Combined axial and pressure buckling of end supported shells of revolution

Combined axial and pressure buckling of end supported shells of revolution G. M. Zintilis Ove-Arup and Partners, London WlP 6BQ, UK

J. G. A. Croll Department of Civil and Municipal Engineering, University College London, London WCIE 6BT, UK (Received October 1981)

A reduced stiffness theoretical analysis of the imperfection sensitive elastic buckling for end supported shells of revolution is extended to the case of arbitrary combinations of axial and radial pressure loading. Depending upon the shell and loading parameters, the potential reductions in load capacity due to imperfections are shown to involve two distinct forms of post-buckling loss of stiffness. Lower bounds in each of these regimes are provided by appropriate reduced stiffness models, and shown by comparisons with available test data to be reliable even for relatively perfect test models. By attributing reductions in load carrying capacity to weakened end support conditions, it is suggested that past interpretations of these tests may have underestimated the deleterious effects of initial imperfections. Key words: shell (structural forms), cooling towers, pressure and axial loading, buckling, boundary conditions, imperfection sensitivity, lower bounds, analysis tests.

Introduction A recent analysis of the pressure load buckling of endsupported rotationally symmetric shells has suggested that the potential degree of imperfection sensitivity is strongly dependent on the precise geometric form. 1 This, it was argued, has important implications for the interpretations of past test data in which the reductions in buckling loads, from those of classical critical load analyses, have been explained in terms of relaxations of end boundary conditions. 2 Consequently, recent attempts to predict wind buckling loads of cooling towers, on the basis of classical critical analysis of an equivalent axisymmetrically loaded shell, may likewise have neglected the important load reductions arising from the effects of imperfections. The following sequel to our earlier analysis considers the same classes of shell but subject to axial loading and combinations of axial and pressure loads. The procedure adopted follows much the same form as that described elsewhere) First, an energy analysis of the 0141-0296/83/03199-08/$3.00 © 1983 Buttezworth & Co. (Publishers) Ltd

critical load spectrum is used to identify those components of the shell's stiffness (or energy) that provide a significant contribution to the initial stability of the shell but whose effects are likely to be eroded in the nonlinear, imperfection sensitive, post-buckling response of the shell. A lower bound approach is then based upon a critical load analysis of a shell from which the appropriate components of stiffness (or energy) have been eliminated. It is shown how the predictions from this reduced stiffness analysis provide an alternative explanation of test buckling behaviour that does not necessitate the adoption of seemingly unrealistic end-support conditions. Pure axial loading is considered first, as the shell response under this form of loading is qualitatively different from that previously described for pure pressure loading. 1 Having identified these two extreme forms of buckling response, a reduced stiffness analysis is described which allows definition of lower bounds to buckling loads for varying combinations of pressure and axial loading.

Eng. Strum., 1983, Vol. 5, July 199

Combined axial and pressure buckling o f end supported shells o f revolution: G. M. Zintilis and J. G. A. Croll

Axial l o a d i n g

XV° = ~ f f NEe'~R dO ds

Classical critical loads The classical critical loads are calculated from the total potential energy (TPE) using the perturbation method of analysis. The shell theory and the form of the finite element solution have been described in some detail earher.l, 3 Table 1 provides a comparison between the present classical analysis and a result 4 for a hyperboloidal model under axial load. Also shown in Table 1 is an indication of the convergence properties of the present solution for the case of pure axial loading. The number of finite elements required to achieve convergence is generally greater for axially loaded shells than it is for pressure loaded shells. This may be due to the somewhat shorter meridional ".¢avelengths of the critical displacement modes and the relative closeness of the magnitudes of the eigenvalues for the axially loaded cases. This latter aspect has the effect of increasing the number of iterations required in the minimum eigenvalue search routine for each mode.

Energy analysis of classical critical modes To identify the importance of the individual components of shell stiffness, the linear strain energy, U, is broken down into its individual membrane contributions:

UM = USM+ U°M+ U~p~

(1)

and bending contributions:

UB = U~+ U°+ U~°

(2)

so that :

U = UM+ UB

(3)

where s and 0 denote the meridional and circumferential coordinates of the shell's middle and surface. The first contributions from the nonlinear strain energy, XV, are, for the present classes of axisymmetric loading composed of the terms:

XV= X(V~ + V~2 + V° + V°2)

(4)

where each arises from the interaction between the prebuckling stresses and strains and the nonlinear components of the critical stresses and strains as follows:

1 t"/" E "R

dOds

(Sa)

sO

XVg2 = 2l f f Ns,,esER dO ds

(5b)

sO

Table I Convergence and verification of present numerical analysis for hyperboloidal model* Minimum critical load a, ( i c m ) (Ibf)

Source (no. of elements)

Pcm

Present (10) (25) (50) (75) Ref. 4 (/)b

59.1 61.1 61.0 61.0 60.2

(7) (7) (7) (7) (/) b

E E a Top edge boundary conditions v T = w T = 0 f o r fundamental state (E) and u T = v T = W T = 0 for critical state; at the base u B = v B = w B = 0 for both states; 3s is free in all cases. b Data not available.

200

Eng. Struct., 1983, Vol. 5, July

(5c)

sO

XV°2 = 2

N;'eoERdO ds

(5d)

sO

These terms are usually written in a combined form and interpreted as belonging to the external load potential. To do so is strictly not consistent and, means that a number of important and analytically useful aspects of axial load buckling have been overlooked. The eigenvalues, X, arise from the linear dependence on load of the fundamental (prebuckling) equilibrium state stresses and strains (denoted by superscript E) assumed in the classical critical analysis. The double prime (") indicates a quadratic dependence of those quantities upon the incremental displacements into the critical modes. The above formulation can be used to write the classical eigenproblem in the form: U + XV= 0

(6)

For a shell with fully clamped ends under axial loading, the classical critical loads, shown by the upper full line in Figure la, are related to the strain energy distribution shown in Figure lb. This shell, (73, was studied s in the test programme. It is reconsidered here together with other test results as an illustration of the relevance of the reduced stiffness analysis method to the test behaviour reported, s The positive energy spectra, as typified by Figure lb, show that the dominant linear membrane energy contributions are due to the meridional terms, U~; the shear term, U ~ , also provides significant contributions, but the circumferential contribution, U° , is negligible. For the shorter circumferential wavelengths (high i) in which bending energy is significant the linear bending energy is dominated by the circumferential term, U°, with the other two terms negligible. The energy distribution analyses also show that the linearized circumferential membrane term, XV°2 is very significant in its contributions to the shell's resistance to buckling. Figure lb shows that for the lowest critical loads for shell C3 the contributions to the positive energy coming from XV°2 are almost as great as those from the entire linear strain energy, U. For the shell of negative Gaussian curvature the remaining three linearized terms are all destabilizing and are dominated by the meridional component XV~z. This component is affected directly by the compressive nature of the fundamental meridional stress resultant, NsE whose maximum magnitude occurs at the shell's throat. The circumferential stress, which is compressive only for shells of negative Gaussian curvature, has its minimum at the throat. The overall nature of the behaviour of these shells is indicated by the significant meridional variation of the fundamental stresses as well as the relatively long wavelengths of the critical modes typified by those shown in Figure le. Such an overall type of behaviour conflicts with the local critical stress criterion for buckling design, s based as it is on tests where the shell model responses were monitored only at the throat parallel circle.

Reduced stiffness critical loads The discussion of the energy analysis of the classic critical modes indicates that the shell's membrane stiffness plays a dominant role in stabilizing the system against buckling.

Combined axial and pressure buckling of end supported shells o f revolution: G. M. Zintilis and J. G. A. Cro#

Rr

Reo= 200

Boundary conditions t ~ eqn (9) to (11) L

50

Classical, Pc I "

4c ~.~ -~ 3c -

!

~

/

C

RpcedUced \G E= 3.450 k N / m 2

2c lO

eqnI (9)

o

h'e° = O. 108

"...~. ~

=22 22 a

v =0.38 t = 1.65mm

BOundary/-;\ ~ conditions '..'~. ~-.-.,,..,~..." 18

3

Rs

3

3

3

I

I

I

I

I

I

I

1

2

3

4

5

6

7

I

i

u\./ /

Ua.~.

/

/ // E vZ 15C

L./.,

c UA

loc

"-. .//./. ....... 50

b

Z I 1

o

I 2

I 3

4

I 6

5

i

,=5>//'~

1.0 0.8 _

~

~

.

.

~" ~.

.

.

.

Throat

..°,. °°.°,."°

:>%"

o.6

"'"•...

~ °'•,.°

~,

0.4

:'~.~..~~= 7

•.

0.2

~

........

,,.•..°°'•

l/. ~.•°,°° f ...... .,°°

o C

1.o

o.6

02

I

o -o2

W,I

I

-0.6

-1.o

Figure 1 Toroid C3 (of reference 5) under axial loading: (a)

Critical load spectra; (b) Strain energy distributions in critical modes; (c) Normal displacements in selected critical modes

This was also the case for shells under hydrostatic pressure. 1 The important difference in the stabilization against axial loading is that the linearized membrane energy term XV°2 now plays a crucial role• From equation (5d) the linearized membrane energy XV°2 can be seen to arise from theproduct of the axisymmetric, prebuckling, hoop strain, eft, and the bands of nonlinear hoop tension, No', associated with deformation into the critical mode. The term XV° is positive, largely as a result of the Poisson expansion occurring when a rotationally symmetric shell is subjected to axial compression. For the cylinder the term XV°2 is entirely the result of the Poisson expansion, while for the doubly curved shell the nonzero hoop stress Ne~ also contributes to the hoop strain• However, for practical levels of meridional curvature the Poisson expansion, that is the positive hoop strain associated with the compressive stress NsE, provides by far the major contribution to e~. Why the stabilizing term XV°2 is so significant is that it is this term that appears to be eliminated in the nonlinear post-critical mode coupling behaviour. The mechanism for this was first described for cylinders by Donnell 6 as a means of identifying the modes that are essential if an effective nonlinear post-buckling analysis is to be performed. This mechanism has been revived recently as part of the derivation of a lower bound analysis for the imperfection sensitive buckling of axially loaded cylinders. 7-9 The mechanics described for cylinders would appear to be shared by the present class of axially loaded doubly curved shells. In the critical mode, wij , depicted in Figure 2a there exist regions of the shell where on account of the radial deformations the parallel circles 'aa' increase in length; nodal circles 'bb' do not undergo any change in length. The associated rt . second-order strains eoii, give rise to second-order hoop stresses N'o'i. whose average values would be the bands of tension sho~wn in Figure 2b. These average components, that are responsible for all the energy k1I°2, may be annihilated by an axisymmetric mode Wo,2j. This secondary axisymmetric mode would have twice the meridional waves, 2j, as the critical mode wij and could develop a linear compressive stress N~ . of amplitude equal to the q~Z/ r . . average nonlinear hoop tensions Nb.. as indicated by Figure 7 tl 2d. It was shown for cylinders that the required postbuckling displacements into the axisymmetric periodic, (0, 2/), secondary modes of Figure 2c are considerably smaller than those of the critical mode, (i,/). Thus, the required increase in the already insignificant meridional bending energy, U~, is small, especially for the long half waves (small j) characterizing the general buckling of the present classes of shell. This form of the mode coupling mechanism in the load displacement space is indicated in Figure 3 which suggests that with increasing imperfections, ~ii, the reductions in buckling load, Pb, capacity will tend to the reduced stiffness minimum critical load, P*m. The tests on cylinders,~ and on doubly curved shells studied in the following, indicate that these reduced stiffness predictions provide close lower bounds to the scatter of test buckling loads, with the implication that the required imperfections need not be large. The reduced stiffness load spectrum, P*. can be obtained from a modified classical critical load analysis by setting, for each mode: XV°2 = 0

(7)

implying that the linearized circumferential membrane

Eng. Struct., 1983, Vol. 5, July 201

Combined axial and pressure buckling of end supported shells o f revolution: G. M. Zinti/is and J. G. A. Cro//

I

that result from a reduced stiffness analysis. This is reflected in Figure 4 showing the effects of variations in the meridional curvature. To estimate the true imperfection sensitivity from the quotient of equation (8) would necessitate finding the complete spectrum of critical loads for each circumferential wave number i. To save time it is therefore recommended that separate classic and reduced stiffness analyses be performed in all cases to ensure that the true imperfection sensitivity is not underestimated.

i

Critical mode__

~

/

C

a

b'~-___ I - - - "

"~

b--

~"

l

i

Ia

Influence of boundary conditions

o

-- -- -Tension N'~I.,j

a

b

I

A careful imposition of boundary conditions on a shell structure can considerably increase the membrane stiffness which, in turn, increases the load-carrying capacity. Conversely, the lack of certain membrane boundary constraints has the opposite effects. It is, of course, the ability of shells to respond by developing membrane action that to a large extent determines their efficiency as load carriers. The models tested and reported in refs 4 and 5 have been argued in refs 1 and 3 to require the kinematic boundary conditions: 0

V E = ~3ET =

Secondory

mode-.,

-

b

. ~

-

~ ~\

/

(9)

b

L°--(

a

w E = u E tan@T

for the axisymmetric, prebuckled, fundamental equilibrium states (superscript E); the subscript T denotes the top edge

-

'

and

.

.

.

.

Axial load

Buckling load

p

Initial stiffness depends upon

.

Pb

/Pcm ~-~

xv~:o /

Buckli ng

k%./l°ad /I

7 -%xvbo

a

Compression N'eo ,2 j' '

c

I / I . . . . - ~ - - ~ - " - ./ ' _ k . -~J-~*

d

Figure 2 Mode coupling mechanism for axially loaded d o u b l y

mode

Wo~j.

energy term, XV2e~is lost in the imperfection sensitive buckling into this mode. The minimum of these reduced loads is the reduced stiffness minimum critical load, P'm, in mode i*cm,which may differ from the classic minimum critical mode, iem. The loci of the classical and the reduced stiffness critical loads for the toroidal model C3 are shown in Figure la by the full and the chain dotted lines respectively. By using the classic critical modes as kinematically admissible approximations of the reduced stiffness critical modes, the quotients:

Eng. S t r u c t . , 1 9 8 3 , V o l . 5, J u l y

I/ACriticol- - mad( --~-

Imperfection ~/117

,

Wj~2

a

b

Figure 3

Relationships between imperfection sensitivity, for unstable axial load buckling, mode coupling, and reduced stiffness critical load

Envelope of Pcm /

I E = 3450 N/mm 2 v = 0.38 R0o = 2 0 0 16 h : 1200 (mm)

(9101

t = 1.O5

(98)

f

/

(11.111(10.9) (9.5j

Boundary conditions

(74) /

.....

J/

....

eqn. (9)-(11)

(8)

may be used to provide approximations of the reduced stiffness critical loads. In these circumstances the energy terms U and XV2e2are those calculated from the (unmodifed) classical analysis. Although this would by-pass the necessity of performing the modified, reduced stiffness analysis some caution is necessary in applying it to other than negatively curved shells. This is because the classical analysis often converges to modes having short meridional half waves, j, for positively curved and 'nearly' cylindrical negatively curved shells. In such modes the importance of XV2°2is considerably less than it is for the longer axial wavelengths

202

t

H/t,2 /r-~

Secondary /- -

curved shells

e*~ = Pc" U / ( U + XV~%)

I

(53) 4 __(63)

...........

/////1"/////

(52) •

"'%

i; 0

L - 006

(51) (711 . . . " (61 •""

(93) ,.'

.,. -Yr

, I

i -O.O4

L

L -002

i

I O

Envelope of Pcm I L t I I I 002 O.O4 0.06

ROo / Rs Figure 4 Effects of curvature variations on classical and reduced stiffness critical loads for toroids with clamped ends, subjected to axial loading

Combined axial and pressure buckling of end supported shells of revolution: G. M. Zintilis and J. G. A. Croll

of the shell. For the non-axisymmetric critical state analysis these boundary conditions become: ur=vr=wr=

= 0

(10)

T

Also, at the base edge (subscript B) the boundary conditions for both analyses would be: u s = vB = wB =

= 0

(11)

Equation (9) allows for the rigid body translation of the top rigid disc in the axial direction while the shell top edge is restrained from displacements normal to the shell's axis and from meridional rotations. In the critical state stability analysis the shell is taken to be fully clamped reflecting the fact that the axial displacements at both ends of the test shells are incapable of developing a periodic form. In the analyses of refs 4 and 5 periodic displacements have been allowed to occur, with the result that the theoretical predictions seem to be unrepresentative of the actual test set-ups. In each case the models were bonded to rigid endplates by cerrobend. The simple supports SS1 and SS24 do not seem to reflect this empirical reality, while Mungan s appears to have recognized this inconsistency in subsequent work m by recommending the use of 'new' and apparently more relevant 'boundary conditions' whose predictions conform with those arising from equations (9)-(11). As in the case of lateral pressure, 1' 3 the most important constraint under axial loading is that on the meridional displacement, u. The effect on the critical loads of using equation (9) for both the fundamental and the critical state analyses is shown by the broken curve in Figure la. This shows the general trends consequent upon the relaxation of the constraint on the in-plane meridional displacement, u, that the critical axial load, and, the classical minimum critical mode number icm , are reduced. This is also shown in Table I of ref. 11. The reductions in the classical mode number, icm, that usually occur when u is relaxed at the boundaries may account for the 'unexplained difference', in conclusion 4 of ref. 2, between the theoretical mode predictions obtained using the less stiff boundary conditions, and the somewhat higher experimentally observed buckling modes. Using such less stiff conditions (Figure la), often results in minimum critical loads that appear to be close estimates of the observed buckling loads. It is possible that this may have led to underestimations of the potential imperfection sensitivity for these shells. It has also led to situations in which predictions from classic bifurcation loads are actually below observed test buckling loads; Figure 2 of ref. 2 demonstrates this phenomenon. The reason advanced for the choice of less stiff boundary conditions is that 'this is the only condition that corroborates the axial load internal pressure ... test data'. 2 This assumes that an internally pressurized shell will have a stable post-buckling behaviour and that, consequently, any reductions in the test buckling loads must be due to boundary flexibilities. In order to check this assumption the present reduced stiffness analyses have been applied to the hyperboloidal model 2.84 which was in the tests subjected to both axial load alone and a combination of axial load with an outward hydrostatic pressure a t p / p a n = 0.51 (p = 0.2 lb/in2). In the first case of axial load alone the reduced stiffness analysis shows that the buckling load could be reduced to 59% of the classical critical load, whereas in the combined load case the reduc-

tion could be as low as 46%. For model 2.8, with empirically realistic end boundary conditions (9)-(11), these predicted lower bounds are in close conformity with the observed buckling for the two loading cases. In contradiction with Veronda and Weingarten2 the present analysis suggests that the potential imperfection sensitivity of axial load buckling increases when internal pressure is present. It seems lz that the reasons for adopting boundary conditions with no in-plane meridional constraint are as unnecessary as they are physically unjustified. The present work will, therefore, adopt the clamped conditions of the test set-up as represented by equations (9)-(11). A more detailed study on the effects of varying boundary conditions is presented by Zintilis. 3 Comparisons with test results The previous sections suggest that the axial load buckling of toroidal and hyperboloidal shells with clamped ends may be imperfection sensitive to a greater extent than that for lateral pressure loading. ~ It will be instructive to consider more extensive comparisons with the test results reported by Mungan. s Table 2 compares the present classic and reduced stiffness predictions with the test buckling loads reported for axial compression alone. In contrast with the conclusions arrived at on the basis of comparisons with classic analysis using meridionally relaxed end boundary conditions, s column 5 shows that the buckling loads are considerably reduced with respect to the classical critical loads obtained using the physically realistic boundary conditions of equations (9)-(11). The reduced stiffness method can be seen to provide theoretical knockdown factors broadly in line with those observed experimentally. The eight reduced stiffness predictions, as the last column shows, are, on average 8% higher than the test buckling loads. This apparently non-conservative average may be a reflection of the effects of using the 'first dimple' criterion for def'ming the buckling loads in tests. Influence o f curvature As in the case of hydrostatic loading 1 the curvature of shells strongly affects their buckling behaviour under axial loading. A typical parametric study for toroidal shells of varying curvatures is summarized in Figure 4. The loci of both the minimum classical and the minimum reduced stiffness critical loads are shown along with the circum-

Table 2 ComParisons between present theoretical predictions and Mungan's test results s for toroidal models under axial load Model No. pba (t: mm) (kN)

Pcm(icm)b

S~ (1.95) Cz (1.95) S= (1.77) S~ ( 1.65) C 3 (1.65) C4 (1.55) S s (1.33) C s (1.33)

25.63 21.72 20.40 17.20 15.03 13.18 10.06 9.18

12.361 12.361 12.361 9.653 9.653 5.867 6.180 5.494

(kN) (6) (6) (6) (6) (6) (6) (7) (7)

p~m/Pcm c (i~m)

Pb/Pcm

P~m/Pb

0.62 0.62 0.61 0.61 0.61 0.61 0.62 0.62

0.48 0.57 0.61 0.56 0.64 0.52 0.61 0.60

1.29 1.09 1.00 1.09 0.95 1.17 1.01 1.03

(6) (6) (6) (6) (6) (6) (7) (7)

a Pb -- test buckling loads (there are no axial buckling loads reported in ref 5 for models C 2 and S4) bjocm(icm) -- "load (mode) corresponding with minimum classical critical analysis. c ~ .~ P~m(lcrn) -- load (mode) ,~orrespondmg with minimum reduced stiffness critical analysis hV22 = 0

Eng. Struct., 1983, Vol. 5, July 203

Combined axial and pressure buckling of end supported shells of revolution: G. M. Zintilis and J. G. A. Croft

ferential wave numbers, i, and meridional half wave numbers, j, associated with the minimizing mode. Critical loads, Pc, in modes other than the minimum have been excluded for the sake of clarity, as in many cases they are very close in magnitude to Peru. The toroidal shells of Figure 4 have been chosen to be of the same dimensions as the S-type models s but with a wall thickness of 1.05 ram. Meridional curvatures, Rs, are varied so that the range of the throat curvature ratio, Roo/Rs, is between --0.07 and +0.07. The boundary conditions used are given by equations (9)-(11) to be identical with those used to model the tests of ref. 5. From Figure 4 it can be seen that the classic load capacities of shells with positive Gaussian curvature are significantly higher than those with negative Gaussian curvature. A plateau of about 15 kN is reached at the cylindrical curvature. The reason for this apparently higher load capacity of positively curved toroids is the enhanced stabilizing effect of the linearized circumferential membrane energy term XV°2 ; to some extent this reflects the tensile hoop stress N0E occurring for positively curved toroids. Conversely, the stabilizing effect of XV°2 for negatively curved shells is reduced, on account of the compressive nature of Nff. However, the penalty for the increased influence of XV°2 for these curvatures in which Roo/Rs > - 0 . 0 3 is that as equation (8) suggests there is a proportionate increase in potential imperfection sensitivity. This is indicated by the lowering of the locus of the reduced stiffness predictions in this range. This juxtaposition of the classical and the reduced stiffness predictions in Figure 4 is interesting in that the reduced stiffness lower bounds to imperfection sensitivity indicate the existence of optimum toroidal curvatures. On account of the considerable variations in potential knock-down factor these optimum geometries would differ considerably from those determined from a classical critical load assessment of buckling capacity.

200 U 150 -

•...a, ~._,_ - - - . 100 ,~ "--

~ .

"/ E

......

5o

Ue

----..

_ -_ . . ~ . ~ .

I'..... ,. ..".,

/ - - ~ -

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

"..,.

C Ld

-

u~7

% %

0

~,,,

• •'-. -50

~

I ---~-'--- Range I

-100 0

02

/Xv2~ • " •"',,. • %~

Tr

TIT

I

I

I

0.4

0.6

0.8

•

10

P/Pcm Figure 5

Combined loading transition f o r t o r o i d C3. Contributions

to stabilizing potential energy in classical critical mode for axial load buckling subjected to y a w i n g superimposed hydrostatic p ressure, p

1.0

Pcm = 21.72 kN C1

Pcm = 14.88kN/m2

0.8 Classical

Transition

C o m b i n e d axial and pressure loading

~

Reduced stiffness critical loads The above has suggested that for pure axial load the loss of stiffness occurring in the post-buckling behaviour is associated with the annihilation of the initially stabilizing nonlinear circumferential energy term XV°2. In contrast, the loss of post-buckling stiffness for shells subjected to pure lateral pressure loading results I from a loss of the linear membrane strain energy UM. For shells subjected to arbitrary combinations of axial loading and lateral pressure, a criterion is required whereby the dominance of either the linearized circumferential membrane energy term XV°2 or the linear membrane energy, UM, is established. This will mean that the post-buckling behaviour is dominated by the axial load or the lateral pressure respectively. This will then enable the implementation of the appropriate stiffness reduction in the analysis. In the case of cylinders, it was shown 7 that axial loading dominated for most load combinations. The transition from the pressure-dominated range to the axial load dominated range was proposed on the basis of the neglect of the greater of the two relevant stabilizing membrane energies, the linear, UM, and the linearized circumferential, XV~2.7 A similar comparison for the typical negatively curved toroid C3 is shown in Figure 5. The point of transition can again be defined to occur when: UM = XVO=

204

E n g . S t r u c t . , 1 9 8 3 , V o l . 5, J u l y

(12)

-q J

.

~u

Lu

..XV° =0

....

~

C

o.21-

/-.. .....

/

0

~ A~

Re oc

0.2

\

,.\ =o

0.4 0.6 Lateral pressure, P/Pcm

Figure 6 Combined axial and pressure load

0.8

10

interaction for toroid

C, of reference 5

as Figures 5 and 6 suggest. Although this criterion may be adequate for cylinders and positively curved shells, considerable areas of doubt must remain for certain shells of negative Gaussian curvature. This is because XV°2 may not be as dominant for shells of negative Gaussian curvature as it is for cylinders. 7 In the case of the negatively curved shells of the present study the buckling behaviour seems to be dominated by the lateral pressure, over a rather wider

Combined axial and pressure buckling of end supported shells of revolution: G. M. Zintilis and J. G. A. Croll range of load combinations, and not by the axial load as for cylinders. A typical range of axial load dominance for shells of negative curvature is 0 < P/Pcm < 0.2 as shown in Figure 5, while for cylinders this may be typically 0 < P/Pcr~ < 0.8.7 In some cases of negatively curved shells XV~2 is smaller than Um even when axial load acts alone. This behaviour is displayed in most of the test models 4 where, in some cases, XV°2 was even slightly destabilizing (negative). In these latter cases any loss of post-critical stiffness would be associated with the loss of linear membrane energy UM. It appears that three distinct regimes of buckling behaviour could occur. These are: (i) a range in which kI?°2 > UM > 0, where axial load dominates and where the reduced stiffness is based upon kF°2 = 0; (ii) a range in which UM > ;kF°2 > 0 where either UM or XF°2 may be eroded; (iii) a range in which UM > 0 > k F~22so that, as for pure pressure loading, the reduced stiffness loads are calculated on the basis of UM = 0. Naturally, the prediction of the precise nature of the post-buckling loss of stiffness requires a full and empirically validated nonlinear analysis. This is perhaps especially true in the uncertain area of loading corresponding with the range (ii) identified above. However, it seems that a criterion based upon the neglect of the larger of UM or k I"°2 provides close agreement with available test data. Comparisons with tests The combined loading behaviour for test models C1, Ca, and $3 is typical of the experimental study reported s and is summarized in Figures 6 to 8. The broken and the full thick interaction curves represent the reduced stiffness and the classical critical loads respectively for boundary conditions given by equations (9)-(11). The three ranges of behaviour identified above are also shown. These interaction diagrams show that the majority of the test results (+) are closely approximated by the present reduced stiffness predictions. The implication seems to be that even the small imperfections in these laboratory specimens are enough to bring the test buckling loads down to the

1.0 Pcm = 15.03 kN

Pcm = 10.15 kN/m 2

C3 0.8

\

\ ""+""..

- onsi io.

aEoo.6 . . . . . . .

\

""-..+

8

\

~

X < 0.2 .

I 0

0.2

Reduced

I

Pcm = 17 20kN Pcm = 9 . 1 4 k N / m 2

$3

isJcol Transition

•i-

....

/

/

Cl

~=o O

°.

'~

-P...

0.,~

Tests

+/ T

,..ml~ ~o f~ge 11 /. / . % . . . =

1Tr

0.2 Reduced

•.. /UM=O "l

I 0

0.2

I

]

".

0.4 0.6 Lateral pressure, P/Pcm

08

'

1.0

Figure 8

Combined axial and pressure load interaction for toroid S3 of reference 5

reduced predictions. Furthermore, it would appear that as the test results are closer to these predictions in the region of axial load dominance, and as the test shells had identical imperfections, the imperfection sensitivity increases for this region. Some comments on the local stress design method 5 would seem pertinent. The throat stress interaction diagrams obtained from the combined load tests on both the C- and S-type models were processed by Mungan 5 so that a single interaction curve was obtained. This seems to distort the true behaviour as the meridional stress is estimated to be up to 30% higher in the C-type models than it is in the S-type models, whilst the hoop stress remains virtually the same for both S- and C-type models. Furthermore, under lateral pressure alone the throat is the location of the minimum hoop and meridional stresses, while for axial loading this is the location of the minimum hoop and of the maximum meridional stresses. Remembering that in the tests the stresses were only measured at the throat, s doubts must arise as to whether this was an optimal choice of location, even assuming that the overall type of shell behaviour can be described in terms of the actions occurring at one parallel circle.

Conclusions

~

'"..

0.4

C.ossica

1.0

""'........./.UM -- 0 I --'.

0.4 0.6 Loterol pressure, P/Pcm

I

~.X

0.8

Figure 7 Combined axial and pressure load interaction for toroid C3 of reference 5

1.0

The reduced stiffness approach for shell buckling has been presented for various geometries and combinations of lateral and axial loading. In order to account for the effects of imperfections and nonlinear mode coupling the appropriate stabilizing membrane stiffness components were assumed to be completely eroded by using a modified form of the linear eigenvalue analysis. The criterion for choosing the stiffness terms to be neglected is based on the relative magnitudes of the stabilizing linear membrane stiffness and the linearized circumferential membrane stiffness. Comparisons with tests seem to indicate that it is imperfections and not boundary flexibilities that account for the scatter and the sometimes considerable reductions in observed buckling loads, when these are compared with the

Eng. S t r u c t . , 1983, V o l . 5, J u l y

205

Combined axial and pressure buckling of end supported shells of revolution: G. M. Zintilis and J. G. A. Croft classical critical load predictions. Previous design proposals in which increased boundary flexibilities have been used to account for these reductions may be grossly nonconservative when shells in reality have flexible boundaries, The reduced stiffness approach is based upon an analysis o f the physics o f buckling and its simplicity allows the separate and important influences o f geometry, loading and boundary conditions to be isolated. As p r o t o t y p e shells are likely to have larger imperfections than laboratory models, it is suggested that the reduced stiffness predictions should provide even more reliable lower bound estimates o f anticipated buckling loads. In contrast with some recent design proposals, the need for purely empirical extrapolations from limited test results is avoided. It would be reassuring if future nonlinear post-buckling analyses could be directed towards even more rigorous validation of the proposed reduced stiffness lower bounds, and in the process assist in the task o f evolving conceptually simple, but reliable and safe, procedures for the prediction o f design buckling loads.

Notation E h i icm ib

/ N P Pc Pcm

P*~m Pb P R Rs, Ro Ro o

modulus o f elasticity shell height circumferential harmonic number classical critical circumferential harmonic number test buckling circumferential harmonic number meridional half-wave number (subscripts as for i) in-plane stress resultants pressure critical pressure minimum critical pressure reduced stiffness minimum critical pressure test buckling pressure axial load (subscripts as for p ) radius of shell parallel circle principal radii o f curvature at the shell's throat circumferential radius of curvature at the shell's throat

206 Eng. Struct., 1983, Vol. 5, July

s t U u, v, w XV z /3 e 0 ?, 4~ v

meridional shell coordinate shell wall thickness linear strain energy orthogonal displacements in the s, 0 and outward normal directions linearized strain energy shell axis coordinate rotation o f the tangents and the normal membrane strains circumferential angle critical eigenvalue (numerically equal to critical load) meridional angle Poisson's ratio

References 1 Zintilis, G. M. and Croll, J. G. A. 'Pressure buckling of end supported shells of revolution', Eng. Struct. 1982, 4, 222 2 Veronda, D. R. and Weingarten, V. I. ' Stability of pressurised hyperbolical shells', J. Mech. Div. ASCE, 1975 (EMS), 663 3 Zintilis, G. M. 'Buckling ofrotationally symmetric shells', PhD Thesis, University College London, Nov. 1980 4 Veronda, D. R. and Weingarten, V. I. 'Stability ofhyperboloidal shells', J. Struct. Div. ASCE, 1975 (ST7), 1585 5 Mungan, I. ' Buckling stress states of hyperboloidal shells', Z Struct. Div. ASCE, 1976 (ST10), 2005 6 Donnell, L. H. 'A new theory for the buckling of thin cylindrical shells under axial compresssion and bending', Trans. ASME, 1934, 56, 795 7 Batista, R. C. 'Lower bound estimates for cylindrical shell buckling', PhD Thesis, University College London, June 1979 8 Batista, R. C. and Croll, J. G. A. 'A design approach for axially compressed unstfffened cylinders', in Stability Problems in Engineering Structures and Components (Eds T. H. Richards and P. Stanley), Applied Science, London, 1979, pp. 377-399 9 Ellinas, C. E., Batista, R. C. and Croll, J. G. A. 'Overall buckling of stringer stiffened cylinders', Proc. Inst. Civ. Engrs. 1981,71 (Pt. 2), 479 10 Mungan, I. ' Buckling stress of stiffened hyperboloidal shells', J. Struct. Div. ASCE, 1979 (ST8), 1589 11 Abel, J. F. et al. 'Buckling of cooling towers', Report No. 79-SM-1, Princeton University, New Jersey, Jan. 1979