Xi. prestressed concrete circular storage tanks and shell roof

Department of Civil Engineering NPIC

XI. GagsþúkragmUl nigdMbUlekagebtugeRbkugRtaMg
Prestressed Concrete Circular Storage Tanks and Shell Roofs

11.1. esckþIepþIm Introduction
CaTUeTA GagragmUlebtugeRbkugRtaMgCabnSMd¾l¥bMputénTMrg;eRKOgbgÁúM nigsMPar³sMrab;sþúksar-
Faturav nigsarFaturwg. kareFVIkarrbs;vaGs;ry³eBlCagknø³stvtSbgðajfa enAeBlEdleKsikSa
KNnavaCamYynwgCMnaj nigkarykcitþTukdak;KYrsm vaGaceFVIkar)an 50qñaM b¤eRcInCagenHedayKμan
karEfTaMFMdMueT.
kic©xMRbwgERbgdMbUgkñúgkareRbIkMlaMgeRbkugRtaMgvNÐeTAelIeRKOgbgÁúMragmUlKWeFVIeLIgeday W.S.
Hewett Edl)anGnuvtþeKalkarN_ tie rod nigeKalkarN_rwtk,al (turnbuckle principle) enAedIm

TsSvtSqñaM 1920. b:uEnþEdkBRgwgEdlmanenAeBlenaHman yield strength TabNas; EdlkMNt;nUv
kugRtaMgTajGnuvtþn_mineGayFMCag 30,000 psi b¤ 35,000 psi ¬ 206.9 eTA 241.3MPa ¦. CakarBit
kMhatbg;eRbkugRtaMgry³eBlEvgd¾FMEdlbNþalBI concrete creep, shrinkage nig steel relaxation
swgEtlubbM)at;kMlaMgeRbkugRtaMg. edaysareRkaymk eKrkeXIjEdkEdlmanersIusþg;x<s; enAkñúg
TsSvtSqñaM 1940 J.M. Crom )anbegáItedayeCaKC½ynUveKalkarN_rMu high-tensile wire CMuvijCBa¢aMg
ragmUlrbs;GagebtugeRbkugRtaMg. taMgBIeBlenaHmk eRKOgbgÁúMsþúkragmUlCag 3,000 RtUv)anksag
eLIgedaymanTMhMGgát;p©itepSg². GagsþúkEdlmanTMhMFMCageKmanGgát;p©itrhUtdl; 300 ft (92m) .
KuNsm,tþid¾cMbgkñúgkareFVIkar nigesdækic©énkareRbIkMlaMgeRbkugRtaMgvNÐkñúgGagebtugBIelI
EdkBRgwgFmμtaKWkarminGnuBaØateGaymansñameRbH. kugRtaMgsgát;vNÐ (circumferential “hugging”
hoop stress in compression) Edlpþl;edaykarrMuEdkeRbkugRtaMgBIxageRkACMuvijGagsþúkragmUlKWCa

bec©keTsFmμCatisMrab;lubbM)at;sñameRbH enAkñúgCBa¢aMgxageRkAEdlbNþalBIsMBaFrbs;sarFaturwg
sMBaF]sμ½n nigbnÞúksarFaturwgEdlGagsþúk. bec©keTsepSgeTotkñúgkareFVIeRbkugRtaMgvNÐedayeRbI
tendon mYy²f<k;Cab;eTAnwgCnÞl;RtUv)aneRbIR)as;y:agTUlMTUlayenAGWur:ubCagenAGaemricxageCIg eday

sarmUlehtuesdækic©kñúgtMbn; nigbec©keTs.
GagsþúkEdleRbIeRbkugRtaMgvNÐ ehIyEdlGaccak;enAnwgkEnøg b¤cak;Rsab;CakMNat;² Edl
rYmmanGagsþúkTwk GagTwks¥úy silo GagsþúkeRbg nigsarFatuKImI eRKOgbgÁúMsßanIybUmeRbgenAkNþal
smuRT (offshore oil platform structure), cryogenic vessel nig nuclear reactor pressure vessels.
eKcat;TukeRKOgbgÁúMTaMgenHCakMralekagesþIg (thin shell) edaysarpleFobd¾EsntUcénkMras;elIGgát;
GagsþúkragmUl nigdMbUlekagebtugeRbkugRtaMg 667

T.Chhay viTüasßanCatiBhubec©keTskm<úCa

p©itrbs;va. edaysarminGnuBaØateGaymansñameRbHeRkamGMeBIbnÞúkeFVIkar eKrMBwgfa shell eFVIkarCa
lkçN³eGLasÞiceRkamGMeBIbnÞúkeFVIkar nigeRkamlkçN³bnÞúkelIs (overload condition).

11.2. eKalkarN_ nigdMeNIrkarsikSaKNna Design Principles and Procedures
11.2.1. kMlaMgxagkñúg Internal Loads
BicarNaBIkareFVIkarrbs;GagragmUlEdlBak;B½n§nwgkarBinitüTaMgsMBaFxagkñúgEdlbNþalBI
sMPar³EdlpÞúkenAxagkñúgmanGMeBIelImuxkat;kMralCBa¢aMgekagragsIuLaMgesþIg (thin-wall cylindrical
shell) nigkMlaMgeRbkugRtaMg radial xageRkA nigeBlxøHkMlaMgeRbkugRtaMgbBaÄrEdleFVIeGaykMlaMgxag

kñúgmanlMnwg. sMBaFxagkñúgCasMBaF radial edk b:uEnþERbRbYltamTisbBaÄrEdlGaRs½yeTAnwgRbePT
sMPar³EdlpÞúkenAkñúgGag. RbsinebIsMPar³CaTwk b¤CaGgÁFaturav sMBaFbBaÄrBRgayeTAelICBa¢aMgGag
manragRtIekaN CamYynwgGaMgtg;sIuetGtibrmaenARtg;)atrbs;CBa¢aMg. sarFatu]sμ½nnwgpþl;sMBaF
edkefrelIkMBs;TaMgmUlrbs;CBa¢aMg. karBRgaysMBaFbBaÄrrbs;sMPar³EdlmanlkçN³dMuRKab; dUcCa
FüÚgfμ b¤RKab;FBaØCatimanragRsedogKñaniwgkarBRgaysMBaFrbs;]sμ½nmkelICBa¢aMgGagEdr. rUbTI
11>1 bgðajBIkarBRgaysMBaDsMrab;krNIénkardak;bnÞúkTaMgbIenH.

Prestressed Concrete Circular Storage Tanks and Shell Roofs 668


RTwsþIeGLasÞicmUldæanrbs; cylindrical shell GnuvtþeTAelIkarsikSaviPaK nigeTAelIkarsikSa
KNnaCBa¢aMgrbs;GageRbkugRtaMg. kMlaMgkg (ring force) bgákMlaMgTajkg (ring tension) enAkñúg
CBa¢aMgsIuLaMgesþIg (thin cylindrical wall) edaysnμt;KμankarTb; (unrestrained) enAxagcugénmuxkat;
edknImYy²eT. TMhMrbs;kMlaMgKWsmamaRteTAnwgsMBaFEdlGnuvtþenAxagkñúg nigKμanm:Um:g;bBaÄrekIt
mantamkMBs;rbs;CBa¢aMgeT. RbsinebIcugrbs;CBa¢aMgRtUv)anTb; (restrained) TMhMrbs; ring force ERb
RbYl ehIym:Um:g;Bt;nwgekItmanenAkñúgmuxkat;bBaÄrrbs;CBa¢aMgGag. dUcenHTMhMrbs; ring force nigm:U
m:g;bBaÄrCaGnuKmn_énkMriténkarTb;rbs; cylindrical shell enARtg;RBMEdnrbs;va ehIyvaRtUv)anKNna
BIRTwsþkMralekageGLasÞic (elastic shell theory).

bnÞúksarFaturavenAelI)atrGiledayesrI Liquid Load on Freely Sliding Base

BIemkanicmUldæan (basic mechanics), ring force KW
pd
F= = pr (11.1a)
2
ehIykugRtaMgkg (ring stress) KW
pd pr
fR = = (11.1b)
2t t
Edl d= Ggát;p©itrbs;sIuLaMg
r = kaMrbs;sIuLaMg

t = kMras;CBa¢aMg

p = sMBaFÉktþaxagkñúgRtg;)atCBa¢aMg = γH

γ = TMgn;maDrbs;sMPar³EdlpÞúkenAkñúgGag.
Tensile ring stress enARtg;RKb;cMnucBIxageRkamépÞrbs;sMPar³EdlpÞúkenAkñúgGagkøayCa

f R = γ (H − y ) = γ (H − y )
d r
(11.2a)
2t t
Edl H CakMBs;rbs;sarFaturav ehIy y CacMgayBIelI)at. Ring force EdlRtUvKñaKW
F = γ (H − y )r (11.2b)
dUckñúgsmIkar 11.1b/ Tensile ring stress GtibrmaenARtg;)atrbs;CBa¢aMgEdlrGiledayesrIsMrab;
y = 0 køayCa
γHd γhr
f R (max ) = = (11.2c)
2t t



bnÞúk]s½μnenAelI)atEdlrGiledayesrI Gaseous Load on Freely Sliding Base

mþgeTotBIeKalkarN_emkanicmUldæan/ tensile ring stress efrKW
pd pr
fR = = (11.3)
2t t
cMNaMfa eKRtUveRbITMhMGgát;p©ittamRTwsþIEdlKitBI centerline EdlmanlkçN³suRkit Etedaysarpl
eFob t / d manTMhMtUceBk dUcenHeKGaceRbIGgát;p©itxagkñúg d )an.

bnÞúkGgÁFaturav nigbnÞúk]s½μnenAelI)atCBa¢aMgEdlTb;
Liquid and Gaseous Load on Restrained Wall Base
RbsinebI)atrbs;CBa¢aMgRtUv)anbgáb; b¤ pinned/ enaHeKlubecal ring tension enARtg;)at.
edaysarkarbgáb;enARtg;)at enaHeKminGacGnuvtþ simple membrane theory rbs; shell EdlbNþal
BIkMhUcRTg;RTayedaysar restraining force enARtg;)atCBa¢aMg. eKcaM)ac;eFVIkarbMElgm:Um:g;Bt;eTACa
membrane stress ehIylMgakecj (deviation) rbs; ring tension enARtg;bøg;kNþalrbs;kMBs;CBa¢aMg

RtUv)anKitCatMélRbhak;RbEhl nigRtUv)anerobrab;enAkñúgEpñk 11.3.
RbsinebIm:Um:g;Bt;bBaÄrenAkñúgbøg;edkrbs;CBa¢aMgenARtg;kMBs;NamYyKW M y / kugRtaMgbegáag
(flexural stress) ebtugrgkarsgát; b¤rgkarTajkøayCa
M y 6M y
ft = f c = = 2 kñúgmYyÉktþakMBs; (11.4)
S t
karBRgaykugRtaMgbegáagelIkMras;rbs;CBa¢aMgGagRtUv)anbgðajenAkñúgrUbTI 11>2.



11.2.2. m:Um:g;Tb; M nigkMlaMgkat; Radial Q enARtg;)atCBa¢aMgEdlrGiledayesrIEdl
o o

ekItBIsMBaFsarFaturav
Restraining Moment M o and Radial Shear Force Qo at Freely Sliding
Wall Base Due to Liquid Pressure
11.2.2.1. Membrane Theory
karsikSaBIkMlaMg nigkugRtaMgenAkñúgCBa¢aMgGagragmUlKμansñameRbHCakarviPaKkMralragekag
sIuLaMgEdleFVIkarCalkçN³eGLasÞic. RbsiinebI shell minxUcRTg;RTayeRkamT§iBlrbs;sMBaFsar-
Faturavxagkñúg eKGacGnuvtþsmIkarlMnwg basic membrane )an. kMlaMgtambeNþayÉktþa N y / kMlaMg
ÉktþavNÐ (circumferential unit force) Nθ nigkMlaMgkat;ÉktþaRtg;p©it N yθ nig Nθ y RtUv)anbgðaj
enAkñúg differential element énrUbTI 11>3 (b). cMNaMfa GBaØtiTaMgbYnenHeFVIGMeBIenAkñúgbøg;rbs; shell.



smIkarlMnwgmUldæanbIsMrab;kMlaMgÉktþaGBaØtiTaMgbYnenHKW
∂Nθ ∂N yθ
+r + pθ r = 0 (11.5a)
∂θ ∂y
∂N y ∂Nθ y
r +r + pyr = 0 (11.5b)
∂y ∂θ
Nθ
+ pz = 0 (11.5c)
r
Edl ∂N yθ = ∂Nθ y KWbNþalmkBIkardak;sIuemRTI. dUcenHcMnYnGBaØtiRtUv)ankat;bnßymkenARtwmbIEdl
bgðajBIeRKOgbgÁúMkMNt;edaysþaTicEdlrgEtnwgkMlaMgedaypÞal;. sMrab;kardak;bnÞúk axisymmetic dUc
bgðajenAkñúgrUbTI 11>3 (d)/ pθ = p y = 0 ehIy p z = p ⋅ f ( y ) . kardak;bnÞúkEbbenHminGaRs½ynwg
θ eT. dUcenH
p z = −γ (H − γ ) (11.6)
ehIydMeNaHRsayrbs;smirk 11.5 KW
N yθ = N y = 0

ehIy Nθ = γ (H − y )r (11.7)

11.2.2.2. RTwsþIbTBt;begáag Bending Theory
karENnaMBIkarTb; (restraint) enARtg;RBMEdnrbs;GagnaMeGayman radial horizontal shear
nigm:Um:g;bBaÄrenAkñúg shell. dUcenH smIkarkMlaMg membrane EdlbgðajenAkñúgEpñkelIkmunRtUv)an
EkERbedaykardak;bEnßmm:Um:g; nigkMlaMgkat;. smIkarEdlEkERbRtUvkMNt;Ca bending theory rbs;
circular shell EdlRTwsþIenHKitTaMgtMrUvkarPaBRtUvKñaénbMErbMrYlrageFob (strain compatibility) enA

kñúgkMhUcRTg;RTayEdlbgáeLIgedaykarekItmankMlaMgkat; nigm:Um:g;xagelI.
m:Um:g;Bt; nig central shear enAkñúgkMralragekagsIuLaMgEdlrgbnÞúk axisymmetric RtUv)an
bgðajedayviucT½rkMlaMg nigviucT½rm:Um:g;enAkúñgrUbTI 11>4. FatuGnnþtUc ABCD bgðajBIcMnucGnuvtþn_
nigTisedArbs;m:Um:g;Éktþa M y eFobnwgG½kS x nig M θ eFobG½kS y / circumferential unit moment
M yθ nig M θ y / kMlaMgkat;EkgÉktþa Q y EdlmanGMeBIenAkñúgbøg;énkMragekagbBaÄr nigEkgeTAnwg

shell axis ehIy unit radial shear Qθ EdlmanGMeBIkat;tamkaM shell enAkñúgbøg;EdlRsbnwg shell.

eFVItMrYlplénm:Um:g; nigkMlaMgkat;enAkñúgrUbTI 11>4 eTAelIkMlaMgenAkñúgrUbTI 11>3 (b) begáIt
)ansmIkarlMnwgxageRkam³



∂Nθ ∂N yθ
+ − Qθ + pθ r = 0 (11.8a)
∂θ ∂y
∂N y ∂Nθ y
r+ + pyr = 0 (11.8b)
∂y ∂θ
∂Qθ ∂Q y
+ r + Nθ + p z r = 0 (11.8c)
∂θ ∂y
∂M y ∂M yθ
r+ + Qy r = 0 (11.8d)
∂y ∂y
∂M θ ∂M yθ
+ r − Qθ r = 0 (11.8e)
∂θ ∂y

edaysarPaBsIuemRTIénkardak;bnÞúk/ N yθ = Nθ y = M θ y = M yθ = 0 ehIyeKGacminKit
dQθ Edlkat;bnßysmIkarDIepr:g;EsüledayEpñk 11.8 mkCasMnMuénsmIkarDIepr:g;EsülFmμta
(ordinary differential equation)
dN y
r + pyr = 0 (11.9a)
dy
dQ y
r + Nθ + p z r = 0 (11.9b)
dy
dM y
− r + Qy r = 0 (11.9c)
dy
CamYynwg central membrane forces N y efr ehIyeKykvaesμInwgsUnü enaHeKGacsresrsmIkar
EdlenAsl; 11.9b nig 11.9c kñúgTMrg;sMrYldUcxageRkamEdlmanGBaØtibI Nθ / Qy nig M y ³


dQ y 1
+ Nθ = − p z (11.10a)
dy r
dM y
− Qy = 0 (11.10b)
dy
edIm,IedaHRsaysmIkarTaMgenH eKRtUvKitBIbMlas;TI nigsmIkarFrNImaRt.

smIkarkMlaMg
RbsiebI v nig w CabMlas;TIenAkñúgTis y nig z enaHbMErbMrYlragÉktþaenAkñúgTisTaMgenHKW
dv
εy =
dy
ehIy εθ = −
w
r
ehIyeKTTYl)an
Ny =
Et
(ε y + μεθ ) = Et⎛ dv w⎞
⎜ −μ ⎟=0
⎜ dy r⎟
(11.11a)
1− μ 2
1− μ ⎝ 2
⎠

nig Nθ =
Et
(εθ + με y ) =
Et ⎛ w dv ⎞
⎜− + μ ⎟
⎜ r dy ⎟
(11.11b)
1− μ2 1− μ2 ⎝ ⎠
Edl pleFobB½rsug
μ=
t = kMras;rbs;CBa¢aMg

BIsmIkar 11.11a
dv w
=μ (11.12a)
dy r
BIsmIkar 11.11b
w
Nθ = − Et (11.12b)
r
smIkarm:Um:g;
edaysarPaBsIuemRTI kMeNagenAkñúgTisvNÐminmanERbRbYleT dUcenH kMeNagenAkñúgTis y RtUv
esμInwg − d 2v / dy 2 . edayeRbIsmIkarm:Um:g;dUcKñasMrab;kMraleGLasÞicesþIg (thin elastic plate) eKTTYl
)an
M θ = μM y (11.13a)
d 2w
M y = −D (11.13b)
dy 2
Edl D = Et 3 / 12(1 − μ 2 ) CaPaBrwgRkajTb;karBt;begáagrbs;kMralekag (shell) b¤kMral (plate)



bBa©ÚlsmIkar 11.12 nig 11.13 eTAkñúgsmIkar 11.10 eKTTYl)an
d 2 ⎛ d 2 w ⎞ Et
⎜D ⎟ + w = pz (11.14)
dx 2 ⎜ dy 2 ⎟ r 2
⎝ ⎠
RbsinebIkMras;CBa¢aMg t efr enaHsmIkar 11.14 køayCa
d 4w Et
D + w = pz (11.15)
dy 2 r2

edayyk β =4 Et
=
(
31− μ2 )
2
4r D (rt )2

smIkar 11.15 køayCa
d 4w pz
4
+ 4β 4 w = (11.16)
dy D

smIkar 11.16 RsedognwgsmIkarEdlTTYlsMrab;Ggát;FñwmEdlmanPaBrwgRkaj D EdlRTedayRKwHeG
LasÞicCab; nigRbQmnwgGMeBIénGaMgtg;sIuetbnÞúkÉktþa p z . dMeNaHRsayTUeTAénsmIkarenHsMrab;
radial displacement kñúgTis z KW
w = e βy (C1 cos β y + C2 sin βy ) + e − βy (C3 cos βy + C 4 sin βy ) + f ( y ) (11.17)
Edl f (y) CadMeNaHRsayTUeTAénsmIkar 11.16 Ca membrane solution EdleGaynUvbM;las;TI
pz r 2
w=
Et

11.2.3. smIkarTUeTAénkMlaMg nigbM;las;TI
General Equations of Forces and Displacements
edayedaHRsaysmIkar 11.17 nigbBa©ÚlnUvnimitþsBaØaxageRkam
Φ (βy ) = e − βy (cos β y + sin βy )

Ψ (β y ) = e − βy (cos βy − sin β y )

θ (βy ) = e − βy cos β y
ζ (βy ) = e − βy sin βy
eKGackMNt;smIkarsMrab; radial deformation tamTis z nigkargakecjbnþrbs;vaRtg;kMBs; y BI)at
CBa¢aMgBIsmIkarsMrYlxageRkamCaGnuKmn_énm:Um:g;ÉktþaRtg;)atCBa¢aMg M o nig radial shear Qo ³
PaBdab w = − 2β13 D [βM o Ψ(βy ) + Qoθ (βy )] (11.18a)

mMurgVil dw = 2β12 D [2βM oθ (βy ) + QoΦ(βy )]
dy
(11.18b)



d 2w
=−
1
[2βM o Φ(βy ) + 2Qoζ (βy )] (11.18c)
dy 2 2βD
d 3w
=
1
[2βM oζ (βy ) − Qo Ψ (βy )] (11.18d)
dy 3 D

GnuKmn_ shell Φ(βy ) / Ψ(βy ) / θ (βy ) nig ζ (βy ) RtUv)aneGayenAkñúgemKuNT§iBlsþg;dar
(standard influence coefficient) éntarag 11>1 sMrab;EdntMél 0 ≤ βy ≤ 3.9 .

BIsmIkar 11.18a bMlas;TI radial Gtibrma b¤PaBdabenARtg;)atCBa¢aMgEdlTb;KW
(w) y = 0 = − 1
(βM o + Qo ) (11.19a)
2β 3 D
ehIyBIsmIkar 11.18b mMurgVil (rotation) Gtibrmarbs;CBa¢aMgRtg;)atkøayCa
⎛ dw ⎞
⎜
⎜ dy ⎟
⎟ =
1
(2βM o + Qo ) (11.19b)
⎠ y = 0 2β D
2
⎝
Edl M o nig Qo Ca restraining moment nig ring shear enARtg;)atdUcbgðajenAkñúgrUbTI 11>1.
sMrab;GagEdlmankMras;CBa¢aMgefr kMlaMgÉktþatamkMBs;CBa¢aMgmandUcxageRkam³
Etw
Nθ = − (11.20a)
r
d 3w
Qy = −D 3 (11.20b)
dy
M θ = μM y (11.20c)
d 2w
M y = −D (11.20d)
dy 2
BIsmIkar 11.18c, 11.18d, 11.20b nig11.20d smIkarsMrab;m:Um:g;bBaÄr nig radial shear tamTisedk
enARtg;)atrbs;CBa¢aMg Edl y = 0 køayCa
γHrt
(M y )y = 0 = M o = ⎛1 − β1 ⎞
⎜ ⎟
( )
⎜ (11.21a)
H⎟⎝ ⎠ 12 1 − μ 2
γrt
(Q y )y = 0 = Qo = −(2 βH − 1)
( )
(11.21b)
12 1 − μ 2
eKGacTTYl)ansmIkarsMrab;m:Um:g;bBaÄrenARtg;kMBs; y BIelI)atCBa¢aMgBI
My = −
1
[βM o Φ(βy ) + Qoζ (βy )] (11.22)
β
bMErbMrYlkMlaMg ring shear force ΔQ y EdlRtUvniwgbMlas;TI radial wy rbs;CBa¢aMgRtg;kMBs; y BIelI
)atenAeBlGagTeT CamYynwgtMélrbs; Qo nig M o EdlbNþalBIkardak;bBa©ÚlsarFaturav nig]sμ½n
eBj dUcbgðajenAkñúgrUbTI 11>5. eKGacsresrsmIkarkMlaMgenHCa



ΔQ y = +
Et
r
( )
wy

b¤ ΔQ y =
Et
2rβ 3 D
[βM o Ψ (βy ) + Qoθ (βy )]

b¤ ΔQ y =
(
61− μ2 ) [βM Ψ(βy) + Q θ (βy )]
o o (11.23)
β 3rt 2
Ring shear Q yenARtg;bøg; y BIelI)atesμInwgplsgrvag ring force sMrab;)atEdlrGiledayesrI
CamYynwg ΔQ y ³
Q y = F − ΔQ y (11.24)



eKcaM)ac;eKarBtamkarkMNt;sBaØaEdleRbIkñúgdMeNaHRdayTaMgenH. viFIEdlgayRsYlKWKUrragEdlxUc
RTg;RTayrbs;CBa¢aMg ehIyeRbIsBaØabUksMrab;lkçxNÐxageRkam³
!> m:Um:g;EdleFVIeGaymankugRtaMgTajenAelIsrésxageRkAbMputénépÞxageRkA.
@> kMlaMg Ring tension radial.
#> kMlaMg thrust EdlmanTisedAcUlkñúgeTArkG½kSbBaÄr. enATIenH eKeRbITisedAdUcKñasMrab;
kMlaMg ring tension edIm,IKUrdüaRkamsMrab;kMlaMgeRbkugRtaMglMnwg (balancing prestressing
forces) enAelIRCugdUcKñanwgkMlaMg ring tension sMrab;kareRbobeFob.

$> clnaCBa¢aMgxagcUlkñúgeTArkG½kSbBaÄr.
%> muMrgVilbRBa©asTisRTnicnaLika.

sMBaFsarFaturavmanGMeBIelICBa¢aMgEdlman)atCaTMr Pinned (Pinned Wall Base, Liquid Pressure)
enAeBl)atCBa¢aMgmanTMr pinned ehIyrgnUvm:Um:g;bnÞúksarFaturav M o = 0 enARtg;)at
2β 3γH (rt )2
Qo = +
(
12 1 − μ 2 )
1/ 2
γH ⎛ rt ⎞
b¤ Qo = + 1/ 4 ⎜ 2 ⎟
[12(1 − μ )]
(11.25)
2 ⎝ ⎠

eKGacKNnarktMélrbs; shell constant β , β 2 , nig β 4 sMrab;eRbIenAkñúgsmIkarelIkmun)any:aggay
BIsmIkarsMrab; β 4 dUcxageRkam³
Et 3( − μ 2 )
1
β4 = = (11.26a)
4r 2 D (rt )2
β 3
=
[3(1 − μ )]2 3/ 4
(11.26b)
(rt )3 / 2
β 2
=
[3(1 − μ )]2 1/ 2
(11.26c)
(rt )
β=
[3(1 − μ )] 2 1/ 4
(11.26d)
(rt )1/ 2

11.2.4. Ring Shear Qo and Moment β 4 Gas Containment
RbsinebIEKmrbs; shell manlkçN³esrIenARtg;)atCBa¢aMg sMBaFxagkñúgbegáItEtkugRtaMg
hoop f R = pr / t ehIykaMrbs;sIuLaMgnwgekIneLIgedayTMhM



rf R pr 2
w= = (11.27)
E Et
dUcKña sMrab; full restraint enARtg;)atCBa¢aMg
(w) y =0 = 1
(βM o + Qo ) (11.28a)
2β 3 D
⎛ dw ⎞
nig ⎜
⎜ dy ⎟
⎟ =
1
(2βM o + Qo ) = 0 (11.28b)
⎠ y =0 2 β D
2
⎝

edaHRsayrk M o nig Qo eyIgTTYl)an
p prt
M o = − β 2 Dw = − =−
( )
(11.29a)
2β 2
12 1 − μ 2
p (2rt )1 / 2
nig Qo = +4 β 3 Dw = +
p
=+
[12(1 − μ )]
(11.29b)
β 2 1/ 4

sMBaFsarFatu]sμ½nmanGMeBIelICBa¢aMgEdlman)atCaTMr Pinned (Pinned Wall Base, Gas Pressure)
RbsinebI)atCBa¢aMgCaTMr pinned ehIyrgm:Um:g;bnÞúk]sμ½n M o = 0 enARtg;)at
⎛ pr 2 ⎞
Qo = 2β 3 D⎜ ⎟
⎜ Et ⎟
⎝ ⎠
1/ 2
⎛ rt ⎞
b¤ Qo =
p
⎜ ⎟
[12(1 − μ )]
(11.30)
2 ⎝ ⎠
1/ 4 2

tarag 11>2 bgðajkarsegçbénsmIkarKNnasMrab;GagsþúksarFaturav ehIytarag 11>3 bgðajBI
taragsegçbRsedogKñasMrab;GagpÞúksarFatu]sμ½n.

11.3. m:Um:g; M nig kMlaMg Ring Shear Q enAkñúgGagsþúksarFaturav
o o
Moment M o and Ring Force Qo in Liquid Retaining Tank
]TahrN_ 11>1³ GagragmUlebtugeRbkugRtaMgRtUv)anTb;eBlelj (full restrained) enARtg;)at
CBa¢aMg. vamanGgát;p©itxagkñúg d = 125 ft (38.1m) nigpÞúkTwkEdlmankMBs; H = 25 ft (7.62m) .
kMras;CBa¢aMg t = 10in(25cm) . KNna (a) m:Um:g;bBaÄrÉktþa M o nigkMlaMg radial ring force Qo
enARtg;)atrbs;CBa¢aMg nig (b) m:Um:g;bBaÄrÉktþa M y enARtg;kMBs; 7.5 ft (2.29m) BIelI)at. eRbIpl
eFobB½rsug μ = 0.2 ehIyTMgn;maDrbs;Twk γ = 62.4lb / ft 3 (1,000kg / m3 ) .



dMeNaHRsay³
(a) enARtg;)atCBa¢aMg
r = × 125 = 62.5 ft (19m )
1
2
t = 10in. = 0.83 ft (0.25m )
BIsmIkar 11.26d
β=
[3(1 − μ )] 2 1/ 4
=
[3(1 − 0.2 × 0.2)]1 / 4 = 0.181
(rt )1 / 2 (62.5 × 0.83)1 / 2
BIsmIkar 11.21a
⎛ 1 ⎞ γHrt
⎜ βH ⎟
M o = −⎜1 −
⎝
⎟
⎠ 12 1 − μ 2 ( )
⎛ 1 ⎞ 62.4 × 25 × 62.5 × 0.83
= −⎜1 − ⎟×
⎝ 0.181× 25 ⎠ 12(1 − 0.04)

= −18,574 ft. − lb / ft (7.68kN .m / m )
BIsmIkar 11.21b
γrt
Qo = +(2 β H − 1)
12 1 − μ 2 ( )
62.4 × 62.5 × 0.83
= +(2 × 0.181× 25 − 1)
12(1 − 0.04 )

= +7,677lb / ft (112kN / m )

(b) enARtg;kMBs; 7.5 ft BI)atCBa¢aMg
y = 7.5 ft
kMBs;Twk = (H − y ) = 25 − 7.5 = 17.5 ft (5.33m)
pleFobkMBs; = ⎛1 − H ⎞ = 1 − 7.5 = 0.7
⎜
⎝
y
⎟
⎠ 25

βy = 0.181 × 7.5 = 1.36
BIsmIkar 11.22
My = +
1
[βM o Φ(βy ) + Qoζ (βy )]
β
BItarag 11.1 sMrab; β y = 1.36
Φ = 0.311
ζ = 0.252



My = +
1
(− 0.181 × 18,574 × 0.311 + 7,677 × 0.252)
0.181
= +4,912 ft − lb / ftt

11.4. kMlaMg Ring Shear Q enARtg;Bak;kNþalkMBs;rbs;CBa¢aMg
y
Ring Force Q y at Intermediate Heights of Wall
]TahrN_ 11>2³ KNna radial ring force Q enAkñúg]TahrN_ 11>1 Rtg; (a) y = 7.5 ft (2.29m)
y

nig (b) y = 10 ft (3.05m) BIxagelI)atrbs;CBa¢aMg sMrab;CBa¢aMgrGiledayesrI.
dMeNaHRsay³
kMlaMg ring force enARtg;)atEdlrGilesrI F = γHr = 62.4 × 25 × 62.5 = 97,500lb / ft
(1,423kN / m ) . BIsmIkar 11.23/ bMErbMrYlkMlaMg ring force KW
6(1 − μ )
ΔQ y = + [βM o Ψ (βy ) + Qoθ (βy )]
β 3 rt 2
BIsmIkar 11.1/ β = 0.181 . dUcenH β 3 = 0.0059
(a) Q y enARtg; 7.5 ft BIelI)atCBa¢aMg

βy = 0.181× 7.5 = 1.36
BItarag 11.1 sMrab; βy = 1.36
Ψ (βy ) = −0.1965

θ (βy ) = +0.0543

6(1 − 0.04 )
ΔQ y = + [0.181(− 18,574)(− 0.1965) + 7,677(+ 0.0543)]
0.0059 × 62.5(0.83)2

= 24,431lb / ft (356kN / m )
BIsmIkar 11.2b/ ring force F = γ (H − y )r = 62.4 × (25 × 7.5) × 62.5 = 68,250lb / ft . dUcenH
Q7.5 = F − ΔQ y = 68,250 − 24,431 = 43,819lb / ft (705kN / m ) dUcbgðajenAkñúgrUbTI 11>6(a).

(a) enARtg;kMBs; 7.5 ft BI)at (b) enARtg;kMBs; 10 ft BI)at.

(b) Q y enARtg;kMBs; 10 ft BI)atrbs;CBa¢aMg

βy = 0.181 × 10 = 1.81

BItarag 11>1 sMrab; βy = 1.81 /



Ψ (β y ) = −0.1984

θ (β y ) = −0.0387

6(1 − 0.04 )
ΔQ y = [0.181(− 18,574)(− 0.1984) + 7,677(− 0.0387 )]
0.0059 × 62.5(0.83)2

= 8,387lb / ft
kMlaMg ring force F = γ (H − y )r = 62.4(25 − 10) = 62.5 = 58,500lb / ft . dUcenH Q10 = F − ΔQ y
= 58,500 − 8,387 = 50,113lb / ft (731kN / m ) dUcbgðajenAkñúgrUbTI 11>6 (b). eRbobeFobCamYynwg

tMél Q y = 50,115lb / ft EdlTTYl)anedayeRbI membrane coefficient enAkñúg]TahrN_ 11>3.

11.5. Cylindrical Shell Membrane Coefficients
eKGackMNt;m:Um:g;Bt;enARtg;kMBs;Nak¾edayBI)atrbs;GagragsIuLaMgBIsmIkarm:Um:g;Bt;sMrab;
Fñwmkugs‘ul (cantilever beam). eKTTYl)ansmIkarenHedayKuNm:Um:g; cantilever edayemKuNEdl
TMhMrbs;vaCaGnuKmn_eTAnwgTMhMFrNImaRtrbs;Gag ehIyRtUv)aneGayeQμaHfa membrane coeffi-
cients. eKGacerobcMsmIkarm:Um:g;mUldæanEdl)anbegáItenAkñúgEpñk 11.2 sMrab;GagsþúkragmUleLIg

vijedaybBa©ÚlemKuN H 2 / dt EdltMNageGayragFrNImaRt nigemKuN γH 3 b¤ pH 2 EdltMNag
eGayT§iBl cantilever sMrab;kardak;bnÞúksarFaturav nigsarFatu]sμ½n.


tMélefr β enAkñúgsmIkar 11.26d CaGnuKmn_én rt b¤ dt Edl d CaGgát;p©itGag. edayeRbI
pleFobB½rs‘ug μ ≅ 0.2 sMrab;ebtug eyIg)an
β=
[3(1 − μ )] 2 1/ 4
=
1.30
=
1.84
(rt )
1/ 2
(rt )
1/ 2
(dt )1 / 2
eKGacsresremKuN 1 / βH EdleRbIenAkñúgsmIkarm:Um:g;Bt;begáagmUldæanénEpñk 11.2 edayeRbItY
(dt / H 2 )1 / 2 edaysar β = 1.84 /(dt )1 / 2 . eKk¾GacsresrplKuN βy eLIgvijedayeRbItY
λ (H 2 / dt ) edayeRbI y = λH Edl y CakMBs;BI)at.
1/ 2

dUcenH eKGacbgðajm:Um:g; M y énsmIkar 11.22 enAkñúgmuxkat;CBa¢aMgEdlmancMgay y BI)at
edayeRbIemKuNrag (form factor) H 2 / dt CamYynwgemKuN cantilever γH 3 b¤ pH 2 dUcxageRkam³
M y = numerical variant × form factor × cantilever facotor

b¤ ⎡
M y = ⎢ variant ×
H2⎤
⎥ × γH or pH
3 2
( ) (11.31)
⎢
⎣ dt ⎥
⎦

emKuNrag (form factor) H 2 / dt CatMélefrsMrab;eRKOgbgÁúMCak;lak;. dUcenH plKuNén variant nig
form factor begáIt)an membrane coefficient C dUcenH smIkar 11.31 køayCa

M y = CγH 3 (11.32a)

sMrab;bnÞúksarFaturav ehIy
M y = CpH 2 (11.32b)

sMrab;bnÞúksarFatu]sμ½n.
tarag 11>4 dl; 11>16 bgðajBI membrane coefficient C sMrab; form factor H 2 / dt Ca
mYynwglkçxNÐRBMEdnEdlniymeRbICageK niglkçxNÐbnÞúkepSg². vakat;bnßykarKNnaCaeRcInEdl
TamTarCaTUeTAenAkñúgkarKNna nigkarviPaK shell edayminman)at;bg;suRkitPaBénlT§pleT. eday
eRbI membrane coefficient sMrab;dMeNaHRsayénkMlaMg nigm:Um:g;GagragmUl vanwgpþl;lT§plRsedog
KñanwglT§plEdlTTYl)anBIdMeNaHRsaym:Um:g;Bt;EdlbgðajenAkñúgEpñk 11>2 nigsMnMuénsmIkarEdl
manenAkñúgtarag 11>2 nig 11>3 .



11.6. Prestressing Effects on Wall Stresses for Fully Hinged, Partially
Sliding and Hinged, Fully Fixed, and Partially Fixed Bases
sarFaturav b¤]sμ½nEdlsþúkenAkñúgGagragsIuLaMgGnuvtþsMBaF radial ecjeRkA γh b¤ p enAelI
CBa¢aMgGag EdlbegáIt ring tension enAkñúgmuxkat;edknImYy²Rtg;srésxageRkAbMputrbs;CBa¢aMgEdl
begáItCasñameRbHEdlminGacGnuBaØat)an. edIm,Ikat;bnßysñameRbHTaMgenHEdlbNþaleGayRCab nig
eFVIeGayeRKOgbgÁúMeFVIkarminl¥ eKRtUvGnuvtþkMlaMgeRbkugRtaMgedkxageRkAEdlbegáIt radial thrust cUl
kñúgEdlGaceFVIeGay radial tension ecjeRkAmanlMnwg. elIsBIenH edIm,IkarBarkarekItmansñameRbH
enAelIépÞxagkñúgrbs;CBa¢aMgenAeBlGagTwkTeT eKRtUvbBa©ÚlkMlaMgeRbkugRtaMgbBaÄredIm,Ikat;bnßykug
RtaMgTajEdlenAsl; (residual tension) eGayenAkñúgEdnénm:UDuldac;rbs;ebtug nigedIm,IbMeBjlkç-
xNÐemKuNsuvtßiPaB.
edIm,IFanakarRbqaMgnwgkarekItmansñameRbHenAépÞxageRkArbs;CBa¢aMgGag eKKYrGnuvtþkMlaMg
eRbkugRtaMgedkFMCagkMlaMgEdlvaRtUvkarbnþicedIm,IeFVIeGaykMlaMg radial ecjeRkAEdlbgáedaysar-
Faturav b¤]sμ½nxagkñúgmanlMnwg GBa©WgehIyvabegáIteGaymankugRtaMgsgát;EdlenAesssl; (residual
compression) enAkñúgGagenAeBlvaeBj. karekIneLIgénkMlaMgeRbkugRtaMgvNÐ (circumferential

prestressing forces) tamry³kareRbIEdkeRbkugRtaMgedkbEnßm nigeBlxøHmanEdkbBaÄrFmμtak¾RbqaMg

nwgT§iBlrbs;sItuNðPaB nigbMErbMrYlsMeNIm (moisture gradient) Edlqøgkat;kMras;CBa¢aMgkñúgbrisßan
minl¥pgEdr.

11.6.1. )atCBa¢aMgrGiledayesrI Freely Sliding Wall Base
enAeBllkçxNÐRBMEdnén)atrbs;CBa¢aMgGacrGiledayesrI enaHeBlGagrgbnÞúkxagkñúg vanwg
minmanm:Um:g;enAkñúgCBa¢aMgbBaÄrEdlbNþalBIbnÞúksarFaturav b¤k¾bNþalBIkMlaMgeRbkugRtaMg eTaHbIenA
eBlGageBjdl;kMBs; H k¾eday. manEt nominal moment d¾tUcb:ueNÑaHekItmanenAeBlGagmineBj
b¤rgeRbkugRtaMgedayEpñk b¤k¾TeT ehIyvaminRtUvkarkMlaMgeRbkugRtaMgbBaÄreT. rUbragxUcRTg;RTay
rbs;GagEdlrGiledayesrIRtUv)anbgðajenAkñúgrUbTI 11>7.
enAeBlEdlkarrGiledayesrICalkçxNÐd¾l¥Edlpþl;nUveRKOgbgÁúMkMNt;edaysþaTic ehIyman
lkçN³esdækic©CaeK b:uEnþeKBi)aknwgTTYl)ankñúgkarGnuvtþCak;Esþg. kMlaMgkkit (frictional force)
EdlekItmanenARtg;)atCBa¢aMgeRkayeBlEdleKdak;eGayGagdMeNIrkareRbIR)as; dUcenHCMerIsenHmin
GaceRbIkar)aneT.



11.6.2. )atCBa¢aMgCaTMrsnøak; Hinged Wall Base
sMrab;CBa¢aMgEdlmantMNsnøak;enA)at kMlaMg radial force GtibrmaEdlbNþalBIsarFaturav
EdlvapÞúk ehIyeRbkugRtaMgenARtg;muxkat;eRKaHfñak;Rtg;cMgay y BIelI)atswgEtesμInwgeRbkugRtaMgkñúg
krNI)atrGiledayesrIRtg;kMBs; y Edr. b:uEnþeKRtUvbBa©Úlm:Um:g;bBaÄr ehIykMlaMgeRbkugRtaMgbBaÄr
køayCacMa)ac;edIm,Ikat;bnßykugRtaMgTajenAkñúgebtugenARtg;épÞCBa¢aMgxageRkA.
rUbragxUcRTg;RTayrbs;CBa¢aMgEdlmanTMr hinged RtUv)anbgðajenAkñúgrUbTI 11>8. cMNaMfa
muxkat;eRKaHfñak;sMrab; ring force mincaM)ac;enARtg;kMBs;dUcKñanwgmuxkat;eRKaHfñak;sMrab;m:Um:g;eT.
edIm,Ikat;bnßysñameRbHEdlGacekItmanrhUtdl;cMnYnGb,brma/ eKcaM)ac;RtUvkar residual
ring compression EdlmantMélGtibrma 200 psi(1.38MPa ) sMrab; wire-wrapped presstressed

tank Edlminman diaphragm nigmantMélGtibrma 100 psi (0.7 MPa ) sMrab;GagEdlman continuous

metal diaphragm. kugRtaMgTajGtibrmaenAépÞxagkñúgrbs;CBa¢aMgminRtUvFMCag 3 f 'c eRkamGMeBIbnÞúk

eFVIkar (working load) dUcEdleGayenAkñúgtarag 11>17 enAkñúgEpñkxagmux. rUbragxUcRTg;RTay
rbs;GacCBa¢aMg nigbMErbMrYlkugRtaMgenAkñúgebtugEdlkat;tamkMras;rbs;muxkat;enAeBlGagTeT nigenA
eBlvaeBj dUcbgðajenAkñúgrUbTI 11>8. sMrab;GagEdlrgeRbkugRtaMgCamYynwg pretensioned tendon
nig post-tensioned tendons kugRtaMgsgát;EdlenAesssl;Gb,brmaKYrmantMéldUcGVIEdlerobrab;enA
kñúgEpñk 11.10.



11.6.3. )atCBa¢aMgrGiledayEpñk nigmanTMrsnøak;
Partially Sliding and Hinged Wall Base
edIm,ITTYl)an partially slinging and hinged wall-base system eKRtUveFVIrn§enAkñúgkMralRT
)atCBa¢aMgy:agNaedIm,IeGayCBa¢aMgGacrGilkñúgGMLúgeBlrgeRbkugRtaMg. eRkayeBlrgeRbkugRtaMg
nigkMhatbg;eRbkugRtaMgedaysar creep, shrinkage nig relaxation, eKRtUvbiTrn§ ehIyCBa¢aMgGageFVI
karCa hinged eRkamGMeBIlkçxNÐ service load. eKRtUvRKb;RKgTMhMénkarrGil eTaHbIvaCa full sliding
b¤ partial sliding k¾eday edIm,IeGayvasßitkñúgkMritGnuBaØatmunnwgeKTTYl)anTMr hinged. karrGil
edayEpñkRbEhl 50% énkarrGileBjelj nigedaymansnøak;enAxagcugrbs;CBa¢aMg vapþl;RbeyaCn_
dl;clnarbs;eRKOgbgÁúMTaMgsMrab;)atrGileBjelj nigTaMgsMrab;)atmansnøak; ehIykarbiTrn§enARtg;
tMN pinned Rtg;)atCBa¢aMgRbqaMgnwgkarelcRCabénsarFaturav b¤]sμ½nKWvakan;EtGaRs½yeTAnwgkMrit
rGil EdleKGnuBaØatcMeBaH full sliding CagcMeBaH anchorage. rUbragxUcRTg;RTayrbs;CBa¢aMgkñúg
GMLúgdMeNIrkarGnuvtþeRbkugRtaMg rYmCamYynwg ring force/ m:Um:g;bBaÄr nigbMErbMrYlkugRtaMgebtugkñúg
kMras;CBa¢aMg RtUv)anbgðajenAkñúgrUbTI 11>9. eRbkugRtaMgbBaÄrEdlRtUvkarsMrab;krNI partial slide-
pinned mantMéltUcCagkrNI fully pinned EdlminmankarrGileRcInNas;.

11.6.4. )atCBa¢aMgbgáb;eBlelj
Fully Fixed Wall Base
PaBbgáb;eBjeljrbs;CBa¢aMgenARtg;)atmann½yfaTb; (restraint) mineGayvilTaMgRsugenA
Rtg;)atCBa¢aMg. eKGacTTYl)anlkçxNÐenH RbsinebIkMNat;TabCageKrbs;CBa¢aMgRtUv)ancak;kñúgeBl
CamYyKñanwgkMral ehIymankarf<k;)any:agl¥eTAkñúgkMral)atEdlmanPaBrwgRkajdUcKña. b:uEnþeKBi)ak
nwgTTYl)anRbB½n§minkMNt;EbbenHNas; ehIyvak¾minmanlkçN³esdækic©pg edaysarépÞ)atGagman
TMhMFM ehIyPaBbgáb;edayEpñkkøayCacaM)ac;. kMlaMg radial tamTisedkEdl)anBIkMlaMgeRbkugRtaMg
nig)anBIsMBaFxagkñúgminmankarERbRbYlBIragRtIekaNsMrab;sarFaturag ragctuekaNsMrab;]sμ½n nigrag
ctuekaNBñaysMrab;GgÁFatuRKab;eT. b:uEnþ restraint EdlbegáItedaykMral)atEkERb ring force nig
bBa©Úlm:Um:g;bEnßmeTAkñúgmuxkat;bBaÄrrbs;CBa¢aMg. edaysarPaBbgáb;enARtg;)at vanwgminmanbMlas;
TIekItmanenARtg;)at b¤kMBUlrbs;CBa¢aMgeT ehIykarERbRbYlénkMeNagrbs;kMBs;CBa¢aMgekItmanenA
eBlGagTeT dUcbgðajenAkñúgrUbTI 11>10. cMNaMfa eKKYrsikSaKNnaCBa¢aMgeGayQrRtg; CamYynwg
kugRtaMgsgát;Edlesssl;Gb,brmaEdlbNþalBIkMlaMgeRbkugRtaMg 200 psi dUckñúgkrNIelIkmun.



eRbkugRtaMgbBaÄrEdlRtUvkarsMrab;GagEdlman)atCBa¢aMgbgáb;eBjeljmantMélFMCagsMrab;krNI
lkçxNÐRBMEdkdéTeTotxøaMgNas;. eKcaM)ac;eFVIeGaymanbMErbMrYlkugRtaMgTajenAkñúg)atCBa¢aMgRtg;épÞ
xageRkAEdlbNþalBIm:Um:g;GviC¢mand¾FMenARtg;)ateGaymantMélFM ¬emIlrUbTI 11>10 a nig b¦ ehIy
bRBa©askMeNagEdlenAEk,rva. eBlxøH vamanlkçN³esdækic©CagedayeRbIEdkBRgwgFmμtaenAEpñkxag
eRkamrbs;CBa¢aMgbEnßmBIelIEdk eRbkugRtaMg edIm,IGaceRbIeRbkugRtaMgbBaÄrtUcCag nigeGayEdkBRgwg
FmμtaTTYlnUvm:Um:g;GviC¢mand¾FMenaH. eKk¾Gackat;bnßykugRtaMgTajenAkñúgebtugedayeRbIkMlaMgeRbkug
RtaMgbBaÄrcakp©itCamYynwgcMNakp©itsmrmü rYmCamYynwgEdkBRgwgFmμtabEnßm. b:uEnþ EdkeRbkugRtaMg



bBaÄrEdleRbIenAkñúgGagsþúkmantMéléfø edaysartMrUvkar anchorage enAxagcug nig)atrbs;CBa¢aMg
Gag. dUcenH eKRtUvkarkat;bnßyeRbkugRtaMgbBaÄrkñúgkarsikSaKNnaedIm,IbEnßmlkçN³esdækic©dl;
karsikSaKNna RbB½n§eRKOgbgÁúMGagTaMgmUl.



11.6.5. )atCBa¢aMgbgáb;mineBjelj Partially Fixed Wall Base
11.6.5.1. karTb;nwgkarvil Rotational Restraint
dUckarbgðajBIxagedIm eKBI)akkñúgkareFVIeGay)ankarTb;eBleljRbqaMgnwgkarvilenARtg;)at
CBa¢aMgNas;. mUlehtusMxan;manbI³ ¬!¦ eKRtUvpþl;nUvPaBrwgRkajcaM)ac;enAkñúgkMral)atGagRtg;kEnøg
RbsBVCamYyCBa¢aMgedIm,ITTYl)ankarbgáb;eBjelj. ¬@¦ clnarbs;dIxageRkamCBa¢aMgGacbgákarvil
rbs;)atCBa¢aMg. nig ¬#¦ eKTamTarkarRb mUlpþúM anchorage sMrab;TaMgeRbkugRtaMgbBaÄr nigTaMgeRb
kugRtaMgvNÐedkrbs;kMNat;CBa¢aMg-)at edaysarCBa¢aMg nig)atrgeRbkugRtaMgdac;edayELkBIKña.
edaysarkMral)atmanépÞFM T§iBlénkarTb; b¤kareFVIeGayrwgrbs;vaRtUv)ankMNt;RtwmbrievN
toe d¾tUcceg¥ótEdleFVIkarCalkçN³ cantilever BI)atCBa¢aMg. CMerIsd¾RtwmRtUvénTTwgrbs; toe b¤ base

ring kMNt;nUvPaBRtwmRtUvéntMélPaBrwgRkajkñúgkarKNnaEdl)anBIdWeRkénkarbgáb;snμt;rbs;)at

CBa¢aMg. rUbTI 11>11 bgðajBIT§iBlénTTwg base ring eTAelImMurgVilrbs;CBa¢aMg nigbMlas;TIrbs;
ring. Epñk (c) rbs;rUbbgðajBIsßanPaBlMnwgEdlcugrbs; ring sßitenAelInIv:UdUcKñanwg)atrbs;CBa¢aMg

b:uEnþlkçxNÐEdlbgðajenAkñúgEpñk (a) nig (b) Bak;B½n§nwgbMlas;TIBIxageRkam)atrbs;CBa¢aMg ehIyva
min)anbMeBjlkçxNÐeT.



eKGacTTYl)anrUbmnþkñúgkaredaHRsayrkTTwg ring base eRKaHfñak;tamry³kareRbIeKalkarN_
tMrYtpl (superposition) edaybUkbBa©ÚlkrNIénCBa¢aMgviledayesrI (freely rotating wall) CamYYynwg
krNICBa¢aMgbgáb;eBjelj dUcbgðajenAkñúgrUbTI 11>12. eKyk
M o = m:Um:g;bgáb;eBjeljtamRTwsþIenARtg;)atCBa¢aMg

M p = m:Um:g;edayEpñkenARtg;)atCBa¢aMgEdlbNþalBI loaded cantilever toe

θ1 = mMurvilesrIrbs;)atCBa¢aMgsMrab;EtTMr pinned EdlRtUvnwgPaBdab Δ1 rbs; stiff unloaded
toe
θ2 = mMurgVil)atCBa¢aMgEdlbNþalBI restraining moment M p EdlRtUvnwgPaBdab Δ 2 én
straight unloaded toe
θ3 = mMurgVileFobxagcugén stiffening toe EdlmanlkçN³Ca cantilever eRkamGMeBIbnÞúkbBaÄr
EdlRtUvnwgPaBdab Δ3 rbs;cug toe EdlbNþalBIbnÞúkbBaÄr.
L = TTwgrbs; stiffening toe.

q = bnÞúkÉktþaEdlGnuvteTAelI stiffening toe = γH Edl H CakMBs;rbs;Gag EdlGgát;p©it

rbs;vaesμInwg d / kMras;CBa¢aMgrbs;vaesμInwg t ehIykMras;kMralrbs;vaesμInwg h .



eKTTYl)anmMurgVilÉktþaθ rbs;CBa¢aMgRtg;)atrbs;vaEdlbNþalBIm:Um:g; M o b:uEnþminmanbMlas;
TI radial BIsmIkar 11.18a edayeGay w = 0 edIm,ITTYl)an Q = −βM . smIkar 11.18b sMrab;mMurgVil
ÉktþakøayCa
Mo Mp
θ1 = θ2 = (11.33)
2βD 2βD
dUcenH eyIg)an
LM o LM p
Δ1 = Δ2 = (11.34)
2βD 2βD
RbsinebIeKcat;Tuk stiffening wall toe Ca cantilever Edlrg transverse load γH / Cantilever
moment Gtibrma M p ehIyPaBdabEdlRtUvKña Δ 3 KW
γHL2 3γHL4
Mp = Δ3 = (11.35)
2 2 Eh 3
eKGacTTYl)anm:Um:g;enARtg;)atCBa¢aMgEdlbgáb;edayeRbI membrane coefficient C BItarag 11>4 sMrab;
form factor H 2 / dt nigRbePTbnÞúk. sMrab;bnÞúksarFaturag
M o = CγH 3 (11.36)
BIsmIkar 11.12(c) rUbragEdlxUcRTg;RTayedaysarbnÞúkeBjKW
Δ1 = Δ 2 + Δ 3
snμt; μ = 0.2 nig β = 2 / dt
edayCMnYs Δ 2 nig Δ 3 BIsmIkar 11.34 eTAkñúgsmIkar 11.35 nig 11.36 enaHeyIg)an
2CH 2
L2 = (11.37)
1+
(t / h )3 (L = 1)
(dt )1 / 2
γHL2
nig Mo =
2
(11.38)

yktY S=
(t / h )3 (11.39)
(dt )1 / 2
tY S énsmIkar 11.39 enHRtUv)aneKeGayeQμaHfa modifying factor sMrab;karbgáb;edayEpñk. em
KuNenHCaTUeTAmantMéltUc ehIybgðajnUvplsgrvagm:Um:g;bgáb;srub M o nigm:Um:g;Tb;edayEpñk
(partial restraint moment) M p . dUcenH
M p = M o (1 − S ) (11.40)

tMélrbs; L enAkñúgPaKEbgénsmIkar 11.37 RtUv)ansnμt;esμInwg 1 sMrab;karsMrYlenAkñúgkarEkERb
emKuN S .


RbsinebItMélrbs; S tUcEmnETn dUckñúgkrNIGagEdlmanGgát;p©itFM ¬Ggát;p©itFMCag 125 ft
eTA 150 ft ¦ smIkarkarsMrab; L nig M p køayCasmIkrsMrab;karbgáb;eBjelj
L2 = 2CH 2
nig M p = CγH 3

11.6.5.2. Base Radial Deformation
kMhUcRTg;RTay radial Δ s én base ring EdlrgkMlaMg radial enAkñúgbøg;rbs;vaGacTTYlBIRTwsþI
rbs;kMralmUlEdlmanRbehagcMp©it. smIkarsMrab;PaBdabrbs;kMralEdlbgðajenAkñúgrUbTI 11>13
(a) KW
d oQ ⎛ d o + d 2
⎜
2 ⎞
Δs = − μ⎟ (11.41)
2hE ⎜ d o − d 2
⎝
2 ⎟
⎠
Edl μ = pleFobB½rs‘ug ~ 0.2 sMrab;ebtug ehIy E Cam:UDuleGLasÞic. kMlaMg radial edkÉktþaEdl
RtUvkarsMrab;begáItbMlas;TIÉktþaenAkñúgkMlagmYltan;KW


2.5hE
Q2 = (11.42)
do
ehIytMélRtUvKñaén radiant thrust EdlGnuvtþeTAelI ring xageRkAKW
2hE
Q3 = (11.43)
do K
⎛d2 + d2 ⎞
Edl K =⎜ o
⎜ d2 − d2
− μ⎟
⎟
⎝ o ⎠
ehIy d = Ggát;p©itxagkñúgrbs; base ring = (do − 2L) .
eKkMNt;PaBrwgRkaj relative rbs;CBa¢aMgedayeRbItYénkMlaMgEdlRtUvkarsMrab;begáItbMlas;TI
ÉktþaenAkñúgCBa¢aMg nigkMral)atBIeKalkar virtual work dUcEdlbgðajenAkñúgrUbTI 11>13 (b) nig
(c). kar BRgayfamBleRbkugRtaMgcenøaHCBa¢aMg nig base slab ring CaGnuKmn_én relative radial

stiffness rbs;va dUcenHeKcaM)ac;RtUvkMNt;PaBrwgRkaj relative. b:uEnþ eKRtUvdwgfa PaBrwgRkajrbs;

base ring enAkñúgGageRbkugRtaMgEdlmankugRtaMgsgát; radial enAkñúgbøg;rbs;vamantMélFMCagPaBrwg

RkajénCBa¢aMgsIuLaMgrbs;GageRkamsMBaF radial xagkñúg. dUcenH kMhatbg;eRbkugRtaMgBIPaBxusKña
énPaBrwgRkajminsMxan;eTsMrab;GagGgát;p©itFM EteKRtUvBicarNavasMrab;GagGgát;p©ittUc.
eKGacTTYlbMlas;TIÉktþa Δ EdlbNþalBIkMlaMg radial Q' EdlminmanmMurgVilenARtg;)at
CBa¢aMgBIsmIkar 11.8 b edayeRbI 2βM = −Q sMrab;mMurgVil dw / dy = 0 . PaBdabÉktþa Δ enAkñúg
smIkar 11.18a køayCa
Q3
Δ=
4β 3 D

b¤ Δ=
Q'
4β 3 D
(11.44)

Et 3
Edl D=
( ) 12 1 − μ 2
edayeRbI μ ~ 0.2 / smIkar 11.44 sMrab;bMlas;TI radial Éktþarbs;CBa¢aMgenARtg;)atCBa¢aMgEdlmin
manmMurgVilkøayCa
3/ 2
⎛t⎞
Q ' = 2 .2 E ⎜ ⎟ (11.45)
⎝d ⎠
Edl E Cam:UDuleGLasÞicrbs;ebtug. BIsmIkar 11.42/ kMlaMg radial EdlRtUvkarsMrab;begáItbMlas;TI
radial ÉktþaenAkñúgkMlagragmUltan;KW
⎛ h ⎞
Q2 = 2.5 E ⎜ ⎟
⎜d ⎟ (11.46)
⎝ o⎠



edaybUk Q' nig Q2 enaHkMlaMgsrubEdlmanGMeBIenARtg;kEnøgRbsBVrvagCBa¢aMg nig)atRtUv)anEbg
EckeTACBa¢aMg nigeTA)atedayQrelIsmamaRténfamBl relative EdlRtUvkarsMrab;begáItbMlas;TI
ÉktþanImYy².
smamaRténkMlaMgsrub Q'+Q2 EdlRtUv)anRTedayCBa¢aMgKW
Q'
R=
Q'+Q2

edayyk R=
1
1 + S1
kMNt; S1 edayCMnYssmIkar 11.45 nig11.46 eTAkñúgsmIkarxagelI eyIg)an
2.5(h / d )
S1 =
2.2(t / d )3 / 2
edaysnμt; d ~ do / b¤
1/ 2
⎛h⎞ ⎛d ⎞
S1 = 1.1⎜ ⎟ × ⎜ ⎟ (11.47)
⎝t⎠ ⎝t ⎠
RbsinebI S1 tUc enaHeKGacyksmamaRténkMlaMgedkEdlepÞrBI)atkMraleTACBa¢aMg ¬suRkitRKb;RKan;¦
100
R= % (11.48)
S1
enAeBlEdl ring xageRkArbs;kMralrgkugRtaMgsgát;eday radial thrust enARtg;EKm eKRtUvEksMrYl
tMélrbs; Q2 EdlTTYlBIsmIkar 11.42 ehIy S1 enAkñúgssmIkar 11.48køayCa
1/ 2
1 ⎛h⎞ ⎛d ⎞
S1 = ⎜ ⎟ × ⎜ ⎟ (11.49)
K⎝t⎠ ⎝t ⎠
Edl BIelIkmun
⎛ do + d 2
2 ⎞
⎜
K= 2 −μ⎟
⎜d −d2 ⎟
⎝ o ⎠
EdlkñúgenaH d CaGgát;p©itxagkñúgrbs; slab ring = d o = 2L ehIy d o CaGgát;p©itxageRkA.

11.7. Recommended Practice for Situ-Cast and Precast Prestressed
Concrete Circular Storage Tanks
11.7.1. kugRtaMg Stresses
eKalkarN_ENnaMTUeTAsMrab;GagsþúkragmUlebtugeRbkugRtaMgEdlcak;enAnwgkEnøgRtUv)anpþl;
eGayeday Prestressed Concrete Institute/ American Concrete Institute nig Post-Tensioning
Institute sMrab;eRCIserIskugRtaMgGnuBaØat/ karkMNt;TMhM kMras;CBa¢aMgGb,brma nigdMeNIrkardMeLIg nig



sagsg;. kugRtaMgGnuBaØatenAkñúgebtug nig shotcrete RtUv)aneGayenAkñúgtarag 11>17. kugRtaMg
GnuBaØatenAkñúgEdkBRgwgRtUv)aneGayenAkñúgtarag 11>18.

11.7.2. emKuNbnÞúk nigersIusþg;tMrUvkar Required Strength Load Factors
eRKOgbgÁúM rYmCamYynwgrcnasm<½n§rbs;va nigeCIgtag KYrRtUv)ansikkSaKNnay:agNaeGay
ersIusþg;KNna (design strength) FMCagT§iBlrbs;bnSMbnÞúkemKuNEdlkMNt;eday ACI 318, ANSI/
ASCE 7-95 b¤GaRs½yelIkarEksMrYledayvisVkrEdlQrelIkarviPaKd¾smehtupl CamYynwglkçxNÐ

xageRkam³



smIkarersIusþg;m:Um:g; nominal M n RsedogKñanwgsmIkarEdleRbIsMrab; linear prestressing
⎛ a⎞
M n = A ps f ps ⎜ d p − ⎟ (11.50a)
⎝ 2⎠
⎛ a⎞ ⎛ a⎞
b¤ M n = A ps f ps ⎜ d p − ⎟ + As f y ⎜ d − ⎟
⎝ 2⎠ ⎝ 2⎠
(11.50b)

enAeBlEdleKeRbI As ehIy
Edl Aps = EdkeRbkugRtaMgbBaÄrkñúgTTwgmYyÉktþa
f ps = kugRtaMgenAkñúgEdkeRbkugRtaMgRtg;ersIusþg; nominal

f y = yield strength rbs;EdkFmμta

11.7.3. tMrUvkarGb,brmakñúgkarKNnaCBa¢aMg
Minimum Wall-Design Requirements
11.7.3.1. kMlaMgvNÐ Circumferential Forces
sarFaturav
kMlaMgedIm Fi = γr (H − γ ) ff pi (11.51a)
ps

karcak;bMeBj (backfill)
kMlaMgedIm Fbi = p(r + t ) (11.51b)

Edl t CakMras;CBa¢aMgsrub.

11.7.3.2. kMras; nigkugRtaMg Thickness and Stresses
kMras;CBa¢aMg (Core Wall Thickness)
Fi
t co = (11.52)
f ci



b:uEnþminRtUvtUcCagkMras;CBa¢aMgGb,brmaEdlmanerobrab;enAkñúgEpñk 11.7.3.6.
kugRtaMgcugeRkayEdlbNþalBIkarcak;bMeBj nigeRbkugRtaMgedIm
Fbi Fi f pe
f = + (11.53)
t t co f pi

11.7.3.3. PaBdab Deflections
PaBdab radial eGLasÞicedImrbs;CBa¢aMgEdlbNþalBIkMlaMgeRbkugRtaMgedImKW
Fi r
Δi = (11.54)
t co Ec
Edl kaMxagkñúgrbs;Gag
r=

t co = kMras;rbs;CBa¢aMgenAxagcug xag)atrbs;CBa¢aMg

Ec = 57,000 f 'c psi (4,700 f 'c MPa ) sMrab;ebtugTMgn;Fmμta nig shotcrete.

PaBdab radial cugeRkay Δf GacmantMélesμInwg1.5 eTA3dgénPaBdabdMbUg. sMrab;lkçxNÐ
eKGacykPaBdab radial EdlGnuBaØatcugeRkaydUcxageRkam
Δf = 1.7 Δ i (11.55)

11.7.3.4. T§iBlTb; Restraint Effects
m:Um:g;Bt;bBaÄrGtibrmarbs;CBa¢aMgEdlbNþalBIkMlaMgkat; radial
M f = 0.24Qo rt co (11.56a)

m:Um:g;enHekItmanenAcMgay
y = 0.68 rt co (11.56b)

BI)at b¤cugEKm
kMlaMgkat; radial sMrab;)atEdlcak;rYmKñaEdlsnμt;favaCatMNsnøak;
t co
Qo = 0.38 Fi (11.57)
r
RbePTénkarlMGitenaHRtUv)aneRbIsMrab;EtGagsþúkEdlcak;enAnwgkEnøgEdlsagsg;edayman
diaphragm enAkñúgCBa¢aMgrbs;vab:ueNÑaH.

11.7.3.5. EdkFmμtasMrab;karf<k;enA)at Mild Steel for Base Anchorage
RbsinebIeKeRbI diaphragm/ eKRtUvbgðÚtEdkxagkñúgEdlmanragGkSr UTaMgGs;cMgay


y1 = 1.4 rt co (11.58a)

BIelI)at. RbsinebIeKmineRbI diaphragm eKRtUvbgðÚtvacMgay
y2 = 1.8 rtco (11.58b)

BIelI)at. cMNaMfa eKRtUvbUkbEnßmRbEvgf<k; (anchorage length) BIelI y1 b¤ y2 . RkLaépÞGb,brma
rbs;EdkbBaÄr nominal enARtg;tMbn;)atKW
As = 0.005t co (11.59)
ehIyeKRtUvbgðÚtvaBI)atnUvcMgay 3 ft b¤
y3 = 0.75 rt co (11.60)

edayykmYyNaEdlFMCag.

11.7.3.6. kMras;CBa¢aMgGb,brma Minimum Wall Thickness
CBa¢aMgcak;enAnwgkEnøg

CBa¢aMgcak;Rsab;

eKRtUvcMNaMfa sMrab;GagEdlrgeRbkugRtaMgCamYynwg tendon, eKENnaMkMras;CBa¢aMgminRtUvtUc
Cag 9in. sMrab;kargarGnuvtþn_.



11.8. RKb;RKgsñameRbHenAkñúgCBa¢aMgrbs;GagebtugeRbkugRtaMgragmUl
Crack Control in Walls of Circular Prestressed Concrete Tanks
nig Preston ENnaMsmIkarxageRkamedayQrelIkargarrbs; Nawy sMrab;TMhMsñameRbH
Vessy

GtibrmaenAépÞxageRkArbs;CBa¢aMgGageRbkugRtaMg³
wmax = 4.1 ⋅10 −6 ε ct E ps I x (11.61)

Edl ε ct = bMErbMrYlrageFobrbs;épÞrgkarTaj (tensile surface strain) enAkñúgebtug
8 ⎛ s 2 s1tb ⎞
I x = grid index = ⎜ ⎟
π ⎜ φ1
⎝
⎟
⎠
s2 = KMlatEdkkñúgTiselx “2”
s1 = KMlatEdkkñúgTisEkgelx “1” ¬Tisedk¦

tb = kMras;karBarEdkKitdl;G½kSEdk

φ1 = Ggát;p©itEdkkñúgTisem “1”
eKGacKNnabMErbMrYlrageFobrgkarTajBI
α t f pi
ε ct = (11.62)
E ps

Edl αt = )a:ra:Em:RtkugRtaMg (stress parameter) ≅ f p / f pi
f p = kugRtaMgCak;EsþgenAkñúgEdkeRbkugRtaMg

f pi = eRbkugRtaMgedImmuneBlkMhatbg;

sMrab;GagEdlsþúksarFaturav TMhMsñameRbHGnuBaØaGtibrmaKW 0.004in.

11.9. karsikSaKNnadMbUg Roof Design
dMbUlsMrab;GagEdlsagsg;kñúgTMrg;ekag (shell dome) b¤ CadMbUlrabesμIEdlRTenAelIssr
xagkñúg. CaTUeTA tMélrbs;dMbUlRbEhlCamYyPaKbIéntMéleRKOgbgÁúMTaMgmUl. kñúgkrNIdMbUlrabesμI
¬eTaHcak;Rsab; b¤cak;enAnwgkEnøg¦ karKNnaeFVItameKalkarN_KNnaRbB½n§kMralebtugGarem: b¤ebtug
eRbkug RtaMgmYyTis b¤BIrTisFmμta dUcEdlerobrab;enAkñúg ACI 318 Code. RbsinebI dMbUlCaRbePT
ebtugeRbkugRtaMgcak;Rsab; ehIyGgát;p©itGagminFM enaHeKmincaM)ac;eRbIssrxagkñúgeT. ebIminGBa©wg
eT tMélrbs;ssrxagkñúgbEnßm nigeCIgtagrbs;vaGacbegáIntMélrbs;eRKOgbgÁúMTaMgmUl.
dMbUlekagmanRbeyaCn_sMrab;GagEdlmanGgát;p©itminFMCag 150 ft edaysarvaminRtUvkar
ssrTMrxagkñúg nigehIyvamanlkçN³esdækic©sMrab;GagEdlbgáb;eRkamdIkñúgkarTb;Tl;nwgbnÞúkcak;bM



eBj (backfill). dUcenH TMrg; shell nigtMNrbs;vaeTAnwgCBa¢aMgGagmanT§iBly:agxøaMgeTAelItMél.
CakareBjniym dMbUl shell RtUv)anRTedayCBa¢aMgGagCamYynwgtMNEdlman flexible completely
ebImindUecñaHeT eKRtUvEksMrYlkarKNnaTaMgCBa¢aMgGag nigTaMg roof dome EdlTak;TgnwgdWeRkénkar
Tb; nigPaBrwgRkaj relative CamYynwgtMélsagsg;bEnßmkñúgeBlCamYyKña.
dMbUlekagragEsV‘rEdlman rise-to-diameter ratio h' / d tUc eKeRcInyktMélRbEhl 1/ 8 .
dMbUlekagEbbenH b¤ axisymmetrical shell begáItkMlaMgedkEdlmanTisecjeRkAenARtg; springing
EdlvaRtUv)anTb;Tl;eday ring beam eRbkugRtaMgEdlKNnay:agRtwmRtUvenARtg;TMr. RbePT ring
beam kMNt;RbtikmμelIs nigm:Um:g;elIsEdlbNþalBIkugRtaMgbEnßmedaypÞal; nigkugRtaMgBt;enARtg;

cugbgáb;enAkñúg shell Ek,r springing. eKGacniyaymüa:geTotfa membrane solution RtUvbMeBjkar
EkERbedaydak;vabEnßmBIelIT§iBlm:Um:g;Bt;EdlkMNt;edaytMrUvkar strain compatibility én bending
theory.

11.9.1. Membrane Theory of Spherical Domes
11.9.1.1. Shell of Revolution
smIkarlMnwg membrane sMrab;kMlaMgedaypÞal;enAkñúg shell of revolution dUcEdlbgðajenA
kñúgrUbTI 11>14 RtUveRbIsMrab;kMNt; kMlaMg meridional Éktþa Nφ / kMlaMg tangential Éktþa Nθ nig
kMlaMg central Éktþa Nφθ nig Nθφ edayeRbItYbnÞúkTMnaj pφ / pθ nig p z . smIkarTaMgenHman³
Meridional:
(
∂ Nφ ro )− N ∂r ∂Nθφ
+ r1 + pφ ro r1 = 0 (11.63a)
θ
∂φ ∂φ ∂θ
∂Nθ ∂ro ∂Nθφ
Tangential: r1 + Nθφ + + po ro r1 = 0 (11.63b)
∂θ ∂φ ∂φ
N φ Nθ
Tis z ³ r1
+
r2
+ pz = 0 (11.63c)

edaysarbnÞúkmanlkçN³sIuemRTI/ RKb;tYTaMgGs;EdlTak;Tgnwg ∂θ RtUv)anbM)at; ehIyeKGacsresr
tYEdlTak;Tgnwg ∂φ eLIgvijCaDIepr:gEsülsrub dφ edaysarKμantYNamYyERbRbYlGaRs½ynwgtY θ .
ehIybgÁúM circumferential load pθ = 0 edaysarkMlaMgpÁÜbkMlaMgkat;RtUv)anbM)at;tambeNþay
meridional nigrgVg;EdlRsb. dUcenH eKGacsresrsmIkar 11.63 eLIgvijCa
d
dφ
( )
Nφ ro − Nθ r1 cos φ + p y r1ro = 0 (11.64a)

N φ Nθ
+ + pz = 0 (11.64b)
r1 r2



11.9.1.2. dMbUlekagragEsV‘r Spherical Dome
smIkarlMnwgrbs; Membrane Analysis
dMbUlekagragEsV‘rmankMeNagefr. dUcenH r1 = r2 = ro . edaysnμt;fa kaMrbs;EsV‘resμInwg a
enaH ro = a sin φ enAkñúgrUbTI 11>14(c) ehIyedayyk p z = wD sMrab;bnÞúkpÞal; enaHsmIkarlMnwg



TUeTA 11.64 køayCa
⎛ 1 ⎞
⎜ 1 + cos φ − cos φ ⎟
Nθ = awD ⎜ ⎟ (11.65a)
⎝ ⎠

nig Nφ = −
awD
1+ cos φ
(11.65b)

Edl wD CaGaMgtg;sIuetrbs;bnÞúkpÞal;kñúgmYyÉktþaépÞ. eyIgeXIjy:agc,as;BIsmIkar 11.65 b fa
kMlaMg meridional Nφ nwgGviC¢manCanic©. dUcenH kMlaMgsgát;nwgekItmantambeNþay meridian
ehIyvanwgekIneLIgenAeBlEdlmMu φ ekIneLIg³ enAeBl φ = 0 / Nφ = −awD / 2 ehIyenAeBlEdl
φ = π / 2 / Nφ = −awD .
kMlaMg tangential Nθ mantMélGviC¢man ¬kMlaMgsgát;¦ sMrab;EttMélkMNt;énmMu φ b:ueNÑaH.
edayyk Nθ = 0 enAkñúgsmIkar 11.65a/ 1/(1 + cosφ ) − cosφ = 0 eKTTYl)an φ = 51o 49' . kar
kMNt;bgðajfa sMrab; φ > 51o 49' kugRtaMgTajekItmanenAkñúgTisEkgnwg meridian. karEbgEckkug
RtaMg meridional Nφ ehIykarEbgEckkugRtaMg tangential Nθ sMrab;TaMgbnÞúkpÞal; wD nigbnÞúk
GefrxageRkA wL RtUv)anbgðajenAkñúgrUbTI 11>15.
RbsinebIbnÞúkxageRkAefr ¬bnÞúkRBil¦ EdleGayGaMgtg;sIuet wL / kMlaMg meridional Nφ
RtUv)anTTYlBIlMnwgénGgÁesrIedayeGaybnÞúkxageRkAesμInwgkMlaMg meridional xagkñúg mann½yfa
− π (d / 2)2 wL = 2π (a sin φ )Nφ . edaysar d / 2 = a sin φ eyIgTTYl)an

wL a
Nφ = − (11.66a)
2
dUcenH Nφ CatMélefrelIkMBs; shell TaMgmUl dUcEdleXIjenAkñúgrUbTI 11.15.
Nθ EdlbNþalBIbnÞúkGefr wL KW

awL ⎛1 ⎞ awL
Nθ = −awL cos 2 φ + = awL ⎜ − cos 2 φ ⎟ = cos 2φ (11.66b)
2 ⎝2 ⎠ 2
sMrab;krNI Nθ = 0 / mMu φ = 45o . dUcenH kugRtaMg shell EdlbNþalBIkMlaMg tangential Nθ sMrab;
φ < 45o CakugRtaMgsgát; Edlkat;bnßysñameRbH. BIkarEbgEckkugRtaMg tangential Nθ eKGacsnñi-
dæan)anfadMbUlrbs;GagsþúkmanlkçN³ flat ¬pleFob h' / d enAkñúgrUbTI 11>15(b) minRtUvFMCag
1 / 8 ¦ EdlebtugTaMgGs;nwgrgkugRtaMgsgát;EdlbNþalBI Nφ nig Nθ enAeBlEdlmMu φ < 51o 49'

sMrab;kMlaMg meridianal nig 45o sMrab;kMlaMg tangential.



dUcEdl)anerobrab;BIxagedIm RbePTTMrenARtg; springing ¬RbsinebIva restrained¦vanwgbegáIt
RbtikmμminkMNt;EdlbgáeGayekItmankugRtaMgedaypÞal; nigkugRtaMgBt;enAkñúg shell Ek,r springing.
dUcenH eKGacGnuvtþ bending theory sMrab; plate nig shell edIm,IkMNt;kugRtaMgBt;. xageRkamenH
Cakarerobrab;BIkarsikSaKNna ring beam eRbkugRtaMgenARtg; springing edIm,IRbqaMgnwgbgÁúMkMlaMg
sgát; meridional edk Nφ EdkeFVIeGayEKmrbs; dome cl½tcUlkñúg.



BIsmIkar 11.65b nig11.66a eKGacsresrkMlaMg meridional Nφ sMrab;bnÞúkpÞal; wD kñúgmYy
ÉktþaépÞ nigbnÞúkGefrBRgayesμI wL kñúgmYyÉktþaépÞRbeyal (unit projected area)
⎛ wD wL ⎞
N φ = − a⎜
⎜ 1 + cos φ + 2 ⎟
⎟ (11.67)
⎝ ⎠

Edl a = d / 2 sin φ CakaMrbs; sheall.
cMNaMfakMlaMg (thrust) Nφ køayCakMlaMgbBaÄrRtg; springing ¬ φ = π / 2 ¦ én hemisphe-
rical dome nigesμInwg W = a / 2(2 wD + wL ) kñúgmYyÉktþaTTwg. cMeBaHtMélepSgeTotrbs; φ / Nφ

manlkçN³eRTt ehIyeKRtUvtMélénbgÁúMedkrbs;vasMrab;karsikSaKNna ring beam eRbkugRtaMgenA
Rtg; springing EdleKGacehAfa shell rim. bgÁúMkMlaMgedkenHKW p = Nφ cosφ . RbsinebI P Ca
kMlaMgeRbkugRtaMgkñúgmYykMBs;FñwmenAkñúg ring beam enaHBIsmIkar 11.1a P = pd / 2 ehIy

P=
d
2
(
Nφ cos φ ) (11.68)

Cak;Esþg RbsinebIeKGacGnuvtþ P edaypÞal;eTAelI dome rim enaHeKGackMNt;kugRtaMgenA
kñúg dome edaysmIkar 11.67. CaTUeTA vamingayRsYleT edaysareKRtUvkarEdkeRbkugRtaMgkñúg
brimaNd¾eRcIn Edl P minGacsßitenAkñúgkMras;d¾esþIgrbs;CBa¢aMg)an ehIykugRtaMgenAkñúgebtugRtg;
tMbn; rim GacmantMélx<s;Nas;. dUcenH eKRtUvkardak; edge beam EdlbMElg shell eGayeTAeRKOg
bgÁúMkMNt;edaysþaTic

kMraldMbYlekagminkMNt;edaysþaTiceRbkugRtaMg
lkçxNÐRBMEdnd¾samBaØbMputEdlTTYl)anenAeBlEdlRbtikmμ edge beam manTisbBaÄr nig
TMrminman restraint dUcbgðajenAkñúgrUbTI 11>16 Edl dome thrust Nφ qøgkat;TIRbCMuTMgn;rbs;Fñwm.
RbsinebIeKkat;vatamExS A − A kMlaMgedk Nφ cosφ eFVIeGayEKm dome cl½tcUlkñúg)ancMgay

Δs =
d
2 Et
(
Nθ − μNφ ) (11.69)

Edl μ= pleFobB½rs‘ug ~ 0.2 sMrab;ebtug
d = ElVg shell (shell aspan)



ehIyeKTTYl)ankMlaMgÉktþa tangential BIsmIkar 11.65 dUcxageRkam
wD d ⎛ ⎞ wL d
⎜ 1 + cos φ − cos φ ⎟ − 4 sin φ (cos 2φ )
1
Nθ = ⎜ ⎟ (11.70)
2 sin φ ⎝ ⎠

Rcasmkvij kMlaMg meridional Nφ eFVIeGay ring beam cl½tecjeRkA)ancMgay
Nφ (cos φ )d 2
Δb = (11.71)
4 Ebh

dUcenHeKRtUvmankMlaMgeRbkugRtaMgRKb;RKan;edIm,Icl½t ring beam cUlkñúgCamYynwgcMgaysrub
ΔT = Δ s + Δ b

dUcenHkMlaMgsrubEdlmanGMeBIelImuxkat; ring beam KW
P=
bh
t
(
Nθ − μNφ +
2
) (
d Nφ cos φ ) (11.72)

Edl h CakMBs; ring beam srub. kareRbobeFobrvagsmIkar 11.72 nig 11.68 bgðajfakMlaMgeRbkug
RtaMgRbsiT§PaBEdlRtUvkarelIkmunmantMélFMCagkMlaMgeRbkugRtaMgRbsiT§PaBEdlRtUvkarelIkeRkay.
TMhMénkarekIneLIgenHmanRbEhlBI 5% eTA10%. lkçxNÐdUcKñamanlkçN³Bit sMrab; dome EdlExS



rbs;kMlaMgBI dome minkat;tamTIRbCMuTMgn;rbs; ring beam ehIyFñwmRtUv)anP¢ab;y:agrwgeTAnwgCBa¢aMg
dUcenAkñúgrUbTI 11>17 (a). eKGacTTYltMélRbhal;RbEhlrbs;kMlaMgeRbkugRtaMgtMrUvkar P eday
begáIntMélrbs; P enAkñúgsmIkar 11.68 cMnYn 10%. kñúgkrNIEbbenH kugRtaMgenAkñúg shell Rtg;tMbn;
springing GacxusBIkugRtaMgEdlTTYl)anBI membrane solution y:agxøaMg ehIyeKRtUveFVIkarEktMrUv

bending solution.

RbsinebIkMlaMgeRbkugRtaMg radial edkenAkñúg ring beam mantMélFMCagtMrUvkar enaHkMhUcRTg;
RTayedaysarkarBt;d¾FMnwgekItmanenAkñúg shell beam dUcbgðajenAkñúgrUbTI 11>17 (b) CamYynwg
karekIneLIgéntMélrbs;kMlaMg tangential Nθ y:agxøaMg ebIeRbobeFobCamYynwgkarekIneLIgénkMlaMg
meridional Nφ . CalT§pl kugRtaMgBt;enAkñúgebtugenARtg;tMbn;EdlT§iBlGacelIskugRtaMgGnuBaØat

GtibrmaeRkamGMeBI service load. RbsinebI kMlaMgeRbkugRtaMgedImmuneBlrgkMhatbg;KW Pi enaHRk-
LaépÞrbs;muxkat;FñwmKW
Pi
Ac = (11.73)
fc

Edl Pi = kMlaMgeRbkugRtaMgedIm P / γ



kugRtaMgsgát;GnuBaØatenAkñúgebtug
fc =

γ = PaKrykugRtaMgEdlenAesssl;.
eKcg;rkSaeGaytMélrbs; f c mantMélTab Rbhak;RbEhlnwg 0.2 f 'c nigminelIs 800 psi
eTA 900 psi eT edIm,IeFVIeGaybMErbMrYlrageFobd¾FMEdlekItmanenAkñúg edge ring beam mantMélGb,-
brma EdlRtLb;mkvijeKGacbegáItkugRtaMgFMenAkñúg shell Rtg;tMbn; springing.
RkLaépÞrbs;EdkeRbkugRtaMgenAkñúg dome ring KW
Pi
A ps = (11.74b)
f pi

Edl CakugRtaMgGnuBaØatenAkñúgEdkrgeRbkugRtaMgmuneBlxatbg;. RbsinebIeKminRtUvkarkMNt;
f pi

A ps CatMélsuRkit enaHeKGacykRkLaépÞEdkeRbkugRtaMg

W cot φ
A ps = (11.74b)
2πf pe

Edl W= bnÞúkefr nigbnÞúkGefrsrubenAelI dome EdlbNþalBI wD + wL
f pe = eRbkugRtaMgRbsiT§PaBeRkayeBlkMhatbg;

eKGacykkMras;Gb,bramrbs; dome EdlRtUvkaredIm,ITb;Tl; buckling dUcxageRkam
1.5 pu
hd = a (11.75)
φβ i β c Ec

Edl a= kaMrbs; dome shell
pu = sMBaFKNnaÉktþaBRgayesμI ultimate EdlbNþalBIbnÞúkefr nigbnÞúkGefr

= (1.2 D + 1.6 L ) / 144
φ= emKuNkat;bnßyersIusþg;sMrab;sMPar³EdlrgkugRtaMgsgát; = 0.65
β i = emKuNkat;bnßy buckling sMrab;bMErbMrYlrbs;épÞragEsV‘rEdlbNþalBIPaBminl¥
β i = (a / ri )2 / Edl ri ≤ 1.4a
β c = emKuNkat;bnßy buckling sMrab; creep/ sMPar³Edl nonlinearity nigsñameRbH
= 0.44 + 0.003WL / b:uEnþminFMCag 0.53 .

Ec = m:UDuleGLasÞicedImrbs;ebtug = 57,000 f 'c psi (4,700 f 'c MPa )



11.10. GagebtugeRbkugRtaMgEdlmanEdkeRbkugRtaMgvNÐ
Prestressed Concrete Tanks with Circumferential Tendons
CMnYseGaykarrMuEdkeRbkugRtaMg (wire or strand) dUcEdl)aneFVIenAkñúg preload system eKeRbI
EdkeRbkugRtaMg (tendons) edkxagkñúg b¤xageRkA. kabeRbkugRtaMgTaMgenHrgkugRtaMgeRkayeBleK
dak;BYkvaenAkñúgCBa¢aMg. Edk post-tensioning bBaÄrRtUv)aneKeRbIkñúgCBa¢aMgCaEpñkmYyénEdkBRgwg
bBaÄr. CBa¢aMgebtugGaccak;enAnwgkEnøg b¤cak;Rsab; ehIyeKcat;TuksñÚlCBa¢aMgCaEpñkmYyénCBa¢aMg
ebtugEdlrgeRbkugRtaMgvNÐ. RbePTsMNg;enHminmaneRbIEdk diaphragm dUcRbePTGagEdlrMuEdk
eRbkugRtaMg ( wrapped-wire prestressing) EdlCBa¢aMgGagGacman b¤k¾KμanEdk diaphragm.
EdkEdlrgeRbkugRtaMgxagkñúgRtUv)ankarBaredaykMras;ebtugkarBardUctMrUvkarrbs; ACI 318
ehIyeKRtUvbMeBjkñúgbMBg; (duct or sheathing) CamYynwgsMPar³EdlkarBarERcH b¤ grouted. eKRtUv
karBarEdk bonded post-tensioned eday portland cement grout dUckarTamTarenAkñúg ACI 318
ehIyeKRtUvkarBarkabeRbkugRtaMgxageRkACamYynwg shotcrete cover EdlmankMras;Gb,brma 1in.
(25mm ) .
dMeNIrkarsikSaKNnaCBa¢aMgmanlkçN³RsedogKñanwgkarsikSaKNnaGagragmUlEdlrgeRbkug
RtaMgedaykarrMuEdkeRbkugRtaMg ehIyvaTamTarnUvkarRtYtBinitüsñameRbHdUcKña. eKRtUvykkugRtaMg
sgát;esssl;Gb,brmaenAkñúgebtugCBa¢aMgeRkayeBlrgkMhatbg;eRbkugRtaMgTaMgGs;esμInwg 200 psi
(1.4MPa ) kñúgkarsikSaKNna enAeBlEdlGagRtUv)anbMeBjdl;nIv:UKNna. RbsinebIGagKμanKMrb eK
RtUvykkugRtaMgesssl;enAcugCBa¢aMgesμInwg 400 psi(2.8MPa ) Edlkat;bnßyCaragbnÞat;rhUtdl;
tMélmYyEdlmintUcCag 200 psi enAcMgay 0.6 Rh BIcugénnIv:UsarFaturav.

RbePT)atCBa¢aMg nigtMNdMbUgragekag
BIkarerobrab;xagelI eyIgeXIjy:agc,as;falkçxNÐRBMEdnenARtg;)atrbs;GageRbkugRtaMgrag
mUl nigenARtg; ring beam support sMrab;dMbUgragekagkMNt;nUvlkçN³énkarGnuvtþ lkçN³esdækic©
nigeCaKC½yénkarsikSaKNnaTaMgmUl. Cavi)ak bTBiesFn_CaeRcInEdlTTYl)anBIkarbegáIttMNeRkam
lkçxNÐTaMgenHKWmantMélxøaMgNas;. karlMGitBItMNRtUv)anbgðajenAkñúgrUbTI 11>18 dl;TI 11>22.



11.11. Step-by-Step Procedure for the Design of Circular Prestressed
Concrete Tanks and Dome Roofs
viFIsakl,g nigEktMrUv (trial-and adjustment procedure) RtUv)anENnaMsMrab;karsikSaKNna
GagragmUlebtugeRbkugRtaMg nigdMbUg shell³
!> eRCIserIsRbB½n§eRbkugRtaMg RbePTEdkeRbkugRtaMg ersIusþg;ebtug nigRbePTTMr EdleyIgGac
rk)anenAkñúgtMbn;.
@> kMNt;sMBaFsMPar³EdlsþúkkñúgGagmanGMeBIelICBa¢aMg γH sMrab;sarFaturav nig p sMrab;sarFatu



]sμ½n. eRbIkarBRgayragctuekaNBñaysMrab;GgÁFaturwgEdlpÞúkenAxagkñúgGag.
kMNt;kMlaMg ring Éktþa F = γ (H − y )r sMrab;)atrGileBjelj Edl r CakaMGag nig y Ca
cMgayBI)at.
#> BItarag 11>4 dl; 11>16 eRCIserIsemKuNm:Um:g;bBaÄreTAtamRbePTbnÞúk niglkçxNÐ
restraint rbs;)atEdlbNþaledaysMBaFsarFaturav

My =+
1
[βM oφ (βy ) + Qoζ (βy )]
β
nigkMNt;kMlaMgTaj ring radial edk
γrt
Qo = +(2 βH − 1)
( ) 12 1 − μ 2
ehIy Q = (F − ΔQ ) EdlbMErbMrYlén
y y

6( − μ ) 2
ΔQ = +
1
y (βM Ψ (βy ) + Q θ (βy ))
o o
β 3 rt 2

nig β = [3(1 − μ )]
2 1/ 4

(rt )1 / 2
Edl μ = 0.2 sMrab;ebtug.
$> kMNt;emKuN membrane C BItarag 11>4 rhUtdl; 11>16. KNnakMlaMg ring F = CγHr .
%> KNnam:Um:g;bBaÄreRKaHfñak;enAkñúgCBa¢aMgEdleRbIemKuN membrane C . smIkarsMrab;m:Um:g;
EdlbNþalBIbnÞúksarFaturavKW
M y = C (γH 3 + pH 2 )

b¤ M y = CpH 2

EdlbNþalBIbnÞúk]sμn½. KNnam:Um:g;enARtg;)at nigRtg;cMnuceRKaHfñak;EdlmancMgay y BI)at.
^> eRCIserIskMlaMgeRbkugRtaMgbBaÄr.
&> KNnakugRtaMgebtugkat;tamkMras;rbs;CBa¢aMgsMrab;lkçxNÐGagTeT nigsMrab;GageBj.
eKGnuBaØatkugRtaMgsgát;tamG½kSesssl;Gtibrma f cv = 200 psi eRkamGMeBIr service load
ehIykugRtaMgTajGtibrma f t = 3 f 'c dUcbgðajenAkñúgtarag 11>17.
*> sikSaKNnaEdkeRbkugRtaMgedk nigEdkeRbkugRtaMgbBaÄrEdlkugRtaMgkMNt;manenAkñúgtarag
11>18.
(> KNnam:Um:g;emKuN M u EdleRbIemKuNbnÞúkEdleGayenAkñúgEpñk 11.7.2. m:Um:g;tMrUvkar M n =
M u / φ Edl φ = 0.9 . KNnaersIusþg;m:Um:g; nominal EdlGacman M n = A ps f ps (d p − a / 2 )



b¤ M n = Aps f ps (d p − a / 2)+ As f y (d − a / 2) . m:Um:g;EdlGacman M n RtUvEtFMCag b¤esμInwg
m:Um:g;tMrUvkar M n .
!0> sikSaKNnaRbEvg L rbs; ring enARtg;)atrbs;CBa¢aMgBIsmIkar
2CH 2
L =
2

1+
(t / h )3
(dt )2
Edl t CakMras;rbs;CBa¢aMg nig h CakMras;rbs;kMral)at.
!!> KNnaPaKryrbs;eRbkugRtaMgenAkñúg)atEdlRtUvepÞreTACBa¢aMgBIrUbmnþ
1
R=
1+ S
Edl S = 1.1(h / t )× (d / t )1 / 2 .
enAeBlEdl rim xageRkArbs; slab ring rgkMlaMgsgát;edaykMlaMg radial Rtg; rim enaHtMél
rbs; S RtUv)anEksMrYleTACa
1/ 2
1 ⎛ h ⎞⎛ d ⎞
S1 = ⎜ ⎟⎜ ⎟
K ⎝ t ⎠⎝ t ⎠
⎛d2 +d2 ⎞
Edl K =⎜ o
⎜d −d
2 2
−μ⎟
⎟
⎝ o ⎠
EdlkñúgenH Ggát;p©itxagkñúg
do =

d = Ggát;p©itkMral ring xagkñúg = d o − 2 L .

!@> RtYtBinitütMrUvkarkMras;CBa¢aMgGb,brma nigKNnaPaBdab radial eGLasÞicedImEdlKμanTb;
(unrestrained initial elastic radial deflection)

Fi r
Δi =
t co Ec
E;dl (
Ec = 57,000 f 'c psi 4,700 f 'c MPa )
t co = kMras;rbs;sñÚlCBa¢aMgenARtg;cug b¤)atrbs;CBa¢aMg
d
r=
2
PaBdab radial cugeRkay Δ f = 1.7Δ i .
!#> f<k;EdkBI)ateTACBa¢aMgEbbNaedayeGayEdkbgðÜscUleTAkñúgCBa¢aMgcMgay y2 = 1.8 rtco
b¤ 3 ft.(0.9m) edayykmYyNaEdlFMCag. dUcKña eKRtUvFanafaEdkbBaÄr nominal Gb,brma
enARtg;tMbn;)atKW


As = 0.005t co
!$> epÞógpÞat;TMhMsñameRbHGtibrma wmax = 4.1×10−6 ε ct E ps I x
Edl ε ct = bMErbMrYlrageFobépÞrgkarTajenAkñúgebtug = (λt f p )/ E ps
f p = kugRtaMgCak;EsþgenAkñúgEdk

f pi = eRbkugRtaMgedImmuneBlxatbg;

λt ~ f p / f pi
8 ⎛ s 2 s1tb ⎞
I x = grid index = ⎜ ⎟
π ⎜ φ1
⎝
⎟
⎠
KMlatEdkkñúgTis “1”
s1 =

φ1 = Ggát;p©itrbs;EdkenAkñúgTis “1”
s 2 = KMlatEdkkñúgTis “2”

tb = kMras;karBarEdkEdlKitdl;p©itrbs;Edk

cMNaMfa TTwgsñameRbHGnuBaØatGtibrma wmax = 0.004in. sMrab;GagsþúksarFaturav.
!%> sikSaKNnadMbUgragekageRkaykareRCIserIsRbePTtMNenARtg;cugrbs;CBa¢aMgGag. kMNt;pl
eFobénkMBs;rbs;dMbUgragekag h' elI)at d rbs;vay:agNamineGay h' / d > 1/ 8 .
KNnakMlaMgeRbkugRtaMg radial tamTisedktMrUvkar P sMrab; edge beam BIsmIkar
bh
P= (Nθ − μNθ ) + φ
d N cos φ ( )]
t 2
w d ⎡ 1 ⎤ w d
Edl Nθ = D ⎢
2 sin φ ⎣1 + cos φ
− cos φ ⎥ − L (cos 2φ )
⎦ 4 sin φ
⎛ wD wL ⎞
N φ = − a⎜
⎜ 1 + cos φ + 2 ⎟
⎟
⎝ ⎠
nig kMBs;srubrbs;Fñwm rim
h=

b = TTwgFñwm rim

wD = GaMgtg;sIueténbnÞúkpÞal;rbs; shell kñúgmYyÉktþaépÞ ¬bnÞúkefr¦

wL = GaMgtg;sIuetrbs;bnÞúkGefrRbeyal

!^> KNnamuxkat; ring-edge beam
Pi
Ac =
fc



Edl kMlaMgeRbkugRtaMgedIm = P / γ
Pi =

γ = PaKrykugRtaMgesssl;
f c = kugRtaMgsgát;GnuBaØatenAkñúgebtug ¬minRtUvFMCag 0.2 f 'c ¦ b:uEnþminRtUvFMCag

800 ~ 900 psi enAkñúg edge beam.

!&> KNnaRkLaépÞrbs;EdkeRbkugRtaMgrbs; edge beam
Pi
A ps =
f si

Edl f si CakugRtaMgGnuBaØatenAkñúgEdkeRbkugRtaMgmuneBlxatbg; b¤
W cot φ
A ps =
2πf pe

RbsinebIeKminviPaKedaysuRkit. kñúgtYcugeRkay W CabnÞúksrubefr nigGefrenAelIdMbUgrag
ekag EdlbNþalBI wD + wL nig f pe CaeRbkugRtaMgRbsiT§PaBeRkayeBlxatbg;.
!*> RtYtBinitükMras;kMraldMbUgekagGb,brmaEdlRtUvkaredIm,ITb;Tl;nwg buckling
1 .5 p u
hd = a
φβ i β c Ec

Edl a= kaMrbs; dome shell
pu = sMBaFKNnaÉktþaBRgayesμI ultimate EdlbNþalBIbnÞúkefr nigbnÞúkGefr

= (1.2 D + 1.6 L ) / 144
φ= emKuNkat;bnßyersIusþg;sMrab;sMPar³EdlrgkugRtaMgsgát; = 0.65
β i = emKuNkat;bnßy buckling sMrab;bMErbMrYlrbs;épÞragEsV‘rEdlbNþalBIPaBminl¥
β i = (a / ri )2 / Edl ri ≤ 1.4a
β c = emKuNkat;bnßy buckling sMrab; creep/ sMPar³Edl nonlinearity nigsñameRbH
= 0.44 + 0.003WL / b:uEnþminFMCag 0.53 .

Ec = m:UDuleGLasÞicedImrbs;ebtug = 57,000 f 'c psi (4,700 f 'c MPa )

rUbTI 11>23 bgðajBI step-by-step flowchart sMrab;CMhanEdlENnaMkñúgkarsikSaKNnaGag
ebtugeRbkugRtaMgragmUl nigdMbUgragekagrbs;va.


Xi. prestressed concrete circular storage tanks and shell roof

Xi. prestressed concrete circular storage tanks and shell roof

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Mais de Chhay Teng

Mais de Chhay Teng (20)

Xi. prestressed concrete circular storage tanks and shell roof