（節(jié)選）橋梁外文翻譯--鋼筋混凝土結(jié)構(gòu)老化的破壞程度概率模型（原文）

1、Reliability Engineering and System Safety 93 (2008) 410–422Probabilistic models for the extent of damage in degrading reinforced concrete structuresB. Sudret?Electricite ´ de France, R Space-variant reliability; Tim

2、e-variant reliability; Degradation models; Concrete carbonation; Rebars corrosion; Random fields; EOLE method1. IntroductionProbabilistic models of concrete degradation have been intensively studied in the past 10 years.

3、 The most important degradation mechanism considered in the literature is the corrosion of the rebars due to chloride ingress in the concrete mass or concrete carbonation. This mechanism is of utmost importance in the ag

4、eing of bridge structures that are submitted to deicing salts, or any structure in a marine environment [1–4]. Authors have focused on the prediction of the initiation time for corrosion and/or the estimation of the resi

5、dual strength of structures. Recent advances in this field have pointed out the necessity of modelling the spatial variability of the model parameters in order to be able to characterize, not only the probability of degr

6、adation, but also the extent of damage[5–7]. This extent of damage is the natural variable that characterizes the global state of damage of the structure, and that may be used in optimizing maintenance policies [5,8–10].

7、 In this paper, a general formulation for spatially variable degradation models is proposed. The so-called point-in- space and space-variant reliability problems are recalled in Section 2 [11]. Then the extent of damage

8、is given a proper definition, from which analytical derivations are carried out in order to compute its first two statistical moments (Section 3). Efficient implementations of these formulae (based on the first order rel

9、iability method (FORM) and Monte Carlo simulation (MCS)) are proposed in Section 4. In order to evaluate the accuracy of the analytical approach, an alternative framework for the direct estima- tion of the extent of dama

10、ge by MCS is proposed. This requires the use of random field discretization techniques and the post-processing of the simulation results. Both approaches (called ‘‘a(chǎn)nalytical’’ and ‘‘field discretization’’ARTICLE IN PRES

11、Swww.elsevier.com/locate/ress0951-8320/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2006.12.019?Tel.: +33 1 60 73 6559; fax: +33 1 60 73 7748. E-mail address: bruno.sudret@edf.fr.the

12、failure state (fully damaged). This is of course a coarse simplification of the real world. Moreover, this does not allow to characterize the extent of damage. To address this issue, additional notation shall be introduc

13、ed. Suppose that the structure under consideration occupies a volume D ? Rd, where d ¼ 1; 2 or 3. The case d ¼ 1 corresponds to modelling beam or arch structures, the case d ¼ 2 to plate or shell structure

14、s. In order to address the problem of spatial variability, the input random vector in Eq. (2) should be replaced by M multivariate scalar random fields gathered in a vector ZðxÞ, where x 2 D is the spatial coor

15、dinate. The probabilistic description of these fields is yet to be specified. Note that in practice the following assumptions usually apply:? The spatial variability of certain components of Z is negligible. They are acc

16、ordingly modelled as random variables. As a consequence, only a small number of scalar random fields have to be specified in practice. However, for the sake of simplicity, the most general notation ZðxÞ is kept

17、 in this section. ? The random field components are homogeneous fields. This is due to the fact that the size of the structure is usually small compared to the scale of fluctuation of the parameters driving the degradati

18、on (e.g. environmental parameters such as surface chloride or carbon dioxide concentration, etc.).In the space-variant context, the limit state function in Eq. (4) should be replaced bygðZðxÞ; tÞ 

19、8; ¯ D ? MðZðxÞ; tÞ. (5)The point-in-space probability of failure is defined in each x 2 Rd as follows:Pf ðx; tÞ ¼ZgðZðxÞ;tÞp0 f ZðxÞðzÞ dz

20、¼ E½1fgðZðxÞ;tÞp0gðz; tÞ?. (6)It is computed by freezing x (i.e. replacing the random field ZðxÞ by the corresponding random vector) and t, and by applying standard time-

21、invariant reliability methods (MCS, FORM/SORM, etc.). Note that if the random field ZðxÞ is homogeneous, then the same reliability problem is posed at whatever the position of the point x under consideration. T

22、hus, the point-in-space probability of failure is independent of x in this case. The space-variant probability of failure is defined, for any subdomain H ? D by [11]:Pf ðH; tÞ ¼ Pð9x 2 H; gðZ

23、0;xÞ; tÞp0Þ¼ P [x2H gðZðxÞ; tÞp0!. ð7ÞThis quantity is the ‘‘spatial’’ counterpart of the so-called cumulative probability of failure in time-variant reliability problems

24、 [15]. When the damage measure is related to serviceability of the structure (e.g. apparition of cracks or rebars loss of diameter)and not directly related to the collapse of the structure, none of the above quantities a

25、re sufficient to characterize the global state of ageing of the structure. Indeed, Eq. (6) is by definition a local quantity (at point x). Eq. (7) refers to the probability that there is at least one point in subdomain H

26、 where the local damage criterion is attained. This probability is likely to be close to one, without meaning that the structure is close to structural failure. In contrast, the extent of damage is of major interest, esp

27、ecially for the comparison of maintenance policies, see e.g. [5,8].3. Extent of damage3.1. DefinitionThe extent of damage is defined at each time instant t as the measure of the subdomain of D in which the local failure

28、criterion is attained:EðD; tÞ ¼ZD 1fgðZðxÞ;tÞp0gðxÞ dx. (8)Note that EðD; tÞ is a scalar random variable since the integral over x is defined for each realization of

29、 the input random field, say zðxÞ. It is positive-valued and is by definition bounded by the volume of the structure in Rddenoted by jDj. Again, due to the monotony of degrada- tion phenomena, each realization

30、of EðD; tÞ, say eðD; tÞ is a continuously increasing function of time.3.2. Mean and varianceBy taking the expectation of Eq. (8) (i.e. with respect to Z), one gets the following expression for the mea

31、n value of the extent of damage:EðD; tÞ ? E½EðD; tÞ? ¼ZD E½1fgðZðxÞ;tÞp0gðx; tÞ? dx. (9)By comparing the integrand of the above equation with Eq. (6), one

32、getsEðD; tÞ ¼ZD Pf ðx; tÞ dx. (10)In case of homogeneous input random field, this integrand is independent of x, as explained above. ThusEðD; tÞ ¼ Pf ðx0; tÞ ? jDj ð

33、Homogeneous caseÞ, (11)where the point-in-space probability of failure is computed at any point x0 2 D. The above equation has the following interpretation: the proportion of the structure where the damage criterion

34、 is attained (i.e. EðD; tÞ=jDj) is, in the mean, equal to the point-in-space probability of failure. This remark has two important consequences:? It is not necessary to introduce the complex formalism of random