Viii. prestressed compression and tension member

T.Chhay viTüasßanCatiBhubec©keTskmú<Ca

VIII. Ggát;rgkarTaj nigrgkarsgát;eRbkugRtaMg
Prestressed Compression and Tension Members

1> esckþIepþIm Introduction
eTaHCaeKeRbIebtugeRbkugRtaMgy:ageRcInsMrab;Ggát;rgkarBt;begáag dUcCaFñwm nigkMralxNÐk¾
eday k¾vaRtUv)aneKeRbIsMrab;Ggát;rgkMlaMgtamG½kS (axially loaded member) dUcCa ssrEvg ¬Ggát;
rgkarsgát;¦ nigGgát;cMNg (tie) sMrab;Ggát;ragFñÚ (arch) nig truss elements ¬Ggát;rgkarTaj¦. b:uEnþ
minTan;maneCIgtageRbIeRbkugRtaMgTajCamun b¤TajCaeRkayeT.
RTwsþI/ karviPaK nigkarsikSaKNnaGgát;rgkarsgát;eRbkugRtaMgKWmanlkçN³RsedogKñanwgGgát;
ebtugGarem:rgkarsgát;Edr. kMlaMgeRbkugRtaMgtamG½kSxagkñúgenAkñúg bonded tendon min)anbegáIt
column action eT dUcenHminman buckling GacekItmaneT elIkElgEtEdkeRbkugRtaMg nigebtugEdl

B½T§CMuvijb:HKñaedaypÞal;tambeNþayRbEvgsrubrbs;Ggát;. edaysarEtEbbenHehIy eTIbPaBcg;ekag
rbs;ebtugenAkNþalRtUv)aneFVIeGayNWtedaysarT§iBlsNþkrbs;EdkeRbkugRtaMgEdlbgáb;tam
beNþayG½kS.
CaFmμtassrEtgEtrgkarBt;begáagbEnßmBIelIbnÞúktamG½kS edaysarbnÞúkxageRkAkMrcMp©it
Nas;. CalT§pl muxkat;ebtugrgkarTajenARtg;RCugEdlenAq¶ayBIExSskmμrbs;bnÞúktambeNþay
CageK. sñameRbHekItmaneLIg b:uEnþeKGackarBarva)antamry³kareRbIkMlaMgeRbkugRtaMgenAkñúgssr.
RbsinebIbnÞúkGnuvtþn_CabnÞúkcMp©it enaHkMlaMgeRbkugRtaMgminmansar³sMxan;eT edaysareKminRtUvkar
eGaykugRtaMgsgát;enAelImuxkat;ebtugekIneLIg.
eKGacBicarNaGgát;rgkarsgát;rgeRbkugRtaMgeBjtambeNþayRbEvgrbs;vaRbsinebIminmankM
hatbg;eRbkugRtaMgenAxagcugrbs;vaeTenaH. RbsinebIekItmankMhatbg;edayEpñk (partial loss) eK
RtUvBicarNakMNat;EdkenAkñúg development zone minmanrgeRbkugRtaMg ehIyeKRtUvKitmuxkat;enA
tMbn;xagcugCamuxkat;ebtugGarem:rgbnÞúkcakp©it.
CaFmμtaGgát;rgkarTajEtgrgEtkMlaMgTajedaypÞal;Etb:ueNÑaH. Ggát;TaMgenHPaKeRcInCag
Ggát;ragCabnÞat; dUcCa railroad ties, restraining tie sMrab; arch bridges, Ggát;rgkarTajenAkñúg
truss nig foundation anchorage sMrab;eRKOgbgÁúMTb;dI. Ggát;rgkarTajk¾GacmanragrgVg; b¤)a:ra:bUl

pgEdr dUcCa witness prestressed circular container b¤ catenary-shaped bridge elements. tYnaTI

Prestressed Compression and Tension Members 492

Department of Civil Engineering NPIC

cMbgrbs;Ggát;rgkarTajKWkarkarBarsñameRbHrbs;vaeRkamGMeBI service load nigGacTb;Tl; service
load xageRkA nig overload. karEdlminmansñameRbHKWkarBarERcHsIuEdk niglkçxNÐbrisßanepSg².

2> Ggát;rgkarsgát;eRbkugRtaMg³ GnþrGMeBIrvagbnÞúk nigm:Um:g;enAkñúgssr nigssrRKWH
Prestressed Compression Members: Load-Moment Interaction in Columns and Piles
edIm,IkMNt; nominal strength rbs;ssreRkamGMeBIbnÞúkcMNakp©itepSg² eKcaMcaMKNnanUv
lT§PaBepSgénbnSMrvag ultimate nominal loads Pn nig ultimate nominal moments M n Edl
eGayeday
M n = Pn ei (8.1)
Edl ei CacMNakp©itrbs;bnÞúkeRkamGMeBIénbnSMrvagbnÞúk nigm:Um:g;epSg². düaRkaménTMnak;TMngrvag
Pn nig M n RtUv)anbgðajenAkñúgdüaRkam interaction énrUbTI 8>1 sMrab;ssrminRsav ¬)ak;edaysar

sMPar³¦ nigssrRsav ¬)ak;edassaresßrPaB¦. enAkñúgssrxøI kar)ak;ekItmanedaysarbnÞúkQaneTA
dl;cMnuc A tambeNþayKnøg OA ehIy concrete arches enARtg;RCugrgkarsgát;. sMrab;ssrRsav
tMélrbs;bnÞúkGtibrmaQaneTAdl;cMnuc B tambeNþayKnøg OBC Edlkat;nwg interaction diagram
enARtg;cMnuc C . GesßrPaBekItmanenAeBlEdlbnÞúkQaneTAdl;bnÞúkeRKaHfñak; (critical load).

karsnμt;CamUldæansMrab;ssrebtugeRbkugRtaMgmanlkçN³RsedogKñanwgkarsnμt;sMrab;ssr
ebtugGarem:Edr. karsnμt;TaMgenaHmandUcxageRkam³
!> karBRgaybMErbMrYlrageFobenAkñúgebtugERbRbYlCalkçN³bnÞat;eTAtamkMBs;muxkat;.

Ggát;rgkarsgát; nigkarTajeRbkugRtaMg 493


@> karBRgaykugRtaMgenAkñúgtMbn;rgkarsgát;manrag)a:ra:bUl ehIyRtUv)anCMnYsedaybøúk
ctuekaNsmamaRtenAkñúgkarviPaK nigkarsikSaKNna.
#> eKsÁal;düaRkamTMnak;TMngkugRtaMg nigbMErbMrYlrageFobrbs;ebtug nigrbs;EdkeRbkugRtaMg.
$> Crushing strain rbs;ebtugedaysarbnSMkarBt;begáag nigkMlaMgtamG½kSenARtg;srésxag
eRkAbMputKW ε c = 0.003in. / in. ehIy crushing strain mFümenARtg;Bak;kNþalkMBs;rbs;
muxkatEdlrgbnÞúktamG½kSCasMxan;KW ε 0 = 0.002in. / in. .
%> eKKitfamuxkat;)ak;enAeBlEdlbMErbMrYlrageFobenAkñúgebtugenARtg;srésrgkarsgát;xag
eRkAbMputQaneTAdl; ε c = 0.003in. / in. b¤ ε 0 = 0.002in. / in. enARtg;Bak;kNþalkMBs;
rbs;muxkat;. cMNaMfa ε c = 0.003in. / in. CatMélEdleRbIenAkñúg ACI Code b:uEnþ code
déTeToteRbItMélFMCagenHKW 0.0035 b¤ 0.0038 .
^> eKsnμt;famanPaBRtUvKñaénbMErbMrYlrageFob (compatibility of strain) rvagebtug
nigEdkeRbkugRtaMg.

TMrg;énkar)ak;k¾manlkçN³RsedogKñanwgkar)ak;rbs;ssrebtugGarem:pgEdr³
!> kar)ak;edaykarsgát;dMbUg (initial compression failure), cMNakp©ittUc. TMrgénkar)ak;enH
ekItmanenAeBlbMErbMrYlrageFobenAkñúgebtugenARtg;RCugEdlrgbnÞúkQandl; ε cu =
0.003in. / in. xN³EdlbMErbMrYlrageFobenAkñúgEdkeRbkugRtaMgEdlenARCugq¶aymçageTot

mantMélTabCag yield strain. cMNakp©it e rbs;bnÞúktamG½kSmantMéltUcCag balanced
eccentricity eb .

@> kar)ak;edaykarTajdMbUg (initial tension failure), cMNakp©itFM. TMrg;énkar)ak;enHbRBa©as
BITMrg;énkar)ak;elIkmun. ebtugenARCugq¶ay yield munebtugEbkRtg;RCugEdlrgbnÞúk. cM
Nakp©it e rbs;bnÞúktamG½kSFMCag balanced eccentricity eb .
#> Balanced state of strain, ε t = 0.002in. / in. , balanced eccentricity. TMrg;enHkMNt;nUv
lkçxNÐéntMélm:Um:g;Gtibrma M nb enAelIExSekagGnþrGMeBIEdlRtUvKñanwg maximum
tensile strain enAkñúgRsTab;rgkarTajesμInwg strain increment Δε ps = 0.0012 eTA

0.002in. / in. bnÞab;BI service load. cMNakp©itrbs;bnÞúktamG½kSRtUv)ankMNt;Ca

balanced eccentricity eb .



cMnucsMxan;bIenAelIdüaRkam interaction KW³
!> M u = 0 EdlRtUvKñaeTAnwg ε 0 = 0.002in. / in. enAeBl)ak;edaysarbnÞúkcMp©it Pu .
TItaMgG½kSNWtKWenAGnnþ.
@> KμankarTajenARtg;srésrgkarTajxageRkAbMputrbs;ebtug ehIy ε cu = 0.003in. / in. enA
Rtg;srésrgkarsgát;xageRkAbMputrbs;ebtug. TItaMgG½kSNWtsßitenAelIsrésrgkarTaj
xageRkAbMput.
#> Pu = 0 nig ε cu = 0.003in. / in. enARtg;srésrgkarsgát;xageRkAbMput. G½kSNWtsßitenA
xagkñúgmuxkat; ehIyRtUv)ankMNt;eday trail and adjustment edaysnμt;kMBs; c .
rUbTI 8>2 bgðajBIkarEbgEckkugRtaMg nigbMErbMrYlrageFobsMrab;krNITaMgbIxagelI.



cMnucEdlenAesssl;enAelIdüaRkam interaction KWsMrab;krNIEdlsßitenAcenøaHdMNak;kal
(a), (b) nig (c) rbs;rUbTI 8>2 b¤BIbnÞúkcMputeTAdl;karBt;begáagsuT§ (pure bending). kñúgkrNIrbs;

ssr pure bending kMNt;sßanPaBEdlminKitBIpleFobrbs;bnÞúktamG½kSemKuN Pu elIm:Um:g;Bt;
begáag M u . karBRgaykugRtaMgrag)a:ra:bUlsMrab;krNI (b) nig (c) RtUv)anCMnYsedaybøúkctuekaN
smamaRt EdlkMBs;rbs;bøúk a = β1c dUckrNIsMrab;FñwmrgkarBt;begáagEdr.



krNIKMrUénGgát;rgkarsgát;KWsßitenAcenøaHdMNak;kal (b) nig (c) rbs;rUbTI 8>2. bMErbMrYl
rageFob kugRtaMg nigkMlaMgsMrab;krNIEbbenHRtUv)anbgðajenAkñúgrUbTI 8>3 sMrab;muxkat;eRKaHfñak;
eRkamsßanPaBkMNt;én ultimate load edaykar)ak;edaysMPar³. kat;düaRkamGgÁesrI (free-body
diagram) enARtg;Bak;kNþalkMBs;rbs;ssrRtg;muxkat; 1-1 muxkat;rbs;Ggát;RtUv)anbgðajenAkñúg

cMnuc (b) rbs;rUb ehIybMErbMrYlrageFob nigkugRtaMgenAeBl)ak;manenAkñúgcMnuc (c) nig (d) erogKña.
bMErbMrYlrageFob ε ce CabMErbMrYlrageFob uniform enAkñúgebtugeRkamGMeBIeRbkugRtaMgRbsiT§PaB
eRkayeBl creep, shrinkage nig relaxation losses EdleGayerogKñadUcxageRkam³
Ccn = 0.85 f 'c ba (8.2a)

T 'sn = A' ps f ' ps (8.2b)

nig Tsn = f ps Aps1 (8.2c)

smIkarlMnwgrbs;kMlaMgKW
Pn = Ccn − T 'sn −Tsn (8.3)
RbsinebIkMlaMgeRbkugRtaMgRbsiT§PaBeRkayeBlkMhatbg;TaMgGs;Ca Pe enaHbMErbMrYlrag
eFobenAkñúg tendon munnwgkarGnuvtþrbs;bnÞúkxageRkAKW
f pe Pe
ε pe = =
E ps (Aps )
− A' ps E ps
(8.4a)

eKGackMNt;karERbRbYlénbMErbMrYlrageFobenAkñúgRkLaépÞEdkeRbkugRtaMg A' ps enAeBl
EdlGgát;rgkar sgát;qøgkat;BIdMNak;kalkMlaMgeRbkugRtaMgRbsiT§PaBeTAdl; ultimate load dUc
xageRkam³
⎛ c − d'⎞
Δε ' ps = ε cu ⎜ ⎟ − ε ce (8.4b)
⎝ c ⎠
⎛d −c⎞
Δε ps = ε cu ⎜ ⎟ + ε ce (8.4c)
⎝ c ⎠
Δε p = Δ ps − ε ce (8.4d)

(
T 'sn = A' ps f ' ps = A' ps E ps ε pe − Δε ' ps )
⎡ ⎛ c − d'⎞ ⎤
b¤ T 'sn = A' ps E ps ⎢ε pe − ε cu ⎜
⎝ c ⎠
⎟ + ε ce ⎥ (8.5)
⎣ ⎦
dUcKña (
Tsn = Aps f ps = Aps E ps ε pe + Δε ps )
⎡ ⎛d −c⎞ ⎤
b¤ Tsn = A' ps E ps ⎢ε pe + ε cu ⎜
⎝ c ⎠
⎟ + ε ce ⎥ (8.6)
⎣ ⎦



edayKitm:Um:g;eFobnwgTMRbCMuTMgn;FrNImaRt cgc rbs;muxkat;eK)an
(8.7)
BIsmIkar 8.2a, 8.5, 8.6 ni 8.7 eKGackMNt; nominal strength Pu nig M u sMrab;cMNakp©it
ei epSg² edIm,Isg;düaRkam interaction P − M sMrab;muxkat;NamYy b¤begáIt nondimensional

series éndüaRkam interaction P − M sMrab;ersIusþg;ebtugepSg². ersIusþg;KNna (design strength)

RtUv)anKNnaBItMél nominal strength
Pu = φPn
nig M u = φM n = φPn e

Edl φ CaemKuNkat;bnßyersIusþg;rbs;Ggát;rgkarsgát;. cMNaMfa tMél Pu nig M u KNnaRtUvman
tMélEk,r EtminRtUvtUcCagtMél Pu nig M u emKuNeT. rUbTI 8>4 bgðajBIdüaRkamTMnak;TMngbnÞúk
nigm:Um:g;EdlmancMNakp©it.



3> emKuNkat;bnßyersIusþg; φ
Strength Reduction Factor φ
sMrab;Ggát;rgkarBt;begáag nigrgbnÞúktamG½kStUc enaHGgát;rgkar)ak;eday tension rein-
forcement eFVIkardl; yield ehIykareFVIkarCalkçN³sVit (ductile) rbs;Ggát;mankarekIneLIg. dUc

enH sMrab;bnÞúktamG½kStUc eKGnuBaØateGaybegáInemKuN φ BIGVIEdleK)antMrUvsMrab;Ggát;rgkarsgát;
suT§. enAeBlEdlminmanbnÞúktamG½kS Ggát;RbQmnwgkarBt;begáagsuT§ (pure flexure) ehIyemKuN
kat;bnßyersIusþg; φ esμInwg 0.90 .
rUbTI 4>45 bgðajBI transition zone EdlenAkñúgenaHeKGacbegáInemKuNkat;bnßyersIusþg; φ BI
0.65 sMrab; tied column eTA 0.70 sMrab; spirally reinforced column eTAdl; 0.90 sMrab; pure

flexure enAkñúg strain limits approach. Balanced limit strain sMrab; compression-controlled state

RtUv)ankMNt;eday limiting strain ε t = 0.002in. / in. b¤pleFobkMBs;G½kSNWt c / dt = 0.60 sMrab;
Ggát;rgkarsgát;. eKGacBicarNatMél φPn = 0.10 f 'c Ag Ca design axial load EdltMé;ltUcCagenH
eKGacbegáIntMél φ edaysuvtßiPaBsMrab;Ggát;rgkarsgát;PaKeRcInEdlsßitenAkñúg transition zone én
rUbTI 4>45. eKGaceFVI interpolation éntMél φ sMrab; transition zone BI limit stain state rgkar
sgát; (ε t = 0.002) eTA limit strain state rgkarTaj (ε t = 0.005) dUcenAkñúgsmIkar 4.36 (a) nig
4.36 (b) dUcxageRkam³

(a) φ CaGnuKmn_énbMErbMrYlrageFob

Tied section³

0.65 ≤ [φ = 0.48 + 83ε t ] ≤ 0.90 (8.8a)

Spirally-reinforced section:
0.70 ≤ [φ = 0.57 + 67ε t ] ≤ 0.90 (8.8b)
(b) φ CaGnuKmn_énpleFobkMBs;G½kSNWt
Tied section³

⎡ 0.25 ⎤
0.65 ≤ ⎢φ = 0.23 + ⎥ ≤ 0.90 (8.9a)
⎣ c / dt ⎦

Spirally-reinforced section:

⎡ 0.20 ⎤
0.70 ≤ ⎢φ = 0.37 + ⎥ ≤ 0.90 (8.9b)
⎣ c / dt ⎦



cMNaMfa balanced strain condition enAkñúgGgát;rgkarsgát;ebtugeRbkugRtaMgmandWeRkminkM-
Nt;x<s;. eKGaceRbIviFIsmrmü (trial and adjustment) edIm,IedaHRsayedaysnμt;tMél Δ ps = 0.0012
eTA 0.0020in. / in. bnÞab;BIrg service load nigedayKNnakMBs;bøúkkugRtaMg a rbs;muxkat;ebtug.
eKRtUvepÞógpÞat;karsnμt;enH ehIyeKRtUvEktMrUv nominal moment sMrab; limit stain condition ε 't =
0.002 eRkayeBlsg;düaRkam interaction rYc. tamviFIenH eKGaceFVIeGaytMélm:Um:g;Gtibrmaman

lkçN³RbesIreLIgsMrab; balanced strain limit state rgkarsgát; EdlsMEdgedaykMBs;G½kSNWt cb
RbsinebIcaM)ac;.

4> dMeNIrkarsMrab;sikSaKNnaGgát;xøIrgkarsgát;ebtugeRbkugRtaMg
Operational Procedure for the Design of Nonslender Prestressed Compression Members
eKGacGnuvtþCMhanxageRkamsMrab;sikSaKNnassrxøIEdlkareFVIkarrbs;vaRKb;RKgedaykar)ak;
edaysMPar³
!> KNnabnÞúktamG½kSxageRkAemKuN Pu nigm:Um:g;emKuN M u . KNnacMNakp©itEdlGnuvtþ
e = M u / Pu .

@> snμt;muxkat; nigRbePTrbs;EdkxagEdlRtUveRbI dUcCa tied b¤ spiral. eKminRtUveRCIserIs
muxkat;tUc.
#> snμt;cMnYn nigTMhMrbs; strand.
$> snμt;fabMErbMrYlrageFobenAsrésrgkarTajxageRkAbMputesμInwgbMErbMrYlrageFobEdlsnμt;
ε ps rbs;EdkeRbkugRtaMg nigbnÞab;mkbnþKNna balanced limit strain axial load Pnb nig
m:Um:g; M nb enARtg; limit strain ε t = 0.002 . CMhanenHk¾GaceGayeKepÞógpÞat;tMélrbs;
emKuNkat;bnßyersIusþg;pgEdr. m:Um:g; M nb )anBItMél strain ε ps EdkeGaym:Um:g;Gtibrma
enAkñúgdüaRkam interaction.
%> snμt;kMBs;G½kSNWt c ehIykMNt;rk Pn nig M n . bnÞab;mkRtYtBinitüPaBRKb;RKan;rbs;mux
kat;Edlsnμt; eday φPn > Pu emKuN nig φM n > M u . RbsinebImuxkat;minGacRTbnÞúkem
KuN b¤vamanmuxkat;FMeBk eKRtUveRCIserIsmuxkat; nigEdkeLIgvijtamry³ trial and adjust-
ment edayGnuvtþCMhan $ nig% eLIgvij edayrYmTaMgkarsg;düaRkam interaction.

^> sikSaKNnaEdkxag (lateral reinforcement).
rUbTI 8>5 bgðajBI flowchart énCMhan trial-and-adjustment kñúgkarviPaK nigsikSaKNna.



5> sg;düaRkamGnþrGMeBIrvagbnÞúk nigm:Um:g;
Construction of Nominal Load-Moment (Pn − M n ) and Design (Pu − M u )
Interaction Diagram
]TahrN_ 8>1³ sg;düaRkam interaction én nominal load-moment sMrab;Ggát;rgkarsgát;eRbkug
RtaMgEdlmanmuxkat;kaer: EdlRCugrbs;vaesμInwg 14in.(356mm) . Ggát;RtUv)anBRgwgeday 7-wire
stress-relieved 270-K strands Ggát;p©it 1 / 2in.(12.7 mm ) cMnYn 8 EdlBak;kNþalenAelIRCugnImYy²

rbs;épÞTaMgBIrEdlRsbnwgG½kSNWtEdlbgðajenAkñúgrUbTI 8>6. düaRkamTMnak;TMngkugRtaMg nigbMEr
bMrYlrageFobsMrab; strain RtUv)anbgðajenAkñúgrUbTI 8>7. kMlaMgeRbkugRtaMgRbsiT§PaBeRkaykMhat
bg;TaMgGs;KW f pe = 150,000 psi(1,034MPa) . elIsBIenH sd;düaRkam design interaction edayeRbI
tMélemKuNkat;bnßyersIusþg;Edlsmrmü. eKeGay
f 'c = 6,000 psi(47.5MPa ) ebtugTMgn;Fmμta

E ps = 29 × 10 6 psi (200 × 103 MPa )

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

ε cu = 0.003in. / in. enAeBl)ak; (failure)
⎛ e2 ⎞
ε ce = 0.0005in. / in. enAeBl Pe eFVIGMeBIenAelImuxkat; = e
P ⎜1 + ⎟
AE ⎜ r2 ⎟
c c ⎝ ⎠
BIrUbTI 8>9.
ε py = strand yield strain ≅ 0.012in. / in.
snμt;tMélsmrmüén ε p nigeFVIkarEktMrUvRbsinebIcaM)ac;.



dMeNaHRsay³
düaRkam nominal strength Pn − M n
!> kMlaMgsgát;tamG½kS³ M u = 0 / c = ∞ ¬eRbI ε cu = 0.003 edaysarvaminGacmankarsgát;tamG½kS
l¥tex©aH¦
kMBs;bøúkrgkarsgát; a = 14in.(356mm) ehIykMBs;RbsiT§PaB d = 14 − 2 = 12in.(305mm) . dUcenH
eyIgman Ccn = 0.85 f 'c ba = 0.85 × 6000 ×14 ×14 = 999,600lb(4,446kN )
BIsmIkar 8.5
⎡ ⎛ c − d'⎞ ⎤
T ' sn = A' ps E ps ⎢ε pe − ε cu ⎜ ⎟ + ε ce ⎥
⎣ ⎝ c ⎠ ⎦
(
A' ps = 4 × 0.153 = 0.612in.2 3.95cm 2 )
BIsmIkar 8.7/ sMrab; E ps = 29 ×106 psi (200 ×10 MPa ) / ε
3
pe = 0.0052in. / in. ehIy ε cu = 0.003
dUcenH
⎡ ⎛∞−2⎞ ⎤
T ' sn = 0.612 × 29 ⋅10 6 ⎢0.0052 − 0.003⎜ ⎟ + 0.0005⎥
⎣ ⎝ ∞ ⎠ ⎦
= 0.612 × 29 ⋅10 6 (0.0052 − 0.003 + 0.0005)

= 47,920lb(213kN )



BIsmIkar 8.6
⎡ ⎛ d −c⎞ ⎤
Tsn = A ps E ps ⎢ε pe + ε cu ⎜ ⎟ + ε ce ⎥
⎣ ⎝ c ⎠ ⎦
⎡ ⎛ 12 − ∞ ⎞ ⎤
= 0.612 × 29 ⋅10 6 ⎢0.0052 + 0.003⎜ ⎟ + 0.0005⎥
⎣ ⎝ ∞ ⎠ ⎦
= 47,920lb(213kN )
BIsmIkar 8.2
Pn = Ccn − T ' sn −Tsn = 999,600 − 47,920 − 47,920

= 903,760lb(4,020kN )
BIsmIkar 8.7

⎛ 14 14 ⎞ ⎛ 14 ⎞ ⎛ 14 ⎞
= 999,600⎜ − ⎟ − 47,920⎜ − 2 ⎟ + 47,920⎜12 − ⎟
⎝2 2⎠ ⎝2 ⎠ ⎝ 2⎠

=0
M
e1 = n = 0
Pn
@> kMlaMgTajsUnüenARtg;épÞrgkarTaj/ c = 14in.
0.05( f 'c −4,000)
β1 = 0.85 − = 0.75
1,000

a = β1c = 0.75 × 14 = 10.5in.(267 mm )

Ccn = 0.85 × 6,000 × 14 × 10.5 = 749,700lb(3,335kN )
⎡ ⎛ 14 − 2 ⎞ ⎤
T ' sn = 0.612 × 29 ⋅ 10 6 ⎢0.0052 − 0.003⎜ ⎟ + 0.0005⎥
⎣ ⎝ 14 ⎠ ⎦
= 55,526lb(247 kN )
⎡ ⎛ 12 − 14 ⎞ ⎤
Tsn = 0.612 × 29 ⋅ 10 6 ⎢0.0052 + 0.003⎜ ⎟ + 0.0005⎥
⎣ ⎝ 14 ⎠ ⎦
= 93,557lb(416kN )
Pn = Ccn − T ' sn −Tsn = 749,700 − 55,526 − 93,557

= 600,617lb(2,672kN )

= 1,502,130in. − lb(170kN .m )



= 2.50in.(63.5mm )
1,502,130
e2 =
600,617
#> karBt;begáagsuT§ (pure bending): Pu = 0
edayecalT§iBlrbs;Edkrgkarsgát; A' ps eyIgman
A ps f ps 0.612 × 240,000
a= = = 2.06in.(52.3mm )
0.85 f 'c b 0.85 × 6,000 × 14

= 2.75in.(69.9mm )
2.06
c=
0.75
⎛ a⎞ ⎛ 2.06 ⎞
M n = A ps f ps ⎜ d − ⎟ = 0.612 × 240,000⎜12 − ⎟
⎝ 2⎠ ⎝ 2 ⎠

= 1,611,274in. − lb
1,611,274
e3 = =∞
0
$> Limit strain condition: Pn / M n / e
snμt;fabMErbMrYlrageFobenAkñúg tensile strand Aps RtUvesμInwg incremental strain Δε p
bnÞab;BIrg service load Pe . edayKittMél Δε p ≅ 0.0014 RtUv)anEkERbeday trial and adjustment
nigBIrUbTI 8>8/ ¬RtIekaNdUc¦ eyIg)an
c ε 0.003
= cu =
(d − c ) Δε p 0.0014
dUcenH c = 8.15in.(207mm) .



a = β1c = 0.75 × 8.15 = 6.11in.(155mm )

Ccn = 0.85 × 6,000 × 6.10 × 14 = 435,540lb(1,937kN )
⎡ ⎛ 8.13 − 2 ⎞ ⎤
T ' sn = 0.612 × 29 ⋅ 10 6 ⎢0.0052 − 0.003⎜ ⎟ + 0.0005⎥
⎣ ⎝ 8.13 ⎠ ⎦
= 61,018lb(271kN )
⎡ ⎛ 12 − 8.13 ⎞ ⎤
Tsn = 0.612 × 29 ⋅ 10 − 6 ⎢0.0052 + 0.003⎜ ⎟ + 0.0005⎥
⎣ ⎝ 8.13 ⎠ ⎦
= 126,509lb(563kN )
Limit strain state sMrab; Pn / M n nig e mandUcxageRkam
Pn = 435.540 − 61,018 − 126,509 = 248,013lb(1,103kN )

= 2,047,838in. − lb(272kN .m )

= 8.26in.(210mm )
2,047,838
e4 = e =
248,013
kUGredaensMrab;krNITaMgbYnBIelIkmunCacMnucGegátenAelIdüaRkam interaction Pn − M n . eK
k¾RtUvKNnacMnucepSgeTotpgEdr edIm,ITTYl)andüaRkamsuRkitEdlRKbdNþb;elIkardak;bnÞúkRKb;Ebb
y:ag. Ca]TahrN_ eKRtUvkMNt;cMnucbEnßmrvagkUGredaenenAcenøaHkrNITIBIr nigkrNITIbI edaysnμt;tM
élbEnßménkMBs;G½kSNWt c nigkMNt; Pn / M n nig e sMrab;tMél c Edl)ansnμt;. tarag 8>1 segçb
BItMélénkUGredaenEdleRbIsMrab;sg;düaRkam interaction Pn − M n k¾dUcCadüaRkam interaction
Pu − M u . BIdüaRkam eyIgeXIjfaGredaenénm:Um:g;GtibrmamantMélEk,rnwg M n = 2,047,838in. − lb

dUcenHkarsnμt; cb = 8.15in. KWepÞógpÞat;.



düaRkam design load-moment (P − M ) . sg;düaRkam interaction P − M sMrab;kUGr-
u u

edaen EdlmanrayenAkñúgtarag 8>1. sMrab;CMhan 7 enAkñúgdüaRkam ssrsßitenAkñúg transition zone
Edl c / dt = 6.0 / 12.0 = 0.50 < 0.60 sMrab; limit balanced strain rgkarsgát;
BIsmIkar 8.9 (a)/ φ = 0.23 + (c0/.25 ) = 0.23 + 0..50 = 0.73
dt
0 25

dUcenH Pu = 101.2 ⋅ 103 × 0.73 = 73.7 ⋅103 lb
M u = 1969.9 ⋅ 103 × 0.73 = 1438.0 ⋅ 103 in. − lb
M u3 sMrab;karBt;begáagsuT§ = φM n3 = 0.90 × 1,611,274
= 1,450,147in. − lb(164kN .m )
¬minGaceRbI)an¦
Pu1 = φPn = 0.65 × 903,760 = 587,444lb

ACI Code TamTareGay design axial load strength Gtibrma φPn sMrab; tied prestressed

column minRtUvFMCag 0.80φPn ehIysMrab; spirally reinforced prestressed columns minRtUvFMCag

0.85φPn eT. dUcenH eyIgman

Pu max = 0.82φPn = 0.80 × 587,444

= 469,955lb(2090kN )
Pu 5 = 0.65 × 826,648 = 537,321(2,574kN )



tMélEdlenAsl;énkUGredaenRtUv)ansegçbenAkñúgtarag 8>1. düaRkam interaction sMrab;
nominal strength (Pn − M n ) ehIy design strength (Pu − M u ) RtUv)anbgðajenAkñúgrUbTI 8>9.

6> sßanPaBkMNt;enAeBl)ak;eday Buckling rbs;ssrEvgeRbkugRtaMg
Limit State at Buckling Failure of Slender (long) Prestressed Columns
RbsibebIpleFobrlas; (slenderness ration) rbs;ssrFMCagEdnkMNt;sMrab;ssrxøI Ggát;
rgkarsgát;nwgekagmunnwgQaneTAdl;sßanPaBénkar)ak;edaysMPar³. bMErbMrYlragenAkñúgépÞrgkar
sgát;rbs;ebtugeRkamGMeBI buckling load RtUvtUcCag 0.003in. / in. EdlbgðajenAkñúgrUbTI 8>10.
ssrEbbenHGacCaGgát;RsavEdlrgbnSMkMlaMgtamG½kS nigkarBt; EdleFVIeGayxUcRTg;RTayxag
nigedaybegáItm:Um:g;bEnßmEdlbNþaledaysarT§iBl PΔ Edl P CabnÞúktamG½kS nig Δ CaPaBdab
rbs;rUbragekagrbs;ssrenARtg;muxkat;EdlBicarNa.

eKmanssrRsavEdlrgkMlaMgtamG½kS Pu enARtg;cMNakp©it e . T§iBl buckling begáItm:Um:g;
bEnßm Pu Δ . m:Um:g;enHkat;bnßylT§PaBrbs;bnÞúkBIcMnuc C eTAcMnuc B enAkñúgdüaRkam interaction én
rUbTI 8>10. m:Um:g;srub Pu e + Pu Δ RtUv)antMNagedaycMnuc B enAkñúgdüaRkam ehIyeKGacsikSa



KNnassrsMrab;m:Um:g;FMCagenH b¤sMrab; magnified moment M c dUcssrxøI.
RbEvgRbsiT§PaB klu EdlbgðajenAkñúgrUbTI 8>11 RtUv)aneRbICa modified length rbs;ssr
EdlKitPaBTb;xagcug (end restraint) EdlxusBITMr pinned. klu tMNageGayRbEvgrbs;ssrEdl
manTMr pinned Fmμta Edlman Euler buckling load esμInwgbnÞúkrbs;ssreRkamkarBicarNa. müa:g
vijeTot vaCacMgayrvagcMnucrbt;rbs;Ggát;kñúgTMrg;ekagrbs;va.

tMélrbs;emKuNRbEvgRbsiT§PaBEdlTb;xagcug (end restraint effective length factor) k
ERbRbYlcenøaH 0.5 nig 2.0 GaRs½yeTAnwgRbePTén restraint dUcxageRkam³
cugssrTaMgsgçag pinned/ minmancl½txag k = 1.0
cugssrTaMgsgçag fixed k = 0 .5

cugmçag fixed nigcugmçageTotTMenr k = 2 .0

cugTaMgsgçag fixed/ Gaccl½txag k = 1 .0

RbePTkrNIEdlbgðajBIragekagrbs;ssrsMrab;lkçxNÐcugepSg² nigemKuNRbEvg k RtUv)anbgðaj
enAkñúgrUbTI 8>11.



sMrab;Ggát;enAkñúgeRKOgbgÁúMeRKag end restraint sßitenAcenøaHlkçxNÐ hinged nig fixed. eK
GackMNt;tMél k Cak;EsþgBI Jackson nig Moreland alignment chart enAkñúgrUbTI 8>12. CMnYs
eGay chart TaMgenH eKGaceRbIsmIkarxageRkamEdl)anesñIeLIgenAkñúg ACI Code commentary
sMrab;KNna k ³
!> Braced Compression members³ eKGacykEdnkMNt;rbs;emKuNRbEvgRbsiT§PaBCatMél
tUcCageKénsmIkar
k = 0.7 + 0.05(ψ A +ψ B ) ≤ 1.0 (8.10a)
nig k = 0.85 + 0.05ψ min ≤ 1.0 (8.10b)

Edl ψ A nig ψ B CatMélrbs;cugrbs; ψ enARtg;cugTaMgBIrrbs;ssr ehIy ψ min CatMél
tUcCageKkñúgcMeNamtMélTaMgBIr. ψ CapleFobénPaBrwgRkajrbs;Ggát;rgkarsgát;TaMg
Gs;elIPaB rwgRkajénGgát;rgkarBt;TaMgGs;enAkñúgbøg;enARtg;cugmçagrbs;ssr.
∑ EI / lu columns
ψ= (8.11)
∑ EI / ln beams
Edl lu CaRbEvgminmanTMrrbs;ssr nig ln Ca clear span rbs;Fñwm.
@> Unbraced compression members restrained at both ends³ eKGacKitRbEvgRbsiT§PaB
dUcxageRkam³
sMrab; ψ m < 2
20 −ψ m
k= 1 +ψ m (8.12a)
20
sMrab; ψ m ≥ 2
k = 0.9 1 + ψ m (8.12b)

Edl ψ m CatMélmFümrbs;tMél ψ enARtg;cugTaMgBIrrbs;Ggát;rgkarsgát;.
#> Unbraced compression members hinged at one end³ eKGacKitRbEvgRbsiT§PaB
dUcxageRkam³
k = 2.0 + 0.3ψ (8.13)
Edl ψ CatMélenAxagcugEdlTb; (restrained end).
eKGacykkaMniclPaB (radius of gyration) r = I g / Ag Ca r = 0.3h sMrab;muxkat;ctuekaN
Edl h CaTMhMrbs;muxkat;ssrEdlEkgeTAnwgG½kSénkarBt;. sMrab;muxkatrgVg; eKyk r = 0.25h .



k> karBicarNaBI Buckling Buckling Considerations
eRKagEdlminmankarBRgwgxag (lateral bracing) dUcCag shear walls, diaphragms b¤
diagonal coupling beams manlkçN³rlas; (flexible) CageRKagEdlman lateral braced. Lateral

flexibility GacbgáeGayeRKOgbgÁúMTaMgmYlbMlas;TItamTisedkRKb;RKan; EdleRKOgbgÁúMGac)at;bg;

esßrPaBedaysarm:Um:g;eFVIeGayRkLab; (overturning moment) bEnßmEdlmantMélFM. lkçN³eFVIkar
enHmaneRKaHfñak;enAeBlEdlssrxøIRTkMral.
ACI 318 Code kMNt;viFIbIsMrab;KNnakMlaMgenAelIssrEvg nigGgát;enAkñúgeRKagEdlTb;Tl;

kMlaMgxag (lateral force) EdlbEnßmBIelIbnÞúkTMnaj (vertical gravity load). b:uEnþ sMrab;kardak;bnÞúk
TMnajEdlminman side-sway enaH first-order analysis EdleRbIemKuNbEnßmm:Um:g; (moment magni-
fycation factors) δ ns KWmanlkçN³RKb;RKan;. sMrab;karbnSMkMlaMgTMnaj nig side-sway forces Edl

bgáeGayemanT§iBl P − Δ / viFITaMgbIenaHKW³
(a) kmμviFIkMuBüÚT½rEdleRbI second-order analysis EdlkMNt;TMhMrbs; overturning

moment bEnßmenAkñúgeRKag.

(b) emKuNbEnßmm:Um:g; (moment magnification factore) EdlKNnaedayEp¥kelIeKal

karN_ first-order lateral displacements nig m:as;enABIelInIv:UnImYy².
(c) Moment magnification relationship EdlmanTMrg;RsedogKñaeTAnwgGVIEdlRtUvkarsMrab;

KNna no-sway magnifier δ ns sMrab;ssrenAkñúg braced frame edayeRbI stability
index Q . eKmincaM)ac;kMNt;bMlas;TItamTisedk (horizontal displacement) enAkñúg

viFIenHeT b:uEnþeKRtUvEtKNnam:Um:g;EdlTb;Tl;nwg lateral forces. viFIenHmanlkçN³sμúK
sμaj nigminsUvsuRkit. viFIEdlmanlkçN³suRkitKWviFI (a) EdleRbIkmμviFIkMuBüÚT½rdUcCa
PCA’s Frame Program, STAAD Pro, CSI Sap 2000 nigkmμviFIdéTeTot.

eKmanssrEvgEdlRbQmnwgbnÞúktamG½kS enARtg;cMNakp©it e . T§iBl buckling begáIt
m:Um:g;bEnßm Pu Δ Edl Δ CabMlas;TIxagGtibrmarbs;ssrrgkarsgát;rvag cugTaMgBIreTATItaMgedIm.
m:Um:g;bEnßmenHkat;bnßy load capacity BIcMnuc C eTAcMnuc B enAkñúgdüaRkam interaction enAkñúgrUbTI
8>10. m:Um:g;srub (Pu e + Pu Δ ) RtUv)antMNagedaycMnuc B enAkñúgdüaRkam ehIyssrKYrRtUv)ansikSa
KNnasMrab; magnified moment M c EdlFMCag dUcssrxøI eday first-order analysis Fmμta.



enAkñúgkarviPaKEbbenH m:Um:g; nigkMlaMgtamG½kSenAkñúgeRKagRtUv)anTTYleday classical
elastic procedures. dMeNIrkarviPaKenHminKitBIT§iBlrbs; lateral displacement Δ eTAelIkMlaMg

tamG½kS Pu nigm:Um:g;Bt; M c eT. dUcenH TMnak;TMngrvagbnÞúk nigbMlas;TI nigTMnak;TMngrvagbnÞúk nig
m:Um:g;KWmanragCabnÞat; (linear). RbsinebIeKKitBIT§iBl P − Δ / second-order analysis køayCacM)ac;
CamYynwg nonlinear relationship énbnÞúkCag lateral displacement (deflection) nigm:Um:g;. ACI
318 - 02 Code GnuBaØateGayeRbI first order analysis b¤k¾ second-order analysis sMrab;ssrEdl

man intermediate slenderness nigeGayeRbI second-order analysis sMrab;ssrEvgEdlman
slenderness ratio FMCagesμI 100. viFI ACI Code EdlminKitT§iBl P − Δ RtUv)aneKeGayeQμaHfa

moment magnification method Edlmanerobrab;enAkñúgcMnucxageRkam.

7> viFIm:Um:g;bEnßm³ karviPaKdWeRkTI1
Moment Magnification Method: First-order Analysis
bnÞúktamG½kSemKuN Pu / m:Um:g;emKuN M1 nig M 2 manGMeBIenAcugssr ehIyPaBdabRtUv)an
kMNt;enAkñúgviFIenHedayeRbI elastic first-order analysis CamYynwg lkçN³muxkat;EdlkMNt;eday
KitT§iBlrbs;bnÞúktamG½kS vtþmanrbs;tMbn;EdlmaneRbHtambeNþayRbEvgrbs;Ggát; nigT§iBlén
ry³eBlénkardak;bnÞúk.
dUcEdl)anerobrab;enAkñúgcMnuc 6xagelI nigtamry³rUbTI 8>10/ m:Um:g; M 2 RtUv)anbEnßmeday
magnification factor δ . ssrrgnUvm:Um:g; M 1 nig M 2 enAxagcugrbs;va EdleKKitfa M 2 FMCag M 1 .

bnÞúktamG½kS Pu nigm:Um:g;emKuN M1 nig M 2 RtUv)anTb;edaylkçN³muxkat;EdleRCIserIsedaykar
viPaK Edlrab;bBa©ÚlTaMgtMbn;EdlmaneRbHtambeNþayRbEvg b¤kMBs;rbs;Ggát;rgkarsgát; nigry³
eBlrbs;bnÞúk. CMnYseGaykarKNnaTaMgenH/ ACI 318-02 Code GnuBaØateGayeRbItMélmFümxag
eRkamsMrab;lkçN³rbs;Ggát;enAkñúgeRKOgbgÁúM³
(a) m:UDuleGLasÞic Ec = 33w1.5 f 'c nigsMrab;ebtugEdlmanersIusþg; 5,000 psi < f 'c < 12,000 psi
c

Ec = (40,000 + 1 × 106 )(wc / 145)1.5
(b) m:Um:g;niclPaB

Fñwm 0.35I g

ssr 0.70 I g

CBa¢aMg ¬KμansñameRbH¦ 0.70 I g



CBa¢aMg ¬mansñameRbH¦ 0.35I g

Flate plates nig flat slabs 0.25I g

(c) RkLaépÞ³ 1.0 Ag

(d) kaMniclPaN (radius of gyration) r = 0.3h sMrab;muxkat;ctuekaN Edl h CaTMhMenAkñúgTis

EdlKitesßrPaB b¤ r = 0.25D sMrab;muxkat;rgVg; Edl D CaGgát;p©itrbs;Ggát;rgkarsgát;.
eKKYrEckm:Um:g;niclPaBCamYynwg (1 + β d ) enAeBlEdl sustained lateral load manGMeBI b¤
sMrab;RtYtBinitüesßrPaB Edl β d CaemKuN creep dUcenH
maximum factored sustained axial load
βd =
total factored axial load
eKsnμt;bnÞúkmanGMeBIenARtg;cMNakp©it (e + Δ ) enAkñúgrUbTI 8>10 edIm,IbegáItm:Um:g; M c . pl
eFob M c / M 2 RtUv)aneKeGayeQμaHfa magnification factor δ . dWeRkrbs; magnification GaRs½y
nwgpleFob slenderness klu / r Edl k CaemKuNRbEvgRbsiT§PaB (effective length factor) sMrab;
Ggát;rgkarsgát; ehIyvak¾GaRs½ynwg stiffness enARtg;tMNéncugrbs;Ggát;nImYy².
eKRtYtBinitü magnification factor tamRbePTrbs; magnified moment δM 2 nig δM1 Edl
manGMeBIenARtg;cugelx 2 nigelx 1 rbs;ssr ¬side-sway rbs;eRKagekItmanb¤Gt;¦. eKKYrcMNaM
faenAkñúgkrNIGgát;rgkarsgát;RbQmnwgkarBt;eFobG½kSemTaMgBIrrbs;va eKRtUvKitm:Um:g;eFobG½kS
nImYy²dac;edayELkBIKñaedayQrelI restraint condition EdlRtUvnwgG½kSenaH.

k> Moment Magnification in Non-Sway Frames
enAkñúgkrNIGgát;rgkarsgát;sßitkñúg non-sway frames (braced frame) eKGacykemKuN
RbEvgRbsiT§PaB k = 1.0 Tal;EtkarviPaKeGaytMéltUcCag. enAkñúgkrNIEbbenH eKkMNt;tMél
k edayEp¥kelItMél EI EdlbgðajenAkñúgcMnucxagelI nig monogram enAkñúgrUbTI 8>12.

eKGacminKitBIT§iBl slenderness RbsinebI
klu ⎛M ⎞
≤ 34 − 12⎜ 1 ⎟
⎜M ⎟ (8.14)
r ⎝ 2⎠
klu =RbEvgRbsiT§PaBrvagcMnucrbt; ehIyeKminGacyk [34 − 12(M1 / M 2 )] FMCagEdlkMNt;én
smIkar 8.14 eT. tY (M1 / M 2 ) mantMélviC¢manenAeBlEdlGgát;ekagedaykMeNageTal (single
curvature) ehIyvamantMélGviC¢manenAeBlGgát;ekagedaykMeNagDub (double curvature) ¬emIlrUb



TI 8>12a). RbsinebI non-sway magnification factor Ca δ ns ehIy sway factor δ s = 0 /
magnified moment køayCa
M c = δ ns M 2 (8.15)

Edl δ ns =
Cm
Pu
≥ 1.0 (8.16a)
1−
0.75Pc
π 2 EI
Pc = (8.16b)
(klu )2
Edl Pc Ca Euler buckling load sMrab; pin-ended column.
eKyk stiffness
0.2 Ec I g + Es I se
EI = (8.16c)
1 + βd
0.4 Ec I g
b¤ EI =
1 + βd
Cm = emKuNEdlTak;TgdüaRkamm:Um:g;Cak;EsþgeTAnwg equivalent uniform moment diagram.
sMRab;Ggát;EdlKμan transverse load ¬rgEtbnÞúkxagcug¦.
M1
Cm = 0.6 + ≥ 0.4 (8.17)
M2
Edl M 2 ≤ M1 nig M1 / M 2 > 0 RbsinebIKμancMnucrbt;enAcenøaHcugrbs;ssr rUbTI 8>12 a (single
curvature). sMrab;lkçxNÐdéTeTot Ggát;EbbenHEdlman transverse load enAcenøaHTMr Cm = 1.0 .

tMélGnuBaØatGb,brmarbs; M 2 KW
M 2, min = Pu (0.6 + 0.03h ) (8.18)

Edl h KitCa in. . sMrab;xñat SI M 2,min = Pu (15 + 0.03h) Edl h KitCamIlIEm:Rt. müa:gvijeTot cM
Nakp©itGb,brmaenAkñúgssrEvgKW emin = 0.6 + 0.03h . RbsinebI M 2,min FMCagmU:m:g;Gnuvtþn_ M 2 eK
KYryktMélrbs; Cm enAkñúgsmIkar 8.17 esμInwg 1.0 b¤edayEp¥kelIm:Um:g;cug M1 nig M 2 EdlKNna
Cak;Esþg.
eRKagEdlBRgwgRbqaMgnwg side-sway b¤BRgwgeday shear wall KYrman lateral deflection tUc
Cag hs / 1500 . enAeBlEdl lateral deflection FMCagpleFobenH eKRtUveFVIeGaym:Um:g;bEnßmEdlbgá
eday side sway mantMélGb,brma nigkat;bnßy lateral drift BIeRKag nigBIssr.



x> Moment Magnification in Sway Frames
sMrab;Ggát;rgkarsgát;EdlminmankarBRgwgRbqaMgnwg side sway eKGackMNt;emKuNRbEvgRb
siT§PaB k BItMél EI EdlbgðajenAkñúgcMnuc 7/ b:uEnþtMélrbs;vaminRtUvFMCag 1.0 eT. eKGacminKitBI
T§iBl slenderness RbsinebI
klu
< 22 (8.19)
r
eKKYrbegáInm:Um:g;cug M1 nig M 2 dUcxageRkam
M 1 = M 1ns + δ s M 1s
M 2 = M 2ns + δ s M 2 s (8.20)
edayeKsnμt;fa M 2 > M1 / enaH design moment
M c = M 2ns + δ s M 2 s (8.21)
Edl M 2ns = m:Um:g;cugemKuNenAxagcugrbs;Ggát;rgkarsgát;EdlbNþalBIbnÞúkEdlbgáeGayman
side-sway EdlminsMxan; vaRtUv)anKNnaedayeRbI first-order elastic frame analysis. M 2 s =

m:Um:g;cugemKuNenAxagcugrbs;Ggát;rgkarsgát;EdlbNþalBIbnÞúkEdlbgáeGayman side-sway Edl
sMxan; vaRtUv)anKNnaedayeRbI first-order elastic frame analysis.
Ms
δsM s = ≥ M s ≤ 2.5 (8.22)
∑ Pu
1−
0.75 ∑ Pc
Edl ∑ Pu CaplbUkénbnÞúkbBaÄrTaMgGs;enAkñúgmYyCan; ehIy ∑ Pc CaplbUk Euler buckling load
¬ Pc sMrab; pin-ended column sMrab;ssrEdlTb;nwg sway TaMgGs;enAkñúgmYyCan; [Pc = π 2 EI /(klu )2 ]
BIsmIkar 8.16b¦ CamYynwgtMél EI EdlTTYl)anBIsmIkar 8.16c b¤ d.
enAkñúgkrNIsMrab;EtGgát;rgkarsgát;EtmYyEdlman
lu 35
>
r Pu / f 'c Ag

eKRtUvsikSaKNnaGgát;sMrab;bnÞúktamG½kSemKuN Pu nig magnified moment M c = δ ns M 2 Edl
M 2 enAkñúgkrNIenHKW M 2 = δ ns M 2ns + δ s M 2 s . krNIenHGacekItmanenAkñúgssrEvgEdlrgbnÞúk

tamG½kSFM enAeBlEdlm:Um:g;GtibrmaGacekItmanenAcenøaHcugrbs;ssr dUcenHm:Um:g;cugmincaM)ac;Ca
m:Um:g;GtibrmaeT.



!> Moment Magnification in sway frames using a stability index Q
enAkñúgviFIenH ¬viFI c enAkñúgcMnuc 6>k¦ code GnuBaØateGaysnμt;ssrenAkñúgeRKOgbgÁúMEdl
BRgwgCa non-sway RbsinebIkarekIneLIgénbnÞúk nigm:Um:g;EdlbNþalBI second-order effect minFM
Cag 5% én first-order end moment. eKGacBicarNaCan;enAkñúgeRKOgbgÁúMCa non-sway RbsinebI
stability index Q enAkñúgsmIkarxageRkamenHminFMCag 0.05
∑ Pu Δ o
Q= (8.23a)
Vu lc
Edl ∑ Pu = bnÞúkbBaÄrsrubenAkñúgmYyCan;
Vu = kMlaMgkat;tamCan; (story shear)

Δ o = first-order relative deflection rvagxagelI nigxageRkamrbs;Can;EdlbNþalBI Vu

lc = RbEvgrbs;Ggát;rgkarsgát;enAkñúgeRKagEdlvagBIG½kSrbs;tMN

Non-sway magnification factor edayeRbItY Q KW
1
δs = ≥ 1.0 (8.23b)
1− Q
enAeBl Q FMCag 0.05 eKRtUvbnþkarKNnaeTA second-order analysis tamry³kareRbIR)as;kmμviFI
kMuBüÚT½r. karviPaKedaykMuBüÚT½rEbbenHGaceGayeKKNnatMélsarcuHsareLIgrbs;m:Um:g; nigtMél sway
Δ o EdlbNþalBIT§iBl P − Δ manPaBsuRkit nigelOn.

eKKYrcMNaMfa stability index Q method manlkçN³sμúKsμaj nigsuRkitsMrab;KNnaT§iBl
P − Δ elIm:Um:g;enARtg;tMNssrenAkúñgeRKagEdlBRgwg.

Casegçb moment magnification method EdlbegáIteLIgdMbUgsMrab; prismatic column eFIV
kar )anl¥CamYynwgssrEdlman slenderness ratio klu / r tUcCag 100 CaBiessRbsinebIeRKag
RtUv)anBRgwg. enAkñúgkrNI unbraced frames Edlman slenderness ration Rbhak;RbEhlKña eKKYr
KitbBa©ÚlT§iBl P − Δ eTAelIm:Um:g; nigPaBdabtamry³ second-order analysis edIm,ITTYl)an
lT§plkan;EtsuRkitCag. karsikSaviPaKGac
!> Gnuvtþ first-order analysis Edl lateral load ¬BI hi BIrUbTI 8>13¦ RtUv)anbUkbnþeday
∑ Pu Δ l enAkñúgCMuénkarKNnamþg² ehIycat;TuklT§plcugeRkayCa second-order result

b¤
@> eRbIkmμviFIkMuBüÚT½r second-order analysis BitR)akd EdlenAkñúgenaHeKeRbIkarkat;bnßy
relative side-sway resistance enAkñúg global stiffness matrix sMrab;Ggát;TaMgBak;B½n§.



8> karviPaKeRKagdWeRkTIBIr nigT§iBl P−Δ
Second-Order Frames Analysis and the P − Δ Effects
Second-order analysis CakarviPaKeRKagEdlrYmbBa©ÚlT§iBlkMlaMgkñúgEdl)anBI lateral dis-
placement (deflection) rbs;ssr. enAeBleKGnuvtþkarviPaKEbbenHedIm,IkMNt; δ s M s enAkñúg non-

braced frame eKRtUvKNnaPaBdabedayEp¥kelI fully cracked section CamYynwgtMél stiffness EI

Edlkat;bnßy. tMélRbhak;RbEhldUckareRbI first-order analysis eRcInCMu ehIykarviPaKGaceFVIeGay
prismatic section kan;EtRbesIreLIg. b:uEnþkarviPaKKYrepÞógpÞat;faersIusþg;EdlrMBwgTukrbs;Ggát;rgkar

sgát;éneRKageRKOgbgÁúMsßitenAkñúgcenøaH 15% énlT§plsMrab;ssrenAkñúgeRKOgbgÁúMebtugGarem:min
kMNt;. lT§plCaragFrNImaRtrbs;Ggát;EdlRtUvviPaKRtUvRsedogKñanwgragFrNImaRtrbs;Ggát;Edl
RtUvsagsg;. RbsinebIGgát;enAkñúgeRKOgbgÁúMcugeRkaymanTMhMmuxkat;xusBIGVIEdlva)ansnμt;kñúgkar
viPaK 10% eKRtUvGnuvtþkarKNnaCafμI.
Second-order analysis CaviFIsarcuHsareLIgénT§iBl P − Δ eTAelIssrRsav EdlrYmbBa©Úl

TaMg shear deformation. dUcenH eKGaceRbIkmμviFIkMuBüÚT½rRbesIrCakarKNnaedayédkñúgkarsikSa
KNnassrRsavrbs;eRKag. b:uEnþ ssrebtugPaKeRcInenAkñúgeRKagsMNg;minRtUvkarkarviPaKEbbenH
eT edaysarpleFob (klu / r ) eRcInEttUcCag 100 .

BicarNassrenAcenøaHBIrCan;KW (i − 1) nig (i ) enAkñúgeRKagEdlbgðajenAkñúgrUbTI 8>13.
snμt;fa lateral displacement Gtibrma b¤ drift enARtg;cugxagelIrbs;cugkMBUlrbs;ssrenAkñúgeRKag



KW xmax nigsnμt;fakMBs;srubrbs;GKarKW hs . Lateral displacement b¤ drift d¾FMrbs;GKarCan;xagelI
bgáeGaymansñameRbHdl;CBa¢aMgdæ b¤kargarbegðIyxagkñúg. EdnkMNt;én lateral deflection Gtibrma
KW hs / 500 . dUcenH karsnμt;d¾l¥KWkareRCIserIs xmax sßitenAcenøaH hs / 350 eTA hs / 500 EdlKitfa
CaFmμta fully braced frame manpleFob drift xmax Gtibrma enAelIkMBs;eRKag hs tUcCag 1 / 1,500 .
RbsinebI xi Ca drift enARtg;nIv:UCan; i nig yi CakMBs;rbs;ssrcenøaHCan; (i − 1) nig (i ) enAkñúg
rUbTI 8>13 a, eKGacsnμt;fa horizontal drift sMrab;Can;KWsmamaRteTAnwgkaer:énpleFobénkMBs; hi
rbs;Can; nigkMBs;srub hs rbs;eRKagTaMgmUl.
2
⎛h ⎞
xi = xmax ⎜ i ⎟
⎜h ⎟ (8.24)
⎝ s⎠
eKGacsegçbdMeNIrkarKNnadUcxageRkam³
!> eRCIserIsmuxkat;rbs;eRKag nig stiffness EI rbs;vaedaytMélRbhak;RbEhl
@> KNna drift (lateral deflection Δi ¦ nig ultimate load Pu,i enARtg;tMN i = 1,..., n rUbTI
8>13.
#> KNnarkkMlaMgtamTisedksmmUl H i BI H i = Pi Δi / hi ¬rUbTI 8>13 b¦.
$> bEnßmtMélEdlTTYl)anenAkñúgCMhan # eTAelI lateral load Cak;EsþgenAelIeRKag.
%> Gnuvtþ frame analysis edayeRbI kmμviFIkMuBüÚT½rEdlsmRsb.
^> Iterative computer program EdleRbI stiffness EI pþl;eGay Δi EdlRtUveRbobeFobCa
mYynwgtMélGnuBaØat xi .
&> RbsinebItMélrbs; Δi TaMgGs; ≤ tMélrbs; xi TaMgGs; enaHeKGacTTYlykdMeNaHRsay
nigkarsikSaKNnaCadMeNaHRsay second-order. RbsinebImindUecñaHeT eKRtUv run kmμviFI
edaybEnßmcMnYnCMuCamYynwg modified stiffness rhUtdl;eKTTYl)anlT§plEdleKcg;)an.
eKGaceRbIkmμviFIkMuBüÚT½repSg²edIm,IKitbBa©ÚlT§iBl P − Δ enAkñúgeRKag side-sway. kmμviFI
TaMgenaHrYmman Strudel, PCA Frame, STAAD Pro ,or CSI Sap 2000 nigkμviFIdéTeTot.

9> Operational Procedure and Flowchart for the Design of Slender Column

!> kMNt;faetIeRKagman side-sway FMb¤Gt;. RbsinebIvaman side-sway FM eRbI magnify-
cation factors δ ns nig δ s . RbsinebIeKecal side-sway, snμt;fa δ s = 0 . bnÞab;mk



snμt;muxkat; rYcKNnacMNakp©itedayeRbIm:Um:gcugEdlFMCageK ehIyRtYtBinitüemIlfava
;
FMCagcMNakp©itGnuBaØatGtibrmab¤k¾Gt;
≥ (0.6 + 0.03h )in.
M2
Pu



@> KNna ψ A nigψ B edayeRbIsmIkar 8.12 b¤ 8.13 nigbnÞab;mkTTYl)an k edayeRbIrUbTI
8>12 b¤smIkar 8.13. KNna klu / r nigkMNt;favaCassrxøI b¤ssrEvg. RbsinebIssr
CassrEvg ehIy klu / r < 100 KNna magnified moment M c . bnÞab;mk edayeRbI
tMélEdlTTYl)an KNnacMNakp©itsmmUl edIm,IKNnassrCassrxøI. RbsinebI klu / r
> 100 Gnuvtþ second-order analysis.

#> KNnassrxøIsmmUl. Flowchart enAkñúgrUbTI 8>14 bgðajBICMhanénkarKNna. smIkar
caM)ac;manenAkñúgcMnuc 2 nigenAkñúg flowchart.

10> sikSaKNnassreRbkugRtaMgEvg
Design of Slender (Long) Prestressed Column
]TahrN_ 8>2³ Square tied prestressed bonded co,umn CaEpñkrbs;eRKagGKar 5 × 3bays Edlrg
nUvkarBt;tamG½kSmYy (uniaxial bending). Clear height rbs;vaKW lu = 15 ft (4.54in.) ehIyvamin
RtUv)anBRgwgRbqaMgnwg sidesway eT. bnÞúkxageRkAemKuN Pu = 300,000lb(1,334kN ) nigm:Um:g;cug
emKuNKW M1 = 425,000in. − lb(48kN .m) nig M 2 = 750,000in. − lb(84.8kN.m) . sikSaKNnamux
kat;ssr nigEdkBRgwgcaM)ac;sMrab;lkçxNÐBIrxageRkam³
!> KitEtbnÞúkTMnajb:ueNÑaH edaysnμt;ecal lateral sidesway EdlbNþalBIxül;
@> ]bma sidesway wind effect bgáeGaymanbnÞúkemKuN Pu = 24,000lb(107kN ) nigm:Um:g;
emKuN M u = 220,000lb(24.9kN .m) . bnÞúkkñúgmYyCan;énssrTaMgGs;enARtg;nIv:UenaHKW
∑ Pu = 4.5 ⋅ 106 lb(20 ⋅ 103 kN ) nig ∑ Pc = 31.0 ⋅ 106 lb(138 ⋅ 103 kN ).

eRbI 270-K stress-relieved prestressing strand Ggát;p©it 1 / 2in. . eKeGayTinñn½ydUcxag
eRkam³
β d = 0.4

ψ A = 1 .0
ψ B = 2 .0
f 'c = 6,000 psi (41.4MPa )
f pu = 270,000 psi (1,862MPa )

f pe = 150,000 psi (1,034MPa )

(
E ps = 28 ⋅ 106 psi 200 ⋅ 103 MPa )



ε cu = 0.003in. / in. enAeBl)ak;
ε ce = 0.0005in. / in. enAeBl Pe eFVIGMeBIelImuxkat;
d ' = 2in.(50.8mm )
f y = 60,000 psi (414MPa )sMrab;Edkkg
düaRkamTMnak;TMngrvagkugRtaMg nigbMErbMrYlragrbs;EdkeRbkugRtaMgKWdUcenAkñúgrUbTI 8>7.
dMeNaHRsay³
BIdüaRkam stress-strain énrUbTI 8>7 EdlRtUvKñanwg f pe = 150,000 psi .
ε pe = 0.0052in. / in.
RsedogKña ε py ≅ 0.012in. / in. BIrUbdUcKña EdlRtUvKñanwg f py = 260,000 psi .
!> sMrab;EtbnÞúkTMnaj (gravity load only)
RtYtBinitüsMrab; no sidesway nigcMNakp©itGtibrma ¬CMhan !¦
edaysareRKagminman sidesway FM/ eyIgyk M 2ns Ca M 2 TaMgmUl ehIyeKyk magnify-
cation factor sMrab; sidesway δ s = 0 enAkñúgsmIkar 8.15. tamry³ trial and adjustment, eyIgGac

snμt;muxkat;ssr nigeFVIkarsikSaviPaK. dUcenH eyIgsakl,gmuxkat; 15in. × 15in.(381mm × 381mm)
dUcbgðajenAkñúgrUbTI 8>15 (a) ehIyeyIg)an
cMNakp©itCak;Esþg = MP2ns = 300,,000 = 2.50in.(63.5mm)
750 000
u

cMNakp©itGnuBaØatGb,rbma = 0.6 + 0.03h = 0.6 + 0.03 × 15
= 1.05in.(2.67 mm ) < 2.50in.
dUcenH yk M 2ns = 750,000in. − lb Cam:Um:g;EdlFMCageKkñúgcMeNam M1 nig M 2 enAelIssr.



KNnacMNakp©itEdlRtUveRbIsMrab;ssrxøIsmmUl ¬CMhan @¦
BI chart enAkñúgrUbTI 8.12 (b)/ k = 1.45 nig slenderness ration KW
klu 1.45 × 15 × 12
= = 58.0
r 0.3 × 15
edaysar 58 > 22 Et < 100 eRbI moment magnification method. eyIg)an
Ec = 33w1.5 f 'c = 33 × 1451.5 6,000 = 4.46 × 106 psi (32 ⋅ 103 MPa )
15(15)3
Ig = = 4,218.8in.4
12
Ec I g / 2.5 4.46 ⋅ 106 × 4,218.3 1
EI = = ×
1 + βd 2.5 1 − 0.4

= 5.34 ⋅ 109 lb. − in 2
(klu )2 = (1.45 × 15 × 12)2 = 68.1× 103 in.2
π 2 EI π 2 × 5.34 ⋅ 109
dUcenH Pc = Euler buckling load = =
(klu )2 68.1 ⋅ 103

= 773,132lb = 773.1kips(3,439kN )
Cm = 1.0 sMrab; nonbraced column. snμt; φ = 0.65 . enaHeyIgman
Cm 1.0
Moment magnifier δ ns = = = 2.07
Pu 300,000
1− 1−
0.75Pc 0.75 × 773,132
Design moment M c = δ ns M 2 ns = 2.07 × 750,000

= 1,552,500in. − lb(184kN .m )

Pn EdlRtUvkar = Pu = 300.,65 = 461,538lb(2053kN )
φ 0
000

M n EdlRtUvkar = = 2,388.462in. − ln (291kN .m )
1,552,500
0.65
cMNakp©it e = 2461,538 = 5.18in.(131mm)
,388,462

sikSaKNnassrxøIsmmUl (equivalent nonslender column) ¬CMhan #¦
ssrsmmYlRtUvRT nominal axial load Pn = 461,538lb nig nominal uniaxial moment
Gb,brma M n = 2,388,462in. − lb .
edIm,IsikSaKNna equivalent nonslender column, eyIgsikSaviPaKmuxkat;ssrEdl)ansnμt;
15in. × 15in. EdleRbI 7-wire stress-relieved strands Ggát;p©it 1 / 2in. cMnYn 5 enAelIépÞnImYy²rbs;mux

TaMgBIrEdlRsbeTAnwgG½kSNWt dUceXIjenAkñúg]TahrN_ 8>1. enaH



(
Aps = A' ps = 5 × 0.153 = 0.765in.2 4.94cm 2 )
Balanced Limit Strain Failure Condition
d = h − 2 = 15 − 2 = 13in.(330mm )
edayeRbobeFobCamYynwg]TahrN_ 8>1 nigedayeRbI trial and adjustment enaHkMBs;G½kSNWtEdl
smRsbsMrab; balanced condition KYrmantMél cb = 8.3in.(211mm) . enaH ab = β1 × cb = 0.75 × 8.3
= 6.23in.(158mm ) .

BIrUbTI 8>3
Ccn = 0.85 × 6,000 × 15 × 6.23 = 476,595lb(2,119kN )
BIsmIkar 8.5
⎡ ⎛ 8.3 − 2 ⎞ ⎤
T 'sn = 0.765 × 28 ⋅ 106 ⎢0.0052 − 0.003⎜ ⎟ + 0.0005⎥
⎣ ⎝ 8.3 ⎠ ⎦
= 73,318lb(385kN )
BIsmIkar 8.6
⎡ ⎛ 13 − 8.3 ⎞ ⎤
Tsn = 0.765 × 28 ⋅ 106 ⎢0.0052 + 0.003⎜ ⎟ + 0.0005⎥
⎣ ⎝ 8.3 ⎠ ⎦
= 158,482lb(704kN )
BIsmIkar 8.2, sMrabk;krNI “balanced” strain limit ¬ c / dt = 0.60 ¦
Pnb = Ccn − T 'sn −Tsn

= 476,595 − 73,318 − 158,482

= 229,310lb(1,020kN )
BIsmIkar 8.7/ sMrabk;krNI “balanced” strain limit ¬ c / dt = 0.60 ¦
⎛ 15 6.23 ⎞ ⎛ 15 ⎞ ⎛ 15 ⎞
M nb = 476,595⎜ − ⎟ = 73,318⎜ − 2 ⎟ + 158,482⎜13 − ⎟
⎝2 2 ⎠ ⎝2 ⎠ ⎝ 2⎠

= 2,103,124in. − lb(237.7kN .m )
M
eb = nb =
Pnb
2,103,124
229,310
Cak;Esþg = 5.18in.
= 9.17in.(233mm ) > e

ssreRbkugRtaMgEdlrgbnÞúkEdlmancMNakp©ittUcnwg)ak;edaykarsgát;. ehIy φ = 0.65 dUckarsnμt.
;
snμt;kMBs;G½kSNWt c = 12in.
a = β1c = 0.75 × 12 = 9.0in.
BIsmIkar 8.1a



Ccn = 0.85 f 'c ba = 0.85 × 6,000 × 15 × 9 = 688,500lb
BIsmIkar 8.5
⎡ ⎛ c − d' ⎞ ⎤
T 'sn = A' ps E ps ⎢ε pe − ε cu ⎜ ⎟ + ε ce ⎥
⎣ ⎝ c ⎠ ⎦
⎡ ⎛ 12 − 2 ⎞ ⎤
= 0.765 × 28 ⋅ 106 ⎢0.0052 − 0.003⎜ ⎟ + 0.0005⎥
⎣ ⎝ 12 ⎠ ⎦
= 68,544lb
BIsmIkar 8.6
⎡ ⎛d −c⎞ ⎤
Tsn = Aps E ps ⎢ε pe + ε cu ⎜ ⎟ + ε ce ⎥
⎣ ⎝ c ⎠ ⎦
⎡ ⎛ 13 − 12 ⎞ ⎤
= 0.765 × 28 ⋅ 106 ⎢0.0052 + 0.003⎜ ⎟ + 0.0005⎥
⎣ ⎝ 12 ⎠ ⎦
= 127,449lb
BIsmIkar 8.2
Pn = Ccn − T ' sn −Tsn
EdlGacman = 688,500 − 68,544 − 127,449
Pn

= 492,507lb > Pn EdlRtUvkar = 461,538lb

dUcenH eyIgbnþeTA trial-and-adjustment CMuTIBIr
snμt;kMBs;G½kSNWt c = 11.2in.
a = β1c = 0.75 × 11.2 = 8.4in.
Ccn = 0.85 f 'c ba = 0.85 × 6,000 × 15 × 8.4 = 642.600lb
⎡ ⎛ 11.2 − 2 ⎞ ⎤
T 'sn = 0.765 × 28 ⋅ 106 ⎢0.0052 − 0.003⎜ ⎟ + 0.0005⎥
⎣ ⎝ 11.2 ⎠ ⎦
= 69,309lb
⎡ ⎛ 13 − 11.2 ⎞ ⎤
Tsn = 0.765 × 28 ⋅ 106 ⎢0.0052 + 0.003⎜ ⎟ + 0.0005⎥
⎣ ⎝ 11.2 ⎠ ⎦
= 132,421lb
EdlGacman = 642,600 − 69,309 − 132,421
Pn

= 440,870lb xN³Edl Pn EdlRtUvkar = 461,538lb O.K.

edaysar moment capacity FMCag M n EdlRtUvkar. karekIneLIgd¾tictYcbMputrbs;kMBs;muxkat;Gac
ykQñHelIPaBxusKñatictYcrvag Pn EdlRtUvkar nig Pn EdlGacman.



BIsmIkar 8.7

⎛ 15 8.4 ⎞ ⎛ 15 ⎞ ⎛ 15 ⎞
= 642,600⎜ − ⎟ − 69,309⎜ − 2 ⎟ + 132,421⎜13 − ⎟
⎝2 2 ⎠ ⎝2 ⎠ ⎝ 2⎠
= 2,467,696in. − lb > 2,338,462in. − lb(678.8kN .m > 250kN .m ) O.K.

e=
2,467,696
448,870
Cak;Esþg
= 5.5 ≈ e = 5.18in.TTYlyk)an
dUcenH TTYlykmuxkat; 15in. × 15in. CamYynwg 7-wire stress-relieved 270-K strand Ggát;p©it 1 / 2in.
cMnYn 5 enARCugnImYy²énépÞEdlenARsbnwgG½kSNWt. bnÞab;mk sikSaKNnaEdkkg (transverse tie)
EdlcaM)ac;.
@> sMrab;bnÞúkTMnaj nigbnÞúkxül; (gravity and wind loading [sidesway])
BIdMeNaHRsaycMnucTI ! eyIgman
Pe = 773,132lb nig U = 1.2 D + 1.0 L + 1.6W . ehIy U = 0.9 D + 1.6W ¬minlub¦.

Pu = (300,000 + 24,000) = 324,000lb / M 2b = 750,000in. − lb nig M 2t = 220,000in. − lb .

RtYtBinitüfa gravity moment RtUvkarm:Um:g;bEnßmb¤Gt;
35 35 l
= = 71.4 > u = 40
Pu 324,000 r
f 'c At 6,000 × 225

dUcenH gravity moment M 2b minRtUvkarm:Um:g;bEnßmeT
BIsmIkar 8.16(b)
1.0 1.0
δs = = = 1.24
∑ Pu 4.5 ⋅ 106
1− 1−
0.75 ∑ Pc 0.75 × 31.0 ⋅ 106
BIsmIkar 8.15
M c = M 2ns + δ s M 2 s = 750,000 + 1.24 × 220,000

= 1,022,800in. − lb
PnEdlRtUvkar =
324,000
0.65
= 498,462lb

MnEdlRtUvkar =
1,022,800
0.65
= 1,573,538in. − lb

cMNakp©it e=
1,573,538
498,462
= 3.16in. < eb = 9.17in. < e Cak;Esþg = 5.18in.



dUcenH vaekItman initial compression failure. ehIy M n = 1,573,538in. − lb
< M n = 2,388,462in. − lb enAkñúgkrNITI !.

lkçxNÐsMrab;krNITI @ Edlman sidesway Gt;lub/ edaysarEtvaenAEt)ak;edaysarkar
sgát;dEdl. m:Um:g;tMrUvkar M n mantMéltUcCagm:Um:g;sMrab;krNITI ! ehIycMNakp©itk¾mantMéltUcCag
krNITI ! Edr. dUcenH TTYlykmuxkat;dUckrNITI ! KW 15in. × 15in. CamYynwg 7-wire stress-relieved
270-K strand Ggát;p©it 1 / 2in. cMnYn 5 enARCugnImYy²énépÞEdlenARsbnwgG½kSNWt.

11> Ggát;rgkarsgát;rgkarBt;BIrTis
Compression Members in Biaxial Bending
k> Exact Method of Analysis



ssrEdlenAkac;RCugrbs;GKarCaGgát;rgkarsgát;EdlRbQmnwgkarBt;BIrTisKWeFobnwgG½kS
x nigG½kS y dUcbgðajenAkñúgrUbTI 8>16. m:Um:g;Bt;BIrTisenHekItmanedaysarbnÞúkminesμIKñaenAelI

ElVgEk,r ehIyCaBiessvaekItmanenAelIssrs<an (bridge pier). ssrEbbenHrgnUvm:Um:g; M xx eFob
nwgG½kS x EdlbegáItcMNakp©it e y nigrgnUvm:Um:g; M yy eFobnwgG½kS y EdlbegáItcMNakp©it ex . dUc
enHG½kSNWtsßitenAelIbnÞat;eRTtEdlpÁúM)anmMu θ CamYynwgbnÞat;edk.
mMu θ GaRs½ynwg interaction énm:Um:g;Bt;eFobG½kSTaMgBIr CamYynwgTMhMénbnÞúksrub Pu . Rk-
LaépÞrgkarsgát; (compressive area) enAkñúgmuxkat;ssrGacmanTMrg;NamYydUcbgðajenAkúñgrUbTI
8>16 (c). edaysarssrEbbenHRtUv)ankMNt;BIeKalkarN_TImYy eKGnuBaØateGayeRbIviFIsaRsþ
trial-and-adjustment enAeBlEdl compatibility of strain RtUv)anrkSaenARtg;RKb;nIv:UTaMgGs;rbs;

EdkBRgwg. eKRtUvkarkarKNnabEnßmeTot edaysarTItaMgrbs; bøg;G½kSNWteRTt nigTMrg;rbs;RkLa-
épÞrgkarsgát;rbs;ebtugGacmanTMrg;bYnxusKña.
rUbTI 8>17 bgðajBIkarBRgaybMErbMrYlrageFob nigbgðajBIkMlaMgenAelImuxkat;ssrctuekaN
EdlrgbnÞúkBIrG½kS. Gc CaTIRbCMuTMgn;RkLaépÞsgát;rbs;ebtug EdlmankUGredaen xc nig yc BIG½kS
NWttamG½kS x nig y erogKña. Gst CaTItaMgpÁÜb (resultant position) rbs;kMlaMgEdkenAkñúgRkLaépÞ
rgkarTajEdlmanTItaMgkUGredaen xst nig yst BIG½kSNWttamG½kS x nig y erogKña. BIsmIkarlMnwgén
kMlaMgxagkñúg nigxageRkA
Pn = 0.85 f 'c Ac + Fsc − Fst (8.25)
Edl Ac =RkLaépÞéntMbn;sgát;EdlRKbdNþb;edaybøúkkugRtaMgctuekaN
Fsc = kMlaMgpÁÜbrbs;Edksgát; (∑ A's f sc )

Fst = kMlaMgpÁÜbrbs;EdkTaj (∑ As f st )

dUcKña BIsmIkarlMnwgénm:Umg;xagkñúg nigm:Um:g;xageRkA
:
Pn ex = 0.85 f 'c Ac xc + Fsc xsc + Fst xst (8.26a)

Pn e y = 0.85 f 'c Ac yc + Fsc y sc + Fst y st (8.26b)

eKRtUvsnμt;TItaMgrbs;G½kSNWtenAkñúgkarsakl,gnImYy² ehIykugRtaMgEdlKNnaenAkúñgEdkBRgwg
nImYy²mansmIkardUcxageRkam
si
f si = Esε si = Ecε c < fy (8.27)
c



x> Load Contour Method of Analysis
viFIEdlpþl;dMeNaHRsayy:agelOnCakarsikSaKNnassrsMrab;plbUkviucT½rén M xx nig M yy
ehIyeRbI circular reinforcing cage enAkñúgmuxkat;kaer:sMrab;ssrenARtg;kac;RCug. b:uEnþ viFIsaRsþmin
pþl;lkçN³esdækic©enAkñúgkrNIPaKeRcIneT. viFIsikSaKNnaepSgeTotEdlepÞógpÞat;edaykarBiesaFKW
karbMElgm:Um:g;BIrTiseGayeTACam:Um:g;mYyTissmmUl )equivalent uniaxial moment) nigcMNakp©it
mYyTissmmUl (equivalent uniaxial eccentricity). bnÞab;mk eKGacsikSaKNnamuxkat;sMrab;kar
Bt;mYyTis ¬dUckarerobrab;BIxagelIkñúgemeronenH¦ edIm,ITb;nwgm:Um:g;Bt;BIrTisemKuNCak;Esþg.
viFIEbbenHBicarNa failure surface CMnYseGay failure planes ehIyCaTUeTAeKeGayeQμaHfa
Bresler-Parme contour method. viFIenHkat; three-dimensional failure surfaces enAkñúgrUbTI 8>18

Rtg;tMélefr Pn edIm,ITTYl)an interaction plane EdlTak;Tgnwg M nx nig M ny . müa:gvijeTot
contour surface S CaépÞekagEdlrYmbBa©ÚlnUvRKYsarrbs;ExSekag EdleKeGayeQμaHfa load contour.

smIkarKμanxñatTUeTA (general nondimensional equation) sMrab; load contour eRkamGMeBI
bnÞúkefr Pn KW
α1 α2
⎛ M nx ⎞ ⎛ M ny ⎞
⎜
⎜ ⎟
⎟ +⎜ ⎟ = 1.0 (8.28)
⎝ M ox ⎠ ⎜ M oy ⎟
⎝ ⎠
Edl M nx = Pn e y



M ny = Pn ex

M ox = M nx Rtg;bnÞúktamG½kS Pn Edl M ny b¤ ex = 0
M oy = M ny Rtg;bnÞúktamG½kS Pn Edl M nx b¤ e y = 0

m:Um:g; M ox nig M oy CaersIusþg;m:Um:g;EdlTb;Tl;smmUltMrUvkar (required equivalent resis-
ting moment strength) eFobG½kS x nigG½kS y erogKña Edl α1 nig α 2 CaniTsSnþEdlGaRs½ynwgrag

FrNImaRtmuxkat; PaKryEdk TItaMgrbs;Edk nigkugRtaMgsMPar³ f 'c nig f y .

eKGacsMrYlsmIkar 8>28 edayeRbIniTsSnþFmμta nigedaybBa©ÚlemKuN β sMrab;tMélbnÞúk
tamG½kSBiess Pn mYy EdlpleFob M nx / M ny KYrmantMéldUcKñanwgpleFob M ox / M oy . kar
sMrYlEbbenHnaMeGayeK)an
α α
⎛ M nx ⎞ ⎛M ⎞
⎜
⎜M ⎟ ⎟ + ⎜ ny ⎟ = 1.0 (8.29)
⎝ ox ⎠ ⎜ M oy ⎟
⎝ ⎠
Edl α = log 0.5 / log β . rUbTI 8>19 bgðajBIdüaRkam contour ABC BIsmIkar 8.27.
sMrab;karsikSaKNna/ eKKitExS contour CabnÞat;Rtg; AB nig BC edaytMélRbhak;RbEhl/
ehIyeKGacsMrYlsmIkar 8.29 CaBIrkrNI³



!> sMrab; AB enAeBlEdl M ny / M oy < M nx / M ox
M nx M ny ⎡1 − β ⎤
+ = 1 .0
M ox M oy ⎢ β ⎥
(8.30a)
⎣ ⎦
@> sMrab; BC enAeBlEdl M ny / M oy > M nx / M ox
M ny M nx ⎡1 − β ⎤
+ = 1 .0
M ox ⎢ β ⎥
(8.30b)
M oy ⎣ ⎦
enAkñúgsmIkarTaMgBIrxagelIenH ersIusþg;m:Um:g;tamG½kSmYysmmUlEdllubCak;Esþg (actual control-
ling equivalent uniaxial moment strength) M oxn b¤ M oyn y:agehacNas;RtUvsmmUleTAnwg

required controlling moment strength M ox nig M oy énmuxkat;ssrEdleRCIserIs.

sMrab;muxkat;ctuekaNEdleKBRgayEdkedaybrimaNesμIKñaRKb;RCugTaMgGs;rbs;muxkat;ssr
enaHeKGacykpleFob M oy / M ox RbEhlesμInwg b / h . enAkñúgkrNIenH eKGacsMrYlsmIkar 8.30
dUcxageRkam
!> sMrab; M ny > b
M h nx


b 1− β
M ny + M nx ≅ M oy (8.31a)
h β

@> sMrab; M ny ≤ b
M h
nx
h 1− β
M nx + M ny ≅ M ox (8.31b)
b β
Controlling required moment strength M ox b¤ M oy sMrab;karsikSaKNnamuxkat;KWCatMélEdlFM
CageKkñúgcMeNamtMélTaMgBIrEdlkMNt;enAkñúgsmIkar 8.31.
eKeRbIdüaRkamenAkñúgrUbTI 8>20 kñúgkareRCIserIs β kñúgkarviPaK nigsikSaKNnassr. Cakar
Bit eKGacniyayfa modified load-contour enAkñúgsmIkar 8.31 CaviFIsMrab;kMNt; equivalent
required moment strength M ox nig M oy sMrab;sikSaKNnassr RbsinebIvargkMlaMgtamTismYy.



K> Step-by-Step Operational Procedure for the Design of Biaxially Loaded Columns
eKGaceRbICMhanxageRkamCaeKalkarN_ENnaMsMrab;sikSaKNnassrEdlrgkarBt;tamTis x
nigTis y . viFIsaRsþsnμt;RkLaépÞEdkenARKb;RCugTaMgbYnrbs;ssrmanbrimaNesμIKña.
!> KNna uniaxial bending moment edaysnμt;cMnYnEdkenAelIRCugnImYy²rbs;ssresμIKña.
snμt;tMélrbs; interaction contour factor β enAcenøaH 0.50 nig 0.70 nigpleFobrbs;
h / b . pleFobenHGacmantMélRbhak;RbEhlnwg M nx / M ny . edayeRbIsmIkar 8.31

kMNt; equivalent required uniaxial moment M ox b¤ M oy . RbsinebI M nx FMCag M ny
yk M ox sMrab;karKNna nigpÞúymkvij.
@> snμt;muxkat;sMrab;ssr nigpleFobEdk ρ = ρ ' ≅ 0.01 eTA 0.02 enAelIRCugnImYy²rbs;
RCugTaMgBIrEdlRsbnwgG½kSénkarBt;rbs; equivalent moment NaEdlFMCag. bnÞab;mk
eRCIserIsmuxkat;dMbUgrbs;Edk nigepÞógpÞat;lT§PaB Pn énmuxkat;ssrEdlsnμt;. sMrab;kar
sikSaKNnaEdlmanlçN³eBjelj eKeRbIbrimaNEdkbeNþaydUcKñaenAelIRCugTaMgbYn.
#> KNna actual nominal moment strength M oxn sMrab; equivalent uniaxial bending
eFobG½kS x enAeBl M ox = 0 . vaRtUvmantMély:agehacNas;smmUlnwg required
moment strength M ox .

$> KNna actual nominal moment strength M oyn sMrab; equivalent uniaxial bending
moment eFobG½kS y enAeBlEdl M oy = 0 .

%> kMNt; M ny edaybBa©Úl M nx / M oxn nigtMélsakl,g β eTAkñúgdüaRkamExSekagemKuN
β énrUbTI 8>20.
^> Gnuvtþ trial and adjustment elIkTIBIr edaybegáIntMélsnμt; β RbsinebItMél M ny Edl
TTYl)anBIkarbBa©ÚleTAkñúg chart tUcCagtMél required M ny . GnuvtþCMhanenHeLIgvij
rhUtdl;tMélén M ny xitCitKña tamry³karpøas;bþÚr β b¤pøas;bþÚrmuxkat;.
&> sikSaKNa lateral ties niglMGitmuxkat;.
Flowchart sMrab;CMhandMbUgkñúgkarkMNt;tMél controlling moment enAkñúg biaxially loaded

column RtUv)aneGayenAkñúgrUbTI 8>21.



12> karBicarNakñúgkarsikSaKNnaGnuvtþn_
Practical Design Considerations
xageRkamCaeKalkarN_ENnaMsMrab;karsikSaKNna nigkartMerobEdkenAkñúgkarsikSaKNna
Gnuvtþn_.

k> EdkbeNþay b¤Edkem Longitudinal or Main Reinforcement
kugRtaMgRbsiT§PaBmFümenAkñúgebtugenAkñúgGgát;rgkarsgát;eRbkugRtaMgminKYrtUcCag 225 psi
(1.55MPa ) . tMrUvkarrbs; code kMNt;pleFobEdkGb,brmaEbbNaeGayGgát;rgkarsgát;Edlrg
kMlaMgeRbkugRtaMgtUc nigmanpleFobEdkminrgeRbkugRtaMgGb,brma1% .


x> EdkxagsMrab;ssr Lateral Reinforcement for Columns
!> EdkcMNgxag Lateral ties

eKRtUvkar lateral reinforcement edIm,IkarBar spalling rbs; concrete cover b¤ local buckling
rbs;EdkbeNþay. EdkBRgwgxagRtUvmanTMrg;Ca ties EdlBRgayedaycenøaHesIμtamkMBs;rbs;ssr.
EdlbeNþayEdlmanKMlatBIKña 6in. KYrRtUv)anTb;eday lateral ties dUcbgðajenAkñúgrUbTI 8>22.
eKRtUvGnuvtþtameKalkarN_ENnaMxageRkamsMrab;kareRCIserIsTMhM nigKMlatrbs; ties:
!> TMhMrbs;EdkcMNg b¤Edjkg (tie) minRtUvtUcCag #3(9.5mm) .
@> KMlatbBaÄrsMrab; tie minRtUvFMCag
(a) 48 dgénGgát;p©itrbs; tie

(b) 16 dgénGgát;p©itrbs;EdkbeNþay

(c) TMhMxagtUcCageKrbs;ssr

rUbTI 8>22 bgðajBIkartMerob tie sMrab;EdkbeNþay 4, 6 nig 8 enAkñúgmuxkat;ssr.

@> EdkkgvNÐ Spirals

RbePTepSgeTotrbs; lateral reinforcement KW spiral b¤ helical lateral reinforcement dUc
bgðajenAkñúgrUbTI 8>23. Spiral manRbeyaCn_BiesskñúgkarbegáIn ductility b¤ PaBrwgrbs;Ggát;
dUcenHeKesñIeGayeRbI spiral sMrab;tMbn;EdlRbQmnwgrBa¢ÜydIx<s;. CaTUeTA ebtugEdlB½T§CMuvij
confined core énssrEdlBRgwgeday spiral Gac spall eRkamGMeBI lateral force minFmμta nigPøam²

dUcCakMlaMgrBa¢ÜydI. ssrRtUvmanlT§PaBTb;Tl;nwgbnÞúkPaKeRcInbnÞab;BIebtugkarBar spall edIm,I
karBarkardYlrlMGKar. dUcenH eKRtUvsikSaKNnaKMlat nigTMhMrbs; spiral edIm,IrkSalT§PaBRTbnÞúk
PaKeRcInrbs;ssr eTaHbICaeRkamlkçxNÐbnÞúkeRKaHfñak;EbbenHk¾eday.



EdlmanKMlatCit²begáIn ultimate-load capacity rbs;ssr. eKRtUv
Spiral reinforcement

eRCIserIsKMlat (spacing b¤ pitch) rbs; spacing edIm,IeGay load capacity Edl)anBI confining
spiral action GacTb;Tl;nwgkar)at;bg;muxkat;ebtugedaysar spalling.

edayeGaykarekIneLIgénersIusþg;EdlbNþalBI confinement esμInwgkar)at;bg; capacity
edaysar spalling ehIybBa©ÚlemKuNsuvtßiPaB 1.2 eyIgTTYl)anpleFob spiral reinforcement
Gb,brma
⎛ Ag ⎞ f'
ρ s = 0.45⎜
⎜− 1⎟ c
⎟ f (8.32)
⎝ Ac ⎠ sy
Edl ρ=
volume of the spiral steel per one revolution
volume of concrete core contained in one revolution
πD 2
Ac = c (8.33a)
4
πh 2
Ag = (8.33b)
4
h= Ggát;p©itrbs;ssr
as = RkLaépÞmuxkat;rbs; spiral

d b = nominal diameter rbs; spiral

Dc = Ggát;rbs;sñÚlebtug (concrete core) EdlKitBIépÞxageRkArbs; spiral

f sy = yield strength rbs; spiral reinforcement



edIm,IkMNt; pitch s rbs; spiral/ KNna ρ s edayeRbIsmIkar 8.33/ eRCIserIsGgát;p©it db
sMrab; spiral/ KNna as nigbnÞab;mkTTYl)an pitch b edayeRbIsmIkar 8.35b xageRkam.
eKGacsresrpleFob spiral reinforcement ρ s dUcxageRkam
asπ (Dc − d b )
ρs = (8.34a)
(π / 4)Dc2 s
dUcenH eKTTYl)an pitch
asπ (Dc − d b )
s= (8.35a)
(π / 4)Dc2 ρ s
4 a (D − d b )
b¤ s= s c
Dc ρ s
2
(8.35b)

EdnkMNt;énKMlat b¤ pitch rbs; spiral sßitenAcenøaH 1in.(25.4mm) eTA 3in.(76.2mm)
ehIyGgát;p©itminRtUvtUcCag 3 / 8in.(9.53mm) . eKRtUvbRBa¢Üsy:agehacNas;mYyCMuknøH RbsinebIeK
mineRbItMNpSar.

#> sikSaKNna Spiral Lateral Reinforcement
]TahrN_ 8>3³ sikSaKNna lateral spiral reinforcement sMrab;ssrebtugeRbkugRtaMgmUlEdl
manGgát;p©it h = 20in.(508mm) nig clear cover dc = 1.5in.(38mm) nigman f y = 60,000 psi
(414MPa ) .
dMeNaHRsay³ edayeRbIsmIkar 8.32
⎛ Ag ⎞ f'
ρ s EdlRtUvkar = 0.45⎜
⎜ − 1⎟ c
⎟ f
A ⎝ c ⎠ sy

edayeRbI spiral #3 Edlman yield strength f y = 60,000 psi eyIgman
clear concrete cover d c = 1.5in.(38mm )
f sy = 60,000 psi

Dc = h − 2d c = 20.0 − 2 × 1.5 = 17in.(432mm )
π (17.0)2
Ac = = 226.98in.2
4
Ag = 314.0in.2
⎛ 314 ⎞ 4,000
ρ s = 0.45⎜ − 1⎟ = 0.0115
⎝ 226.98 ⎠ 60,000
sMrab; spiral #3 / as = 0.11in.2 . dUcenHedayeRbIsmIkar 8.35b eyIg)an


4as (Dc − d b ) 4 × 0.11(17.0 − 0.375)
pitch s = = = 2.20in.(56mm )
Dc ρ s
2
(17.0)2 × 0.0115
dUcenH eRbI spiral #3 CamYynwg pitch 2 14 in. ¬spiral Ggát;p©it 9.53mm CamYynwg pitch 54.0mm ¦.

13> Reciprocal Load Method for Biaxial Bending
viFIenHRtUv)anbegáIteLIgeday Bressler. viFIenHP¢ab;TMnak;TMngrvagtMélbnÞúktamG½kSEdl
cg;)an Pu eTAnwgtMélbIepSgeTotEdl reciprocial eTAnwg failure surface. snμt; S1 CakUGredaen
enAelI failure surface énrUbTI 8>18 EdltMélénbnÞúk nigcMNakp©itCa Pu / ex nig e y . RbsinebI
S 2 CacMnucenAelI compatible reciprocal surface eTAnwgGVIenAkñúgrUbTI 8>18 enaH S 2 nwgkMNt;kUGr-

edaenéncMnucenaHCa 1 / Pu / ex nig e y Edl Pu = φPn EdlCabnÞúkemKuN.
RbsinebI desired axial load Pu eRkam biaxial loading eFobG½kS x nigG½kS y RtUvTak;Tg
eTAnwgtMél Pu EdleGayeday Puy / Pux nig Puo enaH
1 1 1 1
= + − (8.36a)
Pu Pux Puy Puo

b¤ 1
=
1
+
1
−
1
φPn φPnxo φPnyo φPno
(8.36b)

Edl Pux = φPnxo = design strength rbs;ssrEdlmancMNakp©it ex RbsinebI e y = 0
Puy = φPnyo = design strength rbs;ssrEdlmancMNakp©it e y RbsinebI ex = 0

Puo = φPno = axial load design strength tamRTwsþIsMrab;ssrEdlmancMNakp©it ex = e y = 0

M ux = m:Um:g;eFobG½kS x = Pu e y

M uy = m:Um:g;eFobG½kS y = Pu ex

ex = cMNakp©itvas;RsbeTAnwgG½kS y dUcenAkñúgrUbTI 8>24 Edl ex = M uy / Pu = Pu ex / Pu

e y = cMNakp©itEdlvas;RsbeTAnwgG½kS x = Pu e y / Pu

x = TMhMrbs;muxkat;ssrEdlRsbeTAnwgG½kS x

y = TMhMrbs;muxkat;ssrEdlRsbeTAnwgG½kS y



14> Modified Load Contour Method for Biaxial Bending
CMnYseGaysmIkar 8.32, Hsu )anesñInYvsmIkarEdlEktMrUvEdlGacCMnYseGay strength inter-
action diagram nig failure surface rbs;ssrebtugGarem:rgbnÞúkBIrTis dUcenAkñúgrUbTI 8>34. viFI

enHk¾dUcKñanwg reciprocal load method Edr vaTamTarnUvkarKNnaticCagviFIBIrepSgeTot.
smIkar interaction sMrab;bnÞúk nigm:Umg;Bt;eFobnwgG½kSBIrKW
:
1.5
⎛ Pn − Pnb ⎞ ⎛ M nx ⎞
1.5
⎛ M ny ⎞
⎜
⎜P −P ⎟+⎜M ⎟
⎟ ⎜ ⎟ +⎜ ⎟ = 1.0 (8.37)
⎝ no nb ⎠ ⎝ nbx ⎠
⎜ M nby ⎟
⎝ ⎠
Edl Pn = kMlaMgsgát;tamG½kS nominal ¬viC¢man¦ b¤kMlaMgTaj ¬GviC¢man¦
M nx , M ny = m:Um:g;Bt; nominal eFobG½kS x nigG½kS y erogKña

Pno = kMlaMgsgát;tamG½kS nominal Gtibrma ¬viC¢man¦ b¤kMlaMgTajtamG½kS ¬GviC¢man¦

( )
= 0.85 f 'c Ag − Ast + f y Ast

Pnb = kMlaMgsgát;tamG½kSeRkamlkçxNÐ balanced strain
M nbx , M nby = m:Um:g;Bt; nominal eFobG½kS x nigG½kS y erogKña eRkamlkçxNÐ balanced
strain
eKGacTTYl)antMélrbs;sßanPaBbMErbMrYlragkMNt; Pnb nig M nb BI
Pnb = 0.85 f 'c β1cbb + Apsf ' ps − Aps f ps (8.38a)
⎛ ⎞
nig M nb = Pnb eb = Cc ⎜ d − − d "⎟ + C s (d − d '− d ") + Ts d "
⎝
a
2 ⎠
(3.38b)

Edl ab = kMBs;rbs;bøúksmmUl = β1cb = (Aps / f ps )/(0.85 f 'c b)
a = β1c
kugRtaMgenAkñúgEdkrgkarsgát;EdlenAEk,rbnÞúkCageK = f py RbsinebI f ps ≥ f py
f ' ps =

Ts = kMlaMgenAkñúgEdkxagTaj

Step-by-step operational procedure

sMrab;karsikSaKNnassrrgbnÞúkBIrTisGnuvtþeTAtamdMeNIrkarenAkñúgcMnuc 11>K. viFIenHTamTarkar
KNnatickñúgkaredaHRsayssrrgm:Um:g;BIrTis.

k> EdkxagsMrab;ssr Lateral Reinforcement for Columns
]TahrN_ 8>4³ snμt;muxkat;ssrcak;Rsab;enAkñúg]TahrN_ 8>2 CassrxøIEdlrgm:Um:g;BIrTis Etmin
man sidesway. sikSaKNnassrsMrab;m:Um:g;Bt;xageRkam³



M ux = M uy = 825,000in. − lb(93.7kN .m ) nig Pu = 300,000lb(1334kN )
eKeGay³ f 'c = 6,000 psi (41.4MPa ) ebtugTMgn;Fmμta
f pu = 270,000 psi (1863MPa )

f ps = 240,000 psi (1565MPa )

muxkat;RtUv)anBRgwgCamYynwg 7-wire tendon Ggát;p©it 1 / 2in.(12.7mm) cMnYn 5 EdleGay tendon
srubcMnYn 16.
dMeNaHRsay³
Pu = 300,000lbs
M ux = Pu e y = 825,000in. − lb eFobnwgG½kS x
M uy = Pu e x = 825,000in. − lb eFobnwgG½kS y

f 'c = 6,000 psi
f ps = 240,000 psi

dUcenH ex =
M ux 825,000
Pu
=
300,000
= 2.75in.

M uy 825,000
ey = = = 2.75in.
Pu 300,00
x= G½kSRsbeTAnwgRCugxøI b
y = G½kSRsbeTAnwgRCugEvg h

muxkat;ssrKW 15in. × 15in.
b = 15in. h = 15in. d ' = 2.5in.
enAelIRCugnImYy² As = 5 × 0.153 = 0.765in. 2

EdlBRgwgsrub Ast = 16 × 0.153 = 2.448in.2
cMNakp©itEdltUcCageKKW 2.75in. . ]bmafava)ak;edaykarsgát;. sakl,g φ = 0.65 .
Pn Cak;Esþg =
300,000
= 461,538lb
0.65
M n Cak;Esþg =
825,000
= 1,269,231lb − in.
0.65
BI]TahrN_ 8>2/ sMrab;sßanPaBbMErbMrYlrageFobkMNt;rgkarsgát; ¬ ε t = 0.002 ¦
Pnb = 229,310lb

M nb = Pnb eb = 2,103,124in. − lb(237kN .m )


M nb 2,103,124
eb = = = 9.17in.
Pnb 229,310
eb > e = 2.75in. dUcenHkMlaMgsgát;RKb;RKgsßanPaB ehIyemKuNkat;bnßyersIusþg;
φ = 0.65
( )
Pno = 0.85 f 'c Ag − Ast + Ast f ps

= 0.85 × 6,000(225 − 2.448) + 1.53 × 240,000

= 1,502,205lb
edayeRbIsmIkar 8.37 (interaction surface) sMrab; biaxial bending
1.5
⎛ Pn − Pnb ⎞ ⎛ M nx ⎞
1.5
⎛ M ny ⎞ 461,538 − 229,310 ⎛ 1,269,231 ⎞
1.5
⎜
⎜P −P ⎟+⎜M ⎟
⎟ ⎜ ⎟ +⎜ ⎟ = +⎜ ⎟
⎝ no nb ⎠ ⎝ nbx ⎠
⎜ M nby ⎟ 1,502,205 − 229,310 ⎝ 2,103,124 ⎠
⎝ ⎠
1.5
⎛ 1,269,231 ⎞
+⎜ ⎟
⎝ 2,103,124 ⎠

= 0.182 + 0.468 + 0.468 = 1.118 > 1.0
¬muxkat;enH overdesigned bnþicbnþÜc¦
TTYlykkarsikSaKNna Edl
b = 15in. h = 15in. d = 12.5in.
As = 7-wire strand tendon Ggát;p©it 1 / 2in. cMnYn 5 tamRCugnImYy² dUckñúgrUbTI 8>15 Edl
eGay tendon srubTaMgGs; 16 .

15> Ggát;rgkarTajeRbkugRtaMg Prestressed Tension Members

k> kugRtaMgbnÞúkesvakmμ Service-Load Stresses
RbB½n§ nigGgát;rgkarTajdUcCa railroad ties, bridge truss tension members, foundation
anchors sMrab;CBa¢aMgTb;dI nig ties enAkñúgCBa¢aMgén liquid-retaining tank pSMeLIgeday prestressing

strand EdlmanersIusþg;x<s;CamYynwgPaBrwgRkajrbs;ebtug. edaysarEbbenH vapþl;nUversIusþg;Taj

nigkMhUcRTg;RTayEdlfycuH Edlmuxkat;EdksuT§minGacpþl;eGay)andUc sMrab;karRTbnÞúkdUcKña.
eKeRcIneRbIvaCaGgát;cMNg (tie) b¤CaEpñkénRbB½n§eRKOgbgÁúMTaMgmUl.
rUbTI 8>25 eRbobeFonsac;lUtrbs;Ggát;ebtugeRbkugRtaMgkñúgTisénkMlaMgTajCamYYynwgGgát;
eRKOgbgÁúMEdkEdlmanlT§PaBRTRTg;dUcKña. sac;lUtrbs;Ggát;rgkarTajEdl)anBIkarGnuvtþénkMlaMg



xageRkA F xN³Edlsac;lUtrbs; unstressed tendon enAkñúgEpñk (a) EdlbNþalBIkMlaMg F
KW)anBIRTwsþIemkanicmUldæan
FL
ΔL ps = (8.39)
Aps E ps

RbsinebIeKCMnYs tendon eday rolled structural member karERbRbYllkçN³énmuxkat;eFVIeGay
kMhUcRTg;RTay
⎛ A ps E ps ⎞
ΔLs = ΔL ps ⎜
⎜ AE ⎟
⎟ (8.40)
⎝ s s ⎠
Edl As FMCag Aps . dUcenH kMhUcRTg;RTayEdlfycuHy:ageRcInRtUv)anbgðajenAkñúgrUbTI 8>25b.
RkLaépÞebtugbMElgrbs;ebtugenAkñúgrUbTI 8>25 c KW
(
At1 = Ag + n p − 1 Aps ) (8.41)

ehIy RbsinebIbMErbMrYlkugRtaMgenAkñúgkugRtaMgKW


Viii. prestressed compression and tension member

Viii. prestressed compression and tension member

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Viii. prestressed compression and tension member