Probabilistic Life Assessment Of An Impeller With Discontinuities.

This paper discusses and presents a probabilistic method to evaluate the useful life of an impeller with material imperfections. These discontinuities may be on the surface or subsurface. The method combines linear elastic fracture mechanics (LEFM) techniques with Monte Carlo simulation. Siddall (1983) provides a description of Monte Carlo technique in his book on probabilistic engineering design. An example is included in this paper to demonstrate the use of fracture mechanics and Monte Carlo techniques in estimating the reliability of an impeller. It will show the information required, and will present typical results that will 15 PROBABILISTIC LIFE ASSESSMENT OF AN IMPELLER WITH DISCONTINUITIES by Murari P. Singh Chief Engineer & Manager of Engineering Technology George H. Donald Business Manager William E. Sullivan Senior Structural Analyst GE Energy Conmec Bethlehem, Pennsylvania and Jim Hudson Staff Consultant Engineer GE Energy/A-C Compressor Oshkosh, Wisconsin include stress contour plots, assumptions made relating the defects to a range of initial defect sizes, and plots of the probability of reaching a final critical defect size. A similar concept has been employed to assess reliability for various turbomachinery components (Singh, 1991; Jatla, 1990; Thacker, et al., 1990; Newell, et al., 1990; and Tipton, 1991). INTRODUCTION The evaluation of the life for rotating equipment components with surface or subsurface indications has been performed for years using fracture mechanics techniques. Typically, an indication of a defect has been discovered during a nondestructive test of a rotating component. Some conservative assumptions are made regarding the defect size and orientation, data are collected on the relevant material properties, the stress field in the area of the indication is estimated, and a calculation of the crack growth is undertaken using fracture mechanics techniques. The result is the number of stress cycles required for the defect to reach a critical size. This number of cycles is considered to be the useful life of the component. Although this is a reasonable approach, it depends on a lot of assumptions and therefore, out of necessity, is very conservative. By applying ranges and distributions to the various parameters, one can perform a probabilistic evaluation of the defect growth and arrive at the probability of achieving a specific flaw size before the end of useful life is reached. Since, probabilistically, all of the conservative factors are not applied at the same time, this technique results in a more realistic evaluation of the useful life and, because the results include a plot of probability versus life, the risk associated with reaching any particular life can be found. BRIEF THEORETICAL BACKGROUND An indication of a defect is an abnormality found during a nondestructive test (NDT) of a structure or part of a structure. In this case an impeller or part of an impeller (disk, blade, or cover) is the subset of concern. The type of indication depends on the type of NDT being used. The indication would be a reflection of an input signal if ultrasonic inspection (UT) is used, a bleed-out if liquid penetrant inspection (LPI) is used, and a powder pattern if a magnetic particle inspection (MPI) is used. Likewise, the location, and to some extent the geometry, of the indication depends on the NDT being employed. Indications well below the surface of the part are revealed using UT, indications on or just below the surface can be detected using MPI, while only indications on the surface will be detected by LPI. Once an indication is found, specific techniques are used to interpret the location, size, orientation, and severity (e.g., rounded or sharp) of the flaw the indication represents. Generally, UT indications are the most difficult to interpret since one is working with the reflection of an ultrasonic signal. On the other hand, a great deal of information about the flaw is revealed in an LPI indication because the bleed-out that forms the indication is from the indication itself. Once an indication has been detected and interpreted, one needs to know something about the stresses around the indication before any kind of evaluation of the effect of the indication on impeller reliability can be made. For the case being described in this paper, stresses in the impeller were calculated using a finite element analysis (FEA). Lastly, information on the relevant material properties of the impeller is required to determine the effect of the indication on reliability. The required properties are yield strength (0.2 percent), modulus of elasticity, ultimate tensile strength (UTS), and fracture toughness (KIC). Along with these properties, information describing the crack growth rate under an alternating load will be required. The pertinent equation that describes the growth behavior of the material is discussed in the next section. FRACTURE MECHANICS The objective of this section is to provide and discuss briefly an integrated view of the theory of brittle fracture. This is important so that one can appreciate the theory to be used later in this paper. For a detailed discussion of what follows here, one should refer to Lawn and Wilshaw (1975). The treatment will emphasize the basic principles with the fracture model to be used in making a decision in the case of an impeller with reportable imperfections. In adopting this approach a balance is attempted with the following aims: • Basic assumptions and also hypotheses are briefly analyzed to understand the basic setup of the theoretical framework. • To discuss the diverse aspects of fracture theory to establish a physical basis for design of mechanical structure of practical importance, e.g., an impeller with indication. • To justify the use of the theory of fracture mechanics in its own right. Nearly all materials have a tendency to fracture when stressed beyond a critical level. It seems perfectly reasonable to assume that fracture strength is an inherent material property. The idea of a stress-limit is attractive in engineering design; one simply needs to ensure that the maximum stress level in the component does not exceed the stress limit. With an increase in engineering failures, the concept of a critical applied stress approach was questioned. It is easy to see that the inadequacy of the critical stress criterion lay in its empirical nature. For the notion that a solid should break at a characteristic stress level, however intuitively appealing, is not based on sound physical principles. For instance, how is the applied stress transmitted to the inner regions where fracture actually takes place? What is the mechanism of failure? Griffith (1920) in his classical paper considered an isolated crack and formulated a criterion in terms of fundamental energy theorems of classical mechanics and thermodynamics. A worth mentioning work of Inglis (1913) presented the analysis of stress of an elliptical hole in a uniform stress field. He showed that the local stresses about a sharp notch or corner could rise to a level several times that of the applied stress. The Inglis equations gave the first real clue to the mechanism of fracture; the limiting case of an infinitesimally narrow ellipse could be considered to represent a crack. The Inglis equation yields the value of an “elastic stress concentration factor.” The equation showed that this factor, though considerably larger than unity, depends on the shape of the hole rather than the size. If this analysis is indeed to be applicable to a crack system, then why in practice do large cracks tend to propagate more easily than small ones. Such behavior violates the size-independence property of the stress concentration factor. What then is the physical significance of the radius of curvature at the crack tip of a real crack? Griffith (1920) set up a model based in terms of a reversible thermodynamics process. He simply argued that in the configuration that minimized the total free energy of the system, the crack would then be in a state of equilibrium, and thus on the verge of extension. An expression of total energy of the system was developed that contained individual energy terms. The change in the energy was also examined as a result of crack formation. The outer boundary of a cracked body will undergo some displacement such that the applied load will do some work WL. Second, the potential strain energy UE stored in the body will be sensitive to change in system geometry. Third, the surfaces generated by a growing crack require the expenditure of free surface energy US. The total energy of the static system is thus given by: