### Simplified Weight Function for Calculating Stress Intensity Factor in Complicated Stress Distributions

Jirapong Kasivitamnuay
2016 Engineering Journal
Calculation of a stress intensity factor becomes more difficult when crack are subjected to a complicated stress distribution profile. A standard procedure called influence coefficients is inadequate because a stress profile may not be accurately represented by a polynomial function. This paper applies a piecewise linear approximation of stress profile and a weight function method to overcome that restriction. However, the typically adopted weight function, i.e. universal weight function, is
more » ... ght function, is replaced by a weight function, in which its form coincides with the analytical form. Although a new weight function consists of lesser number of terms, it is proved to be accurate when applies to a cracked-cylinder problem, e.g. internal part-through circumferential crack and internal fully circumferential crack under various complicated weld residual stress profiles. Using this simpler weight function and linearized approximation scheme led to a closed-form stress intensity factor solution, which is convenient for programming. Introduction Industrial components such as pressure vessels or pipelines possibly contain a crack-like flaw that originated from a fabrication, e.g. welding or initiated and growth during service. A cracked component has to be assessed for their integrity and remaining useful life, i.e. perform a fitness-for-service (FFS) assessment. Several international recommended practice are developed for conducting FFS assessment, for example, API 579 [1], SINTAP/FITNET [2] etc. A review of the existing FFS recommended practices can be found in literature [3, 4] . One important step in the process of assessment is to determine a crack driving force, namely a stress intensity factor, K. The annex C of API 579 standard [5] provides K-solutions of many cracked components. This standard also adopts a method called influence coefficients for dealing with the case that crack surface is subjected to a complicated stress distribution, e.g. due to weld residual stress or thermal stress. To apply the influence coefficients method, the stress distribution along the component's wall thickness has to be fitted with the 4 th order polynomial function. Note that the stress distribution may be determined from analytical, experimental or computational methods. But, the 4 th order polynomial function sometimes inaccurately describes a stress distribution profile, especially in the case of a weld residual stress [6] . In this situation, a more general approach for computing K such as a weight function method is necessary. Because, it supports any functional form of stress distribution profile. However, the weight function has a singularity at the crack tip. As a result, special numerical and analytical techniques were proposed to minimize an error from the integration of a weight function [7, 8] . However, Wu [9] avoided the effect of singularity by approximating the stress distribution profile with a piecewise straight line. This approach results in the closed-form of K-solution, which is written as a summation of K contributed by each interval of linearized stress. This approach was further applied to many cracked bodies by Glinka [7] and successfully applied to several cases of weld residual stress [10] . Later, Shim [11] extended this approach by using a cubic spline interpolation instead of a linear interpolation and applied to a weld residual stress field. His results conform well to the finite element results. A stress profile between discrete values of stress can be approximated by several approaches. Anderson [12] studied three approximation methods: piecewise constant, piecewise linear and piecewise quadratic. It is concluded that even the quadratic method is the most accurate, but it is difficult to implement when the stress changes abruptly. Thus, he recommended a piecewise linear representation of a stress profile. Note that, application of quadratic method does exist in the literature, for example, a work on a cracked cylinder under thermal transient load by Navabi et al. [13]. Recently, Li et al. [14] proposed a method called segment-wise polynomial interpolation. This method divides the stress profile into several segments. Each segment contained many discrete values of stress that can be fit by the 4 th order polynomial function. The number of terms in weight function is also increased from typically 4 terms to 6 terms for a better accuracy. Even though this technique yields accurate results as compared with the finite element analysis (FEA) results; it requires attention from an analyst in a selection of the appropriate bounds for each segment. As a result, it is difficult to implement in a computer program. Note that API 579 and the above researchers except Wu [9] used the 4-terms weight function in the form that was proposed by Glinka [15] which is usually known as universal weight function (UWF). However, the functional form of a universal weight function does not conform to that of derived from the analytical solution of a displacement field. The present author used to compare the UWF of a semi-infinite plate with edge crack [15] with the most accurate one [16] and found that the UWF deviated about 9%. Whereas, theoretical-form weight function deviated only 1.3% even it has only 3 terms. Results of a cylinder with an internal circumferential crack [17] also showed that if the coefficients in theoretical-form weight function were determined from the same reference cases as UWF, then UWF was less accurate. Therefore, it is interest to further explore the applicability of the simpler, i.e. 3-terms theoretical form weight function. This paper focusses on the application of 3-terms theoretical form weight function and piecewise linear approximation of stress profile to compute K. First, a new closed-form K-solution is derived. Next, this derived equation is applied to a cylinder with an internal part-through circumferential crack and internal fully circumferential crack under complicated stress distributions. Finally, the computed K is compared with the finite element results from the literature [6, 11, 14] to evaluate its accuracy. Part-through circumferential Ri/t = 5, a/c = 1/3  FE results [6] Proposed method Part-through circumferential Ri/t = 5, a/c = 1/3  FE results [6] Proposed method Fully circumferential Ri/t = 5  FE results [6] Proposed method Fully circumferential Ri/t = 5  FE results [6] Proposed method Part-through circumferential Ri/t = 3, a/c = 1/5  FE results [11] Proposed method Part-through circumferential Ri/t = 3, a/c = 1/5  FE results [11] Proposed method