Introduction

Bridges are the crucial components of highway networks. In recent years, there has been growing awareness about the problems associated with the existing old bridges which needs the robust and efficient tools for maintenance and for management of these bridges with limited budgets. A bridge management system provides pro-active early warning or cautionary measures to be undertaken through a set of guidelines/procedures that would also be cost effective towards maintenance of a bridge. BMS assists in determining the optimal time for an agency to execute improvement activities on a bridge, given the availability of funds. The objective of adopting such a system is to improve the overall condition of an agency’s network of bridges, emphasizing the need to maintain the condition of a bridge before lapsing into an unsafe state [1]. Efforts have been made to develop BMS since 1980s. Design of a BMS has grown from the early development of database modules for the storage of elementary information on inventory details, inspection records, actions and costs to become more sophisticated [2–4]. Derivation of a method for converting condition states to meaningful, decision influencing, interpretative results is desirable. A computer driven process can assist in making these decisions, but a flexible approach controlled by engineering judgment should always be used in finalizing decisions [5].

Generally, the field measured data is not beyond the doubt of exactness and representatively has a high degree of uncertainty associated with it. Since past three decades, number of researchers were involved in condition assessment of bridges through both deterministic approach and probabilistic approach [6–9]. But, in many cases, it has been found that the methodologies may not be always very efficient to reflect the condition of existing Reinforced Concrete (RC) bridges.

Human judgment plays an important role in the condition assessment of any structure. This is especially true for the structures which are subjected to very severe loading environments. The process of damage assessment includes interpretation of the data obtained from visual inspection of the structure, in situ measurements and laboratory investigations, if any. Based on the available information, a professional, an authority, an expert, or a group would usually make an assessment on the condition of the structure. A rational approach of damage assessment should be based on a balanced combination of heuristic art and science applicable to the domain. A combination of both numeric (quantitative) and symbolic (qualitative) information can be the basis for the assessment of damage that might have occurred in the structure. Qualitative knowledge (judgement) is used extensively in engineering calculations in order to compensate for the lack of data and for extrapolation outside a given range of data [10–12]. The bridge condition rating is the datum for any bridge management system. The usefulness of a bridge management system and the accuracy of bridge rating rely upon the bridge condition data which constitute subjective judgement and intuition of the bridge inspector. The aim of the bridge condition rating is to evaluate the structural strength and serviceability condition of an existing bridge [13–17].

The aim of the present work is to develop an engineering decision making system to assist bridge inspector/engineer during the inspection and evaluation of the condition of an existing reinforced concrete bridge and to proceed with the maintenance aspects. Towards this, an expert system has been developed for condition assessment of bridges which has the capability to process both symbolic and numerical data pertaining to visual inspection, destructive tests, non-destructive tests and chemical analysis results carried out on the bridge. To enhance the qualitative decision making capability, the knowledge is augmented with the numerical algorithms pertaining to the structural damage identification methods which will qualitatively as well as quantitatively assess the damage by identifying the location and extent of damage in the structure using the numerical methods based on test (static and dynamic) results. Based on the diagnosis and assessment results, the rating of the bridge is performed using a systematic procedure incorporating fuzzy mathematics. The robust system developed as part of this work will empower the bridge inspectors/engineers to carry out the inspection, damage assessment, maintenance and repair/retrofitting of the bridges in a systematic way.

Systematic Approach for Bridge Condition Assessment and Rating

For developing an efficient, robust and reliable decision support system for sustainable bridges, certain actions need to be taken such as an inventory of all structures, inspection planning, periodic inspection of the structures, allocation of required maintenance funds and carrying out maintenance on time. The systematic approach followed towards developing a complete knowledge based decision support system for condition assessment of bridges is shown in Fig. 1. For a complete set of bridges in an object or network level, the approach for maintenance and management starts with general visual inspection which will be followed regularly. If any major distresses were observed during the inspection stage, it goes into the next level to set the priority to carry out the maintenance strategy based on the allocation of budgets, project needs etc. The results obtained from the preliminary stage will be taken into the detailed investigation stage of the expert system which will make the diagnosis of the distress and finds the cause of the distress. These inputs will be supported with the numerical algorithms which will analytically identify the properties of the distress e.g. depth and width of the crack etc. The final assessment results will be taken as input to the rating tool which will rate the condition of the bridge.

Fig. 1
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A systematic approach for condition assessment of bridges

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Development of Knowledge Based Expert System

Condition assessment of bridges is an emerging topic in the present scenario. Expert systems are widely used for the condition assessment of bridges, where the data available is more heuristic in nature than numerical. Expert system is the best method for handling linguistic data. A comprehensive decision making system requires input not only from visual inspection but also from that of confirmatory non-destructive and partially destructive tests [18]. The required data can be collected through detailed inspections on the bridge. Results of these inspections on health/condition of the structure and its components are vital for the condition assessment and form part of the input for the decision support system [19]. A schematic representation of the architecture of the decision support tool for distress diagnosis of bridges is shown in Fig. 2.

Fig. 2
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Architecture of the decision support system

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The system comprises of graphical user interface, which helps the user to provide the input data required for working of the system. As a first step, preliminary assessment is carried out by the system developed in this study based on the observations of visual inspections to evaluate the cause of distress. In the detailed query stage, the user is queried for more details on distress which are based on the data obtained from non-destructive, laboratory and chemical tests. The results thus obtained are checked against those of analytical feedback, wherein the parameters of cracks due to different causes are evaluated numerically. By this process, the system attempts to diagnose the correct cause of distress. Uncertainty associated with the input data is handled using confidence factors. Based on the diagnostic data, the expert system will provide a report on the condition of the bridge as moderately damaged, fairly damaged, fully damaged, or not damaged. Detailed report on reasons for the distress is also given as output. Necessary explanation is provided throughout the consultation process, so that the user can easily understand what each attributes represents.

Knowledge Representation

The knowledge base of the decision support system consists of data pertaining to different types of distresses. Detailed knowledge about major sources of distresses in bridges which are due to cracking, spalling, corrosion, creep, shrinkage, thermal movement, plastic settlement, chemical attack etc. has been incorporated in the knowledge base of the expert system. The causes for the distress suffered by the bridge have been arrived based on the data such as geometric parameters of the bridge, type of bridge, visual observations, results of destructive, non-destructive tests and chemical analysis supplied by the user as given in Table 1. The knowledge base has been represented in an object oriented manner, as it is an efficient method and the knowledge base can be easily updated based on the future needs. A combined backward and forward chaining inference strategy is adopted in the expert system. The goal of the expert system is defined as assessment of distress, and sub-goals are defined as causes for different types of distress. In the expert system, backward chaining strategy is used to find the cause of distress by taking goals one by one for each type of distress and proceeding along the chain of rules associated with that particular goal. Backward chaining strategy is adopted because, in the selection, the set of all feasible distress types are finite and yet the rules for selecting an appropriate cause are numerous. Forward chaining is used for the processing of input from various sources and for the acquisition of knowledge. Forward chaining is adopted for the known facts like general details of bridge and other common visual observations.

Table 1 Details required by input data module of the expert system
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Symbolic-Numeric Processing

Typical distresses which can be observed in bridge decks are shown in Fig. 3. Several knowledge frame nets have been developed for each type of distress which will give the flow of data for assessing the causes of distress. A typical knowledge net is shown in Fig. 4. The symbolic damage assessment is carried out for the bridge deck within the frames of the knowledge base. The symbolic data which is obtained mostly from visual observations as given in Table 2 is supported by the numerical data for various causes of distress as shown in the knowledge frames. The numerical data is obtained mostly from the experiments/tests conducted at field as well in the laboratory. The visual observations are supported by the test results. To perform the assessment, the expert system starts the query for getting the data on visual observation for each component of the bridge viz., girders and slab and prompts the user in identifying the types of damage and also to describe the relevant characteristics of each type of damage during detailed query.

Fig. 3
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Types of cracks (general)

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Fig. 4
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Typical knowledge net for crack diagnosis

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Table 2 Parameters to be considered for evaluation of different types of distress
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Validation of the System

The expert system tool developed in this study has been validated using the assumed data for different types of distresses. The causes for the distress are assumed suitably for the validation. The results obtained from the expert system through the conclusion displays and necessary explanation was provided are shown in Fig. 5 which gives the condition of a particular bridge under consideration. From the validation studies, it has been noted that the expert system could identify the correct reason for the distress and provide satisfactory conclusion on the condition of the bridge. The expert system developed in this study also provides a report on the condition of the bridge as moderately damaged, fairly damaged, fully damaged, or not damaged. Also, the expert system gives a report on the condition of concrete, reinforcing steel and details of cracking.

Fig. 5
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Typical conclusion displays and report on assessment of causes

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Fuzzy Logic Approach for Condition Rating of the Bridges

The aim of the bridge condition rating is to evaluate the structural strength and serviceability condition of an existing bridge [20, 21]. A procedure like fuzzy logic would be useful to handle the uncertainty, imprecision and subjective judgement [22]. It has been found from the existing literature that extensive studies have been carried out to evaluate the condition of different structures using fuzzy logic [17].

But, it is noted that the methods are either too simplistic [22] which would not reflect the proper condition of the structure or very complex [23] that needs a thorough understanding of the methodology and considerable computation time for solving the problem. Further, it is also observed that some of the key issues viz., determination of membership functions, priority vector, final mapping and processing of non-convex fuzzy set which are vital for condition evaluation and rating of bridges using fuzzy logic, did not receive much attention. In this work, a systematic procedure and formulations were developed for condition rating of existing bridges using fuzzy mathematics combined with eigenvector based priority setting technique with emphasis on the above mentioned key issues.

To carry out systematic rating of existing bridges, the essential requirement is the input data from the bridge inspector that consist of the rating and importance factors for the relevant elements of bridge which would reflect the overall condition of a bridge as a whole. The bridge is divided into three major components, namely, ‘deck’, ‘superstructure’ and ‘substructure’. Each component is further divided into number of elements. The deck, superstructure and substructure have 13, 16 and 20 elements respectively as shown in Table 3 [24]. The bridge inspector is required to assess the condition of each element individually. The rating evaluation for that particular component is carried out based on the rating of the constituent elements. This process is repeated for all the three components towards final rating of the bridge.

Table 3 Decomposition of a bridge into elements with observed ratings
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Development of Importance Factors

In a bridge condition evaluation, rating of each element under a particular component does not influence the component’s overall structural condition rating in a similar degree. Some elements play more critical structural role than the others. The importance factors for the elements at various deterioration stages should be evolved from the response of competent bridge inspectors/experts to the inspection questionnaire formulated for opinion survey. In this study, a scale of 1–9 has been considered for rating of the elements. An element with rating value of 9 signifies the best possible condition without distress and the descending rating numbers represent the increased degree of distress. The rating values below 1 reflect the immediate replacement of the same. The fuzzy membership values of structural importance are evaluated for the elements of deck, superstructure and substructure and for a deck these are given in Tables 4, 5. The mean value of the importance of an element increases as the physical condition deteriorates.

Table 4 Mean values of the structural importance for the bridge deck elements for different rating conditions
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Table 5 Mean values of the structural importance for the bridge superstructure elements for different rating conditions
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Fuzzification of Input Data Obtained from Bridge Inspectors

If Rn is a fuzzy set, representing rating of an element (where ‘n’ represents rating number i.e. n = (0,1,…,9), the general form of the membership function can be formed as follows:

$${\text{R}}_{\text{n}} ={\upmu}_{\text{m}} \left({{\text{r}}_{\text{m}}} \right)|{\text{r}}_{\text{m}} \quad \left({{\text{m}} = 0,1,2, \ldots,9} \right)$$
(1)

where, μ(r) is a membership function representing the degree of membership of any fuzzy set and 0 ≤ μ ≤ 1. The function as described in Eq. (1) quantifies of the ambiguity associated with the rating of any element of a bridge.

In this study, the rating membership functions for ‘0’ and ‘1’ are assumed as follows:

$$\begin{aligned} {\text{R}}_{0} & = \, \left\{ {1.00 \, \left| { \, 0, \, 0.76 \, } \right| \, 1, \, 0.55 \, \left| { \, 2, \, 0.35 \, } \right| \, 3, \, 0.16 \, \left| { \, 4, \, 0.00 \, } \right| \, 5, \, 0.00 \, \left| { \, 6, \ldots ,0.00 \, } \right| \, 9} \right\}\;{\text{and}} \\ R_{1} & = \, \left\{ {0.00 \, \left| { \, 0, \, 1.00 \, } \right| \, 1, \, 0.45 \, \left| { \, 2, \, 0.00 \, } \right| \, 3, \, 0.00 \, \left| { \, 4, \, 0.00 \, } \right| \, 5, \, 0.00 \, \left| { \, 6, \ldots ,0.00 \, } \right| \, 9} \right\} \\ \end{aligned}$$

Using fuzzy addition, rating membership functions for ‘2’ is calculated as

$${\text{R}}_{2} = \, \left\{ {0.00 \, \left| { \, 0, \, 0.45 \, } \right| \, 1, \, 1.00 \, \left| { \, 2, \, 0.70 \, } \right| \, 3, \, 0.45 \, \left| { \, 4, \, 0.20 \, } \right| \, 5, \, 0.00 \, \left| { \, 6, \ldots ,0.00 \, } \right| \, 9} \right\}$$

and the rating membership function for ‘3’ is computed as,

$${\text{R}}_{3} = \, \left\{ {0.00 \, \left| { \, 0, \, 0.28 \, } \right| \, 1, \, 0.62 \, \left| { \, 2, \, 1.00 \, } \right| \, 3, \, 0.80 \, \left| { \, 4, \, 0.62 \, } \right| \, 5, \, 0.45 \, \left| { \, 6,0.28} \right| \, 7, \, 0.13\left| { \, 8,0.00 \, } \right| \, 9} \right\}$$

similarly, rating membership functions for ‘4’ through ‘9’ are developed.

Fuzzy membership functions for rating values 0–9, as obtained above, are shown in Fig. 6.

Fig. 6
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Degree of membership of fuzzified rating values

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Fuzzy Resolution Identification Technique

A fuzzy set can be easily decomposed into its level sets or intervals through resolution identity as suggested by the researchers [25]. If A is a fuzzy set of universe (U), then an α-level set or alpha cut of A is a non-fuzzy set denoted by Aα which comprises of all elements of U whose grade of membership in A is greater than or equal to α.

Aα can be expressed in symbolic form as:

$${\text{A}}_{\upalpha} = \{{\text{u}}|{\upmu}_{\text{A}} \left({\text{u}} \right) \ge {\upalpha}\}$$
(2)

In mathematical form, the fuzzy set A can be decomposed into its level sets through the resolution identity such that

$${\text{A}} = \sum\limits_{\alpha = 0}^{1} {\alpha A_{\alpha } } \quad {\text{or}},\quad {\text{A}} = \int\limits_{0}^{1} {\alpha A_{\alpha } }$$
(3)

where, \(\alpha A_{\alpha }\) is the product of a scalar \(\alpha\) with the set \(A_{\alpha }\), and the symbol \(\int\nolimits_{0}^{1} {}\) (or \(\sum\limits_{\alpha } {}\)) is the union of the \(A_{\alpha }\), with \(\alpha\) ranging from 0 and 1.

The minimum (pessimistic) and maximum (optimistic) values of the intervals for a specific level set correspond respectively to the lower and upper limits of fuzzy membership function at that α-level. The set describing the rating of a component at a particular level of α would be

$$R_{\alpha } = \frac{{\sum\nolimits_{i = 1}^{n} {W_{i\alpha } R_{i\alpha } } }}{{\sum\nolimits_{i = 1}^{n} {W_{i\alpha } } }}$$
(4)

where, R iα is the rating value for the ith element at α-level, W iα is the importance value for the i th element at α-level. Therefore, the most pessimistic and optimistic range of the resulting set at each α-level would form all possible combinations using the discretised non-fuzzy values. Hence, the resolution identity technique provides a convenient way of generalizing various concepts associated with non-fuzzy sets to fuzzy sets.

Implementation of the Methodology

In the proposed approach, priority setting values of elements are calculated to evaluate the power of importance of each element in describing the condition of a particular component. The usual techniques available for condition rating combine the rating and importance of elements to arrive at the rating of each component. But, the importance factor, as mentioned earlier, is very much dependent on the prevailing condition (rating) of the particular element. Thus, a minor element with worse condition may unnecessarily reduce the rating value of that component under which the element is grouped. This problem can be tackled by the introduction of power of importance which is independent of the prevailing condition of elements. Computer program has been developed based on the formulations presented in the preceding sections for rating of existing bridges. In the first phase, the program computes the component ratings as well as final rating of a bridge using eigenvector based priority setting approach combined with FWA. Whereas, the second phase of the program determines the final rating of the bridge by processing the rating values of components computed using FWA and the assigned importance of the components over the bridge as a whole using resolution identity technique. Using the above mentioned algorithm, the final rating of the bridge (RR) can be evaluated by processing the rating values of components computed using FWA and the assigned importance of the components over the bridge as a whole.

Computation of Final Rating of the Bridge Using the Unified Approach

As described earlier, the resolution identity technique is adopted in this study to get the final rating of the bridge when the component ratings (from the elemental values) are computed using eigenvector based priority setting technique combined with FWA method. Hence, the basic data considered are the calculated ratings of the components and assigned importance functions for the same. So, the input parameters will be same as presented in Tables 6 and 7. The fuzzy membership functions of rating and importance of different components (deck, superstructure and substructure) thus obtained, are discretised using resolution identity technique. Here, each set is discretised into 11 α-levels (from 0.0 to 1.0 in step of 0.1). At each α-level, there would be 64 combinations to get the most optimistic (maximum) and pessimistic (minimum) range of the fuzzy set at that α-level. For 11 α-levels (as considered in this study) the optimistic and pessimistic ranges of the resultant set is shown in Table 8. For better illustration, the resolution identification of the fuzzy set representing the rating of the deck component of the bridge concerned is shown in Fig. 7a. Further, the membership representation of the Resultant Rating (RR) derived from the pessimistic and optimistic ranges using resolution identity technique is shown in Fig. 7b. The resultant rating of the bridge, as a whole, has been defuzzified, to get the rating value of the bridge. For this particular case, the defuzzification has been executed using the centroidal method and the rating value is obtained as ‘4.6668’. From the result, it is clear that the rating of the bridge falls in between 4 and 5 but closer to 5. It may be the decision maker’s discretion in taking the exact value depending on the practical condition and other factors like the environmental condition, importance of the bridge as a whole on the societal service etc.

Table 6 Computed Fuzzified Rating values for different components of the bridge
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Table 7 Importance membership functions of the components
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Table 8 Most optimistic and pessimistic range of resultant rating at different α-levels
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Fig. 7
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a Resolution identification of the fuzzy set representing rating of the deck component. b RR of the bridge considered in this study

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Conclusions

In the present study, it is aimed to develop an engineering decision making system to assist bridge inspector/engineer during the inspection and evaluation of the condition of an existing reinforced concrete bridge and to proceed with the maintenance aspects. The main component of the decision making system is a Knowledge Based Expert System for condition assessment of reinforced concrete bridges which is capable of handling large data (both symbolic and numeric types) and gives most probable reason for the distress and finally assesses the condition of the bridge. For the expert system, a combined backward and forward chaining inference strategy has been successfully implemented.

The bridge condition rating is the datum for any bridge management system. So, a procedure like fuzzy logic would be useful to handle the uncertainty, imprecision and subjective judgement. It is seen from the present study that as the number of elements of bridge components increase the complexity in arriving at a unique rating number using Fuzzy Weighted Average (FWA) also increases. Hence, a resolution identity method is incorporated in the methodology to take care of the problems that may arise due to non-convexity and requirement of normalisation of the concerned sets. It is also found that the methodology is capable of handling any number of components with any number of divisions of rating values. Thus, the proposed methodology would help the decision maker and the bridge engineer to arrive at a systematic judgement and to formulate methodical steps towards retrofitting, rehabilitation or demolition of bridge in future years.

It is opined that the developed knowledge based decision making system would help the bridge inspectors immensely in selecting the parameters for carrying out proper inspection and to evaluate the condition and rating of the existing reinforced concrete bridges. It is also expected that the methodologies presented in this paper would pave the way for future research in fuzzy logic based condition assessment and rating of structures, in general, and bridges in particular.