Pressure buckling of cantilevered shells of revolution

Thin-Walled Structures 4 (1986) 381--408

Pressure Buckling of Cantilevered Shells of Revolution G e o r g e M. Zintilis Atkins Research and Development, Woodcote Grove, Epsom KT18 5BW, UK

and J a m e s G. A. Croll Department of Civil and Municipal Engineering, University College London, London WC1 6BT, UK

A BS TRA ('T The elastic" buckling of cantilevered shells of revolution is consMered, using both theoretical and experimental modelling. To account.lor the loss of suilJness occurring in the post-buckling response, atut the experimental reductions compared with classical predictions Of critical unti[brm external pressures, the relatively sb~tple reduced stij{lhess method of analysis is developed. Within experimental accuracy, this is s/lowtt to provide a convenient means of establishing lower botmds to buckliHg pressures. An extension to the conservative estimation o! the buckling o[ lton-axisymmetric pressure-loaded cantilevered shells is aZs'o.suggested to he amenable to the reduced stti~hess ana&tical procedure.

NOTATION A E H i

J N P R R~, R,

Throat radius of hyperboloid. Young's modulus of elasticity. Shell height. Circumferential harmonic number. Meridional half-wave number. Membrane stress resultant. Pressure. Radius of shell parallel circle. Principal radii of shell curvature. 381

Thin-Walled Structures 0263-8231/86/$03'50 © Elsevier Applied Science Publishers Ltd. England, 1986. Printed in Great Britain

George M. Zintilis, James G. A. Croll

382

Meridional coordinate of shell surface. Shell wall thickness. /g In-plane displacement in the meridional direction s. U Linear strain energy. UM, UB Linear membrane and bending strain energies. V Linearised strain energy. 142 Normal displacement. 8 Displacement. E Membrane strains. 0 Circumferential angle. X Critical eigenvalue. v Poisson's ratio.

S

t

Subscripts b C' cm s w

Value at buckling. Critical value. Minimum critical value. Value for uniform suction. Value for wind loading.

1 INTRODUCTION Experimental investigations of pressure buckling of shells of revolution were, until recently, dominated by those on end-supported shells under hydrostatic pressure. In civil engineering the interest is more commonly directed towards cantilevered shells under radial axisymmetric and wind pressures. This interest was brought into sharp focus by the collapse of the Ferrybridge ~and Ardeer 2cooling towers and is likely to be revived by the more recent Fidlers Ferry collapse. With more recent and proposed cooling towers often involving even larger and more slender towers it is likely that wind load buckling will provide an increasingly important design consideration. Other shells, such as oil storage tanks, silos and aerospace structures, are also subject to wind buckling considerations. ~ The present work considers the buckling behaviour of cantilevered rotationally symmetric shells under axisymmetric and non-axisymmetric normal pressure. It describes how the reduced stiffness analysis procedure may be adopted as a means of accounting for the imperfection

Pressure buckling of cantileveredshells of revolution

383

sensitivity displayed by these shells. Recent studies of axisymmetric pressure buckling and combined axisymmetric pressure and axial load buckling of end-supported rotationally symmetric shells have suggested that imperfection sensitivity, apparently present in such shells, is strongly influenced by shell geometry. 4._~

2 REVIEW Predictions of wind buckling loads for cantilevered shells on the basis of classical critical analyses of equivalent axisymmetrically loaded shells ~~ have generally not included the reductions of buckling resistance caused by the presence of geometric imperfections inevitably occurring in the as-built shells. Langhaar et al., 1, in an attempt to bridge the gap between predictions and test results, have suggested that such equivalent axisymmetric, or worst pressure, predictions should be reduced to take into account in-plane meridional flexibilities at the base supports. The relevance of such reductions has been discussed for the case of endsupported shells 45 and it is reconsidered below for the case of cantilevered shells. Other recent studies have proposed alternative predictive methods for the wind buckling of shells. One of these is the semi-empirical local critical stress interaction formula, it-t3 As discussed in Refs 4 and 5 this local stress interaction approach is unlikely to represent the overall nature of shell buckling modes, even if it is used for axisymmetrically loaded shells having boundary conditions similar to those for the tests from which this m e t h o d was developed. To approximate the effects of wind loading, Abel, Billington and co-workers ~-~ have proposed another form of bifurcation analysis which approximates the prebuckling effects of wind loading by means of an equivalent axisymmetric stress. For the critical state calculations this m e t h o d assumes that the prebuckling wind membrane stresses on the "worst stressed' meridian apply over the entire shell circumference. Comparisons with the wind tunnel tests on cylinders by Johns et al. 3 seem to suggest that this method is conservative, while comparisons with the wind tunnel tests by Der and Fidler ~4on hyperboloids suggest that the m e t h o d was non-conservative. A potential shortcoming of the worst stressed meridian method is its neglect of the imperfection sensitivity present in the buckling of shells of revolution.

384

George M. Zintilis, James G. A. Croll

The advent of efficient electronic computers has meant that non-linear m e t h o d s of analysis can be used. Such analyses should be able to model the apparently non-linear wind buckling behaviour of this class of shells. However, the results of alternative numerical studies do not yet seem to possess the degree of consistency to recommend them as a substitute for physical testing. Based upon scale-model tests Der and Fidler have suggested a coefficient C = 0.073 3 and power n -- 2.3 in the empirical formula ~' pbw

=

CE(t/A)"

(1~

for which E is the modulus of elasticity, t the nominal thickness and A the throat radius. Chan and Firmin in their non-linear finite element analysis ~5 have proposed the use of the apparently non-conservative coefficient C = 0-148, while O y e k a n in an independent analysis ~6has put forward the seemingly conservative value of C = 0.02. At C = 0.05 Kato and Matsuoka's predictions~7 appear to be a closer representation of the test value, although the index of 2-3 in eqn (1) was not reflected by their predictions. For four of Der and Fidler's models the more recent nonlinear analysis by Chang ~ also appears non-conservative at C = 0.136. A l t h o u g h this is in good agreement with Chan and Firmin's prediction, it differs considerably from Yeh and Shieh's ~9 apparently realistic predictions at C = 0-077 6 which were, however, derived on the basis of a coarse finite element mesh and included no allowance for internal suction due to wind. Although Der and Fidler's test work still remains the most widely used criterion by which predictions are judged, doubts must arise as to the validity of eqn (1) for this complex problem when new shapes and sizes of cooling towers are considered which differ from those used in the wind tunnel tests. ~4Such new-generation cooling towers are likely to represent new statistical populations which may yield a new set of appropriate values for C and n in eqn (1). Moreover, the use of average thicknesses instead of the nominal for the two PVC models tested by Der and Fidler, ~ that is 0-025 in and 0-04 in instead of 0.028 in and 0-044 in, predicts that for the intact models the appropriate form of eqn (1) would be given by C = 0-207 and n --- 2.47. This sensitivity to such changes suggests that such empirical laws should be used with caution. A n o t h e r attempt to overcome the problem of reliable wind buckling predictions for these shells has been described in Refs 20 and 21. Like earlier bifurcation analyses this method was based upon the assumption that it was the average (or uniform) wind pressure component that

Pressure buckling of cantileveredshells of revolution

385

provided the destabilisation. Other non-axisymmetric components of wind loading were considered to provide major imperfections for the bifurcations into each of the circumferentially periodic critical modes. A lower bound to the highly imperfect buckling into each of these modes was considered to be provided by a reduced stiffness bifurcations analysis in which the membrane stiffnesses were considered to have been eroded. The predictions of this method were over-conservative when compared with the results by Der and Fidler, especially for the thinner models. This suggested that the predicted long wavelength modes were not the dominant buckling modes and that with buckling being triggered by shorter circumferential wavelength components of the pressure distribution, it is possible that pressure harmonics--other than the zeroth-may contribute to buckling. However, this early unsuccessful attempt to account for wind load buckling has provided the basis for the reduced stiffness method. In the present work the application of the reduced stiffness method to wind-loaded shells is reported, in the light of the encouraging results reported for this method in other applications.4522 25 The shell stiffnesses and their distributions, it is argued, play an important role in determining the shews response in the presence of imperfections. The validity of this method of incorporating the effects of imperfections is established by reference to past tests and the first author's more recent test programme. Effects of variations in load distribution, tower geometry, boundary conditions, and the effects of these on stability, are considered through appropriate parametric studies.

3 PRESENT ANALYSIS 3.1 Classical critical pressures

The classical critical pressures are calculated from the total potential energy using the method described in detail in Refs 4 and 26. 3.2 Energy and pressure spectra

In order to identify the importance of the various components of stiffness, the linear strain energy U is broken down into its membrane constituents uM

u~+ u~+u~

(2)

386

George M. Zintilis, James G. A. Croll

and its bending constituents u . = uh + u~ + U~

(3)

so that U = UM + UB

(4)

where s and 0 denote the meridional and circumferential coordinates of the middle surface. The linearised strain energy h - V for the present classes of axisymmetric loading is composed of the terms x. v = x(v~+ v~2+ v~+ v~:) where each arises from the interaction between the prebuckling stresses and strains and the non-linear components of the critical stresses and strains as given by

if f ~ =

-

-" ,k-V'.

'ff 5

= 51

f

f

N~.e','.R.~Oiis

(6a)

N'~" E.R.i~Oas

(6b)

N ~ ' e ' e ' "R'iiO&~"

(6c)

No'' " Er~ o'R'aOas

i6d)

The effects of non-axisymmetric pressure are accounted for in a simplified manner, after Ref. 6, so that the above formulation continues to be valid. The eigenvalues h arise from the linear dependence on load of the fundamental (prebuckling) equilibrium stresses and strains (denoted by superscript E) assumed in the classical critical analysis. The double prime indicates a quadratic dependence of strains and corresponding stresses upon the incremental displacements into the critical modes. The above formulation can be written in the form of the classical eigenproblem U+X.V

= 0

(7)

Pressure buckling of cantileveredshells of revolution

387

3.3 Buckling for axisymmetric pressure The classical critical analysis for a typical cantilevered hyperboloid under axisymmetric lateral pressure results in the positive (stabilising) strain energy spectra shown in Fig. l(a), with associated critical loads as in Fig. l(b). The properties of this hyperboloid, tested by Der and Fidler, ~4are )

E=14x106Ibf/in2 v=03 32 U ~ A=4in Zr=3.664in 28 / U~ !Za=-11'916in '~ 24 ~,,~,, ~T =4"2~7irl clamped free

100

f . . . . Boo~\k I, ~ "~

(b)

820

B 6OO

ol

c 400| t W J

\,"X(UM / ,/ / / z . " ."

20 I

~--'"

ur---:-~ 4

.5

u

:,2_O8

x(v~w2%)

"F

7

7

6

7

8(=icm) 9

04-,

> zJ "2 3 3 3 5 6 7 B(=icm)9 Circumferential harmonic

Circumferential harmonic i

z/H .......

............................

. ................ 0 . 8 :

-throat

icm = 8

) i=9

0.4-

0.2-

I

- 1.0

-0.8

[

-0.6

r~-

-0.4

i - - - -

-0.2

i

0 ( C)

0.2

T

0.4

i

i

0.6

0.8

1.0

normal displacemenLw I

Fig. 1. (a) Energy spectra, (b) axisymmetric pressure spectra for Der and Fidler 14 copper hyperboloid, t = 0,008 5 in; (c) selected mode shapes from axisymmetric pressure analysis for Der and Fidler t4 copper hyperboloid, t = 0.008 5 in.

388

George M. Zintilis, James G. A. Croll

listed in Fig. I and represent one of the shell types studied as part of a new test p r o g r a m m e reported in Ref. 26. T h e positive energy spectra, as typified by Fig. t(a), show that the d o m i n a n t linear m e m b r a n e energy contributions are due to the meridional term U~4, with some small contribution from the in-plane shear term U~, while the circumferential term U~4is negligible. For the shorter circumferential wavelengths (high i), in which bending energy is significant, the linear circumferential bending term U ° dominates, with the other two terms small or negligible. The discontinuity in the spectra after m o d e i = 6 is due to the increase in the number of meridional half-waves, j, from 2 to 3, noting that the present analysis operates on the basis of integer numbers of circumferential wave number i. T h r e e selected m o d e shapes for the above hyperboloid, given in Fig. l(c), emphasise the overall nature of the long meridional wavelengths of the shell's response to normal pressure. It is this overall nature of shell behaviour that argues against the use of any local stress criterion for buckling. ~-~3

3.4 Wind buckling for axisymmetric worst stressed meridian The 'worst stressed meridian' analysis, proposed by Abel and coworkers, ~ is used in the present work. This m e t h o d is essentially a bifurcation analysis using the fundamental stress distribution of the ~worst stressed meridian' due to wind loading, which is then assumed to be uniform a r o u n d the shell's circumference. This allows a linear eigenvalue analysis to be performed using eqn (7), where the fundamental stresses and strains of eqns (6) are replaced by those on the "worst stressed meridian'. The critical pressure spectrum of the hyperboloid of Fig. 1, tested by Der and Fidler, ~4resulting from the "worst stressed meridian' analysis, is shown in Fig. 2(a), while the energy spectra associated with this are shown in Fig. 2(b). The fundamental state stresses due to wind loading used in eqns (6) for this analysis are shown in Fig. 3, which also shows the axisymmetric 'worst pressure meridian" distributions for comparison. As an alternative to the 'worst stressed meridian" it has been proposed ~0that an axisymmetric pressure equal to that at the meridian where the pressure is at its maximum could be adopted for a classical bifurcation analysis. The analysis is identical in form with that discussed above for axisym-

Pressure buckling of cantileveredshells of revolution 30-

I E =14xlO61bf/in 2 600" (b) 'v=03 A=4in ZT =3.664in 500za = -11.916in RT=4 257in :lamped free ~ 400

(a)

~20

j.(3~f pc ~ 16

"T:!

;9"11

x~vbv~) / / ......... -_~ . . . . . / < b ~ o "~300

Pcm

12

389

~m

Pbw test)

~08

,~-'~'~\..

Ld 200

~ ' ~ r e d u c e d , p~ j=3

04 .

Nind pressure by 100

B a t c h and Hopley c o e f f i c i e n t s using

0 6 suction

E U

0 6

.

~

7

8

Circumferential

~cm~

,

,

7

8

0

9

10

harmonic

6

i

Circumferential

icm

!

9 10 harmonic i

Fig. 2. Free stream pressure and energy spectra from worst stressed meridian analysis for Der and Fidler H copper hyperboloid, t = 0-008 5 in.

I*O-

~-t0p

~x

L~hr°at

I

I~0 z/H

z/H

k/-axisymmetri¢worst pressure - 'k/ /~( I.G times the freestream pressure

~top

O,B f-wind

~

0.6/0.4

/"

at

=0 °

~ c t [ o n I~axisyrnmetricworst pressure /I ( 1,6 time~ tile free stream pressure )

0,2 "

0

0,6"

)

- I0

0

r l0

(a)

/ J

0=0~ (Batch & Hopley withO.B suction)

~wind at

/ /

'~ 0,2

LSe ii r

/ /

0,4

~

\ O

I

~-throat

(Batch& Hopley

1

r

T

20

30

40

O" 50

,/i

///i L

[2

-8

ba~c i

,

-4

0

4

8

circumferentialN9 [Jbf/in i

rneridional Ns[lbf/in ](tension positive )

(b)

Fig. 3. Axisymmetric and wind pressure stress profiles used for the worst pressure and worst stress meridian bifurcation analysis under I lbf/in: external pressure with 0.6 Ibf/in z suction for the Der and Fidler ~4copper hyperboloid, t = 0-008 5 in.

m e t r i c p r e s s u r e . F i g u r e 3 ( a ) s h o w s that the t w o m e t h o d s result in significantly different fundamental stresses. The dominant energy components, s h o w n in Fig. 2 ( b ) , are t h e m e r i d i o n a l U~4 for t h e l i n e a r m e m b r a n e e n e r g y , a n d t h e c i r c u m f e r e n t i a l

390

George M. Zintilis, James G. A. Croft

U~ for the linear bending energy. The destabilising component is the linearised circumferential membrane energy h. (V~ + V~2). These energy characteristics are c o m m o n to both axisymmetric and wind pressure loadings. The major difference for wind loading is the increased importance of the stabilising influence of the linearised meridional m e m b r a n e energy h(V[ + V[2), as seen by comparing Figs l(a) and 2(b). This reflects the very different distributions of the fundamental meridional stresses shown in Fig. 3(a) for the two cases of loading. The classical critical mode numbers, icm, for the geometries considered in the present work are found to be usually the same for the two types of loading, and where they are different the 'worst stressed meridian' m e t h o d predicts modes that are normally higher than those due to the axisymmetric 'worst pressure meridian'. The meridionai half-wave n u m b e r j is similar for both types of analysis. The comparison of the energy, or stiffness, characteristics for shells due to axisymmetric 'worst pressure meridian' and due to the ~worst stressed meridian' shows that they are qualitatively similar, with the exception of the linearised meridional membrane terms. 3.5 Reduced stiffness critical pressures

The stiffness characteristics of pressure-loaded cantilevered shells appear to be similar to those of end-supported shells reported in Refs 4 and 26. It is argued in these sources, as well as in others, 5'22-2~that imperfections play an important role in buckling for these classes of shells by reducing their load-carrying capacity. It was also argued that these reductions, at small but finite post-critical deformations, are associated with the erosion of some or all of the membrane stiffness contributions, which, for infinitesimally small critical deformations, provide a significant contribution to the shells' resistance to buckling. The linear critical analysis for the present classes of shells requires in-plane displacements involving considerable meridional membrane strain energy. As deformations into these critical modes progress, the relative amplitudes of in-plane to out-of-plane deformations change so that, at moderate deformation levels (of the order of the wall thickness), a non-linear kinematically admissible state can be attained in which most of the m e m b r a n e energy has been eliminated. At this "quasi-inextensional' state the resistance to incremental displacements would be dominated by the bending stiffness of the critical mode.

Pressure buckling of cantilevered shells of revolution

391

As the bending stiffness possesses low non-linearity at moderate deformations, the bending resistance would be virtually the same as it was in the original critical mode. This enables a linear eigenanalysis in which the membrane energy UM is eliminated, to predict a neutral post-buckling response that is a lower bound to the exact non-linear load path. More detailed discussions of the present reduced stiffness approach can be found in Refs 4, 21-24 and 26. The reduced stiffness critical pressures p~ can be obtained by suppressing the linear membrane energy UM, calculated from the classical analysis, in each critical mode i so that

p:

=

{ud(un+ uM)}pc

(s)

where p~ is the classical critical pressure in the corresponding mode. Typical reduced stiffness pressure spectra are shown in Fig. 4 for a cone, a cylinder and a hyperboloid under axisymmetric pressure, and in Fig. 2(a) for a hyperboloid under wind pressure using the 'worst stressed meridian" analysis.

p x I 0 -4

p x I 0 ,1 ( cmitical pmessure )

classical (j = I

2.0

p x I 0 4 ( critical pressure )

[.Jmm]

N/ram ~j 2.5

( critical pressure )

r,l/mrn

8O

/

i

1.5

1.5

GO

4

test 1.0

test 1.0

4

4O

.•

p*. p

or,,-~ reduced 0.5

0.5

-j

.

4

-

5

6

7

8

critical mode ( i )

(a)

test ~

reduced

Curm CO I

Pb: i Icrn~

0

3

4

5

:','] ' c , ~

, 6

7

ttyperboloid }12

0 8

critical mode ( i

(b)

g ...................

p:,,~--reduced I

2O

.'

Cylinder CY I

0 5

daseical

IO0

2.5

4

5

6

7

8

~.

critical mode I i

(c)

Fig. 4. Axisymmetric pressure spectra for models CY1, CO1 and H2 (shell propertms given in Table 3).

392

George M. Zintilis, James G. A. Croll

In the case of the 'worst stressed meridian' analysis the meridional tensile stresses N~ and strains E~ produce the additional stabilising linearised membrane energy terms hV~ and hV~2, For a post-critical mode coupling to annihilate these positive membrane contributions, the coupling mode would need to develop deformations having half the circumferential wavelength. Since the bending energy associated with such a mode is likely to be greater than the membrane energy it annihilates, such a coupling seems most improbable. However, to adequately validate its retention, but eliminate the linear membrane energy UM, it would be necessary to carry out full non-linear analyses. The importance of membrane stiffness in the buckling of shells is that it biases buckling towards the shorter wavelength modes in which bending stiffness is relatively high. Even though imperfections tend to eliminate the stiffness contributions coming from the membrane energy in these modes, it appears that buckling continues to occur in the mode i~, associated with the minimum classical critical pressure p~m. In this mode the reduced stiffness minimum critical pressure given by pcm = {U~/(U~+ UM)Ipcm

i cj)

would be expected to yield a lower bound to imperfection-sensitive buckling. These predictions are shown in Figs 2(a) and 4 along with the test buckling pressures. The closeness of these test buckling pressures and modes (where these are known) to the present predictions seems to indicate that further test comparisons may usefully establish the validity of the method. Before doing so, the behaviour of these shells is considered under varying boundary conditions and curvature. 3.6 Influence of curvature and boundary conditions The importance of both curvature and boundary conditions o n the classical critical and the reduced stiffness loads, as well as their associated m o d e shapes, has been investigated in Refs 4, 5 and 22 for end-supported shells of revolution under lateral and axial loading. A similar study to those of Refs 4 and 5 of varying meridional curvature under axisymmetric pressure is summarised in Fig. 5 for a cantilevered toroidal shell. As in these previous studies, variations in curvature have a pronounced influence on both the classical and the reduced critical pressures. For comparison, the properties of the toroids are the same as those in the studies reported in Refs 4 and 5. The meridional curvature is varied so

Pressure buckling of cantileveredshells of revolution ~ ~7

S

r

classical critical Pc

~

"..

.... .~ ,

~ ~

/al4,2

]. . . . , ~ ~ / ' - - . . \ 51. . . . . . ~ ~cm "....\

"~l.~ %m

a_T

,~,l

2 - H~} ] i

,o_

393

~ -

.... ~ ' / ~

\

(i,j): ( 5 , 3 )

> /

\

v-..'--_I 1:3 ~_~~-~

~ - -~S ~"

\ - '=~*' -' "' -v-~- e' u°Pf Pcm

"reduced

"~m

J-

, , , , , ~\ treating i as a clamped continuous variable - 1MPa V I 0 3 0"-= ', -- ' I 1 J I -0.08 -0-06 -0.04 -0"02 0 0.02 Reo/Rs

eqn(9)

1-

Fig. 5. C u r v a t u r e a n d p r e s s u r e for c a n t i l e v e r e d t o r o i d s t = 1.05 m m ) .

I

0.04

1

0.06

(Ro,, = 200 m m , H = 1200 m m a n d

that its quotient with the throat radius is in the range _+0.07. The full curves represent the loci of the classical critical pressures for different circumferential modes, while the dotted line is the envelope of the m i n i m u m critical pressures peru. The dashed lines are the loci of the reduced stiffness critical pressures pcm* for each circumferential harmonic i, with the discontinuities due to i being taken at integer values only. When /cm is taken to be a continuous variable which relates to the envelope of peru then the reduced stiffness critical pressures would assume the continuous form shown dotted in Fig. 5. The minimum critical pressures for the negatively curved toroids are higher than those of their positively curved counterparts and occur in modes of shorter wavelengths. The imperfection sensitivity, however, also increases with increasing negative curvature, as the contributions from linear m e m b r a n e stiffness are considerably greater for these negative curvatures. Using the reduced pressure capacities as a measure of buckling strength indicates that positive and negative curvatures produce similar effects, with the minimum occurring in the region of the cylindrical curvature. This seems to highlight the advantages of using doubly curved cantilevered shells in resisting lateral pressure. A comparison of the stiffness characteristics for typical doubly curved shells and the cylinder used in the parametric study of Fig. 5 is shown in Fig. 6, where the dominant stabilising energy components are plotted for each harmonic i. At the mode associated with pcm the important m e m b r a n e

George M. Zintilis, James G. A. Croll

394

°°ry

./y

Energy

U

1 // ,,o-:°1 \/!

I

'° I 0

/

/

'

.. I

2

3

4(:icm)5 6 criticat mode, i

Ro~/R,:-O'02

0

I

g

3(=i )4 5 6 7 c%tical mode, i

% /R = 0 (cylinder) o s

0

I

2

3(=i¢~)4 5 d critical mode,

R0 /R o

:~

~= O,OZ

Fig. 6. Energy spectra at selected meridional curvatures.

energy component is Uh, especially for the negatively curved geometries. In the case of the cylinder, UM tends to zero in the higher harmonics. The fundamental stress state stresses also influence the critical state m that the meridional stress N~ is tensile, that is, stabilising for the negatively curved toroids, whereas this is compressive or destabilising for the positively curved toroids. The circumferential stress N Eis compressive for all curvatures. The influences of boundary conditions are reported in Refs 4 and 5 for axisymmetrically loaded end-supported shells. Allowing kinematic boundary conditions other than the in-plane meridional displacement u to be free has little effect on both the classical minimum and the reduced stiffness critical pressures. This behaviour also occurs in cantilevered shells under lateral pressure for all the geometries considered in the present work. For example, allowing the in-plane displacement u to be free at the base for a negatively curved toroid at a curvature ratio of -0.04 resulted in a 63 per cent drop in the minimum critical pressure, and a 58 per cent drop for a positively curved toroid at a curvature ratio of +0-04. In general, support influences in the classical critical pressures are reflected by similar changes in the reduced stiffness pressures. The exception is where the meridional in-plane displacement u is free. In this case the classical and the reduced stiffness predictions are very similar in magnitude, reflecting the very small membrane stiffness in the critical modes. In addition, where the u restraint is removed, the classical critical mode number icmis reduced.

395

Pressure buckling of cantilevered shells of revolution

4 A X I S Y M M E T R I C P R E S S U R E TESTS To assess its validity the predictions of the reduced stiffness method are c o m p a r e d with previously published results 272s and new test results 26 for axisymmetric pressure loading for cantilevered hyperboloids, cylinders and conical frusta. 4.1 Past tests on hyperboloids under uniform suction

There is a limited number of past axisymmetric pressure tests on cantilevered hyperboloidal shells, and these were reported by Veronda and Weingarten 28and by Ewing, 27the latter for a single test carried out by Der and Fidler on one of the hyperboloidal cooling towers used in their wind tunnel investigations. 14 The tests of Ref. 28 are also on hyperboloidal shells, symmetrical about the throat, whose properties are listed in Table 1. The comparison between the test results and the predictions of the present analysis are listed in Table 2, where the elastic modulus used for the Der and Fidler model is taken as 473000 lbf/in 2 and not the apparently excessive 550 000 lbf/in 2 reported in their original paper.~4 This reduced elastic modulus is similar to that proposed in Ref. 6, and a detailed discussion of this appears in Ref. 26. Table 2 shows that all the reduced stiffness predictions p~,, are lower bounds at an average of 81 per cent to the test pressures pb~, and that the classical critical modes are similar to the reported test buckling modes. Also, it may be observed that

TABLE 1 Properties of the hyperboloidsof Ref. 28 Shell

Throat radius

Top elevation Base elevation

Top radius

Thickness

n a

ZT

ZB

RT

t

(1)

(2)

(3)

(4)

(5)

(6)

2 2-8 h 5

2.0 1-98 2-0

4"5 4'5 5"0

-4.5 -4-5 -5.0

2.5 2-475 3-0

0-01 0.009 7 0"01

"All dimensions in inches. ~'As-tested geometry. Elastic modulus E = 460 000 Ibf/in2and Poisson's ratio = 0-37.

396

George M. Zintilis, James G. A. Croll TABLE 2 Cantilevered hyperboloidal models under uniform internal suction

Shell model

(1) Type 2 Type 2.8" Type 5 Der & Fidler (t = 0.04 in)

Uniform internal suction Min. critical Reduction Test buckling pressure ratio p*cm/pcm pressure pu~ pcm (lbf/in 2) (lbf/in 2) and [mode icm] and [mode ib] (2) (3) (4) 0.223 [6] 0.209 [6] 0-182 [7] 1-670 [5]

0.694 0.694 0.733 0.680

0.199" [6] 0.198': [6] 0.145' [6--8] 1-43d [_]h

Ratio pbs/pcm

Ratio p*m/pu~

(5)

IO)

0.892 0.947 0.797 0.856,1

0.778 0-732 0-919 0.794

"Test buckling loads averaged for Types 2 and 5.2~ bTest buckling mode not available for this model. ' E x a c t as-tested geometry and pressure. 'lTest buckling pressure derived from Ref. 27; p,:m based on E = 473 000 lbf/in 2 (326t) MPa) and a Poisson's ratio of 0.4.

1.0 ¸

Po

O~ le//

0.8

~a

06

//

/

@ /

0,4

/ //

02

/--

/

/

Key

+

old t e s t s , t a b l e 2

0

n e w tests, t a b l e 4

Pbs=Pcm

Pcm

Pbs

/

- - ' - - F -

| -

0'2

04

-

• 0.6

.....

I . . . . . . .

0~5

1.0

~lth

Fig. 7. Correlation of experimental and reduced stiffness knockdown factors for cantilevered shells under axisymmetric internal suction.

Pressure buckling of cantileveredshells of revolution

397

the minimum critical pressures pc,, are always upper bounds to the test buckling pressures phi, which average at 87 per cent ofpcm. The scatter of test results and the lower boundedness of per, is more clearly illustrated in Fig. 7. 4.2 New tests on hyperboloids, cylinders and conical frusta under uniform suction As the n u m b e r of past tests on cantilevered shells was considered inadequate, new tests were carried out on PVC hyperboloids, cylinders and conical frusta 26 for further comparison with the present reduced stiffness method. The test apparatus was specially developed so as to obtain the circumferential displacement profiles up to buckling, at various parallel circles of the shell models. These profiles are often considerably different in shape from the large-displacement, post-buckled configurations. Pre-buckling profiles are used to obtain the true initial imperfections and additional displacements so that a discrete Fourier analysis can be used to determine the contributions from each of the circumferential harmonic modes. A schematic arrangement of the test apparatus is shown in Fig. 8. Moduli of elasticity were determined by using cantilevered PVC strips cut from the sheets used to construct the models. The hyperboloids were fabricated using the method described by Der and Fidler. ~ Conical frusta were heated to around 90°C in a hot circulating air oven, and using the I DETAIL 'A

F

p ~shell ~:~

n~det manometer

vBlve

ate

pumpl

(~

l

I P'°tter

andcontro,s

power DETAIL~Ar- Testapparatusinsidethe models Fig. 8. Schematic arrangement of test apparatus.

398

George M. Zintilis, James G. A. Croll TABLE 3

New test series model properties Shell model a (1)

Elastic modulus b E (2)

Throat radius' A (3)

Top elevation C zT (4)

Base elevation c zu (5)

H1 H2 H3 H4 H5 CY1 CY2 CY3 CO1 CO2 CO3

3254 3230 3328 3254 3254 3250 3250 3250 3250 3250 3250

102.80 102-64 102.55 102.64 102.80 106.0 106.0 106.0 160,0 d 144-0a 129-6J

86.64 87.13 90.44 87,13 86.64 375.0 e 275-0 e 175.0" 395.0 e 275-0 ~ 195.0"

- 316-36 -315.87 --312.56 -215.87 -216.36 --------

Top radius' Rr (6)

108.30 107.90 108.30 107,90 108.30 106,0 106.00 106.0 100.00 100.00 100.0

Average thickness' t (7)

%41 0.67 0-90 0.67 {).41 0.24 0.24 0-24 0.24 0-24 i).24

"Key is H = hyperboloids, CY = cylinders, CO = cones.

hUnits in N/ram2; Poisson's ratio = 0.35. 'All lengths in mm. J Base radii for cones, eCone and cylinder heights.

flow p r o p e r t i e s o f this t h e r m o p l a s t i c m a t e r i a l the c o n e s w e r e b l o w n o u t w a r d s o n t o the s a m e m e t a l h y p e r b o l o i d a l m o u l d as was used by D e r a n d F i d l e r . ~ T h e strips u s e d for the c a n t i l e v e r tests w e r e h e a t e d in the s a m e o v e n so as to reflect a n y m a t e r i a l c h a n g e s . T h e t h i n n e r c o n e s a n d c y l i n d e r s w e r e m a d e o f a p p r o p r i a t e l y cut flat sheets o f t r a n s p a r e n t P V C g l u e d at a lap j o i n t a l o n g a shell m e r i d i a n . T h e v a r i a t i o n in the effective elastic m o d u l u s o f the strips with t i m e , d u e to c r e e p , w a s d e t e r m i n e d just p r i o r to testing e a c h m o d e l , so as to c o m p e n s a t e f o r t e m p e r a t u r e , h u m i d i t y a n d ageing effects. T h r e e - m i n u t e cycles f o r l o a d i n g a n d f i v e - m i n u t e cycles for u n l o a d i n g w e r e a d o p t e d as a d e q u a t e f o r fairly c o n s t a n t levels of the elastic m o d u l u s to b e r e a c h e d . T h e v a l u e s o f t h e elastic m o d u l i at the t i m e o f testing are r e c o r d e d in T a b l e 3, w h i c h also lists the g e o m e t r i e s o f the m o d e l s . C o m p a r i s o n s b e t w e e n the p r e s e n t t h e o r y a n d the n e w test results are s h o w n in T a b l e 4 a n d again s u m m a r i s e d in Fig. 7. P r e s e n t r e d u c e d stiffness p r e d i c t i o n s are s e e n to a v e r a g e 82.5 p e r cent o f the test b u c k l i n g

Pressure buckling of cantilevered shells of revolution

399

TABLE 4

New tests on cantilevered hyperboioidal, cylindrical and conical models under uniform internal suction

Lateral buckling pressure (N]mm 2) Shell Model

(/) HI H2 H3 H4 H5 CY1 CY2 CY3 COl C02 C03

Min. critical Test buckling Reduction loadpcm x 10 _4 pressurepb~ x 10 4 ratiOp*m/pcm [icm] [ib]" (2) (3) (4) 13.44 [71 41.82 [5] 81.89 [5] 48.81 [6] 16"55 [7] 1.30 [5] 1.75 [6] 2.79 [8] 1.24 [5] 1.62 [6] 2.47 [7]

10.93 [6] 29.57 [5] 64.28 [5] 37.0 [6] 10'81 [7] 1.246 [5-4] b 2-679 [7] 1-059 [5] 1.50 [6] 2.43 [7]

0.669 0.506 0.636 0.685 0.639 0.701 0-754 0-836 0.651 0.751 0.722

Ratio pbdpcm

Ratio p*m/pbs

(5)

(6)

0.813 0.707 0.785 0.758 0.653 0.958 0.823 0.960 0.854 0.926 0.984

0-822 0.715 0.81/) 0.903 0-978 0.731 0.917 0.870 0.761 0.811 0.737

"pb~ and ib are the test buckling pressures (N/mm 2) and modes. ~'Measured modes are close, with the 5th more dominant.

pressures (last column). These lower bounds are most conservative for hyperboloid H2 and least conservative for H5. But the results suggest that even these carefully fabricated laboratory models are sufficiently imperfect that the considerable scatter and at times severe reductions from classical bifurcation analysis are realistically bounded from below using the reduced stiffness analysis procedure. The test buckling modes ib shown in Table 4 were at first obtained by direct inspection of the recorded circumferential displacement profiles, and subsequently by performing a discrete Fourier analysis on these profiles so as to calculate the contributions in each circumferential harmonic. The largest contribution at all load levels and elevations invariably corresponded with that directly observed by counting the wavenumbers of the profiles. Obtaining displacement profiles was necessary as the suddenness of the snap-through at maximum load, and the smallness of the displacement prior to buckling, prevented visual observation of the modes. Except for models HI, CY2 and CY3, the modes predicted by the classical theory are identical to the experimental

400

George M. Zinfilis James G, A. Croll

modes. The above three models buckled in modes having one less wave than that predicted by the classical theory, which calculates the critical pressures at discrete uncoupled modes and not as continuous functions of wavelengths. Since, in these cases the critical pressures in the modes (icm -- 1), icm and (icm -F l) are very close in magnitude, the true minimum may lie somewhere towards one of the two modes (it= - 1) and (i~m~ l). Furthermore, as the shell in the lower mode (i~m--1) is more imperfection-sensitive by virtue of the higher membrane stiffness, it is quite likely that the imperfections in mode (i~m- 1) may cause the shell to buckle in this lower mode. In most cases considered, the Fourier analysis yields high displacement contributions not only in the buckling mode but also in modes adjacent to it. A typical example showing the behaviour described above is hyperboloid H3, whose true imperfections and displacement profiles at a level near the shell's free top are shown in Fig. 9. The profiles suggest that the meridional seam has had no overriding effect on either the imperfection or the displacement profiles. The two profiles show the mode shapes of the displacements at 98 per cent of the eventual buckling pressure, and of Model 1-t3

f

E'~

2- ~

~"

0

?!-- seam /-displacements at 0.98 Pbs

':'

i

'

/

..'":

".//-.. imperfections

:,I

,'

mm

L

(54°) 0

z = + 62.3 mm

I

l

40

80

[

1

20

I (30

"[

2_00

r

240

........

280

I .....

320

()'~ 3 6 0

Fig. 9. Imperfection and displacement profiles for hyperboloid H3 (properties in Table 3).

Pressure buckling of cantilevered shells of revolution

401

the imperfections at zero pressure. The wall thickness is marked on the displacement axis for comparison. The imperfections and the displacements are in places two or three times larger than the thickness. Figure 10(a) shows the pressure-deflection diagram at a typical point on the circumferential profile of Fig. 9. The onset of buckling is shown to be pronounced--a characteristic of all these cantilevered shells and an indication of the unstable nature of the post-critical equilibrium path. The snap-through was often audible. Figure 10(b) shows the results of the discrete Fourier analysis performed on the two profiles of Fig. 9. Figure 10(b) shows that the main imperfections are in the second and third modes; the zeroth mode represents the radius difference of the actual model from the nominal, and the first mode is the lack of alignment of the axes of the shell and of the displacement probe. The dominant displacement harmonic is the fifth, which corresponds to the predicted minimum critical mode icm.Also of interest is that modes adjacent to the fifth, especially the fourth, also have significant contributions to the displacements at a pressure of 0-98

p b~,. ~SJ 1.0 I/_t Model ti3 at z :: G2.3 mrn ( z ::0 at throat )

Ira,.1 0.8

0.6

01;5

/

:nr:~sbucklir'g

o.5o!/

[ imperfections .."i.. I '

/1 / I

o.41:/t

displacements at 0.98 Pbs

0.25 0

ut 0

,

~ I

~

e, 5 4 " _ 2

disl~lacement Imm (a)

' 0

2

....

4 6

:

8 I0

]

,

12

,

14

16

i

(b)

Fig. 10. (a) Load-deflection diagram of hyperboloid H3 under uniform suction; (b) spectral analysis for imperfections and displacements for H3.

402

George M. Zintilis, James G. A. Croll

5 WIND TUNNEL TESTS 5.1 New tests on cylinders and conical frusta under wind tunnel pressure

As in the case of axisymmetric pressure the quantity of past test data available is limited. The original intention was to perform new tests in the compressed air tunnel of the National Physical Laboratory (NPL), which in providing high air densities and speeds allows achievement of post-critical Reynolds numbers necessary for model pressure distributions to realistically reproduce full-scale conditions. However, the above tunnel was at the time out of commission so that the new test programme was carried out in an open-jet type wind tunnel at the Transport Technology Department, Loughborough University of Technology. Unfortunately, this meant that the hyperboloidal models of Table 3 could not be buckled under the external wind pressure available in this tunnel and that the cones and cylinders were loaded at apparently unrealistically low Reynolds numbers. The objectives of these tests were to find the wind buckling pressures at various levels of internal suction, to obtain the prebuckling circumferential displacement profiles, and to use these to establish the likely dominant modes at buckling; also, to measure the distribution of wind pressure on the model surfaces and to determine its likely effects on the buckling process. The test procedure is identical to that for the uniform pressure tests of the previous section, with the additional element of external wind pressure loading. Given the maximum wind tunnel speed of 25 m/s and the diameters of the models, the Reynolds number was of the order of 4 x 105. Data were also collected at lower speed, about 7.5 re~s, so that the minimum Reynolds number was about 1 x 10~, with the effect that the tests were conducted within the critical range of Reynolds numbers. The tunnel characteristics have significant effects on the interpretation of these new and the o l d 3 results from this tunnel. Flow visualisation techniques using smoke streams and fluorescent dyes showed that the flow patterns over the model surfaces depended on the meridional shapes. The diameter of each parallel circle determined the separation point and turbulence level in the leeward side of the models. The point of separation corresponds to the position of maximum outward pressure. Another important characteristic of the air flow was

Pressure buckling of cantilevered shells of revolution

403

that the pressure along each meridian showed significant variations, and this was verified by measurements. Although the hyperboloids were not buckled it was considered useful to use the flow visualisation methods to confirm the observations for the cones and cylinders. It was found that the flow was indeed heavily dependent on the geometry of model H3, even at high Reynolds number, so that the assumption of a single wind profile a r o u n d the shell circumference would not strictly be valid. A more detailed discussion of the measured pressure distribution on the hyperboloids, cylinders and cones can be found in Ref. 26. For the reasons alluded to above, serious doubts exist as to the relevance of the new cone and cylinder tests listed in Table 3. For this reason they are excluded from our present discussion. The properties of the Loughborough tunnel may also have adverse implications in any extrapolation to full scale of the results reported in Ref. 3.

5.2 Comparison of past wind tunnel tests with predictions There are two important series of past test results on wind-loaded cantilevered shells: those by Der and Fidler on hyperboloids 14and those by Johns et al.3 on cylinders. The present new test results include the cylindrical models CY1, CY2 and CY3 which were similar in form and tested in the same wind tunnel as those of Johns et al. ; it is, therefore, helpful to consider the test conditions for those models as being fundamentally different to those of the hyperboloids of Der and Fidler. t4 The reason for this is the difference in the measured wind pressure distributions obtained from the two tunnels where the tests were conducted. The variability in the pressure distribution coefficients for the Loughborough tunnel is discussed in Ref. 26, noting that the NPL tunnel pressure distribution coefficients, given by BS 4485, are different. Thus, the test results of Ref. 3 are, like the present new test results, excluded from the present comparison with theoretical predictions. The comparison of tests and predictions for the hyperboioids is shown in Table 5. The Batch and Hopley pressure distribution results in lower critical pressures (except for models 5 and 6) but in higher reduction factors than those due to BS4485. This suggests that, although the resistance to buckling of the shells appears to be lower for the Batch and Hopley distribution, the imperfection sensitivity also appears to be lower, as reflected by the lower linear membrane stiffness contribution. The present reduced stiffness predictions for the average thickness Der

0-028 a 0.025 ~ 0.025 ~

0.044 '~

0.040 ~ 0-040" 0.007 a 0"007 a 0.008 5 a 0.0085 a 0-01 a 0.01 a 0.012 J 0.012 J

1 2 3

4

5 6 7 8 9 10 11 12 13 14

1-09 1.09 0.438 0.438 0-747 0.747 1.06 1-06 1.77 1-77

1.09

0.368 0-368 0.368

(3)

(lbf/in 2) icm, ]cm] '~

0-802 [6,2] {l} 0.616 [6,2] {1} 0.689 [6,2] {21 2.26f [5,2] {1} 1.81 [5,2] {l} 1.79 [5,2] {2} 0.944 j [10, 4] { 1 } 1-05 [10,4] {2} 1.4U [9, 4] { 1 } 1-64 [9,4] {2} 2.13 [8,3] {11 2-44 [8, 3] {21 3.20 [8,3] {l} 3.60 [8,3] {2}

{loading type} h (4)

[mode

Test free stream Worst stressed buckling pressure meridian pressure pbw f~m (lbf/in 2)

0-59 0-59 0-55 0.59 0.59 0-55 0-66 0.62 0.62 0.58 0.59 (I.54 0-64 0-59

(5) 1.28 0-99 1-03 1.22 0.98 0.90 1.43 1.48 1.22 1-27 1-18 1.24 1.15 1.20

(6) 0.476[5,2] 0.363[5,2] -1.3215,2] 1.0415,21 -0.552 [8, 3] -0.832 [8, 3] -1.2218,3] -1-8618,3] --

(7)

[icm, jcm]a

pcrn': (lbf/in e)

1-29 0.99 -1.21 0.95 -1-26 --1" 11 -1-15 -1.05 --

(8)

Reduction Prediction to Worst pressure Prediction to factor test ratio meridian minimum test ratio [~*~m/f~cm [~*,,/ph,~ criticalpressure pcm/pb,~

"Circumferential, i~m, and meridional, 2jc~, wave numbers. bType 1 is Batch and Hopley wind distribution coefficient with 0-6 internal suction." Type 2 is BS4485 wind distribution coefficient with 0'6 internal suction. "The axisymmetric worst pressure is 1-6 times the free stream pressure; that is, external windward pressure plus internal suction of 0:6 of the external windward pressure. dNominal thickness with elastic modulus E - 473 000 Ibf/in • and Poisson's ratio of 0-4 for PVC models l~6, and E = 14 × 106 Ibf/in "~and Poisson's ratio of 0-3 for copper models 7-14. Average thickness with elastic properties as above. t T h e present worst stressed meridian analysis results were compared with those of Refs 6 and 7; the comparison is show n in Ref. 26.

(2)

(in)

Shell thickness t

(11

Model

C o m p a r i s o n of wind test results with predictions---tests o n hyperboloids by D e r a n d Fidler ~4

TABLE 5

bq

Pressure buckling of cantilevered shells of revolution

405

and Fidler hyperboloids average at 117 per cent of the test buckling pressures. It may be observed that the copper hyperboloid reduced stiffness predictions average at 127 per cent, while the PVC hyperboloid predictions average at 97 per cent. Also, the non-conservatism of the predictions increases with decreasing thickness for both materials. This phenomenon may be due to the increased susceptibility of the thinner models to vortex shedding vibrations. It was noticed in the new cylinder and cone tests mentioned above that these vibrations may decrease the buckling pressure capacity by an average of 16 per cent for these models. However, the Loughborough tunnel used for these tests may not yield representative results. Another possible explanation is that wind buckling may be controlled by modes lower than the minimum critical, icm.The new tests on cones and cylinders, unrepresentative as they may be, have shown that wind pressure deformation modes are lower than uniform pressure deformation modes, possibly biased into these lower modes by the large load components in harmonics higher than the zeroth (axisymmetric) pressure component. The 'worst pressure meridian' comparison, made by assuming that the sum of windward meridian external and internal pressures is axisymmetric, is also shown in Table 5. The worst pressure meridian predictions for the hyperboloids with average thicknesses are at 109 per cent of the test pressure. The copper hyperboloid predictions are again somewhat more non-conservative than those for the PVC hyperboloids, on average, by about 30 per cent. The worst pressure predictions are expected to be more conservative because, as shown by Fig. 3. the stabilising meridionai tension is much higher for the worst stressed meridian method, and the destabilising circumferential compression is significantly higher for the worst pressure method. Chang j~ has recently compared the predictions of a linear and a full non-linear analysis with four of the tests by Der and Fidler ~ on the two PVC and the two thinner copper hyperboloids~ using the modified properties for the PVC models and internal suction as suggested in Ref. 6. Chang found that under axisymmetric pressure, non-linear predictions are virtually identical to linear predictions. For wind loading, however, the non-linear predictions were found to be lower than the linear by as much as 12 per cent. In contrast with test observations, no snap-through buckling has been observed in either type of loading in the non-linear analysis. Predicted modes are reported as similar for the axisymmetric

406

George M. Zintilis, James G. A. Croll

and the wind analyses, although the circumferential wave numbers appear to be one higher in the non-linear wind analysis for the two PVC models; the meridional mode numbers remain identical in both analyses. Unfortunately, the discrepancy between the PVC and the copper predictions has not been explained to date.

6 S U M M A R Y AND CONCLUSIONS The reduced stiffness method has been applied to the elastic buckling of cantilevered shells of revolution under axisymmetric and nonaxisymmetric normal pressure. The relative simplicity of this linear eigenvalue analysis allows the convenient consideration of the effects of important parameters, such as curvature, boundary conditions, stiffness c o m p o n e n t s and variations in loading. It is suggested that accurate modelling of the non-axisymmetric pressure distribution and boundary conditions is important in establishing reliable predictions to the buckling pressures. It is also argued that the elimination of the linear membrane stiffness in the analysis will account for the combined effects of moderate to large imperfections and non-linear mode coupling. The presence of the postulated stiffness erosion has been shown for the case of axisymmetric axial, lateral or combined loading for end-supported shells. ~~-~:~3 Comparisons with past and new tests show that imperfection sensitivity is also present in cantilevered shells under normal pressure, and that the reduced stiffness method provides a unifying framework for accounting for the imperfection sensitivity displayed in the buckling of these shells.

REFERENCES 1. Report of the Committee of Inquiry into the Collapse of the Cooling Towers at Ferrybridge, CEGB, London, Nov. 1965. 2. Report o[the Committee of Inquiry into the Collapse of the Cooling Tower at Ardeer Nylon Works, ICI, 27 Sept. 1973.

3. Prabhu, S. K., Gopolacharyulu, S. and Johns, D. J., Stability of cantilevered shells under wind loads, J. Mech. Div., ASCE (EM5) (Oct. 1975) 517-30. 4. Zintilis, G. M. and Croll, J. G. A., Pressure buckling of end supported shells of revolution, Eng. Struct:, 4(4) (Oct. I982) 222-32.

Pressure buckling of cantilevered shells of revolution

407

5. Zintilis, G. M. and Croll, J. G. A., Combined axial and pressure buckling of end supported shells of revolution, Eng. Struct., 5(3) (July 1983) 199-206. 6. Abel, J. F. et al., Buckling of Cooling Towers~ Report No. 79-SM-1, Princeton University, Jan. 1979. 7. Cole, P. P., Abel, J. F. and Billington, D. P., Buckling of cooling-tower shells: state-of-the-art, J. Struct. Div., ASCE, 101(ST6) (June 1975) 1185203. 8. Cole, P. P., Abel, J. F. and Billington, D. P., Buckling of cooling-tower shells: bifurcation results, J. Struct. Div., ASCE, 101(ST7) (June 1975) 1205-22. 9. Langhaar, H. L. and Miller, R, E., Buckling of an elastic isotropic cylindrical shell subjected to wind pressure, Proceedings of Symposium on the Theory of Shells to Honor L. H. Donnell, Houston, Texas, 1967. 10. Langhaar, H. L. et al., Stability of hyperboloidal cooling tower, J. Mech, Div., ASCE (EM5) (Oct. 1970) 753-79. 11. Mungan, I., Buckling stress states of cylindrical shells, J. Struct. Div., ASCE, I ~ ( S T I 1) (Nov. 1974) 2289-306. 12. Mungan, I., Buckling stresses of hyperboloidal shells, J. Struct. Div., A S C E (STI0) (Oct. 1976) 2(X15-19. 13. Mungan, I., Buckling stresses of stiffened hyperboloidal shells, J. Struct. Div., ASCE (STS) (Aug. 1979) 1589-604. 14. Der, T. J. and Fidler, R., A model study of the buckling behaviour of hyperbolic shells, Proc. lnst. Civ. Eng., 41 (Jan. 1968) 105-18. 15. Chan, A, S. L. and Firmin, A., The analysis of cooling towers by the matrix finite element method, Pt. 1, A.J.R.A.S., 74(10) (Oct. 1970) 826--35; and Pt. 2, ibid., 74(12) (Dec. 1970)971-82. 16. Oyekan, G. L., Analysis of Hyperbolic" Cooling Towers With Structural Imperfections, Ph.D. Thesis, University of Southampton, March 1978. 17. Kato, S. and Matsuoka, O., Dynamic buckling analysis of cooling tower, Theor. and Appl. Mech., 25; Proceedings of 25th Japan National Congress )br Applied Mechanics 197_5, University of Tokyo Press, 1977. 18. Chang, S. C., An Integrated Finite Element Nonlinear Shell Analysis System With htteractive Computer Graphics, Report No. 81-4, Department of Structural Engineering, Cornell University, Feb. 1981. 19. Yeh, C. H. and Shieh, W. Y. J., Stability and dynamic analysis of cooling tower, J. Power Div., ASCE (PO2) (Nov. 1973) 339-47. 2(i. Croll, J. G. A. and Chilver. A. H., Approximate estimates of the stability of cooling towers under wind loading, Proc. 1ASS on Recommendations,fbr the Structural Design of Hyperbolic or Other Similarly Shaped ('ooling Towers, Brussels, May 1971. 21. Croll, J. G. A., Towards simple estimates of shell buckling loads, l)er Smhlbau (8) and (9) (1975) 243-8 and 283-5. 22. Batista, R. C., Lower Bound Estimates for Cylindrical S/tell Buckling, Ph. D. Thesis, University College London, June 1979. 23. Croll, J. G. A. and Batista, R. C., A design approach for axially compressed unstiffened cylinders, Proceedings ~[" lnstitttte Of Physics ('~m[~,rence ~m

408

24. 25. 26. 27. 28.

George M. Zintilis, James G. A. Croll Stability Problems in Engineering Structures and Components, University College CardifJ] Sept. 1978 (eds T. H. Richards and P. Stanley) Applied Science Publishers, 1979. Ellinas, C. E., Batista, R. C. and Croll, J. G. A., Overall Buckling Stringer Stiffened Cylinders, Report of the Department of Civil and Municipal Engineering, University College London, 1980. Walker, A. C. and Sridharan, S., Analysis of the behaviour of axially compressed stringer-stiffened cylindrical shells, Proc. inst. Cir. Eng., Pt. 2, 69 (June 1980) 447-72. Zintilis, G. M., Buckling of Rotationally Symmetric Shells, Ph.D. Thesis, University College London, Nov. 1980. Ewing, D. J. F., The Buckling and Vibration of Cooling Tower Shells, Part 1, Lab. Report No. RD/L/R 1763; Part 2, Lab. Report No. RD/L/R 1764, CERL, CEGB, England, Nov. 1971. Veronda, D. R. and Weingarten, V. 1., Stability of hyperboloidal shells, .I. Struct. Div.. ASCE (ST7) (July 1975) 1585-602.