**5. Case studies and some typical applications of fuzzy inference system for managing bridges**

As it is mentioned in earlier parts of this chapter and is shown in Fig. 1 the main concerns of bridge management systems are diagnosis of the encountered problems (deteriorations and/or damage detection) and finding the current condition of the bridge structure. It was also discussed that diagnosis and current condition determination accompany with ambiguity. In this section some case studies and applications are presented to show how fuzzy inference system can be used in bridge management issues.

### **5.1 A fuzzy system for concrete bridge damage diagnosis (DIASYN system)**

Bridge management systems (BMSs) are being developed in recent years to assist various authorities on the decision making in various stages of bridge maintenance, which requires, first of all, appropriate preliminary deterioration diagnosis and modeling.

Diagnosis Synthesis (DIASYN) is a fuzzy rule-based inference system for bridge damage diagnosis and prediction which aims to provide bridge designers with valuable information about the impacts of design factors on bridge deterioration.

DIASYN is supposed to be a concept demonstration system for providing the bridge maintenance engineers and the bridge design engineers with assistance to obtain preliminary but important knowledge on individual bridge defects.

The DIASYN system incorporates a fuzzy reasoning process containing a rule base with its acquisition and update facility and a fuzzy inference engine with an explanation facility, and a user interface with option selecting capacity. Fuzzy logic is utilized to handle uncertainties and imprecision involved. The rules are if-then statements that describe associations between fuzzy parameters. Given the required input data, the inference engine evaluates the rules and generates an appropriate conclusion. Users can choose to make diagnoses of new cases or to update the rule base with new training data through the user interface.

Fuzzy Inference System as a Tool for Management of Concrete Bridges 461

The 506 bridge maintenance projects dataset is randomly split into two sample sets: training dataset with 390 projects and testing dataset with 116 projects. Both the training and testing

Inputs to the ANFIS are safety risk rating (SRR), functionality risk rating (FRR), sustainability risk rating (SURR), and environment risk rating (ERR). All inputs range from 0 to 3 with 0 representing *no risk*, 1 *low risk*, 2 *medium risk* and 3 *high risk*. Output to the

With the 390 training dataset, two generalized bell-shaped membership functions are chosen for each of the four inputs to build the ANFIS, which leads to 16 if–then rules containing 104 parameters to be learned. Fig. 7 shows the model structure of the ANFIS that is to be built

> **Rules with** *and* **logical operations**

The developed ANFIS system for bridge risk assessment learns the if–then rules between bridge risk scores and risk ratings from the past bridge maintenance projects and memorizes them for generalization and prediction. It has been observed that ANFIS outperforms artificial neural networks to perform better than multiple regression models (Wang & Elhag, 2007). Differing from artificial neural network, ANFIS is transparent rather than a black box. Its if–then rules are easy to understand and interpret. In this case study the performances of the ANFIS and ANN in modeling bridge risks are compared, where the two models are trained using the same training dataset and validated by the same testing dataset. Comparison shows that the ANFIS has smaller root mean squared error and mean absolute percentage error as well as bigger correlation coefficient for both the training and testing datasets than the ANN model. In other words, the ANFIS achieves better performances than

**Output membership functions**

**Output**

Risk scores (RSs)

ANFIS is the risk scores (RSs) of the 506 bridge projects, which ranges from 5 to 99.

datasets cover all levels and types of bridge risks.

for bridge risk assessment in this study.

**Inputs**

Safety risk rating (SRR)

Functionality risk rating (FRR)

Sustainability risk rating (SURR)

Environment risk rating (ERR)

**Inputs membership functions**

Fig. 7. Model structure of the ANFIS for bridge risk assessment

The fuzzy rules provide associations between observed bridge conditions and damage causes. They are created by a rule generation algorithm that can convert crisp training data into fuzzy statements. The training data are collected from bridge inspection records and formalized into standard vectors. In the operational mode, the system reads a state vector of observed bridge condition and the inference engine performs damage cause implication through evaluation of the rules. The output of this implication procedure is a linguistic variable that describes the possible damage cause with a confident degree. This linguistic variable can be defuzzified by the explanation facility if a crisp output is desired. In the updating mode, new training vectors are input to generate new rules together with the existing training data. New rules, if any, will be installed in the rule base before the system gives a prompting of *updating finished* as output.

Inputs of DIASYN are:


The inference engine in DIASYN basically executes Mamdani's original reasoning procedure. The overall firing strength of the individual rule whose antecedents are connected with an *AND* operator, the intersection, is typically determined by taking the minimum value of the individual firing strengths of the antecedents.

After system training it is ready to be used to diagnose new bridge deterioration case. Two test examples are use for system verification, one for crack diagnosis and one for spalling diagnosis. The input data of the bridge including survey and inspection information which shows that a crack occurs in superstructure with a specific condition mark, and a spalling in support-structure with another given condition mark. The inference results, along with expert opinions indicate that the particular crack was caused by 'loads and its likes' with a confidence degree of 'very true', and that the spalling was caused by 'others' with a confidence degree of very true. Both of the results are in accordance with the expert opinion, which suggests 'overloaded' and 'aging' are the causes of the crack and spalling, respectively (Zhao & Chen, 2001, 2002).

### **5.2 An adaptive neuro-fuzzy inference system for bridge risk assessment**

Bridge risks are often evaluated periodically so that the bridges with high risks can be maintained timely. Modeling bridge risks is a challenging job facing Highways Agencies because good mathematical models can save them a significant amount of cost and time.

In this case study an adaptive neuro-fuzzy system (ANFIS) using 506 bridge maintenance projects for bridge risk assessment is introduced. The system can help British Highways Agency to determine the maintenance priority ranking of bridge structures more systematically, more efficiently and more economically in comparison with the existing bridge risk assessment methodologies which require a large number of subjective judgments from bridge experts to build the complicated nonlinear relationships between bridge risk score and risk ratings.

The fuzzy rules provide associations between observed bridge conditions and damage causes. They are created by a rule generation algorithm that can convert crisp training data into fuzzy statements. The training data are collected from bridge inspection records and formalized into standard vectors. In the operational mode, the system reads a state vector of observed bridge condition and the inference engine performs damage cause implication through evaluation of the rules. The output of this implication procedure is a linguistic variable that describes the possible damage cause with a confident degree. This linguistic variable can be defuzzified by the explanation facility if a crisp output is desired. In the updating mode, new training vectors are input to generate new rules together with the existing training data. New rules, if any, will be installed in the rule base before the system

Design factors, i.e. structural type, span length, deck width, number of spans, wearing

Environmental factors, i.e. humidity and precipitation, climate region, traffic volume,

The inference engine in DIASYN basically executes Mamdani's original reasoning procedure. The overall firing strength of the individual rule whose antecedents are connected with an *AND* operator, the intersection, is typically determined by taking the

After system training it is ready to be used to diagnose new bridge deterioration case. Two test examples are use for system verification, one for crack diagnosis and one for spalling diagnosis. The input data of the bridge including survey and inspection information which shows that a crack occurs in superstructure with a specific condition mark, and a spalling in support-structure with another given condition mark. The inference results, along with expert opinions indicate that the particular crack was caused by 'loads and its likes' with a confidence degree of 'very true', and that the spalling was caused by 'others' with a confidence degree of very true. Both of the results are in accordance with the expert opinion, which suggests 'overloaded' and 'aging' are the causes of the crack and spalling,

Bridge risks are often evaluated periodically so that the bridges with high risks can be maintained timely. Modeling bridge risks is a challenging job facing Highways Agencies because good mathematical models can save them a significant amount of cost and time.

In this case study an adaptive neuro-fuzzy system (ANFIS) using 506 bridge maintenance projects for bridge risk assessment is introduced. The system can help British Highways Agency to determine the maintenance priority ranking of bridge structures more systematically, more efficiently and more economically in comparison with the existing bridge risk assessment methodologies which require a large number of subjective judgments from bridge experts to build the complicated nonlinear relationships between bridge risk

Other factors, such as structure age, function class and location of damages.

**5.2 An adaptive neuro-fuzzy inference system for bridge risk assessment** 

minimum value of the individual firing strengths of the antecedents.

gives a prompting of *updating finished* as output.

surface type, skew angle, etc.,

temperature variations, etc.,

respectively (Zhao & Chen, 2001, 2002).

score and risk ratings.

Inputs of DIASYN are:

The 506 bridge maintenance projects dataset is randomly split into two sample sets: training dataset with 390 projects and testing dataset with 116 projects. Both the training and testing datasets cover all levels and types of bridge risks.

Inputs to the ANFIS are safety risk rating (SRR), functionality risk rating (FRR), sustainability risk rating (SURR), and environment risk rating (ERR). All inputs range from 0 to 3 with 0 representing *no risk*, 1 *low risk*, 2 *medium risk* and 3 *high risk*. Output to the ANFIS is the risk scores (RSs) of the 506 bridge projects, which ranges from 5 to 99.

With the 390 training dataset, two generalized bell-shaped membership functions are chosen for each of the four inputs to build the ANFIS, which leads to 16 if–then rules containing 104 parameters to be learned. Fig. 7 shows the model structure of the ANFIS that is to be built for bridge risk assessment in this study.

Fig. 7. Model structure of the ANFIS for bridge risk assessment

The developed ANFIS system for bridge risk assessment learns the if–then rules between bridge risk scores and risk ratings from the past bridge maintenance projects and memorizes them for generalization and prediction. It has been observed that ANFIS outperforms artificial neural networks to perform better than multiple regression models (Wang & Elhag, 2007). Differing from artificial neural network, ANFIS is transparent rather than a black box. Its if–then rules are easy to understand and interpret. In this case study the performances of the ANFIS and ANN in modeling bridge risks are compared, where the two models are trained using the same training dataset and validated by the same testing dataset. Comparison shows that the ANFIS has smaller root mean squared error and mean absolute percentage error as well as bigger correlation coefficient for both the training and testing datasets than the ANN model. In other words, the ANFIS achieves better performances than

Fuzzy Inference System as a Tool for Management of Concrete Bridges 463

*Objective Data (NDT)*

**Bridge deck fuzzy rating system** (including 162 rules)

Bridge deck fuzzy condition rating

Corrosion probability

**Inspection Results (VI & NDT)**

*Subjective Data (VI)*

Crack width

Spalling

Delamination

Hammer tapping

Fig. 8. Type of inspection results

Crack width

Spalling

Hammer tapping

Corrosion probability

Delamination

Fig. 9. Proposed fuzzy system for bridge deck condition rating.

the ANN model. Therefore, ANFIS is a good choice for modeling bridge risks. Moreover, ANN is a black box in nature and its relationships between inputs and outputs are not easy to be interpreted, while ANFIS is transparent and its if–then rules are very easy to understand and interpret. But the drawback of ANFIS is its limitation to the number of outputs. It can only model a single output. In summary, ANFIS is a good choice and powerful tool for modeling bridge risks (Wang & Elhag, 2008).

#### **5.3 Fuzzy concrete bridge deck condition rating method for practical bridge management system**

Bridge management system (BMS) is a tool for structured decision making and planning/scheduling for bridge infrastructure inspection, maintenance and repair or retrofit. Any BMS is basically constructed based on data stored in inventory and inspection databases. One of the important and crucial efforts in managing bridges is to have some criteria to show the current condition of the elements of bridges based on the results from inspection data. As the results are not precise and are related to the depth and extent of the inspectors' expertise, there are some uncertainties in any evaluation. On the other side condition of bridges are rated linguistically in many cases with some kinds of vagueness in description of the bridge element conditions. Based on these facts in this case study a new fuzzy method is introduced to deal with these shortcomings from the uncertain and vague data. The fuzzy bridge deck condition rating method is practically based on both subjective and objective results of existing inspection methods and tools. The parameters of the model are selected as fuzzy inputs with membership functions found from some statistical data and then the fuzziness of the condition rating is calculated by the fuzzy arithmetic rules inherent in the fuzzy expert system. Since one of the most proven and experienced advantages of fuzzy inference systems is the tolerability for noisy (uncertain and vague) data it is believed that this proposed system can be an alternative method for current rating indices amongst many others which are almost used deterministically.

In this case study Fuzzy Inference System is used to translate the concrete bridge deck inspection results to condition rating (Tarighat & Miyamoto, 2009).

In literature the proposed rating methods are resulted from either visual inspection or nondestructive tests. Here, in order to enhance the capabilities of both methods (visual inspection and nondestructive tests) a hybrid inspection results is used to calculate the condition rating of the concrete bridge deck. Fig. 8 shows the type of inspection results.

The linguistic attributes of observed symptoms are defined as following fuzzy sets.

*A1 = {No; Yes} A2 = {NoCracks; HairlineCracks; WideCracks} A3 = {No; Maybe; Yes} A4 = {Firm; Moderately; Hollow; Very Hollow} A5 = {Low; Moderate; High}* 

Considering Gaussian membership functions for inputs and applying Mamdani's method as fuzzy inference system Fig. 9 shows the proposed system for predicting of bridge deck condition rating. For design of fuzzy inference system 162 rules are defined based on the experts' experience and available facts from previous inspection results. Finally Fig. 10 can be used to convert the crisp condition rating result to linguistic term.

Fig. 8. Type of inspection results

the ANN model. Therefore, ANFIS is a good choice for modeling bridge risks. Moreover, ANN is a black box in nature and its relationships between inputs and outputs are not easy to be interpreted, while ANFIS is transparent and its if–then rules are very easy to understand and interpret. But the drawback of ANFIS is its limitation to the number of outputs. It can only model a single output. In summary, ANFIS is a good choice and

Bridge management system (BMS) is a tool for structured decision making and planning/scheduling for bridge infrastructure inspection, maintenance and repair or retrofit. Any BMS is basically constructed based on data stored in inventory and inspection databases. One of the important and crucial efforts in managing bridges is to have some criteria to show the current condition of the elements of bridges based on the results from inspection data. As the results are not precise and are related to the depth and extent of the inspectors' expertise, there are some uncertainties in any evaluation. On the other side condition of bridges are rated linguistically in many cases with some kinds of vagueness in description of the bridge element conditions. Based on these facts in this case study a new fuzzy method is introduced to deal with these shortcomings from the uncertain and vague data. The fuzzy bridge deck condition rating method is practically based on both subjective and objective results of existing inspection methods and tools. The parameters of the model are selected as fuzzy inputs with membership functions found from some statistical data and then the fuzziness of the condition rating is calculated by the fuzzy arithmetic rules inherent in the fuzzy expert system. Since one of the most proven and experienced advantages of fuzzy inference systems is the tolerability for noisy (uncertain and vague) data it is believed that this proposed system can be an alternative method for current rating

In this case study Fuzzy Inference System is used to translate the concrete bridge deck

In literature the proposed rating methods are resulted from either visual inspection or nondestructive tests. Here, in order to enhance the capabilities of both methods (visual inspection and nondestructive tests) a hybrid inspection results is used to calculate the condition rating of the concrete bridge deck. Fig. 8 shows the type of inspection results.

Considering Gaussian membership functions for inputs and applying Mamdani's method as fuzzy inference system Fig. 9 shows the proposed system for predicting of bridge deck condition rating. For design of fuzzy inference system 162 rules are defined based on the experts' experience and available facts from previous inspection results. Finally Fig. 10 can

The linguistic attributes of observed symptoms are defined as following fuzzy sets.

**5.3 Fuzzy concrete bridge deck condition rating method for practical bridge** 

powerful tool for modeling bridge risks (Wang & Elhag, 2008).

indices amongst many others which are almost used deterministically.

inspection results to condition rating (Tarighat & Miyamoto, 2009).

be used to convert the crisp condition rating result to linguistic term.

**management system** 

*A1 = {No; Yes}* 

*A3 = {No; Maybe; Yes}* 

*A5 = {Low; Moderate; High}* 

*A2 = {NoCracks; HairlineCracks; WideCracks}* 

*A4 = {Firm; Moderately; Hollow; Very Hollow}* 

Fig. 9. Proposed fuzzy system for bridge deck condition rating.

Fuzzy Inference System as a Tool for Management of Concrete Bridges 465

**corroded, contaminated, or repaired area 610%** <sup>3</sup>

**corroded, contaminated, or repaired area 620%** <sup>4</sup>

**corroded, contaminated, or repaired area 625%** <sup>5</sup>

**corroded, contaminated, or repaired area P25%** <sup>6</sup>

Based on Table 3 the condition rating for typical slab and haunch girder under consideration are 6 and 5. Scaling is required to be able to compare the results, therefore the above mentioned numbers should be multiplied by 14.28 to get a 100-based score system. Table 4 provides the comparison. It is shown that results from proposed method can estimate the

> **Numerical value**

**Excellent condition: no contamination; no corrosion; no** 

**Very good condition: minor cracks, no spalls or delaminations; chloride contaminated or repaired areas 62% (of total deck area)** 

**Satisfactory condition: spalls or delaminations 62%; cracked,** 

**Fair condition: spalls or delaminations 65%; cracked,** 

**Poor condition: spalls or delaminations 610%; cracked,** 

**Critical condition: spalls or delaminations 615%; cracked,** 

**Failed condition (total loss of serviceability or functionality): extensive spalling, delamination, repaired areas P30%; maintenance required** 

Administration (FHWA), 1995; Morcous et al., 2003).

**Proposed fuzzy condition rating system** 

> **Linguistically index**

**Slab CR** 75.2 Bad 85.7 Critical

Table 4. Comparison of the condition ratings from two methods.

**fuzzy inference system and practice swarm optimization** 

**Girder CR** 78.9 Bad 71.4 Poor condition 7.5

**5.4 A two stage method for structural damage identification using an adaptive neuro-**

All the above three case studies are of diagnosis and assessment types. As declared earlier another important task of any BMS or BHMS is the possibility to locate damaged area or components. The present case study is about damage detection (Fallahian & Seyedpoor, 2010). In this case study, an efficient methodology is proposed to accurately detect the sites and extents of multiple structural damages. The proposed methodology has two main phases

**Numerical value** 

condition rating very well.

**Deck Component** 

Table 3. Condition rating system for concrete bridge decks (Federal Highway

**Description Condition rating (CR)** 

**Scaled (to 100) Condition Rating based on the selected references** 

> **Linguistically index**

> > condition 10.5

2

7

**Absolute value Difference of the proposed method to the selected references** 

**repaired area** <sup>1</sup>

Fig. 10. Concrete bridge deck condition rating in linguistic terms.

To verify the proposed method an inspected concrete bridge deck is used. The layout of inspection is shown in Fig. 11. The proposed method is applied to the red slab and green haunch girder of the deck shown in Fig. 11.

Fig. 11. Concrete bridge deck inspection layout.

Fuzzy condition rating method facilitates data collection in inspection process. No area calculation is required and it needs only the good judgment of the inspector to find out the condition rating. The inspection data is shown in Table 2. Since the symptoms and deterioration/damages of girders and slabs of a deck are totally similar the proposed model can be used for them during bridge inspection process.


Table 2. Inspection data for typical slab and haunch girder of the reinforced concrete bridge deck.

To compare these results with a well-defined and in-use condition rating method the following seven-state rating scale, which reflects the different damage states associated with chloride-induced corrosion is used (Federal Highway Administration (FHWA), 1995; Morcous, Lounis, & Mirza, 2003). Table 3 provides a summary description of the adopted condition rating system as the benchmark.

To verify the proposed method an inspected concrete bridge deck is used. The layout of inspection is shown in Fig. 11. The proposed method is applied to the red slab and green

Fuzzy condition rating method facilitates data collection in inspection process. No area calculation is required and it needs only the good judgment of the inspector to find out the condition rating. The inspection data is shown in Table 2. Since the symptoms and deterioration/damages of girders and slabs of a deck are totally similar the proposed model

> **Delamination condition**

**(Red Area)** 20 50 90 50 10 75.2

**(Green Area)** 80 70 40 70 90 78.9 Table 2. Inspection data for typical slab and haunch girder of the reinforced concrete bridge

To compare these results with a well-defined and in-use condition rating method the following seven-state rating scale, which reflects the different damage states associated with chloride-induced corrosion is used (Federal Highway Administration (FHWA), 1995; Morcous, Lounis, & Mirza, 2003). Table 3 provides a summary description of the adopted

**Hammer tapping condition**

**Corrosion probability condition** 

**Fuzzy condition rating** 

Fig. 10. Concrete bridge deck condition rating in linguistic terms.

haunch girder of the deck shown in Fig. 11.

Fig. 11. Concrete bridge deck inspection layout.

**Spalling condition**

condition rating system as the benchmark.

**Deck element** 

**Slab** 

**Girder** 

deck.

can be used for them during bridge inspection process.

**Crack width condition**


Table 3. Condition rating system for concrete bridge decks (Federal Highway Administration (FHWA), 1995; Morcous et al., 2003).

Based on Table 3 the condition rating for typical slab and haunch girder under consideration are 6 and 5. Scaling is required to be able to compare the results, therefore the above mentioned numbers should be multiplied by 14.28 to get a 100-based score system. Table 4 provides the comparison. It is shown that results from proposed method can estimate the condition rating very well.


Table 4. Comparison of the condition ratings from two methods.

#### **5.4 A two stage method for structural damage identification using an adaptive neurofuzzy inference system and practice swarm optimization**

All the above three case studies are of diagnosis and assessment types. As declared earlier another important task of any BMS or BHMS is the possibility to locate damaged area or components. The present case study is about damage detection (Fallahian & Seyedpoor, 2010).

In this case study, an efficient methodology is proposed to accurately detect the sites and extents of multiple structural damages. The proposed methodology has two main phases

Fuzzy Inference System as a Tool for Management of Concrete Bridges 467

is performed using the ANFIS within the available input-output data to arrange the structural elements according to their damage potentiality. Essentially, the exhaustive search technique builds an ANFIS network for each damage variable from original ones and trains the network for a little epoch and reports the performance achieved. The step by step summary of the exhaustive search algorithm for determining the MPDE of an unhealthy

b. Randomly generate a number of sample structures having some damaged elements

c. Determine the natural frequencies of the sample structures using a conventional finite

d. Estimate the level of correlation between unhealthy structure and each sample structure

e. Randomly split the sample structures into two sets with some samples for training and

f. Build an ANFIS model for each damage variable as the input and the *MDLAC*(*X*) as the output. This leads to *n* ANFIS models equal to the total number of structural elements.

> <sup>2</sup> 1

where *ac* and *pr* represent the actual and predicted values of the *MDLAC*(*X*), also *nt* is the

a. Sort the structural elements according to increasing their training *RMSE* values and select the first *m* arranged elements, having the least *RMSE* errors, as the reduced

Now it is time to identify damage using optimization algorithms. As mentioned above, the MDLAC index will reach to a maximum value 1 when the structural damage occurs. This concept can be utilized to estimate the damage vector using an optimization algorithm. For this aim, the unconstrained optimization problem with discrete damage variables reduced

> Find 1 2 , ,..., Minimize : () ()

*T = {xr1, xr2, …, xrm}*.

*t i RMSE ac pr*

<sup>1</sup> ( ) *nt*

*i i*

*r r r rm r r d*

*xR i m*

, 1,...,

(12)

*T*

*ri*

where *Rd* is a given set of discrete values and the damage extents *xri (i= 1, ...,m)* can take

The selection of an efficient algorithm for solving the damage optimization problem is a critical issue. Needing fewer structural analyses for achieving the global optimum without trapping into local optima must be the main characteristic of the algorithm. In this study, a

values only from this set. Also, *w* is an objective function that should be minimized.

*X xx x w X MDLAC X*

*n* (11)

g. Calculate the root mean square error (RMSE) for training and testing sets as:

a. Establish the pre-assigned parameters of the intact structure.

by evaluating the *MDLAC*(*X*) index via equation (10).

remaining samples for testing the ANFIS, respectively.

within the allowed space of damage variables *X*.

structure is as follows:

element analysis.

number of training or testing samples.

b. End of the algorithm.

may be stated as:

damage vector, denoted here by *Xr*

combining the adaptive neuro-fuzzy inference system (ANFIS) and a particle swarm optimization (PSO) as an optimization solver. In the first phase, the ANFIS is employed to quickly determine the structural elements having the higher probability of damage from the original elements. In the second phase, the reduced damage problem is solved via the particle swarm optimization (PSO) algorithm to truthfully determine the extents of actual damaged elements.

Structural damage detection techniques can be generally classified into two main categories. They include the dynamic and static identification methods requiring the dynamic and static test data, respectively. Furthermore, the dynamic identification methods have shown their advantages in comparison with the static ones. Among the dynamic data, the natural frequencies of a structure can be found as a valuable data. Determining the level of correlation between the measured and predicted natural frequencies can provide a simple tool for identifying the locations and extents of structural damages. Two parameter vectors are used for evaluating correlation coefficients. A vector consists of the ratios of the first *n*<sup>f</sup> vector natural frequency changes Δ*F* due to structural damage, i.e.

$$
\Delta F = \frac{F\_h - F\_d}{F\_h} \tag{8}
$$

where *Fh* and *Fd* denote the natural frequency vectors of the healthy and damaged structure, respectively. Similarly, the corresponding parameter vector predicted from an analytical model can be defined as:

$$\mathcal{S}F(X) = \frac{F\_h - F(X)}{F\_h} \tag{9}$$

where *F*(*X* ) is a natural frequency vector that can be predicted from an analytic model and *XT = {x1, ..., xi, ..., xn}* represents a damage variable vector containing the damage extents *(xi, i=1, …, n)* of all *n* structural elements.

Given a pair of parameter vectors, one can estimate the level of correlation in several ways. An efficient way is to evaluate a correlation-based index called the multiple damage location assurance criterion (MDLAC) expressed in the following form:

$$\text{MDLAC}(\mathbf{X}) = \frac{\left| \Delta \mathbf{F}^T \mathcal{S} \mathbf{F}(\mathbf{X}) \right|^2}{(\mathbf{F}^T \Delta \mathbf{F}) [\mathcal{S} \mathbf{F}^T(\mathbf{X}) \mathcal{S} \mathbf{F}(\mathbf{X})]} \tag{10}$$

The MDLAC compares two frequency change vectors, one obtained from the tested structure and the other from an analytical model of the structure. The MDLAC varies from a minimum value 0 to a maximum value 1. It will be maximal when the vector of analytical frequencies is identical to the frequency vector of damaged structure, i.e., *F*(*X* ) = *Fd* .

The key point of this case study is that ANFIS concept can be effectively utilized to determine the most potentially damaged element (MPDE) of an unhealthy structure. For this, some sample structures having the damaged elements are randomly generated based on the damage vector *X* as the input and the corresponding *MDLAC*(*X*) as the output. In other words some scenarios are defined for damaged structures. Then, an exhaustive search is performed using the ANFIS within the available input-output data to arrange the structural elements according to their damage potentiality. Essentially, the exhaustive search technique builds an ANFIS network for each damage variable from original ones and trains the network for a little epoch and reports the performance achieved. The step by step summary of the exhaustive search algorithm for determining the MPDE of an unhealthy structure is as follows:


$$RMSE = \sqrt{\frac{1}{n\_l} \sum\_{i=1}^{n\_l} (ac\_i - pr\_i)^2} \tag{11}$$

where *ac* and *pr* represent the actual and predicted values of the *MDLAC*(*X*), also *nt* is the number of training or testing samples.


466 Fuzzy Inference System – Theory and Applications

combining the adaptive neuro-fuzzy inference system (ANFIS) and a particle swarm optimization (PSO) as an optimization solver. In the first phase, the ANFIS is employed to quickly determine the structural elements having the higher probability of damage from the original elements. In the second phase, the reduced damage problem is solved via the particle swarm optimization (PSO) algorithm to truthfully determine the extents of actual

Structural damage detection techniques can be generally classified into two main categories. They include the dynamic and static identification methods requiring the dynamic and static test data, respectively. Furthermore, the dynamic identification methods have shown their advantages in comparison with the static ones. Among the dynamic data, the natural frequencies of a structure can be found as a valuable data. Determining the level of correlation between the measured and predicted natural frequencies can provide a simple tool for identifying the locations and extents of structural damages. Two parameter vectors are used for evaluating correlation coefficients. A vector consists of the ratios of the first *n*<sup>f</sup>

> *h d h F F <sup>F</sup>*

where *Fh* and *Fd* denote the natural frequency vectors of the healthy and damaged structure, respectively. Similarly, the corresponding parameter vector predicted from an analytical

( ) ( ) *<sup>h</sup>*

*F FX F X*

where *F*(*X* ) is a natural frequency vector that can be predicted from an analytic model and *XT = {x1, ..., xi, ..., xn}* represents a damage variable vector containing the damage extents

Given a pair of parameter vectors, one can estimate the level of correlation in several ways. An efficient way is to evaluate a correlation-based index called the multiple damage location

*h*

*F FX*

 

2 . ()

( ) ( . )[ ( ). ( )] *T*

The MDLAC compares two frequency change vectors, one obtained from the tested structure and the other from an analytical model of the structure. The MDLAC varies from a minimum value 0 to a maximum value 1. It will be maximal when the vector of analytical

The key point of this case study is that ANFIS concept can be effectively utilized to determine the most potentially damaged element (MPDE) of an unhealthy structure. For this, some sample structures having the damaged elements are randomly generated based on the damage vector *X* as the input and the corresponding *MDLAC*(*X*) as the output. In other words some scenarios are defined for damaged structures. Then, an exhaustive search

*T T*

frequencies is identical to the frequency vector of damaged structure, i.e., *F*(*X* ) = *Fd* .

*F* (8)

*F* (9)

*F F F X FX* (10)

vector natural frequency changes Δ*F* due to structural damage, i.e.

assurance criterion (MDLAC) expressed in the following form:

*MDLAC X*

damaged elements.

model can be defined as:

*(xi, i=1, …, n)* of all *n* structural elements.

Now it is time to identify damage using optimization algorithms. As mentioned above, the MDLAC index will reach to a maximum value 1 when the structural damage occurs. This concept can be utilized to estimate the damage vector using an optimization algorithm. For this aim, the unconstrained optimization problem with discrete damage variables reduced may be stated as:

$$\begin{aligned} \text{Find} & \qquad \mathbf{X}\_r^T = \{ \mathbf{x}\_{r1}, \mathbf{x}\_{r2}, \dots, \mathbf{x}\_{rm} \} \\ \text{Minimize :} & \qquad w(\mathbf{X}\_r) = -MDLAC(\mathbf{X}\_r) \\ & \qquad \mathbf{x}\_{ri} \in \mathbb{R}^d \quad ; \quad i = 1, \dots, m \end{aligned} \tag{12}$$

where *Rd* is a given set of discrete values and the damage extents *xri (i= 1, ...,m)* can take values only from this set. Also, *w* is an objective function that should be minimized.

The selection of an efficient algorithm for solving the damage optimization problem is a critical issue. Needing fewer structural analyses for achieving the global optimum without trapping into local optima must be the main characteristic of the algorithm. In this study, a

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particle swarm optimization (PSO) algorithm working with discrete design variables is proposed to properly solve the damage problem.

In order to show the capabilities of the proposed methodology for identifying the multiple structural damages, two illustrative test examples are considered. The first example is a cantilever beam discussed in detail and the second one is a bending plate discussed in brief. The numerical results for these examples demonstrate that the combination of the ANFIS and PSO can produce an efficient tool for correctly detecting the locations and sizes of damages induced (Fallahian & Seyedpoor, 2010).
