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Elastic buckling of, and first yielding in, thin torispherical shells subjected to internal pressure
ELASTIC BUCKLING OF, A N D FIRST YIELDING IN, THIN TORISPHERICAL SHELLS SUBJECTED TO INTERNAL PRESSURE
R. W. AYLWARD& G. D. GALLETLY
Applied Mechanics Division, Department of Mechanical Engineering, Universityof Liverpool, Liverpool, Great Britain (Received: 4 September, 1978)
ABSTRACT With the aid of the non-linear shell buckling computer program BOSOR 4, the internal pressures at which elastic circumferential buckling (or wrinkling) take place in thin torispherical shells have been calculated. The maximum equivalent (or effective) stresses in the shells in the axisymmetric pre-buckled state were also obtained," from these, the pressures at which first yielding in the shells commences were determined for
1 <-~
× 103 < 4
The calculations were perjbrmed]br shells with diameter~thickness ratios of 500, 1000 and 2000; other geometric ratios, as detailed in the paper, were also varied. The computations were carried out]or steel shells but the results have been presented in dimensionless form. Utilising the above results it is possible to determine whether a given torispherical end closure will buckle elastically or whether an elastic-plastic analysis of the shell is desirable. Factors which are conducive to elastic buckling are a high yield point, a low modulus of elasticity or a large value of the shell diameter-thickness (D/t) ratio. For steel shells, elastic internal pressure buckling will occur ([or some combinations o]" r / D andRs/D)for D / t = 2000 and%p/E = 3 × 1 0 - 3. For D / t = 1000 and5OO,first yield always precedes elastic buckling for the parameters investigated. The failure mode for these cases is either elastic-plastic buckling or plastic collapse (an axisymmetric mode with large de]brmations). A comparison of the results of linear and non-linear elastic axisymmetric stress analyses of the shells shows that the linear theory sometimes underestimates the first 321
Int. J. Pres. lies.&Piping0308-O161/79/O007-0321/$02.25© Applied Science Publishers Ltd, England, 1979 Printed in Great Britain
322
R. W. A Y L W A R D , G. D. GALLETLY
yield pressure by considerable amounts, Limit pressures obtained from smalldeflection shell theories can be too low in such cases. Also given in the paper are approximate simple expressions whereby the elastic internal buckling pressures o f torispherical shells may be calculated. These expressions should be useJid to designers.
NOMENCLATURE
C D E K
Constant (see eqn. (4)). Diameter of cylindrical portion of vessel. Young's Modulus of Elasticity. Constant (see eqn. (2)). m Exponent (see eqn. (4)). n Circumferential wave number. Pressure. P Buckling pressure. Pcr r Radius of toroidal portion (knuckle) of torisphere. Radius of spherical portion (cap) of torisphere. Rs t Thickness of shell. Constants) ~(see eqn. (2)). Constant Y Poisson s Ratio. d~ Maximum equivalent (or effective) stress using yon Mises yield criterion. O'yp Uniaxial yield stress of material. Note: 10001bf/in 2 = 6.9 MPa.
INTRODUCTION
Very thin ellipsoidal, torispherical or toriconicat shells are used in the aerospace, distilling, nuclear, oil, etc. industries as end closures to cylindrical shells subjected to internal pressure. In the design of these end closures, one of the possible failure modes which has to be considered is circumferential buckling in which the shell deforms into a large number of circumferential waves. The buckling may occur either elastically or elasto-plastically but only elastic buckling will be considered in this paper. The prediction of the elastic buckling pressures of ellipsoidal or torispherical shells subjected to internal pressure is not an easy problem, even when one assumes the shells are perfect. This is particularly the case when the non-linear theory (with finite-deflections or rotations) is used. While there are several digital computer programs available nowadays which will, in theory, calculate the internal pressure at
ELASTIC BUCKLING OF, AND FIRST YIELDING IN, TORISPHERICAL SHELLS
323
which elastic circumferential buckling of the above shell types occurs, one can run into difficulties. The authors have had some experience with two of these computer programs--BOSOR 4 (based on the variational finite-difference method) and MIST 1 (based on the finite element technique). They found (when using the largedeflections options in these programs) that occasionally, through a poor initial choice of the pressure p or circumferential wave number n, the computer programs predicted that buckling would not occur. However, when better initial conditions were used for these geometries, bifurcation buckling pressures would be obtained. In addition, a large amount of computer time (on a CDC 7600 machine) was needed to find the buckling pressure in some cases. In view of the above, the authors thought that a limited parametric study to determine the large-deflection elastic internal buckling pressures of some constantthickness shells would be of value. Having such information available should be useful for (i) a rapid assessment of constant-thickness designs and (ii) in providing initial conditions for calculating the buckling pressure of variable thickness shells. Also, in order to provide the designer with information regarding the likelihood of elastic buckling occurring, the authors have made large-deflection elastic stress analyses of the shells in the axisymmetric pre-buckling state. From these analyses it is possible to determine the pressures at which first yielding of the shell wall occurs. If this latter pressure is much lower than the elastic buckling pressure, then elastic-plastic analyses (axisymmetric collapse and buckling) of the shell will be necessary. A comparison of the results obtained from small-deflection and large-deflection stress analyses shows that, for the thinner shells, the former predicts that first yielding will occur earlier than it actually does. One result from this observation is that the Drucker-Shield values for the limit pressure are likely to be low for the larger D/t values. It should also be noted that the shells investigated herein are assumed to be geometrically perfect and without any residual stresses. In practice, this means that the buckling pressures given later in the paper are likely to be somewhat higher than those applicable to the corresponding as-manufactured vessels (which will usually be out-of-round, contain welding stresses, etc.).
BRIEF RI~SUMf~ OF PREVIOUS WORK
As far as the authors are aware, the first theoretical treatment of internal pressure buckling of perfect torispherical shells was due to Mescall. 2 His analysis was a smalldeflection one and he used the Rayleigh-Ritz technique to calculate the buckling pressures. A few years later, Thurston and Holston 3 published their analysis of the problem. In this, the axisymmetric pre-buckling stress resultants were found from differential equations of equilibrium which are non-linear (due to finite deflections
324
R. W. AYLWARD, G. D. GALLETLY
distributions, as obtained from linear and non-linear shell theories, could differ significantly from each other-~epending on the geometric characteristics of the shell. In the past few years there have been several shell-of-revolution computer programs written which probably have the capability of determining the elastic internal buckling pressures of, say, thin torispherical shells--including a largedeflection pre-buckling stress analysis. Amongst these are: BOSOR 4 (variational finite-differences), STARS and SRA (numerical integration), SATANS (finitedifferences) and MIST 1 (finite elements). The authors have had more experience, at this point in time, with BOSOR 4 than with the other programs. In the remainder of this paper, therefore, only the results obtained from this program will be presented. Details of BOSOR 4 may be found in Bushnell* and comparisons of BOSOR 4 and MIST 1 for free vibration and external pressure buckling problems of shells are given in references 5 and 6. A short paper by the present authors giving the results of preliminary calculations for torispherical and ellipsoidal shells was given at the 3rd SMiRT (Structural Mechanics in Reactor Technology) Conference. 7 Some experimental and theoretical elastic-plastic results for machined torispherical shell models were also given by Galletly. 8 A new method, due to Ranjan and Steele, 9 for determining the stresses in, and the elastic buckling pressures of, torispherical shells subjected to internal pressure, should also be noted. The related problems of buckling and collapse of ellipsoidal shells subjected to internal pressure have been discussed by several authors. (See references 10 and 11 for a review of this work, together with new results).
CHARACTERISTICS OF THE SHELLS INVESTIGATED
A sketch of a torispherical end closure is given in Fig. 1 which also shows the pertinent geometric quantities. The numerical investigations carried out by the authors covered the following ranges of ratios relevant to torispherical shells:
r/D = 0.06, 0-08, 0-10 and 0.15 Rs/D = 0-75, 1-00, 1.25 and 1.50 D/t = 500, 1000 and 2000 being taken into account); the buckling pressures were obtained from non-trivial solutions of a set of eighth order homogeneous partial differential equations. In reference 3 it was also pointed out that the axisymmetric pre-buckling stress Most of the calculations were carried out for steel shells, i.e. using E = 30 × 106 lbf/in 2 (207 GN/m 2) and v = 0-3. However, results have been given in a dimensionless form. It should also be mentioned that some comparable results for torispheres with r/D = 0.20 may be found in reference 7.
ELASTIC BUCKLING OF, AND FIRST YIELDING IN, TORISPHERICAL SHELLS
SPHERC I AL
325
t
TORU~ Sr R~ \
F
o
J ..q
Fig. 1. Geometry of the torispherical shells investigated.
NUMERICAL RESULTS
(a) Elastic internal asymmetric buckling pressures The numerical results for the elastic internal buckling pressures of the torispherical shells are shown in Table 1 where the numbers in parentheses indicate the predicted number o f circumferential waves at buckling. Some of these results (for D/t = 1000 and 2000) are also shown in graphical form in Fig. 2. The computer program MIST 1 was also utilised to calculate some of the elasticpc,'s. In general, the agreement with BOSOR 4 was good (maximum difference -~ 10percent for some D/t = 500 cases, with MIST 1 being higher than BOSOR 4). TABLE 1 NON-DIMENSIONALELASTICBUCKLINGPRESSURES(pcr/E) x 107 FOR THE PARAMETERRANGEUNDERINVESTIGATION r/D 0.06
0.08
0.10
0'15
Rs/D~/t 0.75 1.00 1.25 1-50 0"75 1"00 1.25 1"50 0"75 1.00 1'25 1"50 0"75 1'00 1'25 1.50
2000 13.1(136) 7.71(77) 4.97(68) 3.48(62) 13"6(121) 9.23(81) 6.05(68) 4.26(61) 14.6(109) 9.54(98) 6.92(76) 5.00(51) 19.1(89) 11.3(79) 7.77(73) 5.90(68)
1000
500
62.6(87) 40.4(57) 27.6(50) 20.3(45) 64.5(82) 46.2(75) 33"9(51) 25.3(46) 69'2(76) 48.1(70) 36.9(62) 29.5(48) 95.7(64) 59.9(57) 43.3(53) 34.3(50)
344(54) 233(45) 174(38) 138(35) 349(52) 243(52) 200(47) 172(39) 378(50) 257(49) 209(47) 181(45) -356(42) 275(40) 231(38)
Figures in parentheses indicate the wave number n associated with buckling.
326
R. W. AYLWARD, G. D. GALLETLY Note: 1000 {bfJin 2 = 6-9MPa 100 -
75> D/t = 1000
50-
f 25-
= l o0
o/t
= 2000
=I"L~
r
= 1,50 0--4,
0-1)75
,
j
,
0.10
0'125
0"150
(r/D)
Fig. 2. Non-dimensional elastic internal buckling pressures for differinggeometries.
(b) Maximum equivalent stress curves (for n = 0, i.e. axisymmetric deformations) In order to be able to assess the relevance of elastic internal pressure buckling to any design, it is helpful if one has available the magnitude of the pressure at which first yielding of the shell wall occurs in the axisymmetric pre-buckling state. On the basis of the difference between it and the buckling pressure, one can often decide whether buckling is, or is not, a problem. The maximum equivalent stresses, for various geometric ratios, were determined from the BOSOR 4 program using the large-deflection stress analysis option. These curves are shown in Figs. 3 to 5. For any of the geometries analysed, the nondimensional internal pressure causing initial yield can be read directly from these curves once the parameter %,p/E is prescribed.
(c) Initial yield curves For illustrative purposes, values of arp/E x 1 0 3 = 1, 2, 3 and 4 were selected. Equating these values to the maximum effective stress in the shell divided by E and drawing horizontal lines on Figs. 3 to 5 through the appropriate ordinate 6e/E enables the pressure causing initial yielding of the shell wall to be determined. The resulting curves found by this procedure are shown on Figs. 6 to 8 for D/t = 500, 1000 and 2000.
ELASTIC BUCKLING OF, AND FIRST YIELDING IN, TORISPHERICAL SHELLS
327
/.
//
e0
O
x
/ /
2
/
I
2
I
I
/.
6
(p/E)xl06
/
~f~t7
D/t=500
I
I
2 4 (p/E)xl06
x
I.U
2
1
D=0.08
(p/E)xl06 Fig. 3.
01~
tt/
•~ A
r/D=0-10
r/D=0.15
1
/D'=O.06~
1
D/t=500
I
I
1
2
(p/ElxlO 6
M a x i m u m equivalent stresses versus pressure for torispherical shells with dimensionalised).
D/t = 500 (non-
R. W. AYLWARD, G. D. GALLETLY
328
I//// 1
0
D=015 I
1
1
1,
2 4 (p/E).106 D/t=1000
4
0
r/D=0.10 V
I
I
1 2 (p/E),(106
/*
b
3
3
/
01
O
x
//
~LU2
r/D=O.08 0
Fig. 4.
I
I
,
D=006
1 I
0-5 1 1.5 2 (p/E)*lO 6 D/t=lO00
0
i
0-5 (p/E)x10 6
M a x i m u m equivalent stresses versus pressure for torispherical shells with dimensionalised).
I
1
D/t = 1000 (non-
ELASTIC BUCKLING OF, AND FIRST YIELDING IN, TORISPHERICAL SHELLS
/.
4
3
3
329
~2 x
1 ///
I
r/D=0.15 t
0
1.0 20 (p/E) xlO6
D/t= 2000
r/D=010 I
I
05 10 (p/E)xlO 6
&
~3 x
II/' 1
///
r/D:O 08 I
I
0.5
10
(p/E ) x 106 Fig. 5.
1 ~/D=0.06 Or
D/t=2000
i
1
0-25 0-50 (p/E ) x 106
M a x i m u m equivalent stresses versus pressure for torispherical shells with D/t = 2000 (nondimensionalised).
330
R. W. A Y L W A R D , G. D. GALLETLY
~o/.. X
X
#D=0.15 0
~o'75
\
I
1"00 1-25 1-50 125 ' ' ' °~o 75 1"00 ' ' 150 Rs/D D/t =500 Rs/D
%
,%
3
~o2 x
X
ILl
r/D =0.08 0t,
Fig. 6.
I
'0 •75
I
I
, , , 0 410'.75 1'00 125 1-50 1"00 1'25 150 D/t=500 Rs/D Rs/D
N o n - d i m e n s i o n a l first yield pressure versus crown radius for
D/t =
500.
ELASTIC BUCKLING OF, AND FIRST YIELDING IN, TOR1SPHERICAL SHELLS
3.0
3.0
~
2-0 X
2.0
~1
.0 r/D=0.15
' 1.25 ' 1-50 ' o-o%% 1.00 Rs/D
1.5 ~
~
O0 D/t=1000
Rs/D
1.5
r/D:010~ | 0-0 ~().75 1.00 1-25 150 0-0 ['0-75 1-00 1.25 1.50 O/t =1000 Rs/D Rs/D Fig. 7.
Non-dimensional first yield pressure versus crown radius for D/t = 1000.
331
R. W. AYLWARD, G. D. GALLETLY
332
~
1.0
%
x
1.0
N
r/D=0-1 0.0 t,0.~75 1.130 1.25 1-~50 0.0 10.~75 1.00 1.25 1.50 Rs/D D/t =2000 Rs/D
075
® 0'50 o
¢
0-75
~oxO50
X
_,
,
,
,
00 ,.,75 1.00 1-25 1-50 O0 0-75 100 125 1-50 D/t=2000 Rs/O Rs/D Fig. 8.
Non-dimensional first yield pressure versus crown radius for
D/t =
2000.
ELASTIC BUCKLINGOF, AND FIRST YIELDING IN, TORISPHERICALSHELLS 333 (d) Empirical expressions f o r the elastic internal buckling pressures
The evaluation of the elastic buckling pressures of internally pressurised torispherical end closures using computer programs is both a time-consuming and expensive task for the designer. The authors therefore decided to employ curvefitting techniques to generate a mathematical function which fitted the elastic internal buckling pressures given in Table 1. This did not prove to be an easy task but the following function was derived: Pc-~': Kcq c(2~3(t/D) p E
(1)
where: K = 0"167 × 103 ~ = 1.1(Rs/D) 2 - 1.5(Rs/D ) + 1.0 ~2 = 48"O(r/D) 2 - 6.0(r/D) + 1.0 ~3 = [-35"O(r/D) 2 + 8.0(r/D) - 0.32](Rs/D) 2 + 1"0
/~ = 2"05 + 0"4(Rs/D)
(2)
The function in eqn. (1) always gives a 'safe' prediction of the buckling pressure. The maximum error within the parameter ranges studied is 20 per cent but typical errors are between 0 and 12 per cent. The function proved to be more complicated than anticipated and it was found possible to derive a much simpler expression for the special case of (Rs/D) = 1.0. This is as follows: Pc--r-"= lO013.7r/D + 0"68](t/D) 245 E
(3)
the maximum error being 12 per cent. It should be noted then, in arriving at eqn. (3), the range of r/D considered was 0-06-0-15. If the formula is used for higher values of r/D, the error (on the safe side) can be higher, e.g. for r/D = 0.2, D/t = 1000, the error is about 30 per cent. Again, this expression is always safe, the maximum error being 12 per cent. It is interesting to compare these computational results with the experimental ones of Adachi and Benicek. 12 They fit expressions of the form (using the notation of the present paper):
106~- = C[103 ~] m
(4)
to their results, where C is a constant and m an exponent. For one of their geometries (specimen B, r/D = 0.173, Rs/D = 0-89) the following expression can be derived: Pc__~,= 118(t/D) 2"35 E
Utilising eqn. (1) gives (for r/D = 0-173, Rs/D = 0"89):
(5)
334
R.w.
AYLWARD, G. D. GALLETLY
Pc~ = 126(t/D)Z.41
(6)
E
Ignoring the fact that Rs/D = 0-89 instead of 1-00, eqn. (3) gives:
Pc-z-~= 132(t/D) 245
(7)
E
Although eqns. (6) and (7) are both safe (i.e. lower) fits to the computational results, they are higher (8 and 11 per cent, respectively) than the experimental results. This is not unexpected as experimental buckling results are nearly always lower (often considerably) than theoretical predictions.
DISCUSSION OF RESULTS
From Figs. 6 to 8 it is possible to construct a tabulation of initial yield pressures for torispherical shells. The result is Table 2 and, in this form, it is a simple matter to compare the initial yield pressures with the elastic asymmetric buckling pressures given in Table 1. The cases for which the elastic pc,'s are lower than the first yield pressures are underlined in Table 2. One can see, therefore, for steel shells that: (i) Elastic buckling will not be encountered for D/t = 500 and D/t = 1000 and (ii) For D/t = 2000 and some combinations of riD and Rs/D, elastic buckling can occur with some high yield point steels. For materials with values of arp/E higher than 4 × 10-3 (e.g. aluminium alloys, TABLE 2 NON-DIMENSIONAL FIRST YIELD PRESSURES Q g j E ) × 10 7 FOR THE PARAMETER RANGE UNDER INVESTIGATION
D/t r/D
0.06
0.08
0.10
0.15
~ Rs/D
2000
1000
500
oy/E x 103 ~
1.0
2.0
3.0
4.0
1.0
0-75 1-00 1.25 1-50 0.75 1-00 1.25 1.50 0-75 1.00 1.25 1.50 0-75 1.00 1.25 1.50
2-0 4-2 7.0 -3.8 1.4 3.0 5-0 7-3 2.8 1.1 2.4 4.0 6.0 2.2 0-8 1.9 3.3 5.0 1.8 2.7 5.7 8.7 13.7 5.3 1-7 4.0 6.7 10"_0.0 3-6 1.2 3-0 5.3 8.0 2-7 1.0 2.5 4-3 6-.~7 2-3 3.2 7.0 12.0 - 6.3 1.8 5.0 8-7 12.8 4.3 1.6 3.8 6.7 10.__00 3.3 1.3 3-1 5.6 8.5 2.7 5-3 13-0 20-3 29.7 1 0 . 7 3-3 8.0 i4.0 2_!.~0 6.3 2.3 6.0 11.0 16.7 2.3 2.0 5.0 "9.6' |3.0 4-0
2.0
3.0
8.1 6.2 4.8 4.0 11-0 8.0 6.3 5.0 14.3 10.0 7.7 6.3 23.3 15-3 11.3 9.3
13-0 9.7 8-0 6.7 18.0 13.0 10.3 8.3 23-0 16.3 12-7 10.3 35.3 26.3 20.0 16.7
Underlining indicates that elastic buckling occurs before first yield.
4.0
1.0
-8.0 14.3 6.0 11.3 4.7 9.8 4-0 26,3 1 0 . 3 18.7 7.7 15.0 6.0 12.7 5.0 -14.0 24.0 9-3 18.7 7-0 15.7 6-0 55.3 22.7 38.0 1 3 . 3 30.7 1 0 . 0 25.0 7.7
2.0
3.0
4.0
17.0 12-7 10.3 8-7 22.0 16.7 13.0 11.0 28-3 20.0 15.3 12.7 44-7 29.7 23.0 18.3
26.0 19.7 16.3 14.0 34.3 25.7 30.7 17.7 43.0 32.0 25.3 21.0 72.0 49.0 34-0 30.7
35.7 34.3 26-3 20.3 48.0 36.0 29.3 24-7 61.7 46.7 36.0 30.7 70.0 55.0 45.0
ELASTIC B U C K L I N G O F , A N D FIRST Y I E L D I N G IN, T O R I S P H E R 1 C A L SHELLS
335
plastics, GRP, etc.) elastic buckling is likely to be encountered more often than with steel. The authors have not studied these cases so far. It can be seen from Figs. 3 to 5 that the variation of de with pressure is decidedly non-linear for the larger D/t values. Linear shell theory (obtained from the initial tangents to the curves) predicts that first yield in the shell wall occurs at a pressure which is lower than predicted by the large-deflection theory. Some illustrative results of non-linear shell theory predictions are given in Table 3 for steel and aluminium shells. The Drucker-Shield ~ limit pressures (based on small-deflection theory) are also given as a matter of interest, although D/t = 1000 is rather large for application of their method. It can be seen that cases can arise in which first yield has not occurred whereas a small-deflection calculation indicates that the limit pressure has been reached--see also Esztergar ~3 for a recent discussion of the Drucker-Shield and other plastic methods. TABLE3 FIRST YIELD AND LIMIT PRESSUR~ FOR A TORISPHERICAL SHELL WITH D/t = 1000, R s / D = 1, r / D = 0.2
Cryp
E
- 103
- 106
lbf/in 2
lbf/in 2
Pressure to cause first yieM (Non-linear theory)
30
30 10 30 10
28 37 98 151
80
Drucker-Shield limit pressure 25 25 68 68
N o t e : 10001bf/in 2 = 6.9 M P a .
With reference to elastic-plastic buckling and plastic collapse of internally pressurised dished ends, two papers which discuss these problems for 2:1 ellipsoidal shells are references 10 and 11. Similar calculations for torispherical shells are currently under way and will be published in the near future.
CONCLUSIONS
The elastic internal asymmetric buckling pressures of torispherical shells for several values of D/t, r/D and Rs/D have been computed. The pressures at which first yielding occurs in these shells have also been calculated for several values of the parameter CryJE. Both the elastic Pcr'S and the first yield pressures have been presented in dimensionless form so that they will be applicable to various materials. With regard to steel shells, it is clear from the above two sets of results that elastic buckling only occurs for shells which have large D/t ratios (e.g. 2000) and high yield points (e.g. ayv/E = 3 x 10-3).
336
R. W. AYLWARD, G. D. GALLETLY
Simple approximate expressions for the elastic internal buckling pressure of a torispherical shell are also proposed which should be of use to designers.
REFERENCES
1. SHIELD,R. T. and DRUCKER,D. C., Design of thin-walled torispherical and toriconical pressure vessel heads, Trans. ASME, 83, Series E, (1961), pp. 292-7. 2. MESCALL,J., Stability of thin torispherical shells under uniform internal pressure, NASA TNC-1510 (Collected papers on Instability of Shell Structures), 1962, pp. 671-92. 3. THURSTON, G. A. and HOLSTON, A. A., JR., Buckling of cylindrical shell end closures by internal pressure, NASA CR-540, July, 1966. 4. BUSHNELL, D., Stress, stability and vibration of complex branched shells of revolution, Comp. & Struct., 4 (1974), pp. 399-437. 5. GALLETLV,G. D. and MISTRY, J., The free vibrations of cylindrical shells with various end closures, Nucl. Eng. and Des,, 30 (1974), pp.249 68. 6. AYLWARD,R. W., GALLETL¥,G. D. and MISTRV,J., Buckling and vibrations of shells of revolution A comparison of results obtained by different methods, Proc. Inter. Syrnp. on "Discrete Methods in Engineering', CISE, Milan, Sept., 1974, pp. 288-302. 7. AVLWARD,R. W., GALLETL¥, G. D. and MISTRV, J., Large deflection elastic buckling pressures of very thin torispherical and ellipsoidal shells subjected to uniform internal pressure, Proc. 3rd Inter. S M i R T (Structural Mechanics in Reactor Technology) Con[i, London, Sept., 1975, Paper G2/4, pp. 1 9. 8. GALLEXLV,G. D., Internal pressure buckling of very thin torispherical shells--A comparison of experiment and theory, Proc. 3rd Inter. S M i R T Conj,, London, Sept., 1975. Paper G2/3, pp. 1-10. 9. RANJAN, G. V. and STEELE,C. R., Analysis of torispherical pressure vessels, ASCE Jl, Eng. Mech. Dit,. (Aug., 1976), pp. 643-57. 10. GALLETLY,G. D., Elastic and elastic-plastic buckling of internally pressurized 2:1 ellipsoidal shells, A S M E J. Press. Vess. Tech., 100 (1978), pp. 335-43. 11. GALLETLY,G. D. and AYLWARD, R. W., Plastic collapse and the controlling failure pressures of thin 2:1 ellipsoidal shells, A S M E J. Press. Vess. Tech., 101 (1979), pp. 64~72. 12. ADACHI,J. and BENICEK, M., Buckling of torispherical shells under internal pressure, Experimental Mechanics, 4(8) (Aug., 1964), pp. 217 22. 13. ESZTERGAR,E. P., Development of design rules for dished pressure vessel heads, WRC Bulletin, New York. No. 215, (May, 1976).
R. W. AYLWARD& G. D. GALLETLY
Applied Mechanics Division, Department of Mechanical Engineering, Universityof Liverpool, Liverpool, Great Britain (Received: 4 September, 1978)
ABSTRACT With the aid of the non-linear shell buckling computer program BOSOR 4, the internal pressures at which elastic circumferential buckling (or wrinkling) take place in thin torispherical shells have been calculated. The maximum equivalent (or effective) stresses in the shells in the axisymmetric pre-buckled state were also obtained," from these, the pressures at which first yielding in the shells commences were determined for
1 <-~
× 103 < 4
The calculations were perjbrmed]br shells with diameter~thickness ratios of 500, 1000 and 2000; other geometric ratios, as detailed in the paper, were also varied. The computations were carried out]or steel shells but the results have been presented in dimensionless form. Utilising the above results it is possible to determine whether a given torispherical end closure will buckle elastically or whether an elastic-plastic analysis of the shell is desirable. Factors which are conducive to elastic buckling are a high yield point, a low modulus of elasticity or a large value of the shell diameter-thickness (D/t) ratio. For steel shells, elastic internal pressure buckling will occur ([or some combinations o]" r / D andRs/D)for D / t = 2000 and%p/E = 3 × 1 0 - 3. For D / t = 1000 and5OO,first yield always precedes elastic buckling for the parameters investigated. The failure mode for these cases is either elastic-plastic buckling or plastic collapse (an axisymmetric mode with large de]brmations). A comparison of the results of linear and non-linear elastic axisymmetric stress analyses of the shells shows that the linear theory sometimes underestimates the first 321
Int. J. Pres. lies.&Piping0308-O161/79/O007-0321/$02.25© Applied Science Publishers Ltd, England, 1979 Printed in Great Britain
322
R. W. A Y L W A R D , G. D. GALLETLY
yield pressure by considerable amounts, Limit pressures obtained from smalldeflection shell theories can be too low in such cases. Also given in the paper are approximate simple expressions whereby the elastic internal buckling pressures o f torispherical shells may be calculated. These expressions should be useJid to designers.
NOMENCLATURE
C D E K
Constant (see eqn. (4)). Diameter of cylindrical portion of vessel. Young's Modulus of Elasticity. Constant (see eqn. (2)). m Exponent (see eqn. (4)). n Circumferential wave number. Pressure. P Buckling pressure. Pcr r Radius of toroidal portion (knuckle) of torisphere. Radius of spherical portion (cap) of torisphere. Rs t Thickness of shell. Constants) ~(see eqn. (2)). Constant Y Poisson s Ratio. d~ Maximum equivalent (or effective) stress using yon Mises yield criterion. O'yp Uniaxial yield stress of material. Note: 10001bf/in 2 = 6.9 MPa.
INTRODUCTION
Very thin ellipsoidal, torispherical or toriconicat shells are used in the aerospace, distilling, nuclear, oil, etc. industries as end closures to cylindrical shells subjected to internal pressure. In the design of these end closures, one of the possible failure modes which has to be considered is circumferential buckling in which the shell deforms into a large number of circumferential waves. The buckling may occur either elastically or elasto-plastically but only elastic buckling will be considered in this paper. The prediction of the elastic buckling pressures of ellipsoidal or torispherical shells subjected to internal pressure is not an easy problem, even when one assumes the shells are perfect. This is particularly the case when the non-linear theory (with finite-deflections or rotations) is used. While there are several digital computer programs available nowadays which will, in theory, calculate the internal pressure at
ELASTIC BUCKLING OF, AND FIRST YIELDING IN, TORISPHERICAL SHELLS
323
which elastic circumferential buckling of the above shell types occurs, one can run into difficulties. The authors have had some experience with two of these computer programs--BOSOR 4 (based on the variational finite-difference method) and MIST 1 (based on the finite element technique). They found (when using the largedeflections options in these programs) that occasionally, through a poor initial choice of the pressure p or circumferential wave number n, the computer programs predicted that buckling would not occur. However, when better initial conditions were used for these geometries, bifurcation buckling pressures would be obtained. In addition, a large amount of computer time (on a CDC 7600 machine) was needed to find the buckling pressure in some cases. In view of the above, the authors thought that a limited parametric study to determine the large-deflection elastic internal buckling pressures of some constantthickness shells would be of value. Having such information available should be useful for (i) a rapid assessment of constant-thickness designs and (ii) in providing initial conditions for calculating the buckling pressure of variable thickness shells. Also, in order to provide the designer with information regarding the likelihood of elastic buckling occurring, the authors have made large-deflection elastic stress analyses of the shells in the axisymmetric pre-buckling state. From these analyses it is possible to determine the pressures at which first yielding of the shell wall occurs. If this latter pressure is much lower than the elastic buckling pressure, then elastic-plastic analyses (axisymmetric collapse and buckling) of the shell will be necessary. A comparison of the results obtained from small-deflection and large-deflection stress analyses shows that, for the thinner shells, the former predicts that first yielding will occur earlier than it actually does. One result from this observation is that the Drucker-Shield values for the limit pressure are likely to be low for the larger D/t values. It should also be noted that the shells investigated herein are assumed to be geometrically perfect and without any residual stresses. In practice, this means that the buckling pressures given later in the paper are likely to be somewhat higher than those applicable to the corresponding as-manufactured vessels (which will usually be out-of-round, contain welding stresses, etc.).
BRIEF RI~SUMf~ OF PREVIOUS WORK
As far as the authors are aware, the first theoretical treatment of internal pressure buckling of perfect torispherical shells was due to Mescall. 2 His analysis was a smalldeflection one and he used the Rayleigh-Ritz technique to calculate the buckling pressures. A few years later, Thurston and Holston 3 published their analysis of the problem. In this, the axisymmetric pre-buckling stress resultants were found from differential equations of equilibrium which are non-linear (due to finite deflections
324
R. W. AYLWARD, G. D. GALLETLY
distributions, as obtained from linear and non-linear shell theories, could differ significantly from each other-~epending on the geometric characteristics of the shell. In the past few years there have been several shell-of-revolution computer programs written which probably have the capability of determining the elastic internal buckling pressures of, say, thin torispherical shells--including a largedeflection pre-buckling stress analysis. Amongst these are: BOSOR 4 (variational finite-differences), STARS and SRA (numerical integration), SATANS (finitedifferences) and MIST 1 (finite elements). The authors have had more experience, at this point in time, with BOSOR 4 than with the other programs. In the remainder of this paper, therefore, only the results obtained from this program will be presented. Details of BOSOR 4 may be found in Bushnell* and comparisons of BOSOR 4 and MIST 1 for free vibration and external pressure buckling problems of shells are given in references 5 and 6. A short paper by the present authors giving the results of preliminary calculations for torispherical and ellipsoidal shells was given at the 3rd SMiRT (Structural Mechanics in Reactor Technology) Conference. 7 Some experimental and theoretical elastic-plastic results for machined torispherical shell models were also given by Galletly. 8 A new method, due to Ranjan and Steele, 9 for determining the stresses in, and the elastic buckling pressures of, torispherical shells subjected to internal pressure, should also be noted. The related problems of buckling and collapse of ellipsoidal shells subjected to internal pressure have been discussed by several authors. (See references 10 and 11 for a review of this work, together with new results).
CHARACTERISTICS OF THE SHELLS INVESTIGATED
A sketch of a torispherical end closure is given in Fig. 1 which also shows the pertinent geometric quantities. The numerical investigations carried out by the authors covered the following ranges of ratios relevant to torispherical shells:
r/D = 0.06, 0-08, 0-10 and 0.15 Rs/D = 0-75, 1-00, 1.25 and 1.50 D/t = 500, 1000 and 2000 being taken into account); the buckling pressures were obtained from non-trivial solutions of a set of eighth order homogeneous partial differential equations. In reference 3 it was also pointed out that the axisymmetric pre-buckling stress Most of the calculations were carried out for steel shells, i.e. using E = 30 × 106 lbf/in 2 (207 GN/m 2) and v = 0-3. However, results have been given in a dimensionless form. It should also be mentioned that some comparable results for torispheres with r/D = 0.20 may be found in reference 7.
ELASTIC BUCKLING OF, AND FIRST YIELDING IN, TORISPHERICAL SHELLS
SPHERC I AL
325
t
TORU~ Sr R~ \
F
o
J ..q
Fig. 1. Geometry of the torispherical shells investigated.
NUMERICAL RESULTS
(a) Elastic internal asymmetric buckling pressures The numerical results for the elastic internal buckling pressures of the torispherical shells are shown in Table 1 where the numbers in parentheses indicate the predicted number o f circumferential waves at buckling. Some of these results (for D/t = 1000 and 2000) are also shown in graphical form in Fig. 2. The computer program MIST 1 was also utilised to calculate some of the elasticpc,'s. In general, the agreement with BOSOR 4 was good (maximum difference -~ 10percent for some D/t = 500 cases, with MIST 1 being higher than BOSOR 4). TABLE 1 NON-DIMENSIONALELASTICBUCKLINGPRESSURES(pcr/E) x 107 FOR THE PARAMETERRANGEUNDERINVESTIGATION r/D 0.06
0.08
0.10
0'15
Rs/D~/t 0.75 1.00 1.25 1-50 0"75 1"00 1.25 1"50 0"75 1.00 1'25 1"50 0"75 1'00 1'25 1.50
2000 13.1(136) 7.71(77) 4.97(68) 3.48(62) 13"6(121) 9.23(81) 6.05(68) 4.26(61) 14.6(109) 9.54(98) 6.92(76) 5.00(51) 19.1(89) 11.3(79) 7.77(73) 5.90(68)
1000
500
62.6(87) 40.4(57) 27.6(50) 20.3(45) 64.5(82) 46.2(75) 33"9(51) 25.3(46) 69'2(76) 48.1(70) 36.9(62) 29.5(48) 95.7(64) 59.9(57) 43.3(53) 34.3(50)
344(54) 233(45) 174(38) 138(35) 349(52) 243(52) 200(47) 172(39) 378(50) 257(49) 209(47) 181(45) -356(42) 275(40) 231(38)
Figures in parentheses indicate the wave number n associated with buckling.
326
R. W. AYLWARD, G. D. GALLETLY Note: 1000 {bfJin 2 = 6-9MPa 100 -
75> D/t = 1000
50-
f 25-
= l o0
o/t
= 2000
=I"L~
r
= 1,50 0--4,
0-1)75
,
j
,
0.10
0'125
0"150
(r/D)
Fig. 2. Non-dimensional elastic internal buckling pressures for differinggeometries.
(b) Maximum equivalent stress curves (for n = 0, i.e. axisymmetric deformations) In order to be able to assess the relevance of elastic internal pressure buckling to any design, it is helpful if one has available the magnitude of the pressure at which first yielding of the shell wall occurs in the axisymmetric pre-buckling state. On the basis of the difference between it and the buckling pressure, one can often decide whether buckling is, or is not, a problem. The maximum equivalent stresses, for various geometric ratios, were determined from the BOSOR 4 program using the large-deflection stress analysis option. These curves are shown in Figs. 3 to 5. For any of the geometries analysed, the nondimensional internal pressure causing initial yield can be read directly from these curves once the parameter %,p/E is prescribed.
(c) Initial yield curves For illustrative purposes, values of arp/E x 1 0 3 = 1, 2, 3 and 4 were selected. Equating these values to the maximum effective stress in the shell divided by E and drawing horizontal lines on Figs. 3 to 5 through the appropriate ordinate 6e/E enables the pressure causing initial yielding of the shell wall to be determined. The resulting curves found by this procedure are shown on Figs. 6 to 8 for D/t = 500, 1000 and 2000.
ELASTIC BUCKLING OF, AND FIRST YIELDING IN, TORISPHERICAL SHELLS
327
/.
//
e0
O
x
/ /
2
/
I
2
I
I
/.
6
(p/E)xl06
/
~f~t7
D/t=500
I
I
2 4 (p/E)xl06
x
I.U
2
1
D=0.08
(p/E)xl06 Fig. 3.
01~
tt/
•~ A
r/D=0-10
r/D=0.15
1
/D'=O.06~
1
D/t=500
I
I
1
2
(p/ElxlO 6
M a x i m u m equivalent stresses versus pressure for torispherical shells with dimensionalised).
D/t = 500 (non-
R. W. AYLWARD, G. D. GALLETLY
328
I//// 1
0
D=015 I
1
1
1,
2 4 (p/E).106 D/t=1000
4
0
r/D=0.10 V
I
I
1 2 (p/E),(106
/*
b
3
3
/
01
O
x
//
~LU2
r/D=O.08 0
Fig. 4.
I
I
,
D=006
1 I
0-5 1 1.5 2 (p/E)*lO 6 D/t=lO00
0
i
0-5 (p/E)x10 6
M a x i m u m equivalent stresses versus pressure for torispherical shells with dimensionalised).
I
1
D/t = 1000 (non-
ELASTIC BUCKLING OF, AND FIRST YIELDING IN, TORISPHERICAL SHELLS
/.
4
3
3
329
~2 x
1 ///
I
r/D=0.15 t
0
1.0 20 (p/E) xlO6
D/t= 2000
r/D=010 I
I
05 10 (p/E)xlO 6
&
~3 x
II/' 1
///
r/D:O 08 I
I
0.5
10
(p/E ) x 106 Fig. 5.
1 ~/D=0.06 Or
D/t=2000
i
1
0-25 0-50 (p/E ) x 106
M a x i m u m equivalent stresses versus pressure for torispherical shells with D/t = 2000 (nondimensionalised).
330
R. W. A Y L W A R D , G. D. GALLETLY
~o/.. X
X
#D=0.15 0
~o'75
\
I
1"00 1-25 1-50 125 ' ' ' °~o 75 1"00 ' ' 150 Rs/D D/t =500 Rs/D
%
,%
3
~o2 x
X
ILl
r/D =0.08 0t,
Fig. 6.
I
'0 •75
I
I
, , , 0 410'.75 1'00 125 1-50 1"00 1'25 150 D/t=500 Rs/D Rs/D
N o n - d i m e n s i o n a l first yield pressure versus crown radius for
D/t =
500.
ELASTIC BUCKLING OF, AND FIRST YIELDING IN, TOR1SPHERICAL SHELLS
3.0
3.0
~
2-0 X
2.0
~1
.0 r/D=0.15
' 1.25 ' 1-50 ' o-o%% 1.00 Rs/D
1.5 ~
~
O0 D/t=1000
Rs/D
1.5
r/D:010~ | 0-0 ~().75 1.00 1-25 150 0-0 ['0-75 1-00 1.25 1.50 O/t =1000 Rs/D Rs/D Fig. 7.
Non-dimensional first yield pressure versus crown radius for D/t = 1000.
331
R. W. AYLWARD, G. D. GALLETLY
332
~
1.0
%
x
1.0
N
r/D=0-1 0.0 t,0.~75 1.130 1.25 1-~50 0.0 10.~75 1.00 1.25 1.50 Rs/D D/t =2000 Rs/D
075
® 0'50 o
¢
0-75
~oxO50
X
_,
,
,
,
00 ,.,75 1.00 1-25 1-50 O0 0-75 100 125 1-50 D/t=2000 Rs/O Rs/D Fig. 8.
Non-dimensional first yield pressure versus crown radius for
D/t =
2000.
ELASTIC BUCKLINGOF, AND FIRST YIELDING IN, TORISPHERICALSHELLS 333 (d) Empirical expressions f o r the elastic internal buckling pressures
The evaluation of the elastic buckling pressures of internally pressurised torispherical end closures using computer programs is both a time-consuming and expensive task for the designer. The authors therefore decided to employ curvefitting techniques to generate a mathematical function which fitted the elastic internal buckling pressures given in Table 1. This did not prove to be an easy task but the following function was derived: Pc-~': Kcq c(2~3(t/D) p E
(1)
where: K = 0"167 × 103 ~ = 1.1(Rs/D) 2 - 1.5(Rs/D ) + 1.0 ~2 = 48"O(r/D) 2 - 6.0(r/D) + 1.0 ~3 = [-35"O(r/D) 2 + 8.0(r/D) - 0.32](Rs/D) 2 + 1"0
/~ = 2"05 + 0"4(Rs/D)
(2)
The function in eqn. (1) always gives a 'safe' prediction of the buckling pressure. The maximum error within the parameter ranges studied is 20 per cent but typical errors are between 0 and 12 per cent. The function proved to be more complicated than anticipated and it was found possible to derive a much simpler expression for the special case of (Rs/D) = 1.0. This is as follows: Pc--r-"= lO013.7r/D + 0"68](t/D) 245 E
(3)
the maximum error being 12 per cent. It should be noted then, in arriving at eqn. (3), the range of r/D considered was 0-06-0-15. If the formula is used for higher values of r/D, the error (on the safe side) can be higher, e.g. for r/D = 0.2, D/t = 1000, the error is about 30 per cent. Again, this expression is always safe, the maximum error being 12 per cent. It is interesting to compare these computational results with the experimental ones of Adachi and Benicek. 12 They fit expressions of the form (using the notation of the present paper):
106~- = C[103 ~] m
(4)
to their results, where C is a constant and m an exponent. For one of their geometries (specimen B, r/D = 0.173, Rs/D = 0-89) the following expression can be derived: Pc__~,= 118(t/D) 2"35 E
Utilising eqn. (1) gives (for r/D = 0-173, Rs/D = 0"89):
(5)
334
R.w.
AYLWARD, G. D. GALLETLY
Pc~ = 126(t/D)Z.41
(6)
E
Ignoring the fact that Rs/D = 0-89 instead of 1-00, eqn. (3) gives:
Pc-z-~= 132(t/D) 245
(7)
E
Although eqns. (6) and (7) are both safe (i.e. lower) fits to the computational results, they are higher (8 and 11 per cent, respectively) than the experimental results. This is not unexpected as experimental buckling results are nearly always lower (often considerably) than theoretical predictions.
DISCUSSION OF RESULTS
From Figs. 6 to 8 it is possible to construct a tabulation of initial yield pressures for torispherical shells. The result is Table 2 and, in this form, it is a simple matter to compare the initial yield pressures with the elastic asymmetric buckling pressures given in Table 1. The cases for which the elastic pc,'s are lower than the first yield pressures are underlined in Table 2. One can see, therefore, for steel shells that: (i) Elastic buckling will not be encountered for D/t = 500 and D/t = 1000 and (ii) For D/t = 2000 and some combinations of riD and Rs/D, elastic buckling can occur with some high yield point steels. For materials with values of arp/E higher than 4 × 10-3 (e.g. aluminium alloys, TABLE 2 NON-DIMENSIONAL FIRST YIELD PRESSURES Q g j E ) × 10 7 FOR THE PARAMETER RANGE UNDER INVESTIGATION
D/t r/D
0.06
0.08
0.10
0.15
~ Rs/D
2000
1000
500
oy/E x 103 ~
1.0
2.0
3.0
4.0
1.0
0-75 1-00 1.25 1-50 0.75 1-00 1.25 1.50 0-75 1.00 1.25 1.50 0-75 1.00 1.25 1.50
2-0 4-2 7.0 -3.8 1.4 3.0 5-0 7-3 2.8 1.1 2.4 4.0 6.0 2.2 0-8 1.9 3.3 5.0 1.8 2.7 5.7 8.7 13.7 5.3 1-7 4.0 6.7 10"_0.0 3-6 1.2 3-0 5.3 8.0 2-7 1.0 2.5 4-3 6-.~7 2-3 3.2 7.0 12.0 - 6.3 1.8 5.0 8-7 12.8 4.3 1.6 3.8 6.7 10.__00 3.3 1.3 3-1 5.6 8.5 2.7 5-3 13-0 20-3 29.7 1 0 . 7 3-3 8.0 i4.0 2_!.~0 6.3 2.3 6.0 11.0 16.7 2.3 2.0 5.0 "9.6' |3.0 4-0
2.0
3.0
8.1 6.2 4.8 4.0 11-0 8.0 6.3 5.0 14.3 10.0 7.7 6.3 23.3 15-3 11.3 9.3
13-0 9.7 8-0 6.7 18.0 13.0 10.3 8.3 23-0 16.3 12-7 10.3 35.3 26.3 20.0 16.7
Underlining indicates that elastic buckling occurs before first yield.
4.0
1.0
-8.0 14.3 6.0 11.3 4.7 9.8 4-0 26,3 1 0 . 3 18.7 7.7 15.0 6.0 12.7 5.0 -14.0 24.0 9-3 18.7 7-0 15.7 6-0 55.3 22.7 38.0 1 3 . 3 30.7 1 0 . 0 25.0 7.7
2.0
3.0
4.0
17.0 12-7 10.3 8-7 22.0 16.7 13.0 11.0 28-3 20.0 15.3 12.7 44-7 29.7 23.0 18.3
26.0 19.7 16.3 14.0 34.3 25.7 30.7 17.7 43.0 32.0 25.3 21.0 72.0 49.0 34-0 30.7
35.7 34.3 26-3 20.3 48.0 36.0 29.3 24-7 61.7 46.7 36.0 30.7 70.0 55.0 45.0
ELASTIC B U C K L I N G O F , A N D FIRST Y I E L D I N G IN, T O R I S P H E R 1 C A L SHELLS
335
plastics, GRP, etc.) elastic buckling is likely to be encountered more often than with steel. The authors have not studied these cases so far. It can be seen from Figs. 3 to 5 that the variation of de with pressure is decidedly non-linear for the larger D/t values. Linear shell theory (obtained from the initial tangents to the curves) predicts that first yield in the shell wall occurs at a pressure which is lower than predicted by the large-deflection theory. Some illustrative results of non-linear shell theory predictions are given in Table 3 for steel and aluminium shells. The Drucker-Shield ~ limit pressures (based on small-deflection theory) are also given as a matter of interest, although D/t = 1000 is rather large for application of their method. It can be seen that cases can arise in which first yield has not occurred whereas a small-deflection calculation indicates that the limit pressure has been reached--see also Esztergar ~3 for a recent discussion of the Drucker-Shield and other plastic methods. TABLE3 FIRST YIELD AND LIMIT PRESSUR~ FOR A TORISPHERICAL SHELL WITH D/t = 1000, R s / D = 1, r / D = 0.2
Cryp
E
- 103
- 106
lbf/in 2
lbf/in 2
Pressure to cause first yieM (Non-linear theory)
30
30 10 30 10
28 37 98 151
80
Drucker-Shield limit pressure 25 25 68 68
N o t e : 10001bf/in 2 = 6.9 M P a .
With reference to elastic-plastic buckling and plastic collapse of internally pressurised dished ends, two papers which discuss these problems for 2:1 ellipsoidal shells are references 10 and 11. Similar calculations for torispherical shells are currently under way and will be published in the near future.
CONCLUSIONS
The elastic internal asymmetric buckling pressures of torispherical shells for several values of D/t, r/D and Rs/D have been computed. The pressures at which first yielding occurs in these shells have also been calculated for several values of the parameter CryJE. Both the elastic Pcr'S and the first yield pressures have been presented in dimensionless form so that they will be applicable to various materials. With regard to steel shells, it is clear from the above two sets of results that elastic buckling only occurs for shells which have large D/t ratios (e.g. 2000) and high yield points (e.g. ayv/E = 3 x 10-3).
336
R. W. AYLWARD, G. D. GALLETLY
Simple approximate expressions for the elastic internal buckling pressure of a torispherical shell are also proposed which should be of use to designers.
REFERENCES
1. SHIELD,R. T. and DRUCKER,D. C., Design of thin-walled torispherical and toriconical pressure vessel heads, Trans. ASME, 83, Series E, (1961), pp. 292-7. 2. MESCALL,J., Stability of thin torispherical shells under uniform internal pressure, NASA TNC-1510 (Collected papers on Instability of Shell Structures), 1962, pp. 671-92. 3. THURSTON, G. A. and HOLSTON, A. A., JR., Buckling of cylindrical shell end closures by internal pressure, NASA CR-540, July, 1966. 4. BUSHNELL, D., Stress, stability and vibration of complex branched shells of revolution, Comp. & Struct., 4 (1974), pp. 399-437. 5. GALLETLV,G. D. and MISTRY, J., The free vibrations of cylindrical shells with various end closures, Nucl. Eng. and Des,, 30 (1974), pp.249 68. 6. AYLWARD,R. W., GALLETL¥,G. D. and MISTRV,J., Buckling and vibrations of shells of revolution A comparison of results obtained by different methods, Proc. Inter. Syrnp. on "Discrete Methods in Engineering', CISE, Milan, Sept., 1974, pp. 288-302. 7. AVLWARD,R. W., GALLETL¥, G. D. and MISTRV, J., Large deflection elastic buckling pressures of very thin torispherical and ellipsoidal shells subjected to uniform internal pressure, Proc. 3rd Inter. S M i R T (Structural Mechanics in Reactor Technology) Con[i, London, Sept., 1975, Paper G2/4, pp. 1 9. 8. GALLEXLV,G. D., Internal pressure buckling of very thin torispherical shells--A comparison of experiment and theory, Proc. 3rd Inter. S M i R T Conj,, London, Sept., 1975. Paper G2/3, pp. 1-10. 9. RANJAN, G. V. and STEELE,C. R., Analysis of torispherical pressure vessels, ASCE Jl, Eng. Mech. Dit,. (Aug., 1976), pp. 643-57. 10. GALLETLY,G. D., Elastic and elastic-plastic buckling of internally pressurized 2:1 ellipsoidal shells, A S M E J. Press. Vess. Tech., 100 (1978), pp. 335-43. 11. GALLETLY,G. D. and AYLWARD, R. W., Plastic collapse and the controlling failure pressures of thin 2:1 ellipsoidal shells, A S M E J. Press. Vess. Tech., 101 (1979), pp. 64~72. 12. ADACHI,J. and BENICEK, M., Buckling of torispherical shells under internal pressure, Experimental Mechanics, 4(8) (Aug., 1964), pp. 217 22. 13. ESZTERGAR,E. P., Development of design rules for dished pressure vessel heads, WRC Bulletin, New York. No. 215, (May, 1976).