Probabilistic analysis of stress intensity factor ( SIF ) and degree of bending ( DoB ) in axially loaded tubular K-joints of offshore structures

The stress intensity factor (SIF) and the degree of bending (DoB) are among the crucial parameters in evaluating the fatigue reliability of offshore tubular joints based on the fracture mechanics (FM) approach. The value of SIF is a function of the crack size, nominal stress, and two modifying coefficients known as the crack shape factor (Yc) and geometric factor (Yg). The value of the DoB is mainly determined by the joint geometry. These three parameters exhibit considerable scatter which calls for greater emphasis in accurate determination of their governing probability distributions. As far as the authors are aware, no comprehensive research has been carried out on the probability distribution of the DoB and geometric and crack shape factors in tubular joints. What has been used so far as the probability distribution of these factors in the FM-based reliability analysis of offshore structures is mainly based on assumptions and limited observations, especially in terms of distribution parameters. In the present paper, results of parametric equations available for the computation of the DoB, Yc, and Yg have been used to propose probability distribution models for these parameters in tubular Kjoints under balanced axial loads. Based on a parametric study, a set of samples were prepared for the DoB, Yc, and Yg; and the density histograms were generated for these samples using Freedman-Diaconis method. Ten different probability density functions (PDFs) were fitted to these histograms. The maximum likelihood (ML) method was used to determine the parameters of fitted distributions. In each case, Kolmogorov-Smirnov test was used to evaluate the goodness of fit. Finally, after substituting the values of estimated parameters for each distribution, a set of fully defined PDFs were proposed for the DoB, crack shape factor (Yc), and geometric factor (Yg) in tubular K-joints subjected to balanced axial loads.


INTRODUCTION
Tubular K-joints are frequently adapted in the substructure of offshore jacket-type platforms.Figure 1 shows a tubular K-joint along with the three commonly named positions along the brace/chord intersection: saddle, toe, and heel.Non-dimensional geometrical parameters including α , β , γ , τ , and ζ which are used to easily relate the behavior of a tubular joint to its geometrical characteristics are defined in Figure 1.
Tubular joints are subjected to cyclic loads induced by sea waves and hence they are susceptible to fatigue damage due to the formation and propagation of cracks.Thus, the estimation of the residual life of the cracked joints is crucial.The most commonly used method, to estimate how many cycles a K-joint will sustain before its through-thickness failure, is to refer to an S-N curve (American Petroleum Institute, 2007).When a K-joint is loaded, the hot-spot stress (HSS) range can be obtained through the multiplication of nominal stress range by the stress concentration factor (SCF).Using the S-N curve, the number of cycles can be predicted according to the corresponding HSS range.However, for a K-joint with an initial surface crack, the S-N curve can no longer be applied.In this case, an alternative method to estimate the remaining life of a cracked K-joint is to use fracture mechanics (FM) approach based on the stress intensity factors (SIFs).Moreover, the investigation of a large number of fatigue test results have shown that tubular joints with different geometry or loading type but with similar HSSs often can endure significantly different numbers of cycles before failure (Connolly, 1986).These differences are thought to be attributable to changes in crack growth rate which is dependent on the through-the-thickness stress distribution as well as the HSS.The stress distribution across the wall thickness which is assumed to be a linear combination of membrane and bending stresses can be characterized by the degree of bending (DoB), i.e. the ratio of bending stress to total stress.Deterministic FM analyses typically produce conservative results, since limiting assumptions are to be made on key input parameters.However, some of the key parameters of the problem, such as the SIF and DoB can exhibit considerable scatter.This highlights the necessity of conducting a reliability analysis in which these parameters can be modeled as random quantities.Reliability against fatigue and fracture failure becomes always important in case of random and cyclic excitation (Mohammadzadeh et al., 2014).The fundamentals of reliability assessment, if properly applied, can provide immense insight into the performance and safety of the structural system.The value of SIF is a function of the crack size, nominal stress, and two modifying coefficients called the geometric factor ( g Y ) and crack shape factor ( c Y ).The value of the DoB is mainly determined by the joint geometry.These three parameters exhibit considerable scatter which calls for greater emphasis in accurate determination of their governing probability distributions.As far as the authors are aware, despite the considerable research work accomplished on the deterministic study of SCFs and SIFs in tubular joints (e.g.Bowness and Lee (1998), Lee et al. (2005), Shao and Lie (2005) and Shao (2006) for SIFs; and Wordsworth and Smedley (1978), Efthymiou (1988), Hellier et al. (1990), Morgan and Lee (1998a), Chang and Dover (1999), Shao (2007), Shao et al. (2009), Lotfollahi-Yaghin and Ahmadi (2010), Ahmadi et al. (2011), Lotfollahi-Yaghin andAhmadi (2011), Ahmadi and Lotfollahi-Yaghin (2012), and Ahmadi et al. (2013) for SCFs, among others), no comprehensive research has been carried out on the probability distribution of the DoB and geometric and crack shape factors in tubular joints.What has been used so far as the probability distribution of these parameters in the FM-based reliability analysis of offshore structures is mainly based on assumptions and limited observations, especially in terms of distribution parameters.
In the present paper, results of parametric equations available for the computation of the DoB, g Y , and c Y have been used to propose probability distribution models for these parameters in tubular K-joints under balanced axial loads.Based on a parametric study, a set of samples were prepared for the DoB, g Y , and c Y ; and the density histograms were generated for these samples using Freedman-Diaconis method.Ten different probability density functions (PDFs) were fitted to these histograms.The maximum likelihood (ML) method was used to determine the parameters of fitted distributions; and in each case, Kolmogorov-Smirnov test was utilized to evaluate the goodness of fit.Finally, the best-fitted distributions were selected and are introduced in the present paper.The proposed PDFs can be adapted in the FM-based fatigue reliability analysis of tubular K-joints commonly found in offshore jacket structures.

THE FORMULATION OF SIF IN TUBULAR K-JOINTS SUBJECTED TO BALANCED AXIAL LOADS
The SIF can be calculated as follows: where nom σ is the nominal stress, a is the crack size, g Y is the geometric factor, and c Y is the crack shape factor.Both g Y and c Y are dimensionless quantities.In a tubular K-joint subjected to balanced axial loads, the nominal stress is computed as: (2) where P , d , and t are defined in Figure 1.Geometric factor for a tubular K-joint subjected to balanced axial loads can be calculated using following equation (Shao and Lie, 2005) where 1 θ and 2 θ should be inserted in radians.The expression for crack shape factor is (Shao and Lie, 2005): where T is the thickness of the chord; and a and c are crack dimensions illustrated in Figure 2. The validity ranges for the application of Eqs. ( 3) and ( 4) are as follows: (5)

THE FORMULATION OF DoB IN AXIALLY LOADED TUBULAR K-JOINTS
As mentioned earlier, the degree of bending (DoB) is the ratio of bending stress over total stress expressed as: where B σ is the bending stress component, T σ is the total stress on the outer tube surface, and M σ is the membrane stress component (Figure 3).

PREPARATION OF THE SAMPLE DATABASE
Using MATLAB, a computer code was developed by the authors to generate eight samples for the geometric and crack shape factors, DoBch, DoBch0, DoBch45, DoBch90, DoBch135, and DoBch180 based on Eqs.(3)−( 5) and ( 7)−(13).Values of the size ( n ), mean ( µ ), standard deviation ( σ ), coefficient of skewness ( 3 a ), and coefficient of kurtosis ( 4 a ) for these samples are listed in Tables 1 and 2. According to Table 1, the value of 3 a for both c Y and g Y samples is positive meaning that in both cases, the distribution is expected to have a longer tail on the right, which is toward increasing values, than on the left.Moreover, in both c Y and g Y samples, the value of 4 a is smaller than three which means that, in both cases, the probability distribution is expected to be mild-peak (platykurtic).
As can be seen in Table 2, the value of 3 a for DoBch, DoBch0, DoBch45, DoBch135, and DoBch180 samples is positive meaning that in these cases, the distribution is expected to have a longer tail on the right, which is toward increasing values, than on the left.However, the DoBch90 sample has a negative 3 a value which means that its distribution is expected to have a longer tail on the left.Moreover, in DoBch, DoBch0, DoBch45, DoBch135, and DoBch180 samples, the value of 4 a is smaller than three which means that, in these cases, the probability distribution is expected to be mildpeak (platykurtic).On the contrary, in DoBch90 sample, the value of 4 a is greater than three meaning that, in this case, the probability distribution is expected to be sharp-peak (Leptokurtic).

GENERATION OF THE DENSITY HISTOGRAM USING FREEDMAN-DIACONIS PROCEDURE
For generating a density histogram, the range ( R ) should be divided into a number of classes/cells/bins.The number of occurrences in each class is counted and tabulated.These are called frequencies.Then, the relative frequency of each class can be obtained through dividing its frequency by the sample size.Afterwards, the density is calculated for each class through dividing the relative frequency by the class width.The width of classes is usually made equal to facilitate interpretation.Care should be exercised in the choice of the number of classes ( c n ).Too few will cause an omission of some important features of the data; too many will not give a clear overall picture because there may be high fluctuations in the frequencies.In the present research, Freedman-Diaconis rule was adapted to determine the number of classes: where R is the range of sample data, n is the sample size, and IQR is the interquartile range calculated as follows: where 1 Q is the lower quartile which is the median of the lower half of the data; and likewise, 3 Q is the upper quartile that is the median of the upper half of the data.
For example, density histograms of geometric and crack shape factors are shown in Figure 4; and histograms of DoBch45 and DoBch180 samples are depicted in Figure 5.As it was expected from values of 3 a and 4 a (Tables 1 and 2), all histograms are platykurtic; and in all of them, the right tail is longer than the left one.

PDF FITTING AND THE ESTIMATION OF PARAMETERS BASED ON ML METHOD
In order to investigate the degree of fitting of various distributions to the sample data, ten different PDFs were fitted to the generated histograms.For example, PDFs fitted to density histograms of c Y , g Y , DoBch45, and DoBch180 samples are shown in Figures 6 and 7.It should be noted that the fitted distributions were completely-specified theoretical PDFs.
In each case, distribution parameters were estimated using the maximum likelihood (ML) method.Results are given in Tables 3 and 4. The ML procedure is an alternative to the method of  moments.As a means of finding an estimator, statisticians often give it preference.For a random variable X with a known PDF, ( ) X f x , and observed values 1 x , 2 x , . . ., n x , in a random sample of size n , the likelihood function of θ , where θ represents the vector of unknown parameters, is defined as: The objective is to maximize ( ) L θ for the given data set.This is easily done by taking m partial derivatives of ( ) L θ , where m is the number of parameters, and equating them to zero.We then find the maximum likelihood estimators (MLEs) of the parameter set θ from the solutions of the equations.In this way the greatest probability is given to the observed set of events, provided that we know the true form of the probability distribution.

EVALUATION OF THE GOODNESS OF FIT USING KOLMOGOROV-SMIRNOV TEST
The Kolmogorov-Smirnov goodness-of-fit test is a nonparametric test based on the cumulative distribution function (CDF) of a continuous variable.It is not applicable to discrete variables.The test statistic, in a two-sided test, is the maximum absolute difference (that is, usually the vertical distance) between the empirical and hypothetical CDFs.For a continuous variate X , let x(1), x(1), … , x(n) represent the order statistics of a sample of the size n, that is, the values arranged in increasing order.The empirical or sample distribution function x is a step function.This gives the proportion of values not exceeding x and is defined as: Empirical distribution functions for the c Y , g Y , DoBch, DoBch0, DoBch45, and DoBch180 samples have been shown in Figures 8 and 9. Let    The test criterion is the maximum absolute difference between ( ) 0 F x , formally defined as: Theoretical continuous CDFs fitted to the empirical distribution functions of the c Y , g Y , DoBch, DoBch0, DoBch45, and DoBch180 samples have been shown in Figures 10 and 11.
A large value of this statistic ( n D ) indicates a poor fit.So critical values should be known.The critical values , n D α for large samples, say n > 35, are (1.3581n ) and (1.6276 n ) for a = 0.05 and 0.01, respectively (Kottegoda and Rosso, 2008).

Results of Kolmogorov-Smirnov test for c
Y , g Y , DoBch, DoBch0, DoBch45, and DoBch180 sample data are given in Tables 5−10, respectively.It should be noted that, according to the results of Kolmogorov-Smirnov test, none of considered continuous CDFs was acceptably fitted to the DoBch90 and DoBch135 samples.Hence, no table is provided here for these two samples.
It is evident in Tables 5 and 6 that Gamma and Birnbaum-Saunders distributions have the smallest values of test statistic for c Y and g Y sample data, respectively.Hence, it can be concluded that Gamma and Birnbaum-Saunders distributions are the best probability models for the crack shape factor ( c Y ) and geometric factor ( g Y ) in tubular K-joints under balanced axial loads, respectively.
According to Tables 7−10, that Generalized Extreme Value, Gamma, Log-logistic, and Birnbaum-Saunders distributions have the smallest values of test statistic for DoBch, DoBch0, Do-Bch45, and DoBch180 samples, respectively.Hence, it can be concluded that Generalized Extreme Value, Gamma, Log-logistic, and Birnbaum-Saunders distributions are the best probability models for DoBch, DoBch0, DoBch45, and DoBch180 in axially loaded tubular K-joints, respectively.

PROPOSED PROBABILITY MODELS
Based on the results of Kolmogorov-Smirnov goodness-of-fit test, Gamma and Birnbaum-Saunders distributions are the best probability models for c Y and g Y , respectively (Tables 5 and 6).Moreover, Based on the results of Kolmogorov-Smirnov goodness-of-fit test (Tables 7−10), Generalized Extreme Value, Gamma, Log-logistic, and Birnbaum-Saunders distributions are the best probability models for DoBch, DoBch0, DoBch45, and DoBch180, respectively.The PDFs of these distributions are given by the following equations:

CONCLUSIONS
In the present paper, results of parametric equations available for the computation of the DoB, Yg, and Yc were used to propose probability distribution models for these parameters in axially loaded tubular K-joints.Based on a parametric study, a set of samples were prepared for the DoB, Yg, and Yc; and the density histograms were generated for these samples using Freedman-Diaconis method.
Ten different PDFs were fitted to these histograms.The ML method was used to determine the parameters of fitted distributions; and in each case, Kolmogorov-Smirnov test was utilized to evaluate the goodness of fit.It was concluded that Gamma and Birnbaum-Saunders distributions are the best probability models for Yc and Yg, respectively; and Generalized Extreme Value, Gamma, Log-logistic, and Birnbaum-Saunders distributions are the best probability models for DoBch, Do-Bch0, DoBch45, and DoBch180, respectively.Finally, after the substitution of estimated parameters, a set of fully defined PDFs were proposed which can be used in the FM-based fatigue reliability analysis of axially loaded tubular K-joints.

Figure 1 :
Figure 1: Geometrical notation for an axially loaded tubular K-joint.

Figure 2 :
Figure 2: Crack dimensions a and c through the chord thickness T .

Figure 3 :
Figure 3: Through-the-thickness stress distribution in a tubular joint.

Figure 4 :Figure 5 :
Figure 4: Density histogram of sample data: (a) Geometric factor g Y , (b) Crack shape factor c Y .

Figure 6 :
Figure 6: PDFs fitted to the density histogram of sample data: (a) Crack shape factor c Y , (b) Geometric factor g Y .

Figure 10 :Figure 11 :
Figure 10: Theoretical continuous CDFs fitted to the empirical distribution function of sample data: (a) Crack shape factor c Y , (b) Geometric factor g Y .

Figure 12 :
Figure 12: PDFs proposed for c Y and g Y : (a) Crack shape factor c Y − Gamma distribution, (b) Geometric factor g Y − Birnbaum-Saunders distribution.

Table 1 :
Values of statistical measures for c Y and g Y samples.

Table 2 :
Values of statistical measures for the DoB samples.

Table 3 :
Estimated parameters of PDFs fitted to the density histograms of c Y and g Y samples.Latin American Journal of Solids and Structures 12 (2015) 2025-2044

Table 4 :
Estimated parameters of PDFs fitted to the density histograms of DoB samples.

Table 5 :
Results of Kolmogorov-Smirnov goodness-of-fit test for c Y sample data.

Table 6 :
Results of Kolmogorov-Smirnov goodness-of-fit test for g Y sample data.

Table 7 :
Results of Kolmogorov-Smirnov goodness-of-fit test for DoBch simple.

Table 8 :
Results of Kolmogorov-Smirnov goodness-of-fit test for DoBch0 sample.

Table 9 :
Results of Kolmogorov-Smirnov goodness-of-fit test for DoBch45 sample.

Table 10 :
Results of Kolmogorov-Smirnov goodness-of-fit test for DoBch180 sample.