinebIeKeRbIxñat ksi eKRtUvKuN β eday 0.0316 . ehIysMrab;karrmYl BIsmIkar 12.31
At Tn
= (12.40b)
s 2 Ao f y cot θ

RkLaépÞsrubrbs;EdkRTnug (web reinforcement) KW
Avt Av A
= +2 t (12.40c)
s s s
eKTTYl)anmMu θ BIrUbTI 12>9 edayeRbIkugRtaMgkMlaMgkat; v dUcxageRkam
(a) muxkat;RbGb;³
Vu − φV p T p Ph
V= + (12.41)
φbv d v φAoh
2

(b) muxkat;epSgeTot³
2 2
⎛ Vu − φV p ⎞ ⎛ Tu Ph ⎞
V= ⎜ ⎟ +⎜ ⎟ (12.42)
⎜ φb d ⎟ ⎜ φA 2 ⎟
⎝ v v ⎠ ⎝ oh ⎠
Edl Ph = brimaRtrbs;ExSG½kSén enclosed transverse torsion reinforcement



Aoh = RkLaépÞbiTCitedayExSG½kSén closed torsional reinforcement xageRkAeK
bMput
Ao = gross area B½T§CMuvijeday shear flow path ¬emIlrUbTI 5>45 sMrab;karbgðaj

Ao nig Aoh Edl Ao ≅ 0.85 Aoh ¦

Tu = m:Um:g;rmYlemKuN

φ = emKuNersIusþg;
eKTTYl)antMélrbs; β enAkñúgsmIkar 12.39 sMrab;kMNt;ersIusþg;kMlaMgkat; Vc énebtugsuT§enAkñúgRT-
nugBI chart enAkñúgrUbTI 12>9. edIm,IeCosvagkMueGayEdkbeNþay yield eKRtUvRtYtBinitüfaEdkrgkar
Bt;enAelIépÞrgkarTajRtUvEtsmamaRtedIm,IbMeBjlkçxNÐxageRkam³
2
⎛ 0.45Tu Po ⎞
φ (As f y + A ps f ps ) ( )
M
≥ u + 0.5 N u + cot θ Vu − 0.5Vs − V p 2
+⎜
⎜ 2A ⎟
⎟ (12.43)
dv ⎝ o ⎠
Edl Po = brimaRtrbs; shear flow path
N u = kMlaMgtamG½kSGnuvtþn_/ ykviC¢manRbsinebICakMlaMgsgát;

12.7. AASTHO-LRFD Flexural-Strength Design Specifications vs. ACI
Code Provisions
vamanPaBxusKñarvagviFIrbs; AASTHO-LEFD flexural-strength design specification nig
ACI-318 code provisions. viFI LRFD KWEp¥kelItMélkMNt;rbs;bMErbMrYlrageFobEdlerobrab;enAkñúg

Epñk 12.3 nigRKb;RKgedaypleFobénkMBs;G¾kSNWt c elIkMBs;RbsiT§PaB d e . viFIenHk¾RtUv)aneK
ehA fa unified approach edaysarvaGacGnuvtþ)ansMrab;karsikSaKNnasßanPaBkMNt;cugeRkay
(ultimate) cMeBaHGgát;ebtugGarem: Ggát;ebtugeRbkugRtaMg nigGgát;ebtugeRbkugRtaMgedayEpñk.

eK)anGnuvtþ ACI 318 code strength provision sMrab;kMNt; ultimate design strength f ps enAkñúg
]TahrN_CaeRcIndUcmanenAkñúgEpñk 4.9 nig 4.10. eKGnuvtþvaenAkñúgkarsikSaKNnaGgát;ebtugeRbkug
RtaMgeBjelj nigGgát;ebtugeRbkugRtaMgedayEpñkenAkñúgeRKOgbgÁúMsMNg;. AASTHO standard
specification bc©úb,nñEdlKNnasmamaRtmuxkat;Ggát;ebtugeRbkugRtaMgEdlrgkarBt;KWGnuvtþtam

ACI code provisions. LRFD alternative EdlCaviFI rational design TamTarkarGnuvtþ strain

limits unified procedure. taragTI 12>9 bgðajBIkareRbobeFobCasegçbEdlbgðajPaBxusKñarvag

smIkarEdl)ankMNt;enAkñúgviFITaMgBIrenH.



12.8. viFIsikSaKNnaCaCMhan² Step-by-Step Design Procedure (LRFD)
xageRkamCakarsegçbénCMhankñúgkarsikSaKNna³
!> kMNt;faetIeKRtUveRCIserIskMlaMgeRbkugRtaMgedayEpñk b¤Gt;
@> eRCIserIsm:Um:g;Bt; nigkMlaMgkat;BItarag 12.2(a) nig (b) Epñk 12.7.



#> GnuvtþtamCMhanbnþbnÞab;sMrab;KNnaGgátr; gkarBt;begáagEdlmanerobrab;enAkñúgCMhan 2
rhUtdl;CMhanTI 10 énEpñk 4.13 kñúgCMBUk 4 nig flowchart énrUbTI 12>10 enAeBleRbI
LRFD method sMrab;karKNnaGgát;rgkarBt;begáag. CaTUeTA d v = (d e − a / 2) .

$> kMNt;kMlaMgkat;emKuN Vu EdlbNþalBIbnÞúkGnuvtþn_TaMgGs;enARtg;muxkat;eRKaHEdlsßit
enAcMgay d v b¤ 0.5d v cot θ BIépÞrbs;TMr edayykmYyNaEdlFMCageK Edl
d e = kMBs;RbsiT§PaBdUcbgðajenAkñúgtarag 12>9

= d p RbsinebIeKmineRbIEdkBRgwgFmμta

%> KNnabgÁúMkMlaMgkat;rbs;kabeRbkugRtaMg VP . kugRtaMgkMlaMgkat;emKuNKW
Vu − φV p
v=
φbv d v
kugRtaMgkMlaMgkat;EdlGacman nominal vc = v / h
^> KNna v / f 'c nigsnμt;tMélrbs; θ . karsnμt;dMbUgd¾l¥sMrab;FñwmrgeRbkugRtaMgKW θ = 25o
&> KNnabMErbMrYlrageFobenAkñúgEdkrgkarTajedIm,ITTYl)antMélsakl,gθ nig β enAkñúg
tarag 12>8
⎡ Mu ⎤
⎢ d + 0.5 N u + 0.5Vu cot θ − A ps f po ⎥
εx = ⎢ v ⎥ Fe ≤ 0.002
⎢ (
2 E s As + E ps A ps ) ⎥
⎢ ⎥
⎣ ⎦
Edl f po ≅ 0.70 f pu

kugRtaMgsgát;enAkñúgebtugenARtg;TIRbCMuTMgn;rbs;EdkTajedayKitTaMgkMlaMg
f ce =

eRbkugRtaMgbnÞab;BIkMhatbg; nigbnÞúkGcié®nþTaMgGs;.
RbsinebIbMErbMrYlrageFobenAkñúgEdkrgkarTajGviC¢man eKRtUvKuN ε x edayemKuN Fε ³
E s As + E ps A ps
Fε =
Ec Ac + E s As + E ps A ps

Ac =RkLaépÞrbs;ebtugenAEpñkrgkarTajedaysarBt;begáagrbs;Ggát;
*> bBa©ÚleTAkñúg LRFD rUbTI 12>9 mþgeTot CamYynwgtMél v / f 'c nig ε x RbsinebImMu θ min
mantMélEk,rnwgtMélsnμt;enAkñúgkarsakl,geLIgdMbUgeTenaH edIm,ITTYl)antMélEktMrUv
rbs; β . ebImindUecñaHeT KNna Vc BIsmIkar 12.23 Edl Vc = β f 'c bv d v (lb) b¤ Vc =
0.0316β f 'c bv d v (kip ) edayeRbItMél β Edl)anBI chart enAkñúgrUbTI 12>9.



(> KNna Vs sMrab;EdkRTnug eRkayeBlkMNt;tMélrbs; Vc rYcehIy. kMNt;KMlatrbs;Edk
kMlaMgkat;EdlRtUvKñaBI³
bv s
Av = 0.036 f 'c
fy

!0> enAkñúgtMbn;énkugRtaMgkat;FM FanafabrimaNrbs;EdkbeNþay As nig Aps RtUvbMeBj
smIkarxageRkam³
⎡M N ⎛V ⎞ ⎤
As f y + A ps f ps ≥ ⎢ u + 0.5 u + ⎜ u − 0.5Vs − V p ⎟ cot θ ⎥
⎜φ ⎟
⎣ d vφ φ ⎝ ⎠ ⎦
eKENnaMeGayeFVIkarRtYtBinitüenARtg;épÞén bearing EdlsßitenAelIRbEvgepÞrrbs; strand
EdlminmanekItkMlaMgeRbkugRtaMgRbsiT§PaBeBjelj.
!!> enAeBlmankarrmYlrYmpSMCamYynwgkMlaMgkat; nigkarBt;begáag eKcaM)ac;RtUvGnuvtþtamCM
hanxageRkam
kMlaMgrmYl nominal Tn = 2 Ao At sf y cot θ
bMErbMrYlrageFobenAkñúgEdkrgkarTaj
⎡M ⎛PT ⎞ ⎤
⎢ u + 0.5 N u + 0.5 cot θ Vu2 + ⎜ h u
⎜ 2A ⎟ − A ps f po ⎥
⎟
⎢ dv ⎝ o ⎠ ⎥
εs = ⎢ ⎥ Fe ≤ 0.002
⎢ (
2 E s As + E ps A ps ) ⎥
⎢ ⎥
⎣ ⎦
Edl f po = 0.70 f pu
ersIusþg;kMlaMgkat; nominal
Av f y d v cot θ
Vn = Vc + Vs + V p = β f 'c bv d v + +Vp
s
Edl d v = (d p − a / 2)
EdkkMlaMgkat;
Av Vn − 0.0316 β f 'c bv d v + V p
=
s f y d v cot θ

kMlaMgKitCa kips ehIykugRtaMgKitCa ksi . sMrab;kareRbI lb nig psi ykemKuN 0.0316
ecj.
EdkrmYl



At Tn
=
s 2 Ao f y cot θ

EdkRTnugbiTCitsrub (total web closed ties reinforcement)
Avt Av A
= +2 t
s s s
kugRtaMgkMlaMgkat; v sMrab;TTYl)anmMu θ
(a) muxkat;RbGb;³
Vu − φV p TPh
v= +
φbv d v φAo h 2
(b) muxkat;epSgeTot
2 2
⎛ Vu − φV p ⎞ ⎛ TPh ⎞
v= ⎜ ⎟ +⎜ ⎟
⎜ φb d ⎟ ⎜ φA 2 ⎟
⎝ v v ⎠ ⎝ oh ⎠
edIm,IeCosvagkMueGayEdkrgkarTajbeNþay yield³
2
⎛ 0.45Tu Ph ⎞
φ (As f s + A ps f ps ) ( )
M
≥ u + 0.5 N u + cot θ Vn − 0.5Vt − V p 2
+⎜
⎜ 2A ⎟
⎟
dv ⎝ o ⎠
!@> RtYtBinitükMlaMgkat; interface edk³
vn Acv ≤ φVn
Edl (
Vn = cAcv + μ Avf f y )
⎛ Avf ⎞
vuh ≤ φ ⎜ 0.1 +
⎜
⎟
⎟
⎝ Acv ⎠
Edl Avf =
0.05bv s
fy
¬ f y KitCa ksi ¦
ykersIusþg;kMlaMgkat; nominal CatMéltUcCageKkñúgcMeNam
Vn ≤ 0.20 f 'c Acv
b¤ Vn ≤ 0.80 Acv
Edl c = emKuNs¥it (cohesion factor)

μ = emKuNkkit
Acr = RkLaépÞ interface ebtug = bv I v

Avf = RkLaépÞrbs;EdkkMlaMgkat;Edlkat;tambøg;kMlaMgenAkñúgépÞ Acv

φ = emKuNkat;bnßyersIusþg;
kñúgkrNIEdl vuh / φ > 100 psi eKRtUvkMritEdnkMNt;rbs; Avf .


rUbTI 12>10 bgðajBI flowchart sMrab;CMhanEdlRtUvGnuvtþedIm,IkMNt;ersIusþg;m:Um:g;
nominal sMrab; bonded tendons nig unbonded tendons.

!#> KMlatGnuBaØatGtibrmarbs;EdkkMlaMgkat;RTnug (web shear reinforcement)
s ≤ 0.75h ≤ 24in.
RbsinebI Vs > 4 f 'c bwd / kat;bnßyKMlat 50%
sMrab;KMlat dowel reinforcement
RbsinebI Vu < 0.1 f 'c bv d v / s ≤ 0.8d v ≤ 24in.

RbsinebI Vu > 0.1 f 'c bv d v / s ≤ 0.4d v ≤ 12in.

Edl bv = TTwgénépÞb:HsMrab;kMlaMgkat;edk



12.9. LFRD Design of Bulb-Tee Bridge Deck
]TahrN_ 12>1³ sikSaKNnaFñwm AASTHO-PCI bulb-tee xagkñúgEdlmanTMrsamBaØéns<ankMral
smas EdlKμan skews. FñwmenHmanRbEvg 120 ft (36.6m) . eRKOgbgÁúMEpñkxagelI (superstructure)
pSMeLIgeday pretensioned beam cMnYn 6 EdlmanKMlatBImYyeTAmYycMgay 9 ft (2.74m) EdlKitBI
G½kSeTAG½kS dUcbgðajenAkñúgrUbTI 12>11. s<anmankMras;ebtugcak;enAnwgkEnøgkMras; 8in.(203mm)
CamYynwg wearing surface BIxagelIkMras; 2in. . bnÞúkGefrKNnaKW HL-93 AASTHO-LRFD
fatigue loading.

snμt;fas<anenHsßitenAkñúgtMbn;rBa¢ÜydItUc. eKeGay
kugRtaMgGnuBaØatGtibrma³
kMrals<an f 'c = 4,000 psi ebtugTMng;Fmμta
f c = 0.60 f 'c = 2,400 psi
Fñwm bulb-tee f 'c = 6,500 psi lkçN³muxkat;³
f 'ci = 5,500 psi Ac = 767in.2
f c = 0.60 f 'c = 3.900 psi Service III h = 72in.
f c = 0.45 f 'c = 2,925 psi Service I I c = 545,894in.4

f ci = 0.60 f 'c = 3,480 psi cb = 36.60in.



f t = 6 f 'c = 484 psi ct = 35.40in.

f pu = 270,000 psi S b = 14,915in.3

f py = 0.90 f pu = 243,000 psi S t = 15,421in.3
I c 545,894
f pi = 0.75 f pu = 202,500 psi r2 = = = 712in.2
Ac 767
f y = 60,000 psi WD = 799 plf

E ps = 28.5 ⋅10 6 psi

E s = 29.0 ⋅10 6 psi

dMeNaHRsay³
!> TTwgkMralbMElg (Transformed Deck slab controlling width)
KNnaTTwgsøabbMElg
Ecs = 33w1.5 f 'c = 33 × (1.5)1.5 4,000 = 3,830ksi

enAeBlepÞr Eci = 33(1.5)1.5 5,500 = 4,500ksi
eRkam service Ece = 33(1.5)1.5 6,500 = 4,890ksi
TTwgsøabRbsiT§PaBKWtMéltUcCageKkñúgcMeNam
(i)
1
4
ElVg = 1204×12 = 360in.
(ii) 12h f + kMras;RTnugEdlFMCageK b¤ TTwgsøabxagelI/ b = 12 × 7.5 + 0.5 × 42 = 111in.
1
2
(iii) KMlatmFümrvagFñwm = 9 × 12 = 108in.

dUcenH TTwgsøab = 108in.



pleFobm:UDul ns = Ecs = 3,,830 = 0.78
Ec 4 890
TTwgbMElg (transformed) bm = nsb = 0.78 ×108 = 84in.
@> lkçN³rbs;muxkat;smas
eKmincaM)ac;KitBIkarcUlrYmén deck concrete haunch eTAkñúg I 'c eT edaysarPaBekag
(camber) rbs;Ggát;cak;Rsab;.

A'c = 1,397in.2

h = 80in.
I cc = 1,095,290in.4
cbc = 54.6in. eTAsrésxageRkambMput
ctc = 72 − 54.6 = 17.4in. eTAsrésxagelIbMputrbs;Fñwmcak;Rsab;
ctsc = 80 − 54.6 = 25.4in. eTAsrésxagelIbMputrbs;kMral
1,095,290
S bc = = 20,060in.3
54.6
1,095,290
Sc =
t
= 62,950in.3
17.4
1,095,290
Sc =
ts
= 55,284in.3
25.4 × 0.78
#> m:Um:g;Bt;begáag nigkMlaMgkat; (bending moments and shear forces)
kMral³ WSD1 = 12 × 9 × 150 = 900ib / ft
8

TMgn;bgáan;éd (barrier weight)³ WSD2 = 2barriers(300lb / ft ) = 100lb / ft
6beams

³
2in. future-wearing surface WSD3 =
2
×
48 ft
12 6beams
× 150 = 200lb / ft

bnÞúkGefr (truck load) enAkñúg LRFD Ep¥kelI HL-93 truck fatigue loading.
clear width BIrUbTI 12>12 = 48 ft (14.6cm )

cMnYn lanes = 12 = 4 lanes
48

(a) emKuNEbgEcksMrab;m:Um:g; (Distribution factor for moment)
sMrab; lane load 2 b¤eRcInCagenH emKuNEbgEckbnÞúksMrab;m:Um:g;Bt; (tarag 12>3b)
0.2 ⎛ 0.1
⎛ S ⎞
0.6
⎛S⎞ K ⎞
DFM = 0.075 + ⎜ ⎟ ⎜ ⎟ ⎜ g ⎟
⎝ 9.5 ⎠ ⎝L⎠ ⎜ 12t 3 L ⎟
⎝ s ⎠



kñúgkrNIEdl
KMlatFñwm³ 3.5 ≤ S ≤ 16 Cak;Esþg S = 9.0 ft O.K.

kMrals<an³ 4.5 ≤ Ts ≤ 12 Cak;Esþg Ts = 7.5in. O.K.

ElVg³ 20 ≤ L ≤ 240 Cak;Esþg L = 120 ft O.K.

cMnUnFñwm³ Nb > 4 Cak;Esþg N b = 6 O.K.

e g = cMgayrvagTIRbCMuTMgn;rbs;Fñwm nigkMral
7.5
= + 0.5 + 35.4 = 39.65in.
2
E 4,890
n= c = = 1.28
E sc 3,830
(
K g = n I c + Ac e g
2
)
[
= 1.28 545,894 + 767(39.65)2 = 2,242,191in.4 ]
0. 1
⎛ 9 ⎞
0.6
⎛ 9 ⎞
0.2
⎡ 2,242,191 ⎤
dUcenH DFM = 0.075 + ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ = 0.732 lanes/beam
⎝ 9 .5 ⎠ ⎝ 120 ⎠ ⎢12(7.5) (120 ) ⎥
3
⎣ ⎦
sMrab; design lane load mYy/ BItarag 12>3b
0.3 ⎛ 0.1
⎛S⎞
0.4
⎛S⎞ K ⎞
DFM = 0.06 + ⎜ ⎟ ⎜ ⎟ ⎜ g ⎟
⎝ 14 ⎠ ⎝ L⎠ ⎜ 12t 3 L ⎟
⎝ s ⎠
0. 3 ⎡ 0. 1
2,242,191 ⎤
0. 4
⎛9⎞ ⎛ 9 ⎞
= 0.06 + ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ = 0.499 lanes/beam
⎝ 14 ⎠ ⎝ 120 ⎠ ⎢12(7.5) (120 ) ⎥
3
⎣ ⎦
dUcenH krNI lane load BIr b¤eRcInCagBIr eK)anemKuNEbgEcgm:Um:g;FMCag dUcenHeK)an
DFM = 0.732 lanes/beam
Fatigue mement
m:Um:g;RtUv)anKitsMrab; single design truck EdlmanTMgn;ePøA (axle) dUcKña dUcenAkñúg
sßanPaBkMNt;déTeTot b:uEnþCamYynwgKMlatefr 30 ft cenøaH 32kips axles. eKeRbI lane
factor 1.2 sMrab; fatigue edIm,Ikat;bnßyemKuN DFM Edllub. BItarag 12>2a/ emKuN

bnÞúk (load factor) KW 0.75 ehIyemKuNTgáic (impact factor) (IM) sMrab; fatigue = 15% .
dUcenH m:Um:g;Bt; fatigue truckload køayCa³
M f = (bending moment per lane)(DFM / 1.2)(1 + IM )
⎛ 0.499 ⎞
b¤ M f = (bending moment per lane)⎜
⎝ 1.2 ⎠
⎟(1 + 0.15)



= (bending moment per lane)(0.415)(1.15)

= (0.478)(bending moment per lane)
(b) emKuNEbgEcksMrab;kMlaMgkat; (Distribution factor for shear)
BItarag 12>3 (a)/
sMrab; lane load BIr b¤eRcInCagenH
2
⎛S⎞ ⎛S ⎞
DFV = 0.2 + ⎜ ⎟ − ⎜ ⎟
⎝ 12 ⎠ ⎝ 36 ⎠
kñúgkrNIEdl³
KMlatFñwm³ 3.5 ≤ S ≤ 16 Cak;Esþg S = 9.0 ft O.K.
kMrals<an³ 4.5 ≤ Ts ≤ 12 Cak;Esþg Ts = 7.5in. O.K.
ElVg³ 20 ≤ L ≤ 240 Cak;Esþg L = 120 ft O.K.
10,000 ≤ K g ≤ 7,000,000 Cak;Esþg K g = 2,242,191in.4 O.K.
2
⎛9⎞ ⎛ 9 ⎞
dUcenH/ DFV = 0.2 + ⎜ ⎟ − ⎜ ⎟ = 0.887 lanes/beam
⎝ 12 ⎠ ⎝ 36 ⎠
sMrab; design lane load mYy/ BItarag 12>3a
⎛ S ⎞ ⎛ 9.0 ⎞
DFV = 0.36 + ⎜ ⎟ = 0.36⎜ ⎟ = 0.720 lanes/beam
⎝ 25.0 ⎠ ⎝ 25.0 ⎠
dUcenH krNI lane load BIr b¤eRcInCagBIr eK)anemKuNEbgEcgkMlaMgkat;FMCag dUcenHeK)an
DFV = 0.887 lanes/beam
$> bnSMbnÞúk (Load combinations)
bnÞúkemKuNsrub/ Q = η ∑ γ i qi
Edl η = emKuNEdlTak;TgeTAnwgPaBsVit (ductility)/ GBaØatelIs (redundancy) nigPaB
sMxan;éndMeNIrkar (operational importance)
γ i = emKuNbnÞúk
qi = bnÞúk

yk η = 1.0 sMrab;RKb;karGnuvtþTaMgGs;enAkñúg]TahrN_enH
eFVIkarGegátRKb;bnSMbnÞúkTaMgGs;EdlmanenAkñúgtarag 12>2 (a) nig (b). krNIEdllubKW
dUcxageRkam³
(a) Service I sMrab;kugRtaMgsgát;enAkñúgGgát;ebtugeRbkugRtaMg



Q = 1.0(DC + DW ) + 1.0(LL + IM )
(b) Service III sMrab;kugRtaMgTajenAkñúgGgát;ebtugeRbkugRtaMg
Q = 1.0(DC + DW ) + 0.8(LL + IM )
(c) Strength I sMrab; ultimate strength:
tMélGtibrmarbs; Q = 1.25DC + 1.50DW + 1.75(LL + IM )
tMélGb,brmarbs; Q = 0.90DC + 0.65DW + 1.75(LL + IM )
(d) Fatigue sMrab;RtYtBinitükugRtaMgenAkñúg strands

Q = 0.75(LL + IM )
(fatigue Q CabnSMbnÞúkBiesssMrab;RtYtBinitükugRtaMgTajenAkñúg strands EdlbNþal
BIbnÞúkGefr nig dynamic allowance)
%> kMlaMgkat; nigm:Um:g;Bt;KμanemKuN (Unfactored shear forces and bending moments)
(a) Truck Loads
kMlaMgkat; truck load:
VLT = (shear force per lane)(DFV )(1 + IM )

= (shear force per line )(0.887 )(1 + 0.33)

= 1.180(shear force per lane)kips
m:Um:g;Bt; truck load:
M LT = (moment per lane)(DFM )(1 + IM )

= (moment per lane)(0.732 )(1 + 0.33)

= 0.974(moment per lane) ft − kips
LT = bnÞúkGefrrbs; truck
(b) Lane Loads
sMrab; lane load EdlminmanGnuvtþ dynamic allowance
VLL = (shear force per lane)(DFV )

= (shear force per lane)(0.887 )kips

M LL = (moment per lane)(DFM )

= (moment per lane)(0.732 ) ft − kips
Lane load BIrUbTI 12>4/ bnÞúkenAelIs<anenHdUcbgðajenAkñúgrUbTI 12>12.


^> KNnam:Um:g; nigkMlaMgkat; (Computation of moments and shears)
(a) Lane Loads ¬ DFV = 0.887, DFM = 0.732 ¦

(i) muxkat;TMr³
kMlaMgkat;enARtg;TMrxageqVg (x = 0) BIsmIkar 12.6(a) nigrUbTI 12>12³
VLL =
0.64
(L − x )2 (DFV )
2L
=
0.64
(120)2 (0.887) = 34.1kisp
2 × 120
BIsmIkar 12.6 (b)/ nig DFM = 0.732
0.64( x )(L − x )
M LL = (DFM ) = 0 ft − kip
2
(ii) muxkat;enARtg; 24tf BITMr³
kMNt; VLL nig M LL enARtg; x = 24 ft BITMrxageqVg
VLL =
0.64
(120 − 24)2 (0.887) = 21.8kips
2 × 120
0.64(24)(120 − 24)
M LL = (0.732) = 539.7 ft − kip
2
¬
(b) Truck live loads DFV = 1.180, DFM = 0.974 ¦
eKRtUvKitbBa©Úl Impact factor IM = 33% dUcenH eyIgTTYl)antMél DFV nig DFM FMCag
mun
(i) muxkat;TMr³
BItarag 12>4/
72[(L − x ) − 9.33]
VLT = (DFV )
L


72[(120 − 0.0) − 9.33]
= (1.180) = 78.1kips
120
BItarag 12>5/
72(x )[(L − x ) − 9.33]
M LT = (DFM )
L
= 0 ft − kip sMrab;mU:m:g;Rtg;TMr
(ii) muxkat;enARtg; 24 ft BITMr ³
72[(120 − 24) − 9.33]
VLT = (1.180) = 61.4kips
120
72(24)[(120 − 24) − 9.33]
M LT = (0.974) = 1215.0 ft − kip
120
(c) Fatigue moment enARtg; 24 ft ¬ DFM = 0.478 ¦
BItarag 12>7
72( x )[(L − x ) − 18.22]
Mf = (DFF )
L
BIelIkmun/ DFF = 0.478
72(24)[(120 − 24) − 18.22]
dUcenH Mf =
120
(0.478) = 535.8 ft − kip
(d) Shears and moments EdlbNþalBIbnÞúkefr

bnÞúkEdlRtUvBicarNaKWTMgn;Fñwm ¬WD ¦ bUknwgkMral deck nig haunches ¬WSD1 ¦ nig
wearing surface ¬ WSD3 ¦ EdlRtUvRkalenAeBlGnaKt.

edaysarvaCaFñwmTMrsamBaØ dUcenHkMlaMgkat; nigm:Um:g;tambeNþayElVgKW³
V x = WD (0.5L − x )

M x = 0.5WD x(L − x )
Kitmuxkat;Rtg; 24 ft BITMrxagxageqVg ehIyKNnakMlaMgkat; nigm:Um:g;EdlbNþalBITMgn;pÞal;
WD = 0.799kip / ft ³

V x = 0.799(0.5 × 120 − 24) = 28.8kips

M x = 0.5 × 0.799 × 24(120 − 24) = 920.4 ft − kip
tarag 12>10 nig 12>11 bgðajBIkMlaMg nigm:Um:g;EdlRtUvkarsMrab;sikSaKNnaFñwmxagkñúg.
eKRtUvcMNaMfakarKNnaedayédedIm,ITTYl)antaragEbbenHRtUvkarcMNayeBlevlay:ageRcIn.
eKmankmμviFIkMuBüÚT½rEdlbegáIteLIgedayGKÁnaykdæandwkCBa¢Ún (state DOT) CaeRcIn EdlxøH
manenAkñúg internet dUcCa Washington State DOT Program.



&> sikSaKNnaFñwm bulb-tee eRbkugRtaMgxagkñúg (Computation of moments and shears)
¬!¦ kareRCIserIskabeRbkugRtaMg (Selection of Prestressing Strands)
sMrab;bnSMbnÞúk Service-III/ kugRtaMgsrésxageRkam fb KW³
M D + M S M b + M WS + 0.8(M LT + M LL )
fb = +
Sb S bc
Edl MD = m:Um:g;TMgn;pÞal;EdlKμanemKuN/ ft − kip
M S = m:Um:g;EdlKμanemKuNEdlbNþalBITMgn;kMral nig haunch, ft − kip

M b = m:Um:g;bgáan;édEdlKμanemKuN/ ft − kip

M WS = m:Um:g; wearing surface EdlKμanemKuN/ ft − kip



M LT = m:Um:g;bnÞúk truck EdlKμanemKuN/ ft − kip
M LL = m:Um:g; lane load EdlKμanemKuN/ ft − kip

BIelIkmun/ Sb = 14,915in.3
S bc = 20,090in.3
BItarag 12>10 nig 12>11/ kugRtaMgkNþalElVgxageRkamenARtg;srésxageRkameRkam
GMeBI service load KW
f bc =
1,438.2 + 1,659.6
(12) + 180 + 360 + 0.8(1,830.3 + 843.2) (12)
14,915 20,090

= 2.50 + 1.60 ≅ 4.10ksi(T )
kugRtaMg 4.10ksi(T ) nwgRtUv)anlubbM)at;edaykMlaMgeRbkugRtaMg. kugRtaMgTajGnuBaØatGti-
brma³
f t = 6.0 f 'c psi = 6 6,500 = 484 psi = 0.484ksi

kugRtaMgsgát;eRbkugRtaMgtMrUvkarenARtg;srésxageRkambMput³
f cb = (4.1 − 0.48) = 3.62ksi
snμt;facMgayBITIRbCMuTMgn;rbs;EdkeRbkugRtaMgeTAsrésxageRkambMput = 0.05h
= 0.05(72 ) = 3.6in. yk 4in. / dUcenH ec = 36.6 − 4.0 = 32.6in.

dUcbgðajenAkñúg]TahrN_énCMBUk 4
P ×e
f bp EdlbNþalBIeRbkugRtaMg = e + e c
P
A c S b

b¤ fbp = 767 + P14×,915.6 = 3.62ksi
Pe e 32

eyIgTTYl)an Pe = 1,037kips
snμt;kMhateRbkugRtaMgsrub = 25%
1,037
Pi = = 1,383kips
1 − 0.25
snμt;eRbI 7-wire 270-K low-relaxation strands Ggát;p©it 0.5in. ¬ Aps = 0.153in.2 ¦
cMnYntMrUvkarén strands = 0.153,383 .5 = 44.6strands
1
× 202
eRkayBIkarGnuvtþ trial and adjustment/ eyIgsakl,gmuxkat;Edlman 48 strand dUcbgðaj
enAkñúgrUbTI 12>13. enAeBl strand ticCag 48 eFVIeGaykugRtaMgTajenARtg;srésxageRkam
bMputeRkamGMeBI service load FMCagkugRtaMgGnuBaØatGtibrma ft = 484 psi . Strand cMnYn 20



RtUv)an harp enARtg; 0.4L . dUcenH 36 strands enArkSaPaBRtg;enAkñúgFñwm ¬emIlrUbTI 12>
13¦.

BIsmμtikmμ/ cb = 36.60in. ehIy ct = 72 − 36.60 = 35.40in.
ee = cb − [2 × 70 + 2 × 68 + 2 × 66 + 2 × 64 + 2 × 62 + 2 × 60 + 4 × 8 + 8 × 6

+ 12 × 4 + 12 × 2] / 48

= 36.60 − 19.42 = 17.28in.
ec = cb − [2 × 12 + 12 × 4 + 8 × 6 + 8 × 4 + 2 × 10 + 2 × 12 + 2 × 14 + 2 × 16

+ 2 × 18 + 2 × 20] / 48

= 36.6 − 6.92 = 29.68in.

eKeGay f pi = 0.75 f pu = 202,500 psi

Pi = (48)(0.153)(202.5) = 1,488kips
eRkayeBlGnuvtþnUvkarviPaK step-by-step rbs;kMhateRbkugRtaMg dUcenAkñúgCMBUk 3 Epñk 3.9
kMhatbg;eRbkugRtaMgsrubRtUv)ankMNt;edaymantMélesμInwg 26.4% .
f pe = 202.5(1 − 0.264) = 149.0ksi

dUcenH/ Pe = 1488(1 − 0.264) = 1095.0kips



¬@¦ RtYtBinitükugRtaMgEdlKμanemKuNrbs;ebtug (Check of Concrete Unfactored Stresses)
(a) kugRtaMgenAeBlepÞr
eRbkugRtaMgedIm f pi = 0.7 f pu = 0.7 × 270 = 202.5ksi . karGnuvtþFmμtasnμt;fakMhat
bg; relaxation edImenAeBlrgeRbkugRtaMgmanbrimaNBI 9% eTA10% . eRbIkarkat;
bnßy 10% enAkñúg f pi
Pi = 0.90 × 1,488 = 1,339kips
dUcenH Pi = 0.9(202.5)(0.153 × 48) = 1,338kips
(i) muxkat;Rtg;TMr

BICMBUk 4/ smIkar 4.1(a)
⎛ ee ct ⎞ M D
Pi
ft =− ⎜1 − 2 ⎟ − t
⎝
Ac r ⎠ S
1,338 ⎛ 17.28 × 35.4 ⎞
=−
767 ⎝
⎜1 −
712
⎟ − 0 = −0.25ksi(C )
⎠
/ minrgkarTaj/ O.K.

P ⎛ ec ⎞ M
f b = − i ⎜1 + e 2b ⎟ + D
Ac ⎝ r ⎠ Sb
1,339 ⎛ 17.28 × 36.3 ⎞
=− ⎜1 + ⎟+0
767 ⎝ 712 ⎠
= 3.29ksi(C ) < kugRtaMgGnuBaØat f c = 3.48ksi O.K.

(ii) muxkat;kNþalElVg
1,338 ⎛ 29.68 × 36.60 ⎞ 1,438 ×12
ft =− ⎜1 − ⎟−
767 ⎝ 712 ⎠ 15,421
= 0.917 − 1.119 = −0.202ksi(C ) / minmanrgkugRtaMgTaj/ dUcenH O.K.
1,339 ⎛ 29.68 × 36.6 ⎞ 1,438 ×12
fb = − ⎜1 + ⎟+
767 ⎝ 712 ⎠ 14,915
= −4.513 + 1.157 = −3.356ksi(C ) < kugRtaMgGnuBaØat f 'ci = 5.50ksi O.K.
(b) kugRtaMgenAeBl Service
(i) muxkat;kNþalElVg

BICMBUk 4/ smIkar 4.3(a) nig 4.3(b):
Pe ⎛ ec ct ⎞ MT
ft =− ⎜1 − 2 ⎟ − t ≤ fc
Ac ⎝ r ⎠ Sc
Pe ⎛ ec cb ⎞ M T
fb = − ⎜1 + 2 ⎟ + ≤ ft
Ac ⎝ r ⎠ S cb


Xii.lrfd and stan dard aastho design of concrete bridge

Xii.lrfd and stan dard aastho design of concrete bridge

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