Jan M. van Noortwijk

A comparison of models for measurable deterioration: An application to coatings on steel structures

Steel structures like bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by organic coating systems. This paper focuses on modelling the deterioration of the organic coating layer that protects steel structures from corrosion. Only if there is sufficient knowledge of the condition of the coating on these structures, maintenance actions can be done in the most efficient way. Therefore the course of the deterioration of the coating system and its lifetime, which is also of importance for doing maintenance, have to be assessed accurately. In this paper three different stochastic processes, viz. Brownian motion with non-linear drift, the non-stationary gamma process and a two-stage hit-and-grow physical process, are fitted to two real data sets. In this way we are the first who compare the three stochastic processes empirically on criteria such as goodness-of-fit, computational convenience and ease of implementation. The first data set is based on expert judgement; the second consists of inspection results. In the first case the model parameters are obtained by a least squares approach, in the second case by the method of maximum likelihood. A meta-analysis is performed on the two-stage hit-and-grow model by means of fitting Brownian motion and gamma process to the outcomes of this model.

A Bayesian failure model based on isotropic deterioration

Abstract In this paper, we focus on determining a new failure model for hydraulic structures. The failure model is based on the only information which is commonly available: the amount of deterioration averaged over a finite or an infinite time-horizon. By introducing a prior density for the average deterioration per unit time, we account for uncertainty in a decision problem. Advantages of our Bayesian approach are that we base our probabilistic models on a physical observable quantity, the deterioration, and that the probabilities of preventive repair and failure can be expressed explicitly conditional on the average deterioration. One illustration from the field of hydraulic engineering is studied.

Optimal maintenance decisions for berm breakwaters

Abstract To prevent coastal lines of defence from being affected by severe hydraulic loads from the sea, berm breakwaters can be used. Although berm breakwaters are dynamically stable in the sense that they allow for some rock displacement, they can fail due to severe longshore rock transport. To avoid this type of failure, berm breakwaters have to be inspected and, if necessary, have to be repaired. A decision model is presented enabling cost-optimal maintenance decisions to be determined while taking account of the (possibly large) uncertainties in: (i) the limiting average rate of occurrence of breaches in the armour layer and (ii) given a breach has occurred, the limiting average rate of longshore rock transport. The stochastic process of rock displacement is modelled by a modified generalised gamma process, enabling the uncertainty in these limiting averages to be explicitly taken into account.

Sampling inspection for the evaluation of time-dependent reliability of deteriorating systems under imperfect defect detection

Abstract The paper presents a sampling-inspection strategy for the evaluation of time-dependent reliability of deteriorating systems, where the deterioration is assumed to initiate at random times and at random locations. After initiation, defects are weakening the systems resistance. The system becomes unacceptable when at least one defect reaches a critical depth. The defects are assumed to initiate at random times modeled as event times of a non-homogeneous Poisson process (NHPP) and to develop according to a non-decreasing time-dependent gamma process. The intensity rate of the NHPP is assumed to be a combination of a known time-dependent shape function and an unknown proportionality constant. When sampling inspection (i.e. inspection of a selected subregion of the system) results in a number of defect initiations, Bayes’ theorem can be used to update prior beliefs about the proportionality constant of the NHPP intensity rate to the posterior distribution. On the basis of a time- and space-dependent Poisson process for the defect initiation, an adaptive Bayesian model for sampling inspection is developed to determine the predictive probability distribution of the time to failure. A potential application is, for instance, the inspection of a large vessel or pipeline suffering pitting/localized corrosion in the oil industry. The possibility of imperfect defect detection is also incorporated in the model.

Optimal Maintenance Decisions for the Sea-Bed Protection of the Eastern-Scheldt Barrier

To prevent The Netherlands from flooding, a flood defence system has been constructed, which must be inspected and, when needed, repaired. Therefore, one might be interested in obtaining cost-optimal rates of inspection, i.e. rates of inspection for which the expected maintenance costs are minimal and for which the flood defence system is safe.

OPTIMAL MAINTENANCE DECISIONS FOR DIKES

To protect the Dutch polders against flooding, more than 2500 km of dikes have been constructed. Due to settlement, subsoil consolidation, and relative sea-level rise, these dikes slowly sink “away into the sea” and should therefore be heightened regularly (at present, every 50 years). In this respect, one is interested in safe and cost-optimal dike heightenings for which the sum of the initial cost of investment and the future (discounted) cost of maintenance is minimal.For optimization purposes, a maintenance model has been developed for dikes subject to uncertain crest-level decline. On the basis of engineering knowledge, crest-level decline has been modeled as a monotone stochastic process with expected decline being either linear or nonlinear (i.e., linear after transformation) in time. For both models and for a particular unit time, the increments are distributed according to mixtures of exponentials.In a case study, the maintenance decision model has been applied to the problem of heightening the Dutch “Oostmolendijk.”

Uncertainty in flood damage estimation

Abstract The aim of this article is to incorporate uncertainty into the currently used method for estimating the economic damage of floods in the Netherlands. Using a high‐water information system, this so‐called Standard Method computes the expected flood damage per damage category on the basis of the number of objects subject to flooding, the maximum damage per object, and the relative damage depending on the water depth. Using a probabilistic approach, the Standard Method is extended by taking account of the uncertainties in the maximum damage per object (maximum possible damage per unit object; e.g., house, production facility, meter road) and the damage function (describing the proportion of the maximum damage incurred to an object due to flooding as a function of the water depth). In estimating the damage in a flooded area, it is also important to take account of the spatial dependence between flood damages at different locations. For this purpose, the following three types of spatial dependence are considered: complete spatial dependence, spatial independence, and partial spatial dependence (for which the damage functions are completely dependent for independent water‐depth classes). Using Monte‐Carlo simulation, probability distributions of the flood damage are determined. The new uncertainty‐based model for predicting flood damage is applied in a case study and the results of a sensitivity analysis are reported.

Explicit formulas for the variance of discounted life-cycle cost

Abstract In life-cycle costing analyses, optimal design is usually achieved by minimising the expected value of the discounted costs. As well as the expected value, the corresponding variance may be useful for estimating, for example, the uncertainty bounds of the calculated discounted costs. However, general explicit formulas for calculating the variance of the discounted costs over an unbounded time horizon are not yet available. In this paper, explicit formulas for this variance are presented. They can be easily implemented in software to optimise structural design and maintenance management. The use of the mathematical results is illustrated with some examples.

DETERIORATION AND MAINTENANCE MODELS FOR INSURING SAFETY OF CIVIL INFRASTRUCTURES AT LOWEST LIFE-CYCLE COST

The objective of this paper is to describe and compare deterioration and maintenance models for civil infrastructures. These models can be applied to determine the best maintenance strategy to insure an adequate level of safety at minimal life-cycle cost while taking the uncertainties in the deterioration process into account. The paper discusses the pros and cons of the different models considered.

Optimal maintenance decisions on the basis of uncertain failure probabilities

Due to a lack of data, many maintenance optimisation models have to be initialised on the basis of expert judgment. Rather than eliciting the parameters of a continuous lifetime distribution, experts give more reliable answers when assessing a discrete lifetime distribution. If the prior uncertainty in the probabilities of failure per unit time is expressed in terms of a Dirichlet distribution, Bayes estimates can be obtained of three cost‐based criteria to compare maintenance decisions over unbounded time‐horizons: first, the expected average costs per unit time; second, the expected discounted costs over an unbounded horizon; and third, the expected equivalent average costs per unit time. Illustrates the maintenance model by determining optimal age replacement and lifecycle costing policies, which optimally balance both the failure cost against the preventive repair cost, and the initial cost against the future cost.