Elastic plastic buckling of internally pressurized torispherical vessel heads

Nuclear Engineering and Design 48 (1978) 405-414 © North-Holland Publishing Company

405

ELASTIC PLASTIC BUCKLING OF INTERNALLY PRESSURIZED TORISPHERICAL VESSEL HEADS Guy LAGAE Faculty of Engineering Science, State University of Gent, Belgium and David BUSHNELL Lockheed Missiles and Space Company Inc., 3251 Hanover Street, Palo Alto, California 94304, USA Received 7 December 1977

Comparisons of test and theory are presented for the nonaxisymmetric bifurcation buckling of ten aluminium vessel heads fabricated and tested by Patel and Gill at the University of Manchester in 1976. In the test specimens, the sphere radius was equal to the cylinder diameter and only the torus radius was varied. All specimens were of constant internal cylinder diameter to nominal thickness ratio of 531.5. Meridional variation of thickness was accounted for in the analysis, which was carried out with the BOSOR5 computer program. The analysis includes both material and geometrical nonlinear prebuckling behavior. The results indicate that incipient nonsymmetric buckling can be predicted with reasonable accuracy by means of an eigenvalue formulation.

1. Introduction This paper represents an extension o f the comparisons of test and theory presented in [1 ]. The literature of buckling o f torispherical vessel heads is briefly reviewed in [1] and is extensively referenced in [2]. Such a review will therefore not be repeated here. The purpose of this paper is to give theoretical predictions for bifurcation buckling pressures of torispherical specimens tested by Patel and Gill [3] during the summer of 1976. In [1], [3] and [4] the observed buckling process

Fig. 1. Two of Patel and Gill's specimens after testing.

in internally pressurized torispherical heads is described. The development o f visible buckles in such cases is a process and not the single event predicted by a bifurcation (eigenvalue) buckling analysis. As the pressure in a test specimen is increased above some critical value, a very localized, isolated incipient buckle forms in the knuckle region, invisible to the naked eye b u t detectable by a sensitive probe or a strain gage. The buckle grows slowly at first, and then more rapidly, and suddenly becomes visible. This visible buckle generally covers most of the knuckle region in the meridonal direction but has a very short circumferential wavelength. After formation of the first buckle the pressure can be further increased substantially, causing the formation of other visible buckles in the knuckle region, each one isolated circumferentially from its neighbors. An isolated buckle, generated by circumferential compression in the knuckle region, apparently causes the relief o f this compression within a sector surrounding the buckle, thereby preventing the formation o f the uniform buckle pattern typical of buckled axially compressed cylindrical or externally oressurized spherical shells. Flg. 1, taken from [3],

406

G. Lagae and D. Bushnell / Elastic plastic buckling of torispherical vesselheads

shows two of Patel and Gill's buckled specimens. The analysis here is founded on the assumption that we are interested in the pressure at which the first incipient buckle forms. Therefore, buckling is treated as a single event, predicted by means of an eigenvalue analysis.

2. Analysis Details of the analysis method are given in [5] and [6]. The BOSOR5 computer program [7] was used for the calculations. This program is applicable to any segment or branched, ring-stiffened shell of revolution. It is based on energy minimization with finite difference discretization in the meridional direction and trigonometric variation in the circumferential direction.

2. 2. Nonsymmetric buckling analysis The bifurcation buckling analysis is based on tangential stiffness, defined in equations (22)-(30) in [6]. Bifurcation buckling loads Lcr corresponding to nonsymmetric buckling modes are calculated in the following way: The user of BOSOR5 first selects an initial number of circumferential waves n o which he feels corresponds to the minimum bifurcation load. For this wave number n o the stability determinant IA(n 0, L) I is calculated for each load increment, as shown in fig. 2(a). The load, L, is increased until one or more eigenvalues are detected between two sequential load steps or until the maximum allowable userspecified load has been reached. Under normal circumstances, at this point in the calculations a series of eigenvalue problems of the form [A(n) + XnB(n)]Xn = O,

2.1. Axisymmetric prebuckling analysis In the axisymmetric prebuckling analysis, large deflection effects and elastic-plastic material behavior are simultaneously accounted for by means of a double iteration loop. In the inner loop the nonlinear equations, including terms due to moderately large deflections are solved by the Newton method. Material properties are held constant in this loop. In the outer loop the material properties are updated by means of a subincremental process described in detail in [5]. This subincremental process permits the use of rather large load increments without excessive loss of accuracy in the solution. Plasticity calculations are based on the yon Mises yield criterion and associated flow rule with isotropic strain hardening. Incremental flow theory is always used for the prebuckling analysis. Iterations over the inner and outer ,loops continue at a given load level until the displacement vector converges within a certain prescribed amount. In this way the favorable convergence property of the Newton procedure is preserved, equilibrium is satisfied within the degree of approximation inherent in a discrete model, and the flow law of the material is satisfied at every point in the structure at every load step. The effect of the rotation of the surface on the work done by the pressure during deformation is also included in the analysis.

(1)

is set up and solved, where A(n) = the stiffness matrix corresponding to n circumferential waves of the structure as loaded by L 1 (see following definitions and fig. 2)

I.z

~ L t

o<

Lz

-i 03 LOAD,L

i

(o)

nMtN nCRIT no nMAX

LOAD,L

MAX. LOAD

nMIN CRedIT. n nO nMAX

CIRCUMFERENTIALWAVENUMBER,n (b)

(c)

Fig. 2.(a) Stability determinant at n = n o as function of load L. (b), (c) Eigenvalues as function of number of circumferential waves, n.

G. Lagae and D. Bushnell / Elastic plastic buckling of torispherical vessel heads B(n) = the load-geometric matrix corresponding to

LI

L2 An xn

n

the prestress increment resulting from the load increment L2 - LI = the load state just before the sign change of the stability determinant, or the second-tolast load = the load state just after the sign change of the stability determinant, or the last load = the eigenvalue = the eigenvector = the number of circumferential waves lying in a range n m i n ~ n ~ nmax, w i t h nmi n and nmax provided by the program user. Note that the initial guess n o also lies in the range n m i n ~ n o ~ nma x.

BOSOR5 computes the eigenvalues Xn and eigenvectors x n for nmi n ~ n ~ nmax in wave number increments of nincr (which is also supplied by the program user). Typical results are shown in figs. 2(b) and 2(c). If the precritical behavior is nonlinear, the tangential stiffness varies with the load. An eigenvalue analysis will therefore yield a rigorous solution to the bifurcation problem only if the tangential stiffness happens to be evaluated at the bifurcation buckling load, i.e., if the computed eigenvalue is zero. If the material were elastic and if geometric nonlinearity were mild, the buckling load would be very nearly Lcr = L 1 + Ancrit(Z 2 - L 1 ) .

(2)

However, especially because of the nonlinearity of the material, the tangent modulus of which often changes very steeply for stresses above the proportional limit, the eigenvalue Anerl t c a n n o t be used to calculate Let as in eq. (2) if plastic flow occurs before L 2. In problems involving plastic buckling, the purpose of the series of eigenvalue problems (1)is to find the critical circumferential wave number, ncrit , and to obtain buckling mode shapes. If the situation shown in fig. 2(b) exists, then it is known that L1 ~
407

this additional calculation, and the bifurcation buckling load is identified as being in the load interval LI ~
3. Description of test specimens and BOSOR5 models The specimens tested by Patel and Gill [3] were machined out of aluminium billets and had nominal dimensions as shown in fig. 3 and table 1. Imperfections in profile and thickness were measured as described in [3]. In the BOSOR5 analysis the nominal profile geometries are used. Thickness variations are handled in the followin~ way: for each meridional

408

G. Lagae and D. Bushnell / Elastic plastic buckling of torispherical vessel heads t = 0 • 254

CLAMPED

~'Y'~

(TYPICAL) ~"

SPEC.NO i Cc. ............. I....................... I 2 [. . . . . . . . . . . . .

~,

I/ /

....

3

\

,a

D: 1 3 5

\\,149dio

5 6

(Dimensions in ram)

Fig. 3. Test specimen nominal geometry.

8 9 10

station at which thicknesses were measured, the minimum thickness recorded in a series of measurements at 10 ° intervals around the circumference is used in the axisymmetric BOSOR5 model. The thicknesses between meridional stations where measured values are available are obtained by linear interpolation. Table 2 shows the thickness distributions used in the BOSOR5 analysis. The nodal points are uniformly spaced along each of the spherical and toroidal segments. The first node corresponds to the beginning of the segment and the last node corresponds to the end of the segment. The column headed " z " indicates the axial distance from the crown to the stations in

Fig. 4. Discretized models for the BOSOR5 analysis.

the cylindrical segment for which the thickness was measured. The initial and final z values correspond to the cylinder-torus juncture and the clamped edge, respectively. Fig. 4 shows the discretized models for the 10 specimens. The stress-strain curve for the aluminium is shown in fig. 5. In BOSOR5 this stressstrain curve is modeled as a sequence of straight lines connecting the points listed in fig. 5.

4. Results Table 1 Nominal specimen geometry Specimen no.

Torus Radius (mm) R (Inside)

Head Height (mm) H(Inside)

Sphere-Torus. Junction Angle (Degs.)

1 2 3 4 5 6 7 8 9 10

7.62 10.16 12.7 15.24 17.78 20.32 22.86 25.4 27.94 30.48

22.572 24.107 25.665 27.244 28.847 30.475 32.128 33.808 33.517 37.256

28.04 27.34 26.62 25.87 25.09 24.29 23.45 22.59 21.68 20.74

Fig. 6 depicts the prebuckling axisymmetric deformation of one of the heads and the meridional shape of the bifurcation buckling mode. This meridional shape of the buckling mode varies around the circumference as cos nO, where 0 is the circumferential coordinate and n is the number of full circumferential waves. Bifurcation buckling is caused by the band of axisymmetric circumferential compression that develops in that part of the toroidal knuckle which moves toward the axis of revolution. The experimental and theoretical bifurcation buckling pressures for the 10 specimens are listed in tables 3 and 4. Table 3 results are based on an axisymmetric model in which the thickness of the entire specimen is assumed to be constant and equal to the

409

G. Lagae and D. Bushnell / Elastic plastic buckling of torispherical vessel heads

Maximum Strains Occur Approx. Here

,8

I~Cylinder

.7 0 0

~ ~ T o r o i d a l

Knuckle

,6

x

~Spherical

C. ca m

.5

Normalized Stress-Strain Curve used in BOSOR5 Predicted Maximum Effective Strains at. 2,

pcr

.4

2 u

7

.5 LO

/

.2 / /

,I /

/ 0

0

(Stre-s--s/E)x 102

.°soo 3777

/

4 2 E=6.48x 104MN/m 2 V=/52

t ,I

I .2

Strain (%)

I .3

t .4

.3904 3995 :~997 • :2059 .4182 I i .5 .6 Effective

.ss .40

j .7

,47 60 80 2.00 IO,O0 ~ i .8 .9

1.0

II

12

I5

i4

Strain (%)

Fig. 5. Stress-strain curve and the maximum effective strain at the buckling pressure predicted by BOSOR5 with use of flow

theory.

Z

I

% I

....J (a) DISCRETIZED MODEL......--....

average value in the toroidal knuckle along the meridian corresponding to the circumferential station at which the first buckle occurred in the test. Table 4 results are based on an axisymmetric model in which the thickness varies in the meridional direction as listed in table 2. The table 4 results are incorporated into fig. 7. In fig. 7, PACT PCLEAR

PINCIPIENT (b) PREBUCKLING DEFLECTED / SHAPE (AXISYMMETRIC) /

j~ .

- f

- J

t (c) BUCKLING MODE (VARIES IN THE CIRCUMFERENTIAL DIRECTION AS COS 60 e) --

Fig. 6. Specimen no. 2 (a) Disctetized model; (b) exaggerated

pre-buckling deformation; and (c) Buckling mode.

= pressure at which the first buckle was fully developed, = pressure at which the first buckle could be felt by touching the surface of the specimen, = pressure at which the first buckle was detected by a sensitive probe revolved around the circumference at a station in the toroidal knuckle.

The maximum effective strains at buckling, which occur at the inner fiber in the toroidal knuckle, are shown in fig. 5. The points on the stress-strain curve correspond to the use of the flow theory option in the stability analysis. With the exception of specimens 2 and 9, the tests and BOSOR5 flow theory predictions are within 15% of the points in fig. 7 corresponding to PINCIPtENT. In the tests specimens 9 and 10 did not exhibit the

410

G. Lagae and D. Bushnell / Elastic plastic buckling of torispherical vessel heads

.~

e. o

,a

~

~

Z

G. Lagae and D. Bushnell / Elastic plastic buckling of torispherical vessel heads

8

! oo

c~

oo

~

o o o o o o

o o o o o o

O0

c~

o

~

~

~

~:

o

~

0

,.~

'~1 ~_~ ~

0

i.~ ~ •~ . ~

~

411

412

G. Lagae and D. Bushnell / Elastic plastic buckling of torispherical vessel heads

Table 3 Buckling pressures from test [3] and BOSOR5 with assumption of uniform thickness Bifurcation buckling pressure (MN/m 2) Specimen

1 2 3 4 5 6 7 8 9 10

Thickness (mm)

0.2134 0.2578 0.2459 0.2388 0.2578 0.2845 0.2362 0.2096 0.2380 0.3175

Test [3 ]

BOSOR5 b Bifurcation Pressure

(Incipient buckling pressure)

Flow theory

Deformation theory

Linear elastic

0.272 0.402 0.445 0.411 0.615 0.717 0.810 0.684 No buckling No buckling

0.30(40) a 0.47(40) 0.47 (40) 0.48(40) 0.61(30) 0.85(35) 0.72(35) 0.69(30) 1.00(20) No buckling

0.26(75) 0.40(55) 0.41 (50) 0.45(45) 0.58(35) 0.77(35) 0.68(35) 0.67(35) 0.96(25) No buckling

0.513(40) a 1.00(40) 0.937 (60) 0.894(55) 1.10(50) 1.45(45) 1.00(45) 0.817(45) 1.20(40) 2.64(30)

a Numbers in parentheses indicate circumferential waves. b Thickness assumed to be constant and equal to the average value in the toroidal knuckle along the meridian corresponding to the circumferential station at which the first buckle appeared in the test. Meridian shape assumed to the perfect.

Table 4 Buckling pressures from test [3] and BOSOR5 with meridionally varying thickness

0 - PACT D - PCLEAR •

- PINClPIENT

- B O S O R S - FLOW THEORY

Test [3]

•

o

•

(.9

• - B O S O R S - DEFORMATION THEORY

Bifurcation buckling pressures (MN/m 2)

1.0

BOSOR5 b Bifurcation Pressure

0,~ E]

Z

0

z

.J

<>

Specimen

(Incipient buckling pressure)

Flow theory a

Deformation theory a

"n~' O E

0

(/) o3

0.272 0.402 0.445 0.411 0.615 0.717 0.810 0.684 No buckling No buckling

"'0.7

0.27(45) 0.49(35) 0.42(45) 0.45(45) 0.60(35) 0.82(30) 0.77(35) 0.65(35) 1.15(20) No buckling

0.24(80) 0.39(60) 0.37(55) 0.42(50) 0.58(40) 0.78(35) 0.74(35) 0.63(35) 1.10(25) No buckling

A

nn O Z

[]

(..9

z 0.6

0

.J v o

• 0

[]

,.n 0.5

A

!

0.4

0 []

D

I

•

0.Z

0.2 4

a Numbers in parentheses indicate circumferential waves. b Thickness assumed to vary with the meridional coordinate only, as listed in table 2. Meridian shape assumed to be perfect.

0

•

CL

1 2 3 4 5 6 7 8 9 10

,e

•

1 L I t i2 16 20 24 TORUS RADIUS (ram)

I 28

52

Fig. 7. Variation in buckling pressure with radius of toroidal knuckle: comparisons of test results of [3] with BOSOR5 pre dictions for incipient buckling.

G. Lagae and D. Bushnell / Elastic plastic buckling o f torispherical vessel heads

225

273

ANGULAR POSITION (DEG) 58

. . 322. . 10. .

/

106

413

165

/

rn2 0

0.9 A

~ 0.2

Z~

f/

0.751 0.819 0.886 0.9t2 1.079 1.t40 1. 192

/

"

A

0.."

/

Fig. 8. Development of incipient buckles at the center of the toroidal knuckle in specimen 9.

zx

z 0.3

large buckles corresponding to PACT or PCLEAR in fig. 7. Fig. 8, taken from the data of Patel and Gill, shows the development of incipient buckles in specimen 9. Small amplitude circumferential waves form at about 106 ° at a pressure of about 0.89 MN/m 2 and at 273 ° at a pressure of about 1.14 MN/m 2. These waves never develop into visible buckles, however, and in [3] this specimen is judged not to have buckled at all. The BOSOR5 program predicts bifurcation buckling of specimen 9 at about 1.15 MN/m 2. It appears that if there is bifurcation from the primary axisymmetric state, the post-bifurcation load-deflection curve always has a positive slope in this case. No such incipient buckles formed during the test of specimen

P=1.4 MN/m 2 P=1.2

lo o=O 6

0.8

P: 1 . 4 ~ /

UNDEFORMED

. ~. ~

~-P=(~6

P = 1.4 MN/m2 Fig. 9. C o m p u t e r - g e n e r a t e d p l o t s o f the d e f o r m a t i o n o f

specimen I0 with increasing pressure.

0.2 INNER J---

0,1

0

46" O U T E R ~ - - BOSOR 5 FIBER ~ 0 TEST AT 46"

O.I G2 0.3 0.4 03 HOOP STRAIN (%) AT 4 6 "

Fig. 10. Inner and outer fiber joop strains in specimen 7 at the juncture between the spherical cap and the toroidal knuckle at a circumferential coordinate of 46 °.

10. BOSOR5 predicts that this specimen will fail because of large strains in the cylindrical portion of the head, as shown in fig. 9. This behavior was observed in the test. Specimen 7 was provided with gauges measuring inner and outer surface hoop strain at the sphere-torus juncture. Figs. 10 and 11 show the inner and outer fiber hoop strains as a function of pressure from the Patel and Gill test and as predicted by BOSOR5. The circumferential locations, 46 ° and 80 °, correspond to the two points on the circumference where the thickness was a minimum at the juncture between the spherical cap and the toroidal knuckle. The first buckle appeared at 46 ° , where the circumferential bending increased steeply with increasing pressure, as seen in fig. 10. Of course the BOSOR5 results in figs. 10 and 11 are identical, since the prebuckling model is axisymmetric. The difference between inner and outer fiber hoop strains in the BOSOR5 model arises from the meridonal rotation which is maximum at the sphere-torus juncture. In the test this difference is due

414

G. Lagae and D. Bushnell / Elastic plastic buckling of torispherical vessel heads

1.0 • /

0.9

/

0.8

|0.7

I,x

lu

~

0.6

(isotropic) may not be adequate to describe the actual plastic behavior. A comparison of tables 3 and 4 indicates that reasonably accurate predictions of incipient buckling can be obtained with models in which constant thickness is assumed, the thickness being taken as the average thickness along the toroidal knuckle medidian for which this average is minimum. The quality of the theoretical predictions of incipient buckling as well as the behavior of the test specimens as the pressure is increased above the incipient buckling pressure indicate that these types of vessels are not particularly sensitive to initial imperfections.

..I

I

a,. ,u.i tin

Acknowledgements

- - - - - - BOSOR 5

FIBER 0.1

O

OUTER FIBER

z~

TEST AT 80"

0

BOSOR 5 TEST AT 8 0 °

O.I 0.2 0.3 0.4 0.5 HOOP STRAIN (%)AT 8 0 °

Fig. 11. Inner and outer fiber hoop strains in specimen 7 at the juncture between the sperical cap and the toroidal knuckle at a circumferential coordinate of 80°. both to the meridonal rotation and circumferential bending caused by circumferentially varying thickness and profile.

5. Conclusions There is reasonably good agreement between test and theory for these specimens. Discrepancies may be due to the fact that the actual specimens were nonsymmetric because of circumferentially varying thickness and meridian profile whereas the BOSOR5 models are axisymmetric. Also, the material flow law (associated with von Mises yield surface) and hardening law

Professor S.S. Gill at the University of Manchester Institute of Science and Technology, Manchester, England, very kindly supplied the authors with detailed data on specimen geometry. Figs. 1, 3 and 8 are taken with permission from [8]. Dr. Lagae's effort was sponsored by the National Fund for Scientific Research of Belgium. The support of the Lockheed 1977 Independent Development Program is greatly appreciated. References

[1 ] D. Bushnell and G.D. Galletly, J. Pressure Vessel Tech., 99 (1977) 39. [2] E. Esztergar, "Development of Design Rules for Dished Pressure Vessel Heads", Welding Research Council Bulletin 241, 1976. [3] P.R. Patel and S.S. Gill, Int. J. of Mechanical Sciences, 20 (1978) 159. [4] A. Kirk and S.S. Gill, Int. J. of Mechanical Sciences, 17 (1975) 525. [5] D. Bushnell, Int. J. for Num. Meth. in Eng., 11 (1977) 683. [61 D. Bushnell, Int. J. of Solids Structures, 10 (1974) 1287. [7] D. Bushnell, Computers and Structures, 6 (1976) 221. [8] P.R. Patel, M.Sc. Dissertation, Faculty of Technology, University of Manchester (1976).