Risk Based Inspection Planning for Ship Structures Using a Decision Tree Method

Transcription

1 TECHNICAL PAPER Risk Based Inspection Planning for Ship Structures Using a Decision Tree Method Dianqing Li, Shengkun Zhang, Wenyong Tang ABSTRACT A theoretical framework of risk-based inspection and repair was proposed for ship structures subjected to corrosion deterioration. A repair index was presented to consider reliability updating after repair. The reliability updating after inspection and repair was performed by using the Bayesian updating method. A decision tree was established for selecting the optimal inspection and repair strategy for ship structures. By comparing the expected costs associated with different inspection and repair strategies, the smallest expected cost associated with the inspection and repair strategy can be identified as the optimal one. Based on this, a method was proposed to determine the sensitivities of both optimal inspection and repair strategy. Furthermore, some formulae were derived to analyze the sensitivities. A numerical example was investigated to illustrate the process of selecting the optimal inspection and repair strategy. The results show that the decision tree method is very effective. Furthermore, different values of various costs have significant effects on the reliability and stability of decision results. Introduction To ensure the safety and reliability of ship structures during their service lifetime, inspections are essential and important to evaluating corrosion and fatigue damage and scheduling maintenance or repair. Since ISSC1997, there has been considerable interest in the area of inspections. This is primarily because of advances in inspection technology by which flaws can be reliably found to an increasing extent. Bea and Xu (1997), and Kawano and Augusto (1997) emphasized the importance of the effective inspection strategy. In the coming ISSC 2003, inspection and monitoring of ship structures is proposed as a special subject. It has been found that the research on inspections of ship structures is an important trend. Since risk-based inspection planning has many advantages over traditional inspection planning, it is of significance to investigate riskbased inspection planning for ship structures. Traditional inspection planning is based on prescriptive rules, which do not target inspection efforts to the actual condition or to the importance of a component for the operation of a ship. This may not only incur higher costs but may also result in unnecessary inspections. In contrast with traditional inspection planning, risk-based inspection planning can establish optimized inspection and repair plans for meeting the risk acceptance criterion of a component. The term of risk-based inspection has been used for many years in the oil industry. It was not adopted for ship structures until recent years. For inspections of ship structures, Ma, Orisamolu, and Bea (1997) detailed the steps to be performed in conducting inspections of ship structures. Ma (1998) presented the framework of a risk-based inspection approach for tankers. The risk-based approach uses two parameters, criticality and NAVAL ENGINEERS JOURNAL SPRING

2 Risk Based Inspection Planning for Ship Structures Using a Decision Tree Method susceptibility to rank the inspection priority so that structural details with higher risk receive more attention. This approach, termed priority assessment, provides the basis for developing an optimal inspection strategy. Further details can be found in Ma, Orisamolu, and Bea (1999). Generally, the above three papers mainly focused on a qualitative analysis of inspections for tankers. Landet, Lotsberg, and Sigursson (2000) determined the target failure probability on the basis of a cost-optimal solution to an FPSO. De Souza and Ayyub (2000) proposed a simple decision tree to select the optimal inspection strategy, which did not consider the effects of different inspection and repair methods. Furthermore, they assumed that the failure of component did not occur during the service lifetime if the component was repaired. Their assumption would seem unreasonable. In addition, the sensitivity analysis of the decision results and the problem of reliability updating were not developed in their study. For these reasons, Ayyub et al. (2000) also presented a decision tree for selecting the optimal inspection strategy, which considered the effects of different inspection and repair methods on the selection of an optimal inspection strategy. However, Ayyub et al. (2000) did not take the problem of fatigue reliability updating into account either, which led to the unreasonable result that both no inspection and no repair strategy was the optimal strategy. It is essential that structural reliability should be updated after inspections have been performed on a structure. Furthermore, the potential benefit of inspections cannot be reflected if the reliability updating is neglected. As a result, the problem of reliability updating must be incorporated into the development of inspection and repair planning of ship structures. Besides, since most of existing studies adopt some hypothetical values of various costs because of inadequate data, there exist some uncertainties that are critical in the selection of inspection and repair strategy. The sensitivity analysis of the decision results should be explored for studying the stability and reliability of decision results. Furthermore, all the foregoing studies focused on the problem of fatigue damage for ship structures. Since corrosion is also an important damage form for ship structures affecting the physical life of ship structures, this paper focuses mainly on the inspection and repair planning of ship structures subjected to corrosion damage. This paper presents the framework of riskbased inspection and repair planning for ship structures. By considering the reliability updating, the updated probability of failure and reliability index after inspection and repair are calculated by using the Bayesian updating method. At the same time, a decision tree, considering different inspection and repair methods, is established to select the optimal inspection and repair strategy. Finally, some sensitivity studies are provided as well. Framework of Risk-Based Inspection The risk-based inspection is a method that uses risk analysis to prioritize and manage the inspection planning for structures. The aim of risk based inspection planning is to establish a cost-effective inspection strategy that can be used to document and maintain the target safety level. An effective risk based inspection planning can lower the risk of structures with a specified inspection action. According to the characteristics of the inspection planning for ship structures, Figure 1 presents a general framework to analyze the inspection and repair planning of ship structures. Reliability Updating After an inspection performed on ship structure subjected to corrosion damage, the results can be classified as no corrosion detected, corrosion detected, and corrosion detected and size measured. Each inspection result gives additional information on the in- 74 SPRING 2004 NAVAL ENGINEERS JOURNAL

3 FIGURE 1: Flowchart of Risk- Based Inspection Planning for Ship Structures Subjected to Corrosion Deterioration service condition of the ship structure. The additional information leads to changes of the prior reliability and the basic random variables affecting the reliability. Therefore, it is necessary to update reliability and models of the basic variables through additional information. Jiao (1989) presented a Bayesian updating method to solve the problem of reliability updating. Probability of Detection Model The probability of detection expresses the probability of detecting a flaw of a given size. It is the common measure to evaluate the capability of a non-destructive inspection (NDI) technique. The parameters in probability of detection can be estimated through regression analysis of experimental data. According to Guedes Soares and Garbatov s studies (1996), two NDI techniques are usually applied to inspect hull structures and are used herein for demonstration purposes. These inspection methods are visual inspection (VI) and magnetic particle inspection (MPI). For both inspection methods, Guedes Soares and Garbatov (1996) used the exponential distribution to represent probability of detection, which is written as: (1) where a is the measured corrosion size; λ is the scale parameter of the exponential distribution, which can be estimated through regression analysis of experimental data; and a d is the minimum detectable corrosion size below which the corrosion cannot be detected. Guedes Soares and Garbatov (1996) suggested the following data for the probability of detection distribution. VI: a d = 5.0mm, λ = 1.0mm, and MPI: a d = 1.0mm, λ = 2.5mm. Both techniques are considered as possible inspection methods for inspection planning. Corrosion Model of Ship Structures Corrosion is said to be one of the most dominant factors affecting the physical life of ship structures. There exist some corrosion models to describe the corrosion damage of ship structures. For the purposes of illustration, the corrosion model proposed by Paik, Kim, and Lee (1998) is adopted in this study. The wear of plate thickness because of corrosion may be generally expressed as a function of the time, namely: NAVAL ENGINEERS JOURNAL SPRING

4 Risk Based Inspection Planning for Ship Structures Using a Decision Tree Method (2) where d(t) is the wear of thickness due to corrosion, τ i is the coating life, A,B are coefficients. The safety margin M F(t) modeling failure at time t is formulated as (3) where d crit is the critical allowable thickness of corrosion wastage. Formulae to Update Reliability Regardless of whether corrosion damage is detected or not, each inspection provides additional information to that available at the design phase, which can be used to update the reliability. Different methods can be used to update reliability of ship structures. Event updating, variable updating, and statistical updating are the main categories of updating, that can be used in reliability updating. The selection of a specific updating method mainly depends on available information and additional details of structure. For the purposes of illustration, the event updating is used herein. Hence, the updated probability of failure can be determined by using the following conditional probability. where E is the possible results from an inspection event, M F is the safety margin given by Equation (3). A more detailed (4) introduction of reliability updating can be found in Jiao (1989). Numerical Example A ship structural component subjected to corrosion is considered in the following example. Based on the available data (Ayyub et al. 2002), all the deterministic and random parameters used in this example are listed in Table 1. For the purposes of illustration, the target probability of failure P f min is assumed to be 10-5, and the corresponding minimum acceptable reliability index β min is equal to FORM method is employed for the estimation of time-dependent failure probability and reliability index. The results are plotted in Figures 2 and 3, respectively. It can be seen that a minimum acceptable reliability index β min = 3.95 is reached at the end of a seven year period. Therefore, to ensure the safety of the structure, the first inspection must be performed before the instant t = 7a. It is further assumed that a corrosion size of a = 6mm is detected. The probability of detection can be obtained according to Equation (1). The updated probability of failure and reliability index can be obtained by using the Bayesian updating method. For the convenience of comparison, the updated probability of failure and reliability index are also plotted in Figures 2 and 3, respectively. It can be found from Figures 2 and 3 that the updated probability of failure and reliability index significantly change in comparison with the prior probability of failure and reliability index. Furthermore, these changes become more obvious with time. Therefore, Table 1 Statistical Characteristics of Basic Parameters VARIABLES DISTRIBUTION MEAN COV A (mm/a) Normal B Deterministic 1 τ i /a Deterministic 3 d crit /mm Normal SPRING 2004 NAVAL ENGINEERS JOURNAL

5 FIGURE 2: Time Dependent Probability of failure FIGURE 3: Time dependent reliability index selecting the optimal inspection and repair strategy should consider the problem of reliability updating. Concerning reliability updating after repair, it is assumed that the reliability index increases from β min to β N if a repair is performed after inspection, assuming that the repair is finished in an instant. To consider the effect of repair on the reliability index of the structure, a repair index R is defined in this study. (5) where β 0 is the initial reliability index, β min is the minimum acceptable reliability index, and β N is the reliability after repair. Three NAVAL ENGINEERS JOURNAL SPRING

6 Risk Based Inspection Planning for Ship Structures Using a Decision Tree Method values of R=0.5, R=0.7, and R=0.9 are used to represent three different repair methods herein. At the same time, the bigger the repair index is, the higher the costs associated with repair method used are. It is assumed that a repair is performed at t = 7a by using any one of the three repair methods. For the sake of simplicity, this paper assumes that the material properties are independent before and after repair. Figures 4 and 5 give the updated probability of failure and reliability index after repair by using the Bayesian updating method, along with the prior probability of failure and reliability index. It can be seen from Figures 4 and 5 that the updated probability of failure has a significant decrease in comparison with the prior probability of failure, while the updated reliability index has a significant increase in comparison with the prior reliability index. It is also seen that different values of R significantly influence the updated probability of failure and reliability index. Therefore, reliability updating must be taken into account in selecting the optimal inspection and repair strategy. Decision For Inspection and Repair The decision tree is commonly used to select the optimal inspection and repair strategy for ship structures. Figure 6 presents a decision tree based on the literature (De Souza and Ayyub 2000, Ayyub et al. 2000). This tree illustrates the sequence of decisions and uncertainties involved in the choice among the three possible choices for the structure. These three possible choices are visual inspection (Inspection Method 1), magnetic particle inspection (Inspection Method 2), and no inspection. In the interest of brevity, the result of the MPI method is not given here. The values of various costs presented in Figure 6 are determined according to the literature (De Souza and Ayyub 2000, Ayyub et al. 2000). The costs presented on the decision tree are adopted for the example, which do not correspond to any real case. The probabilities of failure are determined by using the approach described in the foregoing section. In order to analyze the results of the decision tree, the branches related to a given inspection method must be grouped to form a strategy pair. Each pair is composed of one decision branch corresponding to conse- FIGURE 4: Updated Probability of failure after repair 78 SPRING 2004 NAVAL ENGINEERS JOURNAL

7 FIGURE 5: Updated Reliability Index After Repair quences associated with the detection of the corrosion damage and the second branch corresponding to the consequences associated with the non-detection of the corrosion damage. The following pairs are considered for demonstration purposes. Strategy 1-1: Inspection method 1,repair method 1 and non-detection; Strategy 1-2: Inspection method 1,repair method 2 and non-detection; Strategy 1-3: Inspection method 1,repair method 3 and non-detection; Strategy 1-4: Inspection method 1,no repair and non-detection; Strategy 2-1: Inspection method 2,repair method 1 and non-detection; Strategy 2-2: Inspection method 2,repair method 2 and non-detection; Strategy 2-3: Inspection method 2,repair method 3 and non-detection; Strategy 2-4: Inspection method 2,no repair and non-detection; Strategy 3-1: No inspections. Based on the decision tree and the available data presented in Figure 6, the expected costs associated with each of these strategy pairs can be obtained. The results are listed in Table 2. It can be seen from Table 2 that the optimal inspection and repair strategy is the Strategy FIGURE 6: Decision Tree for Selecting Inspection and Repair Strategies NAVAL ENGINEERS JOURNAL SPRING

8 Risk Based Inspection Planning for Ship Structures Using a Decision Tree Method Table 2 Expected Costs for Different Inspection and Repair Strategies Inspection and repair strategy pairs E(C)/K$ FIGURE 7: Risk Profile for Different Inspection and Repair Strategies 1-4,which has the lowest expected cost of 24K$. The worst inspection and repair strategy is the Strategy 2-3 associated with the highest expected cost of 67K$. It can also be seen that the expected costs associated with the strategies for the inspection method 1 are lower than those for the inspection method 2. The reason is that the probability of detection defined by the better inspection method is not high enough to reduce the weight consequential costs related to the structure failure in comparison with its higher costs of inspection. In addition, the expected costs associated with Strategy 1-4 is the lowest among all the strategies for the inspection method 1. The reason is that the higher repair costs incurred using the better repair method are not compensated by the reduction of the weight consequential costs related to the structure failure. Similar conclusion can be drawn for the inspection method 2. It is further seen that the expected costs increase with the increase of the repair costs. This indicates that the increase of the repair costs associated with the better repair method is not compensated by the reduction of the weight consequential costs related to the structure failure. In addition to the analysis of the expected costs, the risk profiles of the nine strategies can be also studied (Ayyub et al. 2000). Figure 7 gives the risk profile for the numerical example studied herein. The upper bound of costs is 300K$, once that beyond this value, the probabilities for all strategies are equal to one. From the comparison among the risk profiles associated with all inspection and repair strategies, it can be found that Strategy 1-4 presents higher probability associated with lower cost, followed by Strategy 1-1. The results show that Strategy 1-4 should be used for the ship structural inspection. Reasonable agreements can be observed between the results from the risk profiles and those from Table.2. Sensitivity Analysis The expected cost is the criterion for selecting the optimal inspection and repair strategy by using the decision tree approach. From the foregoing discussion, it can be observed that the expected costs are closely related to the values of various parameters. Since the values of various costs are hypothetical, there exist statistical uncertainties, which influence the stability and reliability of decision results. Therefore, sensitivities of various costs should be explored. For the convenience of derivation, let E(C) A, E(C) B, and E(C) C be the expected costs associated with the strategy of inspection with repair, the strategy of inspection without repair, and the strategy of no inspection without repair, respectively. Let C 1, C R, and C F be the inspection, repair, and failure costs, respectively. Let C FL, C FM, and C FH be the costs associated with the low, medium, and high consequences of failure. Let P R, P R, and P IR be the probability of failure after repair, after inspection without repair, and after no inspection without repair, respectively. 80 SPRING 2004 NAVAL ENGINEERS JOURNAL

9 According to the available data presented in Figure 6, C I1 = 20 (K$), C 12 = 40 (K$), C R1 = 10 (K$), C R2 = 20(K$), C R3 = 30(K$), P FL = 0.15, P FM = 0.70, P FH = 0.15, C FL = 50(K$), C FM = 100(K$), C FH = 200(K$), P R = , PIR = , PR1 = , P R2 = , P R3 = Using the decision tree presented in Figure 6, we have (6) (7) (8) From Equations (7) and (8), it is easy to find that, if then (9) (10) Equation (10) indicates that regardless of what the values of various costs are, the strategy 3-1 is the optimal strategy in comparison with the strategies 1-1, 1-2, 1-3, 2-1, 2-2, 2-3. This is the same as the result given by Ayyub et al. (2000). However, this is not consistent with the actual case since inspections and repairs are performed during the service lifetime of ship structures. Since Ayyub et al. (2000) did not take into account the reliability updating after inspection, they simply assumed that P IR is equal to PR. Actually, it is impossible that PIR is equal to P R. The reason is that the probability of failure must be updated whether inspections or repairs are performed or not. Furthermore, P R is always lower than PIR. Consider the effects of the values of various costs on the selection of optimal inspection and repair strategy. For the purpose of illustration, the inspection and repair strategies associated with the inspection method 1 are studied herein. Consider the case of the strategy of inspection without repair has the lowest expected cost, followed by the strategy of inspection with repair. The strategy of no inspection without repair has the highest cost. Then, it is essential that the following two conditions should be satisfied. (11) For the convenience of derivation, let C TF = P FL C FL + P FM C FM + P FH C FH. If C TF remains the same, substituting Equations (6) and (7) into Equation (11), gives (12) NAVAL ENGINEERS JOURNAL SPRING

10 Risk Based Inspection Planning for Ship Structures Using a Decision Tree Method (13) By substituting the available data presented in Figure 6 into Equation (13), then (14) Since C R1 = 10(K$) > 2.2(K$), C R2 = 20(K$) > 2.7(K$) and C R3 = 30(K$) > 3.1(K$), Equation (11) holds for all the three repair methods. Reasonable agreements can be observed between the results from the sensitivity analysis and those from Table 2. If C R remains the same, then (15) By substituting the available data presented in Figure 6 into Equation (15), C TF associated with the three repair methods can be obtained as (16) From the available data presented in Figure 6, C TF = 107.5(K$). Since Equation (16) holds for the value of 107.5(K$), Equation (10) can be also obtained. These results are consistent with those from Table 2, as well. Similarly, if C TF remains the same, substituting Equations (6) and (8) into Equation (12), gives From the available data presented in Figure 6, C R1 = 10(K$)<14.9(K$), C R2 = 20(K$) >15.4(K$) and C R3 = 30(K$)>15.8(K$). Therefore, Equation (12) holds for the repair method 1, while Equation (12) does not hold for the repair methods 2 and 3. Reasonable agreements can be observed between the results from the sensitivity analysis and those from Table 2. If C R remains the same, then (19) By substituting the available data presented in Figure 6 into Equation (19), C TF associated with the three repair methods can be obtained as follows. (20) From the available data presented in Figure 6, C TF = 107.5(K$) > 96.2(K$) and C TF = 107.5(K$) < 118.0(K$) and C TF = 107.5(K$) < 139.8(K$). Therefore, Equation (12) holds for the repair method 1, while Equation (12) does not hold for the repair methods 2 and 3. These results are consistent with those from Table 2, as well. According to foregoing discussion, it can be observed that if C TF remains the same, the conditions in Equations (14) and (18) can be reduced to (21) (17) Substituting the available data presented in Figure 6 into Equation (17), gives (18) If C R remains the same, the conditions in Equations (16) and (20) can be reduced to (22) The strategy of inspection without repair is the optimal strategy, if Equation (21) or Equation (22) is satisfied. 82 SPRING 2004 NAVAL ENGINEERS JOURNAL

11 A similar approach can be employed to other cases, such as E(C) A <E(C) B <E(C) C, E(C) C <E(C) A <E(C) B, etc. In the interest of brevity, the results are not given again. The relationship between the expected costs associated with different inspection methods and different repair methods can be further analyzed. First, considering the relationship between the Strategy 1-4 associated with the inspection method 1 and the Strategy 2-4 associated with the inspection method 2. If then (23) (24) From the available data presented in Figure 6, we have C 11 = 20(K$) < C 12 = 40(K$), then Equation (24) holds. So the Strategy 1-4 is always better than the Strategy 2-4. This conclusion agrees with that from Table 2. If (25) C R and P R and remain the same, by considering the relationship between Strategy 1-1 associated with the repair method 1 and Strategy 1-2 associated with the repair method 2, then (26) By substituting the available data presented in Figure 6 into Equation (26), then (27) Since C TF = 107.5(K$) < (K$), Strategy 1-1 is always better than Strategy 1-2. Similarly, Strategy 2-1 is always better than the Strategy 2-2. If C TF, C R1, P R1, and P R2 remain the same, we have (28) Similarly, by substituting the available data presented in Figure 6 into Equation (28), then (29) Since C R2 = 20(K$) > 10.5(K$), the same conclusions can be obtained. All these results are consistent with those from Table 2. Conclusions To keep the probability of failure of ship structures below a specified target safety level, a risk-based inspection and repair plan was developed. A component subjected to corrosion deterioration was investigated to illustrate the application of the proposed method. According to the principle that both the risk costs of failure and the costs of inspection and repair should be minimized, the decision tree approach is used to select the suitable inspection and repair strategy for the component. Finally, from the analysis of the sensitivities for various costs, it is found that the expected costs are significantly influenced by the values of the costs of inspection, repair, and failure. Since the decision results are significantly affected by the reliability of data in the process of selecting the optimal inspection strategy, it is recommended that further work should be focused on the systematic gathering of the costs for performing all relevant activities in advance. REFERENCES Ayyub, B.M., U.O.Akpan., G.F.M.De Souza.,et al.,2000, Risk-based Life Cycle Management of Ship Structures, SR-1407, Ship Structures Committee, Washington D. C. Ayyub,B.M., U.O.Akpan., P.A. Rushton.,et al.,2002, Risk-informed Inspection of Marine Vessels, SSC-421, Ship Structures Committee, Washington D. C. Bea, R.G., T. Xu.,1997. In-service Inspection Programs for Marine Structures, Proceedings of the 16th International Conference on Offshore Mechanics and Arctic Engineering, Yokohama, Japan. De Souza, G,F,M. and B.M. Ayyub., 2000, Risk Based Inspection Planning for Ship Hull Structures, Association of Scientists and Engineers 2000 Technical Symposium, USA. Guedes Soares, C. and Y. Garbatov., 1996, Fatigue Reliability of the Ship Hull Girder Accounting for NAVAL ENGINEERS JOURNAL SPRING

12 Risk Based Inspection Planning for Ship Structures Using a Decision Tree Method Inspection and Repair, Reliability Engineering and System Safety,,51 (3): Jiao, G.Y., 1989, Reliability Analysis of Crack Growth Under Random Loading Considering Model Updating, PhD thesis, Norwegian Institute of Technology, Trondheim, Norway. Kawano,A. and O.B Augusto., 1997, Consideration of the Influence of Independent Factors on Inspection Planning of Structures, Proceedings of the 7th International Offshore and Polar Engineering Conference, 4: Landet,E., I.Lotsberg., G.Sigursson., 2000, Risk Based Inspection of an FPSO, Proceedings of the Annual Offshore Technology Conference, Houston Texas, Ma, K.T., 1998 Tanker Inspection and a Riskbased Inspection Approach, Proceeding of the Eighth International Offshore and Polar Engineering Conference, Canada, 4: Ma, K.T., I.R. Orisamolu., R.G. Bea., 1999, Optimal Strategies Inspection of Ships for Fatigue and Corrosion Damage, Ship Structures Committee, Washington D. C. Ma, K.T., I.R. Orisamolu., R.G. Bea., R.T. Huang., 1997, Towards Optimal Inspection Strategies for Fatigue and Corrosion Damage, Transactions of the SNAME, 105(1): Paik, J.K., S.K. Kim., S.K. Lee., 1998 Probabilistic Corrosion Rate Estimation Model for Longitudinal Strength Members of Bulk Carriers, Ocean Engineering, 25(10): DIANQUING LI is currently completing his Ph.D. at the school of naval architecture and ocean engineering, Shanghai Jiao Tong University. His advisor is Prof. Shengkun Zhang. His dissertation is on risk-based inspection, maintenance, and repair decision making for ship structures. He received his B.S. and M.S. in the school of mechanical and electrical engineering at the Hohai University where his master s thesis was on reliability assessment of hydraulic steel structures. He has published more than forty journal papers. SAVE THE DATE SAVE THE DATE Ship Electric System Control and Reconfiguration Symposium February 16-17, 2005 Jacksonville, FL With Exhibits Theme TBD SAVE THE DATE SAVE THE DATE 84 SPRING 2004 NAVAL ENGINEERS JOURNAL