Elsevier

Structural Safety

Volume 31, Issue 3, May 2009, Pages 234-244
Structural Safety

Modelling and optimizing imperfect maintenance of coatings on steel structures

https://doi.org/10.1016/j.strusafe.2008.06.015Get rights and content

Abstract

Steel structures such as bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by an organic coating system. Unfortunately, the coating system itself is also subject to deterioration. Imperfect maintenance actions such as spot repair and repainting can be done to extend the lifetime of the coating. This paper considers the problem of finding the set of actions that minimizes the expected (discounted) maintenance costs over both a finite horizon and an infinite horizon. To this end the size of the area affected by corrosion is modelled by a non-stationary gamma process. An imperfect maintenance action is to be done as soon as a fixed threshold is exceeded. The direct effect of such an action on the condition of the coating is assumed to be random. On the other hand, due to maintenance the parameters of the gamma deterioration process may also change. It is shown that the optimal maintenance decisions related to this problem are a solution of a continuous-time renewal-type dynamic programming equation. To solve this equation time is discretized and it is verified theoretically that this discretization induces only a small error. Finally, the model is illustrated with numerical examples.

Introduction

Steel structures such as bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by organic coating systems. Unfortunately, the coating system itself is also subject to deterioration and after some time the steel loses its coating and starts corroding. Maintenance can be done to improve the condition of the coating system and by doing so the lifetime of the steel structure is also extended. Typical maintenance actions for coating systems are (local) spot repair, repainting and replacement. Spot repair consists of only painting the most visible corroded parts, while repainting means that the entire surface of the structure is repainted without removing the corrosion completely. Finally, in a replacement action the old coating and all corrosion is completely removed and a new coating is applied. Since in spot repair and repainting some corrosion is not removed these actions can be seen as imperfect. Obviously the replacement action restores the condition of the coating to new and therefore it is a perfect maintenance action. With respect to cost it is obvious that spot repair is the cheapest action, while replacement is the most expensive.

The aim of this study is to find an optimal strategy for imperfect maintenance of engineering structures, in particular steel structures protected by coatings. To this end, a deterioration model that includes the effect of imperfect maintenance is introduced. The above defined actions are then employed to form a maintenance strategy and they are the basis of the optimization model. The objective of this model is to minimize the expected maintenance costs over a finite horizon. Clearly this finite horizon is determined by the economic or technical lifetime of the structure. The decision variables are the maintenance actions to be executed.

To model the deterioration process of the coating a non-stationary gamma process is used, where the state space is the size of the coating area affected by corrosion (in e.g. the number of squared meters). Abdel-Hameed [1] was the first to propose the gamma process as a proper model for continuous time-dependent deterioration. It has increasing sample paths and as such it is a suitable candidate to describe the (monotonic) deterioration of engineering structures, see e.g. Çinlar et al. [4], van Noortwijk and Klatter [21] and Frangopol et al. [5]. In particular, in Heutink et al. [8] and Nicolai et al. [14] the deterioration of coatings on steel structures is modelled by a non-stationary gamma process. A recent overview by van Noortwijk [19] provides more examples of the application of gamma processes in maintenance.

The non-stationary gamma process has also been used to model deterioration in the presence of imperfect maintenance. van Noortwijk and Frangopol [20] give a mathematical description of the lifetime extending maintenance (LEM) model for engineering structures, which was introduced by Bakker et al. [2]. In this model an imperfect maintenance action reduces the amount of deterioration by a fixed amount and afterwards deterioration is again modelled by the same non-stationary gamma process. It shall be argued below that the reduction in deterioration may also be random in practice and secondly that due to imperfect maintenance the parameters of the deterioration process may change. That is, the present application asks for a more complex model.

For steel structures, maintenance actions such as spot repair improve the deterioration rate of the coating only locally, whereas other parts of the surface still deteriorate at the same rate. So, as a whole the deterioration process may increase faster after spot repair than after replacement. The same holds for the repainting action. So, it is necessary to extend the LEM model and therefore we allow for a structural change in the gamma deterioration process after maintenance is done.

In practice (imperfect) coating maintenance is done as soon as the area affected by corrosion exceeds a certain intervention level set by the decision maker. An imperfect maintenance action reduces the size of the affected area by a random amount. This random effect occurs since spot repair and repainting do not cover all corrosion as not all may be visible. Observe all corrosion is removed by a replacement. As the improvement in deterioration is modelled by a nonnegative random variable, the time between two maintenance actions is given by the time the gamma process needs to counterbalance this improvement. This time depends on the parameters of the gamma process and the random improvement. With respect to the latter, we consider generally distributed random improvements in deterioration independent of the gamma process.

In the literature several optimization models for deteriorating systems with imperfect maintenance have been proposed. Newby and Barker [12] extend the general condition-based inspection/replacement model by Newby and Dagg [13] with imperfect repair, where the effect of imperfect repair is fixed. Deterioration is modelled as a (stationary) Lévy process. A failure is defined as the event in which the process exceeds a fixed failure level. Failure is detected only by (aperiodic) inspections. If the system has failed it is correctively replaced and otherwise a partial repair is carried out. In Newby and Barker [12] dynamic programming is used to optimize both the expected average and the discounted maintenance costs with respect to the inspection times.

Castanier et al. [3] proposes a similar condition-based maintenance model as Newby and Dagg [13], but they model stationary deterioration in discrete-time. It is assumed that the effect of imperfect repair is a random function of the observed deterioration. Partial repair and replacement are done as soon as the observed deterioration exceeds certain thresholds. The expected average cost per unit time is minimized with respect to these thresholds. Meier-Hirmer et al. [11] have applied a version of this model to optimize the maintenance of high-speed railway tracks. The above-mentioned LEM model is an age replacement model with the possibility of partial repair. Finally, Liao et al. [10] also consider systems subject to measurable deterioration, where the effect of partial repair is random. As opposed to the other papers, they consider continuous-monitoring and they optimize the availability of the deteriorating system with respect to a preventive replacement threshold.

In the present optimization model maintenance is done as soon as a fixed intervention level is exceeded; no inspections are done. As there are different maintenance actions available for steel structures, the main interest of this paper is in finding the sequence of actions that minimizes the expected maintenance costs over a finite time horizon; extensions with discounting and an infinite horizon will also be discussed. This problem can be formulated as a continuous-time renewal-type dynamic programming equation. Time is discretized to solve this equation and it is shown that the solution of the discrete-time problem is close in the supnorm to the solution of the original continuous-time problem. This is supported by numerical evidence.

The outline of this paper is as follows. In Section 3 a deterioration model for structures with imperfect maintenance is introduced. The associated continuous-time dynamic programming equation describing the optimal maintenance actions is presented and analyzed in Section 4 together with an error analysis for the discretization of this equation. Section 5 reviews techniques presented in Frenk and Nicolai [7] to compute the cumulative distribution function of the time between two maintenance actions. Next, in Section 6, these techniques are employed to solve the optimization problem formulated in Section 4. In Section 7 conclusions are drawn.

Section snippets

Mathematical notation

Unless stated otherwise, random variables and stochastic processes are denoted by boldfaced Roman capitals, distributions by Roman capitals and parameters by small Greek letters.

    N

    set of positive integers 1,2,,

    T

    finite length of total planning horizon

    ρ

    chosen intervention level

    v

    inverse of a strictly increasing continuous function v

    c(a)

    cost of a given maintenance action a

    1A(t)

    indicator function of the set AR

    q(t)

    minimal expected maintenance cost from time T-t up to time T given that at time T-t

Modelling deterioration and maintenance

This section presents a deterioration model for coating systems on corroding structures undergoing imperfect maintenance actions. The deterioration process of the protective coating is given by a non-stationary gamma process and maintenance is done as soon as the size of the affected area exceeds a given level ρ>0 set by the decision maker. Imperfect maintenance yields a random reduction in this size, bringing it back between 0 and ρ. Next, the deterioration process of the coating is again

Maintenance optimization

In this section we introduce a finite horizon optimization model for the maintenance of coating systems protecting steel structures. In Section 4.1 this model (as well as two extensions with discounting) is formulated as a continuous-time stochastic dynamic programming problem. The corresponding renewal-type optimality equation can only be solved by discretizing time. To this end we propose a simple numerical procedure in Section 4.2 and we investigate in detail the accuracy of this procedure.

Computing the cdf of the time between two maintenance actions

To solve the optimization problems introduced in Section 4 we need a fast method to compute the cdf Fa listed in relation (16) of the time between two maintenance actions. Unfortunately, this cdf only has a nice analytical expression in some special cases [7]. In general, evaluating this cdf numerically, e.g. via its two-dimensional integral representation, is time-consuming. However, as we will see it is easy to approximate this cdf.

Note that Fa relates to the cdf of the first time a standard

Numerical examples

Let us illustrate the model and the methods discussed in the previous sections with some numerical examples. All computations are done in MATLAB 7.2 on a Pentium III–2 GHz personal computer. Consider a planning horizon of T=50 time units. We are interested in the optimal maintenance decisions during this horizon. Let the initial gamma deterioration process be given by Xv,λ with v(t)=0.25t2 and λ=1. The intervention level is given by ρ=25. Let A={as,ar,af} be the set of maintenance actions

Conclusions

The life-cycle management of steel structures involves decisions regarding the timing and the type of maintenance of protective coatings. In this paper we have presented a model for optimal maintenance of such coatings on steel structures. The deterioration of coatings is represented by the size of the area affected by corrosion and this is modelled by a non-stationary gamma process. Imperfect maintenance actions such as spot repair and repainting reduce this size by a random amount, whereas

Acknowledgement

The authors would like to thank two anonymous referees whose comments and suggestions on an earlier draft significantly improved the paper.

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