Franck Schoefs

Xavier Aduriz

Olivier Bernard

Comparison of Additional Costs for Several Replacement Strategies of Randomly Ageing Reinforced Concrete Pipes

& Bruno Capra

OXAND, 49 Avenue Franklin Roosevelt, 77210 Avon/Fontainebleau, France Abstract: One of the uses of reinforced concrete pipes (RCPs) is the distribution of aggressive water in industrial systems, for example, in water-cooling systems of nuclear power plants. Some of them carry seawater and can deteriorate with time because of internal corrosion. Because of the low O 2 content of aggressive water, slow corrosion is expected for such applications. If the RCPs are not periodically replaced, they will eventually fail. Replacement strategies for these pipes depend on (1) the risks associated with the failure of the water distribution network, and (2) the costs associated with replacing the pipes, including the removal of existing pipes, installation of new pipes, and associated production losses. Because of the lack of statistical data regarding RCP failure, the development of a risk-based replacement strategy is not an easy task. This article demonstrates how predictive models for the evolution of the failure of RCPs and the associated consequences of failure can be used to develop risk-based replacement strategies for RCPs. An application for the replacement strategies of a network modeled as a system consisting of 228 RCPs is presented as a case study. We focus on the assessment of the number of replaced components that governs the costs. The main objective of this article is to provide a theoretical approach for comparing replacement strategies, based on (1) the results of a reliability study, (2) the representation of the distributions of failed components (binomial distribution), and (3) the decision tree representation for replacement of RCPs. A focus on the scatter of the induced costs themselves is suggested to emphasize the financial risk.

INTRODUCTION

One of the uses of reinforced concrete pipes (RCPs) is the distribution of aggressive water in industrial systems. Some networks of RCPs are used, for example, in seawater transport (water cooling of nuclear power plants) and can deteriorate with time because of internal corrosion [START_REF] Peter-Lazar | Application of probabilistic methods to the analysis of the behaviour of reinforced concrete structures affected by steel corrosion[END_REF]. If the RCPs are not periodically replaced, they will eventually fail.

Inspection, maintenance, and repair (IMR) strategies for such water distribution networks are essentially based on pipe segment replacements. The goal of an optimal IMR strategy for RCP networks is to find when, where, and how to replace the network components. The optimal management strategy (OMS) depends on (1) the risks associated with the failure of the water distribution network, and (2) the costs associated with replacing the pipes, including the removal of existing pipes, installation of new pipes, and associated production losses [START_REF] Luong | Fund allocation for pipe repair maintenance in water distribution networks[END_REF][START_REF] Adey | Risk-based replacement strategies for underground reinforced concrete pipe networks[END_REF]. The OMS should result in the highest reduction of risks and costs for owners of structures. This article does not provide optimization results but instead analyzes the sensitivity of a family of strategies to the lack of knowledge on the number of repairs.

Because of the lack of statistical data for RCP failure, the development of a risk-based replacement strategy is not an easy task. Some authors have addressed this issue in other industrial fields. [START_REF] Estes | Minimum expected cost-oriented optimal maintenance planning for deteriorating structures: Application to concrete bridge decks[END_REF] have used event tree representation to determine OMS of corroded reinforced bridges. Their approach can also be used to update the management strategy from the results of inspection. This article considers inspections at a constant interval. Note that methods based on genetic algorithms allow the consideration of the test interval as a random variable for optimization based on risk and cost criteria [START_REF] Miyamoto | Development of a bridge management system for existing bridges[END_REF][START_REF] Martorell | Alternatives and challenges in optimizing industrial safety using genetic algorithms[END_REF][START_REF] Carlos | A multi-objective genetic algorithm for RAMS+C optimization with uncertain decision variables[END_REF]. The Markov models are commonly used in management systems to model the deterioration of infrastructure assets, such as pipes [START_REF] Micevski | Markov model for storm water pipe deterioration[END_REF][START_REF] Sheils | Development of a two stage inspection process for the assessment of deteriorating bridge structures[END_REF][START_REF] Breysse | Life cycle cost analysis of ageing structural components based on non destructive condition assessment[END_REF] and road bridges (Cambridge Systematics, 1997;[START_REF] Roelfstra | A model of condition evolution of concrete highway bridges[END_REF]. Semi-Markov models have been used to incorporate changes in failure mode that may occur with both time and the number of previous breaks [START_REF] Li | Optimal maintenance related decision making for deteriorating water distribution systems 1. Semi-Markovian model for a water main[END_REF]. The Bayesian networks [START_REF] De Melo | Bayesian networks in software maintenance management[END_REF] can also be used and allow the updating of distribution, especially after inspection. Other alternatives consider nonprobabilistic approaches such as tolerance interval (Martorell et al., 2007a).

Because of the huge number of RCPs in the networks, the reduction of computation time is an important requirement. This article provides a theoretical approach to compare management strategies that meet this challenge, for the long-term horizon. It is based on (1) the results of a reliability study, (2) the representation of the distributions of failed components through a binomial distribution, and (3) the decision tree representation for RCP replacement. The main focus is on assessing the number of components to be replaced. In fact, this parameter directly affects the costs. When the network is composed of several types of components with several ageing laws, the question is which critical components should be replaced and when. To make the procedure more efficient, this number is computed as the number that maximizes the likelihood when considering the distribution of the numbers of failed components. Note that this methodology does not consider shortterm horizons and hydraulic considerations that are needed for sewer networks analysis [START_REF] Dridi | Using evolutionary optimization techniques for scheduling water pipe renewal considering a short planning horizon[END_REF], specific metric taking into account pollutions [START_REF] Nunes | Comparison of variance-reduction and space-filling approaches for the design of environmental monitoring networks[END_REF], or a multicriteria conception based on fuzzy logic [START_REF] Sarma | Life-cycle cost optimization of steel structures[END_REF].

The first part of this article presents the ageing model and the input of the study: the evolution of the probability of failure with time (i.e., the cumulative distribution function [cdf] of times to failure).

Following this, the methodology for assessing the binomial distribution of the number of failed components before replacement is developed. A network modeled as a series system is considered.

This approach is then extended to the case of mixed networks composed of replaced and nonreplaced components. A probabilistic updating is then suggested.

This article concludes with an illustration of a real network with two categories of components and a cost analysis that emphasizes the importance of decision aid tools with estimators based on the whole distribution of costs. It gives a refined analysis complementary to the usual minimization of a cost function [START_REF] Li | A riskcost optimized maintenance strategy for corrosion-affected concrete structures[END_REF].

PREDICTIVE MODELING AND RELIABILITY ANALYSIS OF CHLORIDE-INDUCED CORROSION OF RCP

Definition of RCP

Figure 1 is a schematic of a RCP of a type typically used for industrial applications. Most of these RCPs consist of external and internal concrete layers separated by a cylindrical steel sheet. The external layer is usually reinforced in European countries and prestressed in Asia and the United States. The steel sheet placed between the two concrete layers plays a role in preventing leakage and in resisting the internal fluid pressure.

Local corrosion criterion and reliability analysis

Seawater contains chloride ions (typically 25-30 g/L), which are known to be aggressive against reinforced concrete. In the particular case of a seaside nuclear power station, RCPs are not buried but positioned on supports. They are 1 m in length. RCPs permanently contain seawater except during particular maintenance operations (change of a pipe, for example). It is observed that RCP deterioration with time is linked to internal corrosion [START_REF] Peter-Lazar | Application of probabilistic methods to the analysis of the behaviour of reinforced concrete structures affected by steel corrosion[END_REF] due to the diffusion of chloride ions from seawater to the steel sheet via the internal layer of concrete. A lot of work has been done on the modeling of the penetration of chloride ions into concrete, and models of varying levels of complexity have been developed: based on Fick's second law [START_REF] Collepardi | Quick method to determine free and bound chlorides in concrete[END_REF][START_REF] Andrade | Calculation of initiation and propagation periods of service-life of reinforcements by using the electrical resistivity[END_REF][START_REF] Nilsson | On the uncertainty of service-life models for reinforced marine concrete structures[END_REF][START_REF] Frederiksen | Hetek, a system for estimation of chloride ingress into concrete: Theoretical background[END_REF]Petre-Lazar et al., 2000) and flux equations [START_REF] Tang | Chloride transport in concrete-measurement and prediction[END_REF][START_REF] Frederiksen | Hetek, a system for estimation of chloride ingress into concrete: Theoretical background[END_REF][START_REF] Truc | Prediction of chloride penetration into saturated concrete. Multi-species approach[END_REF][START_REF] Marchand | Predicting the performance of concrete structures exposed to chemically aggressive environment[END_REF]. The software SIMEO TM (SIMEO TM , 2006) has been used to conduct a reliability study to determine the evolution of the probability of corrosion. This software is devoted to reliability computation and simulation in a user-friendly environment (selection of predefined probability density functions, etc.). Only the Monte Carlo algorithm has been used to obtain the results of this article.

The several stages of the corrosion process are described in Figure 2. The first phase of corrosion (Figure 2a) is characterized by chloride penetration from the pipe's internal surface to the steel sheet. This penetration process can be modeled as a stationary diffusion of the chloride brought to the pipe's internal surface by seawater. Taking C s as the chloride concentration on the internal pipe surface, d the internal concrete cover, C 0 the initial chloride concentration of the concrete cover (in this article, C 0 = 0), and D c the chloride diffusion coefficient in concrete, the accumulated chloride concentration on the steel sheet's internal surface can be expressed as a function of time by Equation (1) [START_REF] Peter-Lazar | Application of probabilistic methods to the analysis of the behaviour of reinforced concrete structures affected by steel corrosion[END_REF]:

C (t) = C 0 + (C s -C 0 ) 1 -er f d 2 √ αηD c t (1)
where erf is the mathematical error function, and α and η represent, respectively, the coefficients for the ion interaction and the ion-cement paste interaction in chloride transport in concrete:

α = 1 + 1 4.C s and η = 1 1 + k
with k, the ion-cement paste interaction coefficient.

The estimated time for corrosion initiation is about 5 years according to available data.

When the accumulated chloride concentration C(t) reaches a critical value, corrosion will initiate at the internal surface of the steel sheet (Figure 2b). This critical concentration is defined according to an empirical relationship between chloride concentration and hydroxyl ion concentration in concrete. The selected value in this article is 0.32 mol/L; it corresponds to a concentration [Cl -]/[OH -] = 0.6 and a pH in the concrete of 13.5. Even if this concentration could be considered as random, even stochastic with time, no probabilistic model is available till now. Thus, for simplicity, it is considered as deterministic. Note that probabilistic modeling would modify the probability of failure with time and thus the cost analysis but not the ranking of the strategies. The rate of the loss of steel by corrosion is estimated, in our case, at 55 μm per year according to the feedback on deteriorated pipes that have been investigated (pitting measurements). The rate of the loss of steel section depends on the corrosion current. By this modeling, the average rate of steel loss by chloride corrosion has been calculated for the whole service life of a RCP. In the following sections, it is consider as deterministic because its scatter in confined environment is very fair in comparison to the uncertainty propagation because of Equation ( 1). Once a local hole is present on the steel sheet (Figure 2c), chloride ions can migrate through the external layer of concrete and lead to the corrosion of the external side of the steel sheet. The process is then accelerated because of the nonsaturated condition of the concrete, which is favorable for corrosion development (Figure 2d).

For the purpose of a reliability analysis, probabilistic distributions are considered for modeling the scatter of the values of material parameters in the model, including the concrete cover thickness, the steel sheet thickness, and the corrosion rate. Table 1 summarizes the values and distribution laws taken for the simulations. At each time t, the generated samples allow the calculation of accumulated chloride concentration from Equation (1) through a crude Monte Carlo simulation. Note that the stochastic process is defined as a timedependent function of basic variables: it acts as an analytic transfer function of random variables, indexed with time, through the deterministic Equation ( 1). The limit state is built considering that a RCP is assumed to have failed when there is no more residual steel thickness. Thus, a RCP could have corroded but not yet failed. Taking a zero residual steel sheet thickness as the failure criterion, the time-dependent failure probability is calculated via a coarse Monte Carlo simulation method with 10,000 samples. The resulting distribution of time to failure is presented in Figure 3. We note that the diffusion of ions that leads to no failure with the selected limit state occurs in the first 10 years during which the probability of failure does not increase. We consider this curve as an input in our study. If another RCP is considered, the modeling of Table 1 must be changed. This evolution of the probability of failure describes a survival function. This function is least-square-fitted using several cdfs. When comparing the plots of the cdf fitted to the original cdf, we observe that some functions (see lognormal, for instance) fit quite well during the early stages but diverge from the original distribution after 22 years. Note that this model of cdf can be convenient if the replacement strategy for failed pipes is frequent, in which case only the early stages of the cdf curve are considered. This point is discussed in more detail at the end of this article.

STRUCTURAL ANALYSIS: THE CASE OF RCP NETWORK

Assessment of the failed components distribution in a network modeled as a series system

From the input curve such as in Figure 3, the question is now to assess the number of failed components in a network before and after repair. Note that the representation of a structure as a combination of serial/parallel failure components is widely documented in the literature [START_REF] Hendawi | System reliability and redundancy in structural design and evaluation[END_REF] and is a very useful formal representation for system analysis. An analogy with electronic systems can be made. Note that, in general, this is an approximate representation and may not be an exact depiction of the real physical form of the network. The considered RCP network is modeled as a series system composed of several categories of components, jointed together. This theoretical section of the article deals only with one category to clarify the approach. The final part of this article suggests an illustration with two categories. Several factors affect the initial state and thus the probability of failure:

• RCPs are manufactured by different companies, with different quality control processes; and • the storage conditions (outside or inside, multilayers or single layers) and duration before installation vary between RCPs.

From the feedback of owners, we assume that joints between RCPs have a negligible probability of failure and isolate each RCP from those adjacent to it. We assume that failures/nonfailures of each pipe segment are statistically independent events and the serial representation of the network is perfect. This assumption allows us to simplify the analysis and is conservative when analyzing series systems. Further studies are needed to quantify these factors and how they affect the heterogeneity of the network. Note that in that case the system probability of failure is given by:

P f,s (t) = 1 - n i=1 (1 -P f,i (t))
(2)

where P f ,i (t) is the probability of failure of component i at time t. In this study, the aim is not to compute the probability of failure but to assess the number of failed RCPs at time t. This allows determination of the number of repair or preventive replacements to be planned.

All the RCPs of the network modeled as a series system are supposed to be in a similar state at time t 0 = 0.

The expected number m(t) of failed RCPs at time t is expressed as follows:

m(t) = n × ϕ(t) ( 3 )
where n is the total number of RCPs. Thus, the cumulative failure distribution functions ϕ(t) of each component are the input of the problem (see Figure 3).

In what follows, the series representation is considered to be perfect: the failure of one component does not affect the probability of failure of neighboring components as well as the quality of the water flowing in the network.

Approximation of the distribution of failed RCPs with a binomial distribution

A distribution of failed RCPs is proposed at each time t to determine the number of failed RCPs with a confidence level. By knowing n (total number of RCPs) and ϕ(t), we assume that the distribution of failed components follows a binomial distribution with parameter n and probability p(t) = ϕ(t). The use of such hazard functions in a time-dependent reliability analysis is covered in detail in Mori and Ellingwood (1994a). Under this assumption, the probability of having k failed RCPs at time t is given by the discrete probability distribution (Equation ( 4)):

P f (x = k, t) = n! k!(n -k)! ϕ(t) k (1 -ϕ (t)) (n-k) (4)
We then define a binomial stochastic process. Note that the mean trajectory of this process is an increasing function with time and is not a stationary process: in particular, the first two probabilistic moments (mean and variance) are time-variant functions. For example, the variance equals np(1-p): as t tends to 0 or infinity, p tends, respectively, to 0 or 1, and the variance tends to 0. Figure 4 illustrates four distributions obtained for t = 16, 18, 20, and 22 years from the cumulative fail-ure distribution function presented in Figure 3. Among these four cases, the probability at "t = 20 years" corresponds to the maximum value of the standard deviation and is obtained for a probability of RCP failure of 0.6 (see Figure 3). Note that this distribution is discrete and appears in the figure as continuous because of the high number of RCPs.

For the initial and final corrosion states, the number of failed RCPs is quite deterministic. It is because of the fact that the probability of failure describes a failure criterion with a binary response. Thus, the standard deviation is not increasing with time.

It is now possible to assess the number of failed RCPs with a confidence level of 5% or 95% from these distributions.

PROBABILISTIC UPDATING AFTER REPAIR

Formal representation of maintenance strategies based on event tree

Under the previous assumptions, the event tree representation is used to build maintenance strategies. This formal graphical scheme allows us to represent several paths for maintenance strategy-called scenariosrepresented by a branch that links the bounds of the decision "replace or don't replace" (Figure 5). In this case, the diagram is called a "decision tree," as it represents courses of action: either replacement or nonreplacement. It allows building several branches from the original branch of nonreplaced RCP until the reference period t end is reached. It has been used as a decision aid tool in the offshore industry and bridge maintenance planning [START_REF] Faber | Indicators for inspection and maintenance planning of concrete structures[END_REF]. Note that imperfection in detection can be introduced too in the form of defect detectability function (Mori and Ellingwood, 1994a,b) or the probability of detection and probability of false alarm [START_REF] Rouhan | Probabilistic modelling of inspection results for offshore structures[END_REF][START_REF] Pakrashi | An image analysis based damage methodology, Structure and Infrastructure Engineering: Maintenance, Management, Life-Cycle Design and Performance (NSIE)[END_REF][START_REF] Schoefs | Risk analysis of structures in presence of stochastic fields of deterioration: Coupling of inspection and structural reliability[END_REF]. The cumulative failure distribution function of each component follows the time-dependent probability of failure given in Figure 3.

Management strategy: Choice of the number of replaced RCPs at time t

Over the last decade or so, several papers have covered this issue, including the role of inspection (Bjørnoy et al., 2001;[START_REF] Onofriou | Reliability-based inspection optimisation of complex structures: A brief retrospective[END_REF][START_REF] Ritchie | Probabilistic tools for planning of inspection and repair of corroded pipelines[END_REF]. In this article, only the date of replacement is considered as a maintenance parameter. The number of potential replaced RCPs at time t is determined by the maximum probability of the distribution of the numbers of failed components at this time. This number is supposed to be given by an inspection of the network at time t. All the nonrepaired components are then supposed "as good as new." This is, of course, an assumption to simplify here the formulation of the updating. Thus, under this assumption, the method is consistent with the Bayesian theory.

Distribution of the number of failed RCPs after one action (t 2 > t > t 1 )

Once the number m 1 of replaced RCPs is selected, the accurate size of populations for replaced and nonreplaced (n -m 1 ) RCPs at time t is known. Thus, the probability distribution of the number of replaced RCPs is a binomial probability distribution with parameters m 1 and p(t) = ϕ(tt 1 ). Note that imperfect maintenance can be introduced at this step (S ánchez et al., 2007). The probability distribution of the number of remaining intact RCPs is determined in the same way using a binomial probability distribution with parameters (n -m 1 ) and p(t) = ϕ(t)-ϕ(t 1 ) 1-ϕ(t 1 ) . For numerical simulations, it is important to mention that when t reaches the lifetime of the RCPs, the expression of p(t) becomes numerically undefined. Then, the result ϕ(t)-ϕ(t 1 ) 1-ϕ(t 1 ) -→ ∞ 1 is introduced (ϕ(t) = 1 when t tends to infinity).

The probability of the number of failed RCPs at each time t ≥ t 1 and before the next action is obtained through a summation of products:

P(X = x, t) = x i=0 P 2 (X = i, t) × P 1 (X = x -i, t) (5)
where P 1 (X = x) is the probability to have x nonreplaced RCPs at time t 1 that fail at time t, and P 2 (X = x) is the probability to have x replaced RCPs at time t 1 that fail at time t.

Distribution of the number of failed RCPs after i actions, (t > t i )

There are 2 i populations that are described at time t by only (i + 1) distributions because some of them are complementary. In the following equation, m(i, j) is the number of RCPs of the branch j after i actions. These distributions are determined with the following step-bystep method:

• first, the distribution of the number of nonreplaced RCPs during the whole period t is a binomial probability distribution with parameters:

m(i, 2 i ) and p(t) = ϕ(t -t 0 ) -ϕ(t i -t 0 ) 1 -ϕ(t i -t 0 ) (6)
• second, the distribution of the number of nonreplaced RCPs at time t i but replaced at time t k (1 ≤ k ≤ i -1) is a binomial probability distribution with parameters:

m(i, k) = 2 k-1 -1 j=0 m(i, (1 + 2 j ) 2 i-k ) and p(t) = ϕ(t -t k-1 ) -ϕ(t i -t k-1 ) 1 -ϕ(t i -t k-1 ) (7)
• finally, the distribution of the number of replaced RCPs at time t i is a binomial probability distribution with parameters:

m(i, i + 1) = 2 i-1 j=1 m(i,2 j -1) and p(t) = ϕ(t -t i ) (8)
Then, an expansion of Equation (5) for any value of i writes:

P(X = x) = x a=0 P i+1 (X = a) × • • • × h i=0 P 3 (X = i) × i j=0 P 2 (X = j) × P 1 (X = x -j -i) (9)
where P 1 (X = x) is the probability of having x nonreplaced segments for any value of t i that fail at time t, P k (X = x) is the probability of having x replaced segments at time t k -1 (2 ≤ k ≤ i) that fail at time t, and P i+ 1 (X = x) is the probability of having x replaced segments at time t i that fail at time t. This formalism is well adapted for linear algebra programming such as in SCILAB [START_REF] Scilab | Scilab 4.1.2: The open source platform for numerical computation[END_REF], which is an interactive system whose elementary element is a matrix that does not require any dimensioning. Thus, the explosion of cases of combinations because of the number of components to repair and the number of repairing dates does not pose a problem. Note that the number of actions is also reasonable and therefore makes the computation time convenient.

ILLUSTRATION OF UPDATING ON A REAL RCP NETWORK

Input values, main assumptions, and maintenance strategy

The study of the corrosion process in a real network lies in another curve for describing the evolution of the probability of failure with time than that shown in Figure 3. In fact, in the network selected for the application, we define two sets of RCPs with several behaviors:

• the RCPs located in the ordinary areas where internal corrosion is acting; and • the RCPs located near bends or supports, called "singular" areas where an external corrosion mechanism is superimposed because of differential aeration conditions. This differential aeration takes place anytime when there is a step between concrete cover (supports and bend). The study of existing systems allowed the determination of an equivalent corrosion rate and a corresponding cumulative failure distribution function in this case.

Note that these curves differ from the one in Figure 3 because some parameters of the RCPs are different.

The network is composed of 90 RCPs located in singular areas and 138 located in ordinary areas. In both cases, RCP lifetime is about 60 years (probability of failure about 95%). Figure 6 presents the distribution of time to failure. We observe that the RCPs in singular areas are subject to a faster increase of the probability of failure. Several cdfs are suitable; here, the log-normal cdf has been selected. As mentioned at the beginning of this article, these curves are the input of the methodology.

The management strategy consists of the replacement of failed RCPs. Two types of strategies are then tested:

(1) the date and the number of actions are imposed; and (2) the total number of failed RCPs is limited to an upper-bound value (threshold).

As presented before, the number of potentially replaced RCPs at time t is still determined with the maximum likelihood of failed components at time t. It is supposed that the network was not repaired during the first 20 years. The following sections present the results obtained with the two types of strategies.

Imposed dates of maintenance

Typically, maintenance strategies of operators/owners do not include plans for replacement of RCPs before their guaranteed lifetime is up. Several guaranteed lifetimes are assumed here: 20, 25, 30, 35, and 40 years.

The second assumption is that the owner prefers to plan periodic actions of replacement (here 5 and 10 years) to allocate the costs periodically and simplify the management (Note that some authors have shown that the optimal cost is not obtained with this type of strategy; [START_REF] Breysse | Management strategies and improvement of performances of sewer networks[END_REF][START_REF] Sheils | Comparing efficiency of systematic and conditional maintenance for randomly ageing components[END_REF][START_REF] Breysse | Life cycle cost analysis of ageing structural components based on non destructive condition assessment[END_REF]. The reference period t end is 70 years. The following numerical experimental plan has been programmed. The confidence levels at 5% and 95% on the number of failed RCPs are considered to illustrate the scatter on this number. Figure 7 illustrates the updated curves for RCPs in singular areas obtained for the strategy S32 with five dates of replacement. Note that in this case, the maximum number of failed RCPs is 90: the total number of RCPs in singular areas. A similar curve is obtained for RCPs in ordinary areas. The result shows that this type of strategy drastically reduces the gradient of the plot of the number of failed RCPs with time and thus the expected lifetime. This influence is significant around 55 years. It can be explained by the fact that after four actions of replacement, a large amount of RCPs (87) have been replaced, which represents about 30% of the network. The time needed to reach a totally failed RCP network is increased, after actions of replacement, from 50 years to 110 years. It is interesting to compare the curves at 5% and 95% level of confidence. It appears that the gap between these curves is greater before maintenance (g 1 in Figure 7) than after replacements (g 2 ). The number of RCP replacements with time, in singular and ordinary areas, is computed and is given in Table 2. The scatter of the number of potential replacements can be analyzed through the interval of confidence (5% and 95%). The range of this interval is 13, 13, 6, and 7, respectively, after 2, 3, 4, and 5 repair actions. As replacement actions modify the initial population by introducing new RCPs, the standard deviation of the number of failed RCPs decreases after three maintenance actions compared with the one obtained without any action. It is particularly significant at 60 years. Overall results obtained for the 10 strategies are given in Table 3. The replacements concern both ordinary and singular areas.

We notice that the number of replaced RCPs is very close for the first seven strategies except for strategy S22. For the remaining strategies, the total number of replacements decreases. The number of failed RCPs at the first date of replacement is then very high, and serviceability considerations could be not respected. From results obtained with strategies S12, S21, and S41, we presume that the optimum could be obtained with a first replacement between 26 and 34 years (except 30 years to avoid replacement at 70 years). Note that Tables 2 and 3 cannot already answer the question: "what is the Best Maintenance Strategy of those selected?" because as yet no serviceability constraint has been introduced. Cost and maintenance constraints are introduced at the end of this article.

Another point of interest is the comparison of the number of replaced RCPs in both types of areas (ordinary and singular areas). The evolution of the ratio "replaced RCP in singular areas/replaced RCP in ordinary areas" is thus of interest. The evolution of this ratio for the strategies involving replacement every 5 years (S11, S21, S31, S41, and S51) is plotted in Figure 8. Note that it is interesting to compare this rate with the ratio "special areas/ordinary areas," which is 39% (dashed line in Figure 8). RCPs located in singular areas are replaced first and represent more than 50% of the replaced RCPs, whatever be the policy: this rate remains between 75% and 100% for replacements carried out between 25 and 32 years. Then, when studying the first 70 years, this population has been totally renewed dur-ing the first years of maintenance, and no intervention is required after a given time around 55 years. The more the initial replacement is postponed, the lower is the rate, because a higher number of other RCPs start to deteriorate.

Upper-bound value for the number of failed RCPs

A serviceability constraint is introduced at this point. The target for maintenance policy is "to keep the number of deteriorated components (i.e., the number corresponding to the maximum likelihood) under a limit value considered as acceptable." This strategy is equivalent to the one that requires a limit leakage, for example. To illustrate the sensitivity of this strategy to the acceptable level, four values of acceptable number of failed RCPs are studied: they are expressed as a percentage of the total number of RCPs. The considered strategies are:

• strategy S 1: threshold fixed at 15% of the number of RCPs in ordinary areas, • strategy S 2: threshold fixed at 20% of the total number of RCPs in ordinary areas, • strategy S 3: threshold fixed at 40% of the total number of RCPs in ordinary areas, and • strategy S 4: threshold fixed at 50% of the total number of RCPs in ordinary areas. Figures 9 and 10 present the evolutions of the bounds of confidence interval (5% and 95%) of the number of failed RCPs with time. Let us focus first on the results illustrated by Figure 9 obtained with the strategy S 2 for RCP in ordinary areas with a threshold fixed at 20% (i.e., 28 RCPs among 138). Note that the curve corresponding to the upper bound of the confidence interval (i.e., 95%) does not respect the threshold because the replacement is governed by the number of replaced RCPs corresponding to the maximum likelihood (Table 2). The two curves at the left-hand side correspond to "reference curves" that represent the case with no action (no replacement). It is interesting to underline that after four actions, a lot of segments have been replaced, and the gradient of the graph is very small around 65 years. We represent the evolution after 70 years, even if the criterion is not respected (values exceed the threshold), to illustrate this point. This very fair evolution is more visible in Figure 10 for the strategy S 4: threshold equals 69. The number of deteriorated pipes is then quite constant during 5 years from 60 to 65 years. The conclusions made previously on the difference between the confidence curves at 5% and 95% are similar.

We can conclude that the acceptable level of failure has a great influence on the number of replacements. In terms of cost analysis, another constraint must be included: the discount rate, which makes a postponed replacement more expensive. Note that Table 4 cannot already answer the question: "what is the Best Maintenance Strategy of those selected?" because no consequences of action or inaction were introduced. The cost and maintenance constraints are introduced to this aim in the following section.

COST ANALYSIS OF MAINTENANCE STRATEGIES

Cost assumptions

The objective here is to find the strategy that leads to the minimal cost. This approach is an alternative to the multicriteria analysis (Le [START_REF] Gauffre | A multicriteria decision support methodology for annual rehabilitation programs of water networks[END_REF].

In terms of cost, the number of operations and the number of RCP replacements should be introduced.

In fact, the former implies stopping system utilization, and the latter the procurement and installation of new RCPs (including the cost of replacing deteriorated pipes). Failure costs, meaning financial consequences of failure, are not taken into account in this example, but are included in the software SIMEO TM -Manager (Oxand's internal software; OXAND, Avon, France). The reason of this choice by the others is to focus on the effect of inspection and repair costs on the strategies. In fact, for nuclear stations, failure costs are difficult to assess with accuracy because they depend on the limit of the system (owner, society, etc.) and the estimation and aggregation of indirect costs. In this case, the failure cost is a stochastic process itself and requires deeper investigation to be defined precisely. The following hypotheses and constraints are considered:

• The pumping network is stopped once a year, and no costs arise from this break in operation. During this time, 30 m can be removed. • For the removal of longer sections of the RCP network, it is considered that every 5 m requires a 1-day break in operation, which incurs a cost of 30,490 Euros (because of the loss of services). • The replacement of 6 m costs 25,556 Euros;

• The discount rate of money is 8%.

• It is considered that failed areas are on distinct RCPs, and that a whole RCP of 1-m length must be replaced in the case of corrosion.

Note that several assumptions can be selected and are directly linked with the commercial and financial stakes and functions of the structures [START_REF] Kenn É | Age-dependent production planning and maintenance strategies in unreliable manufacturing systems with lost sale[END_REF] and the supply of human resources (Martorell et al., 2007b).

Figure 11 represents the costs obtained for the strategies presented in the previous section. Strategies with a periodic replacement are more expensive, in particular, strategies S12, S32, and S52. It is more cost-effective to renew often (5 years). The strategies planned using an acceptable level of failed RCPs are less expensive, with the levels considered here.

It is now interesting to analyze the uncertainty of the number of replaced RCPs, which varies depending on the strategy considered. After maintenance, some strategies lead to a very heterogeneous set of RCPs, whereas others lead to a homogeneous set of RCPs. This result is closely linked to the values g 1 and g 2 presented in Figure 7. We can now introduce a risk of extra cost because of a number of replaced RCPs calculated on the basis of the confidence value at 95%. Figure 12 presents the costs for strategies S21 and S51, which are shown to be the less expensive periodic policies on Figure 11. S21 and S51 are very close in terms of total cost, but the risk of overcost for the former is twice that of the latter. At the beginning of this article, the problem of cdf fitting was discussed (Figure 3). Figure 13 illustrates this point on the basis of the cdf given in Figure 3 every 4 years after the network has been in service for 10 years. The difference between two fitted cdfs (normal and gumbel) can reach 25% for the number of RCPs and 24% for the costs after 22 years. This difference is because of the fact that mainly the beginning of the curve-which fits better than the latter section-is concerned.

CONCLUSION

This article provides a theoretical approach to analyze alternative strategies for the management of a RCP network. It is based on (1) the results of a reliability study, (2) the use of random draws (binomial probability distribution), and (3) the decision tree representation to define maintenance scenarios. This method provides an estimation of the evolution of the number of deteriorated RCPs with a confidence level. The curves at 5% and 95% confidence levels are computed and plotted. Two strategies are compared: (1) the date on which actions are undertaken, and (2) the total number of failed RCPs is limited to a threshold. Finally, a cost study is proposed and it shows how to include costs in such a maintenance strategy. The notion of overcost risk is presented as a new criterion for strategy comparison. Finally, one must take care in selecting the original cdf for the fitting, as it can lead to great errors in terms of cost evaluation.

Note that this approach provides an efficient way to predict a decision, assuming that it will be based on the maximum likelihood of the distribution of failed RCPs. By knowing the probability of detection or the probability of false alarm, it can be linked to a riskbased inspection plan. In this case, the decisions of repair and inspection are generally made at the same time. Thus, in our approach, only the possible number of replaced RCPs will change. This method has been implemented in SIMEO-Manager software: it offers a whole customer-oriented package for analyzing maintenance strategies.

Fig. 1 .

 1 Fig. 1. Reinforced concrete pipe (RCP).

Fig. 2 .

 2 Fig. 2. Four stages of internal corrosion of RCP.

Fig. 3 .

 3 Fig. 3. Typical distribution of time to failure (ϕ(t)) of RCP and fitted cumulative distribution functions.

Fig. 4 .

 4 Fig. 4. Distribution of deteriorated segments at four times: t = 16, 18, 20, and 22 years.

Fig. 5 .

 5 Fig. 5. Decision tree as graphical support to represent maintenance strategy.

Fig. 6 .

 6 Fig. 6. Distribution of time to failure in ordinary and singular areas.

  Ten strategies based on regularly spaced replacement are compared:• strategy S11: no replacement before 20 years, periodic replacement every 5 years until 70 years; • strategy S12: no replacement before 20 years, periodic replacement every 10 years until 70 years; • strategy S21: no replacement before 25 years, periodic replacement every 5 years until 70 years; • strategy S22: no replacement before 25 years, periodic replacement every 10 years until 70 years; • strategy S31: no replacement before 30 years, periodic replacement every 5 years until 70 years; • strategy S32: no replacement before 30 years, periodic replacement every 10 years until 70 years; • strategy S41: no replacement before 35 years, periodic replacement every 5 years until 70 years; • strategy S42: no replacement before 35 years, periodic replacement every 10 years until 70 years; • strategy S51: no replacement before 40 years, periodic replacement every 5 years until 70 years; and • strategy S52: no replacement before 40 years, periodic replacement every 10 years until 70 years.

Fig. 7 .

 7 Fig. 7. Evolution of the number of failed RCPs in singular areas (curves at 5% and 95% level of confidence) after five actions at times 30, 40, 50, 60, and 70 years.

Fig. 8 .

 8 Fig. 8. Evolution of the rate of replaced RCPs located in special areas.

Fig. 9 .

 9 Fig. 9. Evolution of the number of failed RCPs in ordinary areas (curves at 5% and 95% level of confidence) after four actions.Acceptable number of failed components = 20%.

Fig. 10 .

 10 Fig. 10. Evolution of the number of failed RCPs in ordinary areas (curves at 5% and 95% level of confidence) after one action.Acceptable number of failed components = 50%.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Total cost of the set of strategies.

Fig. 13 .

 13 Fig. 13. Sensitivity of cost and the number of replaced RCPs to cdf selection.

Table 1

 1 Deterministic or probabilistic modeling of variables for the reliability analysis

	Variable	Description (unit)	Law	Mean value	Standard deviation
	C S	Surface chloride concentration (g/L)	Deterministic	20	/
	D C	Chloride diffusion coefficient (m 2 /s)	Normal	4.2.10 -12	2.2.10 -12
	k	Ion-cement paste interaction coefficient	Deterministic	1.5	/
	d	Concrete cover (mm)	Normal	22.4	0.29
	e	Steel sheet thickness (mm)	Deterministic	2	/

Table 2

 2 Results obtained for the strategy based on fixed dates (S12)

		Number of	Number of	Number of
		replaced	replaced	replaced
	Date of	RCPs (5% RCPs (maximum RCPs (95%
	replacement confidence)	likelihood)	confidence)
	20	0	1	1
	30	15	21	28
	40	40	47	53
	50	15	18	21
	60	5	9	12
	70	17	22

Table 3

 3 Number of replaced RCPs in ordinary and special areas for the 10 strategies

		S11		S12		S21		S22		S31		S32		S41		S42		S51		S52
	d	n	d	n	d	n	d	n	d	n	d	n	d	n	d	n	d	n	d	n
	20	1	20	1
	25	4			25	5	25	5
	30	16	30	21	30	16			30	22	30	22
	35	25			35	25	35	42	35	25			35	46	35	47
	40	22	40	47	40	22			40	21	40	47	40	2			40	69	40	69
	45	13			45	13	45	34	45	13			45	3	45	34	45	12
	50	6	50	18	50	6			50	6	50	18	50	10			50	6	50	18
	55	2			55	3	55	9	55	3			55	17	55	8	55	2
	60	6	60	9	60	6			60	6	60	8	60	47			60	1	60	3
	65	12			65	11	65	14	65	11			65	21	65	12	65	3
	70	16	70	22	70	16			70	16	70	23	70	13			70	13	70	16
	T	123		118		123		104		123		118		119		101		106		106

d = date of replacement; n = number of replaced RCPs at each date; T = total number of replaced RCPs.

Table 4

 4 Number of replaced RCPs for the four strategies

		S 1	S 2		S 3		S 4
	d	n	d	n	d	n	d	n
	36	28
	41	30	39	46	41	58	43	71
	46	31	46	43	52	57
	53	29	59	42
	T	118	131	115		71

d = date of replacement; n = number of replaced RCPs at each date; T = total number of replaced RCPs.

ACKNOWLEDGMENTS

The authors acknowledge Dr. Antoine Rouhan, Dr. Kefei Li, and Yoann Lahorgue for their helpful contributions to this work.


Christophe Danoij

Pierre-Yves Hrcherii

BEHAVIOR OF UN CEMENTED SANDS AND GROUNTED SANDS BEFORE PEAK STRENGTH

Keywords: bender elements, grout injection, improvement, stiffness, strength 111

The results of triaxial tests carried out on both uncemented sands and sands injected by a microfine cement grout are presented in order to highlight and compare the main features of their mechanical behavior. The device used to prepare homogeneous samples of grouted sands in the laboratory is first described. The effect of the grout injection on the strength properties of three granular soils is then shown. By use of bender elements, the different evolutions of the shear modulus Gvh ,max in the small strain domain along isotropic and deviatoric stress paths are presented for both uncemented and grouted sands.

INTRODUCTION

The grout impregnation consists in injecting, under pressure, a fluid grout which seeps through the porous medium. The setting of the microfine cement which composes the grout gives rise to stable bonds between soil particles. Thus, the grout injection involves both a mechanical improvement and a decrease of the soil permeability. The improvement of the mechanical properties resulting from the grout injection has not always been considered in a rational way in structural design: the improvement was greatly underestimated when it was not simply ignored . This could be explained by ambivalence about the quality of the grout impregnation (is the cement content homogeneous in the soil mass?) and by lack of knowledge about the mechanical behavior of grouted sands, particularly in the small and intermediate strain domain (t::;;l0- 3 ). This paper intends to provide material for an answer to this question.

GROUT INJECTION IN THE LABORATORY

Injection Device in the Laboratory

Work site conditions and spatial variability of the geotechnical properties of many soils, even on a local scale, have generally not been conducive to an extensive academic study of the behavior of grouted sands sampled in situ. Therefore, an injection device was developed in the laboratory in order to prepare homogeneous samples of grouted sands. The injection device used in this study (Fig. 1) was inspired by many previous systems described Fig. 1. The injection device in the laboratory in the literature [START_REF] Zebovitz | Injection of fine sands with very fine cement grout[END_REF][START_REF] Benhamou | Comportement rheologique des cou!is de liants hydrauliques ultra-fins destines a !'injection[END_REF][START_REF] Bennabi | Application d ' un produit polymere ala consolidation des sables carbonates[END_REF][START_REF] Tailliez | Etude experimentale du comportement mecanique des sols granulaires injectes[END_REF].

This device is composed of a cylindrical, rigid and transparent plastic tube. It is different from previous devices [START_REF] Zebovitz | Injection of fine sands with very fine cement grout[END_REF][START_REF] Benhamou | Comportement rheologique des cou!is de liants hydrauliques ultra-fins destines a !'injection[END_REF][START_REF] Bennabi | Application d ' un produit polymere ala consolidation des sables carbonates[END_REF][START_REF] Tailliez | Etude experimentale du comportement mecanique des sols granulaires injectes[END_REF] by its height (900 mm) and by its diameter (100 mm) chosen so that it is at least ten times larger than the maximum diameter of the sand particles.

The tube is filled with sand first dried and deposited by successive layers of about 100 mm. Each layer is then manually compacted by means of hammer strokes on the tube. Their number and their intensity determine the final density index Id such as:

I. _ Ymax X Y-Ymin d-~- y Ymax-Ymin (1)
' 1 Associate Professor, Civil Engineering Laboratory of Nantes Saint-Nazaire, Ecole Centrale de Nantes, BP 92101 , 44321 NANTES (Dano@ecnantes.fr). where y is the unit weight of the soil; Ymin its minimum unit weight and Ymax its maximum unit weight (Table 1). These two parameters Ymin and Ymax were experimentally determined following the French Standard NF P 94-059. The repeatability of the filling method was satisfactory enough for density indexes greater than 85%: the scattering of the mean density was lower than 5% from one tube to another. The patented microfine cement grout is then injected upwards with a constant rate of discharge equal to 6 em~ /s. During injection, the mass of the grout injected into the tube, the grout pressure at the level of the pump and the height of the injection front are recorded. A linear evolution of these three parameters as a function of time provides a good indication about the homogeneity of the grouted sand over the full height of the tube [START_REF] Dano | Comportement mecanique des sols injectes[END_REF]. The grouting process is stopped when a volume of grout equal to 1.2 times the initial volume of voids of the sand is injected. Tubes are finally kept in water for at least 28 days.

Additional Con trol of the Injection Process

It is also possible to assess the homogeneity of the grouted sand 28 days after the injection in the laboratory, either by carrying out mechanical tests (destructive unconfined compressive tests, non destructive wave propagation tests) or by obtaining an "image" of the internal structure of the grouted sand. A density profile by gammadensimetry was thus established on a Fontainebleau sand injected by a microfine cement grout characterized by a cement to water ratio C/W of 0.172 (Fig. 2). Its mean unit weight Ymean was 20.4 kN /m The successive unevennesses observed in the profile can be imputed to particle segregation caused by the manual compaction. Indeed, the distance between two density peaks corresponds to the height of a sand layer introduced into the tube (100 mm). However, this heterogeneity remains negligible enough since the variation around the mean unit weight Ymean is only of ± 2%. The injection procedure is also perfectly repeatable [START_REF] Dano | Comportement mecanique des sols injectes[END_REF].

• • • • • A~uvia~ •; • • • Deposits :•--~'--l'>t/ AD I

MECHANICAL BEHAVIOR OF UN CEMENTED AND GROUTED SANDS

Materials

Three soils were examined: -a siliceous sand (NE 34 Fontainebleau sand) previously mentioned; -a first class of alluvial deposits of the Seine River (Paris, France) noted AD I; -a second class of alluvial deposits of the Seine River noted AD II. These alluvial deposits are uncemented silico-calcareous sandy gravel. After being dried, the three soils were subjected to characterization tests in order to determine their respective grain-size distribution curves (Fig. 3) and their minimum and maximum unit weights (Table 1). D , stands for the diameter of the particles corresponding to a passing of x%.

The patented micro fine cement grout contains water, a microfine cement whose particle diameter is less than 12 ,t.tm, an inert charge which increases the stability and fluidity of the grout, and an additive which vouches for the homogeneity of the grout.

Experimental Set-Up

Drained consolidated triaxial tests on dry uncemented and grouted sands were carried out in order to quantify the improvement of their mechanical properties. The evolution of the elastic properties in the small strain domain was more particularly followed. However, such a strain level cannot be reached in conventional triaxial tests because of the error sources listed by [START_REF] Scholey | A review of instrumentation for measuring small strains during triaxial testing of soil specimens[END_REF]. In order to remedy these problems, the load cell was placed inside the triaxial cell and bender elements (piezoelectric transducers) were embedded in the lower and upper platens of the triaxial cell [START_REF] Dyvik | Lab measurements of Gmox using bender elements[END_REF][START_REF] Brignoli | Measurement of shear waves in laboratory specimens by means of piezoelectric transducers[END_REF].

The propagation of the shear wave produced by the bender elements ~}wolves very small strains (e ::-:;10-5 ) into the soil [START_REF] Dyvik | Lab measurements of Gmox using bender elements[END_REF]. Therefore, an elastic behavior can be assumed in order to identify the value of the shear modulus Gvh ,max in the small strain domain:

Gvh ,max=pX v~h.s=pX (~f [START_REF] Baig | Low strain shear moduli of cemented sands[END_REF] where p is the unit mass of the soil; Vvh,s the velocity of the shear wave; L the distance of propagation of the shear wave and Tits travel time. The subscript "vh" stands for a wave vertically propagated with horizontal particle displacements.

For the calculation of the distance L, the penetration length of the bender elements into the soil (2 x 2.5 mm) is subtracted from the full height of the sample (about 200 mm) (Viggiani and Atkinson, I 995). The evolution of the distance L during shearing is followed on an external micrometer. It is well-known that the external measurement of sample shortening is not accurate because of bedding errors. However, in the case of soft uncemented sands, it induces a relative error on the value of the specimen shortening lower than 10%. On the contrary, in the case of stiffer grouted sands, the results must be taken with precaution since the relative error on specimen shortening can reach 30% and even more [START_REF] Scholey | A review of instrumentation for measuring small strains during triaxial testing of soil specimens[END_REF], even if this does not qualitatively question the evolution of the shear modulus presented in the next sections.

Also note that the repeatability greatly depends on the coupling between the grouted sample and the bender elements. Indeed, a perfect contact is necessary between the sample and the transducers to transmit and receive the shear waves without any signal attenuation in addition to the one due to the propagation itself. Therefore, a small cavity is made out on both sides of the specimen. The cavity is filled in with fresh gypsum. When a good contact is obtained, the scattering of the value of Gvh,rnax remains quite low (Fig. 8). It is about 10% in the stress range in which the value of Gvh,max is relatively constant. Finally, volume changes of the sample were determined from the volume of the cell fluid exchanged with a volume-pressure iii e:

4 4 c. c Ill OJ 0 'iii 2 2 - '5 ~ c. Ill .= 0 0 0 Q) c :5 -2 -2 ~ 0 OJ -4 -4 t5• '0 ::s ~ -6 -6 !2.
a. provided that these volume changes are corrected from the change in volume due to the penetration of the piston inside the cell. The determination of the travel time T of the shear wave is less obvious. Indeed, the shear wave is never pure in that near-field effects cause the shear wave to be accompanied by a faster compression wave. The multiple events related to the wave propagation (diffraction and reflection on the grains), the finite length of the transmitter, the parallelism defect between the transmitter and the receiver, the contrast between the acoustic impedances of the soil and the transducers, the spectrum of the input signal can explain the difficulty in interpret the outputing signal. [START_REF] Brignoli | Measurement of shear waves in laboratory specimens by means of piezoelectric transducers[END_REF] have discussed the conditions necessary to attenuate the near-field effects. Following their recommendations, the first deviation of the output signal (point 0 in Fig. 4) corresponds to the arrival of a compression wave due to near-field effects. The shear wave reaches the receiver afterwards, at point 1 in Fig. 4. This corresponds to the first change in the sign of the signal derivative [START_REF] Viggiani | Interpretation of bender element tests[END_REF][START_REF] Jovicic | Objective criteria for determining Gmax from bender element tests[END_REF].

Behavior at Large Strains-Strength Parameters

Figure 5 represents the stress-strain curves obtained by carrying out drained triaxial tests on uncemented and grouted alluvial deposits AD I. Similar trends have been observed for Fontainebleau sand and alluvial deposits AD II. In particular, the maximum strength is reached for lower strains for grouted sands than for uncemented sands. In both cases, the maximum strength depends on the mean stress (Fig. 5).

In the (p'-q) diagram (Fig. 6) where p' is the mean stress and q the deviator stress, the maximum strength envelopes of the alluvial deposits AD I, before and after injection, appear to be linear for confining pressures between 0 and 400 kPa. The friction angle of the grouted sand is slightly greater than the one of the uncemented sand. But the difference can be explained by the compaction of the granular skeleton during the grouting. Indeed, tests on uncemented and grouted Fontainebleau sand with the same density index for the granular skeleton sand reveal that the maximum strength envelopes are almost parallel. This means that the friction angle is hardly changed by the grout injection since low injection pressures do not disturb the granular skeleton. Consequently, the grout injection mainly shows the apparition of a cohesion whose value depends on the nature and the cement to water ratio C I W of the grout as wel1 as the nature, the density and the grading curve of the soil [START_REF] Tailliez | Etude experimentale du comportement mecanique des sols granulaires injectes[END_REF][START_REF] Dano | Comportement mecanique des sols injectes[END_REF]. It was found that the cohesion follows a power law of the cement to water ratio C/W such as:

( C)"' '

c'=m, x W (3)
where m, and m2 are two constant values that mainly depend on the density index.

In both cases, the volumetric behavior is first contracting, then dilating. The values of the Poisson's ratio v identified in the contracting portion of the volumetric strain-deviatoric strain curves are typically between 0.2 and 0.3. The values of the dilation angle 111 are always greater for grouted sands than for uncemented sands, for a given mean stress. For the same density index /d of 85% , the values of the dilation angle remain between 5 and 16 degrees for uncemented sands and between 20 and 30 degrees for grouted sands. These values decrease when the mean stress is increased.

However, it is worth noting that differences between uncemented and grouted sands can be observed in the evolution of the volumetric strain. In grouted sands, the behavior is contracting up to about the maximum strength and then becomes strongly dilatant. In uncemented sands, the dilatant behavior is obtained before the peak deviator stress, for a state of stress corresponding to the characteristic state as defined in Luong (1980). For grouted sands, the maximum rate of volumetric strains no longer coincides with the peak deviator stress, as for uncemented sands. The pre-failure behavior of grouted sands is then mainly contracting.

The values of the tangent modulus £,." were also deduced from the initial part of the stress-strain curves. They correspond to a strain level around 10-3 • For uncemented sands, the tangent modulus clearly depends on the mean effective stress p ' in such a way that:

( p')"

Etan= kX Pb [START_REF] Benhamou | Comportement rheologique des cou!is de liants hydrauliques ultra-fins destines a !'injection[END_REF] where k is the value of the modulus when p' = p 0 and n is a constant. The power n is between 0.5 and 1 depending on the strain level considered for the determination of the tangent modulus: for a strain level equal to 10-3 , the value of the power n is very close to 1. The effect of the mean effective stress is less obvious for grouted sands and can be clarified by the use of bender elements.

Behavior at Small Strains

Previous research studies [START_REF] Hardin | Elastic wave velocities in granular soils[END_REF][START_REF] Iwasaki | Effects of grain size and grading on dynamic shear moduli of sands[END_REF]Hieber, 1996) showed that the shear modulus Gvh,max in the small strain domain for uncemented sands (Fig. 7) follows the empirical correlation:

( p' )" Gvh,max=KXf{e) X Po (5a)
where f(e) is a function of the void ratio. Like (Hieber, 1996), the function f(e) = 1/e is considered.

Tests on grouted sands {Fig. 8) reveal that the ex- Stress deviator q {kPa) Deviatoric stress paths.

Fig. 8. Evolution of the shear modulus G,o,mn in the case of grouted Fontainebleau Sand Table 2. Values of the parameters K and n (Eq. ( 5))

--------------------,-------------------------

( p')n Gvh,max =K' X Po +Go ( 5b
)
where G0 is the value of the modulus at a confining pressure equal to zero and K' a constant. Figure 7 shows the evolution of Gvh ,max in the case of alluvial deposits AD I for isotropic and deviatoric stress paths. Table 2 contains the values of the parameters K and n (Eq. (Sa)) characterizing the three uncemented soils used in this study. The values of the parameter K is typically between 4 and 6 whereas the value of the power n is close to 0.5, as previously stated by many authors [START_REF] Hardin | Elastic wave velocities in granular soils[END_REF][START_REF] Iwasaki | Effects of grain size and grading on dynamic shear moduli of sands[END_REF]Hieber, 1996). However, the power law is accurate only for isotropic loading or unloading and in the contracting domain for a deviatoric stress path. Indeed, beyond the characteristic state, that is to say the state separating the contracting domain from the dilating domain, the values of the shear modulus Gvh,m•• tend to stabilize, then to decrease smoothly (Fig. 7). This shows the influence of the rearrangement of the contacts between particles of the granular skeleton (Shibuya et al., I 992;[START_REF] Kuwano | Shear stiffness anisotropy measured by multi-directional bender element transducers[END_REF]. Along deviatoric stress paths, Eq. (Sa) should be modified in order to distinguish the effect of the major principal stress O'l and the effect of the minor principal stress a3 as usually done in the literature (see for instance [START_REF] Roesler | Anisotropic shear modulus due to stress anisotropy[END_REF].

For grouted sands, the values of the shear modulus Gvh ,max in the small strain domain are at least ten times greater than the values of Gvh,max for the uncemented sands at the same density index. These values mainly de-pend on the density of the granular skeleton (A car and El Tahir, 1986; [START_REF] Lade | Cementation effects in frictional materials[END_REF], on the cement to water ratio of the grout [START_REF] Baig | Low strain shear moduli of cemented sands[END_REF][START_REF] Dano | Comportement mecanique des sols injectes[END_REF] and the filling percentage of voids [START_REF] Schwarz | Effect of preparation technique on permeability and strength of cement-grouted sand[END_REF]. The effect of the curing period, the degree of saturation, the strain rate, and so on was not investigated here but information can be found in the literature relative to cemented sands [START_REF] Shibuya | Elastic deformation properties of geomateria1s[END_REF][START_REF] Barbosa | Effects of stress state during curing on stress•strain behavior of cement-mixed sand[END_REF]. The effect of the overconsolidation ratio was not studied either: [START_REF] Vucetic | Effect of soil plasticity on cyclic response[END_REF] showed that its effect was neg]igible in the case of granular materials without fine content.

However, the evolution of Gvh,max with loading for grouted sands is radically different (Fig. 8). On the one hand, for isotropic stress paths, the dependence of the shear modulus on the mean effective stress p' is much less pronounced. The values of the power n introduced in Eq. {5) are between 0 and 0.1 for sands injected with a grout of cement to water ratio less than 0.235. This observation is similar to the conclusions of experimental programs on cemented sands by Chang and Woods (1987) using a resonant column apparatus and by [START_REF] Baig | Low strain shear moduli of cemented sands[END_REF] using bender elements.

On the other hand, for deviatoric stress paths, the shear modulus Gvh,ma., while relatively constant over a quite restricted strain domain, progressively decreases afterwards. This observation allows us to suppose that damage is here the prevailing mechanism which develops until failure. The formation of vertical cracks on the samples for strains about 2 or 3% tends to confirm this assumption.

Discussion

In the small strain domain, the friction component of the behavior that appears through the dependence of the shear modulus on the mean effective stress for uncemented sands, is strongly concealed by the cementation due to grout injection. The experimental results match the theoretical work by [START_REF] Fernandez | Effect of cementation on the small-strain parameters of sands[END_REF] who showed that the effect of the mean stress could be toned down for small cement contents. This theoretical work also predicts a constant value of the shear modulus in the range of small stresses. The experimental results are also akin to the experimental work by [START_REF] Chang | Effect of confining pressure on shear modulus of cemented sand[END_REF] who concluded that the value of the power n (Eq. ( 5)) for a sand-cement mixture was probably between the value of n for the pure cement paste and the value of n for the granular skeleton. Isotropic compressive tests on pure grout samples with a cement to water ratio equal to 0.172 were performed. The shear modulus G vh,max in the small strain domain is 0.55 GPa (+I-0.05 GPa). This value remains constant over a large domain of mean effective stresses, between 0 and 450 kPa. It was thus verified that: n{grout) ""'0~ n(grouted sand)~ n(granular skeleton)"" 0.5. This finally means that the effect of the grout injection on the stiffness of the granular material is less and Jess significant as the mean effective stress increases.

CONCLUSIONS

From triaxial tests carried out on uncemented sands and on sands injected in the laboratory with a microfine cement grout, the effect of the cementation on the strength and stiffness properties is shown. The use of bender elements enabled distinguishing different evolutions of the shear modulus in the small strain domain along isotropic and deviatoric stress paths. It should be noted that the values of the tangent modulus determined from the triaxial stress-strain curves underestimate the real values of the elastic properties. Improvement ratios defined as the ratio of the shear modulus after injection on its value before injection are greater than 10 whereas, until now, only improvement ratios of about 2 have been considered in structural design. Finally, improvement ratios depend on the mean effective stress, on the soil characteristics and on the cement to water ratio of the grout.

Fig. 3 .

 3 Fig. 3. Grain-size distribution curves of the three soils

Fig. 4 .

 4 Fig. 4. Input and output signals obtained with the bender elements

Fig. 6 .

 6 Fig. S. Comparison of the behavior of alluvial deposits AD I before and afler grout injeetion

Fig. 7 .

 7 Fig. 7. Evolution of the shear modulus G,h ,max in the case of the uncemented Fontainebleau Sand

Table 1 .

 1 Characteristics of the materials

		Material			D!O /liD	Dlo /liD
		Fontainebleau Sand			160	220
		Alluvial deposits AD I			240	410
		Alluvial deposits AD II		340	1300
		Microfine cement			<I	3.5
		80			Fontainebleau Sand
					I =95% ~
	u ._, --E	60			Grout : lntra-J CM/=0.172
	.E :::r:: 'iii en	40
		20
		O ~LULU~~~UiLuLU~~~d±~
		-2 -1.5 -1	-0.5 0	0.5	1.5	2
		((y-y	)ly	)x100

mean mean

Fig.

2

. Density profile of a sample of grollted Fontainebleau sand


The Relative Complexity of Constraints in Co-Predicative Utterances

Bruno Mery1 IUT de Bordeaux, Université de Bordeaux, France & LaBRI -CNRS bruno.mery@me.com

Abstract-We explore the particular variations of semantic felicity in co-predicative utterances the constraints on the combination of facets of polysemous words, and the possible adaptations that can be proposed to existing formal frameworks that support lexical semantics. As the linguistic data is incomplete and disputed, we also include a proposal for a linguistic survey aimed at clearing up many outstanding issues. 1 1 The Generative Approach to Lexical Semantics and Polysemy

The Issue of Polysemy in Compositional Semantics

The inarguable facts that words in human languages can be employed with many different meanings, according to the context they are used in, is seemingly at odds with the principle of compositional semantics inherited from Montague, which presupposes that singular meaningful term can be provided for every lexical item. This can be resolved by Word Sense Disambiguation techniques (associating the most probable meaning to each word before composition, using data refined from corpora). Another approach is to hold polysemy to be an important feature of language, and to re-design the compositional semantics around it. In this approach, the lexicon becomes much more complex than a map of words to meanings, as each lexical entry can be combined with others in order to select the appropriate sense for a word used in a specific context: this is the so-called Generative approach to lexical semantics developed in [START_REF] Pustejovsky | The Generative Lexicon[END_REF] and many subsequent works. We will discuss a few basic features of this approach in Section 1.2.

In the present paper, we will study the specific issue of the felicity of co-predicative utterances. Those constructs are detailed in Section 2.1, the issues of felicity in Section 2.3, and we will focus on the characterisation of constraints on felicity in Section 3, and the necessary adaptations of formal semantics in Section 4.

1.2 GL Qualia, Dots, Processes and Uses [START_REF] Pustejovsky | The Generative Lexicon[END_REF] and subsequent works such as [START_REF] Asher | Lexical Meaning in Context: a Web of Words[END_REF] have detailed the implications on Montagovian semantics of relational polysemy, distinguishing (among many other phenomena) four kinds of meanings that are distinct, but directly related (by opposition to accidental polysemy, which consists of different words that happen to be homonyms).

The qualia are derived from the "modes of explanation" of Aristotle; the idea is that a noun for something can refer to :

1. The thing itself (formal quale, as in long sword), 2. A salient part of the thing (constitutive quale, as in dull sword → dull edge), and, for artificial items, 3. Their creation process (agentive quale, as in master's sword → sword made by a master blacksmith), 4. Their use, for objects that have one (telic quale, as in fine sword → sword fine for fighting).

Pustejovsky argues that such uses are extremely common, and that a competent speaker of a language has access to a qualia, thus justifying that such information should be part of the lexicon itself.

Complex objects, called dot-objects (sometimes written •-objects) are words that denote a singular concept that can be envisioned on two or more different facets. Canonical examples include books (with physical and informational facets), meals (with an event and a food facet), newspapers (with an organisational and a physical aspect), etc. There are many examples which seem to stem from compounding essential and existential information. As the number and type of facets is not determined, contrary to qualia, and as those facets enjoy differentiated individuation conditions, they have proved to be one of the most serious difficulties for formal models of lexical semantics.

In addition, some implicit processes can shift the meaning of a word referring to something to different states: grinding can turn materials into artefacts, animals or plants into food (delicious salmon), packaging can provide implicit containers for masses (there is water and wine on the table) -see [START_REF] Mery | Computing the Semantics of Plurals and Massive Entities using Many-Sorted Types[END_REF] for details. Deverbal nouns are notoriously polysemous between process and result (the construction is at the end of the street) -see [START_REF] Jacquey | Ambiguïtés lexicales et traitement automatique des langues : modélisation de la polysémie logique et application aux déverbaux d'action ambigus en franc ¸ais[END_REF], [START_REF] Real-Coelho | A generative montagovian lexicon for polysemous deverbal nouns[END_REF].

Finally, additional facets can be bestowed upon a lexical item via explicit constructions, which are easier to account for. They include as-phrases (I do not have issues with Mr. X as a lawyer; however, as a candidate. . . ) and for-phrases (this knife is sharp enough for shaving).

The Many Facets of Co-Predicative Sentences

Co-Predications

Co-predicative utterances, phrases or sentences, or co-predications for short, consist in the explicit reference to two (or more) different facets of the same lexical items at the same time. The most classical example is heavy and interesting book, but there are many others. The difficulty of co-predications is that the two predicates apply to different types (here, to physical and informational objects respectively). Lexical semantics frameworks that do not consider this and simply coerce the type of the argument to the one required by the predicate fail to provide a suitable typing for book in such sentences.

Mixing Polysemy Facets

In the original formulation by Pustejovsky, co-predications were mostly studied between facets of dot-objects. However, they can be equally valid between various sources:

two or more qualia:

• good, expensive wine (telic+formal);

• fast blue car (constitutive+formal). . . ; two or more facets of a dot-object:

• red closed door (physical+aperture);

• liberal, picture-less tabloid (organisation+physical). . . ; a facet of a dot-object and a qualia of another facet:

• I have an inspired article in my briefcase (agentive quale of the informational facet + physical facet); process, result, and other facets:

• the translation, printed right after its lengthy completion, is considered bold yet naive (physical and agentive quale of the informative facet of the result + process); an arbitrary number of references to different facets in a coherent discourse:

• The book is 5lbs, and has 400 pages. It has been set in Times 12, with large margins. Its leather cover is aged but sound. Its writing took five years, and the completion of the hundred in-quarto printings four months. [. . . ] It certainly is an interesting read.

Felicitous and Infelicitous Co-Predications

There are thus many possible combinations of facets that can be predicated on simultaneously. However, some co-predications are infelicitous. We have elaborated previously on the case of grinding : * The salmon was lightning-fast and delicious, many examples of process/result alternations: * The construction took three months and stands tall, as well as capital-governments alternations: * Washington is an old American city and has denounced Teheran. 2 3 Characterising Constraints on Felicity

Relaxable and Fixed Constraints

The most evident constraints are the following: facets that can cognitively co-exist for the same object are not constrained in their co-predications, while facets that are only present after a transformative process (grinding, packaging, resultatives) are exclusive to all others.

In addition, a proper name of a polysemous entity (such as a city or place) can be used as a paraphrase for a specific group of people (such as the name of the capital being a proxy for the national government), to the exclusion of all other possible senses. We can thus distinguish between two rough classes of facets: flexible and rigid.

However, the rigidity constraint can be relaxed in some cases. A syntactic split of the components of the co-predications can render some infelicitous phrases acceptable : the salmon, which was very fast, is delicious, or the construction, that took three months to complete, stands tall. Beyond the syntactical separation, the use of different tenses or explicit contrast using yet or but can also relax the constraints on the use of facets that correspond to different, exclusive states of the same entity.

Capital/government (and related) alternation constraints are much more difficult to relax, considering: Washington is an old American city. It is located on the banks of the Potomac. * It has denounced Teheran..

That distinction has given us the three classes of constraints that we have been using until now, flexible facets, semi-rigid facets (the constraints can be relaxed via syntactic means) and rigid facets (the constraints are fixed).

City Names: a Short Case Study

However, this does not cover the whole complexity of constraints on the felicity of co-predications. See [START_REF] Real-Coelho | A generative montagovian lexicon for polysemous deverbal nouns[END_REF] for a case study on deverbal co-predications; we will discuss here the specific case of city names.

City names are highly polysemous, with many possible facets that can be separated in two groups. The first is composed of facets that pertain to the city as a whole: architecture, location, size, atmosphere, climate, population, lifestyle. . . that we will call characteristics, and the second, of facets we will call essentials, that use the name of the city to refer to a specific group of people, such as city/metropolis/regional/national government and sport clubs. In our first approximation, characteristics are flexible and essentials are rigid.

However, the following constructions can allow some of the essentials to copredicate:

lexical proximity -if two facets share the same lexical field, as in Barcelona dominates Europe in football as well as in handball; object identity -if the predication is made on a single direct object, as in Barcelona dominates Europe in football as in architecture; zeugma -if zeugma are considered acceptable, as in Barcelona dominates Europe in football as they do Madrid in politics;

discourse flow -logical relations in the narration can provide felicitous readings, as in Detroit, by lake Michigan, has filed for bankruptcy; the city has become desolate and lifeless. -The above can be daisy-chained in order to co-predicate over seemingly incompatible facets in a long discourse: Bordeaux, struggling with traffic issues due to its growing population and its geographical position on both sides of the Garonne, wishes to build a new bridge in order to ease commuting between those.

Evaluating the Idiosyncrasies of Felicity in Co-Predications

We want to stress that felicity is an highly subjective notion, that varies from speaker to speaker. Additionally, it appears that some of the constraints are idiosyncratic; anecdotical evidence, including separate personal communications from Asher, Lecomte and Luo indicate that some common co-predications are felicitous in English but not in Chinese. While this should not stop formal frameworks to develop mechanisms that integrate constraints on co-predications, establishing a robust catalogue of the actual usages is important.

We would like to propose a set of templates for sentences that can be adapted to valid local city data, with co-predications we consider felicitous, infelicitous, and forced by the various mechanisms evoked above, as well as straightforward predications on a single facet as controls. This survey should be conducted by native speakers of several languages in different linguistic groups (minimally Romance, Germanic and East-Asiatic), and we would like it to be part of collaborative international projects that are being constructed as a result of the recent interest in lexical semantics.

A Proposal for Linear Types and Terms

Constraints on co-predications can be added to current lexical frameworks as an external mechanism3 . However, it is also possible to allow for every possible combination with a redesign of the formalism. Our proposal incorporates a λ -calculus whose terms are typed with formulae of the second-order linear intuitionistic logic.

Linear Λ TY n is adapted from our framework in the following way:

-Simple predications are implemented by linear application (heavy is H ϕ t ).

-Basic morphisms (type transformations) also (selecting the physical facet is done via Λ α. f α ϕ ). -Lexical transformations (giving access to specific facets of each term) are implemented as pairs, the first component being an accessor to the facet (a type transformation, as above) and the second assessing the compatibility of the facet with other transformations.

The present abstract is intended as a discussion material, detailing a specific issue in order to foster collaborations around the subject. Having been prepared by the author in his spare time, it is, by necessity, incomplete. The work presented here is closely related to Λ TY n , a formal framework for lexical semantics developed since

with Christian Bassac, Richard Moot, Christian Retoré and many other researchers at LaBRI and LIRMM. As a team, we would like to thank everyone that has joined in TYTLES and engaged in discussions and research around type-theoretical lexical semantics.

Those examples are subject to personal interpretation and linguistic idiosyncrasies that we will discuss in Section

3.3.

It is sufficient to keep track of the flexible/rigid characteristic of terms, relaxing the constraints when appropriate using a set of rules that detect syntactic or discursive features, and to stop the composition when incompatible co-predications are detected; this is the approach we have proposed so far, see[START_REF] Retoré | The Montagovian Generative Lexicon Lambda Ty n : a Type Theoretical Framework for Natural Language Semantics[END_REF] for a recent synthesis in our Λ TY n framework.

-Additional terms, that modify this compatibility, can be provided by syntax, discourse and pragmatics in order to relax the constraints when appropriate.

A detailed account of this proposal is given in [START_REF] Mery | Lexical Semantics with Linear Types[END_REF].

Summary

While the work presented here is still in progress, we are convinced that the issue of constraints on co-predication is of importance in the establishment of precise natural language semantics. We hope to gather enough linguistic data in order to characterise the extent of the phenomena, and have the necessary formal tools to treat it accurately. Our goal remains to integrate issues of lexical semantics and polysemy directly in the flow of analysis of natural language, from syntax to semantics and logical representation.