Integrity assessment for a tubeplate using the linear matching method

Integrity assessment for a tubeplate using the linear matching method

International Journal of Pressure Vessels and Piping 81 (2004) 327–336 www.elsevier.com/locate/ijpvp Integrity assessment for a tubeplate using the l...

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International Journal of Pressure Vessels and Piping 81 (2004) 327–336 www.elsevier.com/locate/ijpvp

Integrity assessment for a tubeplate using the linear matching method H.F. Chen*, Alan R.S. Ponter Department of Engineering, University of Leicester, Leicester, LE1 7RH, UK Received 17 July 2003; revised 26 February 2004; accepted 26 February 2004

Abstract A series of numerical procedures have been presented recently for the integrity assessment of structures based upon the Linear Matching Method. A typical example of a holed plate has been used to verify these procedures for the evaluation of plastic and creep behaviours of components. In this paper, a more complex 2D tubeplate at the outlet from a typical AGR heat exchanger is analysed for the shakedown limit, reverse plasticity, ratchet limit and creep relaxation based on the application of the Linear Matching Method for a thorough case study. Both a constant material yield stress and a temperature-dependent yield stress are adopted for the evaluation of the ratchet limit. For the evaluation of accumulated creep strains, flow stresses and elastic follow-up factors with differing dwell times at the steady cyclic state, a monotonic creep computation is performed, where the start-of-dwell stress is the rapid cycle creep solution at the beginning of the dwell period. An estimation of the tubeplate lifetime is then obtained by the evaluation of fatigue and creep endurances. q 2004 Elsevier Ltd. All rights reserved. Keywords: Tubeplate; Linear matching; Integrity assessment; Fatigue; Creep

1. Introduction British Energy Generation’s R5 integrity assessment procedure has been widely used for the evaluation of the high temperature response of structures [1], where the comprehensive assessment procedure, including shakedown, ratchetting, fatigue and creep damage, makes use of simplified methods of inelastic analysis, rather than either applying a pessimistic interpretation to an elastic analysis or requiring the cost and complexity of cyclic inelastic computation. However, in most cases, the calculated R5 results are still over-conservative and need to be improved [1]. In recent years, there has been a separate line of development, where standard linear finite element solutions have been used to answer specific design related questions. This can be achieved by generating linear solutions where the linear moduli vary spatially either from element to element or Gauss point to Gauss point. Such methods include the reduced modulus method [2], the GLOSS method [3] and the elastic compensation method [4]. * Corresponding author. Tel.: þ44-116-252-5691; fax: þ 44-116-2522525. E-mail addresses: [email protected] (H.F. Chen); [email protected] (A.R.S. Ponter). 0308-0161/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2004.02.016

These solutions may be used to simulate material nonlinear solutions and have been developed to answer specific design questions involving both time dependent and time independent material behaviour. Although these methods have similar objectives to limit load and the shakedown limit, they tend to approach the problems of the design directly. This has provided a fertile source of methods that may be used within a design office, using the standard tool available to the designer, a linear finite element code. It is this characteristic that has made such methods attractive to industry. Indeed, the need to develop specialist codes for shakedown programming methods has been a major deterrent to their adoption in industry and they have tended to remain as research methods, applied to specific industrial problems. The elastic compensation method, now called the Linear Matching Method [5 – 11], has been applied with considerable rigour to cyclic loading problems where the residual stress field remains constant. This includes classical limit loads, shakedown limits and a class of high temperature creep problems, i.e. rapid cycle creep problems, where the cycle time is small compared with material time scales. A series of numerical techniques based on the Linear Matching Method have also been used for the steady cyclic behaviour associated with complex histories of load

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Table 1 Summary of solution sequence based on the Linear Matching Method (LMM) Stage

Variable

Calculation method

1 2 3 4

Temperature Tðx; tÞ Elastic stresses s^ðx; tÞ Shakedown limit Creep Deformation D1c

5

Plastic strain range D1p

Transient temperature history Transient elastic stress history Elastic shakedown Rapid cycle creep solution, BO method, constant r Reverse plasticity solution

6

Plastic ratchet limit

7

Creep damage: elastic follow-up factor

8

Creep damage: time to creep rupture

Shakedown solution assuming cyclic hardening Creep relaxation solution, starting from rapid cycle solution Creep-reverse plasticity solution method Extended shakedown solution

and temperature where the residual stress field changes during a cyclic state. This includes the plastic strain amplitude and ratchet limit associated with a reverse plasticity mechanism, the creep strain accumulation and elastic follow-up over a creep dwell associated with the creep-reverse plasticity mechanism. In paper [5], the theoretical background of the application of the Linear Matching Method for the assessment of the high temperature response of structures concerned by R5 has been presented and the assessment procedures based upon the Linear Matching Method have been given in detail. A summary of the solution sequence is presented in Table 1. The applicability of the Linear Matching Method on the assessment procedures concerned by R5 has been confirmed by a typical example of a 3D holed plate subjected to cyclic thermal load and mechanical load [5]. This also demonstrated that the Linear Matching Method has both the advantages of programming methods and the capacity to be implemented easily within the commercial finite element code ABAQUS [12]. In this paper, in order to verify the applicability of the Linear Matching method on the integrity assessment of structures with complex geometry and loading conditions, a tubeplate at the outlet from a typical AGR heat exchanger is analysed in detail from the initial transient thermal analysis to the final lifetime estimates. All the results verify the applicability of the methods.

Subsidiary calculation/result

Identify reverse plasticity region for Stage 6 Fatigue cycles to failure N0 from data Factor of safety on mechanical load l Creep endurance limit Dc rom data, hence N0p

Comments Same as R5 Same as R5 LMM LMM estimate ignoring relaxation LMM (not needed in this example) LMM with sy defined by stage 4. Uses standard ABAQUS routine LMM

Evaluate tR

LMM (not required in this example)

region of concern is the upper (hotter) transition radius between the tubeplate and the cylinder. Shutdown of the plant can subject the component to a rapid downshock, the restart being much slower. The assumed fluid temperature histories are shown in Fig. 2. For 300 such shutdowns over a life of around 200 000 h, a realistic average dwell time of 700 h at steady operating conditions is assumed. The material is considered to be Type 316 stainless steel throughout. The material properties adopted in this paper are shown in Table 2. For heat transfer analyses (stage 1, Table 1),

2. Problem description The component is a simplified representation of a tubeplate at the outlet from a heat exchanger. The axisymmetric geometry is summarised in Fig. 1, together with the temperatures, pressures and heat transfer coefficients applying during steady operation. The insulation is represented by a low heat transfer coefficient. The primary

Fig. 1. Tubeplate geometry with temperatures, pressure and heat transfer coefficients.

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329

Table 3 Variation of temperature-dependent yield stress sy ðTÞ with temperature T Temperature (8C) Yield stress (MPa)

Fig. 2. Temperature histories for tubeplate.

the thermal conductivity is 0.01365 W/mm/8C, the specific heat is 465.4 J/kg/8C and the density is 7.966 £ 1026 kg/mm3. For linear elastic analyses (stage 2), Young’s modulus has been taken to be 160 GPa and Poisson’s ratio 0.3. The coefficient of thermal expansion is 15.37 £ 1026/8C. Temperature variation in these material properties is neglected. It is also assumed that the pressure loading and heat transfer coefficients remain constant during the cycle. In order to obtain interaction shakedown and ratchet limit curves (stages 3, 4 and 6), we use a constant yield stress sy0 ¼ 145 MPa as shown in Table 2. In the procedures for obtaining the rapid cycle solutions (stage 4) and the plastic ratchet limit (stage 6), a temperature-dependent yield stress sy ðTÞ is adopted as shown in Table 3. We assume sy ¼ sy0 f ðTÞ ¼ 180ð1 2 0:0005TÞ; with T in 8C. For the rapid cycle solution and elastic follow-up calculations, the secondary creep law adopted is as follows 1_c ¼ B e2Q=ðTþ273Þ sn

ð1Þ

where B ¼ e212:7 ; Q ¼ 19 700 K, n ¼ 5; T is temperature in 8C, s is stress in MPa, and 1_c is creep strain per hour. The solution sequence is shown in Table 1. There are a total of eight possible stages of which seven have been applied in this assessment. Stages 3– 6 and 8 involve application of the Linear Matching Method whereas Stages 1, 2 and 7 involve standard ABAQUS routines.

3. Heat transfer analysis (stage 1) and linear elastic finite element analysis (stage 2) The FE mesh comprised 759 axisymmetric eight-noded quadrilateral elements as shown in Fig. 3a. Fig. 3b shows Table 2 The material properties for Type 316 stainless steel Heat transfer analysis

Thermal conductivity (W/mm/8C) 0.01365 Specific heat (J/kg/8C) 465.4 7.966 £ 1026 Density (kg/mm3)

Linear elastic analysis Young’s modulus (GPa) Poisson’s ratio Thermal expansion (8C21) Inelastic analysis

Constant yield stress sy0 (MPa)

160 0.3 15.37 £ 1026 145

300 153

400 145

500 135

600 126

the refinement near the transition radius. Both the heat transfer analysis and stress analysis were conducted using standard ABAQUS routines. The bottom surface of the lower cylinder was constrained in the vertical direction and a stress applied to the upper surface of the upper cylinder to balance the axial resultant of the internal pressure. The region of interest is the radius between the upper surface of the tubeplate and the upper cylinder. We identify points A and B in the mesh as the points of maximum stress on the inner and outer surface, respectively, in the region of interest and concentrate our attentions at these points. However, it is worth noting that with the present methods complete solutions are evaluated at every stage and critical points for each mode of failure can be found automatically. The point C near the central aperture was used to verify the position of the reverse plasticity mechanism. Table 4 shows the calculated elastic stress components for the nodes of interest. The signed von Mises equivalent values are used, the sign being that of the dominant component. These elastic stress components and von Mises effective stresses for full loading and pure thermal loads applying at different operating times can be further used to assess the integrity of the component based on the current R5 codes [1], where the time-independent residual stresses are chosen by these thermal stresses. For the integrity assessment of some single points in the interested region, the above R5 technique is easily accessible and applicable. However, for the evaluation of the whole structure, the adopted Linear Matching method in this paper shows higher power than the R5 methodology. By the application of the Linear Matching method, both the time-independent residual stress field and the varying residual stress field over the whole of the structure can be determined. The critical points associated with differing modes of failure can be identified automatically. Fig. 4 presents the temperature distributions at the steady state ðt ¼ 0Þ: The temperature distribution at time 100 s is given in Fig. 5. Fig. 6 shows the temperature histories from steady state to 200 s at the inside and outside surfaces of the upper cylinder. We find that the maximum thermal stress in the region of interest appears at time 100 s. Fig. 7 shows the elastic von Mises stress field for the full loads applying at time t ¼ 0 s (steady state). The elastic von Mises stress field for the full loads applying at t ¼ 100 s is presented in Fig. 8.

4. Elastic shakedown analysis (stage 3) The Linear Matching method [5 – 11] was used to calculate the elastic shakedown limit. On the basis of

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Fig. 3. Finite element mesh arrangement.

the calculated elastic stresses with the full thermal and mechanical loads applying at five instances, at t ¼ 0 , 25, 50, 75 and 100 s, an elastic shakedown limit interaction curve was obtained as shown in Fig. 9 for a constant yield stress (145 MPa), where P0 and T0 are the applied mechanical and thermal loads. The applied elastic stress history was subdivided into pressure and thermal components. s^P0 is the von Mises elastic stress corresponding to the applied pressure P0 : s^T0 is the von Mises elastic thermal stress history associated with the applied varying thermal loads T0 : The load factor l was evaluated so that ls^P0 and ls^T0 lies at the elastic shakedown limit, for various proportions of the two components. It can be seen that the applied load point ðP0 ; T0 Þ is just outside the elastic shakedown limit. We will see, as discussed in the next section, the reverse plasticity limit is just exceeded at the edge of the central tubeplate near point

C. In the region of primary interest, near points A and B, the material is within elastic shakedown. The pressure load is significantly less than the limit load for pressure loading, by more than a factor of 2. This initial calculation provides an overview of the relationship between the load history and important strength limits.

5. Rapid cycle solutions (stage 4) This stage involves a first estimate of the creep/plasticity cyclic solution. This is achieved by evaluating the rapid cyclic solution, i.e. a solution that ignores stress relaxation during the dwell period [11]. The simplest and most conservative of such solutions is based on the Bailey – Orowan model and can be expressed as follows.

Table 4 The elastic stress components for nodes of interest Case

Node

sxx ðMPaÞ

syy ðMPaÞ

szz ðMPaÞ

sxy ðMPaÞ

sE ðMPaÞ

Tð8CÞ

T ¼ 100 s full loads

A B

0.243 20.132

187.848 2105.257

243.726 45.6451

7.034 20.136

221.309 2134.012

370.14 405.32

T ¼ 0 s full loads

A B

22.827 20.029

78.192 211.747

53.941 7.717

3.907 0.328

72.356 216.983

563.93 560.98

T ¼ 100 s thermal

A B

3.365 20.103

108.053 282.133

212.721 42.637

2.738 20.526

181.405 2109.829

370.14 405.32

T ¼ 0 s thermal

A B

21.603 11.377

22.936 4.709

20.389 20.062

23.657 9.902

563.93 560.98

T ¼ 20 s thermal

A B

22.672 24.255

54.518 1.668

0.428 20.095

46.711 25.287

531.17 548.41

0.295 0.00067 0.913 20.020

H.F. Chen, A.R.S. Ponter / International Journal of Pressure Vessels and Piping 81 (2004) 327–336

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Fig. 6. Temperature histories for inside and outside surfaces of the upper cylinder.

Fig. 4. The temperature distributions at steady state (time 0) for tubeplate.

For the actual thermal and mechanical loads, the rapid cycle stress history sS ðx; tÞ in the steady cyclic state can be expressed as

sS ðx; tÞ ¼ s^E ðx; tÞ þ rðxÞ; sðsS Þ # S;  n S 1 ðDt Q=ð273ðTþ273ÞÞ 1_ ¼ 1_0 e dt s0 Dt 0

ð2Þ

where s^E ðx; tÞ is the elastic stress field associated with both the varying thermal loads and constant mechanical loads,

Fig. 5. The temperature distributions at time 100 s for tubeplate.

rðxÞ is a constant residual stress field, S is the flow stress, s0 ; 1_0 ; n and Q are material constants and Dt is the cycle time. The inequality in Eq. (2) occurs at some time during the cycle. The Bailey – Orowan model adopted here gives a conservative result; the average inelastic strain rate over a cycle is expressed directly in terms of constant stress and temperature data. The theory assumes that the strain rate is the same as if the maximum stress were maintained during the cycle. This assumption may well be too conservative and may be replaced by any suitable alternative constitutive equation.

Fig. 7. The elastic von Mises stress for the full loads applying at 0 s (steady state).

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Fig. 8. The elastic von Mises stress for the full loads applying at 100 s.

We assume that the flow stress varies, instantaneously, with temperature T in a manner that is identical to the yield stress, i.e. S ¼ S0 f ðTÞ

ð3Þ

where S0 is the value of S if the temperature were instantaneously reduced to 0 8C. The relationship between S0 and the effective inelastic strain rate history 1_ and temperature T is given by Refs. [7,11],

Fig. 10. The rapid cycle solution ðsE þ rÞ for the full loads applying at 0 s (steady state).

Note that the von Mises value of the maximum rapid cycle solution sS corresponds to the Shakedown Reference stress in R5, defined as sSref ¼ ðp=pS Þsy ; where p is the applied load and pS is the shakedown limit. Figs. 10 and 11 show the rapid cycle solution ðsE þ rÞ for the full loads applying at 0 s (steady state) and 100 s, respectively. It can be seen that the highest stresses occur at the edge of the central tubeplate, where the value of the flow

8 91=n 1 ðDt > > ( )21=n >  1 dt _ > < = 1 ðDt Dt 0 n S0 ¼ s0 gðTÞf ðTÞ dt ; > > 1_0 Dt 0 > > ð4Þ : ; gðTÞ ¼ eðQ=ð273ðTþ273ÞÞÞ

Fig. 9. The elastic shakedown limit for tubeplate ðsy ¼ 145 MPaÞ:

Fig. 11. The rapid cycle solution ðsE þ rÞ for the full loads applying at 100 s.

H.F. Chen, A.R.S. Ponter / International Journal of Pressure Vessels and Piping 81 (2004) 327–336 Table 5 The rapid cycle solutions ðsiE þ rÞ at different load instants (MPa) Point

T ¼0s

T ¼ 25 s

T ¼ 50 s

T ¼ 75 s

T ¼ 100 s

A B C

86.24 62.40 185.09

47.10 47.99 88.31

32.16 34.59 0.11

47.64 35.72 78.84

86.25 62.43 185.09

stress S exceeds the yield stress. Elsewhere the yield stress is not exceeded. Table 5 gives the effective stress values for the rapid cycle solutions ðsiE þ rÞ at the points of interest with different load instants. For point A at the inside surface of the cylinder and point B at the outside surface of the cylinder (Fig. 3), the rapid cycle solutions at all instances are less than the yield stress, and both the points A and B lie within shakedown. For point C near the central aperture, the rapid cycle solution exceeds the yield stress. A further numerical phenomenon is drawn from Table 5 that the highest rapid cycle von Mises stresses occur at time 0 and 100 s and are identical. This identifies this region as the location of a reverse plasticity mechanism rather than ratchetting. This solution provides the preliminary solution for the remainder of the assessment.

6. Reverse plasticity solution and fatigue endurance N0 (stage 5) When the stress state of the structure is no longer within the elastic shakedown state and reverse plasticity appears in some local region of the structure, it is found empirically that the low cycle fatigue mechanism occurs and the number of cycles to failure can be determined from the maximum plastic strain range. A good estimate of the maximum plastic strain range may be provided by a recently developed reverse plasticity solution method as shown in Refs. [9,10]. Although the reverse plasticity solution method in Refs. [9,10] is just for two load instances, the method can be extended to the case of multi-load instances. For this specific tubeplate example we now know that both the inside surface point A and outside surface point B lie within shakedown and low cycle fatigue may be ignored. Reverse plasticity does occur at the edge of the central tubeplate but this is not the region of concern in this paper. Hence, for both nodes A and B we assume the number of cycles to failure due to fatigue N0 ! 1 as no plastic strain amplitude occurs at load extremes.

history. It is possible to extend these methods to thermal transient problems. However, in this particular case, we are concerned with the evaluation of conditions away from the reverse plasticity regime where shakedown conditions apply. For such problems, a simplified procedure is possible that makes use of information concerning the location and properties of the reverse plasticity region that is already available from the rapid cycle solution of stage 4. The plastic shakedown analysis procedure comprises two steps. The first step is to solve a rapid cycle solution (stage 4). We assume that the reverse plasticity region then occurs in the regions where the calculated flow stress exceeds the yield stress. As both the yield stress and the flow stress satisfy identical variations with temperature such a region can be evaluated unambiguously. We now increase the yield stress in this region so that reverse plasticity is eliminated

sy ¼ sy0 f ðTÞ sy ¼ S0 f ðTÞ

if S0 , sy0 if S0 $ sy0

ð5aÞ ð5bÞ

The second step is to perform a conventional elastic shakedown analysis using this new definition of yield stress. The effect of locally increasing the yield stress as described above is to take the load point into the shakedown regime. The standard shakedown method may then be used to calculate the capacity of the body to withstand an additional constant load before ratchetting takes place. The resulting shakedown limit is equivalent to assuming complete cyclic hardening. Fig. 12 gives the results of such calculations in the form of an interaction diagram. The complete solution, taking into account the temperature variation of both the yield stress and flow stress is shown as a single ratchet limit value. To provide a more global view of behaviour, the complete interaction diagram is shown for a constant yield stress of 145 MPa. The increase in yield stress in the reverse plasticity region was obtained from a rapid cycle solution with a temperature independent flow stress, i.e. f ðTÞ ¼ 1:

7. Plastic shakedown analysis (stage 6) A general method for evaluation of the plastic strain range and plastic shakedown limit, i.e. ratchetting limit was recently derived [10], although the numerical technique was only applied to problems involving two extremes in the load

333

Fig. 12. The elastic and plastic shakedown limit for tubeplate.

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From Figs. 12 and 13, we also see that the capacity of the tubeplate to withstand an additional internal pressure before ratchetting takes place decreases if the varying yield stress sy ðTÞ and varying flow stress SðTÞ are adopted. The shakedown multiplier lP ¼ s^P =s^P0 reduces from 2.38 to 2.17. The failure mechanism in the tubeplate when an additional constant internal pressure is applied to the ratchet limit is shown in Fig. 14. It can be seen that the ratchet mechanism occurs entirely in the upper cylinder rather than the central tubeplate.

8. Elastic follow-up and creep endurance Dc (stage 7) Fig. 13. The convergence conditions for plastic shakedown analyses.

The elastic thermal stresses were assumed to be linearly scaled values of our basic solution. These calculations, which are rather more elaborate than is necessary for the assessment procedure, show that the load point is near the centre of a very extensive reverse plasticity regime. The convergence conditions for plastic shakedown analyses with and without temperature effects on yield and flow stress are presented in Fig. 13. It can be seen that for both cases the upper bound ratchet limit solutions monotonically decrease and converge after 30 iterations. The applicability of the above procedure was verified.

8.1. Elastic follow-up The term ‘elastic follow-up’ is used to calculate the creep strain accumulated in the stress drop during stress redistribution. The concept of elastic follow-up factor Z allows the determination of the accumulated creep strain during the relaxation process without the necessity of a complete analysis. The process can be described by d1c Z ds þ  ¼0 E dt dt

ð6Þ

where Z is called the elastic follow-up factor, 1c and s refer to effective values and E ¼

3E 3 £ 160 000 ¼ 184 615:38 MPa: ¼ 2ð1 þ nÞ 2 £ 1:3

The value of Z is assessed from a monotonic creep relaxation computation for a creep dwell of 700 h, where the start-of-dwell stress is the rapid cycle solution sS at the beginning of the dwell period. It should be noted that although this option involves an inelastic computation, it is not necessary to analyse large numbers of time increments to obtain a consistent result, and the option therefore, remains much simpler than assessment by full inelastic analysis. Compared with the existing method in Ref. [1], this method seems to represent the most effective and accurate way of estimating the accumulation of creep strain. Comparisons between this method and the correct cyclic solutions for sample problems [13] show that the Z value obtained in this way is conservative but very close to the value of Z for the exact cyclic state, provided the degree of stress relaxation is up to 30% of the initial thermal stress, which is typical of practical cases. Figs. 15– 17 show the relationship between the effective stress and effective creep strain during a creep dwell of 700 h for points A, B and C, respectively. The elastic follow-up factor Z can be calculated by Fig. 14. The failure mechanism for the tubeplate applying an additional constant internal pressure to ratchetting from the applied load point.



D1c E DsrD

ð7Þ

H.F. Chen, A.R.S. Ponter / International Journal of Pressure Vessels and Piping 81 (2004) 327–336

Fig. 15. The stress strain curve for a creep dwell of 700 h for point A ðn ¼ 5Þ:

Table 6 presents the accumulated creep strain, stress relaxation drop and the calculated elastic follow-up factors for points A, B and C, respectively.

Fig. 17. The stress strain curve for a creep dwell of 700 h for point C ðn ¼ 5Þ:

The estimated lifetimes N0p for both inside surface and outside surface are as follows:

8.2. Creep endurance Dc Assuming, pessimistically, that the ductility is independent of the strain rate and equal to a lower bound value, the creep damage per cycle is given as ZDsrD D1c Dc ¼ ¼  L 1L E1

ð8Þ

where D1c is the accumulated creep strain over the creep dwell of 700 h. The lower bound ductility is 1L ¼ 9:4%: Hence, for the inside surface A, Dc ¼ 1:9255 £ 1023 : For the outside surface B, Dc ¼ 5:4574 £ 1024 : 8.3. Estimation of lifetime, N0p Having obtained N0 and Dc ; it is a straightforward calculation to determine the total endurance, i.e. the total number of cycles to failure N0p :  21 1 þ Dc ð9Þ N0p ¼ N0

335

p ¼ N0A

p N0B ¼





1 þ 1:9255 £ 1023 1

1 þ 5:4574 £ 1024 1

21

21

¼ 519

ð10aÞ

¼ 1832

ð10bÞ

9. Creep rupture (stage 8) In order to calculate the creep rupture endurance, creep rupture data should be provided. These can be obtained from the variation of experimental rupture time with tensile stress. In R5, based on the uniaxial creep rupture data, creep rupture is assessed using a rupture reference stress, which is calculated using the primary load reference stress. In paper [14], creep rupture for the notched bar has been assessed using this simplified method. Using LMM technique, a better estimate of creep rupture time tf has been provided by extension of the shakedown theory to include a creep rupture stress, which is a function of the creep rupture time tf [7,11,15]. This technique has been verified by the examples of the Bree problem [7,11] and the holed plate [15]. For this specified example of a tubeplate, the assessment of creep rupture is not required. Table 6 The creep strain, stress relaxation drop and elastic follow-up factor for a creep dwell of 700 h ðn ¼ 5Þ

Fig. 16. The stress strain curve for a creep dwell of 700 h for point B ðn ¼ 5Þ:

Point

D1c

DsrD (MPa)

Z

A B C

1.81 £ 1024 5.13 £ 1025 6.84 £ 1024

17.9469 7.0696 88.6223

1.862 1.340 1.425

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Acknowledgements

Table 7 Key parameters evaluated by LMM Variable

Point A at outside surface

Point B at outside surface

T (8C) at t ¼ 0 T (8C) at t ¼ 100 s s^E (MPa) at t ¼ 0 s^E (MPa) at t ¼ 100 s s^E þ r (MPa) at t ¼ 0 s^E þ r (MPa) at t ¼ 100 s D1p N0 D1c ðn ¼ 5Þ Dc ðn ¼ 5Þ N0p ðn ¼ 5Þ

563.93 370.14 72.356 221.31 83.56 96.04 0 1 0.0181% 1:9255 £ 1023 519

560.98 405.32 16.983 134.01 57.89 65.86 0 1 0.00513% 5:4574 £ 1024 1832

10. Conclusions 1. We have obtained estimates of shakedown and ratchet limits and given much more detailed information about behaviour than an ordinary R5 assessment The key parameters evaluated by the Linear Matching method at two interested locations are shown in Table 7. Although most of results shown in the paper concentrate on some single points, it would be straightforward to automatically identify the critical points associated with differing modes of failure. 2. Although the creep calculation uses the Bailey – Orowan theory, the rapid cycle solution for any other constitutive equations may be substituted The Bailey – Orowan theory gives conservative estimates of creep strain. The monotonic creep computation begins with the rapid cycle solution and this is regarded as a conservative and accurate estimate. 3. Neither fatigue damage nor creep rupture are significant in this example and these aspects of the procedure have not been explored. These results are a first attempt to provide a complete system of calculation for life assessment, consistent with the material data and assumptions of R5 but using more accurate calculation methods.

The authors gratefully acknowledge the support of the Engineering and Physical Sciences Research Council of the United Kingdom, British Energy Ltd and the University of the Leicester during the course of this work. The authors also thank Dr R. A. Ainsworth and David Tipping of British Energy for their provision of materials data and other help.

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