Model uncertainty and Bayesian updating in reliability-based inspection

Structural Safety 22 (2000) 145±160

www.elsevier.nl/locate/strusafe

Model uncertainty and Bayesian updating in reliability-based inspection Ruoxue Zhang, Sankaran Mahadevan * Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235, USA

Abstract In this paper, a Bayesian procedure is proposed to quantify the modeling uncertainty, including the uncertainty in mechanical and statistical model selection and the uncertainty in distribution parameters. The procedure is developed ®rst using a simple example and then is applied to a fatigue reliability problem, with the combination of two competing crack growth models and considering the uncertainty in the statistical distribution parameters for each model. This Bayesian failure probability analysis can be incorporated with information from nondestructive inspections performed on the structure to derive more realistic reliability estimates. The procedure for updating the mechanical model, probabilistic model, distribution parameter statistics and reliability is illustrated for the fatigue reliability problem. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Bayesian method; Uncertainty; Fatigue; Fracture; Reliability; Non-destructive inspection

1. Introduction In recent years, much attention is being paid to the deterioration of nation's infrastructure. One of the important mechanisms of deterioration is the fatigue e€ect on structures subjected to repeated or cyclic load patterns. The design fatigue reliability can be estimated by combining reliability analysis methods with fatigue life or fracture mechanics models. In-service, non-destructive inspections are required at regular intervals and the results can be used for maintenance to mitigate fatigue risk. Moreover, the data obtained from the inspection can be combined with these models to update the reliability estimate during the remaining service life using Bayes' theorem. Several studies have been reported to incorporate the information from inspection to update fatigue reliability. Madsen [1,2] developed the idea of updating failure probability using the information from non-destructive inspection (NDI) with the Bayesian approach. Zhao and Haldar [3,4] extended this method, investigated the e€ect of the uncertainties of detection on updating, * Corresponding author. Tel.: +1-615-322-3040. E-mail address: [email protected] (S. Mahadevan). 0167-4730/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0167-4730(00)00005-9

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and used the updated reliability index for inspection schedule, maintanence and repair decisions. Bayesian updating has also been applied by Byers et al for fatigue reassessment [5]. Zheng and Ellingwood [6,7] incorporated a time-dependent noise term to the fatigue crack growth model to deal with a wide-band load process and considered the interaction of corrosion and fatigue/fracture damage, and used the outcome of NDI to update the distribution of crack size. Uncertainty in engineering analysis arises from three types of sources: (1) Physical uncertainty or inherent variability; this is generally quanti®ed by a probability distribution estimated from observed data. (2) Statistical uncertainty; this refers to the uncertainty in the statistical distribution parameters of the random variables identi®ed in the ®rst source, due to the scarcity in the data. (3) Modeling uncertainty; this includes uncertainty in both probabilistic and mechanical models. In each case, the uncertainty exists in model accuracy as well as model selection. In the case of probabilistic models, the accuracy in distribution parameter estimates is already accounted for by statistical uncertainty. In addition, there are approximations in the computational procedures, and the extent of these approximations is uncertain. In the case of mechanical models, model accuracy relates both to mathematical idealization of behavior, as well as approximations in the numerical solution procedure. Uncertainty in model selection is given special emphasis in this paper, and is addressed with regard to both probabilistic and mechanical models. In the case of probabilistic models, this may refer to the choice of distribution type for the random variables. In the case of mechanical models, this refers to the mechanical or mathematical model adopted to describe the behavior of the system or component (e.g. a model describing the strength of a deteriorated component). In the previous studies of fatigue, all the uncertainty analysis and updating are performed with an assumed fatigue crack growth model. Refs. [1±7] used the Paris' crack-growth model, which is commonly used for explicitly considering the crack size, while Refs. [6,7] adopted a corrosionincorporated model which is based on Paris' model. The statistical uncertainty is considered in these studies. Actually, in time-dependent reliability analysis, part of the uncertainty also derives from the selection of the degradation model (mechanical model) due to the complexity of the problem and lack of experimental data. For the fatigue problem, there exist several empirical equations to describe typical fatigue crack growth behavior in metals [8±12], one of which is Paris' law [8]. Paris' law is most commonly used in fatigue analysis because of its simplicity. It only considers the intermediate region of crack growth and assumes that the crack growth rate depends only on the stress-intensity range. In the present study, we consider two more comprehensive models that account for region II and III of crack growth and incorporate the e€ect of stress-intensity ratio derived by Foreman [9] and Weertman [10]. Both are empirical equations containing di€erent parameters to be obtained through tests. Since there is no literature to demonstrate which one is better, it is dicult to decide which one to use. In this case, we encounter the problem of model uncertainty as it refers to uncertainty in the mechanic model. This problem can be solved with the Bayesian approach because Bayesian analysis provides a framework to incorporate all kind of uncertainties in the analysis. It allows mechanical model uncertainty, probability model uncertainty and statistical uncertainty, to be considered alongside the underlying physical uncertainty. Edwards [13] and Guedes Soares [14] already used the Bayesian approach to combine several competing probability distribution types to describe a random variable. As to the reliability estimation

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problem, this paper shows that the Bayesian framework can not only consider multiple competing distribution types and multiple possible sets of parameters within each distribution, but can also simultaneously consider multiple competing limit state formulations possibly appropriate for the same problem. This bene®t of the Bayesian approach has not been explored in earlier studies and is the subject of this paper. In this paper, the Bayesian approach is used for a more realistic and comprehensive fatigue reliability analysis, with the combination of multiple possible crack growth models as well as the statistical parameters associated with each model. The Bayesian approach can combine preexisting knowledge with subsequent available information and update the prior knowledge of all the uncertainty. As more and more information becomes available, one tends to get more accurate model and parameter distribution information for the problem. Therefore, the Bayesian approach not only facilitates the updating of distribution parameters (as done in earlier studies) but also helps to improve the model selection. 2. Bayesian description and updating of model and parameter uncertainties According to Guedes Soares [14,15], for a particular problem, in principle only one model is the correct one. However in the state of uncertainty about which one is correct, the use of all possible models minimizes the expected losses that would result from choosing the wrong model. Through predicting the extreme values of signi®cant wave height, Guedes Soares demonstrated that the use of all the available information through the Bayesian prediction shows a much smaller variability than the use of individual model [14]. Depending on how the various models are weighted, di€erent ®nal values will be obtained but their range of variation is much smaller than in the case of individual predictions. Edwards [13] also used the Bayesian approach to study a structural system subjected to a loading that could be described by either a normal, a log-normal or a Weibull distribution. It should be indicated that the Bayesian approach is extremely valuable in cases that deal with rare events, since the data available to test the hypothesis about probabilistic models are very scarce and do not allow de®nite conclusions to choose or discard totally one model among several competing models. Thus, in situations where physical, model and statistical uncertainties are all potentially of equal importance, an intuitively appealing and more logical approach would be to use a ``weighted average'' of all possible models, and all possible parameter sets within each model. The method to describe and update these uncertainties is formulated as follows. The Bayesian approach is associated with a particular model Mi and in fact it is conditional on the assumption that the model is the correct one. If various models are possible candidates, the Bayesian probability of event X that incorporates both parameter and model uncertainty is: … m X P…Mi † P…X i ; Mi †f…i Mi †di …1† P…X† ˆ iˆ1

i

in which P…Mi † refers to the model uncertainty, and is the prior probability assigned to model I. i ˆ fi1; i2;:::::: in gT refers to the vector of distribution parameters within Mi if Mi is a probabilistic model; if Mi is a limit state model, i is the vector of random variable in that model. f…i jMi †

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is the prior distribution of parameters within the ith mode and P…Xji ; Mi † is the conditional probability of event X associated with Mi. Let P(Hi) denote the prior probability of model/parameter pair (Mi, i ) P…Hi † ˆ P…Mi †  f…i jMi †

…2†

After getting an observation x, the posterior probabilities are then given by Bayes' theorem as: P…Hi jx† ˆ P…Mi jx†  f…i jMi ; x† ˆ

m P iˆ1

P…xjHi †
f…i jMi †P…Mi † „ P…Mi †  P…xji ; Mi †f…i jMi †di

The posterior probability of Mi is obtained by integrating f…i jMi †: „ P…Mi †   P…xjHi †
f…i jMi †di P…Mi jx† ˆ m „ P P…Mi †  P…xji ; Mi †f…i jMi †di

…3†

…4†

iˆ1

Then the posterior distribution of i in Mi is given by dividing Eq. (3) by Eq. (4): f…i jMi ; x† ˆ „

P…xjHi †
f…i jMi †  P…xjHi †
f…i jMi †di

…5†

It can be seen from Eq. (4) that the parameter updating in model Mi is independent of P…Mi † and is updated in the same way as when only model Mi is assigned to the problem. Furthermore, using the posterior model probability and parameter distribution, we can get the Bayesian expectation of any event E related to Hi : … m X P…Mi jx† Pi …EjHi †f…i jMi ; x†di P…E† ˆ iˆ1



…6†

A simple example is given below to illustrate this updating procedure. 2.1. Example There are two opinions about the distribution of a variable T: . Model 1: Exponential distribution f1 …t l† ˆ leÿlt ; t50, in which l is also a random variable with gamma distribution: f…l† ˆ leÿl ; l50 . Model 2: Gamma distribution f2 …t † ˆ 2 teÿt ; t50, in which parameter is also gamma distributed f…† ˆ eÿ ; 50 Both distributions are obtained from experiments but performed by di€erent researchers. Suppose before any new observation has been taken, we have no idea which model is better, so

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we assign an equal probability to each (such a prior is known as ``uniform'' or ``non-informative''), i.e. …1 …1 f…t† ˆ P…M1 † f1 …tjl†f…l†dl ‡ P…M2 † f2 …tj†f…†d 0 0 …7† 2 6t ˆ P…M1 † ‡ P…M2 † …t ‡ 1†3 …t ‡ 1†4 where P…M1 † ˆ P…M2 † ˆ 0:5, which represents model uncertainty; f…l† and f…† denote the statistical uncertainty of the parameters in each model. After one new observation t=1.5 is obtained, we get fT …1:5† ˆ 0:1792 from Eq. (7). According to Eqs. (4) and (5), the posterior probabilities of the models, and the corresponding parameter density functions may be obtained as: 2 …1:5 ‡ 1†3 ˆ 0:357 0:1792

P…M1 †  P…M1 jt ˆ 1:5† ˆ

6  1:5 …1:5 ‡ 1†4 ˆ 0:643 0:1792

P…M2 †  P…M2 jt ˆ 1:5† ˆ

and f…ljM1 ; t ˆ 1:5† ˆ

l2 eÿ2:5l 0:128

f…jM2 ; t ˆ 1:5† ˆ

1:53 eÿ2:5l 0:2304

Now consider two special cases: 1. Single model: Only model 1 is considered. In that case, only the distribution of parameter l is updated after t=1.5 is observed: f…ljt ˆ 1:5† ˆ

l2 eÿ2:5l 0:128

with the same result as updated in model 1 in the two-model case. 2. No parameter uncertainty: Suppose l and v are deterministic constants, l=v=1, therefore only the model weights are updated: P…M1 jt ˆ 1:5† ˆ 0:4; P…M2 jt ˆ 1:5† ˆ 0:6

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3. Fatigue reliability problem In the previous example, di€erent probability distribution types were incorporated in the overall model and their weights were updated using the Bayesian approach. This belongs within statistical model updating. The same theorem can be applied to physical/mechanical model incorporation and updating. In this section, the fatigue reliability of steel bridge components is considered. Crack growth modeling based on linear-elastic fracture mechanics (LEFM) is addressed for fatigue reliability analysis because it can explicitly incorporate crack size and information from NDI. 3.1. Crack growth model Fig. 1 is a schematic log-log plot of da/dN vs.
K, which illustrates typical fatigue crack growth behavior in metals. In Fig. 1, a is crack size, N is the number of stress cycles, da/dN is the crack growth rate;
K is the stress intensity range. It is observed that the physical process of fatigue can be divided into three phases: the crack-initiation phase (I), the subcritical crack-propagation phase (II) and the critical crack growth phase (III). Paris and Erdogan [16] describe the intermediate region of crack growth with a power law, which became widely known as the Paris law: da ˆ C…
K†m dN

…8†

where C and m are material constants that are determined experimentally. The stress intensity range
K is:
K ˆ Kmax ÿ Kmin

…9†

According to LEFM theory, K can be estimated as: p
K ˆ Kmax ÿ Kmin ˆ F…a; Y† a…max ÿ min †

Fig. 1. Typical fatigue crack growth in metals.

…10†

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where  is the far-®eld stress, F…; Y† is the geometry function to account for the possible stress concentration, Y is the vector of random variables, such as the stress concentration coecient and the dimensions of the specimen under consideration. Paris law only considers the stress intensity range and ignores the e€ect of stress intensity ratio: R ˆ Kmin =Kmax . Many researchers agree that this simple expression is not universally applicable [17]. This conclusion is supported by the observation from Fig. 1 that
K is actually sigmoidal rather than linear when crack growth data are obtained over a suciently wide range. Also, the fatigue crack growth rate exhibits a dependence on the R ratio de®ned above, particularly at the extremes of the crack growth curve. Based on these considerations, Foreman [9] proposed the following relationship: da C…
K†m ˆ dN …1 ÿ R†Kcrit ÿ
K

…11†

where Kcrit is the fracture toughness of the material. The crack growth rate becomes in®nite as Kmax approaches Kcrit. Weertman [10] proposed an alternative equation: da C…
K†m ˆ 2 dN Kcrit ÿ K2max

…12†

Both the Foreman and Weertman equations account for the e€ect of the R ratio, and both are asymptotic to Kmax ˆ Kcrit . They are adopted to calculate the fatigue reliability in the present study. Note that the material constants C and m in Foreman's equation do not have the same numerical values or units as in the Weertman's equation or the Paris equation. Eqs. (11) and (12) are generally expressed as: da ˆ f…
K; R† dN

…13†

This equation can be integrated to estimate the fatigue life. The number of cycles required to propagate a crack from an initial size, a0, to size a is then given by: …a da ˆN …14† f…
K; R† a0 3.2. Fatigue reliability analysis The fatigue failure criterion for a structure subjected to N stress cycles can be de®ned as: ac ÿ aN 40

…15†

where aN is in service crack size corresponding to N stress cycles and can be obtained by Eq. (14); ac is the critical crack size. It can be de®ned as the size of the crack causing failure or a design crack size beyond which the serviceability requirements cannot be satis®ed. Eq. (15) is adopted as

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the limit state function of the fatigue reliability problem considered herein. The corresponding failure probability is de®ned as: Pf ˆ P‰g…Z†40Š ˆ P‰ac ÿ aN 40Š

…16†

To perform reliability analysis, the uncertainties involved must ®rst be de®ned and quanti®ed. In the present study, the sources of uncertainties include: (1) selection of the crack growth model, corresponding to the limit state function; (2) random parameters associated within each crack growth model. As stated earlier, Eqs. (11) and (12) both account for the sigmoidal nature of
K and the e€ect of the R ratio, therefore both are considered candidate models in the present study. Since no literature is available to show which one equation is better or more realistic (it is likely that it is dependent on the application considered), it is dicult to decide which model is to be chosen to obtain more realistic results. This problem can be solved by the Bayesian approach developed in the previous section. The same prior weight (probability of the model being the right one) is assigned to each model, and the failure probability is rewritten as: Pf ˆ P…M1 †Pf1 ‡ P…M2 †Pf2

…17†

where P…M1 † ˆ P…M2 † ˆ 0:5; Pf1 is the failure probability calculated using Eq. (11) and Pf2 is the failure probability using Eq. (12). It should be further noted that the same concept can be applied in selecting the distribution type of the variables associated with each model. In each limit state function corresponding to the two crack growth models, there exist several random variables that are described with probability density functions. If for a certain variable, there are di€erent opinions about its distribution, this becomes a problem of multi-layered model uncertainty. The reliability analysis can be performed with the multi-layered Bayesian description in the same way as in the earlier section, with a multilayered updating procedure when new information becomes available. 4. Model updating with information from NDI During the lifetime of a steel structure, periodic nondestructive inspections (NDI) are essential and important for fatigue damage evaluation and for scheduling maintenance and repair. Also, they provide an opportunity to obtain data on fatigue crack growth in service. The NDI technique has uncertainties in itself. Two sources of uncertainty of ultrasonic NDI, the most commonly used technique are considered herein: detectability and accuracy. Detectability refers to the fact that there is always a crack size ad for a given NDI technique below which a crack can not be detected. Measurement accuracy means that the measured crack size may not represent the actual crack size due to the measurement error. The detailed characterization of these uncertainties has been discussed in [3,7,18], and is not the subject of this paper. In [3], the minimum detectable crack size is considered to have a lognormal distribution with a mean value of 2.54 mm and a coecient of variation (COV) of 0.50. The accuracy of the crack size measurement is also modeled by a lognormal distribution with the measured size as the mean and a COV of 0.25.

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Though NDI contains itself uncertainties, (which have been considered and quanti®ed in earlier studies) and whether any crack is detected or not, each inspection provides additional information and leads to changes of the prior estimated reliability as well as the uncertainty in the model and the basic random variables. The Bayesian method is used in this study to update the information on the model weights and the statistical characteristics of random variables in each model and the updated information is then used to calculate the posterior reliability index, using Eq. (5). Two types of results of an inspection are considered below. 4.1. Case 1: No crack detection This means that the actual crack size aN at the time of inspection (corresponding to N stress cycles) is smaller than the minimum detectable crack size ad for a given inspection technique. The event can be expressed as D corresponding to: D : I40

…18†

where I ˆ aN ÿ ad Using the Bayesian approach, the model weight and probability distribution of the jth random variable xj,i associated with model i are updated as: P…Mi †  P…Ii 40† P…Mi D† ˆ 2 P P…Mi †P…Ii 40†

…19†

iˆ1

FXj;i



 P…X ÿ x 40 T I 40† j;i j;i i xj;i D ˆ P…Ii 40†

…20†

where in Ii, N is obtained from the ith crack growth model, according to Eq. (10) or (11). With these posterior model weights and parameter probability distribution, the failure probability can be easily updated using Eq. (5). However, the failure probability can alternatively be updated directly using the posterior model weight P…Mi D† as: ÿ
ÿ
T T
P g1 40 I1 40 P g 40 I 40 2 2 ‡ P…M2 jD† …21† Pf;up ˆ P…M1 D P…I1 40† P…I2 40† The two updating methods of Eqs. (19)±(21) provide the same result for the updated failure probability. If one is only interested in the failure probability or reliability index of the component, Eq. (21) is preferred, thus avoiding the procedure of updating parameters. However, for the problem of system reliability of large structures, fatigue will cause the reduction of resistance capacity of a single component as well as change the resistance of the entire structure [19]. Hence, to perform the time-dependent system reliability analysis of aging structures, the parameters distribution updating using Eq. (20) will be useful and bene®cial.

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It should be noted that if the outcome of NDI is just crack detection without size measurement, the event is: D : I50 with the similar updating procedure as that of no crack detected, thus is not further described in details herein. 4.2. Case 2: Crack detection with crack size A The event of a crack detected with size A is expressed as DA: aN ÿ A ˆ 0

…22†

Using the Bayesian approach, the model weight and parameter distribution are updated as: @Pi …aN ÿ A40† jaN ˆA P…Mi †  @aN P…Mi DA † ˆ 2 P @Pi …aN ÿ A40† P…Mi † aN ˆA @aN iˆ1 ÿ

FXj;i xj;i jDA




\ @ Pi …Xj;i ÿ xj;i 40 aN ÿ A40†jaN ˆA @aN ˆ @Pi …aN ÿ A40† aN ˆA @aN

…23†

…24†

The updated failure probability is obtained as:

Pf;up

  \ @ Pi gi 40 aN ÿ A40 jaN ˆA 2 2 X X @a ˆ P…Mi jDA †P…gi 40jDA † ˆ P…Mi jDA † N @Pi …aN ÿ A40† iˆ1 iˆ1 jaN ˆA @aN

…25†

Every time an inspection is performed, the reliability and model uncertainty are updated and corresponding decisions are made on inspection schedule, repair, rehabilitation, etc., of the structure. 5. Steel bridge application A simply supported steel bridge is considered in this study. The span has two full-penetration butt welds in its tension ¯ange. The fatigue reliability of these full-penetration butt welds is considered. Following Zhao and Haldar [20], it is assumed that the butt welds are subjected to train

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loading of 112 stress cycles per day. This represents approximately 2,000,000 stress cycles for a design life of 50 years. The width of the tension ¯ange is 1.0 m. To perform reliability analysis, the uncertainty of the basic random variables in the limit state function associated with the two crack models must ®rst be quanti®ed. According to [20], since the crack growth rate is very high near the critical crack size, the e€ect of the critical size ac on the fatigue failure is rather small compared to other random variables involved in the entire fatigue damage process. Thus, ac can be considered to be a deterministic parameter. For illustrative purposes, the critical crack size is considered to be 0.05 m in this example. Then the initial crack size, geometry function, load process and material properties are considered herein. The distributions most frequently suggested in modeling the uncertainty in the initial crack size are the lognormal and the Weibull distributions. The lognormal distribution tends to be more conservative since it predicts a higher probability of large cracks. Agerskov [21] indicated 0.075 and 0.4 mm as the lower and upper bounds of initial crack size of welded joints of steel bridges, while Mohaupt [22] and Albrecht [23] suggested that the mean value of a0 be 0.5 mm. Therefore there is uncertainty in the mean value and variance of a0, which also can be modeled by a ``weighted'' combination of possible probability distributions. Thus, there are two competing distributions for the random variable a0 used in both mechanical models. This becomes a problem of multi-layered model uncertainty. As mentioned in Section 3.2, this problem can be addressed with a multi-layered Bayesian updating procedure. However, for the sake of simplicity, in this example, a0 is modeled by a lognormal variable with mean of 0.5 mm and a COV. of 0.50. As to the geometry function, it is generally recognized that geometry functions are a function of the crack size and a group of other random variables such as the stress concentration coecient and the dimension of the specimen. Closed-form models to express the geometry functions are available only for special cases. The geometry function proposed by Paris [8] to treat the case of center-notched specimens of bridges is used in this study as an illustration: F…† ˆ

1 ÿ 0:5 ‡ 0:372 ÿ 0:0443 p 1ÿ

…26†

where  ˆ 2a=w and w=width of specimen (in this case, it is the width of the tension ¯ange, 1.0 m). It is generally accepted that for bridge structures, the loading process can be approximately modeled as a stationary narrow band Gaussian process, and the corresponding stress follows a normal distribution. A normal distribution of the stress with a mean value of 60 MPa and standard deviation of 25 MPa is assumed in this study for the purpose of illustration. Using random process theory, the density function of maximum and minimum peaks of the stress random process are obtained as:   …max ÿ † …max ÿ †2 exp ÿ …27† f…max † ˆ Var 2Var   …min ÿ † …min ÿ †2 exp ÿ …28† f…min † ˆ ÿ Var 2Var where is the mean stress and Var is the variance of stress.

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For the material property, suppose we perform a group of tests and get the fracture toughness Kcrit of the steel used for the bridge to be 80 MPa m1/2. A group of discrete data on da/dN vs.
K is also obtained through experimentation. With the least-squares ®t method, the material property parameters C and m are estimated for the Foreman and Weertman crack growth model separately. For the sake of comparison, C and m in Paris equation are also estimated. Fig. 2 is the log-log plot of a single test data and the ®tted Paris, Foreman and Weertman curves. Assume that through multiple tests, the mean value and variance of C and m in three models are found and given in Table 1 . It is generally accepted that for steel, C approximately follows a lognormal distribution while m follows a normal distribution. After all the uncertainties are addressed and quanti®ed, Monte Carlo simulation is applied to calculate the failure probability using Eqs. (14) and (16). Three fatigue crack growth models are considered separately and the corresponding failure probabilities are plotted in Fig. 3. To investigate the e€ect of the R ratio on the reliability results in di€erent models, two mean stress values of 60 and 80 MPa are applied and the corresponding failure probabilities is also shown. It is seen in Fig. 3 that there is a considerable di€erence in the failure probability estimate using the three crack growth models with the statistical parameters obtained by the same tests, especially between the Foreman model and Weertman model, both of which are the candidates for our application. It is also found that increasing the mean stress value (i.e. increasing the R ratio) will decrease the fatigue reliability. However, compared to the Weertman model, the failure probability is much less sensitive to the change in R ratio for the Foreman model. Due to the considerable

Fig. 2. Fatigue crack growth test data and the ®tted curves. Table. 1 Statistical characteristics of C and m in each crack growth model

Paris equation Foreman equation Weertman equation

C

COV of C

m

COV of m

2.0710ÿ12 6.8610ÿ10 1.010ÿ7

0.50 0.455 0.59

3.1 2.0 2.12

0.10 0.19 0.16

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di€erence between these two models, choosing one of them will possibly cause either too conservative or too dangerous result. In the situation when it is not known which one is better, Eq. (17) is applied to get the Bayesian estimation of the failure probability. The result is plotted in Fig. 4. Suppose the full-penetration butt weld has been inspected at about N=500,000 and three inspection results are considered respectively: (1) no crack detection; (2) crack size A=0.8 mm is detected; (3) crack size A=1.2 mm is detected. Using Eqs. (18)±(25), the model weights and failure probability corresponding to the number of cycles are updated. The updated failure probabilities for the three cases are also plotted in Fig. 4, along with prior failure probability. It can be observed that the updated failure probability is smaller than the prior one in the case of no

Fig. 3. Failure probability calculated with di€erent crack growth models.

Fig. 4. Update failure probability with di€erent inspection results.

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Table 2 Update of model weight after inspection

P(M1) (Foreman) P(M2) (Weertman)

Before inspection

No crack detected

A=0.8 mm

A=1.2 mm

0.5 0.5

0.501 0.499

0.432 0.568

0.391 0.609

crack detection, which indicates that the structure is more reliable than estimated previously. The failure probability increases when a crack is detected. The larger the crack detected, the more the probability of structural failure. All these results are expected. Table 2 shows the update of model weights after inspection. In the case of no crack detection, model weights almost have no change. This is because that the calculated probability of event D at stress cycle N=500,000 is very large (more than 99%) using either model. Therefore, in Eq. (4), the individual conditional probability of D in the numerator is approximately equal to the denominator, leaving the updated weight equal to the original weight. It is also observed that the weight of the Weertman model increases as the measured crack size increases. This is also an expected result since comparatively the Weertman model is more conservative than the Foreman model. In this example, the two crack growth models with the same complexity are considered as the possible models, for the purpose of illustration. One may argue that since Paris' model is adopted more widely, it should also be included. This is quite easy with the Bayesian approach. In that case, there will be three models instead of two in Eq. (17), with the same procedure of reliability estimation and updating. The updating procedure is repeated when the next inspection is performed, with the result updated the previous time as the new prior value. After some inspections, more and more information is available, the weight shifts to the more accurate model. If the updated result after several inspections shows that the weight of one model is extremely high, this model may be selected as the more realistic one and adopted for the reliability estimation for the remaining life of the structure. Using the information on the updated failure probability, a decision can be made on the next course of action, as to reducing the next scheduled inspection interval, whether to repair or do nothing, for maintenance of the structure in a reliable and economic way. 6. Conclusion This paper developed a multi-layered Bayesian approach to comprehensively include all the available information on physical, model and statistical model uncertainty. The procedure to update the information with new data is formulated. For the fatigue problem considered in this paper, the Bayesian approach combines multiple mechanical models and statistical models and gives a more reasonable, comprehensive prediction of reliability. In spite of the uncertainties existing in itself, NDI provides information to update the quanti®cation of all the uncertainties, including model weight, parameter distributions and failure probability (or reliability index). To the best of the authors' knowledge, the present work is the ®rst study of the combination of di€erent

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competing mechanical models using the Bayesian framework. This is especially useful when different models producing signi®cantly di€erent results are possible candidates, and there is not enough reason to prefer one over the other. The model weights can be updated and tend towards the more accurate model as new information becomes available. Acknowledgements This paper is based on work funded by National Science Foundation under Grant No. 9872342 (Program Director Dr S.C. Liu). The support is gratefully acknowledged.

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