ASSESSMENT OF THE PROBABILITY OF FAILURE OF REACTOR VESSELS AFTER WARM PRE-STRESSING USING MONTE CARLO SIMILATIONS

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1 Int J Fract DOI /s Springer Science+Business Media Dordrecht 2012 LETTERS IN FRACTURE AND MICROMECHANICS ASSESSMENT OF THE PROBABILITY OF FAILURE OF REACTOR VESSELS AFTER WARM PRE-STRESSING USING MONTE CARLO SIMILATIONS V. Brevus, O. Yasniy Ternopil Ivan Pul'uj National Technical University, Ukraine vitaly.brevus@gmail.com, oleh.yasniy@gmail.com R. Moutou Pitti Institut Pascal, UMR 6602 / Université Blaise Pascal / CNRS / IFMA, BP 10448, Clermont-Ferrand, France rostand.moutou.pitti@polytech.univ-bpclermont.fr Y. Lapusta French Institute of Advanced Mechanics, Institut Pascal, UMR 6602 / UBP / CNRS / IFMA, Clermont Université, BP 265, Aubière Cedex, France lapusta@ifma.fr Abstract. The probability of failure of reactor vessel with semi elliptic crack after warm pre-stressing was assessed taken into account the scatter of mechanical properties and loading parameters by Monte Carlo with importance sampling and first order reliability method based on limit state functions from FITNET procedure. The failure assessment diagrams were obtained by means of Monte Carlo simulations for different crack depth and variable internal pressure. Keywords: pressure vessel, crack, stress intensity factor, warm pre-stressing, failure assessment diagram, probability of failure. 1. Introduction. Reactor vessels are important structural elements whose failure can lead to catastrophic consequences. Therefore the problem of integrity assessment of these components is of high importance. One of the methods of increasing resistance of materials to brittle fracture is warm pre-stressing (WPS) after which the stressstrain-state near the crack tip changes: the residual stresses emerge, the crack tip blunts and the strain hardening of material occurs (Chell, 1977). WPS can also arise in situations when the forced cooling of the reactor vessel happens, accompanied by large thermal stresses in the reactor shell (Rintamaa et. al., 2001). WPS of structural elements with cracks consists in their loading under the temperature that exceeds the temperature of fracture mechanism change from brittle to ductile (under the conditions when the material of structural element is in the plastic state).

2 V. Brevus et al. According to the scheme, the loading to the level of stress intensity factor (SIF) K I occurs at elevated temperature T 1, then at the same temperature structure is unloaded, after that it is cooled and loaded finally at temperature T 2 (Fig. 1). The main approaches to the evaluation of brittle strength of structural elements after WPS are based on the concept of "master curve" (Wallin, 2003), which describes the scatter of fracture toughness through the cumulative probability of failure (Wallin, 1984), P = 1 exp K K K K 4, { ( Jc min ) 0 min } f (1) where K Jc elastic fracture toughness, as defined by J c integral; K min lowest value of fracture toughness; K 0 normalized fracture toughness that corresponds to the probability of 63.2%. Figure 1. Temperature dependence of fracture toughness on the part of brittle-to ductile transition and appropriate changes of SIF The approach to the evaluation of structural integrity is well-known. It is the so-called procedure FITNET, which is based on SINTAP procedures and contains the latest European advances in assessing the integrity of structural elements (SINTAP, 1999, FITNET, 2006). Under this procedure, a two-parameter criterion of fracture of solids with cracks is developed the failure assessment diagram (FAD), which allows to consider plasticity in the crack tip. The input parameters that are required for prediction of failure of the structure are the geometrical parameters of the structure and the crack, the parameters of operating loading, and the mechanical properties of the material. The FITNET approach is deterministic and therefore does not allow to take into account the uncertainty and the current variation of loading parameters, the geometry

3 Assessment of the probability of failure of reactor vessels after warm pre-stressing of the structure and of the crack, the mechanical properties of the studied material, and also to obtain the probability of failure, which is an indicator of acceptability of its further exploitation. So an important task is to assess the limit state of structural elements taking into account the variation of the parameters that are used in the failure assessment diagram. The coordinates of the assessed point at FAD, which characterizes the stress state of element with a crack that is depicted in Fig. 2, are defined as (SINTAP, 1999) KIp KIs P σ ref Kr = + + ρ, Lr = =, (2) K K P σ mat mat L where K Ip and K Is are the SIF corresponding to the applied and residual stresses, respectively; K mat (K Ic ) is fracture toughness of the material; is correction for plasticity; P (or ref ) is applied loading (stress); P L (or yield stress 0.2 ) is loading (stress) at the beginning of yielding of the material in a weakened section of the sample. Failure assessment line, K r = f (L r ), that separates the safe region of the structure from the failure region, is determined under the results of tests of samples with cracks on the fracture toughness (SINTAP, 1999). 0.2 Figure 2. Typical failure assessment diagram for a structural element with a crack: 1 - safe region 2 failure line 3 failure region The purpose of this work is to assess the failure probability of the model of reactor vessel after warm pre-stressing on the basis of the failure assessment diagram taking into account the statistical distributions: the depth of the crack a, the internal pressure p, the yield stress 0.2, the ultimate tensile strength U and fracture toughness of the material K Ic. 2. Problem statement. Consider a crack in a solid with a random variation of mechanical properties and geometrical dimensions under random loading. We introduce a limit state function g(x), defined at the p-dimensional space of random

4 variables, where g(x) 0 failure region and g(x)>0 safe region (Dillström, 2000). Here x is a p-dimensional random vector with components x 1, x 2,, x p, which characterizes all system uncertainties and loading parameters. Random vector x corresponds to the probability density function f X (x). Then the failure probability is a multidimensional definite integral. Pf = fx( x ) dx. (3) g ( x) 0 To evaluate the probability of failure, two different limit state functions g (x) are used (Dillström, 2000) gfad ( x) = gfad ( KI c, σ 0.2, a) = ffad Kr (4) g ( x) = g ( σ, σ, a) = L L, max max max Lr Lr 0.2 U r r where σ U ultimate tensile strength; a crack size; L r ratio of the applied stress to yield stress of the material of the structure with the crack. Limit state functions are based on the analysis of the first level in a standardized procedure SINTAP. A reactor shell made of steel 15Kh2MFA (III) after heat treatment is considered. This situation simulates the properties of the material at the end of exploitation and prestressing with complete (Fig. 3) and partial unloading at 423 K and K I = MPa m (Yasniy, 1998). Chemical composition of steel (%) 0.18 C, 0.62 Mo; 0.27 Si; 0.29 V; 0.48 Mn; 2.58 Cr; S; 0.16 Ni; P; Ti (Yasniy, 1998). Mechanical properties of steel 15Kh2MFA (III) at room temperature are shown in Table 1. σ 0.2 σ U δ ψ MPa % Table 1. Mechanical properties of steel 15Kh2MFA (III) A model of the shell is presented in Figure 3. Wall thickness is h = 140 mm and 2R 0 = 4130 mm. A surface defect in a form of semi-elliptic crack on the inner wall of the pressure vessel is considered. The ratio of semi-axles of elliptic crack a/(b/2) is chosen equal to 2/3. The SIF after loading of vessel with pressure p is calculated by the formula (Heliot, 1979). K = σθ πa Y, (5) a 1 a where Y = , 1.2 σ Θ = b a h b 2 pr 0. h V. Brevus et al.

5 Assessment of the probability of failure of reactor vessels after warm pre-stressing Figure 3. The pressure vessel with a crack on the inner wall The internal pressure p, the mechanical properties (σ 0.2, σ U ) and the critical fracture toughness after WPS K mat = K f are considered as random variables, the types of distribution and parameters are presented in Table 2. Input data Distribution type Parameters of distribution p, MPa normal μ p = 32.5; σ p = 3.25; 4.875; 6.5 σ 0.2, MPa lognormal x 0 = 1080, m = 20, σ = 0.4 σ U, MPa lognormal x 0 = 1140, m = 8, σ = 0.6 K mat, MPa m Weibull x 0 = 141, β = 14.39, η = 2.04 Table 2. Types and distribution of input parameters Pressure of hydrostatic test was MPa. The average internal pressure was chosen two times greater the pressure of hydrostatic test. The coefficient of variation of internal pressure p, which for the normal distribution law is defined as cov p =σ p / μ p is chosen equal to 0.1; 0.15; 0.2; here σ p is the standard deviation of the parameter p. The crack depth a was chosen discretely from 16 mm to 30 mm with a step of 2 mm. The probability of failure was determined by Monte Carlo method with importance sampling (MCIS) and first order reliability method (FORM).

6 MCIS allows performing the tests in the most important part of the integration area. The sample is taken near the most probable point (MPP), determined previously by FORM. MCIS method requires fewer simulations than Monte Carlo method. FORM in the part of finding MPP can be implemented by method of iterations and therefore do not require sampling. Therefore, this method is convenient to calculate the probability of failure in case of small values. V. Brevus et al. a Figure 4. Dependence of failure probability P f on a crack depth by alternating pressure for cov p = 0.1 (1) 0.15 (2) 0.2 (3) calculated by MCIS - a) and FORM - b) b 3. Results. The dependencies of failure probability P f of reactor model from crack depth for a randomly distributed pressure for different coefficients of variation cov p of internal pressure (0.1, 0.15; 0.2) were obtained. The probability was calculated by MCIS (Figure 4a) and FORM (Figure 4b). With increasing coefficient of variation for a given crack length, the failure probability increases. The probability of failure of the model of reactor shell, assessed by FORM, in most cases was lower in comparison with the estimation of this parameter by the method of MCIS, with the largest difference observed for shorter cracks. In particular, for crack depth a = 16.0 mm and coefficient of variation of internal pressure cov p = 0.1, the failure probability, obtained by using FORM is approximately two orders of magnitude smaller than that calculated by the method of MCIS. With the increasing of cov p to 0.15 the probability calculated by a method FORM differs from the assessment of probability by MCIS on one order. Besides, FADs were constructed and simulations were performed by Monte Carlo method for different crack depth (Figure 5-1, 2, 3) for normally distributed pressure of μ p = 32.5 MPa for cov p = The number of samples was equal to 10 4.

7 Assessment of the probability of failure of reactor vessels after warm pre-stressing Figure 5. FAD of shell of reactor model under alternating pressure for cov p = 0.05 for different crack depth a: 1 - FAD curve 2 - a = 28 mm, 3-30 mm, 4-32 mm The estimations of probability of failure are given in Table 3. Crack depth, mm Probability of failure Table 3. Probability of failure for different cracks depths by method of Monte Carlo With the increasing of depth of the crack the probability of failure increases. For example, with increasing depth from 28 mm to 32 mm, the probability increases by 16 times. 4. Conclusions. The dependencies of failure probability of reactor shell after warm pre-stressing on the crack depth a were obtained, which take into account the variation of mechanical properties and loading parameters (normally distributed internal pressure for different coefficients of variation cov p ) by the Monte Carlo method with importance sampling and first order reliability method. For crack depth a = 16.0 mm and the coefficient of variation of internal pressure cov p = 0.1, the failure probability, obtained by first order reliability method is of two orders of magnitude smaller than that calculated by the method of Monte Carlo with importance sampling. The failure assessment diagrams with Monte Carlo method for different crack depth were constructed, considering the pressure as normally distributed random variable. With the increasing of crack depth from 28 mm to 32 mm the probability increases by 16 times.

8 V. Brevus et al. REFERENCES Chell, G. G. (1977). The J Integral as a fracture criterion: perhaps it doesn't mean what you thought it meant. International Journal of Fracture 13, Rintamaa, R., Wallin, K., Keinänen, H., Planman, T., Talja, H. (2001). Consistence of fracture assessment criteria for the NESC 1 thermal shock test. International Journal of Pressure Vessels and Piping 78, Wallin, K. (2003). Master Curve implementation of the warm pre stress effect. Engineering Fracture Mechanics 70, Wallin, K. (1984). The scatter in K IC results. Engineering Fracture Mechanics 19, SINTAP: Structural Integrity assessment procedure for European Industry (1999). Report BE British Steel, p FITNET FFS Procedure (2006). Prepared by European Fitness for Service Network FITNET. Dillström, P. (2000). ProSINTAP A probabilistic program implementing the SINTAP assessment procedure. Engineering Fracture Mechanics, 67, Yasniy, P. (1998). Plastically deformed materials: fatigue and crack resistance. World, Lviv, p Heliot, J. (1979). Fissures semi elliptiques axiales de grande longueur, débouchant a l intérieur d un cylindre. Creusot Loire, p. 237.