The purpose of this paper is to apply probabilistic fracture mechanics to the analysis of the influence of remedial actions on austenitic stainless steels piping structural reliability using a single SCC damage parameter. Several papers in the literature [1-8] addressed the probabilistic failure analysis of components subjected to stress corrosion cracking (SCC). Failure probabilities of a piping component subjected to SCC was computed by Guedri et al [9-10] using a Monte Carlo simulation (MCS) techniques.
In This article: A short description of the stress-corrosion cracking model [11-13] is given and the American Society of Materials (ASM) recommendations for computing the stress intensity factors [14] are used in the modified PRAISE code for simulating the initiation and growth of IGSCC cracks.
A parametric approach to characterize IGSCC by a single damage parameter (Dσ), which depends on residual stresses, environment conditions, and degree of sensitization, also described next, followed by numerical examples………... Subsequently, from the results of these examples, it was believed that Dσ could serve as a suitable parameter to summarize results for calculated failure probabilities. and finally, this parameter does provide a useful basis to generalize these results.
2. PROBABILISTIC STRESS-CORROSION CRACKING MODEL
SCC can be intergranular or transgranular in nature depending on the material, level of stress, and environment. The methodology recommended in PRAISE for modeling IGSCC in stainless steel pipe is presented in this section. PRAISE separates the overall time to pipe leak into three steps:
Time to initiate a very small cracks,
Time spent growing the small cracks at an initiation velocity v1,
Time spent growing a larger cracks at fracture mechanics velocity v2 to become through- wall cracks.
2.1 Initiation and growth of cracks
The time to crack initiation under static load conditions has been found to be a function of the damage parameter Dσ of Eq. (1). Therefore, the time to crack initiation tI for a given Dσ is taken to be log-normally distributed. The mean and standard deviation of Log (tI) are given in [15] by:
Mean value of and Standard deviation of (1)
The damage parameter Dσ represent effects of loading, environment and material variables on IGSCC and is given by
(2)
where f1, f2 and f3 are given by
(3)
(4)
(5)
where Pa is a measure of degree of sensitization, given by Electrochemical Potentiokinetic Reactivation in C/cm2, O2 is oxygen concentration in ppm, T is temperature in degrees centigrade, γ is water conductivity in μs/cm, and σ is stress in MPa.
In the above equations, Ci are constants whose values depend on type of material. Values for these constants are presented in Table1.
The growth of very small cracks that have just initiated cannot be treated from a fracture mechanics standpoint. Therefore, an initiation velocity v1 (inches/year), is assigned to newly initiated cracks, and given by
(6)
where J is normally distributed and G is a constant.
Fracture mechanics based crack growth velocity, v2 (inches/year), is given in [15] by
(7)
where K is stress intensity factor, C12, C13 and C15 are constants and C14 is normally distributed.
For AISI 304 austenitic stainless steel [15], J has a mean of 2.551 and standard deviation of 0.4269, and G = 1.3447, C12 = 0.8192, C13 = 0.03621 and C15 = 1.7935; mean value of C14 = -3.1671 and standard deviation of C14 =0.7260.
Priya et al in [16] inferred that expressions given in PRAISE for computation of stress intensity factors for modeling crack propagation need modification. In our modified PRAISE (M-PRAISE) this modification has been introduced by using well-accepted expressions given in ASM Handbook. All cracks considered by M-PRAISE start out as semi-elliptical interior surface cracks, generally circumferentially oriented, as shown in Fig.1. The step-by-step procedure, for computing failure probabilities using the net-section stress as failure criteria and detectable leak rate are shown in Fig.2.
2.2 Multiple and Coalescence of cracks
Welds are divided by PRAISE into 50.8mm segments. Each segment has only one corrosion critical location where a crack can initiate. The multiple cracks that may be present can coalesce as they grow. Linkage of two cracks takes place if spacing between them is less than the sum of their depths. After coalescence of two cracks, the dimensions of modified crack are given by
(8)
where l1 and l2 are lengths of two cracks under consideration, a1 and a2 are crack depths and d is spacing between them.
Cracks in neigh-boring segments may link to form larger cracks. The entire weld is assumed to be inspected at the time of each scheduled inspection. If a crack is detected in the weld, the weld is repaired, and the entire weld regains its integrity (perfect repair.)
2.3 Residual Stresses
Residual stresses influence both crack initiation and propagation. The calculations reported here are concerned with the stress corrosion cracking behavior of tow pipe sizes (size-S and size-L) presented in Table 2. The local residual stresses at the inside surface of pipe size-S is treated as being normally distributed. The through-thickness distributions of stress are assumed to be linear variations between local values sampled for the inner and outer surfaces. For pipe size-L the inner surface had a mean tensile stress of 262 MPa. The through-thickness variation in stress had compressive stresses developing within the inner quarter-wall thickness and changing again to tension stress at greater depths.
For more recent insights into stress corrosion cracking mechanisms; a review of the damage model concluded that these predictions were extremely conservative. Fig.3 compares the original PRAISE cumulative leak probabilities given by Harris [17] and results obtained by Khaleel [18-19], for various values of residual stress adjustment factors and plant loading/unloading frequencies. In our case, to limit the disagreement between predicted and observed leak probabilities, the adjusted residual stress level used was set at 75% of their original values.
2.4 Failure criteria
The net-section stress criterion is applicable to very tough material, and the failure is due to the insufficient remaining area to support the applied loads given by Eqs.(9)-(10) (i.e. net-section stress due to applied loads becomes greater than the flow stress of the material ).
(9)
(10)
where Ri is the internal radius of pipe and h is the pipe wall thickness, the area of cross-section of pipe, and is the area of crack, are the load-controlled component of stress (stress due to dead weight. and operating pressure), and the flow stress respectively.
The flow stress of the material used in Eq.(9) was taken to be normally distributed, with an expected value of 296 MPa and a standard deviation of 29 MPa. For leakage failure, the criterion was that of a crack depth equal to the pipe-wall thickness.
3. definition of SCC damage parameter
The SCC damage parameter Dσ presented in Eq. (2) through Eq. (5) is used to generalize the results of PFM calculations. Dσ values for water conductivity equal to 0.2 μs/cm and various degrees of sensitization, different levels of applied stress, a steady-state temperatures and levels of oxygen are presented in Fig.4.
The damage parameter Dσ is a function of the stress, which consists of both the applied (service-induced pressure and thermal) and residual stresses. The crack-tip stress-intensity factor is given by
(11)
where Kap and Kres are the stress-intensity factors attributable to the applied stress and residual stresses, respectively.
The parametric calculations as presented below consisted of many M-PRAISE runs that covered a range of leak probabilities from 1.0E-04 to 1.0.
4. Numerical Exemples
The parametric calculations concentrated on welds in 304 stainless steel piping and on the cumulative leak probabilities over a 40-year time frame of plant operation. In this section the calculations were focused on leak probabilities because M-PRAISE calculates probabilities of leaks more accurately and economically than probabilities of double-ended pipe breaks.
Table 3 summarizes the matrix of calculations along with the input parameters for the calculations. Base-case (no ISI) M-PRAISE runs were first. These calculations assumed realistic ranges for the various input variables that govern the initiation and growth of IGSCC cracks. The following variables were addressed: O2 content, temperature, coolant conductivity, applied stress, and frequency of heatup and cooldown. The resulting calculations therefore covered the relevant range of the Dσ parameter used to present the results in a generalized manner. This initial set of M-PRAISE runs assumed no in-service inspection and gave calculated 40-year cumulative leak probabilities ranging from 1.0E-04 to 1.0.
The M-PRAISE code has the capability of analyzing the effects of inspection on the piping joint reliability.
The inspection detecting probability is expressed as
(12)
where PND is the probability of nondetection, A is the area of the crack, A* is the area of crack for 50% PND, ε is the smallest possible PND for very large cracks, and ν is the 'slope' of the PND curve.
Based on measured performance for PNNL's mini round robin teams, a range of estimates for A* (crack area for 50% POD) was provided by the NDE experts[20-23].
The second phase of the calculations included simulations of in-service inspection as indicated in Table 4. These calculations addressed only a selected subset of the base-case calculations, but were sufficient to cover the full range of the Dσ parameter.
(13)
Ratios of leak probabilities from the inspection simulation runs and the corresponding base-case runs gave values for improvement factors(Eq. (13)).
5. Results and discussion
A large volume of numerical data was produced by completion of parametric calculations for IGSCC of piping as described by Table.3. This section presents a collection of plots that show trends for pipe-leak probabilities and for the effectiveness of various ISI strategies in leak probabilities.
5.1 Predicted leak probability versus Dσ
The reliability for a large number of welds and fittings in a piping system can be estimated quickly if the results of detailed Monte Carlo simulations are provided in a structured parametric format. Results presented in Figs.(5-6) show a good correlation given by Eq.(14) for the pipe size-S and Eq.(15) for the pipe size-L between 40-year cumulative leak probabilities and Dσ. It was believed that Dσ could serve as a suitable parameter to summarize results for calculated failure probabilities of stainless steel piping.
(14)
(15)
5.2 Improvement factors versus Dσ
The individual data points for improvement factors exhibited some data scatter, which was addressed by using a regression analysis to establish best fit curves (second order polynomial) in terms of improvement factors versus the logarithm (base 10) of Dσ. For clarity, the plots below show only the best-fit curves rather than the individual data points. Fig. 7 to Fig. 8 show predicted improvements in reliability over a 40-year design life for small pipe size that results from in-service inspections performed over the 40-year operating period and describes inspections performed at a Y-year interval and with the first inspection at the Xth year using the notation X/Y. Three different levels of NDE performance are addressed. In Fig. 14 to Fig.16, the inspection method (POD curve) was held constant, and the inspection intervals ranged from 1 to 10 years. Fig. 17 and Fig.18 show the same set of results, but with the curves rearranged to maintain a common inspection interval for each plot, with the individual curves corresponding to different POD curves.
It is seen that POD curves with the better detection probabilities always increase the factor improvement, no matter what inspection interval is used. However, the results for the small pipe size show that even the "Advanced" POD curve requires frequent inspection to achieve meaningful reductions in failure probabilities. The inspections with the "good" and "very good" POD curves (Fig. 14 and Fig.15) give more meaningful enhancement in piping reliability. The greatest benefits are predicted for the "Advanced" NDE technology and procedures, for which an order-of-magnitude improvement on the leak probability can be achieved for an inspection frequency of once per year. This improvement factor of 10 is still significantly less than the theoretical limit of 200 (1/ε = 1/0.005 = 200) for the "Advanced" POD curve. The lower benefits of ISI for IGSCC compared to the benefits for fatigue crack growth can be explained in terms of long incubation periods for stress-corrosion cracking followed by a period of rapid crack growth.
this parameter does provide a useful basis to generalize results for piping-leak probabilities.
6. Conclusions
A probabilistic stress-corrosion cracking model was applied to assess the effect of various inspection scenarios on leak probabilities. This paper has also discussed probability of detection curves and the benefits of in-service inspection in the framework of reductions in the leak probabilities for nuclear piping systems subjected to IGSCC. The results presented in Figs.6-7 show a good correlation between 40-year cumulative leak probabilities and Dσ. this parameter does provide a useful basis to generalize results for piping-leak probabilities.
Acknowledgements
The author would like to thank Dr. Khaleel Mohammad for his support and his invaluable guidance during the course of this work.
