Segmentally Constructed
Prestressed Concrete
Hyperboloid Cooling Tower
Saml H. Rizkalla
Assistant Professor
Department of Civil Engineering
University of Manitoba
Winnipeg. Manitoba
PaulZla
Professor and Head
Department of Civil Engineering
North CarOlina State University
Raleigh. North Carolina
many large capacity power
I nplant
facilities, the natural draft
applied to the hyperboloid natural
draft cooling tower and the
cooling tower in the fonn of a thin method of analysis and design
shell of revolution is often re- which have been developed in
quired to di~sipate a large amount detail by Rizkalla. 1 The structural
of heat. Construction of such large behavior under gravity load, wind
reinforced concrete natural draft pressure and prestressing force is
cooling towers is expensive and examined.
time-consuming.
The analysis is based on the fiThe cost of the structure is nite element method using a trunstrongly influenced by the con- cated conical shell as the basic
struction technique. A possible element. A numerical example is
means of reducing this cost is the included to illustrate the design
technique of segmental construc- method.
tion in which the benefits of both
A conservative cost estimate inprecasting and post-tensioning can dicates that there is a potential
be combined together advantage- saving up to 40 percent of the cost
ously.
of the tower (excluding founThis paper presents the concept dations and columns) by means of
of segmental construction as segmental construction.
Tower Geometry
The geometry of the tower considered herein is the same as that discussed by Gurfmkel and Walser.2 The
middle surface of the shell is defined
by
R2
P
a:.!!.- b- = 1
2
(1)
mental construction is the choice of
the size and the shape of the prefabricated segment. The size should be
convenient for transporting and erection. The shape should be suitable fur
mass production to warrant the repeated use of few standard forms.
For the cooling tower in question,
one can easily visualize a standardized symmetrical unit as shown in
Fig. 1. Therefore, a casting bed can be
in which R. is tile horizontal radius at
any vertical coordinate Y with the origin of coordinates being defined by
the center of the tower throat, a. is the
radius of the throat, and
b =
as
~t;
- ai
(2)
where t. is the top radius and T is the
vertical distance from the throat to the
top of the tower.
Concept of
Segmental Construction
Discussed here are the principles
underlying the prefabricated segment,
the precasting and erection procedures, and the shell-column interaction.
Prefabricated Segment
One of the important factors in seg-
Fig. 1. General layout of cooling tower.
step
A
o
step
8
step
Pref. Seg.
Fig. 2. Symmetrical unit cut away from cooling tower.
constructed in this basic shape for
precasting and the symmetrical unit is
further subdivided into smaller prefabricated segments for ease of handling and erection, as can be seen in
Fig. 2.
Precastlng Procedure
To minimize joint thickness and to
obtain adequate fit between the adjacent segments, it is envisioned that
the segments will be match-cast along
the meridional direction. One casting
bed is required to alternate the casting
operation and to achieve the required
matching.
In casting the prefabricated segments, a typical symmetric unit may
be sectioned into three parts. Starting
from the lower part of the shell, the
first set of prefabricated segments is
cast using the first casting bed as
shown by Step A in Fig. 3.
• AD
After adequate curing of the concrete, the second part of the casting
bed is used to cast the second set
against the first set. This will achieve
the matching along the meridional
joint between the first and the second
set.
The first set will then be removed
from the casting bed for storage and
the second set is then shifted to the
match-cast position on the first casting
bed, ready for the next casting. The
procedure is repeated until all segments are cast for the lower part of the
symmetric unit.
Fig. 3. Precasting sequence of tower.
cumferential direction to assemble the
prefabricated segments into a segmental ring.
In the same manner, the second
segmental ring is assembled and these
two rings are then co.nnected together
by post-tensioning in the meridional
direction. Repeating this procedure,
the structure is then built as an assemblage of the segmented rings.
Segmental
Ring
Fig. 4 illustrates this procedure.
At the successive erection stal
the meridional tendons are tensiOJ
incrementally and the various tend
are terminated and grouted at I
determined levels after a numbel
segmental rings are constructed.
tween the joints, epoxy resin is u
to serve as a lubricant and bedding
joint matching.
Cast in place
Closure strip
Erection Procedure
The tower construction begins with
the erection of the prefabricated segments in the form of a horizontal ring.
A narrow 8 to 10 in. (203 to 254 mm)
closure strip is cast-in-place. Post-tensioning is then applied in the cir-
Meridional
Prestressing
Circumferential
Prestressing
Fig. 4. Installation procedure of tower .
Prefrabricated
segment
Shell-Column Connection
towers in non-seismic areas. The design criteria for wind load based on
the equivalent static wind pressure
distribution have been suggested by
ACI-ASCE Committee 334:5
In general, the shell of the tower is
supported by a group of flexible diagonal columns laying in a conical surface tangent to the shell at the base.
For this investigation, it is envisioned
that a relatively stiff ring girder,
bridging the top of the columns, supports the hyperbolic shell. A groove at in which q (z,8) is the equivalent static
the top of the ring girder supports the normal pressure at a location defmed
first segmental ring, thereby providing by coordinates Z and 8 on the surface
the radial movement of the segmental of the tower:
ring when the circumferential preq 30 = 0.00256 V fo
(4)
stressing is applied. After the application of the prestressing force, the
groove will be filled with a semi-rigid where V 30 is the wind velocity at 30 ft
material so as to constrain the bound- (9.15 m) elevation. Coefficient K. is
ary displacements of the shell except the exposure factor which establishes
the vertical profile of wind pressure
the meridional rotation.
and depends on wind speed and
roughness of the terrain. Coefficients
G and C 6 are respectively the dynamic
Design Considerations
gust factor and the circumferential
A unique feature of the segmentally distribution coefficient.
For each substructure, the vertical
constructed structure is that the design/analysis must be performed for distribution of the wind pressure is
each stage of construction. During determined by Eq. (3). The analysis
construction, each substructure must considers each substructure being
be analyzed as a separate structure for composed of several segmental rings,
the effects of its own weight, con- each of which is, in tum, approxistruction loads, wind load and pre- mated by a series of truncated cones
stressing force. The analysis proceeds of equal height. The variation of the
with the first segmental ring treated as internal stress resultants N" M" N 6
a substructure. As each additional and M 6 is computed along the height
segmental ring is erected a new sub- of each substructure at different reference angles. The maximum tensile
structure is created.
The basic element used in the finite and compressive stress resultants corelement analysis herein is a truncated responding to each substructure are
cone, of which the shell thickness is obtained by developing a stress en-,
proportional to the distance from its velope from the respective stress diagvertex along its generator. 3 Each ele- rams at the reference angle.
By combining the stress envelopes
ment has four degrees of freedom:
three linear displacements (merid- for the stress resultants N, and N 6 of
ional, circumferential and normal) and all substructures, the overall stress
one rotational displacement (merid- envelope for the entire structure is
obtained. Moment diagrams for M,
ional).'
and M 6 corresponding to the
maximum N, and N 6 are then deWind Load
veloped. These stress envelopes and
Wind load is the principal factor moment diagrams represent the most
which governs the design of cooling critical forces developed in the tower
during and after construction. For design purposes, the maximum tensile
forces N, and N 6, and their corresponding moments M, and M 6, are
used to determine the required prestressing forces in both directions.
Similarly, the compressive stresses
in concrete are examined based on the
maximum compressive force envelope
and their corresponding moment diagrams. The maximum transverse and
circumferential shear forces are also
investigated to check the maximum
principal tension in concrete. Finally,
the maximum radial and meridional
displacements are examined to determine the maximum possible movement of the structure during and after
construction.
'e- ·
o
c:!)
.
.
.
:'·.A,~"·~.:.
.;.:......
Symmetric
8=135
....... .
Three Ouarter
o
8=90"
8=45
Half
Quarter
Fig 5. Four loading conditions durir
erection of tower.
Gravity Load
Under gravity load, the analysis of a
segmentally constructed tower is different from that of a cast-in-place
tower. For a segmentally constructed
tower, prefabricated segments are
erected one by one. Thus, the weight
of each individual segment acts as a
vertical distributed load at the top of
the previously erected substructure.
During erection, the effects of both
partial gravity load (resulting from a
partially erected segmental ring) and
full gravity load (resulting from a fully
erected segmental ring) must be investigated. In this study, four different
gravity load cases are considered as
shown in Fig. 5. The last three loadings, being nonsymmetrical with respect to the vertical tower axis, are
approximated by the cosine series for
a uniformly distributed load:
for the wind analysis. The maximUl
tensile and compressive stress resu
tant envelopes for the various 'sul
structures are obtained by the sam
procedure described previously fc
the wind analysis. Under the gravit
load of a complete segmental ring, th
summation of the stresses develope
in the substructure at various stages (
construction represents the fmal stat
of stress due to full gravity load in th
tower.
Meridional PrestreSSing Force
After erection, prefabricated se~
ments are first post-tensioned in th
circumferential direction to form
segmental ring, and then post-ten
sioning is applied to join the segmen
tal rings in the meridional direction
The meridional prestressing force Cal
be replaced by an equivalent outwaI1
pressure resulting from the curvatul1
of the tendons, and an in-plane com
where 8 is measured from the axis of pressive force. Meridional tendons al1
symmetry as seen from Fig. 5.
positioned in the middle surface 0
For each case of the partial gravity the shell, thus eliminating the mo
load, the stress distributions are ob- ments atthe ends of the tendons.
The in-plane compressive force dut
tained at the various locations defined
by the same reference angles as those to the meridional prestressing i~
Design Example
(n-1) ELEM.
I,L---~
____- i -_ _
p
~
p
Fig. 6. Idealized meridional prestressing forces in tower.
idealized as a uniformly distributed equation for the meridional preload acting at the top and bottom of stressing force Ph at ith level:
any substructure, indicated as P, in
Fig. 6. The equivalent outward pres- P j = TH - DH + (20 + Sj-I)t H
sure, however, is idealized as conXI 1-1
centrated loads acting at each nodal
circle of the finite element and is represented by
(7)
Y" = P(sina~ - sina n _ I )
(6a)
X n = P(cosa n _ 1
(6b)
-
cosa,,)
Circumferential
Prestressing Force
The meridional prestressing force is
The circumferential tendons are podesigned to counteract all meridional sitioned so as to encircle the meridtensile stresses induced in the tower ional prestressing tendons. This arduring and after construction. At each rangement would keep concrete
stage of construction, some of the ten- under compression at all times, and
dons will be terminated while others the circumferential membrane forces
will be extended to meet the require- are equal to the applied circumferenments at subsequent stages of con- tial prestressing forces. Thus, the restruction.
quired circumferential prestressing
To determine the required merid- force can be determined easily, based
ional prestressing forces P" Po, ... P n on the tensile hoop stresses due to
at the various levels of the tower, an wind, gravity load and meridional
analysis is first performed for each of prestressing, allowing for a residual
the n-l substructures and the com- compressive stress of 20 ksf (0.96
plete structure under the effect of unit MPa) again.
meridional prestressing force. Then a
Therefore, the circumferential preset of n simultaneous equations is ob- stressing force Pc at any level can be
tained by equating, at each level, the computed as
prestressing force to the net meridional tensile force induced by wind
6M
Pc = N8 + __8 + 20t
(8)
and gravity load plus a small amount
t
of residual compressive stress in concrete, say, 20 ksf(0.96 MPa).
For more details of the analytical
The solution of the simultaneous background, see References 1 through
equations is given by the recursion 4.
1 ,,?
The tower considered herein has
the same geometry as that discussed
by Gurfinkel and Walser.' The shell is
355 ft (108.2 m) high and supported by
a bottom ring girder such that only the
meridional rotational freedom is provided as mentioned previously. The
throat of the tower is 165 ft (50.3 m) in
diameter and is located 60 ft (18.3 m)
below the top of the shell. The thickness varies from 30 in. (762 mm) at the
bottom level to 6 in. (152 mm) at 25 ft
(7.62 m) elevation from the base. At
the top 10 ft (3.05 m) of the shell, the
thickness also varies from 6 to 24 in.
(152 to 610 mm). Other than the top
and the bottom regions, the shell
thickness remains constant at 6 in.
(152.4 mm).
For this investigation, the tower
surface is divided into 20 symmetrical
units around the circumference and
each of these units is subdivided into
25 prefabricated segments (see Fig. 2).
Thus each segment will have a
meridional height of 14.2 ft (4.33 m)
and a circumferential length varying
from 25 to 45 ft (7.62 to 13.72 m).
The actual Structure is idealized as
100 inter-connected conical elements
and the loadings on the structure are
replaced by a set of loads applied at
the inter-connections (nodal circles).
Thus, each segmental ring is approximated by four truncated conical elements. At the boundary, the three
linear displacements are constrained
but not the rotation in the meridional
direction. These boundary conditions,
along with the free boundary condition at the top of the shell, are used in
analyzing all the substructures and the
complete tower under different loading conditions.
Meridional Force and Moment
According to the analytical procedures described above, the tower was
analyzed for wind and gravity load, as
well as unit meridional prestress for
the substructure corresponding to
each of the 25 construction stages. It
was shown that the maximum internal
forces due to gravity load occurred
when the substructure was fully
loaded by a complete segmental ring,
and the maximum tensile force due to
wind always occurred at the zero degree reference angle from the wind-
unit
mer. pres.
dead
1-0
11.3
-1.04
0·96
-0.83
+
49.
L -_ _--'
t.!1
t.!1
M
07
46.9 - - - - K/FT
1
21-4
K/FT
058
K/FT
Fig. 7. Meridional forces due to wind, gravity, and meridional prestress for the 25th
construction stage of tower.
wind
dead
1-9
mer. pres.
j lK/FT
0·3
0·4
ward side of the shell. The meridional
forces due to wind, gravity load, and
unit meridional prestress for the complete tower, are shown in Fig. 7, in
which tensile force is denoted as positive and compressive force as negative.
Similarly, the meridional bending
moments due to the same loadings are
shown in Fig. 8. The distributions of
meridional forces and moments for the
substructures at various stages of construction can be found in Reference 1.
86-1
.II
~I
0-5
~JI~
2·3
K·FT/FT
rmrrirTTr
K-FT/FT
K·FT/FT
Fig. 8. Corresponding meridional moments due to wind, gravity and mendional
prestress for the 25th construction stage of tower.
Table 1. Residual and bending stresses at selected levels of tower [Eq. (7)].
Construction
stage
Height
from
base
(ft)
25
20
15
10
5
2
1
0
355
284
213
142
71
28.4
14.2
2.5
Thickness
Maximum
tensile
force due
to wind
t
loadT
(ft)
(kips)
2
0.5
0.5
0.5
0.5
0.5
1.5
49.03
Gravity
load
forces
D
(kips)
sponding
moment
(kip-ft)
0
6
JO.5
14.8
17
19
20
0
0
0.3
0.1
0.5
0.65
1.2
2.87
0
0
11.325
31.65
43.38
46.948
48.0
49.0
21.36
Maximum
corre-
Bending
stress
S
(ksf)
(20+ S)t
(kips)
0
7.2
2.4
12
15.6
28.8
7.65
0
0
7.6
0.7
1.2
0.8
5.4
21.48
28.64
Note:! 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)].
Construction
stage
25
20
15
10
5
2
1
0
Height
from
hase (ft)
355
284
213
142
71
28.4
14.2
0
Note: 1 ft = 0.305 m; 1 ktp
11;;[
Meridional forces per unit circumferential length
(kips)
1
1.0406
0.967
0.8328
0.6975
0.62
0.6
0.587
= 4.448 kN.
1
0.9288
0.7987
0.6689
0.60
0.58
0.5634
1
0.8611
0.7222
0.65
0.63
0.6088
1
0.8407
0.76
0.73
0.799
1
0.91
0.88
0.8451
Required Meridional
Prestressing Force
the prestressing tendon, three tendons
each consisting of seven Ih-in. (12.7
mm) diameter 7-wire strand are required to provide an actual prestressing force of20.01 kips per ft (89 kN).
Having established the actual prestressing force at level 355 ft (108.2
m), the required prestressing force at
the next level, i.e, level 284 ft (86.56
m) is then obtained. Repeating the
same procedure, the required and actual prestressing forces are determined for each of the remaining
levels.
Review of Compressive Stresses
The various terms of Eq. (7) are
tabulated in Tables 1 and 2. In Table
3, Eq. (7) is used to determine the required prestressing force at the various levels. The computation begins at
the top level of the tower, i.e., level
355 ft (108.2 m) and the required prestressing force is 18.18 kips per ft
(80.87 kN). Based on an effective design stress of 162 ksi (1117 MPa) for
The design criteria call for a
minimum of 20 ksf (0.96 MPa) residual compressive stress at each level
of the tower during and after construction. As shown in Fig. 9, the
shaded area represents the magnitude
of residual compression at all levels of
the complete tower. The slightly
higher residual compressive force at
the lower part of the shell is provided
Table 3. Determination of required prestress force using Eq. (7).
Construction
stage
Height
from
hase
Required prestress force per unit length according
to Eq. (7)
(ft)
25
355
P,
= (11.325 + 7.6)1(1.00406) = 18:18 kips
20
284
PJ
= [31.65 - 20.01(0.967) + 0.7]1(0.9288) = 13.97 kips
15.624
15
213
P,
= [43.38 - 20.01(0.8328) - 15.624(0.7987) +
(0.8611) = 17.859 kips
19.27
10
142
p.
= [46.948 - 20.01(0.6975) - 15.624(0.6689) - 19.27
(0.7222) + 0.8]1(0.8407) = lLl5 kips
12.206
5
71
P,
= [48.0 - 20.01(0.62) - 15.624(0.6) - 19.27(0.65)
- 12.206(0.76) + 5.4]1(0.91) = 10.74 kips
13.69
2
28.4
p.
= [49'- 20.01(0.6) - 15.624(0.58) - 19.27(0.63)
20.01
1.2]1
- 12.206(0.73) - 13.69(0.88) + 21.48]1(0.967)
kips
1
0.967
0.9369
Actual
applied
force
(kips)
= 16.82
or
p.
= (49.03 - 20.01(0.58) - 15.624(0.5634) - 19.27
(0.0688) - 12.206(0.7099) - 13.69(0.845) + 28.64]1
(0.9369) = 22.66 kips
Note: 1 Ii = 0.305 m; 1 kIp = 4.448 kN.
23.60
r
355'
wind + dead + prestress
20.1 K/FT
I
l~/
Iwind
Fig. 9. Residual compresSion at the 25th construction stage of tower.
to counteract the higher bending
stresses in this region.
The most critical compressive force
distribution due to wind load developed at a 60-deg circumferential
angle. Adding to this force distribution the effects of gravity load and
prestressing, the total maximum compressive force distribution is shown in
Fig. 10. Under the combined effect of
the maximum compressive force and
the corresponding meridional bending
moment, the maximum compressive
stress was found to be 1835 psi (12.65
MPa) at 25 ft (7.62 m) from the base of
the shell, which is within the allowable compressive stress for a design
concrete strengthf~ = 5000 psi (34.47
MPa).
Circumferential Forces
and Moments
The application of meridional prestress induces circumferential forces
in the shell. Based on the actual prestressing force established previously,
the induced circumferential force for
each stage of construction was computed. It was shown that the most
critical circumferential force distribution developed at the final stage of
156
construction. The circumferential
forces due to wind were also computed for the various reference angles
during each stage of construction.
Based on the results of these analyses,
the design force envelope is shown in
Fig. 11. An important feature of Fig.
11 is that the design envelope is governed by the peak tensile forces in the
tower at the various stages of erection
rather than the tensile force in the
tower after its completion.
The circumferential forces due to
gravity load were likewise determined
for each intermediate construction
stage. The circumferential force distribution for the complete tower is
shown in Fig. 12. It is observed that
each segmental ring produces a large
circumferential compressive force at
the top edge of the previously completed substructure, and this large
compressive force diminishes very
quickly to a small tensile force at the
lower edge of the previous segmental
ring.
The envelope for the force distribution is similar in shape to that of
the entire tower when it is analyzed
for its own full dead load. Similar envelopes for circumferential moments
/ merid. prest.
/dead/
Fig. 10. Maximum compressive force distribution in tower.
~2==2::.::.4~72~1l=====~===-=-:::--_ _-, h=355
,
I
h=285
h=213'
Design
Envelope
5·454
5·15
7·356
K/FT
Fig. 11. Final deSign tensile force envelope due to wind load at various stages of
erection of tower.
6·24
KlFT
11
355/
comp'
Tower
II
-.l.
17·8
K/FT
Fig. 12. Circumferential force due to gravity load for 25th construction stage of tower.
PCI JOURNAUJuly-August 1980
157
23-046
wind
+
dead + prestress
Table 4. Circumferential forces and moments for different segmental rings along
tower height.
K/FT
Level (ft)
14·7
Location
Segment
No.
1
2
Bottom
part of shell
(+)
Middle part
of shell
Fig. 13. Final design circumferential tensile force envelope in tower due to wind,
gravity, and meridional prestress.
due to wind and gravity load have
been obtained elsewhere.
Required Circumferential
Prestressing Force
various points of the shell. By combining the effects of meridional prestressing, circumferential prestressing,
wind and gravity load, it was found
that the maximum compressive stress
of 1140 psi (7.86 MPa) occurred at the
top of the shell, which is well within
the allowable.
Superposing the effects of meridional prestressing, wind and gravity
load, one can obtain the maximum circumrerential force envelope shown in
Fig. 13. It is seen that the maximum
circumferential force is nearly conEconomics
stant for the middle part of the shell
but varies significantly near the top
Based on conservative cost estiand the bottom of the shell. Therefore, mates of materials and labor in the
the design can be executed for a typi- United States in 1976, the cost of the
cal segmental ring of the middle part 355 ft (108.2 m) segmentally conof the shell, two rings at the bottom structed cooling tower may be
and five rings above the throat level.
itemized as follows:
In Table 4, the design force and
moment are tabulated for the various Materials (concrete, reinsegmental rings. Using Eq. (8), the reforcement, tendons) .... $ 633,000
quired prestressing force per unit Labor (precasting, stresslength is computed in Table 5, in
ing, grouting) ......... . 300,000
which the number and size of tendons Formwork and setup cost . 220,000
for each segmental ring are also tabu- Erection ................ . 500,000
lated.
1,653,000
Contingency, overhead and
Review of Compressive Stress
profit ................. . 678,000
Having determined the circumferential prestressing force, it is necessary to examine the maximum circumferential compressive stress at the
Total
$2,331,000
This estimate may be adjusted to a
0.0
14.2
3 to 20
(typical)
28.4
21
22
23
24
25(a)
25(h)
284
298.2
312.4
326.6
340.8
Top part
of shell
7.356
From
Note: 1 ft = 0.305 m; 1 kip per ft
Corresponding
moment
(kip"ft per ft)
Average
Maximwn
thickness
To
(ft)
fOrce
(kips per ft)
14.2
28.4
2.2
1.0
7.356
6.70
0.285
0.10
284
0.5
7.00
1.14
298.2
312.4
326.6
340.8
355
0.5
0.5
0.5
0.5
1.25
2.00
7.8
9.1
11.2
14.7
19
23.046
1.14
1.0
0.8
0.6
12.01
10.70
= 14.584 kN/m; 1 kip = 4.448 kN.
Table 5. Determination of required Circumferential prestressing forces in tower
using Eq. (8).
Required prestressing force per unit
length
Total
force
P.e=NfJ+
Segment
Location
Bottom part
of shell
Middle part
of shell
Top
part
of
shell
No.
6M.
t
+ 20t (kips)
per
segment
(kips)
Eq. (8)
Circumferential
tendons per
segment
7.356 + 6(0.285)/2.2 + 20(2.2)
6.7 + 6(0.10)/1.0 + 20(1.0)
=
=
52.13
27.3
740.24
387.66 4-(8) 'h-in. 4> strands
3-20
7 + 6(1.14)10.5 + 20(0.5)
=
30.7
435.94 3-(6) 'h-in. 4> strands
21
22
23
24
25(a)
(b)
7.8 + 6(1.14)/0.5 + 20(0.5)
9.1 + 6(1.0)/0.5 + 20(0.5)
11.2 + 6(0.8)/0.5 + 20(0.5)
14.7 + 6(0.6)/0.5 + 20(0.5)
19 + 6(12.01)/1.25 + 20(1.25)
23.056 + 6(10.70)/2.0 + 20(2.0)
= 31.48
= 31.10
= 30.80
= 31.90
= 101.6
= 95.23
1
2
447
441
437
452
1442.7
3-(6) 'h-in. 4> strands
5-(12) 'h-in. 4> strands
(a) At middle of 25th segmental ring whICh controls design.
(b) A~ top e~ge of 25th se!l?'e~tal ring which controls design.
Note. 1 m. - 25.4 mm, 1 kip - 4.448 kN.
total cost of $3,500,000 in 1980, based
on an inflation factor suggested by
Engineering News Record's cost index
of 50 percent increase in materials,
labor and erection and 60 percent increase in furmwork and setup cost.
The cost of a similar cast-in-place
reinforced concrete cooling tower was
quoted as $4,000,000 in 1976 from reliable industrial sources. Using the
same inflation factors, the total cost of
the cast-in-place cooling tower should
be adjusted to $6,000,000 in 1980.
Therefore, it appears that by the
technique of segmental construction, a
saving of almost 40 percent of the cost
of the tower (excluding foundation
and columns) is indicated. In addition,
there is a distinct benefit of savings in
construction time.
CONCLUDING REMARKS
The technique of segmental construction can be advantageously
applied to hyperboloid cooling towers. Analysis and design of such a
tower should be perfonned for each
stage of construction. Circumferential
stresses, induced in the partially completed tower, may exceed the corresponding stresses in the completed
tower, thus dictating the design.
Due to the small meridional curvature for each segmental ring, the circumferential prestressing has little
effect on the meridional prestressing.
Therefore, in design, the total meridional prestressing force should be
detennined first in order that its effect
can be accounted for in the design of
the circumferential prestressing. A
convenient and efficient procedure of
detennining the required meridional
prestressing force at different levels of
the tower is to start from the top level
of the tower and proceed downward
using a recursion relation.
An extension of this study including
an experimental investigation using
small-scale structures, as well as an
examination of dynamic response to
seismic and wind effects, is in progress at North Carolina State University, Raleigh, North Carolina, under
the direction of Dr. Paul Zia.
REFERENCES
1. Rizkalla, S. H., «An Investigation of
Nonsymmetrical Loading," PhD Dissertation, Duke University, Durham,
North Carolina, 1968.
4. Fliigge, W., Stresses in Shells, Springer-Verlag, Berlin, Germany, 1962, p. 78.
5. ACI-ASCE Committee 334, "Concrete
Shell Design and Construction-Design
and Construction of Reinforced Concrete Cooling Tower Shells," Practice
and Commentary (draft), American
Concrete Institute, Detroit, Michigan,
1974.
Segmentally Constructed Hyperboloid
Natural Draft Cooling Tower," PhD
Dissertation, North Carolina State University, Raleigh, North Carolina, 1976.
2. Gurfinkel, G., and Walser, A., "Analysis
and Design of Hyperbolic Cooling
Towers," Journal of the Power Division,
ASCE, 1972, V. 98, No. POI, p. 133.
3. Coffin, G. K., "Finite Element Analysis
of Open Shells of Revolution Under
*
Discussion of this paper is invited.
Please forward your comments to
PCI Headquarters by March 1, 1981.
*
*
APPENDIX A-NOTATION
a o = throat radius of hyperboloid
shell ofrevolution
b = constant parameter for hyperboloid shell of revolution, Eq. (2)
C. = circumferential distribution
coefficient for wind load
D j _, = internal meridional force
due to gravity load at
(i - l)th level
G = dynamic gust factor for
wind load
K. = exposure factor for wind
load
M.= meridional bending moment per unit circumferentiallength
M 9 = circumferential bending
moment per unit meridional
length
N. = meridional force per unit
circumferential length
N 9 = circumferential force per
unit meridional length
n = number of harmonics, or
number of segmental rings
P = meridional prestressing
force
Pc = circumferential prestressing
force
P d = unifonnly distributed gravity load
*
P d = coefficient in cosine series,
Eq.(5)
q(z,8)= equivalent static normal
wind load
Q30 = wind pressure, Eq. (4)
Ro = horizontal radius of hyperboloid shell of revolution at
Y level
S j = bending stress due to wind
and gravity load at ith level
T = vertical distance from shell
top to throat level
T J = internal meridional force
due to wind load atjth level
t = shell thickness
to = radius oftop of tower shell
V 30 = wind velocity at 30-ft elevation
X jJ = internal meridional force at
jth level due to unit meridional prestressing force at
ith level
X.,Y.= component of equivalent
outward pressure of meridional prestressing force
Y = vertical coordinate, Eq. (1)
z = vertical distance measured
from ground level
a. = angle between normal of
shell surface and axis of
revolution
8 = circumferential reference
angle
*
*