Reliability of the ultimate strength of ship stiffened panel subjected to random corrosion degradation

Attentions have been increasingly paid to the influence of the corrosion on the ultimate strength of ship structures. In consideration of the random characteristics of the corrosion of ship structures, the method for the ultimate strength analysis of the ship stiffened panel structure subjected to random corrosion degradation is presented. According to the measured corrosion data of the bulk carriers, the distribution characteristics of the corrosion data for the stiffened panel on the midship deck are analyzed, and a random corrosion model is established. The ultimate strength of the corroded stiffened panel is calculated by the nonlinear finite element analysis. The statistical descriptions of the ultimate strength of the corroded stiffened panel are defined through the Monte Carlo simulations. A formula is proposed on the ultimate strength reduction of the stiffened panel as a function of the corrosion volume. The reliability analysis of the ultimate strength of the corroded deck stiffened panel is performed. It shows that both the corrosion data of the deck stiffened panel and the ultimate strength of the random corroded deck stiffened panel follow the log-normal distribution. The ultimate stress ratio of the stiffened panel is inversely proportional to the corrosion volume ratio.


Introduction
The ultimate strength of ship and offshore structures with corrosion degradation has received intensive attention. Some corrosion models applied in the marine structures have been developed in last decades (Southwell et al., 1979;Melchers, 1999;Yamamoto and Ikagaki, 1998). Guedes Garbatov (1999a, 1999b) developed a model that describes the growth of corrosion wastage by a nonlinear function of time in three phases. The corrosion model of Paik et al. (1998) also categorized the corrosion behavior into three phases. Qin and Cui (2003) proposed a more flexible model for general corrosion of mild steel using a Weibull function. It has the advantage of describing the previously published models by applying certain parameter values.
The general corrosion has been studied widely and compared with the effect of the localized corrosion. Some investigations show that a random surface will lead to a lower ultimate strength compared with uniform corrosion (Silva et al., 2013), while a contradictory conclusion is drawn for low corrosion level (Teixeira et al., 2013). Meanwhile, some numerical studies have been carried out to simulate the pitting corrosion and its influence on the ultimate strength. It is suggested that the effect of the plate aspect ratio and slenderness ratio on ultimate strength reduction can be neglected . However, it is concluded that the aspect ratio can be negligible when the degree of pitting (DOP) is the same, but slenderness ratio governed the collapse behavior Guedes Soares, 2012a, 2012b). With the random nature accounted for, a plate element subjected to random localized corrosion degradation has been investigated recently (Silva et al., 2013(Silva et al., , 2014. Different governing parameters in terms of corrosion and structural features are proposed. Volume loss (Huang et al., 2010;Guedes Soares, 2012a, 2012b), DOP (Nakai et al., 2006b) or the smallest cross-sectional area Islam and Sumi, 2011) are thought to be dominant. However, the global strength of ship hull or offshore structures with corrosion deterioration is almost entirely based on a general decrease of the thickness of the plating (Zhang et al., 2007;Guedes Soares and Garbatov, 1999b;Wang et al., 2008). Although time-consuming, costly and limited for many parameters, the experimental method to assesse the progressive collapse behavior of structural members is always vital in the development and validation of the numer-ical modeling (ISSC, 2006;Nakai et al., 2004Nakai et al., , 2006aSaad-Eldeen et al., 2011. To summarize, it is not hard to find that the ultimate strength of the corroded plate receives more attention than the ultimate strength of the corroded stiffened panel (Dunbar et al., 2004;Khedmati et al., 2012). The objective of the paper is to investigate the ultimate strength of the corroded stiffened panel. The innovation is the random of the corrosion is accounted for in the ultimate strength of the stiffened panel. The distribution of the ultimate strength of the corroded stiffened panel is proposed based on the simulation results. The influence of the corrosion on the ultimate strength of the stiffened panel will be quantified as a function of the corrosion volume. The time variant reliability of the stiffened panel is performed based on the limit state function assumed.

Corrosion data analysis
The stiffened panel investigated is located on the deck of the bulk carriers. There are totally 80 reports involving the measurement of the corrosion of the investigated stiffened panel and corresponding to 15 different ship ages. The measured corrosion data of the stiffened panel are divided into 15 groups in terms of the ship age when the corrosion measurement actions are performed. The statistical analysis on the corrosion data is to be carried out based on any one of the abovementioned 15 ship ages or groups. The mean value and standard deviation of the corrosion data are calculated first for one year based on the measured corrosion data samples collected from the aforementioned reports. Then, the corrosion data between the mean value minus double standard deviation and the mean value plus double standard deviation of the measured corrosion data are regarded as the valid corrosion data of the stiffened panel for each year (Wang et al., 2008). The measured corrosion data and their filtering are given in Fig. 1.
The valid corrosion data will be further applied to find the theoretical distribution of the corrosion data. It can be found that the corrosion data of the plate for the 18th year obey the log-normal distribution with the confidence interval of 95% of the hypothesis testing, as shown in Fig. 2. The corrosion data of the stiffener for the 22nd year also follow the log-normal distribution with the confidence interval of 95% of the hypothesis testing, as shown in Fig. 2. The similar conclusions may be drawn for the corrosion data of the plate and the stiffener corresponding to other years. Consequently, the distribution of the corrosion wastage in the plate and stiffener of the stiffened panel can be theoretically modeled as the log-normal distribution for any one of the years.

Random corrosion model
Paik model in the function of power is adopted, as sho- (1)  where, d c (t) is the corrosion wastage as the function of time t; t c is the coating life, which generally varies from 5 to 10 years (Guo, 2010); k and h are the parameters fitted to the valid measured corrosion data. The mean value and the addition of the mean value and the standard deviation of the corrosion wastage are calculated for each year based on the valid measured corrosion data as shown in Fig. 3. The mean value and the addition of the mean value and the standard deviation of the corrosion wastage are to be respectively fitted to the Paik model, as given in Eq. (1). The coating life may assumedly be constant. As the large scatter the standard deviation shows, the fitting of the addition of the mean value and standard deviation is performed respectively based on a high standard deviation and a low one. Actually, the high standard deviation of the corrosion wastage may be explained that there is some local deep corroded pitting in the structural members. And the low one means the relatively uniform corrosion wastage, for example, the even decrease of the plate thickness. The mean values of corrosion wastage for the plate and stiffener are fitted to the Paik model, with t c =6.5 and h=2/3 (Wang et al., 2008); the additions of the mean value and the standard deviation for the plate and the stiffener are fitted, with t c =5 and h=3/4 (Wang et al., 2008), as shown in Fig. 3. The standard deviation will be derived by the addi-tion of the mean value and standard deviation minus the mean value.
The fitted expressions of the mean value, the high and low standard deviations of the corrosion wastage as a function of time for the plate are given in Eqs.
(2)-(4), and those for the stiffener are given in Eqs. (5)- (7). (2) , and are respectively the mean value, the high and low standard deviation of the corrosion wastage in the plate; , and are respectively the mean value, the high and low standard deviation of corrosion wastage in the stiffener.
Thus, the random corrosion model of the stiffened panel has been established as a log-normal distribution with the time variant mean value and the standard deviation of the corrosion wastage in the plate and the stiffener, as shown from Eq. (2) to Eq. (7).

Ultimate strength analysis of the random corroded stiffened panel
3.1 Finite element modeling of the stiffened panel The length, L, of the stiffened panel is 1600 mm and the width, W, is 800 mm. The thickness of the plate is 10 mm. The stiffener is built as a flat bar, the height of the bar is 320 mm and the thickness is 10 mm. The origin of the right hand coordinates is located at the left bottom corner of the stiffened panel, with the x-axis, and y-axis respectively along the short edge and the long edge of the stiffened panel, as shown in Fig. 4. The boundary condition is taken as follows, the translation along the z-axis is fixed at y=0, y=L, x=0 and x=W, the translation along the y-axis is fixed at y=0, the translation along the x-axis is fixed at both ends, A and B, of the intersection line between the plate and the stiffener. The material model is taken as the elastic-perfectly-plastic model with the yielding strength of 355 MPa, the elastic modulus, 206 GPa, and Poisson ratio, 0.3. The initial deflection of    Guo-qing et al. China Ocean Eng., 2017, Vol. 31, No. 1, P. 11-18 13 the stiffened panel is modelled as the first order elastic buckling mode of the stiffened panel with the nominal amplitude taken as 0.001L, which leads to 1.6 mm. The welding induced residual stress is not taken into account in this study.
The multi-point constraint (MPC) is employed at the two ends to facilitate the application of the external loading. The enforced displacement is applied along the inverse direction of the y-axis at the MPC located at y=L to derive the load bearing curve of the stiffened panel versus the enforced displacement based on the central difference method. The stiffened panel is simulated with the shell element S4R in Abaqus software. Three kinds of mesh sizes are chosen as 80 mm×80 mm, 40 mm×40 mm and 20 mm×20 mm to check the influence of the mesh size on the ultimate strength calculation results of the intact stiffened panel. It is found that the differences of the simulation results by the three mesh sizes are smaller than 0.1%. The mesh size of 20 mm×20 mm is consequently taken in the ultimate strength analysis of the corroded stiffened panel.

Ultimate strength of the randomly corroded stiffened panel
The random corrosion model obtained as log-normal distribution with the time variant mean value and standard deviation, from Eq. (2) to Eq. (7), is applied to the stiffened panel, and the ultimate strength of the stiffened panel subjected to the random corrosion is analysed. Herein, the Monte Carlo (MC) simulation approach is adopted to analyse the statistical characteristics of the ultimate strength of the corroded stiffened panel. The corrosion data of the stiffened panel is generated through MC simulation based on the proposed log-normal model with the corresponding mean value and standard deviation for each year. It is assumed that there are 200 points in the plate and 80 points in the stiffener for one sample. The corrosion data generated through the simulation for one sample of the stiffened panel corresponding to the 25th year with the high standard deviation are shown in Fig. 5 as an example. As it can be seen in Fig. 5, the high standard deviation of the corrosion wastage represents some local deep corroded holes in the stiffened panel, which are highlighted with circles. It should be clarified that one quadrangle of 80 mm×80 mm in Fig. 5 represents a physically corroded hole and it consists of 16 elements of 20 mm×20 mm.
The realizations of the MC simulation are taken respectively 50, 100, 200 and 300. The difference of the statistical descriptors between the realizations of 200 and 300 is smaller than 1%. The volume of the samples is thus taken 300.
The statistical characteristics of the ultimate strength of the stiffened panel subjected to random corrosion are analysed based on MC simulation at the different years of 8th, 10th, 15th, 20th and 25th, which means the different level of the corrosion. The simulation results related to the low standard deviation of the corrosion wastage for the 20th year are given here as an example. The ultimate strength simulation results represent a log-normal distribution, as shown in Fig. 6. Actually, the distributions of the ultimate strength of the randomly corroded stiffened panel for the different years of 8th, 10th, 15th, 20th and 25th all obey the law of the log-normal distribution, whatever the low or high standard deviation of the corrosion wastage. Consequently, it can be reasonably assumed that statistical descriptors of the ultimate strength of the randomly corroded stiffened panel may be modelled as the log-normal distribution throughout the service life of the ship.
Let the ultimate stress ratio, δ σ , be defined as the mean value of the ultimate stress of the stiffened panel, σ u , by the yielding limit of the steel, (σ y =355 MPa). The coefficient of variation (COV) of the ultimate stress is the standard deviation by the mean value of the ultimate stress. The ultimate stress ratio and COV are plotted as a function of time for the low and high standard deviations of the corrosion respectively, as shown in Fig. 7 and Fig. 8. The corrosion loss volume can be calculated through the random corrosion data, and is denoted as ∆V. The corrosion volume ratio, δ V , is defined as the corrosion loss volume, ∆V, by the original volume of the stiffened panel, V. The ultimate stress ratio, δ σ , is drawn as a function of the corrosion volume ratio, δ V , as shown in Fig. 9.  14 FENG Guo-qing et al. China Ocean Eng., 2017, Vol. 31, No. 1, P. 11-18 The ultimate stress ratio of the stiffened panel with low standard deviation of corrosion is very close to the one with uniform corrosion and the ultimate stress ratio of the stiffened panel with high standard deviation of corrosion is much smaller than that with uniform corrosion and the low standard deviation of corrosion, as found in Fig. 9. Consequently, the high standard deviation of corrosion in the stiffened panel will have a considerable influence on the ultimate strength of the stiffened panel. In other words, the local deep corroded pits, which are represented by a high standard deviation of corrosion, exert a greater adverse effect on the ultimate strength of the stiffened panel than the uniform corrosion does. It can also be observed that the ultimate stress ratio is approximately in a linear relationship with the corrosion volume ratio, and the relations between the ultimate stress ratio and the corrosion volume ratio at the respective high and low standard deviation of corrosion are regressed as: ¾ u =¾ y = ¡1:3297¢V=V + 0:6444; (8) ¾ u =¾ y = ¡1:1677¢V=V + 0:7036: 3.3 Empirical formulas The plate ultimate strength is generally evaluated through the plate slenderness ratio, β p , which is written as: where, b is the breadth of the plate, t p is the thickness of the plate, σ y and E are respectively the yielding limit and the elastic modulus of the material. In addition to the slenderness ratio, the column slenderness ratio as defined in Eq. (11) is another important geometric property for a stiffened panel.
is the radius of gyration, where I is the moment of inertia and A is the cross-sectional area as the sum of the plate and the stiffener.
According to Eq. (11), the effective breadth of the plate is applied in the calculation of the moment inertia and crosssectional area to derive the effective column slenderness ratio, λ e , which is used to determine the ultimate strength of the stiffened panel by Paik and Thayamballi (2003). The values are also obtained using Johnson-Ostenfeld formula (Paik and Thayamballi, 2003) and Faulkner formula (Faulkner, 1975) and compared with the results by the proposed Eqs. (8) and (9) in this study, as shown in Fig. 10. It shows that the ultimate stress ratio by Eq. (8) is a little smaller than those by Faulkner (1975) and Paik and Thayamballi (2003), and the ultimate stress ratio by Eq. (9) is between the one by Faulkner (1975) and the one by Paik and Thayamballi (2003), at the relatively small effective column slenderness ratio, λ e . While at the relatively large effective column slenderness ratio, the ultimate stress ratios by both Eq. (8) and Eq. (9) are a little larger than those by Faulkner (1975) and Paik and Thayamballi (2003). However, all of them are smaller than that by Johnson-Ostenfeld formula.
The ultimate strength reduction ratio is defined as the reduction of the ultimate strength, ∆σ u , by the ultimate strength of the stiffened panel without corrosion, σ u . The ultimate strength reduction ratio in terms of the corrosion volume ratio is calculated respectively by Eqs. (8) and (9) and compared with the two empirical formulas for plate structures Huang et al., 2010), as shown in Fig. 11. It can be seen that the influence of corrosion on the ultimate strength reduction of the stiffened panel is weaker than that of the plate under a low corrosion loss condition. However, the influence of corrosion on the ultimate strength re-     Guo-qing et al. China Ocean Eng., 2017, Vol. 31, No. 1, P. 11-18 15 duction of the stiffened panel is stronger than that of the plate in the high corrosion loss circumstance.

Reliability analysis of the randomly corroded stiffened panel
The stiffened panel is located on the deck of the midship. The limit state function of the ultimate strength of the stiffened panel under the compressive loads induced by the still water and wave bending moments in sagging is expressed as (Silva et al., 2014): (12) where, SM(t) is the sectional modulus of midship; σ cr (t) is the ultimate stress of the stiffened panel; M SW is the still water bending moment of the midship; M W is the wave bending moment of the midship; x u , x SW , x W and x s are the uncertainties of the ultimate strength model, the still water bending moment, the wave bending moment, and the nonlinear correction of the wave bending moment in sagging, respectively.
The statistical descriptions of the parameters involved in the limit state function presented by Eq. (12) are assumed here as an example, where, N denotes the normal distribution function and the first and second indicators inside the brackets refer to the mean value and standard deviation, respectively.
The degradation process of the sectional modulus of midship may be assumed as a linear function of time, where, δ 25 is the reduction rate of the sectional modulus of midship, representing the level of the corrosion. It may be determined according to the appropriate corrosion model.
With the present corrosion model, δ 25 is equal to 0.183. The sectional modulus, SM, obeys the normal distribution with the as-built sectional modulus as the mean value, SM » N (SM as-built ; 0:04SM as-built ): (14) The still water bending moment is modeled as a normal distribution. It is assumed that the still water bending moment given by the Classification Societies Rules (IACS, 2006), M SW, CS , is the maximum value with a probability of exceedance of 5%. The large variability in the still water bending moment results in a coefficient of variation of 40%, which gives the mean value of the distribution to be around 60% of M SW, CS : The wave induced bending moment may be modeled as an extreme value following the Gumbel distribution function, where, μ W and σ W are respectively the mean value and standard deviation of the Gumbel distribution, N W is the number of the wave bending moments peaks, ε is the mean square of the wave bending moment. The wave bending moment given by the Classification Societies Rules, M W, CS , is assumed to be the mean value, μ W , of the extreme value distribution, corresponding to 3 hours storm when the number N W is around 1000. The coefficient of variation, σ W /μ W , is approximately 0.09. The stiffened panel in this study originates from the deck of the bulk carrier of the length between perpendiculars 97.3 m, the breadth of 16.6 m, the depth of 8 m and the block coefficient of 0.84, which result to the still water and wave bending moment by Classification Society Rules, as well as the sectional modulus of midship are respectively, 123713.5 kN·m, 209361.3 kN·m and 4.15 m 3 .
The relationship between the probability of failure P f and the reliability index β of the structural element is defined as:  Based on the above analysis on the statistical descriptions of the random variables, the reliability of the ultimate strength of the corroded stiffened panel is performed in terms of different years, 8th, 10th, 15th, 20th and 25th, as shown in Fig. 12.

Conclusions
The reliability of the ultimate strength of the randomly corroded stiffened panel is performed. The following conclusions can be drawn.
The random corrosion wastage of the stiffened panel on the bulk carrier midship deck structures follows the log-normal distribution for certain years. The mean value and the standard deviation for the log-normal distribution of the corrosion wastage of the stiffened panel may be fitted to the corrosion model as a power function of time. The corrosion model may be further distinguished based on different levels of scatter of the corrosion, which is described by different standard deviations of the corrosion wastage.
The ultimate strength of the randomly corroded stiffened panel follows the log-normal distribution. The ultimate stress ratio of the stiffened panel as a linear function of corrosion volume ratio is proposed with the effect of the standard deviation of the corrosion accounted for. The ultimate strength of the stiffened panel in the low standard deviation of corrosion wastage is almost the same as that in uniform corrosion. However, the high standard deviation of the corrosion exerts a considerably adverse effect on the ultimate strength of the stiffened panel. The high standard deviation of the corrosion wastage may be explained as the stiffened panel having some local deep corrosion holes. This illustrates the notable effect of the locally corroded deep holes on the ultimate strength of the stiffened panel.
The limit state function for the ultimate strength of the stiffened panel is given in the state of sagging and the statistical attributes of the random variables are discussed. The reliability of the ultimate strength of the corroded stiffened panel is analysed based on the first order reliability method.