Non-linear lower bounds for shell buckling design

J. Construct. Steel Research 28 (1994) 101-120

Non-linear Lower Bounds for Shell Buckling Design

Ronaldo C. Batista*

Program of Civil Engineering, COPPE/UFRJ, Rio de Janeiro, CP 68506, CEP 21945-970, Brazil

&

Paulo B. Gonqalves

Department of Civil Engineering, PUC-Rio, R. Mq. S. Vicente 225, Rio de Janeiro, CEP 22453, Brazil

A B S T R A C T

Based on rigorous parametric non-linear elastic buckling analyses, the present work underlines the developments towards theoretical lower bounds of existing experimental buckling loads for some of the most practical shell geometries and loading cases, namely circular cylindrical shells under external pressure and~or axial loads and spherical caps under external pressure. Simple equations and formulae are presented and their predictions for buckling loads are compared with available test results and values prescribed by some of the existing design codes. These explicit lower bounds are close and non-conservative estimates of buckling loads of imperfect shells and as such are proposed as a consistent and rational basis for design of these shell structures.

I N T R O D U C T I O N

Buckling analyses and failures

Buckling failures of steel shell structures used in many industrial ap- plications constitute a long list of unsuccessful designs. Recent and near-past progresses in our capability to deal with shells' non-linear

*To whom correspondence should be addressed.

101 J. Construct. Steel Research 0143-974X/93/$06.00 © 1993 Elsevier Science Publishers Ltd, England. Printed in Malta

102 R. ('. Batista. P. B. Gone'aires

problems are mostly through general- or specific-purpose computer pro- grams. However, these computer codes have, to a certain extent, failed t~ predict correctly the buckling behaviour of these structures. The majorit3. of these codes, which present such serious drawbacks, are not developed on the basis of the asymptotic theory of elastic stability, j 3 Bifurcations occurring along linear or non-linear equilibrium paths arising from unloading states can be easily missed; initial post-buckling behaviour and imperfection sensitivity cannot be predicted and buckling failure ~1' imperfect-to-be shells can therefore be overlooked when one uses these general-purpose non-linear finite element computer codes. Moreover, limit loads obtained by these codes, by imputing too general or tot~ specific imperfection shapes, are bound to be misleading and leads the designer into numerical traps.

Shells which display high imperfection sensitive behaviour yield experimental buckling loads that are much lower than classical critical theoretical results. Even when geometrical imperfections are maintained under control during fabrication, small amplitude imperfections with contributions from each of the incipient critical buckling modes will occur inevitably. Even if large imperfection data banks of fabricated shells are used for theoretical estimates of buckling loads, through refined non-linear analyses, it is very likely that additional imperfections will arise accident- ally during installation procedures and service life of the shell.

These additional imperfections in highly sensitive shells constitute a serious drawback of refined calculations which depend on reliability functions based on measured initial distributions that are characteristic of the different fabrication or erection procedures. Put more simply, there is always a great risk that the actual shell loading capacity is much lower than the theoretical prediction from these analyses.

Analytical 4 s or numerical 9 modelling for non-linear analyses, if based on the general theory of elastic stability, have found a better use in enhancing the understanding of the physical process involved in non- linear buckling. These non-linear results have shown how the initial stiffness and load carrying capacity can be undermined by the simultan- eous effects of imperfections and modal coupling and, consequently, can provide a basis to construct simpler but realistic and safe lower bound estimates for buckling loads.

Lower bound approach to design

The lower bound approach is proposed on the general notion that what is needed for design are safe but theoretically well founded lower bounds

Non-linear lower bounds for shell bi~cklin# desion 103

of buckling loads, which for actual moderate to large imperfections would also be reliable estimates.

If higher buckling loads are to be adopted in design, the onus would be upon the designer to ensure 'perfection' of the as-built structure and to provide validation of the higher loads adopted through the use of some sophisticated procedure as, for example, a general-purpose non-linear finite element computer code. There lies a risk 1° for a designer who takes such steps without the total guarantee that the required methodology will be implemented when constructing the 'imperfect-to-be' shell structure. Then the onus is still, tb a large extent, on the designer and constructor to admit certain limited levels and shapes of imperfections, and, having done so, to then concede, through very refined or, even worse, too simplified numerical modelling, the use of 'knocked-down' design buckling loads, which could still be unreliable upper bounds.

In the absence of simple and reliable theoretical estimates for buckling of imperfect shells, engineers have usually used semi-empirical procedures and relied on recommended design curves which are based on the lower limit ot the scatter of selected experimental buckling loads or on statistical evaluations of the probability of these results. Design specifications 11'~2 then prescribe 'knock-down factors' on the classical theoretical predictions to account for the experimental scatter; the severity of these capacity reduction factors varying according to the field of engineering construction.

Hence, because of the above-raised uncertainties, it becomes clear that what is needed for a safe and reliable buckling design are lower bounds founded on the general features of the non-linear elastic and elasto-plastic buckling behaviour of distinct shell structures.

Lower bound buckling concept

A straightforward observation that can be drawn from non-linear elastic stability analyses is that significant non-linearity in the post-crltical response can only occur where geometrical, loading and boundary conditions of the structural system enable substantial changes in the membrane stiffness (or equivalent energy) from that associated with a critical state. The slight non-linearities in bending stiffness occurring in advanced post-critical states are usually assumed to involve deformations whose magnitudes are so large as to be outside any practical range. For shell structures, whose geometry is an optimum shape which withstands the applied pressure through mainly extensional internal stresses, it is com- mon for the critical modes to derive substantial contributions to their initial stabilization from the membrane stiffness of the shell. 4

104 R. C. Batista, P. B. (hmfah~es

It is the importance of this membrane stiffness in the critical modes themselves that gives the idealized perfect shell its apparent advantage as far as critical load levels are concerned. But equally it is the presence ot this membrane stiffness that creates many of the problems in tmderstand- ing an imperfect shell buckling behaviour. For there are many instances m which the combination of initial imperfections and the coupling between the various possible modes can lead to the erosion of the vital stabilizing membrane stiffness in the buckling response. Where this occurs the reduction in membrane stiffness, with critical compound deformations, is associated with a loss of post-critical load carrying capacity.

This leads to the general observation that the non-linearity in post- critical response can only be unstable where the membrane stiffness (or again related membrane energy) contributes to the structure resistance to deformations in the associated compound critical modes. In other words, a change in post-critical membrane stiffness can only lead to a loss of stiffness if there is already positive stabilizing membrane energy,13 ,5 or stiffness, in the critical modes that dominate the initial post-critical response. As to whether or not the post-critical response wilt be unstable, and, if so, to what degree, depends on the prevailing geometrical and physical conditions. Then the success of a lower bound method depends upon the capability of the theoretical model much more than on our ability to define which of the membrane stiffness components will be eroded by imperfections and non-linear coupling between critical and post-critical modes that occur in the initial post-critical response. These features can only be correctly identified through analytical or numerical non-linear models 49 based on the general theory of elastic stability.

Purpose of the present work

Based on rigorous parametric studies using non-linear analytical models, 49 the present work underlines the main developments towards theoretical lower bounds of existing experimental buckling loads for some of the most practical shell geometries and loading cases. These are circular cylindrical shells under external pressure and/or axial loads, and spherical caps under external pressure.

Compact equations and simple formulae are presented and their theoretical buckling estimates are compared with available test results and values prescribed by some of the existing design codes. These lower bounds are close and non-conservative estimates of buckling loads of imperfect shells and as such are proposed as a consistent and rational basis for design of these shell structures. The simplicity of the proposed lower bound expressions could be to the designer an attractive alternative

Non-linear lower bounds for shell bucklino desion 105

to the empirically based 'knock-down factors' applied to classical critical loads.

T H E O R E T I C A L F U N D A M E N T A L S TO L O W E R B O U N D ESTIMATES

The mathematical model to lower bound estimates for shell buckling was firstly derived and applied to thin monocoque circular cylindrical shells under uniform axial compression and/or external pressure. 4 It is based on the energy formulation of the asymptotic elastic stability theory 1-3 and as such differs from previous approaches. 16-19

The lower bound approach to cylindrical shell buckling was then validated by fully non-linear analytical models 5 developed within the framework of the general asympto'tic theory. It has also been extended to stiffened circular cylindrical shells 2° and to spherical shells and caps. a Moreover, it has been used to access a previously reported quasi-inextensional approach to axi-symmetric shells under external pressure. ~4'2~

In the following the general principles and energy criteria for structural elastic stability are summarized so as to establish in a mathematical format the basic ideas behind the lower bound concepts discussed in the previous section.

It should be emphasized that because this lower bound model is based on energy concepts and criteria it could constitute a general approach to thin shells of many distinct geometries.

Energy formulation

To outline the general lower bound energy model, it is considered an imperfect continuous structural system under conservative loads whose total potential energy, V, consists of the sum of internal strain energy, U, and the potential energy of the applied loads, 2V', the latter associated to one load parameter 2.

If the displacement components of the fundamental deformed state (that is, the equilibrium configuration of the imperfect shell whose stability is to be investigated) are called ur(2) and the new configuration at some arbitrary neighbouring state, ue(2)+~(ti, 2), where f are kinematically admissible small incremental displacements and ti are small geometric imperfections, it follows that for stability:

V [uV(2) + ~(~, 2)] > V [uV(2)] (1)

106 R. C. Batista, P. B. Goncah;es

By assuming V as an analytical function in the vicinity of u~:~).j, which is supposed to vary smoothly under increasing load, the left-hand side in eqn (1) can be written in the expanded form:

V [b/F(,,'~) ---~ ~(U, )c)] = V [uF(~)] -[- VI(~; uF; )~)-~- V2(V;/AF; ,~.)

+ V~ff; uV: )0+ V4(~; uF: ;~) ~2)

where Vi are the ith variations of the total potential energy with respect to the incremental displacements, 0, and their derivatives, and have coeffi- cients that contain uV(2) and/or ~ as parameters.

It should be noted that the geometrically imperfect structure is considered to be free of initial stresses when unloaded and that the total displacement components , u, associated to a deformed configuration under load 2 can be described by:

u + a = ur(2) + ~ + ~(a, 2)

or simply:

u=uV(2)+q(~,2) (3)

where it is observed that the deformed configuration of an idealized perfect structure in a fundamental state of equil ibrium is taken as reference. It should be emphasized that this assumpt ion does not represent any loss of generality as it is the incremental displacement ~ that defines the influence of imperfections on the shell behaviour. Put more simply, the relevant imperfections are assumed to be in the shape of incremental critical modes or in shapes that trigger them.

As bending is initially confined to narrow zones adjacent to the supports, the fundamental state is taken as a linear membrane state, which is a valid assumpt ion for shells with small to modera te imperfections. Then u v is a linear function of the load parameter 2 and the expansion (2) can be written as:

v [uF(,~) + ~(a, ,~)] = v EuV(2)] + vlf f ; u F) + , w i f f ) + v ° ( 0

+ ~v~(0 + v°(0 + v°(0 (4)

where the constant term v[uF(,~)] belongs to the fundamental membrane state.

Non-linear lower bounds for shell bucklin O desion 107

As the original deformed configuration of the imperfect shell is an equilibrium state:

Vl(f; ur; 2) = Vl(f; u v) + 2V~(f) ~ 0 (5a)

or more truly:

lim [Va(f; uV; 2)] = 0 (5b) fi ---, O

as eqn (5a) is only approximately satisfied by the fundamental membrane solution due to the presence of small imperfections.

Noting that the imperfections do not affect the initial bending strain energy (or stiffness), and taking into account eqns (4) and (5), the total variation of the total potential energy can be written as:

A V = V [u~(;t) + ~(a; )~)] - V [ur(;t)] = Uu2(u r, 0 + ,~ V'M2(O

+ U u 3 ( 0 + U u , ( 0 + U~2(v) (6)

where UM~ are the membrane energy terms, UB2 is the quadratic bending term which depends on the imperfection-free incremental displacement v, and 2V~t2 is the non-linear 'potential' membrane energy contribution to AV.

The energy criterion of stability requires that:

AV -~- UM2(N F, 0 +/~ V b f 2 ( 1 3 ) 71- UB2(Y) q- UM3(O "~- UM4(O > 0 (7)

Henceforth, the positive quantities in the above equation will be called stabilizing energy contributions, whereas the negative quantities will be called destabilizing energy contributions.

By collecting separately the stabilizing and destabilizing energy contributions the stability criterion reads:

A V = A V ( + ) + A V ( - ) > O (8)

From rigorous parametric non-linear elastic analyses it can be shown 4 8 that, for shells which derive their apparent initial stability from membrane energy (or stiffness), the most important stabilizing (positive) quantities depend on the type of applied load. In the following, these quantities are shown in brackets for each loading case.

108

(A)

R. C. Batista, P. B. Gonfah'es

External loading applied normally to the shell wall, as, for example, external radial pressure on shells of revolution:

A V (+)= UB2-~-(UM2)+(U M4. ) (9a)

. . . . t ) /

A V ( I=2V~M2-.~-AVM2-~-UM3 (9b)

(B)

where the superscripts q and ~ stand for the orthogonal curvilinear coordinates associated with the lines of curvature of the shell reference surface. External loading applied predominantly in a direction ~ through the reference surface of the shell, as, for example, meridional compression on cylindrical shells:

A V ( + ) = UB2 + UM2 -I¢-(UM4.)-~(A~VM2 ) (lOa)

A V ( ) , , ; =AVM2+UM3 (lOb)

(c) External combined loading: combination of cases (A) and (B), as, for example, cylinders and cones under combined external radial pressure p and meridional (or axial) compression P. Here, two separate cases have to be considered according to the classical relationship:

(P/Pc) + (P/Pc) = 1

where the subscript c indicates the classical critical loads:

(C1) for p/pc<~0.6 the non-linear buckling behaviour is dictated by the action of meridional (or axial) load; and the energy distributions are given by eqns (10); and

(C2) for p/pc>0"6 the non-linear buckling behaviour is dictated by the action of radial pressure, and the energy distributions are given by eqns (9).

It should be noted in relation to eqns (9) and (10) that boundary conditions will not affect the energy distributions as far as their signs are concerned, but only their magnitudes.

General non-linear equilibrium equations

The non-linear equilibrium equations for an imperfect shell are derived by applying the stationary value principle to the functional (6):

Non-linear lower bounds for shell buckling design 109

[ " 1 6(AV)-~6 Ua2+ ~ UM~+tV'~2 =0 (II) i=2

Limit loads A limit point occurring along the above imperfect non-linear equilibrium path, emerging from the unloaded state, is detected by the vanishing condition:

(12)

Bifurcation loads Bifurcation points occurring along non-linear equilibrium paths are detected by extremization of the quadratic functional, 1:2, leading to the critical equilibrium equations (Trefftz criterion):

(13)

Non-linear lower bound energy models

Results from fully non-linear equations (eqn (11)) have shown *-s that the combination of unavoidable initial imperfections and the coupling between various critical and post-critical modes lead to the erosion of the vital stabilizing membrane energy quantities shown in brackets in eqns (9a) and (10a).

In these analyses the relevant imperfections and the critical and post- critical modes are simultaneously considered to trace the equilibrium paths of imperfect shells departing from a stress-free unloaded state and to compute the consequent energy distributions at any load level. The obtained results have shown that in the vicinity of the buckling state (eqn (12)) the combined effects of geometric imperfections and non-linear mode coupling produce a situation where these vital stabilizing membrane energies are greatly reduced even for small imperfection amplitudes. Then, by observing this aspect of the buckling behaviour, and extending it beyond the validity range of the mathematical model (valid only for small to moderate imperfection amplitudes) to imperfection levels around the thickness of thin-walled shells, it is admitted that these stabilizing quantities tend to be completely eroded.

Next, eliminating in eqn (11) the most stabilizing terms indicated in eqns (9a) and (10a), the non-linear reduced stiffness equations yielding lower bounds for-shell buckling loads are obtained:

110 R. C. Batista, P. B. Goncalves

(A) For cylindrical shells and spherical caps under external pressure normal to the reference surface, one has:

6 ( A V , ) = 5 [ U B 2 + U M 2 + U M 3 + ) I / , , t ~ ,r -- M2 + A V M2]=O 114)

when the imperfection amplitudes are small and buckling occurs in short wavelength modes, and:

(151

(B)

when the imperfection amplitudes are larger (around the wall thickness) and buckling occurs in long wavelength modes. For cylindrical shells under axial compression, one obtains:

~(A V* ) ~- ~)[UB2 -~ UM2 21- UM3 + )LV~2 + • V542] = 0 (16)

when the imperfection amplitudes are small and buckling occurs in short wavelength modes, and:

6(AV*)-6 UB2+ UMi+,~VM2 =0 i = 2

(17)

(c)

when the imperfection amplitudes are larger (around the walt thickness) and buckling occurs in long axial wavelength modes. For cylindrical shells under combined external pressure p and axial load P, the following equations are obtained:

(C1) for p/pc<~0.6, eqns (16) or (17); and (C2) for p/pc>0.6, eqns (14) or (15).

Linearized lower bound energy models

With the help of the previously cited fully non-linear analyses, identifica- tion of the characteristics of the nature of buckling as well as the confirmation of the existence of lower bounds are possible. These studies confirm the outcome from earlier linearized lower bound predictions, 13 15 that is, the loss of the initially stabilizing membrane energy in a prospec- tive buckling mode that is responsible for the loss of stiffness associated with the non-linear post-buckling of an unstable shell.

Noticing that many large engineering structures can be expected to contain generally large imperfection levels, around the same level of the shell thickness, and that these overall imperfection shapes usually con-


taminate long wavelength critical modes, it appears that a simpler lower bound model can be obtained by linearizing eqns (15) and (17). This approach leads to the following reduced stiffness critical equations:

(A) For cylindrical shells or spherical caps under external pressure normal to the shell wall:

6(v*)- 6[ua2 +,ivY2 ] =o (18)

(B) For cylindrical shells under axial compression:

6(V*)-,51-UB2 + Uu2 + 2V 2 ] = 0 (19)

(c) For cylindrical shells under combined external pressure and axial compression:

(C1) for p/pc<~0.6, eqn (19); and (C2) for pipe>0.6, eqn (18).

Explicit equilibrium equations

Because of the lack of space, the explicit forms of the non-linear and linearized lower bound equations are not reproduced in the present paper. The interested reader will find them in the works cited previously and enumerated in the reference list.

C I R C U L A R C Y L I N D R I C A L SHELLS

In the following, simple equations or formulae derived from numerical results obtained from the present lower bound models for circular cylindrical shells are presented. These theoretical predictions for each loading case are compared with experimental results and estimates from existing buckling design curves.

Monocoque cylinders under axial compression

It has been shown 4'13 that the lower bound critical stress spectrum has a unique minimum occurring for a mode in which the circumferential wavelength is considerably smaller than that in the axial direction. This has enabled the derivation of a very simple analytical equation for lower bound buckling estimates of monocoque cylinders under axial compression. For thin cylinders the elastic lower bound critical loads obtained

112 R. C. Batista, P. B. Gonfalves

from eqn (19) can be expressed in terms of the Batdorf geometric parameter Z over a wide range of shell geometries:

2frEt 2 2" = q 2 a = q [3( 1 __ V 2 ) ] 1 / 2 v=0"3 120)

where r/, for design purposes, can be simplified into the formulae:

q = 2 " 0 Z -3/11 1 0 2 < Z < 5 × 103 (21a)

=4.0Z -5/14 Z > 5 × 103 (21b)

with:

Z /L '~2R V2) 1/2 (22)

The above formulae can be applied to long and short cylinders of simply supported or clamped ends, being valid for L/R >/0"5.

For thin circular cylindrical shells, buckling elastically, the theoretical capacity reduction factor r/ is plotted in Fig. 1 as a function of the geometric parameter Z. One thing to note is that the present theoretical lower bound is a function of not only R/t but also L/R, while past proposals for design do not adequately allow for the rational inclusion of the L/R ratio. 4'11'12 For this reason they may be over-conservative for

1.0

0.9

o.e

"~ 0.7

o.~

Q ~

0.t

1,'. 0.:3

' q (Eqn . 211

, i , , J , i i i i i i

, ~ • . . . ~ o 5

Fig. 1. Capacity reduction factor F/, for present lower bound estimates of elastic buckling of isotropic circular cylindrical shells under axial compression, as a function of the geometric

parameter Z.


EXPERIMENTAL RESULTS 0 - 1.25 ~ L / R ~ t.50

1~ e - 2.50 ~ L / R ~ 3 . 2 0

O.¢j REDUCTION FACTOR 11 DESIGN CURVES It=

. . . . . SNIP 11-23-61 82

0.1a - ! e e EGGS No 2g 88

• . . . . . . . . . . API RP2A 89 ftASME N284 80| • a~ tR / t : )6001

0"7 1 e ~ e 1 I 1 = 1 i API RP2A G9

° e | (RIt<~61 O, Z ~ 2 1 4 )

a6[ -° ~o o •• ' , ' /

o : Qsl- I~ , .o" o o o

I , b . i % •

• 0 e • ,. o • 0 0 • •

0"41-- ~,~k~: ° ° • BOUND FOR L /R , t.5

/ o O

0.2 " o • o

0 'tO = 2 • t0 s R / t

Fig. 2. Comparison of the present lower bound estimates with collected experimental results and recommended design curves.

shells with small L/R ratios, but, on the other hand, could provide non-conservative design loads if used for L/R ratios greater than those for which the design curves might be appropriate. This aspect perhaps is better shown in Fig. 2 where various theoretical and empirical lower bound curves, including the present theoretical lower bounds for L/R = 1.5 and L/R=3.0, are plotted together with two separate collections of experimental results (one for is•tropic shells with 1.25<L/R<~ 1.5 and another for shells with 2.5 <L/R <<. 3.2) as functions of the R/t ratio. It can be easily verified that the present approach allows for safe and not conservative predictions of elastic buckling loads for any R/t and L/R ratios. Comparisons for other L/R bounds can be found in Ref. 4.

Monocoque cylinders under uniform external pressure

Conversely to the classical theory, the lower bound critical stress spectrum has not a minimum3 'z4 The lower bound critical pressures are then


associated with the critical mode. For thin cylinders the elastic lower bound critical pressures obtained from eqn (18) can be expressed over ~ wide range of shell geometries in terms of the Batdorf parameler Z lbr each one of the classical boundary conditions:

rc2D 2* =t/JLc,= L2 R fl (23)

where D( =Et3/12(1-v2)) is the shell flexural stiffness:

(a) Clamped ends

11'71,2 =~oL 5 0 < Z < 3 X 10 3 (24a)

= lz3 :S 3x 1 0 3 < Z < 104 (24b)

=¼Z 1'2 Z> 104 (24c)

(b) Simply supported ends

~]=3Z1'2 5 0 < Z < 3 x 10 3 (25a)

=1Z3/5 3 x 1 0 3 < Z < 104 125b)

~--5Z1/2 Z > 104 (25c]

In Fig. 3, experimental hydrostatic buckling pressures from various sources 4'x4 are plotted against their corresponding Z parameters and compared with the proposed capacity reduction factors # (eqns (24) and (25)). The experimental results have been plotted separately according to whether the end supports could be confirmed as fully clamped (Fig. 3(b)), or whether they more closely resembled the simple support conditions (Fig. 3(a)). It appears from these results that the effects of the end supports are as important as the influence of the imperfections and that the present reduced stiffness estimates provide, in both cases, reasonable lower bounds of the observed elastic buckling loads of relatively short to long thin shells. Comparisons of proposed theoretical and empirical lower bounds 11,12,14. with the present results are shown in Fig. 4.

Ring-stiffened cylinders

Cylinders with widely spaced heavy ring stiffeners that do not distort under bay buckling exhibit shell behaviour between the rings which is


td

id

SCATTER OF EXPERIMENTAL RESULTS / / /

""'--SIMPLY SUPPORTED ENDS (Eqn 25)

i • , , I , , i i * i , i I i i I . . . .

~ ' id z 1o' (e)

. . . . , . . . . . . . i . . . . . . . .

,d CLA PE0 IE, o ",3

$ i I i | ~ i a , n , , , I a . . . . . . . .

to' z ~o' (b)

Fig. 3. Comparison of the lower scatterband of experimental results with the present theoretical results for thin isotropic cylinders under hydrostatic pressure. (a) Shells with

simply supported ends; (b) shells with fully clamped ends.

similar to that of unstiffened cylinders. For lighter stiffeners, it can be shown 6 that the influence of the rings on the local buckling behaviour and imperfection sensitivity of ring-stiffened shells practically disappears for Z >/10 2. Therefore, the above formulae can be used to estimate the local buckling behaviour of these shells in this range.

Stringer-stiffened cylinders

In spite of some significant analytical developments, 2°'22, and indica- tions of safe limit loads for design, 2a the attainment of simple theoretical lower bounds for stringer-stiffened cylinders under axial compression and


~' .~.p IO z

~ . ~ " ; / " A_N RP2A a9 WEINGARTEN I ET lid_ t't'~-O.B) ]

BOUND FOR SIMPLY ! SUPPORTED SHELLS i

td . . . . . . . . . . . . . . . . . . . . . . . . . . l tO I t0 = t0 3 Z. RL_~ U_ v=~ "= IO*

Fig. 4. Comparison of variously proposed theoretical and empirical lower bounds for cylindrical shells under hydrostatic pressure.

external pressure loading still depends on a clearer understanding of the non-linear buckling and post-buckling behaviour of these shells. Even reliable empirical formulae for the buckling design of this class of shell structures still depends on the acquirement of many more experimental results since little definitive information is presently available.

S P H E R I C A L S H E L L S

A reduced stiffness critical pressure would, for shallow spherical caps under external pressure, be defined as that for which all initially stabilising membrane energy in a classical bifurcation analysis has been eroded. This inextensional approach can be validated through a careful interpretation of refined non-linear analyses and experimental investigations of this problem, s,24

Having once eliminated the membrane energy contributions, one obtains the following expression for the reduced critical load parameter: s

2 E t 2 2* = ~/2c, = r/ [3(1 - v 2 )] 1/2R2 (26)

Non-linear lower bounds for shell bucklino desion 117

where 2c, is the classical buckling pressure of a complete spherical shell,

7.34 q= 22 (27)

is the capacity reduction factor, and

2 = [ 1 2 ( 1 - v 2 ) ] l/4aERt] 1/2 (28)

is the spherical cap geometrical parameter. The capacity reduction factor r/is depicted in Fig. 5 as a function of the

geometrical parameter 2 and compared with the lower scatterband of experimental results from a large number of sources) The proposed capacity reduction factor is compared in Fig. 6 with a number of currently employed codes. 11,12 As shown, almost all important rules for the buckling of spherical caps propose constant knock-down factors in the elastic range independent of the shell geometric parameter 2. This quest for simplicity means that in some geometric ranges the buckling estimates are conservative while in others they are potentially unsafe. On the other hand, the present theoretical capacity reduction factor decreases with 2, yielding convincing and safe lower bounds of known experimental results over a wide range of shell geometries.

b-

z o

,3 Q h i

er

~ 0.~ 0

0

t q / r

O.C

SCATTER OF EXPERIMENTAL

L

G 8 t0 G EOMETRIC PARAMETER- k

Fig. 5. Spherical caps under external pressure. Comparison of the capacity reduction factor with the scatterband of experimental results.

118 R. C. Batista, P. B. Gone'aires

ID C A P A C I T Y REDUCTION F A C T O R - ~ "

. . . . ASME Boiler and pressure vessel code ( T/ " 0 2 0 7 }

. . . . . ASME Code case N - 2 8 4 ( T/ , 0 . t 2 4 ) 8 . . . . . . . BS Rules for unfired pressure vessels ( 7/ . 0 3 2 )

i . . . . . ECCsStruEcUtiorOCne?; re~.oc~nen=tions for steel

G E O M E T R I C P A R A M E T E R - X

Fig. 6. Lower bound design curves for spherical caps under external pressure.

C O N C L U S I O N S

By identifying the components of the energy functional which provide a significant contribution in the initial resistance to buckling deformations, and which are lost due to the combined effects of unavoidable initial imperfections and coupling between various critical and post-critical modes, it has been possible to construct a methodology for the calculation of lower bounds estimates of thin-walled shells.

The importance of the present simplified theoretical approach for buckling design of thin-walled shells is demonstrated by its simple and reliable estimation of buckling loads of circular cylindrical shells under external pressure and/or axial loads and shallow spherical caps under external pressure over a wide range of geometric parameters. The quality of the results is enhanced by the favourable comparison of the present lower bound estimates with a large collection of experimental results and some of the existing design buckling codes.

It is suggested that this methodology can provide a realistic basis for the design of shells liable to buckling containing moderate and practically relevant levels of geometric imperfections.

R E F E R E N C E S

1. Koiter, W. T., Over de stabiliteit van het Elastische Evenwitch. PhD thesis, Delft, Holland, 1945. Translation: On the stability of elastic equilibrium, NASA TT-F-10833, 1967.


2. Thompson, J. M. T. & Hunt, G. W., A General Theory of Elastic Stability. John Wiley, London, 1973.

3. Budiansky, B., Theory of buckling and post-buckling behaviour of elastic structures. In Adv. Appl. Mechan., Vol. 14. Academic Press, New York, 1974.

4. Batista, R. C., Lower bound estimates for cylindrical shell buckling, PhD thesis, University College, London, 1979.

5. Antonini, R. C., Influence of modal interaction and geometric imperfections on the buckling of axially compressed cylindrical shells. MSc thesis, COPPE/UFRJ, Rio, 1981 (in Portuguese).

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Non-linear lower bounds for shell buckling design

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