OMAE SPATIAL MODEL FOR CORROSION IN SHIPS AND FPSOS

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1 Proceedings of the ASME 24 33rd International Conference on Ocean, Offshore and Arctic Engineering OMAE24 June 8-3, 24, San Francisco, California, USA OMAE SPATIAL MODEL FOR CORROSION IN SHIPS AND FPSOS Jesus Luque Engineering Risk Analysis Group Technische Universität München Munich, Germany Rainer Hamann DNV GL Hamburg, Germany Daniel Straub Engineering Risk Analysis Group Technische Universität München Munich, Germany ABSTRACT Corrosion in ship structures is influenced by a variety of factors that are varying in time and space. Existing corrosion models used in practice only partially address the spatial variability of the corrosion process. Typical estimations of corrosion parameters are based on averaging measurements over structural elements from different ships and operational conditions, without considering the variability among and within the elements. However, this variability is important when determining the necessary inspection coverage, and it may influence the reliability of the ship structure. We develop a probabilistic spatio-temporal corrosion model based on a hierarchical approach, which represents the spatial variability of the corrosion process. The model includes the hierarchical levels vessel compartment frame structural element plate element. At all levels, variables representing common influencing factors are introduced. Moreover, at the lowest level, which is the one of the plate element, the corrosion process is modeled as a spatial random field. For illustrative purposes, the model is trained through Bayesian analysis with measurement data from a group of tankers. In this application it is found that there is significant spatial dependence among corrosion processes in different parts of the ships, which the proposed hierarchical model can capture. Finally, it is demonstrated how this spatial dependence can be exploited when making inference on the future condition of the ships. INTRODUCTION Time dependent structural degradation (including corrosion, fatigue cracking, mechanical damage) influences the safety of ships. The effect of structural degradation increases with increasing age and often becomes critical when aging vessels with poor maintenance continue operating beyond their design service life (i.e. between 2 and 3 years). Corrosion is one of the most common causes of structural degradation in vessels (Gardiner and Melchers 23). It can occur as uniform corrosion or as localized (pitting) corrosion. Both types of corrosion decrease the load bearing capacity of the structure, making it prone to failures. Significant resources are spent to delay or slow down the deterioration process (Herzberg et al. 2). Inspection, repair and renewal of corroded plates are crucial elements of structural strength maintenance strategies, in order to prevent structural failure and related consequences to crew and environment. The challenge for inspection and maintenance schemes is to put the focus on the critical areas relevant for ship condition. Previous research has led to a set of deterministic relations that predict the amount of corrosion as a function of time. The resulting models range from simple linear models to more complex functions based on an understanding of the chemical and physical processes. An overview on models describing the corrosion process as a function of time is provided in Table. Other models have been proposed for describing the direct effect of the influencing factors (e.g. temperature, salinity, pressure) on the corrosion amount (e.g. Paik et al. 24, Melchers and Jeffrey 27). However, all existing corrosion models are only a simplified representation of reality and cannot include all relevant influencing factors. Additionally, most of those factors are uncertain during the deterioration process. For this reason, probabilistic approaches have been used to complement deterministic models. The uncertainty of the influencing factors is typically included by modelling them as random variables, described through their probability density function (PDF). By combining them with the deterministic mathematical functions for the corrosion loss, a probabilistic corrosion model is obtained (e.g. Guedes Soares and Garbatov 999, Melchers 999, Qin and Cui 23). Several researchers have modeled also the space variability of corrosion loss inside structures. On the one hand, empirical estimations of corrosion rates depending on the location and type of plate element have been obtained using data from previous measurement campaigns (Sone et al. 23, Wang et al. 23a, Wang et al. 23b). On the other hand, probabilistic Copyright 24 by ASME

2 approaches have been proposed to represent the spatial dependence among deteriorated elements using random fields or hierarchical models for deterioration (Guedes Soares and Garbatov 997, Maes et al. 28, Straub 2). However, to the authors' knowledge, no such models have been developed for corrosion in ship vessels or FPSOs. In this paper, a hierarchical Bayesian model is proposed for representing the spatial variability of the corrosion process in vessels. A linear corrosion model (see Table ) is used to describe the time dependence of corrosion loss. Two examples (one of them based on measurement campaigns from real vessels) are presented to illustrate how the results from this model are obtained and can be used for future decisions. Table. Corrosion models Model Linear model (Southwell et al. 978): { corrosion loss time coating life : empirical model parameters Trilinear and power models (Melchers 998): { : empirical model parameters { ( ) scale parameter shape parameter Guedes Soares & Garbatov 999 { ( ) scale parameter long-term loss Qin & Cui 23 { ( scale parameter long-term loss shape parameter ( ) ) Physically-based model for steel immersed in sea water (Melchers 23), with corrosion stages: : Initial corrosion 2: Diffusion controlled 3: Aerobic activity 4: Anaerobic activity Corrosion loss, d Corrosion loss, d Corrosion loss, d (" Corrosion loss, d!#'"!#&" !#%"!#$" Corrosion loss, d!" Power Trilinear Time, t 2 Time, t Time, t!" $" %" &" '" (!" ($" (%" Time, t Time, t SPATIO-TEMPORAL CORROSION MODELS FOR SHIP AND FPSO STRUCTURES Hierarchical Bayesian models Hierarchical models can provide efficient representations of large systems (Raudenbush and Briyk 28). The observed correlation among system elements is incorporated in the model through the definition of multiple levels, which group elements with similar properties. Observable outcomes are typically modeled conditional on a set of parameters, which PDFs are specified in terms of further parameters, known as hyperparameters (Gelman et al. 24). Hierarchical models are commonly used in spatio-temporal systems due to their flexibility for representing dependencies among elements that are physically close or otherwise related. For example, plate elements might be grouped according to their structural element type (e.g. main deck, bulkhead). Those sets can be grouped again based on their spatial location in the vessel (e.g. compartment) and so on. The final goal is that at each level in the hierarchy, each group must contain elements with similar characteristics among them but different from other groups. Hierarchical levels simplify the definition of conditional distributions and the interpretation of the final results. Bayesian Networks (BN) provide a convenient modeling framework that can be applied to graphically represent probabilistic hierarchical structures. A BN is a probabilistic model that consists of a set of random variables (nodes) and directed links, which represent conditional dependencies (Jensen & Nielsen 27). BNs can efficiently perform Bayesian updating, making them suitable for modeling deterioration processes when partial observations from inspections and monitoring are to be included (Straub 29). Ntzoufras (29) and Maes et al. (28) show how hierarchical models are combined with Bayesian models to represent spatio-temporal processes. In hierarchical BNs, stochastic nodes are represented by circles and deterministic nodes by double circles. Single arrows define stochastic conditional dependencies and double arrows deterministic (i.e. mathematical or logical) dependencies between the child node and its parents. Dashed lined squares represent the different levels in the model and they group the corresponding nodes that belong to each level. Indices are used to distinguish among the different elements in a group. The graphical representation of a simple hierarchical Bayesian model with two levels is shown in Fig., wherein deterioration is modeled solely by the corrosion rate. In this example, μ μ_r and σ μ_r are hyperparemeters. Estimation of the posterior distribution of the hyperparameters represents an important step when a hierarchical Bayesian model is solved. Hierarchical structure of the model To represent the spatial dependence of corrosion among plates in a vessel, several hierarchical levels are defined (Fig. 2): 2 Copyright 24 by ASME

3 µ µ_r µ_r R Level (Vessel) Level 4 (Compartment) Level 3 (Frame) Level 2 (Structural Element) Level (Single Plate) µ R,i Compartment, i Fr,, SE,,, SP,,,, SP,,,,N Plate, j Cp, SE,,,M R ij Figure. Example of a hierarchical Bayesian deterioration model (D: deterioration, R: corrosion rate, : mean, : standard deviation) Single plate: Plates correspond to the basic elements and they are the lowest level in the spatial hierarchy. At this level, the variability of the corrosion loss among neighboring plates is described in more detail. In the proposed model, the elements at this level are modeled using random fields, to represent the fact that the correlation in corrosion loss between two elements is a function of their distance. Structural Element: Single plates are grouped according to the structural element type they belong to (e.g. bulkhead, inner bottom, main deck). The reason for this aggregation is that plates from the same structural element type have similar design properties but may exhibit different corrosion loss. This defines the second level of the hierarchical model. Frame: Vessels are usually assembled or repaired frame by frame. Plate elements from those frames may have characteristics that are common among them. Compartment: Structural elements inside a compartment are mostly affected by the same environmental and operational conditions. Due to the common characteristics that are found inside these areas, compartments define the next level in the hierarchy. Vessel: This level corresponds to the complete vessel, which groups all compartments of the structure. Compartments inside a vessel are affected by conditions that depend on the characteristics of the structure itself and its operational profile (e.g. stress distribution, loads, temperature). Fleet: This is the top level, corresponding to the population of vessels that can be described by the same model (e.g. all vessel of a certain type such as bulk carriers or tankers). It is possible to leave out some levels or define additional levels in the hierarchy, depending on the amount and detailing of the available information. For example, a level below compartment can be defined when a certain characteristic (e.g. bottom-middle-upper areas) provides additional information on the corrosion process. In the following, the corrosion prediction for all plates, as well as every thickness measurement, will be uniquely represented by indices (i j k m n) corresponding to the levels in the hierarchy (i.e. i-th vessel, j-th compartment, k-th frame, m-th structural element, and n-th plate). D ij Fleet Vs Vs I Cp, Cp I, Cp I,J Fr,,K Fr I,J, Fr I,J,K Figure 2. Hierarchical structure of the general corrosion model (Vs: Vessel, Cp: Compartment, Fr: Frame SE: Structural Element, SP: Single Plate) Thickness measurements The most common method for estimating corrosion loss is to perform thickness measurements at different times. In a plate, the thickness is typically measured at multiple points (e.g. corners and middle-point) and the average is used to represent the plate s current thickness. The estimated corrosion loss is the difference between its as-built and current thickness, whereas the corresponding linear corrosion rate is estimated by dividing the corrosion loss by the exposure time of the plate to the corrosive environment. The direct estimation of the corrosion rate from thickness measurements is hindered by several factors. Firstly, most plate elements are protected against corrosion and the time when this protection breaks down is not generally known. If the corrosion rate is roughly approximated by dividing the estimated corrosion loss by the vessel age, the corrosion rate might be considerably underestimated. Secondly, a non-negligible percentage of plates have a negative estimated corrosion loss, because their measured thickness is larger than the thickness reported in plans. This situation arises because the built-in plates are thicker than those reported in plans, mainly due to the availability of steel plates when the vessel is built or repaired. The difference between the as-built and gross thickness (i.e. value given in plans) is called thickness margin. SE I,J,K, SE I,J,K,M SP I,J,K,M, SP I,J,K,M,N 3 Copyright 24 by ASME

4 Both coating life and thickness margin directly affect the estimation of the corrosion rate based on measurements. If any of them is not considered, the corrosion rate will be underestimated. For this reason, coating life and thickness margin are important parameters that must be included in the spatial hierarchical Bayesian model of corrosion loss. Corrosion and measurement models A linear corrosion model is used in the proposed hierarchical deterioration model due to its simplicity. However, in principle any of the existing corrosion models of Table could be combined with the proposed spatial model. The linear model assumes that the corrosion loss is zero before the coating breaks. After this point, a constant corrosion rate defines the corrosion loss. The resulting corrosion loss is a function of the coating life, corrosion rate R, and time and is expressed in Eq. (). if ( ) { ( ) R if If w is the plate gross thickness reported in plans and M is its thickness margin, then the actual plate thickness W at time is obtained through Eq. (2). () W( ) w M ( ) (2) Corrosion rate, coating life, and thickness margin are considered as random variables that vary from plate to plate depending on their spatial location. In the following sections, the definition of these random variables and their hierarchical structure are explained in detail. Thickness margin In order to obtain a prior probabilistic model of the thickness margin of a plate element, the following considerations and assumptions are made. The thickness margin is defined as a non-negative random variable. Standards allow small negative margins (up to.6 mm for plates thinner than 2 mm) depending on the quality of the class (Germanischer Lloyd 29) but this possibility is neglected here. Additionally, large margins are less likely than small margins, mainly due to the cost of steel. To account for the possible dependence of margins within a frame, which is due to the fact that plates are typically renewed by frame, they are modeled with a common uncertain mean value in each frame. The prior distribution of the mean frame margin μ M ijkm is the exponential distribution with rate parameter a. The plate margin M ijkmn is modeled as lognormal distributed with mean μ M ijkm and standard deviation σ M. The thickness margin of each plate is constant in time and independent of the spatial location of the plate or other plate margins inside the frame. The hierarchical probabilistic model of the margin is depicted in Fig. 3. Figure 3. Hierarchical representation of the thickness margin Coating life All plate elements in a compartment (i j) are modelled with the same mean coating life μ ij and standard deviation σ (independent of the location). The mean coating life μ ij is described by a lognormal distribution with mean μ μ and standard deviation σ μ. The parameter σ reflects the variability of the coating life among plates within one compartment. Figure 4 summarizes the hierarchical modeling of the coating life. µ µc a µ M,ijkm µ C,ij M ijkmn C ijkmn M Vessel, i Compartment, j Frame, k Struct Elem, m µc C Plate Elem, n Time, p Vessel, i Compartment, j Frame, k Struct Elem, m Plate Elem, n Time, p Figure 4. Hierarchical representation of coating life Previous coating life estimations reported in the literature (e.g. Sone et al. 23) provide percentiles of the coating life for several structural elements. Those values can be used as prior information in the proposed hierarchical Bayesian model, even though they do not take into account spatial dependencies. 4 Copyright 24 by ASME

5 Corrosion rate In the applied linear model, the corrosion rate of a single plate is constant during the life of the plate. The spatial variability of the corrosion rate R is hierarchically defined using variability factors α, α, α 3, α 4 and α. Equation (3) shows how these factors are used to obtain the corrosion rate of a plate. log r ijkmn log α i log α ij log α 3 ijk log α 4 ijkm log α ijkmn log r (3) Factors α i, α ij, α 3 ijk, α 4 ijkm, and α ijkmn correspond to vessel, compartment, frame, structural element, and plate element levels, and r is the base corrosion rate independent of the position of the plate. The set of variability factors α ijkmn with i, j, k and m fixed (i.e. the same vessel, compartment, frame and structural element type) is modeled through a random field. Correlation among the logarithm of their values is represented by the correlation matrix Π ijkm. There exist several correlation models that are based on the distance between points and the correlation length L (Vanmarcke 2). Any of those functions can be used to describe the correlation matrices of the model. Figure presents the hierarchical structure among the variability factors α, α, α 3, α 4, α, the base corrosion rate r, and corrosion rate R. Gray filled circles depict a random field in the model. All variability factors are assumed lognormally distributed (i.e. their logarithm is normal distributed) and their parameters are shown in Table 2. Table 2. Probability distribution of variability factors Factor Distribution Parameters log α i Normal iid μ σ σ, i log α ij Normal iid μ σ σ, i j log α 3 ijk Normal iid μ σ σ 3, i j k log α 4 ijkm Normal iid μ σ σ 4, i j k m log α ijkm [ ] log α ijkm Homogenous Gaussian field P ijkm Number of measured plate elements in vessel i, compartment j, frame k, and structural element m. μ Σ ijkm σ Π ijkm Π ijkm : correlation matrix among plate elements in vessel i, compartment j, frame k, and structural element m. log α ijkmn and log α i j k m n are independent if (i j k m n) (i j k m n ) iid: independent and identically distributed σ log r σ σ σ 3 σ 4 σ () Using equations Eq. (3) and Eq. (), the correlation between the logarithm of corrosion rates from two different points (i j k m n) and (i j k m n ) can be directly obtained. For example, if two elements are in the same vessel but in different compartments (i.e. i i ) then the correlation between their corrosion rates is: ρ(ln r ijkmn ln r i j k m n ) σ σ σ σ 3 σ 4 σ γ (6) where q γ q j= σ j j= σ j (7) The value of γ q from Eq. (7) can be interpreted as the correlation between corrosion rates of plates with the same hierarchical levels from to q, for q { 2 3}. This result comes from the hierarchical definition of the corrosion factors. If two plates are from the same vessel, compartment, frame and structural element (i.e. i i j j k k m m ) then the correlation between their corrosion rates is also affected by the correlation structure defined by the matrix Π ijkm and it is obtained as follows: ρ(ln r ijkmn ln r i j k m n ) σ σ σ 3 σ 4 σ Π ijkm (n n ) σ σ σ 3 σ 4 σ ( γ 4 ) Π ijkm (n n ) γ 4 The previous results are useful when deciding whether inspecting a location or not given a set of measurements from other locations. L 4 4,ijkm 3 3,ijk 2 2,ij r,i (8) Vessel, i Compartment, j Frame, k Struct Elem, m Since the variability factors are lognormally distributed, log r ijkmn is normally distributed with mean and variance according to Eq. (4) and Eq. ().,ijkmn R ijkmn Plate Elem, n Time, p μ log r log r (4) Figure. Hierarchical representation of corrosion rate Copyright 24 by ASME

6 Finally, every thickness measurement is represented by a corresponding random variable Z ijkmnp with mean value W ijkmnp (i.e. the current thickness) and standard deviation σ e. The latter represents the measurement error, which aggregates all sources of errors, of which human and device are the most common types. The distribution of the measurement error is to be independent of the location of the measurement plate. By combining the hierarchical models for the individual model parameters, the complete hierarchical corrosion model is obtained, shown in Fig. 6. a M µ M,ijkm µ µc M ijkm µ C,ijkm µc c 4 3 C ijkmn Figure 6. Hierarchical representation of the full corrosion model Parameter estimation The parameters of the proposed hierarchical corrosion model are learnt from data using Bayesian estimation. We thereby distinguish between (a) learning the hyper-parameters, which describe the fleet-wide model, and (b) learning the parameters that are specific to a vessel, a compartment or any lower-level hierarchical element. Part (a) is referred to as general estimation and it is based on historical measurements from multiple vessels. Part (b) corresponds to the analysis of specific vessels. The estimated probability distributions obtained in part (a) from the analysis of other vessels are here used as prior distributions, which are updated with the new measurements from the specific vessel. It is noted that the differentiation between part (a) and (b) is motivated purely by practical reasons, since it allows analyzing each vessel individually. The Bayesian methodology itself does not require one to make this distinction; rather, a joint model of all vessels can and should be developed. Whenever a new measurement in any of the vessels is available, the entire model of all vessels should ideally be updated. For parameter learning, Markov Chain Monte Carlo (MCMC) is applied. MCMC is a simulation-based method that generates samples of the posterior distribution through a Markov chain whose stationary distribution is the sought,ijkmn 4,ijkm 3,ijk 2 2,ij R ijkmn W ijkmnp r,i Vessel, i Compartment, j Frame, k Struc Elem, m Plate Elem, n Z ijkmnp Time, p e posterior distribution (Gilks et al. 996, Gamerman and Lopes 26). NUMERICAL INVESTIGATIONS In the following sections, two case studies are presented. The first one presents a hypothetical study using simulated data, which allows to assess the performance of the model and the parameter learning against an assumed true model. The second case study presents the learning of the model with thickness measurement from tankers. Based on the learned model, a spatial probabilistic prediction of corrosion in a specific tanker is presented. The Bayesian estimation is carried out with OpenBUGS (Lunn et al. 29). Case study : Simulated scenario To assess the quality of the Bayesian parameter estimation for the proposed hierarchical model, a first example with simulated data is presented. Here, the estimated model can be compared against the true model. The number of simulated vessels, compartments, frames, structural elements, measured plates and measurement campaigns are summarized in Table 3. Several model parameters are assumed to be deterministic in the simulation and their values are shown in Table 4. The resulting hierarchical model is shown in Fig. 7, where deterministic parameters are represented by double circle nodes. Table 3. Size of the simulated hierarchical levels Number of vessels Compartments per vessel Frames per compartment Structural elements per frame 3 Plates per structural element Measurement campaigns after, 8,, 2 years Table 4. Deterministic parameters Margin var. σ M.8 mm Coating var. among compartments σ μ. mm Coating var. inside compartments σ. mm Frame var. factor σ 3 Correlation length L (i.e. correlation matrices Π ijkm I ) Measurement error σ e mm var.:variability, I: Identity matrix The MCMC analysis was carried out with a burn-in period of, and a total of, samples to estimate the joint PDF of the model parameters. Estimated mean values, standard deviations, 9% credible intervals and posterior distributions of the parameters are summarized in Table and Fig. 8. Good agreement between the original and the estimated values for most of the parameters is obtained. The variability factor σ is the only variable in which the true value is not inside the 9% credible interval obtained in the Bayesian estimation. 6 Copyright 24 by ASME

7 a µ µc µc C L r µ C,ij 2,ij,i Vessel, i Compartment, j f(a - ) µ M,ijkm 4,ijkm 3,ijk Frame, k Struc Elem, m Mean margin, a - [mm],ijkmn R ijkmn Plate Elem, n 2 M ijkm M C ijkmn W ijkmt Z ijkmt Time, p e f( 2 ) 8 4 Figure 7. Hierarchical representation of the simulated example Table. Statistics of the stochastic parameters of the model. Estimations Original Value Mean StDev MC error 9% Cred Int Margin Mean a.8 mm e-4 (.76,.89) Coating Mean μ μ 7 years e-3 (6.4,7.4) Corrosion rate and variability factors Vessel var. σ e-3 (.22,.68) Compartment e-3 (.4,.32) var. σ Struct. element e-4 (.26,.32) var. σ 4 Plate var. σ e-4 (.,.6) Base corrosion.3 mm/y e-4 (.9,.32) rate r var.: variability Case study 2: Corrosion in tanker floors This case study is based on thickness measurements of floor plates from four inspected tankers. All tankers in the database have the same structural design, i.e. the spatial locations of plates, structural elements and compartments are the same for all vessels. The general characteristics of the thickness measurement database are presented in Table 6. The structural element Floor is used in this case study because it is the most inspected plate type in the database. It is important to note that floor plates from different compartments are not always physically separated. The compartment numbers correspond to the numbering of the cargo tanks that are above the floor plates. Figure 9 shows where floor plates are located in the vessel and the spatial distribution of the thickness measurements from one campaign. Room variability factor, 2 Figure 8. Estimated PDFs of model parameters of the simulated example Table 6. General information of inspected tankers Tanker Inspection time (years) Structural element Number of compartments Number of measurements T. / 6. Floor 2 28 / 436 T2 4.7 / ---- Floor 2 92 / ---- T3 4.9 / 7.8 Floor 2 27 / 78 T4 4.9 / 7.9 Floor 2 28 / 72 Due to the small number of inspected vessels in the database, the vessel-specific factor is omitted (i.e. σ ) and corrosion rates are assumed to be independent of the frame (i.e. σ 3 ). Moreover, because of the small number of inspection campaigns per vessel, the number of possible combinations of corrosion rate, coating life, and thickness margin that give the same observed corrosion loss is not unique. For this reason, the parameters related to the coating life are not learned, but are fixed to μ μ 6 years, σ μ, σ years. The coefficient of variation δ M (i.e. standard deviation divided by mean value) instead of the standard deviation σ M of the thickness margin inside a frame is defined as a model hyperparameter. Finally, the measurement error is modeled as σ e mm. The estimated posterior distribution and statistics of the parameters are shown in Table 7 and Fig.. The estimated median of the corrosion loss for 2 years old vessels is approximately.26 mm. According to Sone et al. (23), the amount of corrosion in floor plates in single side skin bulk carriers after 2 years has a median of.42 mm, which is similar to the value estimated here. 7 Copyright 24 by ASME

8 a) f(a - ) b) Mean margin, a - [mm] 8 f(r ) Figure 9. a) Location of structural element Floor in vessels; b) set of thickness measurements in one of the measurement campaigns. Table 7. Statistics of the stochastic parameters of the model. Estimations Mean StDev MC error 9% Cred Interval Margin Mean margin a [mm].3.4.3e-3 (.24,.4) Coefficient of variation δ M e-4 (.2,.) Corrosion rate and variability factors Compartment var. σ e-3 (.,.28) Measurement var. σ e-3 (.39,.) Base corrosion rate r e-4 (.9,.32) Analyzing the estimated parameters from Table 7, one can observe that the variation of the corrosion rate among compartments is larger than the variation within them. This result is in agreement with the fact that conditions affecting corrosion are more even among plates within a compartment than between compartments, in particular for plates having the same elevation and orientation. It is also noted that the mean thickness margin of.3 mm represents a non-negligible value, given the low corrosion rate observed in floor plates. Uncertainty in the estimation of the corrosion rate would decrease if the correct initial thickness were reported. Updating the model Once a first estimate of the model parameters for the entire population of vessels (here: tankers) is available, this can be used as a prior probabilistic model for the analysis of a specific vessel. In the following, the results presented in Table 7 are used as prior distributions and two inspection campaigns of an additional vessel are included as observations Base corrosion rate, r [mm/year] Figure. Posterior distribution of parameters of the corrosion model for the tankers example In one of the inspection campaigns, 27% of measurements of floor plates result in a positive thickness diminution (i.e. the measured thickness is smaller than gross thickness), 4% report no diminution, and 9% of measurements correspond to a negative diminution (i.e. the measured thickness is larger than the reported gross thickness). As discussed earlier, negative diminutions are due to large thickness margins that are not reported in plans or due to measurement errors. The amount of measurements with negative diminution in the current tanker is considerably larger than in the general population (33% of measurements had this property in the four tankers analyzed in the previous section). Therefore, the posterior distribution of the thickness margin for this vessel is expected to differ substantially from the prior. Figure shows the locations of the measured plates and the measured thickness differences. It can be clearly seen how measurements from the same frame (i.e. vertical lines) are correlated. Since frames are a level in the hierarchical model, this dependence can be reflected. It is pointed out that in traditional statistical analyses of corrosion loss in ships, the measurements with a negative thickness diminution are commonly neglected, which reduces the number of available measurements and may introduce a bias. The updated mean and standard deviation of all parameters and the PDFs of the mean margin and base corrosion rate are shown in Table 8 and Fig. 2. Major differences between the prior and updated (posterior) distributions are observed for the mean margin a and the base corrosion rate r. 8 Copyright 24 by ASME

9 Table 8. Parameters update Prior distribution Posterior estimations Mean StDev Mean StDev Margin Mean margin a Coefficient of variation δ M Figure. Measurement campaign used for model updating (the diameter of the circle is proportional to the measured thickness diminution) Corrosion rate and variability factors Compartment var. σ Measurement var. σ Base corrosion rate r e-3 f(a - ) Original Update Mean margin, a - [mm] 2 8 f(r ) Original Update Base corrosion rate, r [mm/year] Figure 2. Comparison between the original (prior) and the updated (posterior) distribution The spatial distribution of the margin, coating life and corrosion rate is shown in Fig. 3. Values are shown for all locations at which measurements are available. The resulting mean of the expected margin, coating life and corrosion rate of all measured locations in the ship are.6mm,.9 years, and.2 mm/year, respectively. The results in Fig. 3 show how each random variable is affected by the spatial location of the measured thickness. The spatial pattern is particularly obvious for thickness margins (dependence on the frame) and for corrosion rates (dependence on compartments). Negative diminutions observed in the measurement campaigns (see Fig. ) are explained due to large thickness margins in those plates. Using the proposed model, it is possible to obtain corrosion rates also from those measurements with negative estimated diminutions. Figure 3. Spatial distribution of the expected margin, coating life and corrosion rate per measured location. The color of each bubble represents the mean estimate and the size of the bubble reflects the deviation of the value at this location from the total mean value (i.e. from all measured points). The bigger bubbles are values in the tails of the distribution. 9 Copyright 24 by ASME

10 SUMMARY AND CONCLUSIONS A spatio-temporal corrosion model based on a hierarchical approach is developed for analyzing corrosion in vessels. Corrosion rate, coating life and thickness margin of each plate element are modeled as stochastic parameters. The dependence among them is modeled by a hierarchical model, whereby elements are grouped according to the structural element type, frame, compartment and vessel they belong to. These levels are related to common factors and conditions that affect corrosion. As an example, plate elements from the same compartment are exposed to similar climate factors (e.g. humidity and temperature), stress distribution and protective coating quality, resulting in a high correlation among their corrosion process. This characteristic is used to define the structure of the hierarchical model. With the spatial hierarchical Bayesian model, it is possible to identify current locations with high corrosion rates or inefficient coatings but also to estimate which points are likely to have problems in future years. The hierarchical models were learnt from data using MCMC for two case studies. In the first one, thickness measurements were simulated for several vessels. This allowed verifying the accurateness of the parameter estimation. This example includes both the general estimation and the analysis of a specific vessel. The second case study is based on a set of thickness measurements carried out in four different tankers during a total of seven measurement campaigns. In a first phase, the general parameters were estimated as in the first example, but some simplifications had to be made due to limitations in the number of vessels and campaigns. In the second phase, those results were used as the prior distribution of the model parameters and information from a new vessel was incorporated to update the estimations. The model provides an estimation of the new PDF of the parameters and the corrosion amount that the vessel will have when it will be 2 years old. The latter result is in agreement with values reported in literature for the same type of structural element in similar vessels based on a larger thickness measurement database. The ultimate purpose of the proposed model is to describe the uncertainty of the corrosion process in vessels and to provide quantitative support in decision-making related to inspections. Knowing the corrosion process in a vessel can help to focus inspection on the affected areas and to minimize unnecessary efforts. Such decisions may be made using reliability criteria considering the estimations for corrosion degradation obtained from the proposed model. During the last years, Classification Societies and the International Maritime Organization have tried to incorporate risk-based methods to develop new rules and regulations. The proposed hierarchical Bayesian model provides a new approach to achieve this goal. NOMENCLATURE coating life deterioration amount L correlation length M thickness margin R corrosion rate W current thickness Z observed thickness diminution a margin's rate parameter long-term thickness loss r base corrosion rate time w gross thickness from plans Π correlation matrix α i i-th variability factor δ M coefficient of variation of thickness margin μ mean coating life μ μ mean of mean coating life ρ( ) correlation σ standard deviation of coating life σ M standard deviation of thickness margin σ e standard deviation of measurement error standard deviation of mean coating life σ μ ACKNOWLEDGMENTS This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through Grant STR 4/3- and the Consejo Nacional de Ciencia y Tecnología (CONACYT) through Grant No. 37. REFERENCES Akpan U, Koko TS, Ayyub B, Dunbar TE, 22. Risk assessment of aging ship hull structures in the presence of corrosion and fatigue. Marine Structures, Gamerman D, Lopes H F, 26. Markov Chain Monte Carlo, stochastic simulation for bayesian inference. 2nd Ed. Chapman & Hall/CRC, Florida, USA. Gardiner CP, Melchers RE, 23. Corrosion analysis of bulk carriers, Part I: operational parameters influencing corrosion rates. Marine Structures 6, Gelman A, Carlin JB, Stern HS, Rubin DB, 24. Bayesian Data Analysis. 2nd ed., Florida, USA: Chapman & Hall/CRC Germanischer Lloyd Group, 29. Rules for classification and construction: II Materials and welding. Germanischer Lloyd Aktiengesellschaft, Hamburg. Gilks WR, Richardson S, Spiegelhalter DJ, 996. Markov Chain Monte Carlo in practice. Chapman & Hall/CRC. Grundy P, 23. Bulk carriers. Marine Structures 6, Editorial Copyright 24 by ASME

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