International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Optimization of inspection plans for structures submitted to non-stationary stochastic degradation processes Décatoire, Rodrigue; Elachachi, Sidi Mohammed; Yalamas, Thierry; Schoefs, Franck Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Optimization of Inspection Plans for Structures Submitted toNon-stationary Stochastic Degradation ProcessesRodrigue DécatoireGraduate Student, PHIMECA Engineering, Cournon d’Auvergne, FranceSidi Mohammed ElachachiAssociate Professor, GCE Department I2M, Univ. of Bordeaux, Bordeaux, FranceThierry YalamasCEO, PHIMECA Engineering, Cournon d’Auvergne, FranceFranck SchoefsProfessor, Research Institute of Civil Engineering and Mechanics, UMR CNRS 6183LUNAM Université, Nantes, FranceABSTRACT: The inspection plan optimization of civil engineering structures became a true economicalchallenge in context of limited budget in the last two decades. It is therefore mandatory to optimize theinspection, maintenance and repair plans depending on the evolution of degradation indexes in order toensure the reliability of these structures. Based on simulation of a predictive carbonation model whichare used to identify the degradation index evolution, this paper presents a new methodology which helpsto design an optimal inspection plan, taking into account the spatial variability of the degradation process.The position of the measurements points derives from an adaptative design of experiments built with thedegradation predictions. The optimal time between inspections is thus determined with respect to theinspection, maintenance and failure costs.1. INTRODUCTIONSince the 19th century, the expansion of develo-ped countries and the fast increase of their pop-ulation has lead to the construction of numerousstructures and infrastructures, in order to answerto the needs in energy, in accommodations or inroad and railway infrastructures. Those buildings,mostly built in reinforced concrete, are submitted toseveral degradation phenomena that impair their in-tegrity. However their importance implies that theirsafety is to be insured all the time, on account ofeconomic and social matters. In order to ensure thereliability level of those structures, they are regu-larly inspected and maintained according to IMR(Inspection, Maintenance and Repair) plans pre-viously defined by the stakeholders. However byconsidering the global economic context, the costsinduced by these operations call for an optimizedprocedure in order to reduce the long run moni-toring cost. Many methodology have been deve-loped to solve this problem (e.g. Barone and Fran-gopol (2014); Bucher and Frangopol (2006); Kallen(2007); Straub (2004); Breysse et al. (2009)), byusing Markov chain (e.g. Frangopol et al. (2004);Orcesi and Cremona (2010)), decision tree inStraub and Faber (2005) or performance criteria inOkasha and Frangopol (2010), at different scales :component, structure, or a set of structures scales.Some consider different maintenance methods de-pending on the component (e.g. Okasha and Fran-gopol (2009); Straub and Faber (2005)). Howeverconcerning reinforced concrete, few of them di-rectly address the spatial variability of a componentdegradation (e.g. Schoefs (2007); Schoefs et al.(2009); Sudret (2008)) which is due to the spatialvariability of concrete properties and to the envi-112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015ronmental aggressiveness. This paper thus presentsthe first steps of a new methodology based on de-cision trees for optimizing an inspection plan, ableto deal with such kind of variability. This methodo-logy aims at being:1. Simple to implement,2. Accurate,3. Fast.It helps in determining when and where a structureshould be inspected, considering that it is submit-ted to carbonation process. The future work willbe dedicated to introduce the effect of measurementuncertainties in the process.2. OPTIMIZATION OF INSPECTIONS PLANS2.1. The methodology2.1.1. Degradation indexThe first step of the methodology is to determinea degradation index based on Nsim probabilistic si-mulations of the degradation process. For the sakeof simplicity let us consider a 1D structure indexedby xp ∈ X . Concerning the carbonation process, wewant to avoid the carbonation depth Xc(xp) frombeing greater than the concrete cover b. Yet, itseems logical that a structure will not be maintainedif only one point reaches this criterion since the car-bonation process itself is not critical regarding theserviceability of a structure. Let thus defined an-other criterion which considers that a maintenanceis to be applied if a critical length threshold Lcrit%of the structure reaches the previous criteria. Sincewe are in a probabilistic context, the criterion thusreads:• Failure of a point means Xc(xp)>= b,• If the failed length Lc, representing the per-centage of the structure considered as failedexceeds the given threshold Lcrit , the structureis to be maintained,• The probability that the structure has to bemaintained is supposed to be driven by thecondition statePLc>=Lcrit = NLc>=Lcrit/Nsim, (1)where NLc>=Lcrit is the number of simulatedtrajectories which present a percentage Lc ofthe structure considered as failed higher thanLcrit2.1.2. Adaptative design of experimentsThe second step of the methodology is to deter-mine, at a given inspection date, where to inspectin order to obtain the best evaluation of the degra-dation index. It could have been interesting to putthe inspection location in an optimization process,however it may lead to an intractable optimizationproblem. These locations are thus found with anadaptative design of experiments (ADoE). Since themethodology is based on multiple realizations ofthe degradation process, the location of interest foran inspection are to be defined based on a statisti-cal quantity. In order to be conservative, we choseto work with the 95% quantile of the degradationsimulations. Let us defined the maximum numberof locations Nl that can be inspected (knowing agiven inspection budget). As the methodology isbased on a degradation model, it could be dange-rous to entrust this model entirely by only inspec-ting the locations where a high degradation is pre-dicted. We propose to split the inspections locationsin two parts :• one for the maximum values of the 95% quan-tile Q(xp) which verifiesP(Xc(xp)≤ Q(xp)) = 0.95 (2)• the other for the minimum values of this quan-tile, in order to avoid a false evaluation of theprobability index due to an overconfidence inthe model.For the sake of simplicity, it is defined that half ofthe Nl locations will be chosen in the maximumvalues, the other half in the minimum values, andthe effect of this distribution is not studied. To en-sure a good identification of the degradations sta-tistical properties, the autocorrelation between eachselected point has to be lower than 0.3 in orderto suppose pseudo-independent measurement (seeSchoefs et al. (ress)).The selected points are then extracted from thesimulations, and between them a linear interpola-tion is performed.212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015However, if this proposal ensures that the spacewill be satisfactorily explored, it does not prove thatthe incomplete spatial trajectories will permit to ac-curately estimate the degradation index.An adaptative loop is therefore realized:1. Evaluate the relative difference between thedegradation index derived from the simula-tions and the one derived by the incompletetrajectories,2. If this difference is higher than a given thres-hold εPLc>=Lcrit , and the maximum number oflocations Nl has not been reached, a newinspection point has to be chosen.• Thus for each point compute the criterionI(xi) = εµ(xi)+ εσ (xi), xi ∈ X , (3)• The point which maximizes this criterionis added to the inspection plan,• Start againε•(xi) is the absolute error made at point xi betweenthe estimate of • derived from the incomplete tra-jectories and from the simulations.From this ADoE an estimate P˜Lc>=Lcrit of the em-pirical degradation index PLc>=Lcrit is derived.2.1.3. Decision treeSince we assume an inspection gives a perfectmeasure, after an inspection a stakeholder onlyhave two choices :1. to maintain the structure,2. or to wait for the next inspection.The probability that the structure will be maintainedis P˜Lc>=Lcrit , the complementary probability is theprobability to do nothing. An example of this de-cision tree is shown figure 1 for a plan with twoinspections.Something hidden in this tree, yet obvious, isthat between two inspections the degradation is pro-gressing. If a maintenance is applied, the degrada-tion goes back to the initial state after a delay calledmaintenance time tMa. With the degradation comesa failure probability: the probability that corrosionI 1I 2bCtotbb ∗ PbbNM1− P˜ t2bLc>=LcritCtotba ∗ PbaMP˜t2bLc>=LcritNM1− P˜ t1Lc>=LcritI 2aCtotab ∗ PabNM1− P˜ t2aLc>=LcritCtotaa ∗ PaaMP˜t2aLc>=LcritMP˜t1Lc>=LcritFigure 1: Decision tree with two inspections. M corres-ponds to a maintenance action, NM to no maintenance,I• to inspection number • realized at time t•. Ctot◦ is thetotal cost induced by the branch ◦appears. Since no value is available in the literature,the failure probability per year is computed byPf (level) = 1×10−6+level ∗P(Lc ∈ Intlevel), (4)with level goes from 0 to 4, Int = ([0%,10%[,[10%,20%[, [20%,30%[, [30%,40%[,[40%,100%[) is a vector of intervals of thedegradation index possible values and P(Lc ∈ Inti)is the probability that the degradation index iswithin Inti.2.1.4. Modeling costsInspection cost According to the tree (see figure1) the expected inspection costs writes :E[CIn]=CInm∑i=1PitIn∑j=1nI j(1+ r)t j (5)where• CIn is the cost of an inspection,• tIn is the number of inspection date,• nI j < Nl is the number of measures performedat year t j,• r is the discount rate of money,• m= 2tIn is the number of possible branches,• Pi is the probability of the branch i.312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Maintenance cost The expected maintenancecost writes:E[CMa]=CMam∑i=1PitIn∑j=11i j(1+ r)t j+tMa (6)where• CMa is the maintenance cost,• 1i j is equal to 1 if a maintenance action is de-cided on branch i at time t j, 0 otherwise.Failure cost The expected failure cost writesE[CF]=CFm∑i=1Pittot∑j=1∏5l=1 Pf (l)∗P(Lc ∈ Intl)(1+ r)t j(7)2.1.5. Optimization problemThe aim of the methodology is therefore to mini-mize the expected total cost which writesE[Ctot]= E[CIn]+E[CMa]+E[CF](8)To act on this cost, the only variables are theinspection dates. For the sake of simplicity, andsince it is in a better agreement with the stakehold-ers habits, the step between two inspections is setas a constant. Therefore there is only one variableleft to optimize: the time between two inspections∆I.A global scheme of the methodology is presentedfigure 2.3. APPLICATION3.1. Description3.1.1. The bridgeThe application uses a model of a reinforced con-crete bridge column located on the edge of a road,in a rainy environment. We assume that the baseof the column is regularly splashed by the cars pas-sing on the soaked road. The top of the column isprotected from the rain by the deck here supposedto be perfectly waterproof. A basic illustration ofthe column is shown figure 3. We only considerthe saturation rate of the concrete cover as a one-dimensional non-stationary random field since it isDegradation simulationsInspection, maintenance and failure costsDegradation indexDegradation thresholdChoice of a time between two inspectionsNext inspection : ADoEExtract the inspected points degradation from the simulations at the inspection dateCompute an estimate of the probability of maintenance PAdd measurements points if necessary to increase the accuracyMaintain the whole structureDo nothingsimulation1-PPUntil the time limit is reachedFigure 2: Global scheme of the methodologyFigure 3: Basic illustration of the columndirectly correlated to the water present in the con-crete cover which is not homogeneously splashedby the cars over the column height. This randomfield of the saturation rate is simulated with theKarhunen-Loeve expansion with the following pa-rameters:• the mean is equal to 100% from the base of thecolumn up to 1 meter,• the mean is equal to 60% from the top of thecolumn down to 3 meters,• the mean decreases linearly from 100% to60% from 1 meter to 3 meters,• the variance is a constant equal to 5%,• the autocorrelation is modeled by a quadraticexponential function with a correlation para-meter of 1 meter.412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Ten trajectories belonging to this stochastic field areplotted on figure 4.3.1.2. The degradationThe degradation considered in this paper is the car-bonation. To simulate the carbonation process wechose to use the DuraCrete model (see DuraCrete(2000)) since its use is recommended by the Eu-ropean Union. Based on the DuraCrete report, theparameters of the model are set in order to simulatea C25 concrete. This model readsXc(t) =√kekcktCstRcarb∗(t0t)n(9)where ke is derived from the relative humidity de-termined by the saturation rate (thanks to a realisticisotherm BSB of a C25 concrete). An illustrationof the resulting fields of the carbonation depth after20 years of natural carbonation is presented figure5.0 1 2 3 4 5Height (m)405060708090100S r(%)SrFigure 4: 10 sample paths of the saturation rate.3.2. DataThe data used to simulate the carbonation processare summarized in the table 1. They are chosenas deterministic variables in order to isolate the ef-fect of the spatial variability of the saturation rateon the inspection plan. 1000 simulations have beenlaunched to minimize the statistical bias. The car-bonation depth progression is computed every year.The different costs, the maximum number of points0 1 2 3 4 5Height (m)0.0000.0050.0100.0150.0200.0250.0300.035X c(m)Xc  after 20 yearsFigure 5: 10 sample paths of carbonation depth com-puted with the DuraCrete model.n = 0.1t0 = 1 yearkc = 4.94kt = 0.983Rcarb = 7.05e−5Table 1: Parameters value for the carbonation Du-raCrete model (see DuraCrete (2000)). They corre-spond to a mean C25 concrete.that can be measured at an inspection date and thethreshold for the ADoE are referenced in the table2.b = 1cmr = 0.02Nl = 25Lcrit = 30%ttot = 60yearsεPLc>=Lcrit = 5%E[CIn]= 5E[CMa]= 100E[CF]= 500Table 2: Parameters of the inspection plan optimization512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20153.3. Results3.3.1. Analysis of the autocorrelationThe autocorrelation of the carbonation depth il-lustrated figure 6 appears to be non-stationary eventhough it was not the case for the autocorrelation ofthe saturation rate. It obviously affects the ADoE,however since there is no proof that this model cor-rectly transfers spatial correlation, we will not drawany conclusion on this point and will assume thatit represents the reality. Yet, the use of this ma-0 20 40 60 80020406080Autocorrelation matrix of Xc  at 20 years0.000.150.300.450.600.750.90Figure 6: Autocorrelation matrix of the carbonationdepth after 20 years. A distance of 20 points is equal toone meter.trix proves that the methodology is able to deal withcomplex non-stationarity.3.3.2. An example of ADoEThe figure 7 illustrates what would be the ADoEif the first inspection was run after 10 or 20 years.From the original design of experiment, only 6measurements are performed. Yet, to ensure a goodquality in the estimate of the degradation index Lc,the adaptative part adds 4 points at 10 years, and1 point at 20 years. Without any optimization pro-cedure, it proves to be efficient since only 10 and7 points are inspected when the maximum allowednumber of inspections was around 25 points. ThisADoE is then computed for each inspection date ofevery branch of the decision tree.3.3.3. Optimization of the inspection planThe predictive part of the methodology is the mini-mization of the expected total cost, with ∆I as the0 1 2 3 4 5Height (m)0.0000.0050.0100.0150.0200.0250.030X c(m)Two inspection plans at 10 and 20 years95\% quantile - 20 yearsDoE points - 20 yearsAdded point - 20 years95\% quantile - 10 yearsDoE points - 10 yearsAdded point - 10 yearsFigure 7: Illustration of the adaptative design of exper-iment.only optimization variable. Therefore, the objectivefunction has been computed for every ∆I from 5years (meaning 4096 branches in the decision tree)to 25 years (respectively 4 branches) for a totaltime prediction ttot of 60 years. The results tooktwo hours to be computed and are presented figure8. The different costs present a classical behavior(e.g. Sheils et al. (2010)), many inspections leadto a higher maintenance cost and a low failure costwhen a long time without inspection leads to theopposite. The variations observed in the inspectioncost are an effect of the ADoE which helps to in-troduce the effect of the spatial variability. In thisexample, the optimum time between two conse-cutive inspections is 30 years. This methodologytherefore proves itself to be usable in the case of anon-stationary degradation process.4. CONCLUSIONSThe optimization of inspection plan is a wide sub-ject which has been intensely studied throughoutthe last decade. Many methodologies, based onprobabilistic studies, markov chains, decision trees,bayesian framework, ..., have been proposed, yetonly a few of them deal with the spatial variabilityof the degradation a stakeholder can observe on itsstructure. The methodology proposed in this paperis a first step to address for this issue. As simpleas possible, it takes into account the spatial variabi-lity without increasing the complexity nor the com-612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20155 10 15 20 25 30 35 40∆I0100200300400500600700800Cost unitE[CIn ]E[CMa ]E[CF ]E[Ctot ]Figure 8: Evolution of the inspection, maintenance,failure and total costs with respect to the time betweentwo inspections. The minimum total cost is obtained fora time of 30 years.putational time of the cost optimization problemcommonly solved in the non-spatial case. Yet itremains a first step. The next one will be to takeaccount for the measurement error closely relatedto the inspection cost, and for real costs of mainte-nance and failure.5. REFERENCESBarone, G. and Frangopol, D. (2014). “Life-cycle main-tenance of deteriorating structures by multi-objectiveoptimization involving reliability, risk, availability,hazard and cost.” Structural Safety, 48, 40–50.Breysse, D., Elachachi, S., Sheils, E., Shoefs, F., andO’Connor, A. (2009). “Life cycle cost analysis of age-ing structural components based on non-destructivecondition assessment.” Australian journal of struc-tural engineering, 9, 55–66.Bucher, C. and Frangopol, D. (2006). “Optimiza-tion of lifetime maintenance strategies for deteriorat-ing structures considering probabilities of violatingsafety, condition and cost thresholds.” ProbabilisticEngineering Mechanics, 21, 1–8.DuraCrete (2000). “Final technical report: Generalguidelines for durability design and redesign.” Reportno., The European Union - Brite EuRam III - ContractBRPR-CT95-0132 - Project BE95-1347/R17.Frangopol, D., Kallen, M.-J., and van Noortwijk, J.(2004). “Probabilistic models for life-cycle perfor-mance of deteriorating structures: review and futuredirections.” Progress in Structural Engineering andMaterials, 6, 197–212.Kallen, M.-J. (2007). “Markov processes for main-tenance optimization of civil infractructure in thenetherlands.” Ph.D. thesis, Delft University of Tech-nology, Delft University of Technology.Okasha, N. and Frangopol, D. (2009). “Lifetime-oriented multi-objective optimization of structuralmaintenance considering system reliability, redu-dancy and life-cycle cost using ga.” Structural Safety,31, 460–474.Okasha, N. and Frangopol, D. (2010). “Novel ap-proach for multicriteria optimization of life-cycle pre-ventive and essential maintenance of deterioratingstructures.” Journal of Structural Engineering, 136,1009–1022.Orcesi, A. and Cremona, C. (2010). “Optimization ofbridge maintenance strategies based on multiple limitstates and monitoring.” Engineering Structures, 32,627–640.Schoefs, F. (2007). “Risk analysis of structures in pres-ence of stochastic fields of deterioration: flowchart forcoupling inspection results and structural reliability.”International Forum on Engineering Decision Mak-ing.Schoefs, F., Clément, A., and Nouy, A. (2009). “Assess-ment of roc curves for inspection of random fields.”Structural Safety, 31, 409–419.Schoefs, F., Tran V, T., Bastidas-Arteaga, E., Villain, G.,and Derobert, X. (Inpress). “Stochastic characteriza-tion of random fields from ndt measurements: a twostages procedure. Structural Safety, in press.Sheils, E., O’Connor, A., Breysse, D., Schoefs, F.,and Yotte, S. (2010). “Development of a two-stageinspection process for the assessment of deteriorat-ing infrastructure.” Reliability Engineering and Sys-tem Safety, 95, 182–194.Straub, D. (2004). “Generic approaches to risk basedinspection planning for steel structures.” Ph.D. thesis,Swiss Federal Institute of Technology Zürich, SwissFederal Institute of Technology Zürich.Straub, D. and Faber, M. (2005). “Risk based inspectionplanning for structural systems.” Structural Safety,27, 335–355.Sudret, B. (2008). “Probabilistic models for the extent ofdamage in degrading reinforced concrete structures.”Reliability Engineering & System Safety, 93, 410–422.7

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