# Stochastic modelling of corrosion fields used for the reliability calculation of metallic port structures

Jérôme Boéro, Humberto Yáñez-Godoy, Bruno Capra, Franck Schoefs

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Jérôme Boéro, Humberto Yáñez-Godoy, Bruno Capra, Franck Schoefs. Stochastic modelling of corrosion fields used for the reliability calculation of metallic port structures. 2nd International Conference on Applications Heritage and Constructions in Coastal and Marine Environment, MEDACHS'010, 2010, La Rochelle, France. hal-01008956

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# Stochastic modelling of corrosion fields used for the reliability calculation of metallic port structures

Boéro J.* — Yañez-Godoy H.* — Capra B.* — Schoefs F.**

* OXAND S.A., 49 avenue Franklin Roosevelt, F-77210 Avon
jerome.boero@oxand.com
humberto.yanez@oxand.com
bruno.capra@oxand.com

** Institute in Civil and Mechanical Engineering (GeM), UMR 6183, Nantes Atlantic University, CNRS, 2 rue de la Houssinière, F-44322 Nantes cedex 3
Franck.Schoefs@univ-nantes.fr

**ABSTRACT.** This article summarises research led during the second phase of the GEROM (Risk management of maritime and river harbour structures) project in order to quantify the risks associated with vulnerable structures identified during the initial phase. A risk analysis is applied here to steel sheet-pile seawalls. For this purpose, a stochastic (spatial-temporal) model of steel corrosion is proposed, based on a statistical analysis of data collected from several French ports in multiple zones that have been exposed to marine environment for varying lengths of time. The predictions obtained from the corrosion model are then integrated into numerical simulations to carry out a time-dependent reliability analysis of corroded harbour structures. Two coupled approaches between a mechanical deterministic finite element model and a probabilistic model are used. The results obtained are then compared and analysed to optimize the maintenance of the structures under consideration.

**KEY WORDS:** management, risk, optimization, maintenance, probabilistic modelling, ageing, corrosion, impact, mechanical behaviour, harbour structures.

## 1. Introduction

This article summarises research carried out in the framework of the GEROM (risk management of marine and river harbour structures) project. The main objective of the project is to develop a risk management methodology to assist owners in establishing decision-making procedures through maintenance master plans. The project is split into two phases: the initial phase consists of a study into maintenance management practices for harbour infrastructure and a preliminary risk analysis to identify vulnerable structures
(ageing and stakes); the second phase then consists in quantifying the risks associated with the vulnerable structures identified in the initial phase.

The initial phase has been described in several papers (Boéro et al., 2009a & 2009b). The present article is concerned only with the second phase, specifically the risk analysis of steel sheet-pile seawalls undergoing corrosion.

For this purpose, a stochastic (spatial-temporal) model of steel corrosion is proposed, based on a statistical analysis of data collected from several French ports in multiple zones that have been exposed to marine environment for varying lengths of time. At the preliminary stage, an in-depth understanding of the following issues is necessary:

- interpretation of the experimental data and physico-chemical phenomena which allow spatial corrosion to be treated essentially as a one-dimensional problem, according to the depth of the structures under consideration (Boéro et al., 2009c). An analysis of corrosion mechanisms shows that the process can be modelled by 5 independent random variables corresponding to 5 different exposure zones, each one based on a different random context;

- probabilistic modelling for taking into account the variability of the phenomena. Having taken the preceding points into consideration, the stochastic time-dependent corrosion model presented in this paper is based on the temporal evolution of the probability distribution parameters defining steel loss of thickness in each exposure zone.

Predictions resulting from the corrosion model are then used to perform a time-dependent reliability calculation for corroded harbour structures. Two approaches, coupled between a deterministic finite element model (Cast3M) and a probabilistic model, are used: Monte-Carlo simulations and Spectral Stochastic Finite Element Method (SSFEM) (Ghanem et Spanos, 2003; Berveiller, 2005).

Finally, the results obtained through the different approaches are compared and applied to optimize the maintenance of the harbour structures under consideration.

## 2. Modelling of stochastic corrosion fields in steel harbour structures based on feedback from French ports

The proposed corrosion model takes into account the spatio-temporal aspects of corrosion on steel sheet piles seawalls or cofferdams. It is based, after data cleansing, on a statistical analysis of over 35000 measurements taken from several French ports (Boéro et al, 2009c).

The model is based on the general hypothesis that corrosion can be considered to be a decoupled phenomenon on the $R^2$ plane of a structure; that is, in the length direction x and along the depth z. The spatial aspect of corrosion in the x-direction is represented by the deterministic tendency $T(x,z_j,t)$ which evolves in time t and depends on the exposure zone $z_j$ (tidal zone, immersion zone, etc.). The form of the deterministic tendency (linear, sinusoidal, etc.) is unique for each asset. It depends principally on the environmental conditions to which the structure is exposed (marine currents, effluent discharge, etc.). In the absence of evidence demonstrating spatial correlation (correlation length less than
20cm), the random variables may be considered as being independent between two points in the x-direction, with a previously obtained deterministic tendency (Boéro et al., 2009c). Corrosion may be considered as being uniform in the z-direction within a given exposure zone $z_j$ (tidal zone, immersion zone, etc.). The corrosion distribution $c(z_j,t, \theta)$ is therefore assumed to be independent of $x$, dependent on the time $t$ and discretized along the depth $z$ by exposure zone $z_j$. Thus the predictive corrosion model takes the following form [1]:

$$c(x, z_j, t, \theta) = T(x, z_j, t) + c(z_j, t, \theta) \quad [1]$$

$c(x, z_j, t, \theta)$: loss of thickness as a function of the x-coordinate and the time $t$, for the exposure zone $z_j$ (mm)

$T(x, z_j, t)$: deterministic tendency with respect to a constant mean corrosion level throughout the x-direction

$c(z_j, t, \theta)$: random variable representing the loss of thickness with time $t$ for the exposure zone $z_j$ (mm). It is important to emphasise that the statistical parameters for the random variable $c(z_j, t, \theta)$ may vary throughout the length $x$ of the structure, depending on the heterogeneity of the environment (concentrated discharges, etc.). After identifying the heterogeneous zones, the random variable $c(z_j, t, \theta)$ representing loss of thickness, can be discretized in the x-direction.

The Gamma probability distribution is used to characterize the variability of the corrosion $c(z_j, t, \theta)$, since it is the theoretical probability law that can be best fitted, in terms of maximum likelihood, to the empirical distribution of steel loss of thickness. The Gamma distribution is characterized by a shape parameter $\alpha$ and a scale parameter $\beta$ which depends on the mean and standard deviation of steel loss of thickness and which evolves, according to the present model and in the absence of other information on the stochastic structure, in a spatio-temporal manner. The statistical analysis of empirical data taken from within each exposure zone may be refined using the least-squares method to produce better-adjusted temporal evolution laws for the mean and standard deviation of loss of thickness.

In agreement with the literature, the average thickness loss reaches a maximum value in zone of low seawater level due to the differential aeration which is established between the tidal zone, very oxygenated, and the immersion zone (Memet, 2000).

<figure><img src="image_2.png" /><figcaption>Figure 1. Temporal evolution laws of the mean of thickness loss.</figcaption></figure>
Legend: zS = Splash zone; zT = Tidal zone; zL = Low seawater level zone; zI = Immersion zone; zM = Mud zone.

Steel thickness loss distributions obtained from the corrosion model for a 25 year exposure period in the marine environment are shown in figure 2.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Steel thickness loss distributions obtained from the corrosion model for the different zones over a time period t=25 years.</figcaption></figure>

Spatio-temporal corrosion predictions can thus be very easily used in reliability analysis, as well as in traditional engineering for the re-evaluation of corroded structures through the choice of quantiles on the distribution of steel loss of thickness for example.

# 3. Simulation of the influence of corrosion on the mechanical behaviour of a steel sheet piles seawall

## 3.1. Description of the studied problem

The objective of this exploratory study is to integrate the previously-defined spatio-temporal corrosion fields into an analysis of the mechanical behaviour of a quay using the finite element method. The example is representative of current situations and considers a wharf with U shaped piles, anchored through one level of passive tie rods. A recapitulation of the principal geometrical and mechanical characteristics of the quay is shown in figure 3.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Geometrical and mechanical characteristics of the quay.</figcaption></figure>
A load $q$ of 50kPa is applied to the quay, and a tidal range of 5m is considered. The water table level is not equal to the sea level, since the latter corresponds to the Mean Low Water Springs (MLWS).

## 3.2. Finite element model

It is assumed that the quay is sufficiently long that plane deformation conditions apply to calculations carried out on cross sections. In terms of soil-fluid interaction, only the Archimedes buoyancy force on the soil is considered. The force of flow generated by viscous fluid friction on the soil grains is considered negligible, taking into account only the difference in elevation between the sea and the water table level. The construction phases of the quay forces are not taken into account.

The soil is modelled as a pulverulent, homogeneous material that behaves in a perfectly elasto-plastic manner (Mohr-Coulomb criterion). The steel (present in the main wall, the pile anchorage and the tie-rods) is assumed to behave in an isotropic and linearly elastic manner. The elements of interface, laid out around the main wall, are described by the Coulomb friction criterion. The angle of the interface is equal to $2/3 \phi'$ (approximately 23.3 degrees), and the contact cohesion is negligible. The resistance of the tension elements is also zero, which allows for relative movement between the nodes of the various interfaces.

## 3.3. Impact of corrosion on the mechanical behaviour of the quay

In the example studied, the steel sheet piles seawall is topped by a reinforced concrete beam. Therefore, the steel sheet piles seawall is only in contact with the tidal zone, the zone of low seawater level, the immersion zone and the mud zone. The study was carried out for 4 different exposure times ($t=0, 10, 25$ and $50$ years). As a first step, the influence of corrosion on the bending moment in the seawall was analysed, using the 95% fractile of steel loss of thickness, as predicted by the probabilistic model in each exposure zone along the depth of the seawall (cf. figure 4). The simulations were carried using the Cast3M finite element computer code developed by the Atomic Energy Commission (CEA) (http://www.cast3m.cea.fr).

<figure><img src="image_5.png" /><figcaption>Figure 4. Bending moment in the steel sheet-piles seawall.</figcaption></figure>
Time-dependent corrosion causes a decrease in the geometric characteristics of the steel sheet piles seawall, thus making the wall more flexible. A reduction in bending moment of around 20% of the initial state over 50 years is observed on the wall towards the seaside excavation. On the other hand, the normal strength in the tie-rods increases slightly, in this case in the order of 2.5%. The bending moments in the zone of embedment and at the level of the passive tie-rods are little affected by corrosion. The zone where corrosion is most severe does not correspond with the most mechanically loaded zone: the bending moment reaches a maximum 43% of the maximum value, which is situated in the submerged zone.

These aspects are further developed in the following section, where predictions from the probabilistic corrosion model are used to carry out a time-dependent reliability analysis to evaluate the structural safety of corroded harbour structures.

# 4. Time-dependent structural reliability of a corroded steel sheet piles seawall

## 4.1. Definition of the relative limit-state criteria

An evaluation of the failure probability of old structures, sometimes designed with very different knowledge bases, is always a delicate problem. A complete reliability analysis can be envisaged, but the objective here is to measure the mechanical performance loss of a quay due to corrosion, as compared with the initial performance. For this purpose, the (deterministic) maximum design stress $\sigma_R$ is calculated and the following limit state is considered [2]:

$$G(c_{z_T}, c_{z_L}, c_{z_I}, c_{z_M}) = \sigma_R - \sigma_S(c_{z_T}, c_{z_L}, c_{z_I}, c_{z_M}) \quad [2]$$

$\sigma_R$: maximum admissible stress in the steel sheet piles seawall at time $t_0$ (MPa)

$\sigma_S(c)$: stress in the steel sheet piles seawall at time $t$, along the depth $z$, whilst considering the vector of basic random variables

$$c = \{c_{z_T}, c_{z_L}, c_{z_I}, c_{z_M}\}, \text{ avec } c_{z_j} = c(z_j, t, \theta)$$

In the current problem, only the steel loss of thickness due to corrosion is considered to be a random process. Consequently, the random variables characterising the four exposure zones, and for the three time periods studied, are represented by a Gamma probability law, whose parameters may be determined from the proposed predictive corrosion model. Stresses along the depth $z$ are obtained by the 2D deterministic mechanical model previously presented. The probability of relative performance loss may then be calculated using the expression [3]:

$$P_{f,r} = P(G \le 0) = P(\sigma_S(c) \ge \sigma_R) \quad [3]$$

It must also be emphasised that other limit-states may be considered, for example: maximum admissible horizontal displacement at the head of the seawall.
## 4.2. Probabilistic methods used to evaluate the probability of mechanical performance loss

Two numerical approaches, coupling a deterministic finite element model (Cast3M) and a probabilistic model, are used to calculate the probability of mechanical performance loss of the quay due to corrosion: Monte-Carlo simulations (Metropolis and Ulam, 1949) and a non intrusive Spectral Stochastic Finite Element Method (SSFEM) (Polynomial Chaos expansion) (Ghanem and Spanos, 2003; Berveiller, 2005).

The non intrusive SSFEM (projection method) consists of firstly developing each basic random variable into a series of orthogonal polynomials. In this study, Laguerre's polynomials are used to measure the probability associated with the basic random variables, the latter being based on the Gamma probability laws. The mechanical response of the system under consideration is then projected onto polynomial chaos basis, in order to reduce the calculation of the response coefficients to an integral, which may be evaluated by quadrature method. For this purpose, recourse must be made to deterministic finite-element calculations. An adequate post-treatment of the developed coefficients is then carried out to evaluate the response of the system and to allow the failure probability to be calculated (Berveiller, 2005).

Finally, the Monte-Carlo simulations method is used as a reference, as it allows a precise determination of the failure probability to be obtained.

## 4.3. Comparative analysis of the probabilistic methods used and explanation of the results for maintenance optimisation

The probability of performance loss $P_f$ is calculated for the three exposure periods studied ($t=10, 25$ and $50$ years). The time-dependent structural reliability is an instantaneous reliability, an approach that is considered sufficient insofar as the limit-state function $G$ is strictly decreasing and where only the random variables are time-indexed: no process is therefore considered. The variation in $P_f$ along the depth of the steel sheet piles seawall is shown in figure 5 for the SSFEM. Using the Monte-Carlo simulations method (MC) the probabilities of performance loss of the steel sheet piles seawall at the level of the tie-rods (-3.5 m) and at the level of the maximum bending moment in the immersion zone (-8.5 m) are calculated (cf. table 1).

<figure><img src="image_6.png" /><figcaption>**Figure 5.** Evolution of the probability of performance loss along the depth of the quay after 10, 25 and 50 years of exposure to the marine environment.</figcaption></figure>
<table><caption>**Table 1.** Probabilities of relative performance loss of the steel sheet piles seawall at the level of the tie-rods and at the level of the maximum bending moment for the two approaches.</caption><thead><tr><th>Probabilistic method</th><th>t (years)</th><th>P<sub>f,r</sub> tie rods</th><th>Nc</th><th>P<sub>f,r</sub> Max. bending moment (Immersion zone)</th><th>Nc</th></tr></thead><tbody><tr><td rowspan="3">SSFEM (PC order = 4)</td><td>10</td><td>0,0001</td><td>625</td><td>0,9570</td><td>625</td></tr><tr><td>25</td><td>0,0030</td><td>625</td><td>0,9994</td><td>625</td></tr><tr><td>50</td><td>0,0082</td><td>625</td><td>1,0000</td><td>625</td></tr><tr><td rowspan="2">MC</td><td>10</td><td>-</td><td>-</td><td>0,9651</td><td>3555 (ε &lt; 5 %)</td></tr><tr><td>50</td><td>0,0074</td><td>20819 (ε≈15,7%)</td><td>1,0000</td><td>4940 (ε &lt; 5 %)</td></tr></tbody></table>

The probabilities of performance loss obtained using the SSFEM and the Monte-Carlo simulations method are similar. The advantage, in terms of computational cost, of the SSFEM in this study (4 basic random variables and 5 Gaussian integration points for the quadrature method), over that of the Monte-Carlo simulations method, should be emphasised: the slow convergence of the Monte-Carlo method requires a large number of iterations, and therefore a large number of Nc calculations on the limit-state function in order to obtain an accurate and reliable result.

In general terms concerning maintenance optimisation of the studied harbour structures, the loss of mechanical performance around the level of maximum bending moment in the immersion zone of the seawall (-7 to -10m) is practically certain, when one considers limit-state criteria. This is the most highly loaded section. The mechanical characteristics of the seawall are dimensioned accordingly. As a result, this section is particularly sensitive to the effects of corrosion and the safety margin is systematically impacted. In other zones, the effect of corrosion on mechanical performance loss is negligible, since there is a large supplementary safety margin due to the homogeneity of mechanical characteristics over the depth of the steel sheet pile seawall for reduced load levels.

This analysis, which takes into account the 'ageing – mechanical behaviour' combination, therefore identifies a single zone in which thickness loss measurements should be carried out for the studied limit-state criterion. It is, however, important to emphasise that the optimal maintenance strategy must take into account all limit-state criteria: holes of sheet-piles, maximum admissible horizontal displacement at the head of the seawall, etc.

## 5. Conclusion

This article has presented a global vision of research carried out to date in the framework of the GEROM project. The first phase produced an assessment of maintenance practices for the French port, brought to light the non-uniformity of the asset base and the
significance of the associated maintenance constraints. It is essential to integrate these
aspects into a risk management, or any other, methodology in order to insure its robustness.

The second phase was focused on the vulnerable harbour structures identified in the first phase. Stochastic corrosion fields were modelled based on feedback from French ports. The resulting predictions were then integrated into a finite element model in order to carry out a sensitivity analysis of the mechanical behaviour of a quay undergoing corrosion.

Due to the large spatio-temporal variability of corrosion, and its influence on the stress-
state and resistance of the structure, a mechanical reliability study was carried out using
several approaches, coupled between a deterministic finite element model of the quay and a
probabilistic model. The results of this analysis allowed to identify the most critical zone
according to the depth of the quay, in view of the limit-state criterion that is being
considered.

Finally, a study on the integration of other basic random variables, such as geotechnical
soil characteristics, loads and environmental actions, is currently underway, all in view of
further optimizing time-dependent maintenance actions.

Acknowledgements

The GEROM project was carried out in GIS MRGenCI (contact: Franck.Schoefs@univ-
nantes.fr) and financed by Oxand. The authors would like to thank the CEA for their
permission to use the Cast3M software.

**6. Bibliography/References**

Berveiller M., Éléments finis stochastiques : approches intrusive et non intrusive pour des analyses de fiabilité, PhD thesis, Université Blaise Pascal – Clermont II, Clermont-Ferrand, France, 2005.

Boéro J., Schoefs F., Capra B., Rouxel N., “Technical management of French harbour structures. Part 1: Description of built assets”, *PARALIA*, 2009a, Published on line December 21th 2009, Vol. 2, p 6.1-6.11.

Boéro J., Schoefs F., Capra B., Rouxel N., “Technical management of French harbour structures. Part 2: Current practices, needs–Experience feedback of owners”, *PARALIA*, 2009b, Published on line December 21th 2009, Vol. 2, pp 6.13-6.24.

Boéro J., Schoefs F., Melchers R., Capra B., “Statistical Analysis of Corrosion Process along French Coast”, *ICOSSAR’09, paper n°528, Mini-symposia MS15 “System Identification and structural health Monitoring”*, 2009c, Osaka, Japan, September 13-19, 2009.

Ghanem R., Spanos P., “Stochastic finite elements: a spectral approach”, revised edition. *New York: Dover*, 2003, 222 p. ISBN 0-486-42818-4.

Memet, J.B, La corrosion marine des structures métalliques portuaires : étude des mécanismes d'amorçage et de croissance des produits de corrosion, PhD thesis. Université de La Rochelle, France, 2000, 164 pp.

Metropolis N., Ulam S., "The Monte Carlo Method", *Journal of the American Statistical Association*, vol. 44, no 247, September 1949, p. 335-341.
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