Thin-Walled Structures 5 (1987) 145-155

Stresses in Thin Spherical Shells with Imperfections. Part II: Influence of Local Imperfections*

Luis A. Godoy and Fernando G. Flores

Departamento de Estructuras, F.C.E.F. y N., Universidad Nacional de Cordoba, Argentina

(Received 16 April 1986; accepted 22 August 1986)

ABSTRACT

The changes in stress resultants in thin spherical shells, associated with a local imperfection introducing curvature errors in all directions, are investigated. An axisymmetric finite element model of the shell and imperfection is employed to carry out the linear elastic analysis. Parametric studies have been performed, to identify the main parameters controlling the response, for the case of internal pressure. The results are compared with those obtained in Part I for axisymmetric imperfections, and bounds for maximum elastic stress resultants are established to cover the possibility of both local and axisymmetric imperfections.

NOTATION

mit, mf2 Changes in bending moment resultants due to imperfection. nit, nf2 Changes in membrane stress resultants due to imperfection. p Internal pressure. r Radius of spherical shell.

Radius of imperfect shell. r/ t. Shell thickness. Angular coordinate along parallel circle.

*Part I appeared in Thin-Walled Structures,S (1987) 5-20.

145

Thin-Walled Structures 0263/8231/87/$03·50 © Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain

146 Luis A. Godoy, Fernando G. Flores

OJ Angular extent of imperfection. ~ Amplitude of imperfection. ~; Maximum amplitude of imperfection. ¢> Angular coordinate along the meridian. ¢>; Angular extent of imperfection.

1 INTRODUCTION

The stress redistributions that occur in thin spherical shells due to the presence of axisymmetric imperfections (AI) were investigated by the authors in Part 1. 1 Although important for the conceptual understanding of mechanical behaviour, a purely axisymmetric deviation from perfect geometry is unlikely to occur in practice, and in most engineering situations an imperfection introduces local curvature errors in two principal directions. However, the question remains in what respect the stress fields due to AI and local imperfections (LI) will differ from each other.

Flugge's studies on imperfect spheres concentrated- on a banded imperfection. For local rather than AI, Flugge assumed that 'the essential features of the stress disturbance will be similar' . 2 However, it will be shown in this paper that some essential features are very different, i.e. the linear relationship between amplitude of imperfection and stress resultants does not hold true for local imperfections.

Because of the computational effort required to study stresses in locally imperfect shells, only a small number of results have been published in the literature, and they were obtained in the context of cooling tower shell analysis. Ellinas et al. 3 and Langhaar and Boresi4 investigated the influence of imperfections in which the circumferential curvature error dominates the meridional one. For the mathematical modelling of the problem, an equivalent load techniqueS has been proposed as a simplified way of performing the linear analysis in imperfect shells and reducing the computa­tional effort. The technique has been validated for banded imperfections, but further research is needed to investigate the limits of applicability for non-axisymmetric imperfections. Comparisons between a finite element discretization of an LI and the corresponding equivalent load representation have been presented for cooling towers. MlThe results show that for LI larger than the shell thickness, a first order equivalent load is not adequate, due to the interaction between both curvature errors. For LI with amplitude equal to twice the thickness, a second order equivalent load produces good estimates of membrane stress resultants, but the banding moments have errors of about 80%. And for imperfections equal to four times the thickness, even the membrane stress resultants show significant errors.

147 Stresses in spherical shells with imperfections. Part II

Thus, it seems that the applicability of the equivalent load technique is limited to LI of small amplitude.

In view of the above, a direct, rather than equivalent load representation of the imperfection has been used here; and the influence of LI in spherical shells is examined using the finite element method to model the geometry of the imperfect shell.

2 FINITE ELEMENT MODEL OF THE SHELL

The radius ri of a shell with an imperfection of amplitude gis given by

(1)

As in Part I, a cosine function has been used to model the imperfection, but now in the form

~i [ (2mp)] [ ( 21T(J ) ] (2)~ ="4 1+ cos --;;;: 1+ cos T

in which cf>i is the central angle of imperfection in the meridional direction, Oi is the angle in the circumferential direction, and gi is the maximum amplitude of imperfection.

Some restrictions on the load and imperfection considered have been adopted in the present work in order to simplify the studies on locally

Fig. 1. Notation and positive convention for spherical shell with local imperfection.

148 Luis A. Godoy, Fernando G. Flores

imperfect spheres. First, the loads are assumed to produce uniform pressure normal to the shell. Second, the LI considered has the same curvature errors in all directions. For the class of load and imperfection described, the shell can still be analysed as a shell of revolution, with the advantage that the finite element code used in Part I can also be applied to the stress analysis of locally imperfect spherical shells. Figure 1 shows the basic parameters used in the definition of the geometry of an imperfect sphere. The type of imperfection studied in Part II is representative of local damage that is likely to occur in the shell after the construction is completed.

3 STRESSES IN AN IMPERFECT SHELL

To highlight the main features of the behaviour of a spherical shell with LI, a particular shell with r = 28 ()()() mm and t = 30 mm has been analysed, with an imperfection of amplitude ~i = - t and central angle CPi = 30°. In Fig. 2, membrane and bending stress resultants ni2 and mil are plotted. Because of

nilkN/mmJ m~ fkNmm/mmJ-0.2 -0.1 0 0.1 0.2 o 0.1 0.2 OJ r---~----~~~~--~

I I I

I I

I /

i / .I /

/ .I/o

/!//

/ / 10·/ i

I I I i : i \ \.

" \."'.'\ ~.~

ZO°'~()Dmm I 56000 mm. q,

(a) (b)

Fig. 2. Changes in stresses for a spherical shell, r = 28000 mm; modulus of elasticity E = 2·1 X 105 N/mm2; Poisson's ratio v = 0·3; P = 0·05 N/mm2; with inward local imperfection CPi = 30°. (a) and (b) n22 and mit at different levels; -- {i = - 3t; - - ­

{i = - 2t; - . - . - {i = - t.

149 Stresses in spherical shells with imperfections. Part II

the symmetry in load and geometry, the other resultants ntl and miz yield the same values as n!z and m'l\ in Fig. 2. For this rather long imperfection with gi = - t, the membrane stresses and moment resultants follow a banded pattern, much in the same way as in the AI, but with higher values. For example, the maximum values for niz are 57 and 53 N/mm; and for mtl are 73 and 97 kN mm/mm for the LI and AI, representing differences of7·5% in niz and of33% in mtl'

But if the amplitude of the LI is increased fromtto 3t, Fig. 2 shows that the stress and moment changes are not linear with gi' For example, this increase by a factor of 3 in the amplitude of the imperfection produces increments in the maximum values of n!2 of 258%, and of 404% for mil' This nonlinear dependence of stress changes on amplitude of imperfection will be seen to be even more severe for short imperfections. Note that an equivalent load

. model would be unable to detect this kind of nonlinearity. The stress changes are strongly dependent on the wavelength of the LI.

This is illustrated by the studies shown in Fig. 3, in which the central angle 4>i is increased from 3° to ISO. The distribution of n!2 and mtl along a meridian is represented in Fig. 3(a) and (b), and it can be seen that the shape of the diagram changes with 4>i. The banded pattern in n!2 illustrated in Fig. 2 is seen to be representative of long imperfections and is similar to that found to occur in AI. But in short imperfections the sign at the centre is reversed, with an inward LI producing compressive rather than tensile stresses. This effect is quite different from what was observed in AI. Changes in mil are also seen to occur in short LI.

It is important to study how these maximum values of n!z and m'l\ change with 4>i. The variations are plotted in Fig. 3 (c) and (d). A shell with gi = - 3t shows maximum values of n!z and mil at approximately 4>i = 10°, and shows a strong decrease for 4>i< 10°. It must be noticed that although similar behaviour was detected in n!z for AI, in the LI mil also shows a maximum depending on 4>i. To understand the existence of such maximum values, it must be considered that an LI which extends over a very small area would remain unapparent in the shell, and would not change the stress distri­bution; on the other hand, if the imperfection affects a very large area, the curvature errors are small and small values of stress resultants integrated over a large area would provide the necessary contribution to equilibrium. A maximum value is reached between both situations, that is, for imper­fections of intermediate length.

Figure 4 shows the stress changes for shells with different radius r to thickness t ratios. An increase in t reduces the parameter R = r/t, and affects the moments mil' The results in Fig. 4(a) indicate that the membrane stress resultant n!z remains almost unaffected by a reduction in R from 933 to 467. However, the small modifications in n!z are accompanied by large

20

~

E 'i,'K.

'\ E'. \\E --E,\ E

~~~;I. EZ OJI \ z '\..!II:: I. ..lc:: I, \.

:l:N ~

N , \ .~c I' E I \

I-< 56000 mm.1\ ;11'1 I •

0 '\ .-. -----,;.-/

\ \'--~ // / -.-.:::-­0\' / / .t+ .-'" -":'"\;_/ / ,. ",'/

/ /

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~i I ,)./

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\ i-10 \ i \. i

./

0 4 8 12 16 20 0 4 8 12 16 ¢i ~i 20

(a) (b)

~30 E .€E EE

~- E z Z _r ..lc::

N *N 11::= C E

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..--~ ,,, \

\ \

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, ,

~'''''' , ,..

i~ ......., ­\ , ,/ ........... . ,- -, ....""

0 0 0 4 8 12 16 20 0 4 8 12 16 20

~i !f>i (c) (d)

Fig. 3. Influence of angular extent of local imperfection for the shell described in Fig. 2, P = 0·05 N/mm2. (a) and (b) n~2 and mfl at different levels, gi = -3t; - ... - ¢i = 3°; -- ¢i = SO; ---- ¢i = 7°; -. - . - ¢i = 100; -- ¢i = 15°. (c) and (d) Maximum nh

andm'hasafunctionof¢i;Ogi = -3t; .gi = -2t;Agj = -to

10

o~------~~----------~ Ii I

I

'Il . I

. I

I IC'J90mm. I I

. I0;

t ////./ ~

I. I< 56000 mm. )1 1/

/ .

/ /

/

/

/

.I./

/ . I \, I

./

. .......~~

~

20

-0,1 0 0,1 0,2 -0,6 -0,3 0 0,3 0,6 ni2[ kN/mm] m11 [kNmm/mm]

(a) (b)

30E E E E /k-._ ............. --E Z-- E ...... ::1 K oX:

z

.NN ...-,

'."­'" ...... _-_., .;:::: 20rI c

~e,\ ~.

10 0

'// /.

"./ //

.-",,/ .I

.- .- / /-1 ~

l .... ~; ,. / ./ /

-10 \. .Y .-­

0 4 8 12 16 0 4 12 16 20¢j 20 ¢i

(c) (d)

Fig. 4. Influence of shell parameter rlt for the shell described in Fig. 2 with an inward imperfection, gi = - 90 mm; <Pi = 200;p = 0·05 N/mm2. (a) and (b) n~2 and mil at different levels; -'- t = 30 mm; --- t = 40 mm; -- t = 60 mm. (c) and (d) n~2 and mtl as a

function of <Pi; At = 30 mm; • t = 40 mm; Ot = 60 mm.

152 Luis A. Godoy, Fernando G. Flores

changes in mil, which is inversely proportional to R. From the results, it can be seen that the relationship between mil and R in Fig. 4(b) is nonlinear. The dependence of maximum stress changes on the angular extent <Pi of the LI has also been investigated, and the results are summarized in Fig. 4(c) and (d). As shown previously in Fig. 3( c) and (d), maximum values of ni2 and mil are identified for certain values of <Pi.

4 COMPARISON WITH AXISYMMETRIC IMPERFECTIONS

The results in Part I for AI, and here for LI show different stress changes for two possible families of imperfections in spherical shells. One of the main features highlighted by the results is the dependence of maximum stress resultants on the imperfection characteristics, as reflected by the amplitude ~i and the angular extent <Pi' The results shown in Fig. 3(c) and (d) of this paper, and shown in Fig. 5(c) and (d) of Part I are plotted together in Fig. 5, in order to compare maximum values as a function of imperfection parameters for internal pressure p = 0·05 N/mm2. The membrane stress changes ni2 in Fig. 5(a) indicate that for short wavelengths, the AI yields higher values than the corresponding LI. For <Pi of approximately 10--13°, the curves intersect, and for <Pi > 14° it is the LI that produces higher values. If the bending moments are compared, again the AI yields higher values for short wavelengths. For <Pi = 5-7° the maximum due to an LI is equal to that given by the AI. A peak value of mil is obtained in LI, which is not present in AI. The solid curves in Fig. 5(a) and (b), for values of ~i = - 3t, - 2t and - t, show maximum values of ni2 and mil for both AI and LI.

However, for the design engineer it would be important to have some more general picture of the maximum stresses that can be expected for a given shell, without having to specify what particular imperfection produces it. For a shell with a maximum deviation of 3t, one could thus obtain the maximum values of ni2 and mil that can occur. If the wavelength is restricted to <Pi> 5° (that is, if very short imperfections are excluded), it can be seen from Fig. 5(a) and (b) that both nizand mil show clear maximum values. For niz the maximum is associated with the AI, whereas for bending moments the maximum is due to the LI. Although both values do not occur for the same imperfection, they represent bounds on stress redistributions for the cosine shape imperfection.

The results shown in Fig. 5 have been obtained for a shell with r/t = 933. Similar results can be found for thicker shells. Maximum membrane and bending stress resultants for a given amplitude and for different spherical shells have been plotted in Fig. 6. For example, a shell with r/t = 500 under uniform internal pressure p = 0·05 N/mm2, having an unspecified imper­

Stresses in spherical shells with imperfections. Part II 153

2 E E E

..§ -z oX: E

E .~ Zc: =. -,r 25,0

12,5

o 5 10 15 ~. 20 I

(b)

Fig. 5. Comparison between local and axisymmetric imperfection, for the shell described in Fig. 2 with p = 0·05 N/mm2; .• , • , • axisymmetric imperfection; t::. , 0 , 0 local

imperfection.

O+---~----~----~--~

15 ~. 20 I

o 5 10 (a)

~. =-2t• I

p=0,05 Nlmm2

•°1'-------... __-­:.1----­r----

o~ E E E E z =.

20-=t=

10

~.:: t

o~--~----~--~----~--~----~400 500 600 700 800 900 1000

R= r It

Fig. 6. Maximum values of n~2 and mIl for shells with cosine shape imperfection, and !/Ji 2::: 5°, for p = 0·05 N/mm2.

154 Luis A. Godoy, Fernando G. Flores

fection with maximum amplitude equal to the shell thickness, would yield maximum values of ni2 = 0·37 kN/mm and mil = 15 kN mm/mm. It is interesting to consider that for the range of shells studied, linear relations exist between maximum stress and moment resultants and the slenderness of the shell R. The dependence on U t is, however, nonlinear.

5 CONCLUSIONS

In this paper, the influence of localized imperfections of the stresses in thin spherical shells has been investigated. Although the behaviour for this kind of imperfection is similar to the banded imperfection studied in Part I, in the sense that both produce concentrations of stresses which only affect the zone of imperfection, there are some distinctive features which are summarized as follows:

(1) The changes in membrane and bending stress resultants are related to the amplitude of the imperfection in a nonlinear way.

(2) For a given amplitude of local imperfection, the largest changes in stress resultants occur for <Pi = 5-10°. Contrary to the case of banded imperfections, in the local imperfection a maximum value is identified as a function of the wavelength <Pi.

(3) An increase in the thickness of the shell produces only small reductions in the hoop membrane stresses.

(4) For a given amplitude and extent of imperfection, a local imperfec­tion yields higher values for stress resultants than the banded imper­fection in the long wavelengths range; whereas the reverse holds true for short imperfections.

On the basis of these results, it seems reasonable to establish bounds on stress resultants for a given shell and maximum amplitude of imperfection. These bounds have been defined in the present work by straight lines that approximate the relations between maximum stresses and shell slenderness.

It must be noted that the finite element model used in the present analysis is restricted to linear strain---displacement relations, and as such the results represent upper bounds to stresses in imperfect shells. Furthermore, the studies discussed in Parts I and II of this paper concentrated on the elastic behaviour of spherical shells with imperfections, and the elasto-plastic behaviour which is characteristic of steel structures has not been included in this first analysis. It is recognized that the present work should be extended to theelasto-plastic range of material behaviour for particular shell applications before it can be used for engineering design.

155 Stresses in spherical shells with imperfections. Part II

REFERENCES

1. Godoy, L. A. and Flores, F. G., Stresses in thin spherical shells with imperfections. Part 1: Influence of axisymmetric imperfections, Thin-Walled Structures,S (1987) 5-20. .

2. Flugge, W., Stresses in Shells, Springer-Verlag, Berlin, 1963,364-8. 3. Ellinas, C. P., Croll, J. G. A. and Kemp, K. 0., Cooling towers with

circumferential imperfections, 1. Structural Div., ASCE, 106 (1980) 2405-33. 4. Langhaar, H. L. and Boresi, A. P., Effect of out-of-roundness on stresses in a

cooling tower, Thin-Walled Structures, 1 (1983) 31-54. 5. Croll, J. G. A., Kaleli, F., Kemp, K. O. and Munro,J., A simplified approach to

the analysis of geometrically imperfect cooling tower shells, Engineering Structures, 1 (1979) 92-8.

6. Moy, S. S. J. and Niku, S. M., Finite element techniques for the analysis of cooling tower shells with geometric imperfections, Thin-Walled Structures, 1 (1983) 239-63.

7. Gould, P. L., Hang, K. J. and Tong, G. S., Analysis of hyperbolic cooling towers with local imperfections, Proc. Second Int. Symp. on Natural Draught Cooling Towers, Springer-Verlag, Berlin, 1984,397-411.

8. Godoy, L. A., Stresses in shells of revolution with geometric imperfections: Evaluation of two numerical tools, Proc. XXI South American Conf. of Structural Engineering, Rio de Janeiro, Brazil, 1981, Vol. 2, 237-57 (in Spanish).