Opblazen lantaarnpaal

Opblazen lantaarnpaal

Citation for published version (APA):

Vosmer, J. (1982). Opblazen lantaarnpaal. (TH Eindhoven. Afd. Werktuigbouwkunde, Laboratorium voor mechanische technologie en werkplaatstechniek : WT rapporten; Vol. WT0537). Technische Hogeschool Eindhoven.

Document status and date: Gepubliceerd: 01/01/1982

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376 _{J. E. Harding and P. J.}

DOIl'ling

7. HARDING, J. E.,

H~B8S,

R. E. and NEAL, B. G., Ultimate load

behavi~ur

plates

un~er combm~

direct and shear in-plane loading, Steel Plat:; Structures, an InternatIOnal Symposium Crosby Lockwood St I L d

1977. , a p es, on on,

8.

HARlDINSG'bJ·~·

and H ODBS, R. E., The ultimate load behauiour o/box girder web pane s. u mJUed for publication.

9. HORNE, M. R., DOWLING, P. J. and OGLE, M. H., Re ort toSteerin G

:~~l:~::rr;;~~es,

Committee B116/3, British StandIrds Institutio;'

L:'fo~:

21

A Design Approach for Axially Compressed

Unstiffened Cylinders

R. C. BAT/STAt

Unh'ersitade Federal do Rio de Janeiro

AND

J. G. A. CROLL

University Col/ege, London SUMMARY

The classical critical load analysis of circular cylindrical shells under the action ofuniform axial compression is re-examined. It is shown, through the inspection of the potential energy, which of the components of the membrane stiffness are most likely to be eroded during the imperfection-sensitive, non-linear mode interaction that occurs in the buckling of these shells. Based upon this reassessment a simplified critical load analysis is outlined which, in contrast with the classical theory, is shown to predict a critical load associated with a unique critical mode that appears to agree with the buckling modes observed in both the present experimental programme and those described in the literature.

Apart from providing close agreement with observed buckling mod~s, the simplified theory is shown to have other advantages which should commend it to the designer. First, in contrast with other theories to account for its 'perplexing behaviour', the present simplified theory does not involve the lengthy solution of highly non-linear equations. Second, and possibly most significant, when proper account

is

taken of all the appropriDte parameters describing the behaviour of the shel" the method provides lower bound estimates of reported buckling loads. Properly extended therefore, the method could provide the

basis

of a simple but safe design procedure.

t

Presently engaged in research towards a Ph.D. at University College. London.

378 R. C. Batista and J. G. A. Croll

INTRODUCTION

~he axially compressed circular cylindrical shell, although one of the slI?plest ge.ometric forms, and under the action of such a simple

aXlsymmet~lc load, displays a buckling phenomenon whose physical understandmg and theoretical solution still present formidable obstacles. !hese o~stacles, and the evident importance of cylindrical shell structures

In the al~craft and space industries, and more recently in the marine platform md~stry, have led to considerable scientific effort being directed towards findlOg an explanation of its somewhat 'perplexing' behaviour. Many papers have been written covering both experimental and theoretical

treatm~nts, but ~t is probably still true to say that none can yet be considered to provIde a satisfactory solution. Two excellent reviews of the subject are

~ found in refs. [1,2].

f Th: discrepancies between the classical analytical predictions and the \

' ex~nme.!l1a1 values, and the scatter of these experimental results are enormous.

or

the factors that have been advanced as possible reasons for both the discrepancies and scatter, the effect of initial imperfections has come to ~ acce~ted as the major cause. There seems to be general \.. a.gree~ent tn the hterature [3-6] that end conditions (at least for practical

situatIOns). have a negligible influence on the experimental buckling load and behaViour of a circular cylindrical shell under axial compression. This has been substantiated by recent theoretical investigations [7,8I, which show that when the shell behaviour is dominated by the influence of initial iz:'perfections, the effect of different boundary conditions (provided the circumferential and radial displacements are supressed) can be disregarded in the analysis.

Nevertheless, the studies that have been made on the effect of initial imperfections, although having achieved considerable success [7,8), would seem to be unsuitable for incorporation into engineering practice. In reference. [81 the. relevant experimentally measured initial imperfection modes WIth amphtudes smaller than the shell's thickness were taken into acc0ll;nt in a theoretical analysis, and good agreement between expenmental buckling loads and the resulting critical loads were obtained But

applic~tion o~

this det:rministic approach is currently limited by

th~

lack of avaIlable tmperfection data corresponding to various fabrication methods. Even when measurements of imperfection spectra of fun scale structures are ~\'ailable, it is far from clear how the specifications of design tolerances, which may r~uire consideration of the imperfection-sensitivity and control of many different possible combinations of imperfection

A Design A.pproach/or A.xially Compressed unstiffened Cylinders 379

modes, will be made available to engineering prac:ice.

S~

that in the absence, in the literature, of simple and reliable theoretical

estlm~tes

for the buckling load of axially compressed thin cylindrical shells, engmeers to a large extent still rely on empirical formulae. . ' ..

Due to the expected difficulty in controlling

a~d meas~nng

the Imttal imperfections and pre-buckling deformations whIle carrymg out tests on full-scale structures, and also because of the

difficultie~

of a

subsequ~nt

analysis that takes into account gross initial

imperfectto~s

and couphng between modes, alternative lines of research have been

d~rected

towards solutions that could provide lower bounds for the

expenm~ntal

results. Some attempts [9,10] have been made in this direction

~~

usmg advanced post-critical calculations to predict the minimum

~o.st-cntlca~

loads, and to use these as the lower limits for imperfection-sensitive

bu.c~hng

loads .. But based on ref. [II] it has been argued [1] that these post-cnttcal calculations are not as relevant as was at first thought. Not least of

th~

probl:ms associated with this method is the exceptionally.

difficul~

analYSIS, posslbl.y requiring consideration of very many

hi~~ly mt:r~ctlve

modes, that IS

needed reliably to determine these post-cntlcal

ml~lmum

loads.

To overcome this practical obstacle an alternattve approach

h~s .be~n

proposed [12,13], which attempts to define the physical

character~St1cs

In

an advanced post-critical state so that they can be used as the baSIS of. an equivalent eigenvalue analysis. The approach is.based

~n

the

~~neral

notIOn that for shells buckling into modes that denve thetr stabthty from the pres:mce of significant membrane energy in the

cri~ical.mode,

any

eff~ct

which undermines the membrane stiffness would gtve nse to substantial reductions in the buckling loads. These reductions in

buckli~g

loads. would be due to the tendency for both imperfections and mode mteractlons to eliminate the membrane energy. In ref. [121 this approach

w~s

interp:e.ted as simply setting the whole of the

membra.n~

strain .energy In

~he cnttc~l

modes equal to zero, which implies a

quasl-mexte~slonal solutIO~

of

.th~s

problem. For the case of the axially loaded cybnder such a

slm~hstlc

approach clearly could not account for many of the known behaVIOural

characteristics. . ' .

The following, then, is an outline of a more systematic

apphca~tOn

of thIS general philosophy to the analysis of the axially loaded cyhnder. The classical critical modes are first re-examined to isolate those

~omponent~

of the membrane stiffness most likely to be

erod~

in the

non-l~near

buckhn.g response. Based on this, a simplified analysts proposed 10 ref. (14

1

1S

developed and shown to compare

fav~urably

with

~he

results .of new

.an~

previously reported experimental evtdence for thiS shell. Fmally, It IS

380 R, C. Batista and J,G, A. Croll

suggested how the simplified method co Id .

for future design of unstiffbned I'd-u prOVIde a. safe and rational basis cy m ers under aXIal load.

CLASSICAL CRITICAL MODES RE-EXAMINED The energy criterion requires that f, .•

exist. the total potential energy

V

or

s~a~lbty

of a structural system to external energy shall be . ' , ' conslstmg of the sum of internal and displacement

co~ponents ~;~nru:

at the state of

e~uilibrium.

If the configuration of the shell

whoseest~~'I~m~ntalbe

st~te

(th,at IS, the equilibrium and the new configuration at I I bY .IS to

l~vestlgated)

are called, u

F

h

some ar Itrary neighbou . . . F '

W _ ere u are kinematically admissibl .

nn~

state, U'

+

u,

follows that for stability e smalllOcremental dIsplacements, it

V(uF

+

u)

>

V(uF)

U ' (1)

SlOg Taylor's expansion th h

written as eorem, t e left-hand side in eqn. (I) may be

V(uF

+

u)

==

V(uF₎

₊

_t:5V

₊

_t:52_V

_{+ ...}

₌

_-

_VF

₊

_VI

₊

_V

2

+ . . .

(2)

where, VI' V_{2 " ••} ,arethefirst second . ,

e~ergy,

that are linear, quadratic .' ....

~anatlons

of the

to~al

potential displacements, u, and their

derivati~es'

., with respect. to the Incremental as parameters, ' and have coeffiCients that contain uF

In the following, the fundamental stat f .

cylindrical shell is taken as a memb eo an aXially compressed circular of the load arameter A and th rane ,state so that u

F

is a linear function can be rewntten as [' : 16] e expansIOn (2) for the present loading case

V[uFO.)

+

u1

=

_-

V[UF(A)]

₊

r':( I U)+AV, ₁(U)+J!i(U)+A Vi(u)+"1(u)+V:(u)

(3)

where, the constant term

V[uF(..t»)

state. As the original configurat'

~longs

to the fundamental membrane Ion IS a state of equilibrium

. , VI (u, A)

==

J-1(u)

+

..tV~

(u) = 0 '

which IS satisfied by the membrane solution (4)

A necessary condition for stabilit . hat .

potential energy must be

y~st

the second variation of the total non-negattve, requiring

V2(u, A)

==

J!i(u)

+

AVi(u) :i!: 0' (5)

A Design Approach/or Axially Compressed Unstiffened Cylinders 381

C"

The critical case of equilibrium occurs if Vz(u,l)

==

0, for one or more linearly independent displacement fields,

u.

The condition of stationarity at

the critical point, (6)

yields the eigenvalue problem whose solution gives the critical modes,

u

=

u·, and the critical load coefficients )'c'

As the energy criterion for elastic stability is based on potential energy considerations, it will assist later discussion to observe how the different parts oBhe energy contribute to the second variation of the total potential energy aboutthecritical point.

V

2(ue, Ae}

==

O. Thus, breaking

V

z down into

its constituent parts, this critical condition may be rewritten as

V₂

==

VlD

+

Vi~

+

ViM

+

~M

==

0

or more completely

V

2

==

Via

+

ViD

+

U2~

+

Vi~

+

ViM

+

Ae

ViM

+

ViM

+

A.v2~

==

0 (7)

where,

U~a. ~D' U2~,

are respectively the axial, circumferential and twist bending strain energy contributions, ViM'

U~M' Ui~

are the axial. circumferential and shear membrane strain energy contributions, and

ViM'

J1M

are the axial and circumferential membrane potential energies consisting of the two parts

ViM ::=

U1

M

+

A. V Z

1

M

V'iM

::=

ViM

+

),.v;6M

In these membrane potential energies,

Af

Vi"M,

A.

V

Z8M _{are the axial and}

circumferential components of what is sometimes referred to as the load potential.

For the present loading case it is well known that the condition of stationarity (6) yields the classical eigenvalue problem, associated with the quadratic form in the identity (7), whose solution shows that many very close gitjcalload coefficients,

A.,

occur for

~ily

of critical modes.

~.

Investigation of the energy contributions for all these cntlcal modes has been made in ref. [14], but for reasons ofspace only the most salient features of this analysis are presented here. In Fig. 1 the energy contributions for critical modes with one half axial wave

U

=

I) are plotted as a function of

the circumferential wave number, i. This shows that:

(a) the membrane energy is largely dominated by its axial membrane strain energy contribution,

Uh..

with the circumferential membrane strain energy.

ViM'

negligible;

382

R. C. Batista and J. G. A. Croll

w

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9

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_lkl1.v2j.l03 C REt I

-2

.

-10

.

t

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_{, \}

7

, II -11

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1

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-12

FIG. I. Energycontribuf 7)

Ions toeqn. ( .j = I half axial wave; sheUgeometry L/R"" 2'88

R/I '= 300. ' .

A Design Approach for Axially Compr~ssed Unstiffened Cylinders 383

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O.

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r

umber of clrc~f. waves· .i.

I ,

-10

A

=

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O.

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3!

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I-Flo. 2. Energy contributions to eqn. (7). j = 20 halfaxial waves; shell geometry, Li R ~ 2 88,