M. J. N. PriestleyReader in Civil EngineeringUniversity of CanterburyChristchurch, New Zealand

Analysis and Design ofCircular PrestressedConcrete Storage Tanks

P restressed concrete circular tanksare widely used as water supply res-

ervoirs, sewage digesters, and for stor-age of such diverse materials as acid, oil,cement, hot effluent from pulp andpaper factories, and other applications.

Most tanks are cylindrical and groundsupported but elevated tanks are usedfor water supply in areas where local to-pography is not suitable for natural pro-vision of adequate head for town supply,or where industrial requirements de-mand short-term provision of largequantities of high pressure water. Suchtanks are visually prominent, andaesthetic considerations frequentlydictate that more complex shapes thansimple elevated cylinders be utilized.

Analysis of cylindrical tanks underaxisymmetric loads, such as fluid or gaspressure, is covered by standard texts onthe design of circular reservoirs, and isonly briefly summarized here. Loads arecarried by two contributing mecha-nisms: hoop stress and bending of the

wall along the surface generators. Fig. Iillustrates the radial deflection of apinned-base ground supported cylindri-cal tank under internal fluid pressure,and defines the nomenclature to be usedhereinafter.

Consideration of compatibility re-quirements results in the characteristicequation for radial deformation:

where 4(h) is the loading function [4(h)= pgh for fluid ]oading, where p is themass density and g is the accelerationdue to gravity 1, K = E, 11(1 – v), I –t3/12, and:

4 I a=R2t2

is the characteristic of the equation,where R is the tank radius, t is the wallthickness, and v is Poisson's ratio for theconcrete.

64

Eq. (1) has the typical solution:

^h

y=PI+[e y (Asin^2+Bcos--1

y2 ( ah ah+ e C sin v2 + D cos 2̂ (2)

where PI is the load-dependent par-ticular integral, and A, B, C, D are con-stants of integration dependent onboundary conditions at the wall baseand top.

Once Eq. (2) is solved for y, the coin-plete stress distribution in the tank maybe calculated as follows:

Hoop Tension

Since circumferential length andradius R are related by the constant 2ir, aradial deformation y causes a circumfer-ential length increase of 21ry. The cir-cumferential strain will thus he:

er, = 2ims 12-irR = y /R

and hence hoop tension stress is givenby:

f„= E, y/R

where E, is the concrete modulus of

elasticity.

Vertical Bending

From the beam equation, verticalbending moments are found after dou-ble differentiation of Eq. (2), as:

E, I d'Ly(4)

1 – v'zdh2

if moments have been calculated forunit wall width, then surface bendingstresses in the vertical direction can befound from:

(5)

SynopsisAspects of the analysis and aesign

of prestressed concrete storage tanksare discussed, and a simple analogyfor the analysis of circular prestressedtanks is described.

The method, which is suitable foruse with small microcomputers, oreven the larger programmable calcu-lators, is capable of modelling bothcylindrical tanks, and tanks with dou-ble curvature under a wide range ofloading, including dead load, fluid orgas pressure, thermal load, and pre-stressing.

The significance of actions often ig-nored in tank design, including shrink-age and swelling of the walls, creepredistribution of prestress, and ther-mal effects are examined in somedetail.

Comparisons are given betweenresults predicted by the frame analogyand more sophisticated analyticalmethods for a ground supported cy-lindrical reservoir, and an elevateddoubly curved tank.

Surface Circumferential StressSince the loading and structure are

both axisymmetric, radial Iines con-necting points on the inside and outsidesurfaces of the tank wall must remainradial after application of loading. Thiseffectively means that transverse Pois-son's ratio strains from vertical bendingmoments cannot develop, and in conse-quence, the wall is in a state of planestrain for vertical bending. Conse-quently, circumferential Poissonstresses, vf,,, are developed in the walls.

These Poisson stresses are commonlyignored in design, but can be quite sig-nificant, particularly when high bendingmoments develop from base fixity. Final

PCI JOURNAUJuly-August 1985 65

i pgH

(al Continuous WinklerFoundation

Pin-endedstrut

E , ^I

Area Ai

bl Discretised Foundation I c ] Detoil of Equivalent Strut- Frame Analogy

for i zonNinklerlounda

circumferential stresses at the wall sur-face are thus:

fC = fA ± JV

R t2 (6)

Solution of Eq. (1) is tedious and un-suitable for design office usage. Con-sequently, it is common to use design

charts or tables, such as those developedby Creasy' or Ghali, 2 which typicallygive vertical bending moment and hooptension force as functions of the tankshape factor HEIRt and fraction of wallheight x = h/H. Although such tablesand charts are extremely useful, they arelimited in applicability.

Generally, loading must he either hy-drostatic pressure or uniform (gas) pres-

Fig. 1. Pinned-base tank under fluid pressure.

Fig. 2. Beam on elastic foundation and frame analogy simulations of cylindrical tank wall.

66

sure over the full height (i.e,, partiallyfilled tanks are not considered), orbending moment or shear force appliedto top or bottom of the wall. Tank wallsmust be vertical and generally of uni-form thickness, although Ghali 2 in-cludes data for tapering walls.

FRAME ANALOGY

Cylindrical Tanks WithAxisymmetric Loading

In the introduction it was establishedthat the response of cylindrical tankwalls under rotationally symmetricloads involved load sharing betweentwo mechanisms: hoop tension and ver-tical bending. Eq. (1) is developed fromthe compatibility requirement that ra-dial deformations of the two mecha-nisms must be identical at all points.Behavior can be described as basic ver-tical beam action, where radial defor-mations are further constrained by thestiffness of hoop action.

The analogy to Beam on ElasticFoundation (BEF) analysis3 is obvious,and has been used in the past to gener-ate design tables for tanks. Verticalbending of the tank wall is representedby beam action, and the elastic radialstiffness of hoop action is representedby the spring stiffness of the Winkler'foundation.

Consider a unit height of tank wallsubjected to pressure p, and freed fromcantilever action. From considerationsof simple statistics, the radial expansionswill be:

pR2

y W, (7)

Fig. 2a shows a unit circumferentiallength of tank wall supported by a hori-zontal Winkler foundation. Deflection ofthis foundation under pressure p willbe:

y = p/k (8)

where k is the subgrade modulus in Nlms(Ibs/in 3),

Thus, Fig. 2a can exactly model thewall behavior, providing the subgrademodulus is given by:

tE ,

k=R 2 (9)

A more convenient method of simula-tion, capable of modelling thicknessvariation of the tank walls and complexloading, is to replace the continuousfoundation of Fig. 2a by the discretesystem of Fig. 2b, where the tank wall isdivided into a number of vertical beamelements whose connecting nodes aresupported by lateral pin-ended strutsfrom a rigid foundation.

This system can be solved by simpleframe analysis, or relaxation programs.Beam members are given the local ver-tical bending stiffness properties of thesection of wall represented, while strutproperties model the radial stiffness ofthat portion of wall extending midway toadjacent nodes above and below, shownshaded in Fig. 2b, and in detail in Fig.2c for a tapering wall.

Thus, if unit circumferential wallwidth is again considered, simulation ofhoop stiffness requires:

2pR 2 _ p AhI, (10)_ (t, + t2 )E, A1E,

Hence, if for convenience the strutmodulus of elasticity E, is set equal toE,, then:

A, R (t,+t2)Ah

I, 2R2 (11)

Either the cross-sectional area A1, orthe length 1, of the strut (or both), may bevaried to obtain the required similitude.Loads applied to the vertical beam mustbe commensurate with the width of walladopted for analysis (e.g., unit width inthe above development). Vertical bend-ing moments in the tank wall are di-rectly modelled by the moments de-

PCI JOURNAL/July-August 1985 67

I;

Ntp

OO40

Fig. 3. Example: 10,000 m 3 (2.65 million gallon) cylindrical waterreservoir. (Note: 1 mm – 0.03937 in.)

veloped in the beam members, and cir-cumferential stresses are found from thenode radial displacements, using Eq.(6).

Loading applied to the frame mem-bers corresponds to the loads applied tothe width of wall modelled, and willgenerally be applied as resultant jointloads without significant loss of accu-racy. Greater accuracy will result by ap-plication of the actual distributed loadsto the beam members as member loads.

As an example in the use of the frameanalogy, the 10,000 m 3 (2.65 milliongallon) cylindrical water reservoirshown in Fig. 3 is modelled under a 9.5m (31.2 ft) hydrostatic pressure dis-tribution. Fig. 4a represents the framesimulation, based on ten vertical beammembers, modelling a width of 1 m(39.37 in.) in the circumferential direc-tion. Since the wall thickness is constantat 225 mm (8.86 in.), the vertical beamstiffness is:

1 x 0.2253ra =

12

Ib = 0.949 x 10-3 m' (2282 in.'

For all except the top strut (i.e., Mem-bers 12 to 20 in Fig. 5a), the strut lengthis arbitrarily chosen as l = 2 m (6.5 ft).Therefore, from Eq. (11):

2 x 0.225 x 1Ar18.725'

= 1.283 x 10-a mz (1.99 in,2)

For Member 11, to keep memberproperties the same as Members 12 to20, the length is increased to 4 m (13 ft),since the tributary wall height is 0.5 m(1.64 ft).

A standard computer program used forroutine analysis of structural frames wasused to solve the problem, with hydro-static loading simulated by equivalentjoint loads at beam nodes, as shown inFig. 4a. Results from this simulation areshown plotted as discrete points in Fig.4b, and compared with results from ananalysis using Creasy's tables,' whichare plotted as continuous curves.Agreement between the two methods is

68

to within the precision of Creasy's coef-ficients.

This simple comparison indicates theaccuracy of the proposed method forcases that can be modelled by existingtables, and gives confidence in the useof the analogy for load cases that areoutside the range of applicability ofcharts and tables.

PRESTRESS SIMULATIONThe frame analogy can also be used to

calculate the stresses in the tank wallsby prestressing. Consider the case of atank constructed of precast wall panels,and stressed by tendons in internalducts as shown in Fig. 5. If F is thetendon force, then the radially inwardline load applied to the tank by a singletendon, as shown in Fig. 5a, is:

P - FIR (12)

The stresses induced by prestress canbe found by analyzing the tank wallunder the loads P at the appropriateheights, as shown in Fig. 5b. Alterna-tively, if the spacings is a small fractionof the wall height, the prestress may besimulated as an equivalent radially in-wards pressure of

FP=Prs = S (13)

In the case of a tank stressed by wirewrapping, this would be the appropriateapproach. However, for a tank pre-stressed by individual tendons of highprestress force, the local stresses in-duced by the discrete nature of the pre-stress line loading may be significant, orthe stresses induced during stressing

—Gm -{

0.2kN^0.5m

II10

5.Ik N^ Me,2 Beam Members Creasey's Tables1 -10 I .. Frame Analogy

14.7k Unit width, . a 83 I =0.949;103m4

24.5kNi 14I34.3kN-L 4

t5

6Strut MembersH=fpm 44.2k L 16 11-20 = HooP-

6 A =1,283 K143m2 Vertical- outside50.OkN-a 17 inside -I

1 outside63.8NN^

I73.5kN^ 19

I83.4kN-

920 /^^

-6 -4 -2 C 2 L 6p9H=93.2kN/m Compression Tension Stress (MPo)2m

(a) Frame Simulation of Tank in Fig 3 fb) Stresses

Fig. 4. Comparison of stresses under hydrostatic load predicted by frame analogy andCreasy's tables for 10,000 m 3 (2.65 million gallon) reservoir.(Note: I m - 39.37 in., 1 MPa = 145 psi.)

PCI JOURNAL`Juiy-August 1985 69

Stressedtendon

Equivalentinwardspressure

non

(al Plan P- F (bl Elevation

Fig. 5. Forces applied to tank wall by prestressing tendons.

operations by stressing a single tendonmay be of interest. In such cases theprestress should he modelled by dis-crete radial loads to the frame analogy.

As an example of this, the framesimulation of Fig. 4a was used to inves-tigate stresses induced by a tendon lo-cated 3.5 m (11.1 ft) above the base andstressed to 860 kN (193.4 kips). FromEq. (12), with R = 18.5 m (60.7 ft), P =46.5 kN/m (3.19 kips/ft), and for a modelrepresenting a unit circumferentiallength, as in Fig. 4a, this represents theradial load to be applied at the mid-height of Member 7 (see Fig. 4a).

Results from the frame analysis areshown in Fig. 6. Note that the base con-dition has been assumed pinned, as forthe hydrostatic Ioading. The high localbending stresses resulting from the pre-stress line load are apparent in Fig. 6,and the Poisson's ratio stresses from thevertical bending have a significant in-fluence on the peak hoop stresses.

It is clear that simulation of prestressas a series of radial loads will provide amore realistic simulation than that pro-vided by an equivalent pressure, usingtables or design charts. This method ofanalysis also enables the prestress de-sign to be optimized since variation inindividual tendon positions can be eas-ily modelled.

CREEP DISTRIBUTIONOF PRESTRESS

It is common for codes of practice torequire a residual compression in thewalls of prestressed tanks under fluidloading. This implies a requirement forcircumferential prestress at the tankbase.

When thermal stresses are consid-ered, the required level of circumfer-ential prestress at the tank base can beas high as 4 to 5 MPa (580 to 725 psi),since it has been shown that circumfer-ential bending stresses of significantmagnitude occur due to ambient tem-perature gradients through the tankwall.

Circumferential precompression can-not he achieved if the prestress isapplied with the wall base pinned orfixed, since radial, and hence circumfer-ential strain cannot develop. Con-sequently, it has been common practiceto stress some or all of the circumfer-ential cables with the wall free to slideradially, and then pin the base to im-prove performance under fluid pressure.This stressing technique is particularlybeneficial when tanks are subject toseismic loading.

It is not often recognized that the ac-tion of pinning the base after prestress is

70

1aE

v8xJ.II

/6Vertical -Hoop-

inside-- outsidetside nside

4 ` .46.5kN

\ f

-2.0 -1.0 0 1.0 2.0Compression Tension Stress (MPa)

Fig. 6. Stresses induced in 10,000 m 3 (2.65 milliongallon) reservoir by single prestressing tendonstressed to 860 kN. (Note: 1 kN = 225 Ibs, 1 MPa =145 psi, I in = 39.37 in.)

applied causes a redistribution of pre-stress due to restraint of creep deflec-tions. Consider the radial deflection ofthe walls under circumferential pre-stress, as shown in Fig. 7. If all the pre-stress is applied with the base free toslide, an initial elastic radially inwardsdisplacement occurs.

If the walls remain free to slide, creepeffects would gradually increase thedisplacements as shown. However, ifthe base is pinned after stressing, fur-ther radial displacement at the basecannot occur, while creep displace-ments higher up the wall are largely un-affected. The consequence is the de-velopment of a radially outwards reac-tion which will cause vertical bendingand a reduction in circumferential pre-stress at the base.

The problem of creep redistributionclue to structural modification can besolved by a rate of creep analysis . 6 Thesolution indicates that the final behavior

can be considered to he identical to thatwhich would result from an elasticanalysis considering a portion 18 1 of theprestress to be applied in the initialstructural form (sliding base) with theremainder jar = 1 - f3; of the prestressbeing applied in the modified form(pinned base). From the rate of creepanalysis, the following formula can beused:

-cm, -m o (14)Rr = e p

where cb is the creep function set up attime of prestressing, and 0, and 0a arethe values of di at time of pinning thebase and after all creep distribution hasoccurred.

Typically, 4, - ¢, = 0.9, giving F3, _0.40 and /3r = 0.60. That is, final pre-stress will be as though 60 percent of theprestress was applied in the pinned basecondition, and only 40 percent in theinitial sliding base condition.

PCI JOURNAL/July-August 1985 71

WallHight Wall

Height

is) RADIAL DEFLECTIONS

I deflectionase pinned•prestressled

Final baflactionif base free tosI da

)¢flection

Initial

Final(after losses)

Prestress

(5i STRESS INDUCED IN WALLS

V4

a100

NdL

Wallbase

Fig. 7. Redistribution of circumferential prestress due to pinning base after prestressing.

CYLINDRICAL TANKSWITH ROTATIONALLY

UNSYMMETRICAL LOADINGIt is not commonly realized that loads

that are not rotationally symmetric canoften be analyzed with adequate accu-racy by use of a rotationally symmetricanalysis. Of particular interest is thecase of seismic loading, Fig. 8 illustratesseismic pressure distributions inducedin a cylindrical tank.

The circumferential distribution ofpressure follows a cosine distributionwith maximum pressure increase anddecrease occurring at opposite ends ofthe diameter parallel to the earthquakeexcitation. The vertical distribution ofpressure on the wall at any location con-sists of components from impulsive re-sponse and convective response (slosh-ing) of the tank and its contents.Methods for calculating the magnitudesof these pressure distributions havebeen fully described by Jacobsen ? andHousner.8

For analytical purposes it is commonto approximate the curved pressure dis-tributions of Figs. 8h and Sc by linear

approximations, as shown. However, itwould initially appear that a full shellanalysis would be necessary to modelthe effect of the circumferential varia-tion of pressure. In fact, a solution ofsufficient accuracy for maximumstresses may be obtained by assumingthe maximum pressure distribution (p.in Fig. 8a) to be distributed with rota-tional symmetry around the tank.

This is because at the location ofmaximum pressure, the rate of variationof pressure in the circumferential direc-tion is small, and stresses induced byshell distortion resulting from pressuredifferences along adjacent generatorsare negligible. This being the case, theframe analogy approach or standard de-sign tables or charts may be used to pre-dict maximum stresses induced byearthquake loading.

To provide justification for the aboveassertion, the cylindrical tank of Fig. 3was analyzed using Housner s8 method,for a maximum ground acceleration of0.4 g. The resulting pressure distributionp,, at the tank generator subjected tomaximum pressure increase could berepresented with adequate accuracy by

72

po Pressure

ag

a) Crcumterential Di5lnbution lb l Vertical Distribution Icl Vertical Distribution,pe = pocose Impulsive Pressure Convective Pressure

Fig. 8. Seismic pressure distribution in cylindrical tanks.

the linear pressure distribution shownin Fig. 9a, with pressures on a generatorat angle 9 to the critical generator givenby:

ps = po cos 6 (15)

The tank was analyzed using theframe analogy simulation of Fig. 4a, as-suming the pressure p. to be constantaround the tank circumference, and alsoby use of a plate bending finite elementsolution, using the ICES STRUDLPBSQ2 element. The finite elementanalysis modelled one quadrant of thetank taking advantage of s ymmetry andantisymmetry about the axes paralleland perpendicular to ground accelera-tion, respectively, and used 300 rectan-gular elements (10 up the wall height by30 round the quarter circumference).The finite element analysis used thesame linear vertical pressure distribu-tion at the critical section, but reducedpressure in accordance with Eq. (15) forsections angle 0 from the critical section.

Fig. 9b compares results predicted bythe full finite element analysis and theframe analogy, for the critical section.The agreement is very close, and ocr-

tainly justifies the use of the rotationalsymmetry approximation. The reductionin computer costs, and data preparationand results interpretation in using thesimple frame analogy in preference tothe finite element analysis is, of course,substantial.

TANKS WITHCURVED GENERATORS

Design tables and charts cannot coverthe infinite variety of shapes possible forelevated tanks with curved generators,and sophisticated analytical techniques,such as full shell, or finite elementanalyses are generally adopted. It is,however, possible to apply the frameanalogy simulation, developed above forcylindrical walls, to noncylindrical wallswith only minor modification.

Fig. 10 describes the simulation of thebowl of an elevated tank by a frameanalogy. In Fig. 10a, the wall is dividedinto a number of straight meridionalbeam members, which will carry axialload as well as bending moments andshears. The radial stiffness of the hoop

PCI JOURNAUJuly-August 1985 73

tension mechanism is again simulatedby radial pin-ended struts of equivalentaxial stiffness.

In developing the analytical methodfor cylindrical walls in the previoussection, a unit circumferential width ofwall was adopted. Consideration ofrotational symmetry shows this is in-appropriate for noncylindrical walls,and requires use of a slice of constantangle for the full height of the wall.Any angle 0 may be used, providedloads are also based on the same an-gular slice.

Beam Properties

Fig, 10b shows a typical beam mem-ber ij, of length L, and thickness vary-

ing from tr at i to t at j. For an angularslice of r¢ radians, the widths at i and jare:

b, _ (AR1; b; = , (16)

respectively, where R, and R; are theradii to the center of the wall at i and j,respectively.

Most computer frame analysis pro-grams allow variation of section prop-erties along the length of a member,However, provided the difference insection between the member ends is nottoo large, the average properties may beused, namely:

Moment of Inertia:

= 12 (b,+b,) ^r`^^ 2

F-7-10= Full F E. Analysis• • Frame Anelo y s

E

O_1N

VeicaI -inside HOOPinsideoutside outside is

2

---_-4 -3 -2 -1 0 1 2 3 4

Compression Tension Stress (MPal

(b 1 Stresseslol Equivalent Linear

Pressure Distribution

Fig. 9. Seismic stresses for 10,000 m 3 (2.65 million gallon) reservoirpredicted by frame analogy and full finite element analysis. (Note: 1 kPa= 0.145 psi, 1 MPa = 145 psi, 1 m = 39.37 in.)

74

Axis Axis Ic

Pin-andadstrut s

13 t^

(a) Simulation by Be ams and Struts ( b) Typical Becm Member

Fig. 10. Frame analogy simulation of noncylindrical elevated tank.

Cross Section Area:tt+tt

A ^= ("+") 2 (18)

Note that because of the noncylindri-cal shape, axial forces will develop inthe meridional as well as circumferen-tial directions, making correct simula-tion of the beam area essential.

Strut PropertiesThe radial strut j to the node joining

adjacent beam members ij and jk mustmodel the circumferential stiffness ofhalf the beam ij and half the beam jk.Putting strut modulus of elasticity equalto that of the walls, the approach used todevelop Eq. (11) yields:

A, = 1 [ bu to • Lo + b,kzt,k • Llk 1 (19)1, 2 R r, J

where A, and 1, are the area and length,respectively, of the strut at Node j, andb i,, t{,, Ro are the average width, thick-ness and radius for Member ij, etc. Notethat forL;j = L,,, = 9h; b0 = b,K = 1; andR e = = R, Eq. (19) reduces to the

value given by Eq. (11) for a unit cir-cumferential width of tapering cylindri-cal shell.

LOADING

Loads applied to the nodes of a tanksimulation, such as those shown in Fig.lda, must be based on the width of themembers adjacent to the nodes. Fordead load the simulation is straightfor-ward: provided beam thickness t andwidthb do not vary excessively betweenadjoining members, the vertical IoadP,,at Node j may be approximated as foI-lows:

1f'»s = Pc g bs tAU+L1k) (20)

where p g is the concrete unit weight.Loads applied by fluid pressure may

be distributed to nodes as shown in Fig.11a. The distributed load w lj per unitlength acts perpendicular to themember axis, thus inducing equivalentforces F x and F„ at the nodes. The mag-nitude of w, depends on member widthas well as water head h.

PCI JOURNALIJuly-August 1985 75

h^x f Pbo

Myra

Insides Outsian

PMJMBi ie

MfiB

Mies

Pie - PEA l

-Pie

Mje

IAM^g= MfMD-M1iGPgh^

Thus, at Node j:

tv i = prgh J b, (21)

where pfg is the fluid unit weight.Thermal loading requires special con-

sideration, both because of its signifi-cance as a stress-inducing load case,"and also because of the greater analyti-cal complexity. Figs. lib and Ile de-scribe a technique for analysis, based onseparation of a temperature changethrough the wall thickness of the tank

into an average (uniform) temperaturechange 9A , and a differential tempera-ture 8n, relativeto a thermally stress freetemperature 8 r, as illustrated in Fig. 12.

The method of analysis is similar tothat developed by Priestley5 for thermalanalysis of cylindrical tanks. First, de-formations associated with 8A and BA arerestrained, and the thermal restraintforces and moments necessary to pro-vide this restraint are calculated (Fig.lib), Thermal stresses in this restrainedcondition are calculated. Where the re-

{ a) Fluid Loading (b) Thermal Restraint Forces (c) "Release" Thermal Forcese0 78i eo}ei

Fig. 11. Equivalent joint loads for frame simulation of elevated tank.

aEF^ N

C d

a

Stress free aer temp

DistancetW

Actual distribution

ty^

Average change Differantfa! change

Fig. 12. Components of tank wall temperature change.

76

straint forces are not balanced by inter-nal equilibrium, equal and opposite re-lease forces must be applied to thestructure, and the stresses resulting fromanalysis under these forces are added tothe restraint stresses to obtain the finalstress state. The various steps are de-scribed in the following.

Step 1— Average TemperatureChange: 0A

Vertical expansion is unrestrained andinduces no stress. Restrain the radialthermal expansion associated with 8,y,inducing circumferential compressionstress of:

fs = E r a &, (22)

This will require radially inwards re-straint forces at the nodes of:

A,ErPt = jr (aOAR,)(23)

whereA f andl i are the area and length ofthe strut at i. These forces are "unbal-anced" and will be removed in Step 3.

Step 2 — DifferentialTemperature Change: 6D

Restrain circumferential and merid-ional curvature associated with the dif-ferential gradient. Because of the biaxialstress state, this induces circumferentialand meridional bending stress of:

fm = fc = I

a Oy(24)

Assuming average properties forMember ij may be used without excesserror, the vertical fixing moments neces-sary to induce this restraint will be:

E[ ifp1–v

2EI jja6^,

(1–v)te (25)

Out of balance moments will occur atjoints between members because ofdifferent moments of inertia, thickness,and possibly temperature gradient be-tween members. As with cylindricaltanks, the circumferential restraint mo-ments are self-equilibrating.

Step 3 — Release of RestraintForces

Fig. lib shows the direction of re-straint forces P i and M, imposed to re-strain the thermal effects resulting froman increase in outside surface tempera-ture relative to inside, and stress-free,temperatures.

These restraint or fixed-end forcesmust now be released. Forces applied tothe structural model to simulate this re-lease are shown in Fig. lie, that is, theradial restraint loads are reversed in di-rection, and joint moments AM A = — (I41)k— M}f) are applied to joints.

Stresses resulting from analysis underthese forces are added to the restraintstresses from the average thermalchange [Eq. (22)] and those from re-straint of differential thermal change[Eq. (24)], as appropriate, to obtain totalthermal stresses.

Where thermal stresses result fromdiurnal ambient temperature changes inthe wall, creep reduction of thermalstress will be negligible. However,when the stress results from storage ofhot or cold fluids for considerableperiods of time, the magnitude of ther-mal stress will he appreciably reducedby creep.

Shrinkage and SwellingThe influence of shrinkage and

swelling on the stress state in reservoirwalls is similar to thermal effects. If thewalls shrink due to loss of moisture be-fore filling, there will be a tendency forinward movement which will be re-strained at the base, unless it is free to

PCI JOURNALJJuIy-August 1985 77

Esn

Creep compensatedz shrinkage {monolithic wall/base

Tank filled

50 I 10C i 1 200 AGE (days)

castbase

Concrete E Ern

restrained Esn

7 ^y' - In situ monolithic walls

--- Precast walls

Fig. 13. Effective shrinkage and swelling strains for tank walls.

slide. For circular tanks, restraint will bein the form of a radially outward basereaction inducing hoop tension and ver-tical bending in the lower parts of thewall.

Since shrinkage develops slowly, thestresses induced are subject to somecreep relaxation. Fig. 13 shows the de-velopment of shrinkage strain over aperiod of 100 days after casting, as Line1. If the wall is monolithicall y con-structed with the base, the creep-com-pensated (or equivalent elastic) strain isshown by Line 2. A rate of creepanalysiss gives the creep-compensatedstrain as:

1 — e^E r — Exh (26)

where 0 is the value of the creep func-tion at the time considered. If the wall isinitially free to slide radially, no stressresults from shrinkage until the wallbase is pinned, subsequent to stressing.Assuming this occurs when the concreteof the walls is 50 days old, Line 3 repre-sents the development of shrinkage-

induced creep—compensated strain (seeFig. 13).

When the tank is filled, the concreteabsorbs water and regains its shrinkagevery rapidly. The effect is a swellingstrain that is again resisted at the base,inducing hoop compressions near thebase, and vertical bending moments ofopposite sign to those resulting fromshrinkage. Although the initial swellingmay be considered an elastic strain be-cause it occurs so quickly, the stressesinduced are gradually relaxed by creep.

The relationship that applies in termsof equivalent elastic strains, can beshown to be:

E,' = e Esh (27)

where 0 is a new creep factor relating tothe age of the concrete at the time offilling the reservoir.

The net effect of shrinkage followedby swelling is illustrated in Fig. 13 by aninstantaneous creep-compensated strainchange of E,h from Lines 2 to 4, or Lines3 to 5, followed by a gradual decrease ofeffective strain.

78

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U

22 22 23f1 RL 7.00m it 11 21

208 x7 20 19I ' 6 4 19 17II g to

14 15

12 '7 77 13Y II 1O i 6 76 11

II 8 5 15 06 < 1"7

1a 7 5 Scolt I m)

^2 1? 0 1 2 3 4 5

f 12

Joan! Cc-ord^natesx{m, y(m)

t(m) ember I(m4) A(m2)1 4,0 0.0 0.60 1 2,0E-3 0.1702 5.0 0.80 0.20 2 3.66E-4 0.1103 6.58 0.80 - 3 4.33E-4 0.1304 6.0 1.51 0.20 a 5.00E-4 0.1505 8.47 1.51 - 5 5.67E-4 0.1706 7.0 2.15 0.20 6 6.33E-4 0.1907 9.97 2.15 - 7 6.88E-4 0.20658 8.0 2.76 0.20 B 7.22E-4 0.21659 11.39 2.76 - 9 7.33E-4 0.220

10 9.0 3.40 0.20 10 7.21 E-4 0.216211 12.71 3.40 - it 6.84E-4 0.205212 10.0 4.13 0.20 12-22 0.00 0.0113 15.43 4.13 -14 10.65 4.80 0.2015 16.86 4.80 - E = 35 GPa16 11.00 5.50 0.20 v = 0.1817 18.84 5.50 -18 11.00 6,12 0.20 Member properties19 18,14 6.12 - based on 0.1 radian slice20 10.62 6.80 0.2021 16.14 6.80 -22 9.90 7.50 0.2023 19.76 7.50 -

Fig. 14. Frame simulation of elevated tank. (Note: 1 m = 39.37 in., 1 GPa = 145,000 psi.)

Stresses induced by these strains maybe calculated by analogy to an equiva-lent average temperature change of:

88 = F,/a (28)

since both 6, and e, will produce identi-cal effects.

Typical values suggested in a recentNew Zealand Draft Codes for design ofconcrete water retaining structures, for200 mm (7.87 in.) precast prestressedwalls are 50 x 10- 11 and 160 x 10-6 for

effective shrinkage and swelling strains,respectively.

The approach outlined above forthermal stress may be used to calculatethe stresses induced in noncylindricaltanks, but for more conventional groundsupported cylindrical tanks, publisheddesign tables for thermal stresses' maybe used. Generally, shrinkage stresseswill be found to influence only thelower one-quarter of tank wall heightand are much less than those resultingfrom thermal stress.1°

PCI JOURNAL/July-August 1985 79

Table 1. Joint loads for elevated tank in Fig. 14. (Note: 1 kN = 225Ibs, 1 kNm = 738 lb-ft.)

Water Pressure 'Thermal Load

joint FX FY M FX FY M(kN) (kN) (kNm) (kN) (kN) (kNm)

2 22.85 -30.34 0 266.0 0 27.104 21.72 -32.26 0 84.5 0 -2.866 20.75 -33.22 0 82.5 0 -2.868 20.67 -33.06 0 82.5 0 -2.86

10 21.61 -31.60 0 84.9 0 -2.8212 19.58 -23.24 0 64,5 0 -2.3514 15.50 -11.59 0 60.1 0 -1.4516 10.78 - 3.23 0 49.1 0 -0.4718 6.42 1.56 0 53.9 0 0.5220 2.02 I.21 0 67.4 0 1.5722 0.02 0.02 0 35.1 0 29.20

EXAMPLE OF USE OFFRAME ANALOGY FOR

ELEVATED TANKS

As an example of the use of the frameanalogy for tanks of noncylindricalshape, the elevated prestressed concretereservoir of Fig. 14 was simulated andanalyzed under hydrostatic, thermal andprestress loading. For simplicity, thebowl alone was analyzed, and was con-sidered fixed at Joint 1 (see Fig. 14),though a more detailed simulation couldhave included the bowl base, internal 3m (39.4 in.) radius shaft, and main col-umn as well. Full details of the simula-tion, corresponding to a radial slice of0.1 radians are included in Fig. 14.

Table I included joint loads appliedto simulate the effects of hydrostaticwater load with the free surface at ele-vation y = 7.0 m (23 ft), and thermalrelease forces and moments corre-sponding to a linear temperature gra-dient with outside surface at 20C (68°F)above the inside surface (and stress free)temperature.

Fig. 15 compares radial and verticaldisplacements under hydrostatic load-

ing predicted by the frame analogy withresults from an axisymmetric finite ele-ment analysis using the program SAP4"with elements corresponding to theframe members 1 to 11 (see Fig. 14). Forease of plotting, the vertical axis of Fig.15 is the straightened meridionalgenerator of the tank. Displacements area suitable basis for comparing themethods, since hoop stresses are prop-ortional to radial displacements, andmeridional bending stresses are depen-dent on the curvature of the displace-ments.

Examination of Fig. 15 reveals veryclose agreement for radial displace-ments, and good average agreement forvertical displacements, though peakvertical displacements of the finite ele-ment solution exceed the correspondingframe values by up to 4 percent. Itshould be recognized that the simula-tion of Fig. 14 is rather coarse, particu-larly in the upper portion of the bowlwhere meridional curvature is high, anda simulation with more joints in thisregion should yield more accurate re-sults.

Fig. 16 plots total thermal stresses

80

0

fns+de

E

I)UC0U)

mC_

C61U

OC0

I

1

0 4 8 12 16 20Displacement i m ^ 10 4)

Fig. 15. Displacements of elevated tank (Fig. 14) underhydrostatic loading, (Note: 1 mm = 0.3937 in.)

10— Circumferential

--- f c8 _

outsideSurface

w6 ---C

C

(_)48

I ^ 2-II I

Bost

- 8 -6 -4 -2 0 2 4 5Compression Stress Tension Stress (MPa1

Fig. 16. Thermal stresses for 20"C (68°F) gradient for elevated tankin Fig. 14. (Note: 1 m = 39.37 in., 1'C = 1.8F, 1 MPa = 145 psi.)

PCI JOURNALIJuly-August 1985 81

Loa dNcde {kN)22 77

20 14418 11316 11314 13812 176

10 $-195

190

190

195

251

-8 -6 -4 -2 0 2 Compression Stress (MPa) Tension

Fig. 17. Circumferential prestress stresses for elevatedtank in Fig. 14. (Note: 1 m = 39.37 in., 1 MPa = 145psi, 1 kN = 255 lbs.)

(i.e., restraint plus release stresses) re-sulting from analysis under the jointloads of Table 1. The temperature gra-dient of 20°C (68° F) with the outsidesurface hotter than the inside is likely tooccur frequently as a result of solar radi-ation on the east or west surface in theearly morning or late evening, re-spectively, and peak gradients are likelyto exceed 30°C (86°F). 5

As with seismic loading, the gradientwill not be rotationally symmetric.However, as with seismic loading, er-rors resulting from analyzing for thecritical section as though the local gra-dient is distributed with rotationalsymmetry are not particularly significantsince the temperature gradient variesonly slowly round the circumference. Itis perhaps of interest that the stressesinduced by the comparatively moderate20°C (68°F) gradient are substantiallyhigher than those induced by hydro-static loading.

As a final example, stresses induced in

the bowl by circumferential prestressingare examined. It is assumed that thebowl is constructed of precast elements,circumferentially stressed together atground level, and then lifted up a previ-ously constructed stem by stressingjacks, and fixed to the top of the stem.With this mode of construction, the bowlbase will not be radially constrainedduring the prestressing operation, andcircumferential precompression will beinduced over the full wall height of thetank.

It is further assumed that the dis-tribution of applied circumferential pre-stress force up the bowl wall is based onan intended prestress level of 8 MPa(1160 psi) at all heights. It is of interestto investigate the extent to which thedouble curvature of the bowl redistrib-utes the actual prestress in the wallsfrom the intended level.

Using the frame analogy simulation ofFig. 14, the required nodal loads arefound as follows. The circumferential

82

prestress force F per unit meridionallength will be:

F= 8t (29)

where t is the average shell thickness atthe location considered. Hence, fromEq. (13), the equivalent radial pressurewill be:

p = R (30)

Thus, at Node j, the radially inwardsload will depend on the distance be-tween nodes, and the width of wallsimulated:

P _ (Le+L,r^) . (ba+b,k) (31)j=p 2 2

_ 8 (10+t,k).(L +Uk).(bU+blk)R f 2 2 2

(32)

To analyze under the changed baseconditions, Joint 1 in Fig. 14 must bereleased for rotation and x-direction dis-placement, and a strut member of length1.18 m (3.87 ft) added at Joint 1 to repre-sent the stiffness of the lower half ofMember 1.

The joint loads given by Eq. (32), andthe resulting circumferential and me-ridional stresses are plotted against astraightened meridional generator inFig. 17. It will be seen that the averagecircumferential precompression doesnot vary greatly from the required 8 MPa(1160 psi) over the wall height, thoughthere is a small redistribution of pre-stress from the top and bottom of thebowl into the stiffer central region ofdouble curvature.

The meridional bending stresses re-sulting from circumferential prestressare modest. A Poisson's ratio component

from these stresses should be added tothe average circumferential surfacestresses to obtain the circumferentialsurface stresses induced by the appliedprestress.

Note that if the bowl was fixed to thestem before most of the creep associatedwith prestressing had occurred, argu-ments presented above about prestressredistribution resulting from structuralmodification would apply and a reduc-tion in prestress at the base of the bowlcould be expected as radial creep de-flections were restrained.

CONCLUSIONSA frame analogy method for simulat-

ing circular prestressed concrete tankswas developed and shown by compari-son with more sophisticated methods tobe able to accurately predict the be-havior of such tanks under a wide vari-ety of loading cases. The method is sim-ple to use, and suitable for both cylin-drical and noncylindrical (doublycurved) tanks.

Although the method assumes rota-tional symmetry of both the tank and itsloading, a comparison with results pre-dicted by a full finite element analysisunder seismic loading, where the pres-sure distribution varied round the cir-cumference, indicated that the methodcould be used with adequate accuracyfor nonrotationally symmetric loading.

ACKNOWLEDGMENTSThe finite element solutions included

in Fig. 7 and Fig. 11 were provided byDr. J. H. Wood and Drs. P. J. Moss andA. J. Carr, respectively. Their permis-sion to use the results is gratefully ac-knowledged.

PCI JOURNAtlJuly-August 1985 83

REFERENCES

1. Creasy, L. R., Prestressed Concrete Cy-lindrical Tanks, John Wiley and Sons,New York, N.Y., 1961.

2. Ghali, A., Circular Storage Tanks andSilos, SPON, London, England, 1979.

3. Hetenyi, M., Beam on Elastic Founda-tions, University of Michigan Press, AnnArbor, Michigan, 1946.

4. Winkler, E., Theory of Elasticity andStrength (in German), Prague, Czechos-lovakia, 1867, 184 pp.

5. Priestley, M. J.N., "Ambient ThermalStresses in Circular Prestressed Con-crete Tanks," AC! Journal, V. 31, No. 10,October 1976, pp. 553-560.

6. Neville, A. M., Creep of Concrete: Plain,Reinforced and Prestressed, Pitman,Bath, England.

7. Jacobson, L. J., "Impulsive Hydrody-namics of Fluid Inside a CylindricalTank, and of Fluid Surrounding a Cylin-drical Pier," Bulletin, Seismological So-

ciety of America, V. 39, 1949, pp. 189-204.

S. Housner, G. W., 'Dynamic Pressures onAccelerated Fluid Containers," Bulletin,Seismological Society of America, V. 47,1957, pp. 15-35.

9. DZ 3106:1984, "Draft Code of Practicefor Concrete Structures for the Storage ofLiquids," Standards Association of NewZealand, Wellington, 1984.

10. Priestley, M. J. N., Vessey, J., and North,P. J., "Concrete Structures for the Stor-age of Liquids," New Zealand ConcreteConstruction, V.29, February 1985.

11. Bathe, K. J., Wilson, E. L., and Peter-son, F. E., "SAP IV — A StructuralAnalysis Program for Static and Dy-namic Response of Linear Systems,"Earthquake Engineering ResearchCenter, Report No. EERC 73-11, Uni-versity of California, Berkeley, June1973.

NOTE: Discussion of this paper is invited. Please submityour comments to PCI Headquarters by March 1, 1986.

84

APPENDIX - NOTATIONA { = cross section area of frame analogy

strutb { = width of shell corresponding to ra-

dial slice 0 radiansE, = modulus of elasticity of concreteE, = modulus of elasticity of frame

analogy strut= circumferential stress= average hoop stress

fm = meridional stress in doubly-curvedshellfu = vertical stress in cylindrical shell

F = prestress tendon forceh = height, measured down from top

edge of shellI 1, = moment of inertia of frame analogy

beamk = subgrade modulus of equivalent

Winkler foundationl^ = length of frame analogy strutLe= length of frame analogy beam

member for doubly-curved shellM = momentp = pressure on tank wall

P = radial line load applied by cir-cumferential tendon to tank wall

P, = joint load for frame analogyH i = radius of tank at Node is = spacing between adjacent tendons

in tank wallt = shell thicknessy = radial deflection of tanka = coefficient of thermal expansionE, = creep compensated shrinkage

straineh = hoop strainE,n = unrestrained shrinkage strain0 = angular slice of doubly-curved

shell for frame analogy; also creepfactor

= unrestrained thermal curvaturev = Poisson's ratio for concretepC = mass density of concretepr = mass density of fluid9A = average temperature change of

wall from stress-free temperature

BD = differential temperature change atsurfaces of wall relative to 0A

PCI JOURNAUJuly-August 1985 85