Equivalent nozzles in thermomechanical problems

EQUIVALENT NOZZLES IN THERMOMECHANICAL PROBLEMS

F. CESARI

CNEN, Bologna, Italy

& T. K. HELLEN Berkeley Nuclear Laboratories, CentralElectricity Generating Board, Berkeley, Glos., Great Britain (Received: 4 September, 1978)

ABSTRACT

Axisymmetric nozzle-sphere geometry models for finite element analysis are described which are equivalent to three-dimensional nozzle-cylinder intersections in the sense of giving the same maximum equivalent stress. This equivalence is obtained by varying the radius of the sphere. Both mechanical and thermal load cases typical of actual operating conditions are considered, although it is shown that it is not possible to use the same equivalent model on both types of load. The implications are that cheaper computing costs couM be obtained for expensive non-linear calculations provided the loss in accuracy of the results through using the equivalence model was not too severe.

INTRODUCTION

During the stress analysis of pressure circuits such as those which occur in nuclear vessels, it is frequently necessary to study nozzle-cylinder intersections. For elastic analysis, it is now a straightforward venture to produce accurate results with mechanical or thermal loads using three-dimensional finite element analysis. In fact, automatic mesh generation procedures are available to help minimise the effort required by the analyst. Although computer costs are not large for elastic material behaviour, the extension of the analysis into non-linear plasticity or creep can become very expensive. Consequently, any technique which can solve these non-linear problems at 309 Int. J. Pres. Ves. & Piping 0308-0161/79/0007-0309/$02.25 © Applied SciencePublishers Ltd, England, 1979 Printed in Great Britain

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F. CESARI, T. K . H E L L E N

cheaper cost is of interest, particularly if the loss in accuracy which may be incurred can be predicted. The purpose of the present work is to replace the threedimensional model by a two-dimensional one, in this case a nozzle-sphere intersection. Although geometrically different, the main characteristics, such as maximum stress, can sometimes be shown to behave in a similar manner between the two models. This similarity is investigated and a variable parameter representing the ratio of the radius of the sphere divided by the radius of the cylinder it replaces is investigated to find the best results fit. The present work is restricted to elastic analysis and deals with one geometry. In order to establish whether well-defined parameters exist for a series of nozzle-cylinder intersections, a suitable parametric survey would have to be carried out. For each, a repeat of the analyses detailed here would be required. The comparison o f two- and three-dimensional models in non-linear analysis also awaits investigation. The present work is intended to initiate analysis procedures for such parametric surveys.

T H R E E - D I M E N S I O N A L A N A L Y S I S OF T H E N O Z Z L E - C Y L I N D E R

INTERSECTION

The particular geometry studied consisted of a cylindrical vessel of radius R c = 191.4rnm and thickness T = 6-7mm, with a nozzle of radius r = 24.675 mm and thickness t = 1.350 mm (Fig. 1). Experimental results for this geometry subject to internal pressure have already been determined. 1 For the finite element analysis,

z

--I

t--t

r i I

R¢ r

I__

m

Fig. I.

.

Nozzle d i m e n s i o n s .

,------.m-

EQUIVALENT NOZZLES IN THERMOMECHANICAL PROBLEMS

311

symmetry considerations, both for the pressure and thermal loading cases, allowed consideration of only one-quarter of the complete structure (Fig. 2). The modelling required 96 three-dimensional isoparametric elements, each with 60 degrees of freedom. A total of 661 nodes, or 1983 degrees of freedom, was involved. The mesh was considered completely adequate for the mechanical loading case and sufficient for the thermal case. In view of the more complicated equivalent loading system inherent in practical temperature cases, one would require a finer mesh for an equivalent accuracy to the pressure case but such action was not considered necessary for the present exercise. Both loading cases were analysed using the BERSAFE system. 2 In the mechanical load case, the maximum value of the stress concentration

Fig. 2.

Three-dimensional mesh for nozzle-cylinder intersection.

F. CESARI, T. K. HELLEN

312

factor,f, was obtained at the inside surface at the union of the nozzle and the main vessel's axial direction (in the cylinder-plane intersection). With reference to the hoop stress in the main vessel, the value calculated was 3.37, which is in good agreement with the experimental results' and Money's empirical formula 3 which gives a value of 3.5. This verifies the assertion of the mesh being very adequate for this kind of loading. The total computing time required was six minutes of CPU time on an IBM 370/168 computer. In the thermal load case, the temperature distribution considered was obtained from a transient typical of an accident in a sodium-cooled fast reactor. Figure 3

T(°C) 540

NozzLe of PEC vessel, 520

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480

Data

460

E = 1,7.10 5 N / r a m 2

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:

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380

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2o Fig. 3.

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30

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~

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so

60

W/ram 2*C

t(sec)

Thermal transient history.

shows the temperature trend for the sodium inside the vessel with an insulating argon temperature constant at 525 °C. The maximum value of the equivalent stress was found to be at time 33 sec in the inside surface at the union of the nozzle and the vessel's axial direction, with a magnitude of 113 N/mm 2. This corresponds to a temperature drop, AT, in the sodium, where: AT-

~r(1- v) E~

- 25 °C

and E, e, v are Young's modulus, the coefficient of linear expansion and Poisson's ratio, respectively. Including the temperature transient analysis, the total CPU time was 12 min.

EQUIVALENTNOZZLESIN THERMOMECHANICALPROBLEMS

313

TWO-DIMENSIONALANALYSISOF THE NOZZLE-SPHEREINTERSECTION An axisymmetric finite element representation of a cylinder-sphere intersection has been made. The dimensions of all geometrical parameters except the radius R~ of the sphere were kept identical to the three-dimensional model. Defining ~ = Rs/Rc, several axisymmetric meshes, with varying values of ~, were used to assess the maximum stress concentration factor. The analyses were also performed usinlg both three-node (linear displacement) and six-node (quadratic displacement) triangular elements without changing the mesh (Fig. 4). In each case there were 799 elements. The mechanical load consisted of the same internal pressure as used in the threedimensional case. The resulting curves of maximum stress concentration factor against ~ for both types of finite element are shown in Fig. 5. For the six-noded elements, the required value of ~ is slightly above 3 whilst for the three-noded elements (which are therefore the less accurate results) ¢ is 3.5. The equivalent stress contours for ~ = 1 are shown in Fig. 6 along with a three-dimensional section plot

Fig. 4. Mesh for cylinder-sphereintersection (~ = 1).

314

F. CESARI, T. K. H E L L E N

3-node elements=475 nodes 739 elements CPU time - 6' 6-nocle elernents=1750nocles 739elements CPU t i m e - 1 2 ' f:

°

6-node ~/ etement/S~

I

I ~/37node /~/'~-~ement

I

i

|

1

2

3

= R$1Rc

Fig. 5. Mechanical load.

through that section containing the highest SCF. It is seen that the general behaviour of the time sets of stress contours agrees well. The contour values vary in a manner consistent with the different ~ values used, so that ~ = 1 results are between one-third and a quarter of the three-dimensional results.

!

~'

3

2

4

Key

1 2 3

0.33 0.40 0.47

6 7 8 9 10

0.60 0-67 0.74 0,81 O' 87 0-94

Fig. 6(a). Equivalent stress contours for two-dimensional mesh with ~ = 1.

EQUIVALENT NOZZLES IN THERMOMECHANICAL PROBLEMS

315

Key Equi~ stress

r-1

II 1

factor, f 0.40

I

2

0.75

,l

3

/-,

1.09 1 ./.3 1 -78 2-12 2 4,7 2.81

5 6

7 8

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3

7 6

Fig. 6(b).

5

z

4

3

Equivalent stress contours for three-dimensional mesh.

In the thermal load cases, using the same temperature distribution at time 33 sec as before, Fig. 7 shows the variation of the maximum stress as a function of ~ for the mesh of three-noded triangles. In Fig. 8, the maximum stresses found from both types of element are plotted against ~; in this case it is seen that the two sets of results remain close together, as in the mechanical load case, but they only approach the three-dimensional results very slowly, indicating equivalence with a large value of (,-, 10).

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[=3

60

11o 5o 4O

Fig. 7.

I

I

I

I

2o

30

4o

50

~.

t ,

Sec.

Thermal transient with three-noded element.

)

316

F. CESARI, T. K. HELLEN

3-D solution

110

100

~ 6-node

elements

3-node

elements

90

80

70

E

60

z x o

50

z,¢

30

20

10

t 1

Fig. 8.

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I 3

I "

I 5

M a x i m u m equivalent stress (t = 33 sec) with thermal load.

EQUIVALENT NOZZLES IN THERMOMECHANICAL PROBLEMS

317

CONCLUDING REMARKS

The use of a two-dimensional axisymmetric nozzle-sphere intersection for a threedimensional nozzle-cylinder intersection to produce equivalent maximum stress concentration factors has been shown to be valid for the internal pressure case. The sphere had to have a radius some three times greater than the cylinder. For the thermal case, however, the values of ~ in this range gave maximum stresses some 20 per cent lower than in the three-dimensional case. Values of ~ ~ 10 would be required for the same order of accuracy as in the mechanical loading case, implying a large spherical radius such as to make it effectively a fiat plate. Hence, although again a useful result is obtained by extrapolation in the thermal case, it is not possible to identify a single two-dimensional approximation suitable for both mechanical and thermal loading. For non-linear analysis of nozzle-cylinder intersections, suitable parametric surveys will be required to ascertain whether or not usable ¢ parameters can be obtained for both mechanical and thermal loading. Such a survey would inevitably be expensive in computing costs, but potential savings would be effected if welldefined parameters like ~ were identified for subsequent analyses of geometries within the framework of the survey.

ACKNOWLEDGEMENTS

This paper is published by permission of the Central Electricity Generating Board.

REFERENCES 1. FAVRETTI,CURIONI,Ricerca Estensimetrica Sul VesselDel Reattore PEC, Inst. Mecc. Appl. Facolta di Ingegneria Universita di Bologna, 1974. 2. HELtEN, T. K. and PROTHEROE,S. J., The BERSAFE Finite Element System, Comp. Aided Design, 6 (1974), pp. 15-24. 3. MONEY,H. A., Designing flush cylinder to cylinder intersection to withstand pressure, 68-PVP-17, ASME Paper, 1968.