Modelling creep of pressure vessels with thermal gradients using

Theta projection data

M. Law*, W. Payten, K. Snowden

Australian Nuclear Science and Technology Organisation (ANSTO), Materials Division, PMB 1, Menai 2234, Australia

Received 7 October 2001; revised 11 April 2002; accepted 11 September 2002

Abstract

Pressure vessels are often exposed to through-wall temperature gradients. Thermal stresses occur in addition to pressure stresses. The

resulting creep response is calculated using the Theta projection creep algorithm within a finite element code. It was found that the stress and

temperature dependence of the creep response may lead to complex stress evolution.

q 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Creep; Theta projection; Temperature variation

1. Introduction

The effect of thermal gradients on thick walled pressure

vessels can be significant. If large thermal gradients exist in

steady-state operation, thermally induced stresses [1] will

exist in addition to the pressure stresses (Fig. 1). The creep

response of the wall depends on the local stress, tempera-

ture, and previous creep history. The combination of the

altered initial stress state and the altered creep response may

lead to differing final stress states through time and a

reduced creep life to that calculated for vessels using a

constant secondary creep rate law.

The Theta projection method [2] has been widely used as

its advantages and flexibility become appreciated. The

Theta projection equation is an attempt to both empirically

fit the strain–time behaviour of a material during creep, and

to provide an insight into the processes occurring during

creep [3]. The equation is an expression of creep response

over time

1c ¼ u1ð1 2 e2u2tÞ þ u3ðeu4t

2 1Þ; ð1Þ

where 1c is the creep strain, t is the time, and the u terms are

experimentally determined constants. Each Theta term is

itself a function of temperature and stress, of the form

log ui ¼ Ai þ Bisþ CiT þ DisT ; ð2Þ

where s is the stress, T is the temperature and i ¼ 1; 2; 3; 4:

The log ui values derived from testing vary approximately

linearly as functions of stress and temperature [4]. If the ui

functions can be defined on the basis of short term tests at

high temperature, then values for long term creep data can

be predicted [5,6].

2. Methods

The u parameters used in this work were obtained as part

of an extensive creep testing program of ex-service 2 14

Cr–

1Mo steels. The experimental details have been reported

elsewhere [7]. Briefly, the creep specimens were tested at

temperature in a vacuum to reduce possible oxidation

effects and their extensions were continuously monitored.

The creep curve information was fitted to the Theta equation

using the method of Evans and Wilshire [2].

3. Solution method

To implement the Theta equation within the FEA

program (EMRC NISA II), a strain rate procedure was

used. The first derivative of the Theta equation

d1=dt ¼ u1u2 expð2u2tÞ þ u3u4 expðu4tÞ ð3Þ

is used to calculate the creep rate and the incremental creep

strain, for the time-step and element (which in turn depends

0308-0161/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.

PII: S0 30 8 -0 16 1 (0 2) 00 1 00 -X

International Journal of Pressure Vessels and Piping 79 (2002) 847–851

www.elsevier.com/locate/ijpvp

* Corresponding author. Tel.: þ61-2-9717-9102; fax: þ61-2-9543-7179.

E-mail address: mlx@ansto.gov.au (M. Law).

on the local stress, temperature, and prior creep). This

process leads to the use of a time hardening formulation.

The Theta coefficients given in Table 1 were derived

from eight creep tests at stresses from 40 to 60 MPa, and at

temperatures from 645 to 715 8C using regression analysis

and Eq. (2).

4. Creep in a uniform cylindrical pressure vessel

For comparison with the cases modelled later with

temperature variation, a creep analysis was performed on a

cylindrical pressure vessel (i.d. ¼ 50 mm, w.t. ¼ 25 mm)

with no temperature variation. Details have been given

previously elsewhere [8].

The analysis was also carried out using the Norton

equation for the minimum creep rate

_1min ¼ Ksn ð4Þ

where K and n are experimentally derived material

constants. The analysis assumed constant secondary creep

at the minimum creep rate and a time hardening formu-

lation. The temperature was taken as 928 K (645 8C) and the

average hoop stress found in the model under elastic

conditions was 32.08 MPa. At this stress with K ¼

2:3995 £ 10220 and n ¼ 7; the Norton equation predicts

a similar minimum creep rate to that predicted by the Theta

method (8.39 £ 10210 m/m/s).

4.1. Results

The Norton model redistributed to values predicted by

the Bailey equations [9]. The Theta model redistributes

more quickly than the Norton model (Figs. 2–4) with the

final values being different.

It is not expected that the Theta solution should be equal

to the Norton-based relaxed stress state. The Norton model

is based on the power law relation between stress and creep

Fig. 1. Elastic thermal stresses; inner wall 10 8C hotter than outer wall.

Table 1

Theta coefficients

u1 u2 u3 u4

A 2106.55 18.966 2284.16 263.664

B (MPa21) 2.7948 22.754 3.6464 8.53 £ 1022

C (K21) 1.11 £ 1021 24.45 £ 1022 2.79 £ 1021 4.88 £ 1022

D (K21 MPa21) 22.98 £ 1023 3.10 £ 1023 23.66 £ 1023 1.77 £ 1025

Fig. 2. First principal stress in cylindrical pressure vessel during creep.

M. Law et al. / International Journal of Pressure Vessels and Piping 79 (2002) 847–851848

rate, whereas the Theta equation shows an increasing rate of

minimum creep rate against stress in a log–log plot (Fig. 5).

The faster redistribution of the Theta model compared to

that of the Norton model (even though they predict similar

minimum creep rates) was attributed to the fact that the

Theta model describes primary and tertiary stages of creep

in which the creep rates are higher than the secondary creep

rate. A constant secondary creep rate is an essential feature

of the Norton model. It is the higher creep rates of the Theta

model that leads to the faster stress redistribution. The

variation in stress late in life (Figs. 3 and 4) after

1.5 £ 107 s, is due to the onset of tertiary creep in the

Theta model.

5. Modelling of creep under temperature variation

Theta projection and FEA were used to analyse the creep

behaviour of cylindrical pressure vessels with two tempera-

ture variations, with the inner wall 5 and 10 8C higher than

the outer wall. The resulting temperature and stress

dependence of the minimum creep rates are shown in Fig. 6.

Similar modelling was performed by Loghman and

Wahab [10] on a cylindrical pressure vessel which showed a

significant variation of the von-Mises stresses with time.

The closed form solution employed by them is not

applicable to complex geometries or multiple material

models such as welded joints, whereas the FEA procedure

can be used in these situations. The details and results are

discussed below.

5.1. Results

Thermally induced compressive hoop stresses exist at

the point where the temperature is highest; the combination

of the thermal stresses, the pressure loading, and differing

creep rates across the wall thickness due to temperature

variation gives rise to varying stress profiles through time.

5.1.1. Inner wall 58 higher than outer wall

With the inner wall 58 hotter than the outer wall, the inner

wall is under thermally induced compression in addition to

the pressure stresses, and the highest hoop stresses are at

Fig. 3. First principal stress in cylindrical pressure vessel using Theta creep

model.

Fig. 6. Effect of temperature and stress on minimum creep rate.

Fig. 5. Comparison of minimum creep rates from Theta and Norton models.

Fig. 4. von-Mises stress in cylindrical pressure vessel using Theta creep

model.

M. Law et al. / International Journal of Pressure Vessels and Piping 79 (2002) 847–851 849

the outer wall. The von-Mises stresses (Fig. 7) are highest

at the inner wall and the creep rates are highest here also

initially. The inner wall reaches tertiary creep at approx.

1 £ 107 s.

These higher creep rates at the inner wall offload stresses

onto the mid wall and outer wall. As the von-Mises stress is

the driving stress for creep, the creep strain rates follow

these von-Mises stresses and the first principal stresses are a

result of these creep strains Fig. 8.

The final state of the von-Mises stresses is similar to

that reported by Loghman and Wahab [10], but there

was insufficient data in that paper to replicate the

results.

5.1.2. Inner wall 108 higher than outer wall

The thermally induced compressive hoop stress at the

inner wall is greater than the case described above. The

combination of the thermal stress, the pressure loading, and

differing creep rates across the wall thickness due to

temperature variation gives rise to the following stress

profiles through time (Figs. 9 and 10).

The final state of the von-Mises stresses is similar to that

seen in the þ5 8C temperature case, though, rather than the

simple reversal seen in that example (Fig. 7), there is an

earlier double reversal (Fig. 10, also shown in Fig. 11 in log

time for clarity).

6. Discussion

The stress redistributions show complex behaviour. In

addition to the expected creep induced redistribution, based

on the initial thermal and pressure induced stress states,

there are two linked factors, which add to the complexity of

the stress response. Firstly, the temperature variation across

Fig. 7. Theta projection modelling of þ58 thermal case, von-Mises stresses.Fig. 9. Theta, þ108 thermal case, first principal stresses.

Fig. 10. Theta, þ108 thermal case, von-Mises stresses.

Fig. 8. Theta, þ 58 thermal case, first principal stresses. Fig. 11. Theta, þ108 thermal case in log time.

M. Law et al. / International Journal of Pressure Vessels and Piping 79 (2002) 847–851850

the wall thickness alters the creep rates so that, at the same

stress, the inner wall experiences faster creep rates than that

of the outer. Thus, the inner wall relaxes offloading stress

onto the outer wall. At these higher stresses, the outer wall

then completes its strain redistribution; this then offloads

stress onto the inner wall. Thus, it is some time before the

stresses stabilise. The difference in relative creep rates

between the inner and outer walls is a function of

temperature, local von-Mises stress (Figs. 6 and 11) and

time.

Secondly, the temperature variation means, the relative

difference in the creep rates may vary through time. For

example, the material in one part of the vessel may be

entering its tertiary phase while another may still be

exhibiting a decreasing or constant creep rate (primary or

secondary creep). These two factors are shown in Fig. 12,

where the minimum creep rate occurs at the inner wall at

approximately 6 £ 106 s and at the outer wall at 1.2 £ 107 s.

At 1.0 £ 107 s, the inner wall has entered its tertiary creep

phase, while the outer wall still exhibits a decreasing creep

rate.

The problems of interest to plant owners are situations

where failures occur, these may involve additional bending

or axial loads, temperature variation, non-uniform geome-

try, or non-uniform materials such as welds or dissimilar

metal joints. Due to these complications, simple analytical

methods are not applicable and well validated numerical

solutions are indicated.

7. Conclusion

The redistribution of stresses by creep in a pressure

vessel was modelled and compared to results predicted by

the Norton equation. The differences found were attributed

to the more complex nature of the Theta projection.

The cases modelled included thermal variation across the

wall of a pressure vessel. The stress evolution was similar to

that found in previously published work. However, more

complex interactions were noted as a result of coupled

thermal variation of creep rates and thermally induced

stresses.

Acknowledgements

The authors are grateful to the CRC for Welded

Structures, Pacific Power, and the Australian Institute for

Nuclear Science and Engineering (AINSE) for support for

this work.

References

[1] Timoshenko S, Goodier J. Theory of elasticity. New York: McGraw

Hill; 1951. p. 413.

[2] Evans R, Wilshire B. Creep of metals and alloys. London: Institute of

Metals; 1985.

[3] Ule B, Rodic T, Sustar T. Modification of u projection creep law by

introducing mean stress term. Mater Sci Technol 1997;13:555–9.

[4] Parker J. Prediction of creep deformation and failure for 1/2Cr–1/4V

and 2-1/4Cr–1Mo steels. ASME J Pressure Vessel Technol 1985;107:

279–84.

[5] Snowden K, Merhtens E. Analysis of creep data for remaining life

estimation. Proceeding of the Remaining Life Assessment on High

Temperature Plant in Australia, Lucas Heights; 1990.

[6] Evans R, Beden I, Wilshire B. Creep life prediction for 1/2Cr–1/

2Mo–1/4V ferritic steel. Creep and fracture of engineering materials

and structures, Swansea, UK: Pineridge Press; 1984. p. 1277–90.

[7] Ripley M, Snowden K. Remaining creep life assessment of welds in

steel pressure vessels. Proceedings of the Structural Integrity:

Experiments, Models and Applications. 10th Biennial conference on

fracture, Berlin, vol. 1; 1994. p. 761–6.

[8] Law M, Payten W, Snowden K. Finite element analysis of creep using

Theta projection data. Int J Pressure Vessel Piping 1998;75:437–42.

[9] Greenbaum G, Rubinstein M. Creep analysis of axisymmetric bodies

using finite elements. Nucl Engng Des 1968;7:379–97.

[10] Loghman A, Wahab M. Creep damage simulation of thick walled

tubes using the Q projection concept. Int J Pressure Vessel Piping

1996;67:105–11.

Fig. 12. Variation of creep rate at inner and outer walls for þ58 case.

M. Law et al. / International Journal of Pressure Vessels and Piping 79 (2002) 847–851 851