Finite element dynamic bifurcation buckling analysis of torispherical head of BWR containment vessel subjected to internal pressure

Nuclear Engineering and Design 133 (1992) 245-251 North-Holland

245

Finite element dynamic bifurcation buckling analysis of torispherical head of BWR containment vessel subjected to internal pressure * N. M i y a z a k i a, S. H a g i h a r a a, T. U e d a a, T. M u n a k a t a a a n d K. S o d a b

"Department of Chemical Engineering Faculty of Engineering, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812, Japan b Department of Fuel Safety Research, Japan Atomic Energy Research Institute, Tokai, lbaraki 319-11, Japan Received 24 June 1991

In this paper the bifurcation buckling pressure for the torispherical head of the Mark II type BWR containment vessel subjected to dynamically applied internal pressure is calculated, using a finite element program for a dynamic analysis. Three kinds of dynamic loadings, that is, step loading, ramp loading and pulse loading are considered in the present analysis. The minimum bifurcation buckling pressure is predicted for the respective loadings. The minimum bifurcation buckling pressure for dynamic loading is much lower than the bifurcation buckling pressure for static loading.

1. Introduction A torispherical shell is known to buckle with circumferential waves in a knuckle region, when it is subjected to excessive internal pressure [1,2]. The circumferential waves observed are caused by bifurcation buckling. A torispherical shell is utilized as a head of a B W R containment vessel. It may buckle due to overpressure caused by severe accidents. Dynamic loading may act on a containment vessel due to postulated hydrogen or steam explosion. Larger deflection is induced under dynamic loading than the deflection predicted by a static analysis [3-5]. In general, the bifurcation load is lower in the case of dynamic loading than in the case of static loading. Therefore, it is important to obtain the bifurcation load for a torispherical head due to dynamic loading from the viewpoint of estimating safety margin of a B W R containment vessel under severe accident conditions. In the present paper, we calculate the bifurcation buckling pressure for the torispherical head of the Mark II BWR containment vessel, using a finite element program for a dynamic analysis. Three kinds of dynamic loadings, that is, step loading, ramp loading

* Presented at SMiRT-11 Conference, Division J.

and pulse loading are considered in the present analysis. The minimum bifurcation buckling pressure is predicted for the respective loadings.

2. Method of analysis An elastic-plastic bifurcation buckling analysis under dynamic loading consists of two parts, that is, a pre-buckling dynamic deformation analysis and a bifurcation buckling analysis.

2.1. Pre-buckling dynamic deformation analysis In the pre-buckling dynamic deformation analysis of the axisymmetric mode, we used the finite element equation of motion. The geometrical nonlinearity was taken into account in the strain-displacement relations, and the J2 flOW theory with the isotropic strain hardening rule was used for the stress-strain relations in the plastic range [6]. Then, the finite element equation of motion of a total system is given as follows:

MAd'+ CAd + K A d = A F + A F * ,

(1)

where M is a mass matrix, C is a damping matrix, K is a stiffness matrix, A F is an incremental nodal force

0 0 2 9 - 5 4 9 3 / 9 2 / $ 0 5 . 0 0 © 1992 - E l s e v i e r S c i e n c e P u b l i s h e r s B.V. A l l rights r e s e r v e d

246

N. Miyazaki et al. / Finite element dynamic bifurcation buckling analysis

vector, A F * is a residual force vector, 3 d is an incremental nodal displacement vector, A d is an incremental nodal velocity vector, and ad" is an incremental nodal acceleration vector. The stiffness matrix K is composed of the followings: K~°) :elastic small displacement stiffness matrix, K(0) K )d p(0 :plastic stiffness matrix, :initial displacement stiffness matrix, K}°) :initial stress stiffness matrix, where the superscript (0) indicates the axisymmetric mode. The Newmark-/3 method was applied to the time integration of equation (I). Then, the following equation is derived from equation (1):

= ar + aF* +

(4

)

5-;M + 2C a. + 2Md;~,

(2)

where the subscript n indicates the value relevant to the time step n. We can obtain the incremental nodal displacement vector at the time step n by solving equation (2). The total nodal displacements are obtained by accumulating the incremental ones at each time step. The incremental nodal acceleration vector and the incremental nodal velocity vector can be calculated from the incremental nodal displacement vector. 2.2. Determination of bifurcation buckling To examine whether the bifurcation buckling with n circumferential waves occurs or not, we stop time marching [7] and add the following infinitesimal displacement 3U to the equilibrium state obtained from the pre-buckling deformation analysis.

where K ~') is the stiffness matrix relevant to n circumferential waves and 6U (n)= {~u ~n) ~3w~) ~c~}. This condition can be replaced by the following equation: det[K ~")] = 0 ,

(5)

where det[ ] is the determinant of a matrix. It is assumed that loading condition continues even after the infinitesimal displacement 6U is given to the system, which is so-called consistent loading [8]. In obtaining K (n), we need the stress-strain relations in the plastic range. The incremental form of the J2 deformation theory [9] was adopted as the plastic theory in the present bifurcation buckling analysis, because it gives more appropriate results than the J2 flow theory, as pointed out by Bushnell, and Roche and Autrusson [8,10].

3. Results and discussion

Figure 1 shows the dimensions of the torispherical shell of the head employed in the Mark II type BWR containment vessel [11]. The dimensions of the analyzed one are as follows: Crown radius of the shell: R S= 8.734 m, Knuckle radius of the shell: r = 1.657 m, Diameter of the attached cylindrical shell: D = 9.652 m, Height of the attached cylindrical shell: L = 2.244 m, Thickness of the shell: t = 0.024 m.

I

~U= {~u ~t, ~w} = {6u ~) cos nO 6w (~) cos nO 6v (n) sin nO},

(3)

where the infinitesimal displacement components u, w and c, denote the meridian in-plane, meridian out-ofplane, and circumferential displacements, respectively, and 0 is the circumferential angle. The superscript (n) indicates a quantity relevant to the bifurcation buckling with n circumferential waves. The bifurcation buckling with n circumferential waves can be predicted when the following equation has a nontrivial solution. K(n)~ U (~) = 0,

(4)

Fig. 1. Dimensions of a torispherical head.

247

N. Miyazaki et al. / Finite element dynamic bifurcation buckling analysis

J

xl 0 -2 ,3

t

. . . .

i

. . . .

i

. . . .

. . . .

2~_ STEPLOAD Point A

E I--Z

f

i

• Bifurcation Point

"~ L.ul 0 .<

1

-

--J

~0 c~

1.037MPa

--~

~

i

i

J

0

I

r

i

i

i

5

I

i

i

10

i

i

I

15

i

i

i

i

2O

TIME (ms)

Fig. 2. Locations and directions of the displacements plotted in the followingfigures.

Fig. 4. Radial displacement of the point A versus time curves for step loading.

A bilinear approximation of the stress-strain curves was used in the present analysis. The Young's modulus E and the Poisson's ratio u, the yield stress O-y,the rate of strain hardening H ' and the density p were determined from the data of Isozaki et al. [12]. Young's modulus: E = 1.90 x 105 MPa, Poisson's ratio: u = 0.3, Yield stress: o-y = 3.98-102 MPa, Rate of strain hardening: H ' = 1.50 × 103 MPa, Density: p = 7800 k g / m 3. A half of the shell was divided into 180 axisymmetric thin shell elements. A finite element mesh was used at the knuckle region.

Three points were selected to show the results of analysis, as shown in fig. 2, where the points A, B and C are the middle of the attached cylindrical shell, the middle of the knuckle region and the top of the torispherical shell, respectively. The arrows in fig. 2 indicate the positive direction of the displacement at the respective locations. Three kinds of dynamic loadings, that is, step loading, ramp loading and pulse loading were considered in the present analysis as the internal pressure applied to the torispherical shell. The schematic figures of these loadings are given in fig. 3 where Tr and T o denote the ramp time and the pulse width, respectively. The displacement versus time curves for the points A, B and C are respectively shown in figs. 4, 5 and 6 in the case of step loading. As shown in these figures, the minimum bifurcation buckling pressure is 1.037 MPa for step loading, and the bifurcation buckling can not be induced by the 1.0 MPa

Step Load

Ramp Load

Pulse Load

(D

E

a-

I

I

I

I

f

mr

Time

Time Fig. 3. Three kinds of dynamic loadings.

°0

Tp Time

248

N. Miyazaki et al. / Finite element dynamic bifurcation buckling analysis

X10-2

xl 0-2

0.0 r \ v

-0.5

STEPLOAD Point B

3

\\

. . . .

i

. . . .

t

. . . .

t

RAMP LOAD (lOms) Point A

• Bifurcation Point

. . . .

L

.

.

.

.

• BifurcationPoint

k--

l---

z taA

1.7MPO

21

~ -1,0

(D ,< ....-I CL

g -1.5

O

1.3MPa

V?

-2.0 0

7,

5

,,

10 TIME (ms)

, 15

--1

i

20

Fig. 5. Radial displacement of the point B versus time curves for step loading.

internal pressure. It seems that the displacements at the point A, which is the middle of the attached cylindrical shell, oscillates regularly, while those at the other points oscillate irregularly. The period of the regular oscillation is in good agreement with the natural period of the attached cylindrical shell calculated by the simplified method, 6.61 ms [13]. The bifurcation buckling seems to be induced by the large negative displacement of the knuckle region, as shown in fig. 5. We confirmed large plastic zone extended in the knuckle region at the bifurcation buckling point in each case. Figures 7, 8 and 9 show the displacement versus time curves for the points A, B and C, respectively, in the case of the ramp loading with the ramp time T, = 10 ms. As shown in these figures, the minimum

.

.

,

i

.

.

.

.

~

5

.

.

.

.

r

.

.

.

.

i

10 15 TIME (ms)

,

,

,

i

2O

25

xl 0 -2

RA ;' o D

0.o

........ B

~. -0.5 E

• BifurcationPoint

Z -1.0 w hi

o< -1.5 \~',, 1.627MPa

-2.0 -2.5

0

1 . 7 M P O ~ "£% ......... 10 15 20 TIME (ms)

5

25

Fig. 8. Radial displacement of the point B versus time curves for the ramp loading with Tr of l0 ms.

xl 0 -2 .

.

t

.

.

.

.

t

.

STEP LOAD P Point C / ~

.

.

.

i

.

.

.

.

6

• Bifurcation Point vE

1.3MPa

~3

,

Fig. 7. Radial displacement of the point A versus time curves for the ramp loading with T, of l0 ms.

x10 -2 5

,

0

5

.

.

.

.

t

.

.

.

.

i

Pal nt C

.

.

.

.

i

.

.

.

.

i

.

.

.

.

/ / ~ ~

///I

F-

/~

~_J

~4

1.0MPO ,

Ld O3

62

• BifurcationPoint

1 ,

0

5

,

,

i

i

i

1'0 TIME (ms)

i

i

15

20

Fig. 6. Radial displacement of the point C versus time curves for step loading.

0

5

10

15 TIME (ms)

20

25

Fig. 9. Radial displacement of the point C versus time curves for the ramp loading with T~ of 10 ms.

N. Miyazaki et al. / Finite element dynamic bifurcation buckling analysis xlO 6 3.0

~ .

.

.

.

n

.

.

.

.

~

.

.

.

.

u

.

.

.

Table 2 Minimum bifurcation buckling pressure, buckling time and number of circumferential waves at buckling for pulse loading

.

STEP LOAD Point B

0.0

E t5 13v

2 -5.o

-9.0

0

. . . . . . . . . . . . . . 5 10 TIME (ms)

' .... 15

20

Fig. 10. Circumferential membrane force of the point B versus time curves for step loading.

Table 1 Minimum bifurcation buckling pressure, buckling time and number and circumferential waves at buckling for ramp loading. T, (ms)

Pm (MPa)

t b (ms)

n

0 5 10 15 20 25 30

1.04 1.24 1.63 1.76 1.66 1.69 1.77

7.15 10.87 14.55 25.24 28.10 31.86 38.17

33-37 26 25 24 23 24 23

Tr: ramp time, Pro: minimum bifurcation buckling pressure, th: buckling time, n: number of circumferential waves at buckling.

5.0

. . . .

i

. . . .

i

. . . .

r

. . . .

i

. . . .

i

Tp (ms)

Pm (MPa)

t h (ms)

n

1 2 3 4 5 10 15 20 25

4.41 1.27 1.19 1.10 1.15 1.41 1.50 1.65 1.75

2.760 5.985 5.743 5.956 6.558 7.627 9.900 12.27 15.07

3 15 15 15 20 29 26 26 26

Tp: pulse width, Prn: minimum bifurcation buckling pressure, tb: buckling time, n: number of circumferential waves at buckling.

bifurcation buckling pressure is 1.627 MPa. T h e same trends as in the case of step loading can be found in this case. T h a t is, the regular oscillation which is corresponding to the natural period of the a t t a c h e d cylindrical shell is f o u n d at the point A, as shown in fig. 7, and the bifurcation buckling occurs w h e n the large negative displacement is induced at the knuckle region, as shown in fig. 8. Figure 10 shows the circumferential m e m b r a n e force at the knuckle region versus time curves. T h e bifurcation buckling occurs w h e n the circumferential m e m b r a n e force at the knuckle region is negative. Table 1 shows the m i n i m u m bifurcation buckling pressure, Pro, the buckling time, tb, and the n u m b e r of circumferential waves at the buckling, n, for r a m p loading. Step loading is a special case of ramp loading, that is, Tr = 0. T h e m i n i m u m bifurcation buckling pressure versus the ramp time Tr are shown in fig. 11 for

5.0

. . . .

RAMP LOAD

•

4.0

4.0

"~" 3.0

"~ 3.0

O_

Static

E2.0 1.0'

0

PULSE LOAD

Stoti c

d2.o

13_

0.0

1.0

............................. 5 10 15 (ms)

20

25

249

30

]-r

Fig. 11. Minimum bifurcation buckling pressure for ramp loading.

°°°

0 . 0

.

0

.

.

.

i

5

,

.

,-.

,

10

.

.

.

.

i

.

.

15 Tp (ms)

.

.

i

20

.

.

i

i

I

25

i

i

i

i

30

Fig. 12. Minimum bifurcation buckling pressure for pulse loading.

250

N. Miyazaki et al. / Finite element dynamic bifurcation buckling analysis i

x l 0 -2 2

. . . .

,

. . . .

,

. . . .

,

i

~

i

. . . .

,

STATIC

1.o

Point A E

E

p..-

7 k~A W 0

3

0.5

-J dL

• Bi furcati on Point PULSE LOAD ( S m s )

0.0

1.1MPo

I --~

i

i

i

i

I

0

,

,

i

I

I

5

,

i

~

J

I

10

i

i

i

i

15

I

I

I

I

Cylinder l Knuckle l

2O

Distance

TIME (ms)

from

I

I

SphericolCop

I

cldmped edge

Fig. 13. Radial displacement of the point A versus time curves for the pulse loading with To of 5 ms.

Fig. 15. Radial amplitude of the bifurcation buckling mode for the static loading.

ramp loading. The bifurcation buckling pressure and the circumferential waves obtained from a static bifurcation buckling analysis are 1.91 MPa and n = 22-24, respectively. As shown in fig. 1 I, the minimum bifurcation buckling pressure for the dynamic loading are lower than the bifurcation buckling pressure for static loading. W h e n the ramp time becomes longer, the minimum bifurcation buckling pressure for ramp loading approaches to the static bifurcation buckling pressure. On the other hand, it approaches to the minimum bifurcation buckling pressure for step loading, Tr = 0, when the ramp time decreases to zero. Table 2 shows the results for pulse loading. The minimum bifurcation buckling pressure versus the pulse width, Tp, for pulse loading are shown in fig. I2. It is found from the figure that the minimum bifurcation buckling pressure for pulse loading gradually increases with the increase of the pulse width after it has the minimum

point around Tp = 4 ms and approaches to the static bifurcation buckling pressure. On the other hand, the minimum bifurcation buckling pressure tends to infinite when the pulse width approaches to zero. Figures 13 and 14 show the time variations of the displacement at point A in the cases of pulse loadings with a pulse width Tp of 5 ms and 10 ms, respectively. The displacement oscillation is damped gradually in the case of Tp = 10 ms. On the other hand, the displacement is not damped in the case of To = 5 ms and it is larger than that in the case of Tp = 10 ms. Figures 15, 16 and 17 show the bifurcation buckling mode for static loading, for the pulse loading with Tp = 5 ms and for the pulse loading with Tv = 10 ms, respectively. It is found that the circumferential waves are observed at the knuckle region for static loading and for the pulse loading with Tp = 10 ms, but they are observed at the middle of the attached cylindrical shell for the pulse loading with T o = 5 ms. It is supposed that in the last

x l 0 -2 2

. . . .

,

,

,

,

,

,

. . . .

,

1.o

. . . .

PULSE L O A D ( S m s )

Poi nt A E 1

_l.5MPa ~ / / 1 . 4 0 6 M P a

z

g o.5

g

d_ t23

--1

• BifurcationPoint PULSE LOAD(1 0ms)

- 2

J

i

~

i

i

5

i

i

i

i

0.0 i

i

i

10 TIME (ms)

, i

~

i

15

i

i

i

I

i

20

Fig. 14. Radial displacement of the point A versus time curves for the pulse loading with To of l0 ms.

I

Cylinder

I

I Knuckle

Distonce from

I

I

I

I

I

SphericaICcp

clamped edge

Fig. 16. Radial amplitude of the bifurcation buckling mode for the pulse loading with Tp of 5 ms.

N. M~vazaki et aL / Finite element dynamic bifurcation buckling analysis

251

References

1.0

LSE LOAD(1Ores)

0.5

0.0 I

I

I

I

I

I

I

Cylinder I Knuckle [ SphericGICap Distance from clamped edge Fig. 17. Radial amplitude of the bifurcation buckling mode for the pulse loading with To of 10 ms.

case the bifurcation buckling is due to the large deformation to the inward radial direction of the attached cylindrical shell caused by resonance because the pulse width Tp = 5 ms is nearly corresponding to the natural period of the attached cylindrical shell.

4. Concluding remarks We calculated the bifurcation buckling pressure for the torispherical head of the Mark II type B W R containment vessel, using a finite element program for a dynamic analysis. Three kinds of dynamic loadings, that is, step loading, ramp loading and pulse loading are considered as applied internal pressure in the present analysis. The minimum bifurcation buckling pressure for dynamic loading is much lower than the bifurcation buckling pressure for static loading. Especially, the minimum bifurcation buckling pressure for step loading is about a half of the bifurcation buckling pressure for static loading. When the ramp time or the pulse width becomes longer, the minimum bifurcation buckling pressure for dynamic loadings approaches to the static bifurcation buckling pressure. In the case of pulse loading, the minimum bifurcation buckling pressure tends to infinite when the pulse width approaches to zero.

[1] D. Bushnell, Plastic buckling of various shells, ASME Journal of Pressure Vessel Technol. 104 (1982) 51-72. [2] G.D. Galletly, A simple design equation for preventing buckling in fabricated torispherical shells under internal pressure, ASME Journal of Pressure Vessel Technol. 108 (1986) 521-526. [3] J.A. Stricklin, J.E. Martinez, J.R. Tillerson, J.H. Hong and W.E. Haisler, Nonlinear dynamic analysis of shells of revolution by matrix displacement method, AIAA Journal 9 (1971) 629-636. [4] S. Nagarajan and E.P. Popov, Non-linear dynamic analysis of axisymmetric shells, Internat. J, for Numer. Methods in Engrg. 9 (1975) 535-550. [5] R.E. Ball and J.A. Burt, Dynamic of shallow spherical shells, ASME Journal of Appl. Mechs. 40 (1973) 411-416. [6] N. Miyazaki, S. Hagihara, S. Tanaka and T. Munakata, Elastic-plastic creep buckling analysis of a cylindrical shell under axial compression by the finite element method, Technology Reports of the Kyushu University 60 (1987) 193-200. [7] N. Akkas, Asymmetric buckling behavior of spherical caps under uniform step pressure, ASME Journal of Appl. Mechs. 39 (1972) 293-294. [8] D. Bushnell, Bifurcation buckling of shells of revolution including large deformations, plasticity and creep, Internat. J. of Solids and Struct. 10 (1974) 1287-1305. [9] N. Miyazaki, S. Hagihara and T. Munakata, Elastic-plastic creep buckling analysis of a partial spherical shell by the finite element method, in: Computational Plasticity, Part I (Pineridge Press, Swansea, 1989), pp. 687-698. It0] R.L. Roche and B. Autrusson, Experimental tests on buckling of torispherical heads and methods of plastic bifurcation analysis, ASME Journal of Pressure Vessel Technol. 108 (1986) 138-145. [11] L.G. Greimann, F. Fanous, A. Wold-Tinsae, D. Ketalaar, T. Lin and D. Bluhm, Reliability analysis of steel-containment strength, NUREG/CR-2442 (1982). [12] T. Isozaki, K. Soda and S. Miyazono, Structural analysis of a Japanese BWR Mark-I containment under internal pressure loading, Nucl. Engrg. Des. 104 (1987) 365-370, [13] T. Isozaki, K. Soda and S. Miyazono, Structural analysis of Japanese PWR steel containment under inner pressure loading, Nucl. Engrg. Des. 126 (1991) 387-393.