Axisymmetric snap buckling of imperfect, tapered circular plates

Compum @ P~mon

a Smcrrns

Vol. 9. pp 551-558

Press L:d.. 1978.

Printed in Great Britain

AXISYMMETRIC SNAP BUCKLING OF IMPERFECT, TAPERED CIRCULAR PLATES G. J. TURVEY Department of Engineering,University of Lancaster, Bailrigg, Lancaster LAI 4YR. England (Received

12 Deecmber

1977: received for publication

3 April

1978)

hstraei-The application of the Dynamic Relaxation method to the analysis of the axisymmetric snap buckling behaviour of uniformly loaded, imperfect, tapered circular plates is briefly described.It is then used to carry out a limited study of the effects of: (1) imperfection magnitude,(2) thicknesstaper ratio and (3) support conditionson the plate response.Two types of responseare distinguished:(a) “snap-through”and (b) “smooth-transition”.The former type of responseis found to occur only when the imperfection amplitude is sufficientlylarge. Furthermore. the results of the study show that decreasingthe taper ratio and clampingthe plate boundaryboth serve to raise the snap buckling load and that the latter measure is relatively more efficient in this respect.

NOMENCLATURE

The present study, like that of Ref.[ 111, is concerned with the response of uniformly loaded, tapered, imperfect plates in the large deflection regime. Whereas in the earlier work, the real plate imperfections were idealised in the form of a single-wave, which was sympathetic to the direction of loading, the imperfections used in thepresent context, though of a similar form, are antipathetic to the applied loading. Thus as the lateral pressure is increased from zero, the plate begins to flatten and undergoes compression. If the compressive stress exceeds a certain critical value, which depends on: the magnitude of the imperfection, the plate taper ratio (defined in the next section) and the boundary conditions, then the plate may exhibit a type of instability known as “snap-through” behaviour. The primary object of this paper is to examine the effect or influence of the imperfection magnitude, taper ratio and the boundary conditions on the plate response. However, in order to limit the com@ttational effort, but nevertheless provide results of practical significance, only single-wave type imperfections, linear thickness tapers and clamped and simply supported boundary conditions are considered. These are discussed in greater detail in the next and subsequent sections of the paper.

plate diameter extensionalstiffness flexural stitfness radial and tangential strains Young’s modulus plate thicknessat r = 0 plate thicknessat an arbitrary radius, r radial and tangentialcurvatures radial and tangential stresscouples radial and tangential stressresultants lateral load dimensionlesslateral load arbitrary radius reference radius in-plane displacement total deflection dimensionlesstotal deflection initial deflection initial deflection at the plate centre

dimensionless initial deflection at the plate centre plate thickness taper ratio

u
Poisson’sratio membrane stress at the plate centre dimensionless membrane stress at the plate centre bendingstressat the plate centre

0.

dimensionless bending stress at the plate centre differentiation with respect to r

kcb(= ~,~r~~E-‘h,-~)

tNlTlAL-ON is assumed that the plate possesses an initial imperfection in the form of a single-wave, which is antipathetic or convex, when viewed in the direction of the applied lateral pressure. This imperfection is described by the following equation, It

LNTRawmoN

The continuing trend towards the use of thinner plate than hitherto in steel structures has accentuated the need to base design on large deflection and postbuckling theory and to take account of initial imperfections. Whilst there is a considerable literature on the large deflection and postbuckling response of imperfect plates (see, e.g. Refs. [l-lo]), it pertains almost exclusively to plates of uniform thickness. This situation is unsatisfactory from the point of view of the designer, since the use of variable thickness or tapered plates may be justified. Circular plates are widely employed in structural fabrications and it is, indeed, not unusual for them to be tapered in thickness in order to satisfy a particular design criterion. However, it is only recently that large de&&on results for variable thickness, imperfect, circular plates have become available to the designer [ 111.

W0= wocos (&I)

(1)

in which w. is the imperfection amplitude at the plate centre and r = O+fa. Three values of the dimensionless imperfection amplitude (w,Jh,) viz: - 0.5, - 1 and - 2 have been selected for the parameter study presented herein. (N.B. the negative signs indicate that the imperfection isantipathetic-positive signs referring to those that are sympathetic). TAIglltATlO

of the high cosrof machining operations, it is only likely that step or linear tapers would be employed in practice. Therefore, the present study concentrates on only one of these, namely the latter. It is convenient to

551

In view

552

G.I.TURVEY

describe different degrees of taper in terms of a parameter, a, which will be referred to as the taper ratio. Accordingly, the plate thickness at an arbitrary radius, r, is given by the following equation, h, = hdl - Zara-‘)

in which A, = Eh,(l - v2)-’ and D, = A&*/12. 3. Compatibility equations

(2) e,= r-l14

in which ho is the thickness at the plate centre and positive a values describe plates. which reduce in thickness from the centre outwards and vice versa. Again, three values of the parameter, a. namely:-a,0 and +b have been chosen for the purposes of the parameter study, since they span the likely range of practical taper ratios.

UWMMEIWCIWERFET

cIRCuLAR PLATE eQuATiONS

The nonlinear equations, which describe the large deflection and/or “snap-through” response of the plate are presented here not as a pair of coupled, nonlinear, ordinary differential equations, but rather in their separated forms of: equilibrium, constitutive and compatibility equations. The reason for this is that they are utilised in the solution procedure, Dynamic Relaxation (DR). in precisely this form. The three sets of equations ar egiven below as, 1. Equilibrium equations

k,zmwr" k, = -r-‘w,

BOUNDARY CONDITIONS I.

Plate centre (r = 0)

Since the behaviour of the plate is assumed to be axisymmetric regardless of the magnitude of the applied loading, then all the quantities: w, M,, A4,. N, and N, exhibit axial symmetry with respect to the plate centre. The remaining condition at r = 0 is that the in-plane displacement vanishes so that, u = 0.

M;‘+r-‘(2M;-M;)+N,(w”+r-‘w’)+N;w’+q=O

(6)

2. Plate edge (r = ia) Both clamped and simply supported conditions are considered along this edge and in each case, full in-plane restraint is assumed. Thus the simply supported condition implies, u= w,=h4,=0

N;+r-‘(N,-N,)=O

(5)

(7)

and the clamped condition implies, (3)

u=w,=w;=o.

(8)

in which w = w’t w, and wI = additional deflection, 2. Constitutive equations

SOLUnON PROCEDUILE

&, = A,(e, + ve,) N, =A,(ve,+e,) M, = D,(k, t vk,) M, = D,(vk, + k,)

(4)

There are a number of methods which might be employed to solve eqns (3)-(g). In this instance, the author has chosen to make use of the DR method, which is basically a finite difference iterative procedure, because it is easy to code and is well suited to this type of problem. No details of the method are given here, since it is adequately described elsewheref12], but the

(a)

Axisymmctric

snap buckling of imperfect.

tapered circular plates /

Fig.

I. Central deflection vs lateral pressure for simply supported and clamped circular plates (a = - 0.5.0 and +0.5). (a) GQ= - 2, (b) & = - 1 and (c) GJ*= - 0.5.

present application does make use of ratio&y

.&culated fictitious densities[l3] in an effort to optimise the large deflection and postbucMing computations.

PROGRAMklmFlCATtoN

Before considering the principal features of the compu~d results in detail, it should be emph~ised that although the problem under consideration bears a strong similarity to that of a spherical cap subjected to a uniform radial pressure, it is nevertheless a different prob. lem by virtue of the shape of the initial imperfection and also the direction of the lateral loading. These dissimilarities have ted to some ditficulty in locating suitable alternative solutions for program verification purposes. Indeed, the author has been unable to find such solutions. A~o~~y, the onty program check has been to run the program for a number of positive imperfection values and reproduce selected results of Ref. II 11.

Throughout the study, the compu~tions have been carried out with respect to a I@ interval interlacing finite diffcren~e mesh extending over the plate radius. Previous work has shown this degree of mesh refinement to yield results, which are suficiently accurate for engineering applications. The parameter study is restricted in the sense that only one load case and a very limited range of plate geometries and im~~ection amplitudes is considered. It is further restricted in so far as the quantities of interest, namely the deflection, membrane and bending stress at the plate centre are traced for positive load in~~ments only. Unloading paths are not speci&ally studied, since they. are considered to be practically less significant. Nevertheless, for those cases of the present study, which do not exhibit “sn~~ou~ behaviour, the htading and unloading paths are identical. Figures i(a)_(c) present plots of the plate centre

554

G. J. TURVEY

(b)

S-

,(I) DC’ 0.5.

b--

3-m B

(11

5hPlY

(2)

Clomprd.

I -t-_,,

I

I

,’ f / ’ IJ I/ I /I ’ I 1 i ’ / ,I I /

/I / / I' ' ' 3' '/ '/ /'J I r-T 0 10

l-

----a-_

,‘;z1 I

~Pport*d.

2-

0.

G-2

-C

-4:

;

q

(c)

I loo

Fig. 2. Central membrane stress vs lateral pressure far simply supported and clamped circular plates (; n = - 0.5. [b) (I = 0 and (c) a = + 0.5.

= - 2). (a)

Axisymmetric

snap buckling of imperfect,

i&B--05

ot-45, 4

(1)

Simply

tapered circular plates

Supported

(a)

5r (2) 4-

3-In QL

(1)

SWY

(2)

Clanped

Suwott~

2-

W l-

I xx)

-1

-

-2-

5r

X2) I

v&-

oc-05.

I

-05

4

i I

t 3-

(0

simply srpport@d

(2)

clamped

I I

-

-m T 2-

I’,/ ’ /

(c)

,/',/

l-

_* OF=

(1) /Ii ‘/ ,/1 ,’ /’ ’ /’

_-=_e-=___-' 1

*MC _Y_'

' I' /' ,' ,' I lo

';

I loo

I

-1

-2L

t

Fii. 3. Central membrane

stress vs lateral pressure for simply supported and clamped circular plates (I&, = -0.5). (a) o = -0.5, (b) 0 = 0 and (c) (I = +0.5.

G. J. TURVEY

556

/-

4L'-@5+~--2 12 -

lo-

T.--_---(2)

I (1)

simply

(2)

Cl~

sup(rortcd

,A

1 r--L--

-’

8-

q.” (al

6-

6

(b)

qb c

I

n

I

loo

i

1

=-05.wo--2

(1)

Sinply

(2)

ClamFed

,(2) /‘,’

SuFported _. r-_-----c_. eM

i-

-2

--r

J

lcoo

I I_/* ,

.' .'

/'/ill

/',' ,I,' /'

(c)

*-

L

Fig. 4. Central bending stress vs lateral pressure for simply supported and clamped II = -0.5, (b) (I = 0 and (c) (I = +O.J.

circular

plates

(3 = -2).

(a)

Axisymmetric

snap buckling

of imperfect.

tapered circular

7-

plates /cl)

/RI /

=--M,W--05 6-

/

//

S-

(1)

Simply

(2)

cm

./'

Swtaortcd

/ /I

,.'

L' ,//' 9:

,,/'I.

3-

./

f

(a)

.’ ’

/’

/

2-

/ /

ii A.'

/

l/

./'

,

/

,

_,_.=<--

y

0

04 ,

I

1

I

l0

x-0,

I

lo0

'i

xxx)

w;-o-5

6

5

(1)

Simply

Supported

(2)

Clamped

4

4” 3

(b)

2

1

0

1

10

ac- 0.5

6-

5-

(1)

SimplYsuppwtcd

(2)

CkIwed

3

I

100

( w--o.5

loo0

/(2) 'Al) I/ '/ '/

//

2

I’,’ ,‘/’

4-b 7

/ /’

,y ,/

3,’ ,’ /’ .’ , .’

/’

2-

.’

/’

/‘/I / ’

/’

w

,’ ,’

_@

,s’_.’ l-

c/'

,,-5 Oh 01 Fii. 5.

/'/ .L--

,‘/ 4H

e-3 /' 1'

I

I

1

x)

v

1 100

Central bending stress vs lateral pressure for simply supported and clamped circular 4 = -0.5, (b) = = 0 and (c) (I = +0.5.

CASVd. 9 No. bc

I tO0

plates (I?,, = -0.5).

(a)

G. J. TURVEY

558

deflection for imperfection amplitudes of -2. - 1 and -0.5 respectively. These figures show that the “snap througfi” behaviour is accentuated as the imperfection amplitude increases. Indeed, it is necessary for the imperfection amplitude to exceed a certain minimum value for “snap-trout’ to occur at all (see Fig. Ic). These figures also demonstrate that the snap buckling load may be raised by clamping the plate boundary and/or decreasing the plate taper ratio. In general, it appears that the former choice would be more efficient. In Figs. Z(a)-&) the central memb~ne stress is plotted for plates, which exhibit pronounced “snap-through” behaviour. The two curves on each of these iigures show the same genera) trends, namely that the membrane stress is ini~~iy compressive. This compressive stress gradually increases in value until the snap buckling load is reached, whereupon it drops rapidly in value-indicating a dynamic change in stress state-and then rapidly becomes tensile. It may also be observed that the compressive membrane stress increases more rapidly in simply supported plates, but is of greater rn~nitu~ in clamped plates. These stresses also increase with decreasing taper ratio. After “snap-through” and for the load range considered, the ten&e stresses are lower in clamped plates. By contrast, Figs. 3(a)-(c) show the variation in the central membrane stress for plates which do not “snap throu%” (4 = -0.5). These curves are smooth t~ou~out the load range considered-the membr~e stress being initially compressive and then rapidly becoming tensile as the lateral load increases in value. However, for this case, the compressive peak value of the memb~ne stress appears to be practic~y independent of the taper ratio. The bending stress at the centre of the plate is plotted in Figs. 4(a)-(c) for plates with an imperfection amplitude of - 2. The curves show large changes in stress abode, which are associated with “sna~t~ou~‘. Some of the plates, notably those with a negative taper ratio, exhibit negative bending stresses prior to “snapthrough”. This appears to be associated with the development of a local flat~~g of the plate. At loads slightly in excess of the snap buckling load, the bending stress (now positive) reduces moderately, before beginning to increase again. The final set of figures, Figs. S(a)-(c), depict the plate centre bending stress for plates, which do not “snap throu& (A= -0.5). In a# cases, the stress is positive throughout the load range and at the lower load levels. the clamped plate bending stresses are always smaller than those of the simply supported plate. This is because the antipathetic initial imperfection is more readily overcome by the applied loading in the latter case. CONCWSIONS

main conclusions of this study, which has considered the magnitude of the antipathetic initial imperfection. the plate thickness taper and the plate boundary The

conditions on the “snap-through” response, may be summarized as follows: (1) “Snap-through” behaviour occurs provided the magnitude of the initial imperfection exceeds a certain value. For the plates considered herein, this value lies in the range, -0.5 to - 1. (2) The snap buckling load increases as the magnitude of the initial imperfection increases and the taper ratio decreases. (3) Clamping the plate boundary raises the snap bucking load and is more e&ient in doing so than decreasing the plate taper ratio. (4) The membrane and bending stress at the plate centre are characterised by targe changes in magnitude and/or sign at the snap buckling load. Otherwise, when the imperfection amplitude is of insufhcient magnitude to permit “snap-through”, they are smooth functions of the lateral pressure. Ac~~wi~g~~ts-~e author wishes to record his indcbtedness to the Department of Engineeringfor providing computing facilities and also to his father, Mr. George John Turvey. for his skill in+e preparationof the figures.

I. J. M. Coan, Large-deflection theory for plates with small initial curvature loaded in edge compression.3. Appl. Mech. u&Z),143-151 (1951). _ i. N. Yamaki, Post-buckling behaviour of rectangular plates with small initial curvature loaded in edge compression.i. A@. Me&. 27(2), 335-342 I196OJ. 3. H. Nyiander, initialty detIected thin plate with initial deflection atline to additional deflection. Inf. Associution Sri&e Strucruti Engng 11,3&3?4 (195It. 4. H. Nylander, Die Durchschbaglast van Platten. dstemichisches Ingenieur-Archiu 9(2-3), 181-l% (19SS). 5. 1. Hlavacek, Einguss der Form der A~~gs~mmung auf das A~~~t~ dtr g~~kten rtcht~~gen Platte. Afro Technica CSA V 7(t), 174-205 (1962). 6. A.. C. WaJker. The post-buckling behaviour of simplysupported square plates. 77th~ Aeromurical Quunedy B(3), 203-222 (1969). 7. T. M. Roberts and 5. G. Ashwtll, The use of finite element midincrementstiffnessmatrices in the post-bucklinganalysis of imperfect structures.Inr. 1. So/ids Sl~crures 7(7). 805-823 (1971). 8. K. R. Rushton, Large degection of plates with initial curvature. Int. I. Me& sci. lZ(12). 103?-I051 11970). 9. K. R. Rushtan. guckiing of lateralty loaded plates having initial curvature. 1st. J. Mech. Sci. 14(10). 6fJ-680 (1972). 10. R. G. D~WSOII and A. C. Watker. Post-bucklingof geometrica& imperfect plates. Froc. Am. Sot. Cio. Engn. f. Structurd Dia 9qsm), 75-94 (1972). 11. G. J. Turvey, Variabk thicknessimperfect circular plates at large detlections. Proc. Conj. Non~Linaar Pmblenk Stress Analysis. UtIiVCrSitY of I)urham (&nt. 1977). 12. A. $. Day, An imroduction to dynamic relaxation. The Engineer 219(l), 21~221 (l%S). $3. A. C. Casseil and Hobbs R. E.. NumericaJ stability of dynamic relaxation analysis of non-linear structures. int. I. Numer. Merh. EngRg 1q(6), 1407-1410 (1976).