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Indicators for inspection and maintenance planning of concrete structures
Structural Safety 24 (2002) 377–396 www.elsevier.com/locate/strusafe
Indicators for inspection and maintenance planning of concrete structures M.H. Fabera, J.D. Sorensenb,* a Swiss Federal Institute of Technology, Zu¨rich, Switzerland Department of Building Technology, University of Aalborg, Sohngaardsholmsvej 57, 9000 Aalborg, Denmark
b
Abstract Based on an idea introduced by Benjamin and Cornell (1970. Probability, statistics and decision for civil engineers. New York: McGaw Hill) and previous works by the authors it is demonstrated how condition indicators may be formulated for the general purpose of quality control and for assessment and inspection planning in particular. The formulation facilitates quality control based on sampling of indirect information about the condition of the considered components. This allows for a Bayesian formulation of the indicators whereby the experience and expertise of the inspection personnel may be fully utilized and consistently updated as frequentistic information is collected. The approach is illustrated on an example considering a concrete structure subject to corrosion. It is shown how half-cell potential measurements may be utilized to update the probability of excessive repair after 50 years. Furthermore in the same example it is shown how the concept of condition indicators might be applied to develop a cost optimal maintenance strategy composed of preventive and corrective repair measures. # 2002 Published by Elsevier Science Ltd. Keywords: Condition indicators; Concrete; Reliability; Corrosion; Bayes; Degradation; Half-cell potential measurements; Assessment; Maintenance; Inspection; Planning
1. Introduction Condition control of existing structures by means of inspection and testing forms a corner stone in the maintenance management of structures. It is generally accepted that the information collected by inspection and testing is the basis for the estimation of the condition of the structure and moreover for the estimation of the future development of the condition of the structure; i.e. the future structural degradation.
* Corresponding author. Tel.: +45-96-35-85-81; fax: +45-98-14-82-43. E-mail address: [email protected] (J.D. Sorensen). 0167-4730/02/$ - see front matter # 2002 Published by Elsevier Science Ltd. PII: S0167-4730(02)00033-4
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Ideally the principle for the planning of inspection and testing activities is to minimize the service life economical risks, i.e. the sum of the costs of inspection and testing, the costs due to future maintenance and strengthening works and the costs due to future failures. To be able to plan inspection and testing efforts optimally in the above-mentioned sense it is necessary that the effect of inspections and tests can be related to those particular states of the structure, which impact the service life economical risks for the structure; i.e. the results of the inspections and tests must be related to the service life economical risks. Traditionally in applications such as bridge management the results of inspections and tests are expressed in linguistic terms such as ‘‘corrosion has initiated in xx% of the structure’’, ‘‘the membrane is defect’’, etc. This information is then processed and results in an overall grading by assigning a certain number (e.g. 1, 2, 3,..) to the overall condition of the structure. The further treatment of such information for the purpose of estimating the service life economical risk is less than obvious, and even though some systematic may be developed on this basis there is no theoretical basis ensuring that the information is consistently utilized for the purpose of maintenance management. Based on an idea introduced by Benjamin and Cornell [1] and previous works by the authors (Faber and Sorensen [2]) it is demonstrated how condition indicators may be formulated for the general purpose of quality control and for assessment and inspection planning in particular. The formulation facilitates quality control based on sampling of indirect information about the condition of the considered components. This allows for a Bayesian formulation of the indicators whereby the experience and expertise of the inspection personnel may be fully utilized and consistently updated as frequentistic information is collected. The approach is illustrated on an example considering a concrete structure subject to corrosion where it is shown how half-cell potential measurements may be utilized to update the probability of excessive repair after 50 years.
2. Information sampling As a basis for the further derivations it is assumed that the defect rate can be used as a measure of the general condition of a series of components. Defect rate is here understood in a broad sense as the rate of occurrence of any state of the considered components, which is of special interest. Components are considered for which the unknown defect rate is , i.e. the probability that a considered component is defective is and the probability that it is not defective is 1. On the basis of sample (inspection) results and subjective information the estimate of the defect rate can be updated and decision-making regarding the inspection and maintenance of the components may be undertaken on this basis. 2.1. Random sampling The usual sampling technique from classical quality control is to test (inspect) samples, which are taken randomly from the considered population. Here this approach is shortly described within a Bayesian statistical framework utilizing, to the highest degree possible, all available information.
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It is assumed that a well defined population with constant defect rate is considered. n components are selected randomly from the population and inspected with the result that nF (4n) components are defective. Letting the random variable Y model the number of defect components nF in a random sample of size n, Y is binomial distributed with probability density function given by n y fY ðy j Þ ¼ ð1 Þny ð1Þ y However, the defect rate is not known with certainty and must hence be considered as an uncertain variable . Initial information regarding the defect rate is modelled by the prior probability density function f 0 ðÞ, see e.g. Lindley [3]. If no initial information is available or if the initial information is very uncertain a diffuse prior may be used, i.e. f 0 ðÞ / 1. Through the prior probability density function it is possible to incorporate subjective information regarding the general defect rate. Having inspected a sample of n components with the result that nF components are defective, this information can be utilized to update the available information regarding . The updated probability density function (the posterior probability density function) for can be derived on the basis of Bayes formula. The posterior probability density is denoted f 00 ð j Y ¼ nF Þ and expresses the updated knowledge about the general defect rate. The posterior probability density function is given by 1 f 00 ð j Y ¼ nF Þ ¼ fY ðnF j Þ f 0 ðÞ c
ð2Þ
Ð1 where c is a normalizing constant such that 0 f 00 ð j Y ¼ nF Þd ¼ 1 The posterior probability density function for the defect rate may readily be used in the decision analysis, see e.g. Raiffa and Schlaifer [4] either on its own or by using it together with Eq. (1) whereby the predictive probability density function for the number of defective components may be achieved as ð1 n y ð3Þ ð1 Þny f 00 ð j Y ¼ nF Þd fY ðy j Y ¼ nF Þ ¼ y 0 2.2. Sampling based on indicators Inspection and maintenance planning of engineering systems such as structural systems, pipelines and process systems is complicated by the fact that the systems often are very large, in terms of the number of components. Furthermore, the components of the systems often belong to a larger number of different populations. Application of classical quality control procedures for the quantification of the effect of inspections on such systems is thus hardly practical and hence not used in practice. In practice, inspections are performed on a rather limited number of components. The components inspected are typically selected on the basis of criteria such as criticality, expected condition, inspectability and experience. Following such schemes little possibility is left to evaluate the effect of the sample size (number of inspected components) and even less possible to quantify and justify the rationale behind the inspection scheme.
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In the following it is hence attempted to outline a statistical framework for an alternative approach based on Bayesian statistics and so-called indicators. Initially the case is considered where knowledge is included in the estimate of the defect rate. Letting Di denote that indicator no. i is inspected and indicates a defect. It is assumed that prior information may be represented by the probabilities: PðF j Di ¼Þpi : the probability that the inspected component is defective (F) given the observation no. i indicates a defect. that indicator P F j Di Þ ¼ qi : the probability that the inspected component is defective (F) given the observation that indicator no. i does not indicate a defect. The number of components for which the indicator has been inspected and found to indicate a defect is denoted ND and the number of components inspected where the inspected indicator does not indicate a defect is denoted ND . On the basis of these probabilities and information regarding the total number of inspected components the probability density function for the defect rate may be updated. The updated probability density function is denoted f 000 ðÞ. In f 000 ðÞ both prior information regarding and results of random inspections may be included. f 000 ðÞ is the updated probability density function for the general defect rate when all information is included (prior information, results of random samples and results of targeted samples). The probability density function may be updated approximately by ð4Þ f 000 ð j Di Þ ffi c f 00 ð j F Þpi þ f 00 ð j F Þð1 pi Þ for the case where the inspection indicates a defect and f 000 ð j D i Þ ffi c f 00 ð j F Þqi þ f 00 ð j F Þð1 qi Þ
ð5Þ
for the case where the inspection indicates that there is no defect, c is a normalising constant. As input for the updating information regarding the probabilities pi and qi is required. If these are not known with certainty Bayesian statistics may also be applied to model this. pi and qi are then modelled as uncertain variables and their uncertainty, modelled by the prior probability density functions, may be updated when information is available. The derivation of Eqs. (4) and (5) may be realised from T T T T P D F þ P D F Pð j DÞ ¼
¼
PðDÞ T T P D j F Pð j FÞ PðFÞ þ P D j F P j F P F PðDÞ PðD j FÞ Pð j FÞ PðFÞ þ P D j F P j F P F
ffi
PðDÞ ¼ PðF j DÞ Pð j FÞ þ P F j D P j F
ð6Þ
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The approximation introduced in Eqs. (4) and (5) may thus be expressed by the assumptions that PðDi j F \ ¼ Þ ffi PðDi j FÞ
ð7Þ
P Di j F \ ¼ ffi P Di j F
ð8Þ
which means that it is assumed that the probability of finding an indication of a defect, given that the component is defective (or not defective) is independent of the defect rate. This assumption appears to be non-restrictive for practical applications. If inspection results are available the probabilities f 00 ð j FÞ and f 00 j F may be determined by using f 00 ð j Y ¼ nF Þ as prior probability density function: f 00 ð j FÞ ¼
1 00 f ð j Y ¼ nF Þ c’
f 00 j F ¼
ð9Þ
1 ð1 Þ f 00 ð j Y ¼ nF Þ 1 c’
ð10Þ
Ð1 where c0 ¼ 0 f 00 ð j Y ¼ nF Þd is a normalising constant. If no inspection results are available f 0 ðÞ may be used as prior probability density function instead of f 00 ð j Y ¼ nF Þ. First the situation where the observation of ND indicate that that the component is defect is considered. The ND indicators are denoted Di ; i ¼ 1; ND . The updated probability density function is determined recursively by pi ð1 pi Þð1 Þ 000 ðiÞ f ð j D1 ; :::; Di Þ ¼ ci 0 þ f 000 ði1Þ ð j D1 ; :::; Di1 Þ; i ¼ 1; ::; ND ci 1 c0i ð11Þ where ci is determined such that c0i
ð1 ¼
Ð1 0
f 000 ðiÞ ð j D1 ; ::; Di Þd ¼ 1 and c0 i is a normalising constant:
f 000 ði1Þ ð j D1 ; :::; Di Þd
ð12Þ
0 0
Furthermore there is f 000 ð0Þ ðÞ ¼ f 00 ð j Y ¼ nF Þ or f 000 ð0Þ ðÞ ¼ f ðÞ. Secondly the situation is considered where observations of ND components do not indicate that the components are defective. The ND indicators are denoted D j ; j ¼ 1; ND . The updated probability density function is determined recursively by ! q ð1 q Þð1 Þ j j 000 ðj1Þ 1 ; :::; D j1 ; j ¼ 1; ::; N j D f f 000 ð j Þ j D 1 ; :::; D j ¼ cj 0 þ D cj 1 c0j ð13Þ
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In the above derivations it has been assumed as a prerequisite that the considered components belong to the same population. 2.3. Updating knowledge about indicators If inspections of the condition of interest are conducted fully for the components for which one or more indicators have been found to indicate a defect the result of the inspections can be used to update the probabilities pi ¼ PðF j Di Þ and qi ¼ PðF j D i Þ as described in the foregoing and by letting pi and qi be modelled by the random variables Pi and Qi . If more populations are considered pi and qi may be assumed to be the same for the same indicators in the different populations. In this case it is possible to update the statistical knowledge regarding pi and qi on the basis of one population and to use this updated knowledge on the other populations. If on the other hand pi and qi are different for a given indicator for different populations the improved knowledge regarding pi and qi can only be used for the indicator in the inspected population.
3. Inspection and maintenance planning for concrete structures The concept of indicators for quality control purposes as described in the previous section may be utilised in a wide range of applications where the quality or the reliability of components belonging to a larger population is considered. In civil engineering applications one such situation arises in connection with the maintenance management of concrete structures. Inspection and maintenance planning should be performed such that the expected total service life costs E½CT are minimised, i.e. E½CT ¼ E½CI þ E½CR þ E½CF
ð14Þ
where E½CI , E½CR and E½CF are the expected inspection costs, repair costs and failure costs respectively. The expected costs are to be determined as the probability that the cost-inducing event will occur multiplied with the cost consequences given the event. The expected total service life costs are to be minimised by means of planning the inspections for condition control and by means of choice between different repair strategies. Decision variables are thus
inspection method indicator of damage part or percentage of structure to be inspected repair method criteria for repair.
A large variety of approaches exist to assess the expected total service life costs. However, it is important that the approach selected resembles the characteristics of the degradation process and the way inspections and maintenance decisions are carried out in practice. In the following it is
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therefore investigated how the theory developed in Section 2 can be adapted for the purpose of inspection and maintenance planning for concrete structures. 3.1. Consequences of degradation and criteria for repair Deterioration of concrete structures in the context of inspection and maintenance planning may appropriately be described in terms of the different phases of the deterioration, which are associated with different consequences. Considering degradation due to corrosion, the first phase is the initiation phase at the end of which de-passivation occurs at the outer layer of reinforcement. Thereafter simultaneously with the de-passivation of the next layers of reinforcement the propagation phase follows in various steps. The characteristics and duration of these steps depend in general on the specifics of the concrete, the reinforcement, the concrete cover thickness, the humidity and the exposure conditions. At first the corrosion is initiated and corrosion products build up on the surface of the reinforcement. After a certain increase of the volume of the corrosion materials small cracks of the concrete cover will occur and again after some time the corrosion products may disperse through the cracks and become visible on the surface of the structure. If the corrosion products continue to build up on the surface of the reinforcement the concrete cover may eventually fall off (spalling) leaving the structure even more exposed to the environment and thus accelerating the deterioration processes for the next layer of reinforcement. Ultimately the degradation may proceed until the load carrying capacity or the serviceability of the structure is reduced beyond acceptability. However, the consequences of degradation may also be reduced by means of condition control together with maintenance and repair activities. Due to the characteristics of the different phases of the degradation process the cost efficiency of maintenance and repair activities will be highly dependent on when and how these are implemented. If the progress of deterioration is detected at an early stage it may be possible by means of minor and inexpensive repairs to reduce the risk of major and expensive future repair works. For a large part of the normal reinforced concrete structures the load carrying capacity and thus the reliability of the structures in regard to the ultimate limit state condition will only be reduced beyond acceptability after a very significant degree of degradation has become visually observable. Such severe signs of deterioration are usually not acceptable alone for cosmetic reasons. Thus early signs of degradation such as cracking and colouring together with the results of various NDE techniques usually comprise the information basis about the condition of the structures for decision making in regard to the implementation of maintenance and repair activities. In practice repair decisions are often taken alone on the basis of the visually observable condition of the structures. For concrete structures where the likely location of degradation is known, i.e. structures with hot spots, the criteria for implementing a repair could thus simply be the occurrence of visible degradation. Similarly for concrete structures with distributed degradation, i.e. where in principle ‘‘all spots are hot’’ the criteria for implementing a repair could be the occurrence of visible degradation on more than a certain critical percentage of the surface of the structure. Given that the criterion for repair is fulfilled the degraded area(s) is repaired according to a pre-determined repair strategy. In the following it is shown how inspection and maintenance strategies for the two cases (1) localised degradation and (2) distributed degradation may be formulated using the concepts of condition indicators described in Section 2.
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3.2. Inspection and maintenance planning—localised degradation For concrete structures where the critical degradation is localised the planning of inspection and maintenance activities may be performed by assessing the expected service life costs of Eq. (14) by consideration of the event tree illustrated in Fig. 1. In Fig. 1 CV refers to the event of visual corrosion at the time of the inspection, CI refers to the event of initiated corrosion at the time of an inspection. ID refers to the event of a positive indication of initiated corrosion at the inspection. CV, CI and ID refer to the complementary events, respectively. The branching in the event tree takes place between the performed inspections. The event tree in Fig. 1 only illustrates how event trees may be developed for inspection and maintenance planning purposes. It is assumed that visual inspections are carried out at regular intervals, i.e. at times indicated by tV . If at any time during the life of the structure visual degradation is observed a major corrective repair R1 will be implemented. Furthermore it is assumed that the decision problem is to identify the first time where in addition to a visual inspection also a NDE of the structure should be carried out in order to detect and facilitate a minor preventive repair R2 . In the illustrated event tree it is for simplicity assumed that the branches leading to repairs do not contribute further to the expected service life costs. This assumption is in effect equivalent to assuming that repair is perfect, which surely is not the case in practice, however, data and experience still need to be collected and processed before a more realistic modelling of the durability of repaired concrete structures may be formulated. Given that probabilistic models are formulated for the event of initiation of degradation as well as for the event of visual degradation as a function of time the different repair probabilities in the event tree may be calculated using standard tools from structural reliability such as FORM/
Fig. 1. Illustration of an event/decision tree for inspection and maintenance planning of concrete structures subject to localised deterioration.
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SORM and simulation. However, due to the fact that NDE techniques only provide an indication about the real state of the structure the branching probabilities in the event tree after the time where the NDE is performed must take into account the quality of the applied NDE method. The quality of the applied NDE method may appropriately be expressed in terms of the probability that the NDE method indicates degradation given that degradation has initiated, i.e. PðID j CIÞ together with the probability that the NDE method indicates degradation given that degradation has not initiated, i.e. PðID j CIÞ. When assessing the probabilities of repair in the event tree at inspection times after a NDE has been performed the quality of the NDE has to be accounted for. Consider as an example the event of a major repair R1 at a visual inspection following immediately after an inspection using NDE. The branch in the event tree is considered where degradation in fact has initiated at the time of the NDE inspection but where no indication of initiated degradation was found (see figure 1). In this case the probability of the repair R1 at time tV3 can be assessed through \ \ \ \ P R1 tV3 ¼ P CV tV3 CV tP1 ð15Þ CI tP1 ID tP1 CVðtp1 Þ ¼ \ \ \ P CV tV3 CI tP1 CVðtp1 Þ P ID tP1 j CI CV tP1
Based on event trees as illustrated in Fig. 1 and probability assessments of the type in Eq. (15) the overall service life costs associated with inspection and maintenance may be calculated and the optimal strategy for combining preventive and corrective repairs may be established. 3.3. Inspection and maintenance planning—distributed degradation For concrete structures where degradation can be expected at any location with equal probability the spatial characteristics of the degradation process play an important role. One way to consider this aspect is to model the structure not as one large whole but rather as an assembly of many individual components, either as small structural (area) components or as zones of larger structural elements. The probabilistic characteristics of the parameters governing the degradation of the structure should also be modelled in this light. Studies in e.g. Hergenro¨der [5] indicate that such components could have a typical dimension of about 2 2 m. In order to assess the probability that a given percentage of the structure has achieved a critical degree of degradation it is in principle necessary to evaluate the following expression
PðdðtÞ 5 n=NÞ ¼ P
K mi N;n \ [
gj ðX; tÞ 4 0
!
ð16Þ
i¼1 j¼1
where d(t) is the fraction of the structure exhibiting visual degradation at time t and n is the number of components constituting the considered critical fraction of the structure. N is the total number of components, KN;n is the number of different combinations of components constituting
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the critical percentage or more of the structure and mi is the number of components involved in the i’th combination. Finally gj ðX; tÞ is the limit state describing the critical degradation for component j. The probabilistic modelling of the parameters entering into Eq. (16) should include not only the uncertainties which may be assessed by examination of one of the components of the structure but include also the variability of the probabilistic characteristics of the parameters from component to component. Evaluation of Eq. (16) is possible using standard methods and software for system analysis but not least due to the number of intersections and unions the required numerical effort is rather large. It may also be possible to establish closed form solutions for the case of equi-correlated components, or sets of equi-correlated components. In the following however, for the purpose of simplification, the areas of the structure, which can be divided into components with the same probability of critical degradation, are considered separately. Furthermore, it is assumed that the components are independent. For one area with N components Eq. (16) then simplifies to PðdðtÞ 5 n=NÞ ¼ 1 Bðn 1; N; ðtÞÞ
ð17Þ
where Bðn 1; N; ðtÞÞ is the cumulative Binomial distribution and ðtÞ is the probability of failure of the individual components at time t. ðtÞ may in fact be considered as a defect rate for the components constituting the considered area of the structure and is associated with uncertainty due to the random variations from component to component of the probabilistic parameters describing the degradation process. If the probability distribution function fYðtÞ ððtÞÞ for the uncertain defect rate ðtÞ is known the probability that a certain degree of degradation occurs may then be obtained by ð1 PðdðtÞ 5
CRIT Þ
¼1
Bð
CRIT N
1; N; ðtÞÞ fðtÞ ððtÞÞdðtÞ
ð18Þ
0
For the purpose of illustration a simplified event tree is illustrated in Fig. 2, containing the principal features of the inspection and maintenance planning problem. It is assumed that visual inspections are performed at the same time as potential measurements only one time during the life of the structure. The decision problem is then to identify the first time tP an inspection should be carried out and how large a fraction INSP of the structure should be inspected by NDE measurements. If at an inspection it is (visually) observed that degradation has taken place over a critical fraction CRIT of the surface of the structure a major corrective repair R1 is implemented. If at an inspection it is indicated by NDE that degradation has initiated over a critical fraction CRIT of the structure a minor corrective repair R2 is implemented. CRIT is assumed to be a criteria given by the owner of the structure as a measure of the acceptable deterioration for the structure. The decision variables thus reduce to the parameters tP , INSP and CRIT . The repair probabilities indicated in the event tree in Fig. 2 may be assessed using the principles of the condition indicators derived in Section 2. The probability of the event of a major repair at the time of the inspection is found applying Eq. (18) where the probability of a major repair is evaluated on the basis of the prior probabilistic modelling of the defect rate (probability of visual corrosion at the time of visual inspection) for the individual components of the structure, i.e. before any inspections are performed. In the event tree illustrated in Fig. 2 it is assumed that a
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Fig. 2. Illustration of an event tree for inspection and maintenance planning of concrete structures subject to distributed deterioration.
major repair may be implemented at the end of the service life. This assumption is of course not realistic, however, the results may still be meaningful and useful if by the service life a period of time relevant for budgeting of overall maintenance costs is selected. The probability of a major repair at the time of the first inspection shall thus be assessed by evaluation of the probability that a certain critical number of components at the potential measurement inspections are found to exhibit visual corrosion mCRIT ¼ N CRIT , where N is the total number of components of the structure. This probability may be evaluated by ð1 PR1 ¼ 1
BðmCRIT 1; N; CV Þ fYCV ðCV ÞdCV
ð19Þ
0
where fYCV ðCV Þ is the prior probability density function for the probability of visual degradation. The probability of a minor repair at time tp may be assessed by evaluation of the probability that a certain critical number of components at the time of the NDE inspections are found to exhibit initiated degradation nCRIT ¼ m CRIT , i.e. ð1 PR2 ¼ 1
B nCRIT 1; INSP N; CI PðID j CIÞ þ ð1 CI Þ P ID j CI fYCI ðCI ÞdCI
ð20Þ
0
Finally given that a certain number of components nI are found to exhibit initiated corrosion at the time of the potential measurement inspections the conditional probability of a major repair at the end of the service life may be evaluated as
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ð1 PR1 jnI ¼ 1 0
BðmCRIT 1; m; CV Þ f 000 YCV ðCV j nI ÞdCV
ð21Þ
where f 000 CV ðÞ is the posterior probability density function for the probability of visual degradation at the end of the service life. In order to assess the posterior probability density function in Eq. (21) by utilisation of Eqs. (11) and (13) we need to establish p ¼ PðCV j ID Þ and q ¼ PðCV j ID Þ. To this end it is useful to consider the event tree shown in Fig. 3. From Fig. 3 it is seen that p and q may be assessed as T T P CV CI P ID j CI P CV CI P ID j CI q ¼ P CV j ID ¼ þ with K1 K1 h i h \ i \ \ \ CV þ P CI CV þ P ID j CI P CI CV þ P CI CV K1 ¼ P ID j CI P CI ð22Þ T T P CV CI PðID j CIÞ P CV CI P ID j CI p ¼ PðCV j ID Þ ¼ þ with K2 K2 h \ \ i \ i h \ CV þ P CI CV þ P ID j CI P CI CV þ P CI CV K2 ¼ PðID j CIÞ P CI ð23Þ As we do not know the number nI we need to integrate PR1 jnI as given in Eq. (21) out over all possible values of nI weighted by their probabilities. This integration may be performed as PR1 ¼ 1
ð m INSP 1 X i¼1
PR1 ji b i; INSP m; CI PðID j CIÞ þ ð1 CI ÞP ID j CI fYCI ðCI ÞdCI
ð24Þ
0
In Eq. (24) we thus integrate the conditional repair probability out over the possible NDE inspection outcomes, which we assume are generated from our prior model of the uncertain probability of initiated corrosion at the time of the inspection. The various fractions defined or chosen for the area to be inspected, the critical area for the minor repair and the critical area for the major repair shall be chosen such that the corresponding numbers of components are integers.
4. Example For the purpose of illustrating the principles outlined in Section 3 an example is presented considering first the situation where a structure is subjected to localized degradation and secondly the situation where a structure is subjected to distributed degradation. It is assumed that the degradation process is corrosion due to the ingress of chlorides and that the effect of the corrosion is mainly related to the serviceability condition of the structure.
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Fig. 3. Event tree relating the indication of the time of NDE with the condition of the structure at the end of the service life.
The time until corrosion becomes visual is modeled as the time till initiation of corrosion TI plus a certain corrosion propagation time TP , i.e. TV ¼ TI þ TP . In the following the time till initiation TI is assessed assuming that the ingress of chlorides through the concrete cover d may be described by a diffusion process with diffusion coefficient D and that corrosion initiation will take place when the concentration of chlorides exceed a certain critical concentration CCR . The time till initiation may thus be written as, see e.g. Engelund et al. [6] d CCR 2 1 erf 1 TI ¼ 4 D CS
ð25Þ
where CS is the concentration of chlorides on the surface of the concrete. The propagation time till corrosion becomes visual is for simplicity modelled as a Log-normal distributed random variable with mean value and standard deviation as shown in Table 1. Based on the safety margin M ¼ gðX; tÞ ¼ XI TI þ TP t
ð26Þ
where XI is a model uncertainty associated with the corrosion initiation time and the probabilistic model given in Table 1 the probability of corrosion initiation as well as visual corrosion may be readily calculated using e.g. FORM/SORM methods, see Madsen et al. [7].
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Table 1 Modeling of parameters in the corrosion limit state Description (units)
Distribution
Mean value
Standard deviation
Cover thickness (d) (mm) Diffusion coefficient (D) (mm2/year) Surface concentration (CS) (wt.% of concrete) Critical concentration (CCR) (wt.% of concrete) Model uncertainty (XI) Propagation time (TP) (years) Stat. uncertainty ( D ) (mm2/year) Stat. Uncertainty ( CS ) (wt. % of concrete)
Log-normal Log-normal Log-normal Log-normal Log-normal Log-normal Normal Normal
55.0 D CS 0.15 1.0 7.5 40.0 0.4
11.0 10.0 0.08 0.05 0.05 1.88 4.0 0.04
Due to statistical uncertainty the mean value of the diffusion coefficient D and the surface concentration CS are assumed to be uncertain themselves. As a part of the planned condition monitoring of the structure half-cell potential measurements are performed on a regular basis. It is the aim to utilize the measurements in order to update the probability of repair and thus the expected future maintenance costs. In order to quantify the effect of the performed inspections the indicator corresponding to the half-cell potential measurements needs to be established. According to tests performed by Marschall [8] the probability distribution function for the half-cell measurement may be described by a normal distribution function. Given that corrosion has initiated the mean value and standard deviation are 0.354 and 0.08 V, respectively. Given that no corrosion has initiated the mean value and the standard deviation are 0.207 and 0.0804 V, respectively. A central question concerns what measurement result to assign with an observation of ‘‘corrosion’’ or ‘‘no corrosion’’, i.e. the choice of the indicator. Choosing somewhat arbitrarily as an indicator for corrosion initiation a potential reading corresponding to the upper 10% quantile value for the measurements given corrosion this value corresponds to 0.2515. We then have potential P ID j CI ¼ 0:10; P ID j CI ¼ 0:71 where CI denotes the event of corrosion initiation at the time where the measurement is performed and ID the event of no indication. The probability of getting an indication of no-corrosion initiation given that corrosion exists is 0.1 and the probability of getting an indication of no corrosion initiation given that no corrosion exists is 0.71. Correspondingly we have PðID j CIÞ ¼ 0:90; P ID j CI ¼ 0:29 where ID denotes the event of an indication of corrosion. Later on the significance of the choice of the indicator will be assessed further. 4.1. Localised corrosion The inspection and maintenance planning for the first 30 years of the life of the structure is considered and it is assumed that visual inspections are performed in connection with ordinary maintenance activities every 5 years. The branching probabilities in the event tree in Fig. 6 are calculated using FORM/SORM systems reliability analysis using in principle the safety margin given in Eq. (26) with and without the propagation time depending on the considered branching event. The random variables as defined in Table 1 are used for the probabilistic modelling. The inspection costs entering into Eq. (14) are evaluated as the costs of performing visual inspections every 5 years and the costs of the half-cell potential measurements to be performed only once
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during the considered 30 year period. The costs of repair R2 are assumed associated with the costs of performing a preventive measure such as an exchange of only the concrete cover. These costs have been assessed to US$ 1250 per square meter. The failure costs in Eq. (14) are here considered to be the costs due to major repairs, i.e. of type 1. These have been assessed to US$ 5500 per square meter. The costs of visual inspections and the costs of half-cell potential measurements have been assessed to US$ 10 and US$ 100 per square meter, respectively. It is assumed that all costs are assessed in terms of their net present value and a real rate of interest net of inflation equal to 4% is used. The expected total costs have been evaluated as function of the time of the half-cell potential measurement and the results are shown in Fig. 4. From Fig. 4 it is seen that the cost optimal time to perform the half-cell potential measurement is after 20 years. The evaluation of the expected costs as shown in Fig. 4 have been evaluated using the value of 0.2551 V as an indication of initiated corrosion, as discussed previously. However, this value may not be optimal and therefore a study has been performed in order to identify the cost optimal choice of the indicator value. In Fig. 5 the total expected costs of the optimal inspection plan are shown as function of the indicator value. From Fig. 5 it is seen that in the present example indicator values in the order of 0.4 V are cost optimal, however, this depends on the actual condition of the considered structure together with the relative costs of the different repairs and not least the assumed real rate of interest. For the present case the optimum choice of indicator value indicates that it is optimal to avoid erroneous interpretations in the situation where there is in fact no initiated corrosion, whereas it is not so important to make such errors when corrosion has initiated. For structures in poorer condition this effect could be reversed. 4.2. Distributed corrosion Further it is assumed that the considered structure in terms of its corrosion characteristics may be modelled as an assembly of 100 components and when any 50 of these components exhibit initiated corrosion a major repair will be performed.
Fig. 4. Total expected costs as function of time till half-cell potential measurement.
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Fig. 5. Total expected costs for optimum inspection plan as function of the choice of indicator value. Note that the absolute value of the indicator value is given.
Fig. 6. Strength of the indicators p and q as function of the inspection time.
The ensemble of the 100 components may thus be considered as 100 components with a common failure probability (evaluated on the basis of the physical uncertainties only), which is uncertain in itself due to the remaining statistical and model uncertainties. Assuming for the sake of illustration that the individual components may be assumed statistically independent the probability of having more than 50% of the components exhibiting visual corrosion may be assessed through ð1 PðnF ðtÞ 5 50Þ ¼ 1
Bð50 1; 100; ðtÞÞ fðtÞ ððtÞÞdðtÞ
ð27Þ
0
where Bð50 1100; ðtÞÞ is the cumulative Binomial distribution (see Faber and Rostam [4]) and ðtÞ is the uncertain probability of visual corrosion of the individual components at time t determined by simulation and nested FORM/SORM analysis on the limit state equation given in Eq. (26). fðtÞ ððtÞÞ is the prior or posterior PDF for the probability of visual corrosion depending on
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the state of knowledge. The strength of the indicator p ‘‘no corrosion initiation’’ and q for ‘‘corrosion initiation’’ for the half-cell potential measurement are shown as a function of time in Fig. 6. The prior probability density function for the probability of visual corrosion for one component after 50 years is shown in Fig. 7 (Prior). Half-cell potential measurements are performed for 10 components after 25 years with the result that a) ‘‘no corrosion’’ and b) ‘‘corrosion’’ has initiated. Based on the indicator for ‘‘no corrosion initiation’’ and ‘‘corrosion initiation’’ using half-cell potential measurements the posterior probability density function for the uncertain probability of visual corrosion is evaluated using Eqs. (11)–(13) and illustrated in Fig. 7. The probability of extensive repair, i.e. more than 50% of the structure exhibiting visual corrosion is evaluated using Eq. (24) and illustrated in Fig. 8 as function of the time where the inspection is performed. From Fig. 8a and b it is seen that the inspection using half-cell measurements will not provide any knowledge in regard to the development of excessive visual corrosion until after about 10 years. This is because the probability of initiation of corrosion is negligible until this point in time. Thereafter the indication of no-corrosion and corrosion respectively is seen to have a significant impact on the probability of more than 50% of the concrete surface exhibiting visual corrosion.
Fig. 7. Posterior PDF for the probability of visual corrosion of one component given (a) no indication, (b) indication.
Fig. 8. (a) No indication of corrosion. (b) Indication of corrosion.
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Fig. 9. The strength of the indication (a) no-corrosion and (b) corrosion for different choices of the indicator.
Fig. 10. Probability of repair as function of time for different choices of the indicator.
Returning now to the question regarding the choice of indicator the strength of the indicators p and q is depicted in Fig. 9 for different choices (5, 10 and 25%) of the quantile values in the probability distribution function for the hall-cell potential reading given corrosion has initiated. From Fig. 9 it is seen that the strength of the indicator depends significantly on the choice of the indicator. This dependency is also carried over to the updated probability of excessive repair as shown in Fig. 10. The optimal choice of the quantile value in the distribution function for the half-cell potential reading given corrosion, which is associated with an indication of corrosion, may be solved along the lines as illustrated for the case of localised corrosion. This requires the simultaneous consideration of the effect on the updated probability of excessive repair as shown in Fig. 10, but also consideration of the maintenance activities, which may be implemented in order to reduce the risk of excessive repair after 50 years.
5. Discussion and conclusions A Bayesian approach to assessment and inspection planning has been formulated on the basis of a general methodology for indicator based quality control. The approach allows for the con-
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sideration of problems where the components have uncertain defect rates and where tests are performed on condition indicators rather than the condition of interest directly. The theoretical derivations have been based on the assumption that the probability of achieving an indication of a defective (or non-defective) component given that the components is defective (or non-defective) is independent on the defect rate for the considered components. This assumption appears to be fulfilled for most practical applications, at least in cases where testing and inspections are automated and/or largely independent of human interaction. In cases where the quality of the inspections is highly influenced by human behaviour it is imaginable that the quality of inspections might be influenced by the defect rate due to the effect that if defective components are rare then the awareness of the inspector might decrease. This situation might, however, be seen as a quality control problem of the inspection itself and is thus not really a limiting factor for the proposed methodology. A further assumption made concerns the requirement that defect rates may be defined for well defined populations of components. In engineering applications this requirement necessitates careful considerations in regard to failure modes, deterioration mechanisms, boundary conditions etc. It will thus often be necessary to perform a differentiation of the considered components into two or more populations in order to fulfil this requirement. The approach may readily be utilized for the optimal planning of tests, inspections and maintenance activities of components within the framework of Bayesian decision analysis. The Bayesian formulation of condition indicators is especially well suited to consistently incorporate subjective knowledge in the test and inspection planning and the treatment of defect rates as being uncertain facilitates the development of quality control schemes for populations of components for which little if any frequentistic information is available. The approach is demonstrated on an example considering the assessment and inspection and maintenance planning of a concrete structure. From the example it is seen that the concept of Bayesian indicators might be a very useful tool providing important decision support in connection with the planning of inspection and repair measures. A number of further potential application areas for the developed methodology are yet to be explored. These include the important aspect of quantitative probabilistic modelling of the quality of inspections and tests as a function of the inspection coverage such as e.g. the quality of inspections of pressure vessels subject to corrosion processes. As in the case where corrosion deterioration of concrete structures is considered it is, however, required that information in regard to the spatial correlation of the deterioration process is available in order to perform such investigations. The spatial correlation structures of different deterioration processes furthermore appear to be rather case dependent and much work (experimental) is lying ahead before robust models for this purpose may be established. References [1] Benjamin JR, Cornell CA. Probability, statistics and decision for civil engineers. NY: McGraw-Hill; 1970. [2] Faber MH, Sorensen JD. Bayesian sampling using condition indicators. In: Proc. to the ICOSSAR, July 2001, Newport Beach, CA. [3] Lindley DV. Introduction to probability and statistics from a Bayesian viewpoint. Cambridge: Cambridge University Press; 1976.
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[4] Raiffa H, Schlaifer R. Applied statistical decision theory. Cambridge, Mass: Harward University Press, Cambridge University Press; 1961. [5] Hergenro¨der M. Zur statistischen Instandhaltungsplanung fu¨r bestehende Bauwerke bei Karbonatisierung des Betons und mo¨glicher Korrosion der Bewehrung‘‘ Berichten aus dem Konstruktiven Ingenio¨rbau. TU Mu¨nchen, 4/92, 1992 [in German]. [6] Engelund S, Sørensen JD, Sørensen B. Evaluation of repair and maintenance strategies for concrete coastal bridges on a probabilistic basis. ACI Materials Journal; 1998. [7] Madsen HO, Krenk S, Lind NC. Methods of structural safety. Englewood Cliffs, NJ: Prentice-Hall; 1986. [8] Marschall SJ. Evaluation of instrument-based, non-destructive inspection methods for bridges. MSc thesis, University of Colorado, Boulder, Colorado; 1996.
Indicators for inspection and maintenance planning of concrete structures M.H. Fabera, J.D. Sorensenb,* a Swiss Federal Institute of Technology, Zu¨rich, Switzerland Department of Building Technology, University of Aalborg, Sohngaardsholmsvej 57, 9000 Aalborg, Denmark
b
Abstract Based on an idea introduced by Benjamin and Cornell (1970. Probability, statistics and decision for civil engineers. New York: McGaw Hill) and previous works by the authors it is demonstrated how condition indicators may be formulated for the general purpose of quality control and for assessment and inspection planning in particular. The formulation facilitates quality control based on sampling of indirect information about the condition of the considered components. This allows for a Bayesian formulation of the indicators whereby the experience and expertise of the inspection personnel may be fully utilized and consistently updated as frequentistic information is collected. The approach is illustrated on an example considering a concrete structure subject to corrosion. It is shown how half-cell potential measurements may be utilized to update the probability of excessive repair after 50 years. Furthermore in the same example it is shown how the concept of condition indicators might be applied to develop a cost optimal maintenance strategy composed of preventive and corrective repair measures. # 2002 Published by Elsevier Science Ltd. Keywords: Condition indicators; Concrete; Reliability; Corrosion; Bayes; Degradation; Half-cell potential measurements; Assessment; Maintenance; Inspection; Planning
1. Introduction Condition control of existing structures by means of inspection and testing forms a corner stone in the maintenance management of structures. It is generally accepted that the information collected by inspection and testing is the basis for the estimation of the condition of the structure and moreover for the estimation of the future development of the condition of the structure; i.e. the future structural degradation.
* Corresponding author. Tel.: +45-96-35-85-81; fax: +45-98-14-82-43. E-mail address: [email protected] (J.D. Sorensen). 0167-4730/02/$ - see front matter # 2002 Published by Elsevier Science Ltd. PII: S0167-4730(02)00033-4
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Ideally the principle for the planning of inspection and testing activities is to minimize the service life economical risks, i.e. the sum of the costs of inspection and testing, the costs due to future maintenance and strengthening works and the costs due to future failures. To be able to plan inspection and testing efforts optimally in the above-mentioned sense it is necessary that the effect of inspections and tests can be related to those particular states of the structure, which impact the service life economical risks for the structure; i.e. the results of the inspections and tests must be related to the service life economical risks. Traditionally in applications such as bridge management the results of inspections and tests are expressed in linguistic terms such as ‘‘corrosion has initiated in xx% of the structure’’, ‘‘the membrane is defect’’, etc. This information is then processed and results in an overall grading by assigning a certain number (e.g. 1, 2, 3,..) to the overall condition of the structure. The further treatment of such information for the purpose of estimating the service life economical risk is less than obvious, and even though some systematic may be developed on this basis there is no theoretical basis ensuring that the information is consistently utilized for the purpose of maintenance management. Based on an idea introduced by Benjamin and Cornell [1] and previous works by the authors (Faber and Sorensen [2]) it is demonstrated how condition indicators may be formulated for the general purpose of quality control and for assessment and inspection planning in particular. The formulation facilitates quality control based on sampling of indirect information about the condition of the considered components. This allows for a Bayesian formulation of the indicators whereby the experience and expertise of the inspection personnel may be fully utilized and consistently updated as frequentistic information is collected. The approach is illustrated on an example considering a concrete structure subject to corrosion where it is shown how half-cell potential measurements may be utilized to update the probability of excessive repair after 50 years.
2. Information sampling As a basis for the further derivations it is assumed that the defect rate can be used as a measure of the general condition of a series of components. Defect rate is here understood in a broad sense as the rate of occurrence of any state of the considered components, which is of special interest. Components are considered for which the unknown defect rate is , i.e. the probability that a considered component is defective is and the probability that it is not defective is 1. On the basis of sample (inspection) results and subjective information the estimate of the defect rate can be updated and decision-making regarding the inspection and maintenance of the components may be undertaken on this basis. 2.1. Random sampling The usual sampling technique from classical quality control is to test (inspect) samples, which are taken randomly from the considered population. Here this approach is shortly described within a Bayesian statistical framework utilizing, to the highest degree possible, all available information.
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It is assumed that a well defined population with constant defect rate is considered. n components are selected randomly from the population and inspected with the result that nF (4n) components are defective. Letting the random variable Y model the number of defect components nF in a random sample of size n, Y is binomial distributed with probability density function given by n y fY ðy j Þ ¼ ð1 Þny ð1Þ y However, the defect rate is not known with certainty and must hence be considered as an uncertain variable . Initial information regarding the defect rate is modelled by the prior probability density function f 0 ðÞ, see e.g. Lindley [3]. If no initial information is available or if the initial information is very uncertain a diffuse prior may be used, i.e. f 0 ðÞ / 1. Through the prior probability density function it is possible to incorporate subjective information regarding the general defect rate. Having inspected a sample of n components with the result that nF components are defective, this information can be utilized to update the available information regarding . The updated probability density function (the posterior probability density function) for can be derived on the basis of Bayes formula. The posterior probability density is denoted f 00 ð j Y ¼ nF Þ and expresses the updated knowledge about the general defect rate. The posterior probability density function is given by 1 f 00 ð j Y ¼ nF Þ ¼ fY ðnF j Þ f 0 ðÞ c
ð2Þ
Ð1 where c is a normalizing constant such that 0 f 00 ð j Y ¼ nF Þd ¼ 1 The posterior probability density function for the defect rate may readily be used in the decision analysis, see e.g. Raiffa and Schlaifer [4] either on its own or by using it together with Eq. (1) whereby the predictive probability density function for the number of defective components may be achieved as ð1 n y ð3Þ ð1 Þny f 00 ð j Y ¼ nF Þd fY ðy j Y ¼ nF Þ ¼ y 0 2.2. Sampling based on indicators Inspection and maintenance planning of engineering systems such as structural systems, pipelines and process systems is complicated by the fact that the systems often are very large, in terms of the number of components. Furthermore, the components of the systems often belong to a larger number of different populations. Application of classical quality control procedures for the quantification of the effect of inspections on such systems is thus hardly practical and hence not used in practice. In practice, inspections are performed on a rather limited number of components. The components inspected are typically selected on the basis of criteria such as criticality, expected condition, inspectability and experience. Following such schemes little possibility is left to evaluate the effect of the sample size (number of inspected components) and even less possible to quantify and justify the rationale behind the inspection scheme.
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In the following it is hence attempted to outline a statistical framework for an alternative approach based on Bayesian statistics and so-called indicators. Initially the case is considered where knowledge is included in the estimate of the defect rate. Letting Di denote that indicator no. i is inspected and indicates a defect. It is assumed that prior information may be represented by the probabilities: PðF j Di ¼Þpi : the probability that the inspected component is defective (F) given the observation no. i indicates a defect. that indicator P F j Di Þ ¼ qi : the probability that the inspected component is defective (F) given the observation that indicator no. i does not indicate a defect. The number of components for which the indicator has been inspected and found to indicate a defect is denoted ND and the number of components inspected where the inspected indicator does not indicate a defect is denoted ND . On the basis of these probabilities and information regarding the total number of inspected components the probability density function for the defect rate may be updated. The updated probability density function is denoted f 000 ðÞ. In f 000 ðÞ both prior information regarding and results of random inspections may be included. f 000 ðÞ is the updated probability density function for the general defect rate when all information is included (prior information, results of random samples and results of targeted samples). The probability density function may be updated approximately by ð4Þ f 000 ð j Di Þ ffi c f 00 ð j F Þpi þ f 00 ð j F Þð1 pi Þ for the case where the inspection indicates a defect and f 000 ð j D i Þ ffi c f 00 ð j F Þqi þ f 00 ð j F Þð1 qi Þ
ð5Þ
for the case where the inspection indicates that there is no defect, c is a normalising constant. As input for the updating information regarding the probabilities pi and qi is required. If these are not known with certainty Bayesian statistics may also be applied to model this. pi and qi are then modelled as uncertain variables and their uncertainty, modelled by the prior probability density functions, may be updated when information is available. The derivation of Eqs. (4) and (5) may be realised from T T T T P D F þ P D F Pð j DÞ ¼
¼
PðDÞ T T P D j F Pð j FÞ PðFÞ þ P D j F P j F P F PðDÞ PðD j FÞ Pð j FÞ PðFÞ þ P D j F P j F P F
ffi
PðDÞ ¼ PðF j DÞ Pð j FÞ þ P F j D P j F
ð6Þ
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The approximation introduced in Eqs. (4) and (5) may thus be expressed by the assumptions that PðDi j F \ ¼ Þ ffi PðDi j FÞ
ð7Þ
P Di j F \ ¼ ffi P Di j F
ð8Þ
which means that it is assumed that the probability of finding an indication of a defect, given that the component is defective (or not defective) is independent of the defect rate. This assumption appears to be non-restrictive for practical applications. If inspection results are available the probabilities f 00 ð j FÞ and f 00 j F may be determined by using f 00 ð j Y ¼ nF Þ as prior probability density function: f 00 ð j FÞ ¼
1 00 f ð j Y ¼ nF Þ c’
f 00 j F ¼
ð9Þ
1 ð1 Þ f 00 ð j Y ¼ nF Þ 1 c’
ð10Þ
Ð1 where c0 ¼ 0 f 00 ð j Y ¼ nF Þd is a normalising constant. If no inspection results are available f 0 ðÞ may be used as prior probability density function instead of f 00 ð j Y ¼ nF Þ. First the situation where the observation of ND indicate that that the component is defect is considered. The ND indicators are denoted Di ; i ¼ 1; ND . The updated probability density function is determined recursively by pi ð1 pi Þð1 Þ 000 ðiÞ f ð j D1 ; :::; Di Þ ¼ ci 0 þ f 000 ði1Þ ð j D1 ; :::; Di1 Þ; i ¼ 1; ::; ND ci 1 c0i ð11Þ where ci is determined such that c0i
ð1 ¼
Ð1 0
f 000 ðiÞ ð j D1 ; ::; Di Þd ¼ 1 and c0 i is a normalising constant:
f 000 ði1Þ ð j D1 ; :::; Di Þd
ð12Þ
0 0
Furthermore there is f 000 ð0Þ ðÞ ¼ f 00 ð j Y ¼ nF Þ or f 000 ð0Þ ðÞ ¼ f ðÞ. Secondly the situation is considered where observations of ND components do not indicate that the components are defective. The ND indicators are denoted D j ; j ¼ 1; ND . The updated probability density function is determined recursively by ! q ð1 q Þð1 Þ j j 000 ðj1Þ 1 ; :::; D j1 ; j ¼ 1; ::; N j D f f 000 ð j Þ j D 1 ; :::; D j ¼ cj 0 þ D cj 1 c0j ð13Þ
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In the above derivations it has been assumed as a prerequisite that the considered components belong to the same population. 2.3. Updating knowledge about indicators If inspections of the condition of interest are conducted fully for the components for which one or more indicators have been found to indicate a defect the result of the inspections can be used to update the probabilities pi ¼ PðF j Di Þ and qi ¼ PðF j D i Þ as described in the foregoing and by letting pi and qi be modelled by the random variables Pi and Qi . If more populations are considered pi and qi may be assumed to be the same for the same indicators in the different populations. In this case it is possible to update the statistical knowledge regarding pi and qi on the basis of one population and to use this updated knowledge on the other populations. If on the other hand pi and qi are different for a given indicator for different populations the improved knowledge regarding pi and qi can only be used for the indicator in the inspected population.
3. Inspection and maintenance planning for concrete structures The concept of indicators for quality control purposes as described in the previous section may be utilised in a wide range of applications where the quality or the reliability of components belonging to a larger population is considered. In civil engineering applications one such situation arises in connection with the maintenance management of concrete structures. Inspection and maintenance planning should be performed such that the expected total service life costs E½CT are minimised, i.e. E½CT ¼ E½CI þ E½CR þ E½CF
ð14Þ
where E½CI , E½CR and E½CF are the expected inspection costs, repair costs and failure costs respectively. The expected costs are to be determined as the probability that the cost-inducing event will occur multiplied with the cost consequences given the event. The expected total service life costs are to be minimised by means of planning the inspections for condition control and by means of choice between different repair strategies. Decision variables are thus
inspection method indicator of damage part or percentage of structure to be inspected repair method criteria for repair.
A large variety of approaches exist to assess the expected total service life costs. However, it is important that the approach selected resembles the characteristics of the degradation process and the way inspections and maintenance decisions are carried out in practice. In the following it is
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therefore investigated how the theory developed in Section 2 can be adapted for the purpose of inspection and maintenance planning for concrete structures. 3.1. Consequences of degradation and criteria for repair Deterioration of concrete structures in the context of inspection and maintenance planning may appropriately be described in terms of the different phases of the deterioration, which are associated with different consequences. Considering degradation due to corrosion, the first phase is the initiation phase at the end of which de-passivation occurs at the outer layer of reinforcement. Thereafter simultaneously with the de-passivation of the next layers of reinforcement the propagation phase follows in various steps. The characteristics and duration of these steps depend in general on the specifics of the concrete, the reinforcement, the concrete cover thickness, the humidity and the exposure conditions. At first the corrosion is initiated and corrosion products build up on the surface of the reinforcement. After a certain increase of the volume of the corrosion materials small cracks of the concrete cover will occur and again after some time the corrosion products may disperse through the cracks and become visible on the surface of the structure. If the corrosion products continue to build up on the surface of the reinforcement the concrete cover may eventually fall off (spalling) leaving the structure even more exposed to the environment and thus accelerating the deterioration processes for the next layer of reinforcement. Ultimately the degradation may proceed until the load carrying capacity or the serviceability of the structure is reduced beyond acceptability. However, the consequences of degradation may also be reduced by means of condition control together with maintenance and repair activities. Due to the characteristics of the different phases of the degradation process the cost efficiency of maintenance and repair activities will be highly dependent on when and how these are implemented. If the progress of deterioration is detected at an early stage it may be possible by means of minor and inexpensive repairs to reduce the risk of major and expensive future repair works. For a large part of the normal reinforced concrete structures the load carrying capacity and thus the reliability of the structures in regard to the ultimate limit state condition will only be reduced beyond acceptability after a very significant degree of degradation has become visually observable. Such severe signs of deterioration are usually not acceptable alone for cosmetic reasons. Thus early signs of degradation such as cracking and colouring together with the results of various NDE techniques usually comprise the information basis about the condition of the structures for decision making in regard to the implementation of maintenance and repair activities. In practice repair decisions are often taken alone on the basis of the visually observable condition of the structures. For concrete structures where the likely location of degradation is known, i.e. structures with hot spots, the criteria for implementing a repair could thus simply be the occurrence of visible degradation. Similarly for concrete structures with distributed degradation, i.e. where in principle ‘‘all spots are hot’’ the criteria for implementing a repair could be the occurrence of visible degradation on more than a certain critical percentage of the surface of the structure. Given that the criterion for repair is fulfilled the degraded area(s) is repaired according to a pre-determined repair strategy. In the following it is shown how inspection and maintenance strategies for the two cases (1) localised degradation and (2) distributed degradation may be formulated using the concepts of condition indicators described in Section 2.
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3.2. Inspection and maintenance planning—localised degradation For concrete structures where the critical degradation is localised the planning of inspection and maintenance activities may be performed by assessing the expected service life costs of Eq. (14) by consideration of the event tree illustrated in Fig. 1. In Fig. 1 CV refers to the event of visual corrosion at the time of the inspection, CI refers to the event of initiated corrosion at the time of an inspection. ID refers to the event of a positive indication of initiated corrosion at the inspection. CV, CI and ID refer to the complementary events, respectively. The branching in the event tree takes place between the performed inspections. The event tree in Fig. 1 only illustrates how event trees may be developed for inspection and maintenance planning purposes. It is assumed that visual inspections are carried out at regular intervals, i.e. at times indicated by tV . If at any time during the life of the structure visual degradation is observed a major corrective repair R1 will be implemented. Furthermore it is assumed that the decision problem is to identify the first time where in addition to a visual inspection also a NDE of the structure should be carried out in order to detect and facilitate a minor preventive repair R2 . In the illustrated event tree it is for simplicity assumed that the branches leading to repairs do not contribute further to the expected service life costs. This assumption is in effect equivalent to assuming that repair is perfect, which surely is not the case in practice, however, data and experience still need to be collected and processed before a more realistic modelling of the durability of repaired concrete structures may be formulated. Given that probabilistic models are formulated for the event of initiation of degradation as well as for the event of visual degradation as a function of time the different repair probabilities in the event tree may be calculated using standard tools from structural reliability such as FORM/
Fig. 1. Illustration of an event/decision tree for inspection and maintenance planning of concrete structures subject to localised deterioration.
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SORM and simulation. However, due to the fact that NDE techniques only provide an indication about the real state of the structure the branching probabilities in the event tree after the time where the NDE is performed must take into account the quality of the applied NDE method. The quality of the applied NDE method may appropriately be expressed in terms of the probability that the NDE method indicates degradation given that degradation has initiated, i.e. PðID j CIÞ together with the probability that the NDE method indicates degradation given that degradation has not initiated, i.e. PðID j CIÞ. When assessing the probabilities of repair in the event tree at inspection times after a NDE has been performed the quality of the NDE has to be accounted for. Consider as an example the event of a major repair R1 at a visual inspection following immediately after an inspection using NDE. The branch in the event tree is considered where degradation in fact has initiated at the time of the NDE inspection but where no indication of initiated degradation was found (see figure 1). In this case the probability of the repair R1 at time tV3 can be assessed through \ \ \ \ P R1 tV3 ¼ P CV tV3 CV tP1 ð15Þ CI tP1 ID tP1 CVðtp1 Þ ¼ \ \ \ P CV tV3 CI tP1 CVðtp1 Þ P ID tP1 j CI CV tP1
Based on event trees as illustrated in Fig. 1 and probability assessments of the type in Eq. (15) the overall service life costs associated with inspection and maintenance may be calculated and the optimal strategy for combining preventive and corrective repairs may be established. 3.3. Inspection and maintenance planning—distributed degradation For concrete structures where degradation can be expected at any location with equal probability the spatial characteristics of the degradation process play an important role. One way to consider this aspect is to model the structure not as one large whole but rather as an assembly of many individual components, either as small structural (area) components or as zones of larger structural elements. The probabilistic characteristics of the parameters governing the degradation of the structure should also be modelled in this light. Studies in e.g. Hergenro¨der [5] indicate that such components could have a typical dimension of about 2 2 m. In order to assess the probability that a given percentage of the structure has achieved a critical degree of degradation it is in principle necessary to evaluate the following expression
PðdðtÞ 5 n=NÞ ¼ P
K mi N;n \ [
gj ðX; tÞ 4 0
!
ð16Þ
i¼1 j¼1
where d(t) is the fraction of the structure exhibiting visual degradation at time t and n is the number of components constituting the considered critical fraction of the structure. N is the total number of components, KN;n is the number of different combinations of components constituting
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the critical percentage or more of the structure and mi is the number of components involved in the i’th combination. Finally gj ðX; tÞ is the limit state describing the critical degradation for component j. The probabilistic modelling of the parameters entering into Eq. (16) should include not only the uncertainties which may be assessed by examination of one of the components of the structure but include also the variability of the probabilistic characteristics of the parameters from component to component. Evaluation of Eq. (16) is possible using standard methods and software for system analysis but not least due to the number of intersections and unions the required numerical effort is rather large. It may also be possible to establish closed form solutions for the case of equi-correlated components, or sets of equi-correlated components. In the following however, for the purpose of simplification, the areas of the structure, which can be divided into components with the same probability of critical degradation, are considered separately. Furthermore, it is assumed that the components are independent. For one area with N components Eq. (16) then simplifies to PðdðtÞ 5 n=NÞ ¼ 1 Bðn 1; N; ðtÞÞ
ð17Þ
where Bðn 1; N; ðtÞÞ is the cumulative Binomial distribution and ðtÞ is the probability of failure of the individual components at time t. ðtÞ may in fact be considered as a defect rate for the components constituting the considered area of the structure and is associated with uncertainty due to the random variations from component to component of the probabilistic parameters describing the degradation process. If the probability distribution function fYðtÞ ððtÞÞ for the uncertain defect rate ðtÞ is known the probability that a certain degree of degradation occurs may then be obtained by ð1 PðdðtÞ 5
CRIT Þ
¼1
Bð
CRIT N
1; N; ðtÞÞ fðtÞ ððtÞÞdðtÞ
ð18Þ
0
For the purpose of illustration a simplified event tree is illustrated in Fig. 2, containing the principal features of the inspection and maintenance planning problem. It is assumed that visual inspections are performed at the same time as potential measurements only one time during the life of the structure. The decision problem is then to identify the first time tP an inspection should be carried out and how large a fraction INSP of the structure should be inspected by NDE measurements. If at an inspection it is (visually) observed that degradation has taken place over a critical fraction CRIT of the surface of the structure a major corrective repair R1 is implemented. If at an inspection it is indicated by NDE that degradation has initiated over a critical fraction CRIT of the structure a minor corrective repair R2 is implemented. CRIT is assumed to be a criteria given by the owner of the structure as a measure of the acceptable deterioration for the structure. The decision variables thus reduce to the parameters tP , INSP and CRIT . The repair probabilities indicated in the event tree in Fig. 2 may be assessed using the principles of the condition indicators derived in Section 2. The probability of the event of a major repair at the time of the inspection is found applying Eq. (18) where the probability of a major repair is evaluated on the basis of the prior probabilistic modelling of the defect rate (probability of visual corrosion at the time of visual inspection) for the individual components of the structure, i.e. before any inspections are performed. In the event tree illustrated in Fig. 2 it is assumed that a
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Fig. 2. Illustration of an event tree for inspection and maintenance planning of concrete structures subject to distributed deterioration.
major repair may be implemented at the end of the service life. This assumption is of course not realistic, however, the results may still be meaningful and useful if by the service life a period of time relevant for budgeting of overall maintenance costs is selected. The probability of a major repair at the time of the first inspection shall thus be assessed by evaluation of the probability that a certain critical number of components at the potential measurement inspections are found to exhibit visual corrosion mCRIT ¼ N CRIT , where N is the total number of components of the structure. This probability may be evaluated by ð1 PR1 ¼ 1
BðmCRIT 1; N; CV Þ fYCV ðCV ÞdCV
ð19Þ
0
where fYCV ðCV Þ is the prior probability density function for the probability of visual degradation. The probability of a minor repair at time tp may be assessed by evaluation of the probability that a certain critical number of components at the time of the NDE inspections are found to exhibit initiated degradation nCRIT ¼ m CRIT , i.e. ð1 PR2 ¼ 1
B nCRIT 1; INSP N; CI PðID j CIÞ þ ð1 CI Þ P ID j CI fYCI ðCI ÞdCI
ð20Þ
0
Finally given that a certain number of components nI are found to exhibit initiated corrosion at the time of the potential measurement inspections the conditional probability of a major repair at the end of the service life may be evaluated as
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ð1 PR1 jnI ¼ 1 0
BðmCRIT 1; m; CV Þ f 000 YCV ðCV j nI ÞdCV
ð21Þ
where f 000 CV ðÞ is the posterior probability density function for the probability of visual degradation at the end of the service life. In order to assess the posterior probability density function in Eq. (21) by utilisation of Eqs. (11) and (13) we need to establish p ¼ PðCV j ID Þ and q ¼ PðCV j ID Þ. To this end it is useful to consider the event tree shown in Fig. 3. From Fig. 3 it is seen that p and q may be assessed as T T P CV CI P ID j CI P CV CI P ID j CI q ¼ P CV j ID ¼ þ with K1 K1 h i h \ i \ \ \ CV þ P CI CV þ P ID j CI P CI CV þ P CI CV K1 ¼ P ID j CI P CI ð22Þ T T P CV CI PðID j CIÞ P CV CI P ID j CI p ¼ PðCV j ID Þ ¼ þ with K2 K2 h \ \ i \ i h \ CV þ P CI CV þ P ID j CI P CI CV þ P CI CV K2 ¼ PðID j CIÞ P CI ð23Þ As we do not know the number nI we need to integrate PR1 jnI as given in Eq. (21) out over all possible values of nI weighted by their probabilities. This integration may be performed as PR1 ¼ 1
ð m INSP 1 X i¼1
PR1 ji b i; INSP m; CI PðID j CIÞ þ ð1 CI ÞP ID j CI fYCI ðCI ÞdCI
ð24Þ
0
In Eq. (24) we thus integrate the conditional repair probability out over the possible NDE inspection outcomes, which we assume are generated from our prior model of the uncertain probability of initiated corrosion at the time of the inspection. The various fractions defined or chosen for the area to be inspected, the critical area for the minor repair and the critical area for the major repair shall be chosen such that the corresponding numbers of components are integers.
4. Example For the purpose of illustrating the principles outlined in Section 3 an example is presented considering first the situation where a structure is subjected to localized degradation and secondly the situation where a structure is subjected to distributed degradation. It is assumed that the degradation process is corrosion due to the ingress of chlorides and that the effect of the corrosion is mainly related to the serviceability condition of the structure.
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Fig. 3. Event tree relating the indication of the time of NDE with the condition of the structure at the end of the service life.
The time until corrosion becomes visual is modeled as the time till initiation of corrosion TI plus a certain corrosion propagation time TP , i.e. TV ¼ TI þ TP . In the following the time till initiation TI is assessed assuming that the ingress of chlorides through the concrete cover d may be described by a diffusion process with diffusion coefficient D and that corrosion initiation will take place when the concentration of chlorides exceed a certain critical concentration CCR . The time till initiation may thus be written as, see e.g. Engelund et al. [6] d CCR 2 1 erf 1 TI ¼ 4 D CS
ð25Þ
where CS is the concentration of chlorides on the surface of the concrete. The propagation time till corrosion becomes visual is for simplicity modelled as a Log-normal distributed random variable with mean value and standard deviation as shown in Table 1. Based on the safety margin M ¼ gðX; tÞ ¼ XI TI þ TP t
ð26Þ
where XI is a model uncertainty associated with the corrosion initiation time and the probabilistic model given in Table 1 the probability of corrosion initiation as well as visual corrosion may be readily calculated using e.g. FORM/SORM methods, see Madsen et al. [7].
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Table 1 Modeling of parameters in the corrosion limit state Description (units)
Distribution
Mean value
Standard deviation
Cover thickness (d) (mm) Diffusion coefficient (D) (mm2/year) Surface concentration (CS) (wt.% of concrete) Critical concentration (CCR) (wt.% of concrete) Model uncertainty (XI) Propagation time (TP) (years) Stat. uncertainty ( D ) (mm2/year) Stat. Uncertainty ( CS ) (wt. % of concrete)
Log-normal Log-normal Log-normal Log-normal Log-normal Log-normal Normal Normal
55.0 D CS 0.15 1.0 7.5 40.0 0.4
11.0 10.0 0.08 0.05 0.05 1.88 4.0 0.04
Due to statistical uncertainty the mean value of the diffusion coefficient D and the surface concentration CS are assumed to be uncertain themselves. As a part of the planned condition monitoring of the structure half-cell potential measurements are performed on a regular basis. It is the aim to utilize the measurements in order to update the probability of repair and thus the expected future maintenance costs. In order to quantify the effect of the performed inspections the indicator corresponding to the half-cell potential measurements needs to be established. According to tests performed by Marschall [8] the probability distribution function for the half-cell measurement may be described by a normal distribution function. Given that corrosion has initiated the mean value and standard deviation are 0.354 and 0.08 V, respectively. Given that no corrosion has initiated the mean value and the standard deviation are 0.207 and 0.0804 V, respectively. A central question concerns what measurement result to assign with an observation of ‘‘corrosion’’ or ‘‘no corrosion’’, i.e. the choice of the indicator. Choosing somewhat arbitrarily as an indicator for corrosion initiation a potential reading corresponding to the upper 10% quantile value for the measurements given corrosion this value corresponds to 0.2515. We then have potential P ID j CI ¼ 0:10; P ID j CI ¼ 0:71 where CI denotes the event of corrosion initiation at the time where the measurement is performed and ID the event of no indication. The probability of getting an indication of no-corrosion initiation given that corrosion exists is 0.1 and the probability of getting an indication of no corrosion initiation given that no corrosion exists is 0.71. Correspondingly we have PðID j CIÞ ¼ 0:90; P ID j CI ¼ 0:29 where ID denotes the event of an indication of corrosion. Later on the significance of the choice of the indicator will be assessed further. 4.1. Localised corrosion The inspection and maintenance planning for the first 30 years of the life of the structure is considered and it is assumed that visual inspections are performed in connection with ordinary maintenance activities every 5 years. The branching probabilities in the event tree in Fig. 6 are calculated using FORM/SORM systems reliability analysis using in principle the safety margin given in Eq. (26) with and without the propagation time depending on the considered branching event. The random variables as defined in Table 1 are used for the probabilistic modelling. The inspection costs entering into Eq. (14) are evaluated as the costs of performing visual inspections every 5 years and the costs of the half-cell potential measurements to be performed only once
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during the considered 30 year period. The costs of repair R2 are assumed associated with the costs of performing a preventive measure such as an exchange of only the concrete cover. These costs have been assessed to US$ 1250 per square meter. The failure costs in Eq. (14) are here considered to be the costs due to major repairs, i.e. of type 1. These have been assessed to US$ 5500 per square meter. The costs of visual inspections and the costs of half-cell potential measurements have been assessed to US$ 10 and US$ 100 per square meter, respectively. It is assumed that all costs are assessed in terms of their net present value and a real rate of interest net of inflation equal to 4% is used. The expected total costs have been evaluated as function of the time of the half-cell potential measurement and the results are shown in Fig. 4. From Fig. 4 it is seen that the cost optimal time to perform the half-cell potential measurement is after 20 years. The evaluation of the expected costs as shown in Fig. 4 have been evaluated using the value of 0.2551 V as an indication of initiated corrosion, as discussed previously. However, this value may not be optimal and therefore a study has been performed in order to identify the cost optimal choice of the indicator value. In Fig. 5 the total expected costs of the optimal inspection plan are shown as function of the indicator value. From Fig. 5 it is seen that in the present example indicator values in the order of 0.4 V are cost optimal, however, this depends on the actual condition of the considered structure together with the relative costs of the different repairs and not least the assumed real rate of interest. For the present case the optimum choice of indicator value indicates that it is optimal to avoid erroneous interpretations in the situation where there is in fact no initiated corrosion, whereas it is not so important to make such errors when corrosion has initiated. For structures in poorer condition this effect could be reversed. 4.2. Distributed corrosion Further it is assumed that the considered structure in terms of its corrosion characteristics may be modelled as an assembly of 100 components and when any 50 of these components exhibit initiated corrosion a major repair will be performed.
Fig. 4. Total expected costs as function of time till half-cell potential measurement.
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Fig. 5. Total expected costs for optimum inspection plan as function of the choice of indicator value. Note that the absolute value of the indicator value is given.
Fig. 6. Strength of the indicators p and q as function of the inspection time.
The ensemble of the 100 components may thus be considered as 100 components with a common failure probability (evaluated on the basis of the physical uncertainties only), which is uncertain in itself due to the remaining statistical and model uncertainties. Assuming for the sake of illustration that the individual components may be assumed statistically independent the probability of having more than 50% of the components exhibiting visual corrosion may be assessed through ð1 PðnF ðtÞ 5 50Þ ¼ 1
Bð50 1; 100; ðtÞÞ fðtÞ ððtÞÞdðtÞ
ð27Þ
0
where Bð50 1100; ðtÞÞ is the cumulative Binomial distribution (see Faber and Rostam [4]) and ðtÞ is the uncertain probability of visual corrosion of the individual components at time t determined by simulation and nested FORM/SORM analysis on the limit state equation given in Eq. (26). fðtÞ ððtÞÞ is the prior or posterior PDF for the probability of visual corrosion depending on
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the state of knowledge. The strength of the indicator p ‘‘no corrosion initiation’’ and q for ‘‘corrosion initiation’’ for the half-cell potential measurement are shown as a function of time in Fig. 6. The prior probability density function for the probability of visual corrosion for one component after 50 years is shown in Fig. 7 (Prior). Half-cell potential measurements are performed for 10 components after 25 years with the result that a) ‘‘no corrosion’’ and b) ‘‘corrosion’’ has initiated. Based on the indicator for ‘‘no corrosion initiation’’ and ‘‘corrosion initiation’’ using half-cell potential measurements the posterior probability density function for the uncertain probability of visual corrosion is evaluated using Eqs. (11)–(13) and illustrated in Fig. 7. The probability of extensive repair, i.e. more than 50% of the structure exhibiting visual corrosion is evaluated using Eq. (24) and illustrated in Fig. 8 as function of the time where the inspection is performed. From Fig. 8a and b it is seen that the inspection using half-cell measurements will not provide any knowledge in regard to the development of excessive visual corrosion until after about 10 years. This is because the probability of initiation of corrosion is negligible until this point in time. Thereafter the indication of no-corrosion and corrosion respectively is seen to have a significant impact on the probability of more than 50% of the concrete surface exhibiting visual corrosion.
Fig. 7. Posterior PDF for the probability of visual corrosion of one component given (a) no indication, (b) indication.
Fig. 8. (a) No indication of corrosion. (b) Indication of corrosion.
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Fig. 9. The strength of the indication (a) no-corrosion and (b) corrosion for different choices of the indicator.
Fig. 10. Probability of repair as function of time for different choices of the indicator.
Returning now to the question regarding the choice of indicator the strength of the indicators p and q is depicted in Fig. 9 for different choices (5, 10 and 25%) of the quantile values in the probability distribution function for the hall-cell potential reading given corrosion has initiated. From Fig. 9 it is seen that the strength of the indicator depends significantly on the choice of the indicator. This dependency is also carried over to the updated probability of excessive repair as shown in Fig. 10. The optimal choice of the quantile value in the distribution function for the half-cell potential reading given corrosion, which is associated with an indication of corrosion, may be solved along the lines as illustrated for the case of localised corrosion. This requires the simultaneous consideration of the effect on the updated probability of excessive repair as shown in Fig. 10, but also consideration of the maintenance activities, which may be implemented in order to reduce the risk of excessive repair after 50 years.
5. Discussion and conclusions A Bayesian approach to assessment and inspection planning has been formulated on the basis of a general methodology for indicator based quality control. The approach allows for the con-
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sideration of problems where the components have uncertain defect rates and where tests are performed on condition indicators rather than the condition of interest directly. The theoretical derivations have been based on the assumption that the probability of achieving an indication of a defective (or non-defective) component given that the components is defective (or non-defective) is independent on the defect rate for the considered components. This assumption appears to be fulfilled for most practical applications, at least in cases where testing and inspections are automated and/or largely independent of human interaction. In cases where the quality of the inspections is highly influenced by human behaviour it is imaginable that the quality of inspections might be influenced by the defect rate due to the effect that if defective components are rare then the awareness of the inspector might decrease. This situation might, however, be seen as a quality control problem of the inspection itself and is thus not really a limiting factor for the proposed methodology. A further assumption made concerns the requirement that defect rates may be defined for well defined populations of components. In engineering applications this requirement necessitates careful considerations in regard to failure modes, deterioration mechanisms, boundary conditions etc. It will thus often be necessary to perform a differentiation of the considered components into two or more populations in order to fulfil this requirement. The approach may readily be utilized for the optimal planning of tests, inspections and maintenance activities of components within the framework of Bayesian decision analysis. The Bayesian formulation of condition indicators is especially well suited to consistently incorporate subjective knowledge in the test and inspection planning and the treatment of defect rates as being uncertain facilitates the development of quality control schemes for populations of components for which little if any frequentistic information is available. The approach is demonstrated on an example considering the assessment and inspection and maintenance planning of a concrete structure. From the example it is seen that the concept of Bayesian indicators might be a very useful tool providing important decision support in connection with the planning of inspection and repair measures. A number of further potential application areas for the developed methodology are yet to be explored. These include the important aspect of quantitative probabilistic modelling of the quality of inspections and tests as a function of the inspection coverage such as e.g. the quality of inspections of pressure vessels subject to corrosion processes. As in the case where corrosion deterioration of concrete structures is considered it is, however, required that information in regard to the spatial correlation of the deterioration process is available in order to perform such investigations. The spatial correlation structures of different deterioration processes furthermore appear to be rather case dependent and much work (experimental) is lying ahead before robust models for this purpose may be established. References [1] Benjamin JR, Cornell CA. Probability, statistics and decision for civil engineers. NY: McGraw-Hill; 1970. [2] Faber MH, Sorensen JD. Bayesian sampling using condition indicators. In: Proc. to the ICOSSAR, July 2001, Newport Beach, CA. [3] Lindley DV. Introduction to probability and statistics from a Bayesian viewpoint. Cambridge: Cambridge University Press; 1976.
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[4] Raiffa H, Schlaifer R. Applied statistical decision theory. Cambridge, Mass: Harward University Press, Cambridge University Press; 1961. [5] Hergenro¨der M. Zur statistischen Instandhaltungsplanung fu¨r bestehende Bauwerke bei Karbonatisierung des Betons und mo¨glicher Korrosion der Bewehrung‘‘ Berichten aus dem Konstruktiven Ingenio¨rbau. TU Mu¨nchen, 4/92, 1992 [in German]. [6] Engelund S, Sørensen JD, Sørensen B. Evaluation of repair and maintenance strategies for concrete coastal bridges on a probabilistic basis. ACI Materials Journal; 1998. [7] Madsen HO, Krenk S, Lind NC. Methods of structural safety. Englewood Cliffs, NJ: Prentice-Hall; 1986. [8] Marschall SJ. Evaluation of instrument-based, non-destructive inspection methods for bridges. MSc thesis, University of Colorado, Boulder, Colorado; 1996.