risk based inspection planning for ship structures using...

IntroductionTo ensure the safety and reliability of shipstructures during their service lifetime,inspections are essential and important toevaluating corrosion and fatigue damage andscheduling maintenance or repair. SinceISSC1997, there has been considerable inter-est in the area of inspections. This is primari-ly because of advances in inspection technol-ogy by which flaws can be reliably found toan increasing extent. Bea and Xu (1997),and Kawano and Augusto (1997) empha-sized the importance of the effective inspec-tion strategy. In the coming ISSC 2003,inspection and monitoring of ship structuresis proposed as a special subject. It has beenfound that the research on inspections ofship structures is an important trend. Sincerisk-based inspection planning has manyadvantages over traditional inspection plan-ning, it is of significance to investigate risk-based inspection planning for ship structures.

Traditional inspection planning is based onprescriptive rules, which do not targetinspection efforts to the actual condition orto the importance of a component for theoperation of a ship. This may not only incurhigher costs but may also result in unneces-sary inspections. In contrast with traditionalinspection planning, risk-based inspectionplanning can establish optimized inspectionand repair plans for meeting the risk accep-tance criterion of a component. The term ofrisk-based inspection has been used for manyyears in the oil industry. It was not adoptedfor ship structures until recent years.

For inspections of ship structures, Ma,Orisamolu, and Bea (1997) detailed the stepsto be performed in conducting inspections ofship structures. Ma (1998) presented theframework of a risk-based inspectionapproach for tankers. The risk-basedapproach uses two parameters, criticality and

Risk Based Inspection Planningfor Ship Structures Using aDecision Tree Method■ Dianqing Li, Shengkun Zhang, Wenyong Tang

ABSTRACTA theoretical framework of risk-based inspection and repair was proposed for ship structures subjectedto corrosion deterioration. A repair index was presented to consider reliability updating after repair. Thereliability updating after inspection and repair was performed by using the Bayesian updating method. Adecision 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 smallestexpected 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 andrepair strategy. Furthermore, some formulae were derived to analyze the sensitivities. A numerical exam-ple was investigated to illustrate the process of selecting the optimal inspection and repair strategy. Theresults show that the decision tree method is very effective. Furthermore, different values of various costshave significant effects on the reliability and stability of decision results.

T E C H N I C A L P A P E R

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susceptibility to rank the inspection priorityso that structural details with higher riskreceive more attention. This approach,termed priority assessment, provides the basisfor developing an optimal inspection strategy.Further details can be found in Ma,Orisamolu, and Bea (1999). Generally, theabove three papers mainly focused on a qual-itative analysis of inspections for tankers.Landet, Lotsberg, and Sigursson (2000)determined the target failure probability onthe basis of a cost-optimal solution to anFPSO. De Souza and Ayyub (2000) proposeda simple decision tree to select the optimalinspection strategy, which did not considerthe effects of different inspection and repairmethods. Furthermore, they assumed that thefailure of component did not occur duringthe service lifetime if the component wasrepaired. Their assumption would seemunreasonable. In addition, the sensitivityanalysis of the decision results and the prob-lem of reliability updating were not devel-oped in their study. For these reasons, Ayyubet al. (2000) also presented a decision tree forselecting the optimal inspection strategy,which considered the effects of differentinspection and repair methods on the selec-tion of an optimal inspection strategy.However, Ayyub et al. (2000) did not takethe problem of fatigue reliability updatinginto account either, which led to the unrea-sonable result that both no inspection and norepair strategy was the optimal strategy.

It is essential that structural reliability shouldbe updated after inspections have been per-formed on a structure. Furthermore, thepotential benefit of inspections cannot bereflected if the reliability updating is neglect-ed. As a result, the problem of reliabilityupdating must be incorporated into thedevelopment of inspection and repair plan-ning of ship structures. Besides, since mostof existing studies adopt some hypotheticalvalues of various costs because of inadequatedata, there exist some uncertainties that arecritical in the selection of inspection andrepair strategy. The sensitivity analysis of the

decision results should be explored forstudying the stability and reliability of deci-sion results. Furthermore, all the foregoingstudies focused on the problem of fatiguedamage for ship structures. Since corrosionis also an important damage form for shipstructures affecting the physical life of shipstructures, this paper focuses mainly on theinspection and repair planning of ship struc-tures subjected to corrosion damage.

This paper presents the framework of risk-based inspection and repair planning for shipstructures. By considering the reliabilityupdating, the updated probability of failureand reliability index after inspection andrepair are calculated by using the Bayesianupdating method. At the same time, a deci-sion tree, considering different inspection andrepair methods, is established to select theoptimal inspection and repair strategy. Finally,some sensitivity studies are provided as well.

Framework of Risk-Based Inspection The risk-based inspection is a method thatuses risk analysis to prioritize and managethe inspection planning for structures. Theaim of risk based inspection planning is toestablish a cost-effective inspection strategythat can be used to document and maintainthe target safety level. An effective risk basedinspection planning can lower the risk ofstructures with a specified inspection action.According to the characteristics of theinspection planning for ship structures,Figure 1 presents a general framework toanalyze the inspection and repair planning ofship structures.

Reliability UpdatingAfter an inspection performed on ship struc-ture subjected to corrosion damage, theresults can be classified as no corrosiondetected, corrosion detected, and corrosiondetected and size measured. Each inspectionresult gives additional information on the in-

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

7 4 SPRING 2004 NAVAL ENGINEERS JOURNAL


service condition of the ship structure. Theadditional information leads to changes ofthe prior reliability and the basic randomvariables affecting the reliability. Therefore,it is necessary to update reliability and mod-els of the basic variables through additionalinformation. Jiao (1989) presented aBayesian updating method to solve the prob-lem of reliability updating.

Probability of Detection ModelThe probability of detection expresses theprobability of detecting a flaw of a givensize. It is the common measure to evaluatethe capability of a non-destructive inspection(NDI) technique. The parameters in proba-bility of detection can be estimated throughregression analysis of experimental data.According to Guedes Soares and Garbatov’sstudies (1996), two NDI techniques are usu-ally applied to inspect hull structures and areused herein for demonstration purposes.These inspection methods are visual inspec-tion (VI) and magnetic particle inspection(MPI). For both inspection methods, GuedesSoares and Garbatov (1996) used the expo-nential distribution to represent probabilityof detection, which is written as:

(1)

where a is the measured corrosion size; λ isthe scale parameter of the exponential distri-bution, which can be estimated throughregression analysis of experimental data; andad is the minimum detectable corrosion sizebelow which the corrosion cannot be detect-ed. Guedes Soares and Garbatov (1996) sug-gested the following data for the probabilityof detection distribution. VI: ad = 5.0mm, λ= 1.0mm, and MPI: ad = 1.0mm, λ = 2.5mm.Both techniques are considered as possibleinspection methods for inspection planning.

Corrosion Model of Ship StructuresCorrosion is said to be one of the most dom-inant factors affecting the physical life ofship structures. There exist some corrosionmodels to describe the corrosion damage ofship structures. For the purposes of illustra-tion, the corrosion model proposed by Paik,Kim, and Lee (1998) is adopted in this study.The wear of plate thickness because of cor-rosion may be generally expressed as a func-tion of the time, namely:


FIGURE 1:Flowchart of Risk-Based InspectionPlanning for ShipStructures Subjectedto CorrosionDeterioration


(2)

where d(t) is the wear of thickness due tocorrosion, τi is the coating life, A,B are coef-ficients.

The safety margin M F(t) modeling failure attime t is formulated as

(3)

where dcrit is the critical allowable thicknessof corrosion wastage.

Formulae to Update Reliability Regardless of whether corrosion damage isdetected or not, each inspection providesadditional information to that available atthe design phase, which can be used toupdate the reliability. Different methods canbe used to update reliability of ship struc-tures. Event updating, variable updating, andstatistical updating are the main categoriesof updating, that can be used in reliabilityupdating. The selection of a specific updat-ing method mainly depends on availableinformation and additional details of struc-ture. For the purposes of illustration, theevent updating is used herein. Hence, theupdated probability of failure can be deter-mined by using the following conditionalprobability.

(4)

where E is the possible results from aninspection event, MF is the safety margingiven by Equation (3). A more detailed

introduction of reliability updating can befound in Jiao (1989).

Numerical ExampleA ship structural component subjected tocorrosion is considered in the followingexample. Based on the available data (Ayyubet al. 2002), all the deterministic and ran-dom parameters used in this example arelisted in Table 1.

For the purposes of illustration, the targetprobability of failure Pf min is assumed to be10-5 , and the corresponding minimumacceptable reliability index βmin is equal to3.95. FORM method is employed for the esti-mation of time-dependent failure probabilityand reliability index. The results are plottedin Figures 2 and 3, respectively. It can be seenthat a minimum acceptable reliability indexβmin = 3.95 is reached at the end of a sevenyear period. Therefore, to ensure the safety ofthe structure, the first inspection must be per-formed before the instant t = 7a. It is furtherassumed that a corrosion size of a = 6mm isdetected. The probability of detection can beobtained according to Equation (1). Theupdated probability of failure and reliabilityindex can be obtained by using the Bayesianupdating method. For the convenience ofcomparison, the updated probability of fail-ure and reliability index are also plotted inFigures 2 and 3, respectively.

It can be found from Figures 2 and 3 thatthe updated probability of failure and relia-bility index significantly change in compari-son with the prior probability of failure andreliability index. Furthermore, these changesbecome more obvious with time. Therefore,



Table 1Statistical Characteristics of Basic Parameters

VARIABLES DISTRIBUTION MEAN COVA (mm/a) Normal 2.1 0.01

B Deterministic 1 —τi/a Deterministic 3 —

dcrit/mm Normal 40 0.20


selecting the optimal inspection and repairstrategy should consider the problem of reli-ability updating.

Concerning reliability updating after repair, itis assumed that the reliability index increasesfrom βmin to βN if a repair is performed afterinspection, assuming that the repair is fin-ished in an instant. To consider the effect of

repair on the reliability index of the struc-ture, a repair index R is defined in this study.

(5)

where β0 is the initial reliability index, βmin isthe minimum acceptable reliability index,and βN is the reliability after repair. Three


FIGURE 2:Time DependentProbabilityof failure

FIGURE 3:Time dependentreliability index


values of R=0.5, R=0.7, and R=0.9 are used torepresent three different repair methods here-in. At the same time, the bigger the repairindex is, the higher the costs associated withrepair method used are. It is assumed that arepair is performed at t = 7a by using anyone of the three repair methods. For the sakeof simplicity, this paper assumes that thematerial properties are independent beforeand after repair. Figures 4 and 5 give theupdated probability of failure and reliabilityindex after repair by using the Bayesianupdating method, along with the prior prob-ability of failure and reliability index.

It can be seen from Figures 4 and 5 that theupdated probability of failure has a signifi-cant decrease in comparison with the priorprobability of failure, while the updated reli-ability index has a significant increase incomparison with the prior reliability index.It is also seen that different values of R sig-nificantly influence the updated probabilityof failure and reliability index. Therefore,reliability updating must be taken intoaccount in selecting the optimal inspectionand repair strategy.

Decision For Inspection and RepairThe decision tree is commonly used to selectthe optimal inspection and repair strategy forship structures. Figure 6 presents a decisiontree based on the literature (De Souza andAyyub 2000, Ayyub et al. 2000). This treeillustrates the sequence of decisions anduncertainties involved in the choice amongthe three possible choices for the structure.These three possible choices are visual inspec-tion (Inspection Method 1), magnetic particleinspection (Inspection Method 2), and noinspection. In the interest of brevity, the resultof the MPI method is not given here. The val-ues of various costs presented in Figure 6 aredetermined according to the literature (DeSouza and Ayyub 2000, Ayyub et al. 2000).The costs presented on the decision tree areadopted for the example, which do not corre-spond to any real case. The probabilities offailure are determined by using the approachdescribed in the foregoing section.

In order to analyze the results of the decisiontree, the branches related to a given inspec-tion method must be grouped to form astrategy pair. Each pair is composed of onedecision branch corresponding to conse-



FIGURE 4:Updated Probabilityof failure after repair


quences associated with the detection of thecorrosion damage and the second branchcorresponding to the consequences associat-ed with the non-detection of the corrosiondamage. The following pairs are consideredfor demonstration purposes.

Strategy 1-1: Inspection method 1,repairmethod 1 and non-detection;



Strategy 1-4: Inspection method 1,no repairand non-detection;




Strategy 2-4: Inspection method 2,no repairand non-detection;

Strategy 3-1: No inspections.

Based on the decision tree and the availabledata presented in Figure 6, the expectedcosts associated with each of these strategypairs can be obtained. The results are listedin Table 2.

It can be seen from Table 2 that the optimalinspection and repair strategy is the Strategy


FIGURE 5:Updated ReliabilityIndex After Repair

FIGURE 6:Decision Tree forSelecting Inspectionand Repair Strategies

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1-4,which has the lowest expected cost of24K$. The worst inspection and repair strat-egy is the Strategy 2-3 associated with thehighest expected cost of 67K$. It can also beseen that the expected costs associated withthe strategies for the inspection method 1 arelower than those for the inspection method2. The reason is that the probability ofdetection defined by the better inspectionmethod is not high enough to reduce theweight consequential costs related to thestructure failure in comparison with its high-er costs of inspection. In addition, theexpected costs associated with Strategy 1-4is the lowest among all the strategies for theinspection method 1. The reason is that thehigher repair costs incurred using the betterrepair method are not compensated by thereduction of the weight consequential costsrelated to the structure failure. Similar con-clusion can be drawn for the inspectionmethod 2. It is further seen that the expectedcosts increase with the increase of the repaircosts. This indicates that the increase of the

repair costs associated with the better repairmethod is not compensated by the reductionof the weight consequential costs related tothe structure failure.

In addition to the analysis of the expectedcosts, the risk profiles of the nine strategiescan be also studied (Ayyub et al. 2000).Figure 7 gives the risk profile for the numeri-cal example studied herein. The upperbound of costs is 300K$, once that beyondthis value, the probabilities for all strategiesare equal to one. From the comparisonamong the risk profiles associated with allinspection and repair strategies, it can befound that Strategy 1-4 presents higherprobability associated with lower cost, fol-lowed by Strategy 1-1. The results show thatStrategy 1-4 should be used for the shipstructural inspection. Reasonable agreementscan be observed between the results from therisk profiles and those from Table.2.

Sensitivity AnalysisThe expected cost is the criterion for select-ing the optimal inspection and repair strate-gy by using the decision tree approach. Fromthe foregoing discussion, it can be observedthat the expected costs are closely related tothe values of various parameters. Since thevalues of various costs are hypothetical,there exist statistical uncertainties, whichinfluence the stability and reliability of deci-sion results. Therefore, sensitivities of vari-ous costs should be explored.

For the convenience of derivation, let E(C)A,E(C)B, and E(C)C be the expected costs asso-ciated with the strategy of inspection withrepair, the strategy of inspection withoutrepair, and the strategy of no inspectionwithout repair, respectively. Let C1, CR, andCF be the inspection, repair, and failure costs,respectively. Let CFL, CFM, and CFH be thecosts associated with the low, medium, andhigh consequences of failure. Let PR, PR

—, andPIR— be the probability of failure after repair,after inspection without repair, and after noinspection without repair, respectively.



Table 2Expected Costs for Different Inspection and Repair Strategies

Inspection and repair strategy pairs1-1 1-2 1-3 1-4 2-1 2-2 2-3 2-4 3-1

E(C)/K$ 29 35 41 24 51 59 67 44 32

FIGURE 7:Risk Profile forDifferent Inspectionand Repair Strategies

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According to the available data presented in Figure 6, CI1 = 20 (K$), C12 = 40 (K$), CR1 = 10(K$), CR2 = 20(K$), CR3 = 30(K$), PFL = 0.15, PFM = 0.70, PFH = 0.15, CFL = 50(K$), CFM =100(K$), CFH = 200(K$), PR

— = 0.0349, PIR— = 0.2956, PR1 = 0.0144, PR2 = 0.00971, PR3 = 0.00643.

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

(9)

then

(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. Thisis the same as the result given by Ayyub et al. (2000). However, this is not consistent with theactual case since inspections and repairs are performed during the service lifetime of shipstructures. Since Ayyub et al. (2000) did not take into account the reliability updating afterinspection, they simply assumed that PIR

— is equal to PR—. Actually, it is impossible that PIR

— isequal to PR

—. The reason is that the probability of failure must be updated whether inspectionsor repairs are performed or not. Furthermore, PR

— is always lower than PIR— .

Consider the effects of the values of various costs on the selection of optimal inspection andrepair strategy. For the purpose of illustration, the inspection and repair strategies associatedwith the inspection method 1 are studied herein. Consider the case of the strategy of inspec-tion without repair has the lowest expected cost, followed by the strategy of inspection withrepair. The strategy of no inspection without repair has the highest cost. Then, it is essentialthat the following two conditions should be satisfied.

(11)

(12)

For the convenience of derivation, let CTF = PFLCFL + PFMCFM + PFHCFH. If CTF remains the same,substituting Equations (6) and (7) into Equation (11), gives



(13)

By substituting the available data presentedin Figure 6 into Equation (13), then

(14)

Since CR1 = 10(K$) > 2.2(K$), CR2 = 20(K$) >2.7(K$) and CR3 = 30(K$) > 3.1(K$), Equation(11) holds for all the three repair methods.Reasonable agreements can be observedbetween the results from the sensitivityanalysis and those from Table 2.

If CR remains the same, then

(15)

By substituting the available data presentedin Figure 6 into Equation (15), CTF associat-ed with the three repair methods can beobtained as

(16)

From the available data presented in Figure6, CTF = 107.5(K$). Since Equation (16) holdsfor the value of 107.5(K$), Equation (10)can be also obtained. These results are con-sistent with those from Table 2, as well.

Similarly, if CTF remains the same, substitutingEquations (6) and (8) into Equation (12), gives

(17)

Substituting the available data presented inFigure 6 into Equation (17), gives

(18)

From the available data presented in Figure6, CR1 = 10(K$)<14.9(K$), CR2 = 20(K$)>15.4(K$) and CR3 = 30(K$)>15.8(K$).Therefore, Equation (12) holds for the repairmethod 1, while Equation (12) does nothold for the repair methods 2 and 3.Reasonable agreements can be observedbetween the results from the sensitivityanalysis and those from Table 2.

If CR remains the same, then

(19)

By substituting the available data presentedin Figure 6 into Equation (19), CTF associat-ed with the three repair methods can beobtained as follows.

(20)

From the available data presented in Figure6, CTF = 107.5(K$) > 96.2(K$) and CTF =107.5(K$) < 118.0(K$) and CTF = 107.5(K$) <139.8(K$). Therefore, Equation (12) holdsfor the repair method 1, while Equation (12)does not hold for the repair methods 2 and3. These results are consistent with thosefrom Table 2, as well.

According to foregoing discussion, it can beobserved that if CTF remains the same, theconditions in Equations (14) and (18) can bereduced to

(21)

If CR remains the same, the conditions inEquations (16) and (20) can be reduced to

(22)

The strategy of inspection without repair isthe optimal strategy, if Equation (21) orEquation (22) is satisfied.




A similar approach can be employed toother 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 ofbrevity, the results are not given again.

The relationship between the expected costsassociated with different inspection methodsand different repair methods can be furtheranalyzed. First, considering the relationshipbetween the Strategy 1-4 associated with theinspection method 1 and the Strategy 2-4associated with the inspection method 2.

If

(23)

then

(24)

From the available data presented in Figure6, we have C11 = 20(K$) < C12 = 40(K$), thenEquation (24) holds. So the Strategy 1-4 isalways better than the Strategy 2-4. Thisconclusion agrees with that from Table 2.

If

(25)

CR and PR and remain the same, by consider-ing the relationship between Strategy 1-1 asso-ciated with the repair method 1 and Strategy1-2 associated with the repair method 2, then

(26)

By substituting the available data presentedin Figure 6 into Equation (26), then

(27)

Since CTF = 107.5(K$) < 213220(K$), Strategy1-1 is always better than Strategy 1-2.Similarly, Strategy 2-1 is always better thanthe Strategy 2-2.

If CTF, CR1, PR1, and PR2 remain the same, wehave

(28)

Similarly, by substituting the available datapresented in Figure 6 into Equation (28), then

(29)

Since CR2 = 20(K$) > 10.5(K$), the same con-clusions can be obtained. All these results

are consistent with those from Table 2.

ConclusionsTo keep the probability of failure of shipstructures below a specified target safetylevel, a risk-based inspection and repairplan was developed. A component subject-ed to corrosion deterioration was investi-gated to illustrate the application of theproposed method. According to the princi-ple that both the risk costs of failure andthe costs of inspection and repair should beminimized, the decision tree approach isused to select the suitable inspection andrepair strategy for the component. Finally,from the analysis of the sensitivities for var-ious costs, it is found that the expectedcosts are significantly influenced by the val-ues of the costs of inspection, repair, andfailure. Since the decision results are signifi-cantly affected by the reliability of data inthe process of selecting the optimal inspec-tion strategy, it is recommended that fur-ther work should be focused on the system-atic gathering of the costs for performingall relevant activities in advance.

REFERENCES

Ayyub, B.M., U.O.Akpan., G.F.M.De Souza.,etal.,2000,” Risk-based Life Cycle Management ofShip Structures,”SR-1407, Ship StructuresCommittee, 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 InspectionPrograms for Marine Structures,” Proceedings ofthe 16th International Conference on OffshoreMechanics and Arctic Engineering, Yokohama,Japan.

De Souza, G,F,M. and B.M. Ayyub., 2000,”RiskBased Inspection Planning for Ship HullStructures,”Association of Scientists andEngineers’ 2000 Technical Symposium, USA.

Guedes Soares, C. and Y. Garbatov., 1996,”FatigueReliability of the Ship Hull Girder Accounting for



Inspection and Repair,” Reliability Engineeringand System Safety, ,51 (3): 341-351.

Jiao, G.Y., 1989, ”Reliability Analysis of CrackGrowth Under Random Loading ConsideringModel Updating, “PhD thesis, NorwegianInstitute of Technology, Trondheim, Norway.

Kawano,A. and O.B Augusto., 1997,“Consideration of the Influence of IndependentFactors on Inspection Planning of Structures,”Proceedings of the 7th International Offshoreand Polar Engineering Conference, 4:167-173.

Landet,E., I.Lotsberg., G.Sigursson., 2000, “RiskBased Inspection of an FPSO,” Proceedings ofthe Annual Offshore Technology Conference,Houston Texas, 739-748.

Ma, K.T., 1998 “Tanker Inspection and a Risk-based Inspection Approach,” Proceeding of theEighth International Offshore and PolarEngineering Conference, Canada, 4:504-512.

Ma, K.T., I.R. Orisamolu., R.G. Bea., 1999,“Optimal Strategies Inspection of Ships forFatigue and Corrosion Damage,” Ship StructuresCommittee, Washington D. C.

Ma, K.T., I.R. Orisamolu., R.G. Bea., R.T. Huang.,1997, “Towards Optimal Inspection Strategiesfor Fatigue and Corrosion Damage,”Transactions of the SNAME, 105(1):99-125.

Paik, J.K., S.K. Kim., S.K. Lee., 1998 “ProbabilisticCorrosion Rate Estimation Model forLongitudinal Strength Members of BulkCarriers,” Ocean Engineering, 25(10): 837-860.



DIANQUING LI is currently completing his Ph.D. at the school of naval architecture and oceanengineering, Shanghai Jiao Tong University. His advisor is Prof. Shengkun Zhang. His dissertation is onrisk-based inspection, maintenance, and repair decision making for ship structures. He received hisB.S. and M.S. in the school of mechanical and electrical engineering at the Hohai University where hismaster’s thesis was on reliability assessment of hydraulic steel structures. He has published morethan forty journal papers.

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