Assessment of thermal stresses and ratchetting in reactor vessels

ht. J. Pres. Ves. & Piping 61 (1995)41l-425

0308-0161(94)00118-9 ELSEVIER

ElsevierScienceLimited Printedin NorthernIreland 030%0161/95/$09.50

ASSESSMENTOFTHERMALSTRESSES ANDWYTCHETTING IN REACTORVESSELS R J du Preez, Dept. University

of Mechanical

Engineering

of Pretoria

June 14,1994 Abstract Reactor vessels are often subject to “hot spots” developing at certain positions on the shell, which, if they are of a cyclic nature, may result in strain growth and fatigue failure if the amplitude of the cyclic strains are not restricted. The paper deals with an investigation of a particular vessel with the object of determining the characteristics of “hot spots” that would cause cyclic plasticity or ratchetting in order to determine the limiting criteria to avoid or restrict such effects onder continuous operational conditions. The basis of the investigation is finite element analysis. Temperature dependent elasto-plastic material properties are taken into account and the mesh is refined in selected “hot spot” positions in order to obtain accurate strain and stress states under cyclic loading conditions. The assessment.criteria are discussed which are the basis of recommendations for continuous operation under cyclic conditions. 1

INTRODUCTION

This paper deals with the investigation of stresses caused by localized “hot spots” in the shell of a reactor vessel, using finite element analysis methods. The main objective was to establish maximum tolerable temperature variations in the shell as determined by the stress/strain limits set by safety requirements of the reactor. The occurance of “hot spots” at certain positions on the shell, may, if they are of a cyclic nature, result in strain growth and fatigue failure if the amplitude of the cyclic strains are not restricted. The analysis therefore determines stress and strain histories in and around the “hot spots” due to cyclic loading from internal pressure and thermal expansion. As an example, a particular vessel is investigated with the object of determining the characteristics of “hot spots” that would cause cyclic plasticity or ratchetting. The main results are limiting criteria which avoid or restrict such effects onder continuous operational conditions.

4If

412

R. J. du Preez

2

SCOPE

OF THE INVESTIGATION

The investigation was divided into three parts: (a) The developement a finite element model for the reactor shell, (b) elasto-platic analyses for the worst location of the “hot spot” for average temperatures of 350 and 180 “C and “hot spot” temperatures varying from 350 to 600 ‘C, with no temperature gradient across the shell wall thickness, (d) elasto-plastic analyses for the worst location “hot spot” for selected cases including a temperature gradient across the shell wall thickness caused by external cooling.

3

FINITE

ELEMENT

MODEL

The reactor shell WAS modelled with QUAD4 thin shell elements, Since the nozzles were excluded, the shell was axisymmetrical, and it was assumed that the “hot spot3 could be positioned in one vertical plane. The model could therefore be restricted to a 180” sector of the reactor (with the plane of symmetry cutting through the centers of the “hot spots”). A typical “hot spot” as modelled is show in Fig, 1. The model extended only over the cylindrical part and the upper conical part of the vessel, with the “hot spot” some distance below the junction of the two parts. At the lower boundary the shell was assumed to be fixed vertically and against tangential rotation, but free to move radially. At the upper boundary the shell was assumed to be fixed against tangential rotation, but free to move radially. On the plane of symmetry displacements normal to the plane, and rotation vectors in the plane were fixed. The total loading of the shell consisted of two parts, a constant internal pressure and a constant temperature distribution with superimposed “hot spots”. The amplitude and distribution of the temperature within a “hot spot” was described by a sine function in terms of a radius r from the centre of the circular “hot spot” and in terms of a temperature amplitude in the “hot spot” of AT, as shown in. Fig 2.’ For the elasto-plastic analysis three cycles of thermal loading was defined in terms of linear variations from lower bound to upper bound values of maximum temperatures in the “hot spots”. The material stress-strain characteristics were approximated as bi-linear curves defined in terms of the elastic modulus, the Poisson’s ratio, the yield stress, and the plastic modulus (slope of the g - e curve beyond the yield point), as a function of temperature. The characteristics used are shown in Fig. 3. A summary of the material properties is given in Table 9.1.

4

ELASTIC-PLASTIC

ANALYSIS

An initial elastic analysis indicated that the stresses around a “hot spot” were higher than the yield stress of the material at the corresponding temperatures. It was therefore appropriate to perform elastic-plastic analyses to obtain more accurate solutions for the stress levels in and around the “hot spots”. The basic assumptions for the analyses were as follows: l

The elastic-plastic behaviour of the material could be described with the stressstrain characteristics as shown in Fig. 3, including the temperature dependency as

Thermal stresses and ratchetting

413

Figure 1. Mesh

Hot spot temperature function I

io’ ....:........................ .:..tso.dat”

. .. . . .. . . .. . ... . . .. .. .. .. .. .. . .i . .. .. .. .. .. .. . .. . .. .. . . .. .. .. . . .

,............................... i ................................

.:._.............................

............................................................

.............................. .............................

.......................... \

\ 50

Figure 2.

100

150 radial distance [mm]

200

250

300

414

K. J. du Preez

340 ,

!

320

i

Stress-strainctirves for material . I I I I I ;:ml’ + i2(yc i i.........................i ... ...... .. f . . ...__ ;-,:r.t _......__............... i.,....,.._............... %!K..:t: “ma’ .Q-::” ; .4*: *.* Fmq ..*....- 1 *.-:

1

t 300

...................

i. ......................

.: +. . .... .. ..,*. c.: 7....... i.. ...................... j .*. *.

. ........................

$. .......................

/. ........................

.Cm5:...+:~ rm6” -)lt.-

..,

i /.p 44 “c i .*.; i.e. ... .~.~.~...~....~...~...~..~~~~~..~~~~~~~~~~~~~~~~~~~~~~ ........................................ i................. ,-z.<:+. ....................... i......................... j...................... i.’ d.: : .-. _.-. ; *.- ..j/.:: .* * ... : -.-. : ................. .......................... .....j.. ...................... i......................... i.. .................... . 26CI - ...................... ,.*.r .-.A’ ...................... .................... .-i:.,:z:. /-’ / _*.* : .**_a..“....i” ............j......................... .I: ............. ........-.K.~:I:t:::“:. /........................ 240 IL .....................i./* ......... .-- .._(......jIL.::y.~.~Ol~~ ................. -.; ................ ‘i” : . ..e ; .e.- ._.*.-. : “...... ....:........ i........................ . ........................ ......................... . .................. .... :‘........................i.. ...................... .&. .................... .................. 220 -. ..................... .............! * .............. ......... .‘Y ...*;.................... : ____..!; ------- *...-- 4 4t7 *c ............. ......................... . ...................... ..~..........~:.:...l...-.;.,j”‘.‘.~‘.:.‘.:. .........i’....................... 4...................... 200 ,.::::a.-- .............. ........................ **.>.---. ..*.J---------i ;*.i***-----t 48& i........................ . ........................ ......................... .&,,* ........... .................:.:.::.+ ,I.^ .. :.-:.::: ....... j......................... j......................... 180 -___*..; ----*+---------; +------: ***-+--------: j ........................ i........................ ...........,_---~;;‘-.‘;“‘“‘--“.f j--**------<-----........................ T........................ i’: ....................... 160 -__**--4, :" ........................... ......................I.........................~ ........................ i......................... i ...................... 140 -. .....................
280

120 0

0.005

0.01

Material

0.025

0.03

properties Temp

Yield stress ASME

“C

w4

20 93

275 254

149 204

242 232

260

224

316

Yield stress Adjusted w4 242

Elastic modulus

Expansion coeff.

(MPa/lOOO)

(*lo-y

224 213

205 200 197

204

192

12.67 13.91

190

14.72

217

197 191

186

15.39

371

208

183

181

15.89

427

199 188

175

176

165

170

16.23 (16.70)

181 172

159 152

068) (165)

(136)

(160)

have been graphically

extrapolated.

482 510 538 600 Note:

0.02

plastic strain ep

Figure 3.

9.1

0.015

values in brackets

9.76 11.50

(16.82) (17.15) (17.40)

0.035

0.04

Thermal stresses and ratchetting

shown. These simplified bi-linear relationships plastic strain problems.

415

have proven to be adequate for small

b The hardening and cyclic behaviour of the material can be described by a simple kinematic hardening law. This is a reliable assumption for relatively small plastic strains. o The deformations and strains are so small that non-linear deformation and follower force effects can be neglected. This is again a valid assumption, since the relative radial displacements at a hot patch are in the order of a few millimeters, ie. much less than the shell thickness. o The usual associative flow rule for metals apply. 4.1

Load

cycles

A typical load cycle as used is shown in Fig 4-a and 4-b. It is described in nine “steps” as follows: Step 1 : Application

of full internal pressure p.

Step 2 : Heat up of whole reactor to a uniform operational

temperature

Step 3 : Further heat up in local “hot spot” to maximum temperature center of “hot spot”. Step 4 : Cooling down inside “hot spot” to average temperature Steps 5 to 8 : Repetition

T. of T + AT in

T,

of steps 3 and 4 two times.

Step 9 : Cooling down to ambient temperature

and removal of pressure.

A few trial runs indicated clearly that cyclic plastic deformation and associated ratchetting could be adequately determined with the load cycle selected above. 4.2

Results

of a typical

elastic-plastic

analysis

A number of cases were analyzed for different “hot spot” sizes and temperatures, as will be described below. For each case a comprehensive set of results were determined and presented in graphical form: 1. The cyclic loading in terms of average and “hot spot” temperatures. 2. The radial displacement the “hot spot”. 3. The principal

at the centre of the ‘
inside and outside surface stresses (srr , sZz).

4. The inside and outside von Mises stress intensities. 5. The inside and outside principal

total strains (err, ezz).

416

R. J. du Preez Pressure cycle

T i

1

“press

at” ff .

0.8

i5 ‘ti s

0.6

....................

0.4

. ..._........ i... .. ..... .... . ....................

F?? 2 L h

....................

0

0

1

2

J3

1

I

4

Figure 4a. 1.8

I

load step

5

Temperature cycle I I

...................

6

7

8

I

I

I

6

7

8

9

1.6 1.4 1.2

$ E I iii 5 is !2

1 0.8 0.6 0.4 0.2 0 0

1

Figure 4b.

2

3

4

load step

5

9

411

Thermal stresses and ratchetting

(residual) Figure 5.

1

1

elem

97,

I

3

pe22

= plastic

Figure 6. Case H

z n

2 -

4

5

?frain

1

6

7 -

7

(l=inside,

I -

8

9

2-outside)

R. J. du Preez

418

6. The inside and outside plastic strains (pe,,,pe,,). 7. The inside and outside plastic strain magnitudes intensities. 8. The inside and outside stress-strain relationships For a few selected cases the deformations graphically presented. The following cases were analyzed: Case

(pe), or equivalent

plastic strain

(sri vs err and sZ2 vs ezz)

of the reactor and the “hot spot” were also

HS Size

T

mm

“C

A

400

350

600

600

B C

400 400

350 350

550 500

D E F G H

400 400 400 400 400

350 350 180 180 180

550 550 400 380 350

550 500 550 450 400 380 350

I J

500 500

350 350

550 so0

550 500

K L

600 600

350 350

550 sod

550 500

TtAT inside outside

For Case B and Case H the plastic strain histories are shown in Figs. 6 and 7. The stress-strain diagrams for these cases are shown in Figs. 8 and 9. The residual deformations of the shell in the “hot spot” area are also graphically presented in Fig. 5.

5

BACKGROUND

TO ASSESSMENT:

RATCHETTING

It is now well established [2,3] that for certain combinations of stresses induced by thermal cycling and primary (mechanical) stress incremental growth of strain per cycle, called ratchetting, can occur. The incremental growth of strains may lead to distortion or fracture unless the accumulated strain is kept within allowable limits. The Bree diagram, Fig. 10, defines a number of stress regimes which are useful for the discussion of elastic-plastic behaviour under thermal and primary loading. In the case considered here the stress due to mechanical loading (pressure) uP is constant. For varying temperature stresses bt the instantaneous stress point will lie on a vertical line, as shown for example on the figure. At low cyclic thermal stress levels the point lies in regime E, and the response of the structure is ratchet-free. In regimes S, and S, a shakedown situation is achieved where there is plastic deformation only for the first cycle, thereafter the situation is again elastic.

Them1

-1

elem

L

91,

stresses and ratchet@

2

1

pe22

= plastic'gfrain

4

5’

419

6

7

5

(l=inside,

c)

2=outside)

Figure 6a.

reformer7,

elem

97,

pe =

Figure 7a. Case A

plastic

s'zain

magn.

(l=inside,

2=outside)

420

R. J. du Preez

0

reformer8,

elem

97,

1

7

pe =

plastic

Figure 7b. Case B

-1

Figure 8a. Case A

s%?ain

magn.

I 7

(l-inside,

2-outside)

Thermal stresses and ratchet&g

421

-1 0

elem

,

2

97 (outside)

Figure 8b. Case A

Figure 9a. Case H

‘

: DLialr’

, ,.10..-1,

R. J. du Preez

0

elem

L

97

*

3

I

5

‘

,

I

9

rtr,,n

(outside)

10 ,*,o**-3,

Figure 9b. Case H

ut = Thermal stmss up = Pressure stress cry= yield stress R,, R,, P, S,, S,, 8~ E are stressregimes &Y

ut

ut (ut = up, = u2 Y %

up=ut=u2

/ RI

0

u,=2u

Y

Y

0.5 uY

E: Regime of no creep ratcheting or plastic ratcheting (note however that creep due to up can exist.) S,, S,, P & E: Regimes of bounded creep/plastic ratcheting analyzed in reference 28 R, & R,: Regimes of progressive ratcheting

Figure 10. Stress regimes (taken from Bree [22]).

II

Thermal stresses and ratchetting

423

In regime P cyclic plasticity occurs for all cycles, but there is no growth in the plastic strain. The inner fibers of the shell are taken through a hysteresis loop with every heating and cooling cycle but the deformation is constrained by the outer layers where the cyclic thermal strains are too small. In regimes R,, R, ratchetting occurs with unbounded growth in strain. For the relatively low temperatures of the reactor shell, creep of the shell and creep ratchetting can be disregarded. We are therefore only concerned with plastic ratchetting. A study of the strain results for the cases investigated shows that they produce situations which fall within the shakedown, S, cyclic-plasticity, P, and ratchetting, R regimes, depending on the temperature levels chosen. 6 6.1

DISCUSSION Radial

OF THE

displacements

RESULTS in “hot

spot”

The results indicate step-wise changes in radial displacement in the center of the “hot spot”. In all cases there is a permanent growth of “hot spot” radial displacement in the first thermal cycle (steps 2 to 3). During the second and further cycles there may be further growth if plastic ratchetting occurs. The residual deformation after the 9 steps may therefore be determined only by the first thermal cycle, or by all the thermal cycles. For Case A the residual deformation is shown in Fig. 5. 0.2

Strain

results

Figs, 6-a and 6-b show the step-wise changes in the surface strains in the center of the “hot spot”. The plastic strains pe22 shown are circumferential strains. For cycles where plastic flow occurs the lines within cycles are curved,iwhereas they are straight for no plastic flow. The residual strains after removal of temperature and pressure are given by the values at step 9. The graphs of equivalent plastic strain pe, Figs. 7-a and 7-b show the regime for each surface, ie. elastic (zero plastic strain), shakedown (initial plastic strain in the first thermal cycle, thereafter constant), cyclic plasticity (initial plastic strain in the first thermal cycle, thereafter cyclic variation of plastic strain between constant upper and lower bounds), or ratchetting (growth of plastic strain in every cycle). 6.3

Stress-strain

results

Figs. 8 and 9 give the stress-strain curves for the center of the “hot spot” on both surfaces. With these the hysteresis cyclic behaviour can be studied. 6.4

Summary

of results

A summary of results is given in Table 9.2 for all the cases considered. The presence of ratchetting is indicated as well as the incremental plastic strain per cycle (after the first cycle).

R. J. du Preez

424

9.2

Summary Case

0.5

of results

from

HS Size mm

Av Temp “C

A B

400 400

C D E

400 400 400

350 350 350

F G H

400 400 400

180 180 180

I

500

350

J

500

350

K

600

350

L

600

350

Assessment

350 350

the elasto-plastic

analyses

HS Temp

Cyclic Plasticity

“C ins. outs.

Ae,l (%-range) ins.

Plast. Ratchetting AQ

(%/cycle) outs.

600

600

.42

.08

550 500 550

550 500 500

016

.04

.14

.005

550 400

450 400

.12 .ll

.005

380 350

380 350

.05

.005

550 500

550 500

.19 .02

.02 .005

550 500

550 500

.20 .03

.02 .005

criteria

The acceptability of a particular “hot spot” size and temperature judged according to various criteria: a Elastic regime - acceptable without

amplitude

may r be

restriction.

l

Shakedown regime - acceptable without

restriction.

l

Cyclic plasticity regime - acceptable, but plastic strain in a cycle should be restricted.

l

Ratchetting regime - only acceptable with a restriction strain during the lifetime of the vessel.

on the total accumulated

The limit on cyclic plastic strain can be set by low cycle fatigue considerations [4]. The relationship between plastic strain range and the number of cycles to failure in logarithm coordinates is a straight line with a slope of - f, largely independent of the type of material and the temperature. Typically failure occurs after 100 cycles at a plastic strain range of 7% and after 1000 cycles at 2%, Fig. 11. The limits on accumulated strain as specified in ASME Set III Div I rules for the design of components at elevated temperature service, can be summarized as follows [53: 1. strains averaged through

the thickness, 1%

2. strains at the surface, due to an equivalent thickness, 2% 3. maximum

linear distribution

of strain through the

local strains, 5%

For this particular reactor it was suggested that a condition which is within the shakedown regime should be selected as a basis for continuous operation, and “hot spots” which fall in the cyclic plasticity and ratchetting regimes should not be allowed for prolonged neriods of operation.

Thermal stresses and ratchetting

‘7 SUMMARY

AND

In summary the following

425

CONCLUSIONS comments, conclusions and recommendations

can be made:

1. The initial elastic analyses indicated that the worst “hot spot” position was just below the junction of the lower cylinder and the upper conical part of the reactor vessel. This is mainly due to the local stiffening effect in the shell at the junction. 2. The maximum elastic stresses occured for the larger “hot spots” (600 mm) with the higher temperature differentials (200 “C), and exceeded the yield stress of the material at the elevated “hot spot” temperatures. 3. Cyclic plasticity occurs on the inside only for all cases where AT > 150°C except case H, with a maximum plastic strain range of .42% at AT = 250°C. 4. Plastic ratchetting

occurs only on the outside and for all cases except C, E, and H.

5. The only cases for which no cyclic plasticity or plastic ratchetting occur are C, with AT = 150°C) and H, with AT = 170°C but at an average temperature of only 180 “C. 6. For longer periods of operation, case H seems to provide an acceptable “hot spot”, ie. 180 “C average with 350 “C peak temperature in the “hot spot”. The techniques applied here have proven themselves as useful and effective in inderstanding cyclic stress/strain behaviour in reactor shells, and in determining limiting criteria for safe long term operations.

8

REFERENCES

[l ] RJ du Preez, GAPJ van Zijl, Modelling of Mechanical properties of ASTM A285 Gr C under Cyclic Temperature

Conditions,

ISE Report 90/02.

[2 ] J Bree, Elastic-plastic behaviour of thin tubes subjected to internal pressure and intermittent high-heat fluxes with application to fast-nuclear reactor fuel elements, J Strain Anal., 2, 1967. [3 ] FJ Beer, Plastic growth of a pressurized shell through interaction of steady pressure with cyclic thermal stresses, Thermal stresses and thermal fatigue, ed. DJ Littler, Butterworths, 1971. [4 ] JF Tavernelli, LF Coffin, A compilation and interpretation tests on metals, Trans ASM, Vol 51, 1959. [S ] Criteria for the design of elevated temperature of ASME PVC., ASME, 1976.

of cyclic strain fatigue

Class 1 components in Set III Div 1,