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ACI JOURNAL

TECHNICAL PAPER

Title no. 76-8

Specifying Tolerance Limits for Meridional Imperfections in Cooling Towers

by J. G. A. Croll and K. 0. Kemp Following the recent collapse of a cooling tower shell in Great Britain, believed to have been primarily due to the combined effects of geometric imperfections and vertical cracks, the design implications of geometric imperfections in the meridional profiles of cooling towers are reassessed. It is suggested that present and proposed tolerance recommendations are not sufficiently linked to the specific nature of the shell and its reinforcement, and if followed could result in serious overstressing. On the basis of a simplified approach to the analysis of a geometrically imperfect shell, tentative, code oriented, and rational tolerance specifications are presented. While retaining simplicity these recommendations do allow the tolerances to be related to the specific conditions which apply at any position of the "as designed" tower, and enable the complete specification of meridional tolerances prior to the commencement of construction. Keywords: cooling towers; concrete construction; reinforced concrete; reinforcing steels; shells (structural forms); specifications; stresses; structural design; tolerances.

Received Sept. 15, 1977, and reviewed under Institute publication policies. Copyright © 1979, American Concrete Institute. All rights reserved, including the making of copies unless permission in writing is obtained from the copyright proprietor. Discussion closes Apr. 1, 1979.

ACI JOURNAL/January 1979

139

James G.A. Croll is a lecturer in civil engineering at University College London and a graduate of the University of Canterbury, Christchurch, New Zealand, B.Eng. 1964, Ph.D. 1967. Engineering experience includes consulting on the design and construction of cooling towers and participation in both the British Standard and International Association for Shell and Spatial Structures Committees for drafting recommendations for the design and construction of cooling towers. Professor K.O. Kemp is Chadwick Professor and Head of the Department of civil engineering at University College London. He is a graduate of the University of London, B.Sc(Eng) 1947, Ph.D. 1960. He was a member of the Committee of Inquiry into the collapse of the cooling tower at Ardeer Nylon Works, Ayrshire, Scotland in 1973 and acts as a consultant on the design and construction of reinforced concrete cooling towers.

INTRODUCTION The classical literature on shell analysis and design provides little guidance to the engineer as to whether or not deviations from the designed shell geometry are likely to be important. Even when geometric imperfections are treated the advice given is far from clear. Flugge, for example, has considered the effects of axi-symmetric discontinuities in the meridional curvature of a hemisperical dome (1). Using a membrane theory he showed that very severe disturbances would be introduced in the circumferential stresses in the vicinity of the imperfection. But surprisingly, Flugge implies that these stress concentrations are not important since when bending is included the circumferential stress "makes only a feeble attempt to follow the •••• high peaks" that are predicted by the membrane analysis (reference (1) pp 367). In a later section of this same work it is concluded, on the basis of this example, that deviations of the order of magnitude of the wall thickness in cylindrical shells "are not a matter of great concern" (reference (1) pp463). Since the implications of Flugge's advice would appear to have influenced initial attempts to understand the role of meridional geometric imperfections in cooling towers, it is important that the veracity of such assertions should be re-examined. The influence of an axi-symmetric imperfection is perhaps most clearly illustrated by the case of an axially loaded circular cylinder shown in Fig. 1. If N is the meridional stress resultant in the perfect shell, then the 8 out-of-balance force that would be produced when a band imperfection of amplitude g. is introduced could be carried in only two ways. Either ~

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ACI JOURNAL/January 1979

the out-of-balance moment that would exist on a typical imperfect element shown in Fig. 2(b) would be carried by developing meridional moments M (as would occur in a column) or it would be equilibrated by ~he tangential components of moment T.d, shown in Fig. 2(c), which results from the adjacent bands of hoop tension T and compression C indicated in Fig. 2(b). As in any linear shell response the fact that the membrane stiffness is very much higher than the bending stiffness will mean that the shell has a predilection to resist this out-of-balance force by developing membrane stress in which the contribution from T.d/r is substantially greater than that from M • The dominance of the membrane stresses in providing th~ resistance to the out-of-balance moment will increase as the shell radius to thickness ratio, r/t is increased. This does not of course imply that bending stresses will not be high; merely that the greater proportion of the out-of-balance load will be carried by the development of circumferential stresses, so that hoop tensions and compressions play a vital role in the behaviour of meridionally imperfect shells. Contrary to Flugge's suggestion, there is certainly no way that the bending action M could replace the contributions of the circumferential s~resses in an elastic system with practical r/t ratios. The importance of this hoop action has been demonstrated more recently by both Soare and Calladine. Soare (2), considering a rotationally symmetric imperfection in the meridian of a hyperbolic cooling tower, shows that under both gravity and wind loading the "hoop forces are strongly influenced". His Figs. 5 to 8 make quite clear that for an approximation of the imperfections measured on one particular tower, the hoop tensions induced are of a magnitude greater than even the maximum meridional compression occurring in the tower. However, despite his warning that "generally •••• constructional imperfections of the order of the shell thickness may increase considerably the hoop forces in the shell" (reference (2) pp 378) Scare's results did not make the design impact that they would seem to justify. In part this is possibly due to Scare's use of membrane theory which apparently allows Flugge's assumption to be invoked, namely that the high peaks in hoop stresses would be effectively eliminated by the inclusion of bending terms in the analysis. That this is not the case however, has been recently demonstrated by Calladine (3) who, along with Heyman (4) in an early article, has also drawn attention to the importance of hoop actions. Unlike Scare's analysis Calladine's simplified theory did include bending terms.

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THE ROLE OF IMPERFECTIONS IN THE COLLAPSE OF A COOLING TOWER A more graphic demonstration of the importance of imperfection generated hoop actions is to be found in the conclusions of the report of the committee of inquiry investigating the collapse of a cooling tower at Ardeer in Scotland in 1973 (5). It is fortunate that this tower was being built in the rather sensitive post Ferrybridge failure atmosphere, since both the construction and postconstruction records are unusually thorough .These indicated that considerable surface imperfections existed in the shell, particularly below the throat, and that over the seven year life of the structure, a number of well defined meridional cracks had developed. One of these cracks had been observed, just prior to the collapse, to reach a level at which the imperfections in the shell had their most severe magnitudes. Two independent calculations carried out for the committee of inquiry indicated that the hoop tensions developed in the vicinity of these imperfections could be resisted by the concrete up to the design wind speed of 120m.p.h. However, if a discrete vertical crack were to pass across this imperfection, as seemed to have occurred in the Ardeer tower, then yield of the 0.15% horizontal steel at the level of the imperfection would be anticipated at a wind speed of between 60 and 80m.p.h. (27 to 36m/s). Collapse occurred at an estimated wind speed of between 70 and 80m.p.h. (31 and 36m/s). Further information concerning this collapse, together with details of the calculations, can be found in reference (6). Although other possibilities for the cause of collapse were investigated, the committee of inquiry concluded that the presence of surface imperfections played a major role in the "tension failure of the circumferential reinforcement" which was in turn considered to be "the most probable initiating cause of the collapse of the Ardeer tower" (reference (5) pp 14). As a consequence of this finding, it was recommended that "for future cooling towers, tolerances on imperfections should be specified". In making these specifications the committee recognised the need for "rational procedures to be developed" which would not only take account of the "shape and size of the tower" but would also "be related to the minimum percentage of circumferential reinforcement" (reference (5) pp 16-17).

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ACI JOURNAL/ January 1979

EXISTING RECOMMENDATIONS OF GEOMETRIC TOLERANCES Using some erection procedures the imperfections in the constructed meridians of concrete cooling towers can be substantial. There is certainly growing evidence to suggest that those found in the Ardeer tower were not altogether extraordinary. But despite the apparent difficulties of producing "perfect" meridians and the known sensitivity of hoop stresses to meridional imperfections, design recommendations which recognise the need for strict control of geometric tolerances are only just starting to appear. A working group of The International Association for Shell and Spatial Structures has just proposed interim guidelines on acceptable geometric deviations (7). Firstly, the maximum error in the slope of the meridian should not exceed +1.5%. Secondly, within this slope the absolute maximum radial error should not exceed +0.10. or the value (measured in m) ~/47.5 where r. and t are the regional values of the horizofttal radius §nd the wall thickness of the shell in metres. These tolerances are associated with specified minimum percentages of 0.25% high tensile steel or 0.35% mild steel in both directions at a spacing not exceeding twice the shell thickness. Similar recommendations are made in the new draft British Standard on the design of cooling towers (8). In this document, it is specified that the maximum error in meridional slope should not exceed +1%, subject to a maximum error in horizontal radius of 15cm with a specified allowance for survey inaccuracies which could increase the permitted measured error to at least +10cm. The specified minimum reinforcement is 0.20% of the concrete cross sectional area in both directions. In both these recent design recommendations the specified tolerances are based very largely in intuition, and neither would seem to take adequate account of the particular shell characteristics. The specified minimum steel percentages in the two recommendations are significantly different with the British Standard adopting the lower level. It would appear that there remains a need for simple but rational specifications of the permissible geometric tolerances in meridional errors.

THE INFLUENCE OF MERIDIONAL IMPERFECTIONS Two approaches to the simulation of axi-symmetric meridionally imperfect shells have been used by the authors. In the first approach, the geometry of the imperfect tower is modelled exactly as a series of piece-wise continuous second order shells of revolution, and the appropriate equations for shell bending are solved using a finite difference discretisation (9). In the

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143

second approach the geometric imperfection is simulated by a normal pressure distribution chosen so that it is statically equivalent to the out-of-balance forces that would be present when the membrane stresses of the perfect shell are considered to act on the geometrically imperfect shell (10). The advantage of the latter, equivalent load, approach is that it enables the effects of axi-symmetric and non axi-symmetric imperfections to be investigated with relatively minor modifications to standard shell bending programmes. It has the disadvantage, however, that the relationship of the modification to the stresses to those that would actually occur is uncertain. Clearly, if it can be shown that the two methods predict substantially the same results, then the equivalent load method would allow considerable computational economies. Also, it will be shown in the following section that the equivalent load method provides a clear means of delineating those parameters which have the most important influence on the behaviour of the imperfect shell. To compare the two approaches the shell, whose geometry is a slightly idealised representation of the Ardeer tower shown in Fig. 3(a), is considered to have the axi-symmetric inward imperfection shown in Fig. 3(b). Also shown in Fig. 3(b) are the hoop stress resultants in the imperfect shell under gravity load. The specific weight of the shell of uniform thickness 15.25cm, is taken as 23.3kN m-3. It can be seen that the modifications in stress are confined to the zone of the imperfection itself, and that for this particular imperfection the two methods predict almost identical disturbances in behaviour. The difference in the maximum magnitude of predicted using the two methods is in the example of F1g. 3 only 1%. More detailed comparisons, reported in reference (10) show that for other imperfections, and for imperfections at various locations in the shell, the equivalent load method provides an accurate basis for the analysis of meridional imperfections.

Ne

Ne

Making use of the equivalent load approach the changes in hoop stress resultants are shown in Fig. 4 for the shell geometry, and imperfection position and magnitude shown in Fig. 3, but for a representative range of imperfection profiles. These changes have been normalised so that they correspond with a unit meridional tensile stress resultant in the perfect shell and do not include any hoop stresses from the perfect shell response. Fig. 4 shows that for the inward imperfections considered the changes in tensile hoop stress results n9 are in the range

o.54o < lnel < 1.127

144

(1)

ACI JOURNAL/January 1979

while the change in compressive hoop stress resultant is in the range

o.413

< tn 6 t < o.6o4

(2)

To cover the possibility of both inward and outward imperfections, the unit changes in hoop tensile stress resultant n9 could, therefore, lie in the range

(3)

AN APPROXIMATE TREATMENT OF MERIDIONAL IMPERFECTIONS Provided the maximum imperfection, gi' is small in comparison with the wavelength of the imperfection, hi, it can be shown that for the symmetrically disposed piece-wise continuous circular arcs shown in Fig. 5 the curvature error is

(4) where rs and r; are the meridional radii of curvature of the perfect and imperfect shells respectively. The out-of-balance force associated with the meridional tensile stress resultant at the level of the imperfection,Ns., acting along the imperfect meridian is then statically ~ equivalent to a normal pressure p, (positive inwards) given by

(5) With this normal pressure the change in the hoop stress resultant could be approximated by the simple ring hoop stress prediction, (6) In this expression ri is a measure of the horizontal radius of

the perfect shell at the level of the imperfection. Combining Eqs. (4) to (6) results in the approaiaation of the changes in hoop tension,

(?) where the ring hoop stress for a unit meridional stress resultant is

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16ri gi

(8)

hi 2 For the imperfection and shell considered in the example reported in Fig. 4, for which ri = 24.5m, gi = 0.305m and hi = 12.6m, the unit change in hoop stress is (9)

On the basis of the approximate solution of the band in Fig.5 the absolute range of the changes in stress indicated in Eq. (3) may be normalised with respect to this unit change in hoop stress resultant as,

na

(10) That is, the maximum hoop stress in some unspecified imperfection profile would be expected to lie between 0.60 and 1.54 times the hoop tension predicted on the basis of a smooth circular imperfection of the same maximum amplitude using simple ring theory. It may be observed in Fig. 4 however, that the ring stress = 0.732 is a very close estimate of the change in hoop stress that occurs for the imperfection whose profile does in fact consist of three piece-wise continuous circular arcs.

ne

PROPOSED RECOMMENDATIONS ON MERIDIONAL TOLERANCES The range of possible changes in hoop tension indicated in Eq. (10) is representative of the situation that occurs when an imperfection is situated at other positions in this tower and within other practical cooling towers. Taking the most pessimistic prediction of n9 as the rounded off value, ( 11)

ne = 1.5oiie then the total circumferential stress resultant Ne that will occur in the imperfect shell will be a combination of the change n8 .Nsi and the perfect hoop stress resultant N9 . at 1 the levei of the imperfection. If this total stress resultant is to be limited to the resultant N~ at which the circumferential reinforcement yields, then

(12) Making use of Eqs. (8) and (11) the inequality Eq. (12) may be re-written in the form

T:

1 24N

s.

1

146

.r. 1

(N~ - N9 )

(13)

i

ACI JOURNAL/January 1979

Inequality Eq. (13) then provides a convenient measure of the tolerance limits on the meridional curvature errors, expressed in terms of: the perfect stresses Ne· , Ns· that would exist at the level of the imperfection; th~ hori~ontal reinforcement, expressed in the form of its yield stress resultant, N~ • In other words, the permissible tolerance on meridional curvature errors T is expressed entirely in terms of quantities that are determined at the design stage. This tolerance T would, of course, depend on the specific shell, its loading and the height above shell soffit. As an example of how information available at the design stage could be used with Eq. (13) to prepare tolerance to be achieved in construction, consider the perfect shell whose overall dimensions are given in Fig.3(a). Suppose further that this shell is to be designed for wind speed V = 53.5m/s (120m.p.h.) for which the associated circumferential and vertical pressure distributions are taken in conformity with BS 4485 (8), and that the mild steel horizontal reinforcement is the minimum of 0.35% specified in reference (7), has a uniform distribution over the shell height and has a yield stress of 340MN/m2. For this shell the information necessary to specify tolerances T is shown in Table 1 along the critical meridian at +70° to the wind direction. The meridian at roughly +70° to th; wind direction will generally be critical, for at this location the gravity and wind meridional compressive stresses are additive, yielding the maximum values of Nsi and therefore minimum of T in Eq. (13). The curvature tolerance T can be seen from Table 1 to reach its minimum magnitude at around 1/10 the shell height, or just above what would normally be the soffit ring beam. At 4 th~s level the curvature tolerance limit ofT< 3.36 x 10m- gives rise to the slope limitations indicated by line aa in Fig. 6. Also shown in Fig. 6 are the corresponding tolerances for these same quantities as specified in the !ASS guidelines (7). The line bb represents the 1.5% slope limitations, while cc is the critical maximum absolute error ~i < 5cm that would be permitted at this position. It is apparent that for imperfections with wave lengths less than about 12.5m the IASS guidelines require considerably less stringent tolerances than those to prevent steel yield specified by Eq. (13). For very small wavelengths of 5m or less, it is possible that the consequences of yield of the reinforcement would not be as serious as for the longer wavelengths in which stress redistribution may not be able to accommodate local overload. However, for the reasons alluded to in the introduction, the potential for stress redistribution for a band imperfection is not great. Until considerably more is known about the ductility characteristics for reinforced concrete shells in this situation it would seem that the IASS guidelines should certainly not be regarded as conservative. On the other hand, they may

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147

provide unnecessarily conservative and perhaps unrealistic tolerances for the longer wavelength meridional imperfections. Similar conclusions may be reached on the tolerances recommended in the draft British Standard (8). For the 0.2% minimum horizontal steel permitted by this standard at the same location as the previous example, curvatu~e errors must be less than the tolerance T < 2.02 x 10-~m- 1 • Again, it can be seen in Fig. 7 that for almost the entire range of practical hi the requirements of the British Standard appears to overestimate the permissible tolerance for the minimum steel percentage specified. The discrepancies between existing tolerance recommendations and those proposed here become even more serious as the overall size of the shell is increased. Assuming, for example, that all dimensions of the shell including the wall thickness are simply scaled in the same proportion from those given in Fig. 3(a), then the effects of varying overall shell height H are shown in Fig. 8(a) and (b). In Figs. 8 the tolerances are again considered at 1/10 the shell height for the case of ~-35% horizontal reinforcing steel of yield stress 340MN/m • It can be seen in Fig. 8 how for a given imperfection wavelength hi the !ASS recommendations would allow an increase in both the permissible slope ~i/(hi/2) (provided it is less than 1.5%) and radial error~ i(provided it is less than 0.1m) as the shell height is increased. This is in marked contrast with the present proposals, where to prevent steel yield the reverse would need to be the case. Thus, as the overall height H is increased the !ASS recommendations, and also of course those of the BS 4485, appear to become increasing nonconservative. For a 200m shell height the maximum permissible amplitude of radial error, for an imperfection wavelength hi = 10m, is ~i < 9.27cm in the !ASS document, whereas that indicated on the basis of Eq. (13) is ~i < 1.79cm. Associated with these increasing discrepancies is an increase in the wavelength hi for which the !ASS recommendations start to provide a conservat~ve tolerance limit as compared with those of Eq. (13). Again for the 200m shell the wavelength at which the !ASS recommendations are more conservative is hi= 24m compared to~= 11.5m for a 100m high tower. Had the comparisons of Figs. 8 included the increases in the design wind velocity that would be necessary with increasing height, H, the indicated deficiencies of existing recommendations would be even greater. Considering the vertical distribution of meridional tolerances, Fig. 9 shows that for towers with overall dimensions scaled from Fig. 3(a) the !ASS minimum radial deviations show an almost opposite trend to those of Eq. (13).

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For a typical wavelength hi = 10m, the IASS recommendations indicate that tolerance should be more stringent in the upper regions of the shell, while the present proposals suggest that the most critical zone occurs over the lower third of the shell. Clearly other considerations such as aesthetics or the effects of circumferential curvature errors will limit radial errors towards the top of the shell. In the above, the IASS recommendations have been used as a standard of reference for discussing the present proposals. This has been done precisely because these recommendations would appear to represent the most comprehensive guidance available to the engineer. But it would appear that even they may not be adequate in many practical situations, particularly as overall tower dimensions are increased. It is perhaps relevant to observe that the recent report (11) of ACI/ASCE Task Committee on Concrete Shell Design and Construction dealing with reinforced concrete cooling towers, makes no recommendations as to acceptable tolerances. With these latter guidelines explicitly directed towards cooling towers of height greater than 100m, this could represent a serious omission. CONCLUSIONS With the effects of geometric imperfections now considered to have provided an important contribution to the collapse of the Ardeer cooling tower in Scotland, initial results from a continuing reappraisal of the influences of meridional constructional errors have been presented. By examining the changes in hoop tensions that occur for a range of axi-symmetric imperfection profiles in a typical hyperboloidal shell, it has been shown how these changes can be conveniently related to the stress that would result from a simple ring analysis. The ring analysis is based on the equivalent load that would result from a piece-wise continuous assemblage of circular arcs with the same maximum error in horizontal radius. In this way, a simple design formula is derived which enables a tolerance parameter T to be expressed in terms of the static stresses and geometry of the perfect shell, and the yield strength of the horizontal reinforcement in the region of the imperfection. By comparing tolerances based on the proposed design formula with the tolerances specified in available design and construction guidelines, it is suggested that over certain regions of the shell the existing recommendations may not be conservative for short wavelength meridional errors, and over conservative for longer wavelengths. It is also suggested that as tower dimensions are increased the potential importance of meridional imperfections increases while existing guidelines permit decreasing stringent tolerance specifications.

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ACKNOWLEDGEMENTS The authors wish to express their appreciation to Imperial Chemical Industries for their support in carrying out aspects of this work and for permission to use sections of material prepared for the Committee of Inquiry into the collapse of the Cooling Tower at Nylon Works, Ardeer,Stevenston, Ayrshire. They would also like to acknowledge their debt to Mr. F. Kaleli whose computer programmes were used as the basis of the numerical studies. REFERENCES

1. Fliigge, W., "Stresses in Shells", Springer-Verlag, 2nd Printing (1963),p 367 and p.463. 2. Soare, M., "Cooling Towers with Constructional Imperfections", Concrete, November (1965), pp369-379. 3. Calladine, C.R., "Structural Consequence of Small Imperfections in Elastic Thin Shells of Revolution", Int. J. Solids & Structures, vol. 8 ( 1972), pp679-697. -4. Heyman, J., "On Shell Solutions for Masonry Domes", Int. J. Solids & Structures, vol. 3 (1967), pp227-241. 5. "Report of the Committee of Inquiry into the Collapse of the Cooling Tower at Ardeer Nylon Works, Ayrshire, on Thursday 27 September 1973", Imperial Chemical Industries Ltd., Petrochemicals Division (1974) Millbank, London SW1P 3JF. 6. Kemp, K.O. and Croll, J.G.A., "The Role of Geometric Imperfections in the Collapse of a Cooling Tower". The Structural Engineer, vol. 54, No. 1, January (1976) pp33-37. 7. "Recommendations for the Design of Hyperbolic or other similarly shaped Cooling Towers", Edited by !ASS Working Group, Brussels 1977. 8. B.S. 4485 (draft) Part 4, "Structural Design of Cooling Towers", British Standards Institute, 2 Park Street, London W1A 2BS. 9. Croll, J.G.A., Kaleli, F. and Kemp, K.O, "Behaviour of Geometrically Imperfect Cooling Towers", In process ASCE Engineering Mechanics Division. 10. Croll, J.G.A., Kaleli, F., Kemp, K.O. and Munro, J., "A Simplified Approach to the Analysis of Geometrically Imperfect Cooling Tower Shells", Engineering Structures, vol. 1, No. 1, September (1978). 11. AC~ASCE Committee 334 '~einforced Concrete Cooling Tower Shells - Practice and Commentary", ACI Journal, vol. 74, No. 1, (1977), pp22-31.

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Table 1. Information required to calculate tolerance limits of Eq. (13). for perfect shell of Fig. 3 subject to gravity and 54mVs wind load.

Height above Soffit*

Horizontal Radius

z

r

Hoop Stress Dead and Wind Load

Ne.

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 o.8 0.9 1.0

*

z

Meridional Stress Dead and Wind Load

Tolerance Factor Eq.(13)**

Ns.

T

1

(m)

(kN m- 1 )

(kN m- 1 )

(m-1)

39.5 36.1 32.8 29.8 27.2 24.9 23.2 22.2 21.9 22.5 23.7

-165 -14.1 -19.2 -25.9 -32.5 -37.9 -35-9 -22.1 + 2.5 +29.2 +69.3

-740 -675 -621 -581 -550 -515 -452 -368 -209 - 65 0

4 .93x1o_-4 4 3.36x10 4 4.11x1o: 4 5.01x10 4 6.01x1o- 4 7 .14x10- 4 8.65x1o-4 10.40x1o-4 16.40x1o-4 35.80x10-

(height above soffit level/overall height of shell 0.0035 x t xfy (t

ACI JOURNAL/January 1979

= 0.153m,

fy

= z/H)

= 340MN.m- 2 )

151

Figure 1.

Axi-symmetric Meridional Band Imperfection

N5

(a)

T.d T.

"'-r;

""'-~I

~~

c (b)

(C)

Figure 2. Statical consideration for out-of-balance forces when stress N~ of (a) the perfect shell acts on (b) a band imperfect1on. Moment T.d statically equivalent to the combined effect of C and T.

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ACI JOURNAL/January 1979

47.4m

I

~

l {

15·25cm ri

(a)

height above soffit z(m)

=24.5m

7lam

I

height above soffit z I m I

--

0

geometric imperfection

+0.5

~ (mJ

N; lkNtm)

(b)

Figure 3. (a) Shell geometry and assumed positions of axi-symmetric band imperfection. (b) Comparison of total hoop stress resultants for geometric and equivalent load simulation of shell with band imperfection under gravity loads.

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153

straighf/c

cui r

~-

rcosinP ~{ straigh t_~---~~o-AI'If---+.,___-----~,oiF---+-~ ~--

u/artstraight --t--="'ff~IL----1--+.."llf:Hf'lr- ~ ••••••• cJr ular 1 ~-

---+---"'~~~~"tfl'li&:ar-1-oo/

.. -

ic

~-x-

-1.0

-0.5

0 n~

"

+0.5

+1.0

(N/m)

Figure 4. The influence of variations of meridional imperfection profiles on the changes in hoop stress resultant ne resulting from a meridional tensile stress resultant in the perfect tower of Nsi = 1N/m •

154

ACI JOURNAL/January 1979

p

=N

s;

16r;

n,~

~

p

~

~ 14-.,

fl. I

-....

-""

-

..._

p ,.

,.

~

Figure 5. Imperfection band assumed for calculating the out-of-balance loads in the approximate treatment used as a basis for specification of design tolerances.

ACI JOURNAL/January 1979

155

~i

h;

c

1.5

\. '

("/.)

1.0

/

b

v

I

2~·

~

-' h;