Segmentally ConstructedPrestressed ConcreteHyperboloid Cooling Tower

Sami H. RizkallaAssistant ProfessorDepartment of Civil EngineeringUniversity of ManitobaWinnipeg, Manitoba

Paul ZiaProfessor and Head

Department of Civil EngineeringNorth Carolina State University

Raleigh, North Carolina

'n many large capacity powerplant facilities, the natural draft

cooling tower in the form of a thinshell of revolution is often re-quired to dissipate a large amountof heat. Construction of such largereinforced concrete natural draftcooling towers is expensive andtime-consuming.

The cost of the structure isstrongly influenced by the con-struction technique. 'A possiblemeans of reducing this cost is thetechnique of segmental construc-tion in which the benefits of bothprecasting and post-tensioning canbe combined together advantage-ously.

This paper presents the conceptof segmental construction as

applied to the hyperboloid naturaldraft cooling tower and themethod of analysis and designwhich have been developed indetail by Rizkalla.' The structuralbehavior under gravity load, windpressure and prestressing force isexamined.

The analysis is based on the fi-nite element method using a trun-cated conical shell as the basicelement. A numerical example isincluded to illustrate the designmethod.

A conservative cost estimate in-dicates that there is a potentialsaving up to 40 percent of the costof the tower (excluding foun-dations and columns) by means ofsegmental construction.

146

Tower Geometry

The geometry of the tower consid-ered herein is the same as that dis-cussed by Gurfinkel and Walser. 2 Themiddle surface of the shell is definedby

Ro_Y2a b2

1 (1)

in which Ro is the horizontal radius atany vertical coordinate Y with the ori-gin of coordinates being defined bythe center of the tower throat, ao is theradius of the throat, and

b= to°T a2 (2)0

where t o is the top radius and T is thevertical distance from the throat to thetop of the tower.

Concept ofSegmental Construction

Discussed here are the principlesunderlying the prefabricated segment,the precasting and erection proce-dures, and the shell-column interac-tion.

Prefabricated Segment

mental construction is the choice ofthe size and the shape of the prefabri-cated segment. The size should beconvenient for transporting and erec-tion. The shape should be suitable formass production to warrant the re-peated use of few standard forms.

For the cooling tower in question,one can easily visualize a standard-ized symmetrical unit as shown inFig. 1. Therefore, a casting bed can be

172-1

295----HOne of the important factors in seg- Fig. 1. General layout of cooling tower.

PCI JOURNAUJuly-August 1980 147

0M

J

U0

c0C-)

60 ^^

R9'

295

-p

Pref. Seg.

Fig. 2. Symmetrical unit cut away from cooling tower.

constructed in this basic shape forprecasting and the symmetrical unit isfurther subdivided into smaller pre-fabricated segments for ease of han-dling and erection, as can be seen inFig. 2.

Precasting ProcedureTo minimize joint thickness and to

obtain adequate fit between the adja-cent segments, it is envisioned thatthe segments will be match-cast alongthe meridional direction. One castingbed is required to alternate the castingoperation and to achieve the requiredmatching.

In casting the prefabricated seg-ments, a typical symmetric unit maybe sectioned into three parts. Startingfrom the lower part of the shell, thefirst set of prefabricated segments iscast using the first casting bed asshown by Step A in Fig. 3.

After adequate curing of the con-crete, the second part of the castingbed is used to cast the second setagainst the first set. This will achievethe matching along the meridionaljoint between the first and the secondset.

The first set will then be removedfrom the casting bed for storage andthe second set is then shifted to thematch-cast position on the first castingbed, ready for the next casting. Theprocedure is repeated until all seg-ments are cast for the lower part of thesymmetric unit.

Erection ProcedureThe tower construction begins with

the erection of the prefabricated seg-ments in the form of a horizontal ring.A narrow 8 to 10 in. (203 to 254 mm)closure strip is cast-in-place. Post-ten-sioning is then applied in the cir-

148

step step stepM C5 I(IIrISTORAGE LU

v YARD F_

:i__

ISt2 2nd3rd

CASTING SET CAST. SET CAST. SET CAST. SET CAST SETMATCH. POS. MATCH. POS.

Fig. 3. Precasting sequence of tower

cumferential direction to assemble theprefabricated segments into a seg-mental ring.

In the same manner, the secondsegmental ring is assembled and thesetwo rings are then connected togetherby post-tensioning in the meridionaldirection. Repeating this procedure,the structure is then built as an as-semblage of the segmented rings.

Fig. 4 illustrates this procedure.At the successive erection stages,

the meridional tendons are tensionedincrementally and the various tendonsare terminated and grouted at pre-determined levels after a number ofsegmental rings are constructed. Be-tween the joints, epoxy resin is usedto serve as a lubricant and bedding forjoint matching.

Cast in place PrefrobricatedClosure strip segment

Meridional CircumferentialPrestressing Prestressing

Fig. 4. Installation procedure of tower.

PCI JOURNAUJuly-August 1980 149

Shell-Column ConnectionIn general, the shell of the tower is

supported by a group of flexible diag-onal columns laying in a conical sur-face tangent to the shell at the base.For this investigation, it is envisionedthat a relatively stiff ring girder,bridging the top of the columns, sup-ports the hyperbolic shell. A groove atthe top of the ring girder supports thefirst segmental ring, thereby providingthe radial movement of the segmentalring when the circumferential pre-stressing is applied. After the applica-tion of the prestressing force, thegroove will be filled with a semi-rigidmaterial so as to constrain the bound-ary displacements of the shell exceptthe meridional rotation.

Design Considerations

A unique feature of the segmentallyconstructed structure is that the de-sign/analysis must be performed foreach stage of construction. Duringconstruction, each substructure mustbe analyzed as a separate structure forthe effects of its own weight, con-struction loads, wind load and pre-stressing force. The analysis proceedswith the first segmental ring treated asa substructure. As each additionalsegmental ring is erected a new sub-structure is created.

The basic element used in the finiteelement analysis herein is a truncatedcone, of which the shell thickness isproportional to the distance from itsvertex along its generator. 3 Each ele-ment has four degrees of freedom:three linear displacements (merid-ional, circumferential and normal) andone rotational displacement (merid-ional) 4

Wind LoadWind load is the principal factor

which governs the design of cooling

towers in non-seismic areas. The de-sign criteria for wind load based onthe equivalent static wind pressuredistribution have been suggested byACI-ASCE Committee 334:5

q(z,0) = GCOK3g 30 (3)

in which q (z, 0) is the equivalent staticnormal pressure at a location definedby coordinates z and 0 on the surfaceof the tower:

q 30 = 0.00256 V30(4)

where V30 is the wind velocity at 30 ft(9.15 m) elevation. Coefficient K z isthe exposure factor which establishesthe vertical profile of wind pressureand depends on wind speed androughness of the terrain. CoefficientsG and C B are respectively the dynamicgust factor and the circumferentialdistribution coefficient.

For each substructure, the verticaldistribution of the wind pressure isdetermined by Eq. (3). The analysisconsiders each substructure beingcomposed of several segmental rings,each of which is, in turn, approxi-mated by a series of truncated conesof equal height. The variation of theinternal stress resultants N 8 , M 3 , Neand M B is computed along the heightof each substructure at different refer-ence angles. The maximum tensileand compressive stress resultants cor-responding to each substructure areobtained by developing a stress en-velope from the respective stress diag-rams at the reference angle.

By combining the stress envelopesfor the stress resultants N. and N B ofall substructures, the overall stressenvelope for the entire structure isobtained. Moment diagrams for M$and M B corresponding to themaximum N. and N B are then de-veloped. These stress envelopes andmoment diagrams represent the mostcritical forces developed in the tower

150

during and after construction. For de-sign purposes, the maximum tensileforces N, and N B , and their corre-sponding moments M., and M B , areused to determine the required pre-stressing forces in both directions.

Similarly, the compressive stressesin concrete are examined based on themaximum compressive force envelopeand their corresponding moment dia-grams. The maximum transverse andcircumferential shear forces are alsoinvestigated to check the maximumprincipal tension in concrete. Finally,the maximum radial and meridionaldisplacements are examined to deter-mine the maximum possible move-ment of the structure during and afterconstruction.

Gravity LoadUnder gravity load, the analysis of a

segmentally constructed tower is dif-ferent from that of a cast-in-placetower. For a segmentally constructedtower, prefabricated segments areerected one by one. Thus, the weightof each individual segment acts as avertical distributed load at the top ofthe previously erected substructure.

During erection, the effects of bothpartial gravity load (resulting from apartially erected segmental ring) andfull gravity load (resulting from a fullyerected segmental ring) must be in-vestigated. In this study, four differentgravity load cases are considered asshown in Fig. 5. The last three load-ings, being nonsymmetrical with re-spect to the vertical tower axis, areapproximated by the cosine series fora uniformly distributed load:

P a = ; p a(n)cos(nO) (5)

where 0 is measured from the axis ofsymmetry as seen from Fig, 5.

For each case of the partial gravityload, the stress distributions are ob-tained at the various locations definedby the same reference angles as those

0=180 0=135

Symmetric Three Quarter

0=90 0=45

Half QuarterFig 5. Four loading conditions duringerection of tower.

for the wind analysis. The maximumtensile and compressive stress resul-tant envelopes for the various sub-structures are obtained by the sameprocedure described previously forthe wind analysis. Under the gravityload of a complete segmental ring, thesummation of the stresses developedin the substructure at various stages ofconstruction represents the final stateof stress due to full gravity load in thetower.

Meridional Prestressing ForceAfter erection, prefabricated seg-

ments are first post-tensioned in thecircumferential direction to form asegmental ring, and then post-ten-sioning is applied to join the segmen-tal rings in the meridional direction.The meridional prestressing force canbe replaced by an equivalent outwardpressure resulting from the curvatureof the tendons, and an in-plane com-pressive force. Meridional tendons arepositioned in the middle surface ofthe shell, thus eliminating the mo-ments at the ends of the tendons.

The in-plane compressive force dueto the meridional prestressing is

PCI JOURNAL/July-August 1980 151

P P

(n) ELEMEI(n 1) ELEM

fP

n

PFig. 6. Idealized meridional prestressing forces in tower.

idealized as a uniformly distributedload acting at the top and bottom ofany substructure, indicated as P. inFig. 6. The equivalent outward pres-sure, however, is idealized as con-centrated loads acting at each nodalcircle of the finite element and is rep-resented by

Y„ = P(sina n – sina n _1 ) (6a)

X„ = P(cosa n_1 – cosa n) (6b)

The meridional prestressing force isdesigned to counteract all meridionaltensile stresses induced in the towerduring and after construction. At eachstage of construction, some of the ten-dons will be terminated while otherswill be extended to meet the require-ments at subsequent stages of con-struction.

To determine the required merid-ional prestressing forces P 1 , P2 , ... P.at the various levels of the tower, ananalysis is first performed for each ofthe n-1 substructures and the com-plete structure under the effect of unitmeridional prestressing force. Then aset of n simultaneous equations is ob-tained by equating, at each level, theprestressing force to the net merid-ional tensile force induced by windand gravity load plus a small amountof residual compressive stress in con-crete, say, 20 ksf (0.96 MPa).

The solution of the simultaneousequations is given by the recursion

equation for the meridional pre-stressing force Pi , at ith level:

Pi= - D_1 + (20 + Ss-i)ta_1P X i 2 -^

n

_ .t=i +1 PiX t -i (7)Xii-i

CircumferentialPrestressing Force

The circumferential tendons are po-sitioned so as to encircle the merid-ional prestressing tendons. This ar-rangement would keep concreteunder compression at all times, andthe circumferential membrane forcesare equal to the applied circumferen-tial prestressing forces. Thus, the re-quired circumferential prestressingforce can be determined easily, basedon the tensile hoop stresses due towind, gravity load and meridionalprestressing, allowing for a residualcompressive stress of 20 ksf (0.96MPa) again.

Therefore, the circumferential pre-stressing force P, at any level can becomputed as

P, = N B + 6Me + 20t (8)t

For more details of the analyticalbackground, see References I through4.

152

Design Example

The tower considered herein hasthe same geometry as that discussedby Gurfinkel and Walser. 2 The shell is355 ft (108.2 m) high and supported bya bottom ring girder such that only themeridional rotational freedom is pro-vided as mentioned previously. Thethroat of the tower is 165 ft (50.3 m) indiameter and is located 60 ft (18.3 m)below the top of the shell. The thick-ness varies from 30 in. (762 mm) at thebottom level to 6 in. (152 mm) at 25 ft(7.62 m) elevation from the base. Atthe top 10 ft (3.05 m) of the shell, thethickness also varies from 6 to 24 in.(152 to 610 mm). Other than the topand the bottom regions, the shellthickness remains constant at 6 in.(152.4 mm).

For this investigation, the towersurface is divided into 20 symmetricalunits around the circumference andeach of these units is subdivided into25 prefabricated segments (see Fig. 2).Thus each segment will have ameridional height of 14.2 ft (4.33 m)and a circumferential length varyingfrom 25 to 45 ft (7.62 to 13.72 m).

The actual structure is idealized as100 inter-connected conical elementsand the loadings on the structure arereplaced by a set of loads applied atthe inter-connections (nodal circles).Thus, each segmental ring is approxi-mated by four truncated conical ele-ments. At the boundary, the threelinear displacements are constrainedbut not the rotation in the meridionaldirection. These boundary conditions,along with the free boundary condi-tion at the top of the shell, are used inanalyzing all the substructures and thecomplete tower under different load-ing conditions.

Meridional Force and MomentAccording to the analytical proce-

dures described above, the tower wasanalyzed for wind and gravity load, aswell as unit meridional prestress forthe substructure corresponding toeach of the 25 construction stages. Itwas shown that the maximum internalforces due to gravity load occurredwhen the substructure was fullyloaded by a complete segmental ring,and the maximum tensile force due towind always occurred at the zero de-gree reference angle from the wind-

unitwind dead mer. pres.

1.0 86—I

31.

-104 i 3

6 - - - - 105 0 96 ^-° LO

41 .8/- — - -- -4148 -- 0 .83 M

07 0

49 - - - - - --- 21.4 058K/FT K/FT K/FT 145—

Fig. 7. Meridional forces due to wind, gravity, and meridional prestress for the 25thconstruction stage of tower.

PCI JOURNAL/July-August 1980 153

wind dead mer.pres.

j l K/FT 86--^1.9 0.3

0.4

Li compTower

0.5 - 2.3 0.04K•FT/FT K•FT/FT K•FT/FT

Fig. 8. Corresponding meridional moments due to wind, gravity and meridionalprestress for the 25th construction stage of tower.

Table 1. Residual and bending stresses at selected levels of tower [Eq. (7)].

Maximumtensile Gravity Maximum

Height Thick- force due load corre- , BendingCon- from ness to wind forces sponding stressstruction base t load T D moment S (20 + S) tstage (ft) (ft) (kips) (kips) (kip-ft) (ksf) (kips)

25 355 2 0 0 0 0 020 284 0.5 11.325 6 0.3 7.2 7.615 213 0.5 31.65 10.5 0.1 2.4 0.710 142 0.5 43.38 14.8 0.5 12 1.2

5 71 0.5 46.948 17 0.65 15.6 0.82 28.4 0.5 48.0 19 1.2 28.8 5.41 14.2 1.5 49.0 20 2.87 7.65 21.480 2.5 49.03 21.36 0 0 0 28.64

Note: 1 ft = 0.305 m; 1 kip = 4.448 kN; 1 kip-ft = 1.356 kN • m; 1 ksf= 0.048 MPa.

Table 2. Internal meridional forces due to unit meridional prestress force [Eq. (7)].

Constructionstage

Heightfrombase (ft)

Meridional forces per unit circumferential length(kips)

25 355 120 284 1.0406 115 213 0.967 0.9288 110 142 0.8328 0.7987 0.8611 15 71 0.6975 0.6689 0.7222 0.8407 12 28.4 0.62 0.60 0.65 0.76 0.91 11 14.2 0.6 0.58 0.63 0.73 0.88 0.9670 0 0.587 0.5634 0.6088 0.799 0.8451 0.9369

Note: 1 it = 0.305 m; 1 kip = 4.448 kN.

154

ward side of the shell. The meridionalforces due to wind, gravity load, andunit meridional prestress for the com-plete tower, are shown in Fig. 7, inwhich tensile force is denoted as pos-itive and compressive force as nega-tive.

Similarly, the meridional bendingmoments due to the same loadings areshown in Fig. 8. The distributions ofmeridional forces and moments for thesubstructures at various stages of con-struction can be found in Reference 1.

Required MeridionalPrestressing Force

The various terms of Eq. (7) aretabulated in Tables 1 and 2. In Table3, Eq. (7) is used to determine the re-quired prestressing force at the vari-ous levels. The computation begins atthe top level of the tower, i.e., level355 ft (108.2 m) and the required pre-stressing force is 18.18 kips per ft(80.87 kN). Based on an effective de-sign stress of 162 ksi (1117 MPa) for

the prestressing tendon, three tendonseach consisting of seven ½-in. (12.7mm) diameter 7-wire strand are re-quired to provide an actual prestress-ing force of 20.01 kips per ft (89 kN).

Having established the actual pre-stressing force at level 355 ft (108.2m), the required prestressing force atthe next level, i.e, level 284 ft (86.56m) is then obtained. Repeating thesame procedure, the required and ac-tual prestressing forces are deter-mined for each of the remaininglevels.

Review of Compressive StressesThe design criteria call for a

minimum of 20 ksf (0.96 MPa) re-sidual compressive stress at each levelof the tower during and after con-struction. As shown in Fig. 9, theshaded area represents the magnitudeof residual compression at all levels ofthe complete tower. The slightlyhigher residual compressive force atthe lower part of the shell is provided

Table 3. Determination of required prestress force using Eq. (7).

Height Actualfrom applied

Construction base Required prestress force per unit length according forcestage (ft) to Eq. (7) (kips)

25 355 P, = (11.325 + 7.6)/(1.00406) = 18:18 kips 20.01

20 284 P5 = [31.65 - 20.01(0.967) + 0.7]/(0.9288) = 13.97 kips 15.624

15 213 Ps = [43.38 - 20.01(0.8328) - 15.624(0.7987) + 1.21/(0.8611) = 17.859 kips 19.27

10 142 P4 = [46.948 - 20.01(0.6975) - 15.624(0.6689) - 19.27(0.7222) + 0.81/(0.8407) = 11.15 kips 12.206

5 71 P5 = [48.0 - 20.01(0.62) - 15.624(0.6) - 19.27(0.65)- 12.206(0.76) + 5.41/(0.91) = 10.74 kips 13.69

2 28.4 PB = [49 - 20.01(0.6) - 15.624(0.58) - 19.27(0.63)- 12.206(0.73) - 13.69(0.88) + 21.481/(0.967) = 16.82kips

orP6 = (49.03 - 20.01(0.58) - 15.624 (0.5634) - 1927

(0.0688) - 12.206(0.7099) - 13.69(0.845) + 28.641/(0.9369) = 22.66 kips 23.60

Note: 1 It = 0.305 m; 1 kip = 4.448 kN.

PCI JOURNAL/July-August 1980 155

86

complete355 tower

X145 ---^merid. prest. dead

Fig. 9. Residual compression at the 25th construction stage of tower.

to counteract the higher bendingstresses in this region.

The most critical compressive forcedistribution due to wind load de-veloped at a 60-deg circumferentialangle. Adding to this force distribu-tion the effects of gravity load andprestressing, the total maximum com-pressive force distribution is shown inFig. 10. Under the combined effect ofthe maximum compressive force andthe corresponding meridional bendingmoment, the maximum compressivestress was found to be 1835 psi (12.65MPa) at 25 ft (7.62 m) from the base ofthe shell, which is within the allow-able compressive stress for a designconcrete strength f f = 5000 psi (34.47MPa).

Circumferential Forcesand Moments

The application of meridional pre-stress induces circumferential forcesin the shell. Based on the actual pre-stressing force established previously,the induced circumferential force foreach stage of construction was com-puted. It was shown that the mostcritical circumferential force distribu-tion developed at the final stage of

construction. The circumferentialforces due to wind were also com-puted for the various reference anglesduring each stage of construction.Based on the results of these analyses,the design force envelope is shown inFig. 11. An important feature of Fig.11 is that the design envelope is gov-erned by the peak tensile forces in thetower at the various stages of erectionrather than the tensile force in thetower after its completion.

The circumferential forces due togravity load were likewise determinedfor each intermediate constructionstage. The circumferential force dis-tribution for the complete tower isshown in Fig. 12. It is observed thateach segmental ring produces a largecircumferential compressive force atthe top edge of the previously com-pleted substructure, and this largecompressive force diminishes veryquickly to a small tensile force at thelower edge of the previous segmentalring.

The envelope for the force distri-bution is similar in shape to that ofthe entire tower when it is analyzedfor its own full dead load. Similar en-velopes for circumferential moments

156

wind i- dead + prestress

I I L__J 98596.5

windmend. prest. ^deaddFig. 10. Maximum compressive force distribution in tower.

22.472

h=285'-D7 .116-

.164 —h=213'DesignEnvelope 5.454 \--- h=142'

515 'ri h=71

7.356 -- -'K/FT

Fig. 11. Final design tensile force envelope due to wind load at various stages oferection of tower.

—6.24 K/FT

86

355comp

er

17 . 8 K/FT

Fig. 12. Circumferential force due to gravity load for 25th construction stage of tower.

PCI JOURNAL/July-August 1980 157

wind + dead + prestress23. 046 K/FT

Fig. 13. Final design circumferential tensile force envelope in tower due to wind,gravity, and meridional prestress.

due to wind and gravity load havebeen obtained elsewhere.

Required CircumferentialPrestressing Force

Superposing the effects of merid-ional prestressing, wind and gravityload, one can obtain the maximum cir-cumferential force envelope shown inFig. 13. It is seen that the maximumcircumferential force is nearly con-stant for the middle part of the shellbut varies significantly near the topand the bottom of the shell. Therefore,the design can be executed for a typi-cal segmental ring of the middle partof the shell, two rings at the bottomand five rings above the throat level.

In Table 4, the design force andmoment are tabulated for the varioussegmental rings. Using Eq. (8), the re-quired prestressing force per unitlength is computed in Table 5, inwhich the number and size of tendonsfor each segmental ring are also tabu-lated.

Review of Compressive StressHaving determined the circumfer-

ential prestressing force, it is neces-sary to examine the maximum cir-cumferential compressive stress at the

various points of the shell. By com-bining the effects of meridional pre-stressing, circumferential prestressing,wind and gravity load, it was foundthat the maximum compressive stressof 1140 psi (7.86 MPa) occurred at thetop of the shell, which is well withinthe allowable.

Economics

Based on conservative cost esti-mates of materials and labor in theUnited States in 1976, the cost of the355 ft (108.2 m) segmentally con-structed cooling tower may beitemized as follows:

Materials (concrete, rein-forcement, tendons) .... $ 633,000

Labor (precasting, stress-

ing, grouting) .......... 300,000Formwork and setup cost . 220,000

Erection................. 500,000

1,653,000Contingency, overhead and

profit.................. 678,000

Total $2,331,000

This estimate may be adjusted to a

158

Table 4. Circumferential forces and moments for different segmental rings alongtower height.

Level (ft) Average Maximum CorrespondingSegment thickness force moment

From ToLocation No. (ft) (kips per ft) (kip-ft per ft)

Bottom 1 0.0 14.2 2.2 7.356 0.285part of shell 2 14.2 28.4 1.0 6.70 0.10

Middle part 3 to 20 28.4 284 0.5 7.00 1.14of shell (typical)

21 284 298.2 0.5 7.8 1.1422 298.2 312.4 0.5 9.1 1.0

Top part 23 312.4 326.6 0.5 11.2 0.8of shell 24 326.6 340.8 0.5 14.7 0.6

25(a) 340.8 355 1.25 19 12.0125(b) 2.00 23.046 10.70

Note: 1 ft = 0.305 m; 1 kip per ft = 14.584 kN/m; 1 kip = 4.448 kN.

Table 5. Determination of required circumferential prestressing forces in towerusing Eq. (8).

Required prestressing force per unitlength Total

6M, forceP, = N, + - + 20t (kips) per Circumferential

Segment t segment tendons perLocation No. Eq. (8) (kips) segment

Bottom part 1 7.356 + 6(0.285)/2.2 + 20(2.2) = 52.13 740.24of shell 2 6.7 + 6(0.10)/1.0 + 20(1.0) = 27.3 387.66 4-(8) Y2-in. (A strands

Middle part 3-20 7 + 6(1.14)/0.5 + 20(0.5) = 30.7 435.94 3-(6)'/2-in. 0 strandsof shell

21 7.8+ 6(1.14)/0.5+ 20(0.5) = 31.48 447Top 22 9.1 + 6(1.0)/0.5 + 20(0.5) = 31.10 441 3-(6)' -in. 0 strandspart 23 11.2 + 6(0.8)/0.5 + 20(0.5) = 30.80 437of 24 14.7 + 6(0.6)/0.5 + 20(0.5) = 31.90 452shell 25(a) 19 + 6(12.01)/1.25 + 20(1.25) = 101.6 1442.7

(b) 23.056+ 6(10.70)/2.0 + 20(2.0) = 95.23 5-(12)'/2-in.0 strands

(a) At middle of 25th segmental ring which controls design.(b) At top edge of25th segmental ring which controls design.Note: 1 in. = 25.4 mm; 1 kip = 4.448 kN.

total cost of $3,500,000 in 1980, basedon an inflation factor suggested byEngineering News Record's cost indexof 50 percent increase in materials,labor and erection and 60 percent in-crease in formwork and setup cost.

The cost of a similar cast-in-placereinforced concrete cooling tower wasquoted as $4,000,000 in 1976 from re-liable industrial sources. Using the

same inflation factors, the total cost ofthe cast-in-place cooling tower shouldbe adjusted to $6,000,000 in 1980.Therefore, it appears that by thetechnique of segmental construction, asaving of almost 40 percent of the costof the tower (excluding foundationand columns) is indicated. In addition,there is a distinct benefit of savings inconstruction time.

PCI JOURNAL/July-August 1980 159

CONCLUDING REMARKS

The technique of segmental con-struction can be advantageouslyapplied to hyperboloid cooling tow-ers. Analysis and design of such atower should be performed for eachstage of construction. Circumferentialstresses, induced in the partially com-pleted tower, may exceed the corre-sponding stresses in the completedtower, thus dictating the design.

Due to the small meridional curva-ture for each segmental ring, the cir-cumferential prestressing has littleeffect on the meridional prestressing.Therefore, in design, the total merid-ional prestressing force should bedetermined first in order that its effect

can be accounted for in the design ofthe circumferential prestressing. Aconvenient and efficient procedure ofdetermining the required meridionalprestressing force at different levels ofthe tower is to start from the top levelof the tower and proceed downwardusing a recursion relation.

An extension of this study includingan experimental investigation usingsmall-scale structures, as well as anexamination of dynamic response toseismic and wind effects, is in prog-ress at North Carolina State Univer-sity, Raleigh, North Carolina, underthe direction of Dr. Paul Zia.

REFERENCES

Nonsymmetrical Loading," PhD Dis-sertation, Duke University, Durham,North Carolina, 1968.Flugge, W., Stresses in Shells, Spring-er-Verlag, Berlin, Germany, 1962, p. 78.ACI-ASCE Committee 334, "ConcreteShell Design and Construction—Designand Construction of Reinforced Con-crete Cooling Tower Shells," Practiceand Commentary (draft), AmericanConcrete Institute, Detroit, Michigan,1974.

1. Rizkalla, S. H., "An Investigation ofSegmentally Constructed HyperboloidNatural Draft Cooling Tower," PhDDissertation, North Carolina State Uni- 4versity, Raleigh, North Carolina, 1976.

2. Gurfinkel, G., and Walser, A., "Analysis 5•and Design of Hyperbolic CoolingTowers,"Journal of the Power Division,ASCE, 1972, V. 98, No. POI, p. 133.

3. Coffin, G. K., "Finite Element Analysisof Open Shells of Revolution Under

Discussion of this paper is invited.Please forward your comments toPCI Headquarters by March 1, 1981

160

APPENDIX A-NOTATIONa o = throat radius of hyperboloid

shell of revolutionb = constant parameter for hy-

perboloid shell of revolu-tion,. Eq. (2)

C 0 = circumferential distributioncoefficient for wind load

D i_1 = internal meridional forcedue to gravity load at(i-1)th level

G = dynamic gust factor forwind load

K 0 = exposure factor for windload

M 8 = meridional bending mo-ment per unit circumferen-tial length

M 0 = circumferential bendingmoment per unit meridionallength

N 8 = meridional force per unitcircumferential length

N B = circumferential force perunit meridional length

n = number of harmonics, ornumber of segmental rings

P = meridional prestressingforce

P, = circumferential prestressingforce

Pa = uniformly distributed grav-ity load

Pa= coefficient in cosine series,Eq. (5)

q(z,O) = equivalent static normalwind load

q 30 = wind pressure, Eq. (4)R o = horizontal radius of hyper-

boloid shell of revolution atY level

S i = bending stress due to windand gravity load at ith level

T = vertical distance from shelltop to throat level

Tf = internal meridional forcedue to wind load at jth level

t = shell thicknesst,, = radius of top of tower shell

V30 = wind velocity at 30-ft eleva-tion

X = internal meridional force atjth level due to unit merid-ional prestressing force atith level

X,,, Y„= component of equivalentoutward pressure of merid-ional prestressing force

Y = vertical coordinate, Eq. (1)z = vertical distance measured

from ground levela,= angle between normal of

shell surface and axis ofrevolution

B = circumferential referenceangle

PCI JOURNAL/July-August 1980 161