Despite the great achievements accomplished so far in the field of damage detection, the need for a more comprehensive damage detection approach that effectively functions despite several real-world obstacles still exists. Among different methods suggested so far, model-updating holds the most appeal; however, its accuracy and efficiency are seriously challenged when the problem size grows as it fails to solve the problem with an increase in the number of variables. In this paper, a two-step approach is suggested that reduces the problem size in model updating-based damage detection methods by identifying potentially damaged elements using a static strain-based damage identification index over the first step. Therefore, only a few unknown variables will be introduced to the next step, which involves an iterative model-updating process that includes a novel damage-sensitive objective function to determine damage severities in the located elements of the previous step. In this stage, a meta-heuristic optimizer, the Equilibrium Optimizer, is utilized to detect the unknown variables, i.e., damage values. This optimization algorithm is used given that it has not been applied to damage detection problems before. The method is also tested on a number of numerical examples to be validated and the effects of external factors such as measurement noise can be taken into account. Apart from that, a comparative study is also carried out to compare the efficacy of the present work with previous references. Interestingly, the method is capable of detecting damaged elements with great speed and accuracy, and the quality of the results has not been seriously affected in the presence of perturbing factors. Moreover, information from even a limited number of first modes has been enough to solve the problem, which means that the approach does not require all vibration modes. Finally, an experimental verification study is also carried out to demonstrate the capability of the method in detecting damaged elements in real structures. It is shown that the method can detect the damaged element successfully, which is indicative of the practicality of the suggested technique in real-world applications.

1. Introduction

Yao was one of the first researchers to introduce different definitions for structural damage, according to whom this term is attributed to any deficiency in physical characteristics of a structure as a result of several factors such as external loading and environmental conditions [1]. This author also claims that structures can even have initial defects as a result of poor construction or design even before external loading. Later on, Frangopol and Curley outlined that these defects would adversely affect the normal performance of any structure in its operational life, leading to abrupt failures in the most severe cases [2]. According to these definitions, damage does not necessarily indicate complete failure, but rather a gradual deterioration that could either lead to malfunction or an eventual unpredicted failure, if not detected timely. This is the main reason for the advent of Structural Health Monitoring (SHM), which refers to a set of strategies applied for long-term inspection of civil, aerospace, and mechanical engineering structures [35]. SHM, in fact, includes a range of strategies from identifying structural characteristics, known as structural identification, to the long-term measurement of the remaining service life of existing structures [6]; however, damage detection (DD), which refers to the process of detecting characteristics of damage such as location and severity, is known as the most essential step for maintaining existing structures to the most of their capabilities [7]. Excluding destructive tests to investigate structural damage, nondestructive DD techniques can be divided into two categories including local and global methods [7]. Local DD methods such as ultrasound and magnetic particles can only locally identify damaged structural members provided that the approximate vicinity of the affected section is both known a priori and accessible [8, 9]. Therefore, vibration-based global methods were put forward by researchers which could detect even the smallest damages in the most complex structures without the aforementioned limitations [10]. Several comprehensive reviews have been carried out so far on DD methods based on vibration characteristics, some of which can be found in Refs. [1116].

A wide range of vibration-based DD approaches have been developed in the literature, among which finite element model-updating (FEMU) has been of great significance over the recent years [17, 18]. In this increasingly appealing technique, a baseline finite element (FE) model of the structure is simulated first, after which the modal characteristics of the structure are iteratively updated in several attempts to adjust the parameters of the model to those of the real structure [19]. Regarding applications of FEMU in structural DD, it can be said that this technique attempts to adjust the parameters of an initial FE model of the potentially damaged structure with real measured parameters [18]. In fact, this technique formulates the DD problem in the form of an inverse problem where an objective function is pre-defined to minimize the difference between numerical data and experimentally measured parameters of the structure to accomplish the model adjusting purpose. Meanwhile, the minimization process is mostly carried out with the aid of a meta-heuristic optimization algorithm to help speed up the otherwise extremely time-consuming process of finding the optimal answer [19]. In this regard, several papers have been published so far, most of which have defined their objective functions based on the modal characteristics of the structure. As it is impossible to mention all published papers on this topic, some of the most recent studies are reviewed in the following. Many researchers have used modal parameters such as natural frequencies, mode shapes, flexibility matrix, and/or a combination of these parameters to establish the aforementioned objective function. The use of these parameters has several advantages; for instance, natural frequencies and mode shapes can be easily calculated from dynamic measurements recorded via some installed sensors on the structure, and modal flexibility is simply obtained according to these two modal parameters. In addition, measuring and investigating changes in these parameters can provide useful information on the existence and extent of structural damage since such defects can considerably affect these modal parameters [20]. In 2016, Khatir et al. proposed a model updating-based DD approach in which natural frequencies were used to define an objective function, and the inverse problem was solved via the bat algorithm (BA) [21]. In their research, both damage locations and severities were estimated in some numerical examples, while also taking the effects of measurement noise into account. Xu et al. utilized residual error of frequencies as damage-sensitive parameters to diagnose defects in structures [22]. In their study, the method was numerically validated using a simply supported beam as well as a truss, while also being experimentally tested on a laboratory structure. Their inverse problem was solved via a modified Cuckoo search (MCS) algorithm. Later on, Huang and Lu developed a DD technique by means of a nonlinear objective function based on acceleration responses of the structure, where the optimization problem was dealt with through a Big Bang and Big Crunch (BB-BC) algorithm [23]. What makes their study stand out is that they have considered relatively large damage extents in their numerical study, and the results have been promising in the presence of noise. Kaveh and Dadras defined an inverse DD problem based on an objective function built upon modal characteristics of the structure, where the thermal exchange optimization (TEO) algorithm was used to solve the problem, and the method was tested in several comparative studies [24]. In another attempt by Dabbagh et al. in 2018, an objective function was defined on the basis of natural frequencies and mode shapes, and the Imperialist Competition Algorithm (ICA) was suggested to solve the iterating model-updating process [25]. They have tested their suggested approach on several numerical structures and have also determined that the results would not be significantly affected by measurement noise. In 2020, Chen and Yu defined a DD objective function based on modal parameters including natural frequencies and mode shapes to diagnose damage in structures [26]. They proposed a hybrid algorithm which was a mix of the Ant-Lion Optimizer (ALO) and a modified Nelder-Mead algorithm to solve the inverse optimization problem defined. They validated their suggested technique via a set of benchmark structures (numerically) and a laboratory structure to further evaluate the method. Dinh-Cong et al. carried out a DD study on full-scale structures within a model updating-based framework in which an objective function was defined based on the flexibility matrix [27]. In their research, the structures under study were modeled in the Structural Analysis Program2000 (SAP2000), and the optimization problem was solved with the aid of an enhanced symbiotic organisms search (ESOS) algorithm, while also taking the impacts of measurement noise into account. It must be mentioned that they also had to create a link between SAP2000 and MATLAB in order to carry out the iterative model updating-based optimizing algorithm. Most recently, Su et al. defined a multi-parameter objective function based on natural frequencies, mode shapes, and modal flexibility of the structure to detect damaged elements [28]. To solve the optimization problem, they employed a modified directional bat algorithm (MDBA) with an improved updating mechanism to search for the optimal answer. Meanwhile, they tested their suggested approach on beam-type and truss-type bridges while also comparing the results with those obtained via other optimization algorithms to further show the efficacy of their proposed optimizer. Modal Strain Energy (MSE) is yet another modal parameter that has been widely used in the literature as a damage-sensitive factor to detect damage in structures. Extensive reviews on the use of this modal parameter have been carried out in Refs. [29, 30]. An outstanding advantage of this parameter is that it can be obtained using the stiffness matrix and the squared mode shapes of the structure [30]. Apart from this, information from only a limited number of modes is enough to measure this parameter, which makes it quite practical in case complete modal information is not available [31, 32]. Cha and Buyukozturk introduced a multi-objective DD framework for 3D steel structures where MSE was used to generate the objective function [33]. A hybrid genetic algorithm (GA) was suggested in their study to solve the optimization problem with information from a limited number of modes in the presence of measurement noise. Kaveh and Zolghadr proposed a DD approach based on MSE to define an objective function, where the optimization problem was solved via a tug-of-war (TWO) optimization algorithm [34]. They tested their method on a set of numerical examples to examine the efficacy of their approach. Huang et al. employed modal flexibility and modal frequency strain energy to construct an objective function to diagnose defects in structures [35]. The moth-flame optimizer was suggested to help optimize the problem under consideration, followed by ultimately testing the method on three numerical examples and two experimental structures. In another study, Huang et al. developed a DD scheme in which an objective function was formulated using the MSE of the global structure, after which the structure was decomposed into some substructures (using the substructure technique, more details can be found in Ref. [36]) [37]. An improved Whale optimization algorithm (IWOA) was then introduced to complete the DD process, followed by testing the proposed approach on a series of numerical and experimental examples. However, this parameter has some major drawbacks; as only a limited number of modes are utilized to establish this factor, DD results will be considerably affected if the information from these low-order modes is inaccurate, and it is highly sensitive to noise [29]. Several efforts have been made so far to overcome the shortcomings associated with this parameter as the benefits it brings about far outweigh the drawbacks. Among different approaches suggested so far in this respect, two-step methods that combine the use of this parameter with other techniques have been holding so much appeal. Many researchers combined MSE with different DD approaches, among which only those related to the use of MSE in combination with model updating-based techniques are reviewed in this paper. The general idea of these approaches is to overcome the mentioned problems. Vo-Duy et al. published a research article in 2016 to propose a two-step procedure for DD in laminated composite structures. Over the first step, they employed MSE to locate potentially damaged members, while they used an optimization algorithm in the next step which determined the extent of damage in each of the potentially damaged elements identified in the previous stage [38]. In fact, they had defined an objective function based on mode shapes of the structure for the second step of their suggested approach. They finally tested their method on several numerical structures while also taking the effects of noise into consideration. Seyedpoor and Montazer proposed a two-step DD method for truss structures where a number of potentially damaged members were identified over the first step based on a flexibility-based index [39]. The mentioned index was defined based on the changes in strain, calculated using the flexibility of the structure. Over the next step, an optimization framework was defined to evaluate damage severities using a differential evolution algorithm (DEA) in the form of an inverse DD problem. The objective function defined in the second step was based on the mode shapes of the structure. Another two-step DD approach was developed by Dinh-Cong et al. in which a normalized MSE damage index was suggested to find the locations of some potentially damaged elements [40]. A teaching-learning-based optimization (TLBO) was then used in a second step to evaluate the extent of damage in the previously identified damaged elements, which would also exclude false locations possibly identified mistakenly over the first step. The objective function used in the second step was defined based on modal flexibility. The method was tested on a set of truss structures to validate their method. In 2021, Fan developed a two-step approach for DD in structures [41]. This author first determined potentially damaged locations using an MSE index, named the cross-modal MSE, in the first step. Afterward, an optimization-based approach would determine damage severities in the second step, for which a trust region-based optimization method (TRM) was utilized, and the objective function was formulated based on natural frequencies and mode shapes. It must be noted that there have been a variety of other two-step DD methods in the literature that have used a combination of different techniques to diagnose structural faults. As an example, Abbasnia et al. proposed a two-step method in 2016 that used Wavelet transform and model-updating in two different steps for the purpose of DD [42]. The utilized Wavelet transform, which was applied to the axial components of the mode shapes, predicted potentially damaged elements in the first step, whereas a model-updating approach was used in the next step to determine the extent of damage. Moreover, Mirzaei et al. suggested a two-step DD approach for DD in large truss structures using the Wavelet transform in the first step to predict damaged locations in the structure [43]. Then, a model-updating process was proposed to calculate damage severities in the next step with the aid of an optimization algorithm. They tested their suggested method on several large trusses to show the accuracy of the suggested approach.

It is worth mentioning that many other researchers have proposed different methods for detecting other types of structural anomalies such as landslides or ruts and cracks induced in pavement asphalt. By way of example, Xiang and Wang introduced an asphalt-mastic packaged optical fiber sensor to monitor strain distribution in asphalt pavement [44]. For that purpose, they employed a strain analysis on the fiber sensors to improve the design of this protective layer. Another popular technique that is increasing in popularity is the use of artificial intelligence and soft computing. For instance, Ma et al. proposed the use of a mutual information-based measure for input-variable selection in combination with a vector regression technique [45]. Their method was able to forecast seepage-driven landslides. In another paper published by ma et al. in 2020, a prediction model was developed by means of computational intelligence to foresee landslide displacements while also considering uncertainties [46]. Their proposed intelligent technique had two stages where the first stage used a copula analysis for input variable selection, and the second stage applied a vector regression technique for density prediction. As effective as such methods are, more explanations about them would be out of the main scope of the present paper.

In this paper, a two-step method is proposed for DD in structures using a combination of model-updating and a damage index. This is done for two simple yet important reasons. Firstly, model-updating, which is known as an efficient and simply implemented DD method, would operate too slowly in the case of large-scale structures, which is inevitable as the number of problem variables increases with the number of finite elements; therefore, to make use of the advantages of this approach, this deficiency must be somehow handled. Secondly, this paper tries to use static strain energy because of its convenience to locate damaged elements; however, unfortunately, this parameter is unable to determine the extent of the identified damage. As a result, a two-step method based on a combination of these two approaches is proposed to reduce the problem size in the model-updating process and also to make the most of the advantages of the static strain energy as an efficient damage index. To explain more, static strain energy is used to define a damage index in the first step. This parameter is chosen due to its high sensitivity to structural damage. However, in order to measure static data, the structure must be subject to static excitation, which is quite complicated and impractical. Accordingly, in this paper, static data are computed using modal information to circumvent the complexity of statically exciting the structure. In other words, a two-step method is suggested that uses the advantages of both of these mentioned approaches while also overcoming their shortcomings. To clarify more, the locations of potentially damaged elements are identified using a damage index defined based on static stain energy in the first step. Afterward, an iterative model-updating procedure is carried out on the previously introduced suspicious elements to calculate the extent of the damage. In this way, the likely errors resulting from the damage index of the first step would be effectively tackled in the next step as the method will calculate damage severities in false locations as zero. On the other hand, the problem variables will significantly decrease for the model-updating part of the approach, increasing the accuracy and efficiency of the method. In this stage, the model-updating process includes a damage-sensitive objective function defined based on static strain energy which is aimed at finding the severity of damage in the previously identified damaged elements. Moreover, a meta-heuristic optimization algorithm, Equilibrium Optimizer (EO), aids the process to find the values of unknown variables. To explain what makes this work stand out from previous studies, the following items must be mentioned. Firstly, this two-step method, which quite successfully addresses the shortcomings of both the combined methods and makes the most of their advantages, has not been used for detecting structural damage before. More importantly, previous studies that have used static strain energy have considered static excitation to calculate this parameter, which is very complicated and impractical; however, the present study uses modal data to define this index in order to overcome the mentioned complexity. Apart from this, the novel objective function defined for the mode-updating process is minimized by means of EO, which has not been applied to DD problems before.

2. Materials and Methods

As it was said before, the DD approach proposed in this paper consists of two stages. The first step includes defining a static strain energy–based damage index based on which damaged locations are identified. Over the course of the next step, these damaged members will be introduced as input unknown variables to a model-updating problem which attempts to determine the extent of damage in these potential locations. The details of both steps are presented in more detail in the following subsections.

2.1. Locating Potentially Damaged Elements

In this step of the proposed approach, a damage index is defined using static strain energy (SSE) which is aimed at finding possibly damaged elements in the structure. This parameter is chosen as the damage-sensitive index in this study due to the aforementioned advantages it brings. However, as was mentioned earlier, static excitation of structures is a tedious task, making it quite impractical for real problems. To tackle this issue, this parameter is extracted from the modal information of the structure. The reason why this paper insists on using SSE is its simplicity and accuracy. The first step to define this index is the equation of static displacement, as shown by (1). In fact, static displacement under an arbitrary load can be related to modal flexibility as follows:

Here, is a unit static load applied to all degrees of freedom of the structure, and is the modal flexibility matrix, which can be calculated using the first m modes of the structure and as follows:where is the square root of the diagonal frequency matrix using m lower modes, and is the mode shape matrix using m lower modes. Herein, the static strain energy of a structure is referred to as static strain energy (SSE), and it is obtained as (3):where is the matrix of static displacements of element e, and is the stiffness matrix associated with element e. For the sake of computational convenience, is normalized with respect to , which is the square root of the sum of squared static displacements of all elements. Accordingly, the normalized static strain energy of element e is shown by , and it is obtained as follows:where:

Ultimately, a damage index called is defined based on the abovementioned parameters to locate damaged elements in the structure. If the value obtained for an element is more than , the element must be considered suspicious to damage.

In this equation, as locations of damaged members are not known a priori for real cases, the element stiffness matrix of the undamaged structure is applied here to estimate .

2.2. Quantifying Damage Extent

At this stage, the potential damage locations have been already introduced via the previous step. Therefore, damage severities in these elements must be calculated using an accurate technique. By far, the number of variables in the DD problem for this second step has been considerably reduced, making it much easier to solve the problem. To briefly explain the idea lying behind model updating-based DD techniques, it should be said that the DD is defined in the form of an inverse problem where the unknown variables are damage severities. The method is an iterative problem-solving scheme. Here, an objective function is formulated and the problem is solved with the aid of a meta-heuristic optimization algorithm to finally yield the extent of damage in the suspicious elements introduced through the previous step.

There is a likelihood that the damage index defined in the previous step mistakenly identifies some undamaged elements as suspicious; however, the model-updating process in this step will determine their associated damage severities as zero.

The objective function defined as well as the optimization algorithm is described in more detail in the following.

2.3. Objective Function

The key step in a model updating-based DD approach is defining an objective function using damage-sensitive parameters which are quite insensitive to measurement noise at the same time. This function, in fact, measures the correlation between experimental and numerical model parameters, as a result of which, the unknowns of the problem will be found. In this respect, the novel objective function is defined based on the damage-sensitive parameter .where is calculated by (8):where:

In (7), MAC is the modal assurance criterion, which is a well-known correlating function that scales the correlation degree between modal parameters of the objective function [47].

When MAC is obtained as 1, it means that there is a complete correlation between experimental and numerical data, and when it equals to 0, it means that there is no correlation. Therefore, when equals to zero, a complete correlation is accomplished between the data, which means that the optimizer has been able to accurately minimize the objective function and estimate the values of the unknown variables.

2.4. Equilibrium Optimizer (EO)

Based on the equation of dynamic mass balance used in a control volume, Faramarzi et al. developed a new meta-heuristic optimization algorithm called the Equilibrium Optimizer (EO) [48]. This equation is an ordinary first-order differential equation defined as follows:where is the rate of change in mass in volume V, C shows concentration inside the volume, and V indicates control volume. On the other hand, Q represents the volumetric flow in and out of the control volume while shows concentration in the equilibrium state and G is the mass generator rate inside the control volume. After differentially solving (11) for C, the following equations are obtained:

As it can be seen in the abovementioned equations, F is an exponential term that maintains a balance between exploration and exploitation search processes of the algorithm. Equation (15) includes three terms for the updating process imposed on each particle.

Similar to other meta-heuristic optimizers, EO starts with an initial population, namely, the initial concentration of particles. Here, represents the number of particles which is described as follows:where is the initial concentration of particles, and , are the maximum and minimum values of dimensions, respectively. is a random vector between 0 and 1. After the initial population is generated, the particles are updated iteratively according to their corresponding solutions using the following equation and are stored as the best-so-far answers:in which and are the current and new solution vectors associated with each particle, and is a random concentration vector selected randomly from the equilibrium space. is a random vector between 0 and 1, and F is an exponential term defined by equation (18):where and are constants controlling the exploration and exploitation features, determines the direction of exploration and exploitation ( is a random vector between 0 and 1). Moreover, and show the current and the maximum number of iterations.

G is called the generation rate in equation (17) defined in equation (19):

In this equation, GP is the generation probability, and and are random vectors between 0 and 1.

Each iteration includes calculating the value of the objective function for each particle, after which the equilibrium space is updated with new best-so-far solutions. To avoid local minima, the optimization process has a saving memory.

To summarize the algorithm for the DD problem in this paper, the following steps must be taken:

(1)Define all associated variables and the objective function
(2)Read the associated data (structural parameters)
(3)Initialize the concentration of particles using a random set of control variables equation (16)
(4)Assign the fitness of the equilibrium of particles a large number
(5)Set control parameters as:
(6)If the number of current iterations is smaller than the desired number of iterations, calculate the value of the objective function for the initial particles. Use the following loop for this purpose:
While T < Tmax
 For i = 1: number of particles (NPC)
  Calculate the objective function value of the particle i
  If F(Ci) < F(Ceq1)
   Replace Ceq1 with Ci and F(Ceq1) with F(Ci)
  Else, if F(Ci) > F(Ceq1) and F(Ci) < F(Ceq2)
   Replace Ceq2 with Ci and F(Ceq2) with F(Ci)
  Else, if F(Ci) > F(Ceq1) and F(Ci) > F(Ceq2) and F(Ci) < F(Ceq3)
   Replace Ceq3 with Ci and F(Ceq3) with F(Ci)
  Else, if F(Ci) > F(Ceq1) and F(Ci) > F(Ceq2) and F(Ci) > F(Ceq3) and F(Ci) < F(Ceq4)
   Replace Ceq4 with Ci and F(Ceq4) with F(Ci)
  End (If)
 End (For)
Cave = (Ceq1 + Ceq2 + Ceq3 + Ceq4)/4
The equilibrium space is Ceq.S = {Cave, Ceq1, Ceq2, Ceq3, Ceq4}
Generate saving memory if T > 1
(7)Randomly select a particle from the equilibrium space and calculate the objective function and repeat from step 6.
2.5. Measurement Noise

In real-world problems, measurements of the modal information are always contaminated with some levels of noise. To make the results of this study more realistic, the impacts of these perturbations are also included in the model. In this study, measurement noise is applied to experimental natural frequencies using the following equation [49]:

In this equation, denotes natural frequency after applying noise, and is the same parameter before noise.

Without taking the effects of noise into consideration when recording modal data from a higher number of modes, the results are more accurate. On the contrary, when the influence of noise is considered, the higher modes are more prone to be affected.

3. Numerical Study

In order to evaluate the efficiency of the suggested approach, the method is tested on several numerical examples in this section. It is attempted to test the suggested approach on different types of structure with different damage cases to evaluate the method. Different levels of measurement noise are also applied to show the efficacy of the suggested approach in real circumstances. It must be mentioned that all DD problems in this paper are solved using information from a limited number of modes to show the capability of the method in case of incomplete modal data. The optimization algorithm used in this paper is EO, as it was mentioned earlier, with 1000 iterations. The stability of the optimization algorithm is also demonstrated later in this section by showing the results from several consecutive runs. In addition, a comparative study is also carried out to compare the effectiveness of the proposed method by this paper with another similar study. In the comparative example, the performance of the present damage index has been compared with another damage index proposed by a previous publication. Moreover, to show the consistency of the optimization algorithm selected for the paper (i.e. EO) and to demonstrate how consistent the objective function is with other optimizers, another optimization algorithm has also been compared with EO. Finally, an experimental verification study is also presented in the final part of this section to validate the practicality of the suggested method in real applications. To accomplish this, an experimental setup is selected from a previously published reference, and the proposed method is used to detect the damaged element(s) of the mentioned structure.

3.1. 40-Element Beam

The first numerical example studied in this section is a beam, as shown in Figure 1. The beam is 8 m in length, with a rectangular cross-sectional area of , and it is made of steel material with and .

A damage scenario is considered for this beam, the details of which can be found in Table 1.

As can be seen from the table, there are seven damaged elements with different damage severities in this scenario. To assess structural damage in these structural members using the proposed DD of this paper, a number of suspicious elements are first identified using the SSEBI. The results are obtained considering two different noise levels in measured natural frequencies, 2% and 4% to be exact. To make the results even more realistic, modal information from only a limited number of modes is used to obtain the results. DD results of the first step for this scenario are shown in Figure 2.

As it can be seen, seven elements are identified as potentially damaged. Therefore, only the damage extents of these few elements will be used as unknown variables to be determined through the next step, instead of taking all elements into consideration. According to the information given before in Table 1, all elements have been identified accurately so far, with no false locations. However, even if an element had been identified falsely as damaged, the extent of damage in those false locations would be determined as zero through the next step.

After introducing damaged members through the first step, it is time to determine damage severities in these elements using the model-updating procedure. This is done with the aid of the meta-heuristic optimizer introduced earlier, EO. The results are, again, obtained using a limited number of modes only, to show how capable the technique is in case of incomplete modal information while also taking two different levels of measurement noise into consideration. Finally, DD results associated with the second step for this scenario are indicated in Figure 3.

As it can be observed in the figure, the method has estimated the extent of damage in these elements with great accuracy, and neither incomplete modal information nor measurement noise has affected the accuracy of the results.

Figure 4 shows the convergence rate of the optimization algorithm in the second step for the DD problem of the beam.

As it can be seen, the final values of the objective function in the case where 2% noise is applied has been better than that of the case with 4% noise, which shows how perturbations of this kind can affect the method. Apart from this, Figure 4(b) shows that the convergence rate has been slightly better when modal information from the first 10 modes has been utilized; however, this has changed by the end of the optimization process where it has been overtaken by the results of the case using only 4 modes. This shows that an increase in measurement noise affects higher modes to a greater extent, proving the accuracy of the suggested approach of the present work using only a limited number of modes.

3.2. 35-Element Planar Truss

The structure simulated in this section is a 35-element 2D truss shown in Figure 5. The truss is made of steel material with and . The cross-sectional area of the horizontal, vertical, and diagonal elements of the truss are, respectively, , , and .

As shown in Table 2, a damage scenario is assumed for this structure including five damaged members. To test the efficacy of the method in the vicinity of the supports, element 3, which is attached to the simple support to the left end of the truss, is considered damaged.

Once again, damaged elements are identified first using the SSEBI. This is done considering two different noise levels–2% and 4% noise in measured frequencies–so that the impacts of this factor could be seen on DD results. The results associated with these two cases are shown in Figures 6(a) and 6(b). All of these cases are performed using modal information from the first four modes of the truss.

As Figure 6 clearly illustrates, while all damaged elements have been identified correctly, element 31 has been mistakenly introduced as damaged in this step.

After identifying suspicious elements over this step, step 2 is carried out to determine the extent of damage in these elements. Similar to the previous example, the problem is solved with the aid of the EO algorithm with 1000 iterations. In this particular example, the second step is repeated in four consecutive runs to show the stability of the optimization procedure. Accordingly, DD results for the second step including the results from 4 runs for two noise levels of 2% and 4% are listed in Tables 3 and 4, respectively.

The first result worth noting in the tables is that the damage severity associated with element 31, which was introduced as damage by mistake in the previous step, has been obtained as zero, which shows how the second step can address the probable mistakes of the damage index in step 1.

It can also be clearly seen in the tables that the results in four consecutive runs are quite similar with minor discrepancies. This is indicative of the accuracy of the method in determining damage extents for the previously identified damaged elements. Apart from this, although the meta-heuristic optimization algorithms utilize a random nature to search for optimal answers, the final results obtained have been completely stable, which shows how reliable the method is.

3.3. Frame

The frame investigated in this example is shown in Figure 7. The beams have a cross-sectional area of and , whereas the columns have a cross-sectional area of and . The frame is made of steel material with and .

The damage case considered for this structure includes three damaged elements as listed in Table 5. Two damaged columns and a damaged beam element have been assumed here.

To begin with, suspicious elements are first diagnosed via the SSEBI. Similar to the previous examples, this is done for two cases considering 2% and 4% noise in natural frequencies to make the results more realistic. The results of this stage are shown in Figure 8.

As can be seen, three suspicious elements have been detected exactly, with no false locations.

After this stage, damage severities in these locations are determined through the model-updating step, with the aid of EO. In order to show the stability of the method and to further evaluate the accuracy, similar to the previous example, this stage is repeated four times. Table 6 shows the DD results for this stage. However, the average of damage severities predicted over these four consecutive runs are calculated and reported in the table. Moreover, the standard deviation of all the results is also included in the table to show the stability of the method in detecting the results.

As it can be seen, the results are quite close to the real damage assumed for the elements. Apart from this, the standard deviation of the results is a negligible figure, showing the stability of the results obtained. In fact, regardless of the number of modes used or the presence of measurement noise, the method has always yielded stable results, which further explains the reliability of the method in DD problems.

4. Comparative Study–15-Element Cantilever Beam

In this section, the method proposed by this paper is compared with that of a previous publication with a similar approach. More details about the selected paper for this comparative study can be found in Ref. [50]. The paper chosen to be compared with the present study uses a two-step method for DD where the first step locates damaged elements by means of a damage index defined based on modal strain energy, which is called MSEBI in the main reference. In the second step, the mentioned reference uses model updating-based process to determine damage extents with the aid of the particle swarm optimization (PSO). The structure selected for the comparison purpose is a 15-element cantilever beam with two damaged elements, as shown in Figure 9.

The details of the considered damage scenario can be found in Table 7. According to the main reference, the length, thickness, and width of the beam are respectively 2.74, 0.00635, and 0.0760 m. The mass density of the beam is 7860 kg/m2, whereas the modulus of elasticity of the beam is 210 GPa. Identifying damaged locations is performed using the damage index proposed in Ref. [50] and the proposed method of the present paper in order that their performances can be compared. It must be noted that the DD problem is solved using modal information from the first mode and the first five modes of the structure to demonstrate the results more comprehensively.

This example has been investigated without considering the effects of measurement noise in order to compare the effectiveness of both methods more clearly under ideal noise-free conditions. DD results are obtained using the first mode and the first five modes of the beam using both of the methods. For this case, the results of the first steps of the compared methods, which identify the damaged elements, are indicated in Figure 10.

In Figure 10, MSEBI, shown by black color, refers to results obtained by the damage index defined in the compared reference, whereas SSEBI, shown by grey color, is associated with the results from the proposed method of the present study. As it can be seen, while both methods have successfully identified the damaged elements, MSEBI has mistakenly detected element 14 as damaged in the case where 1 mode is used to obtain the results.

In order for damage severities to be determined, the second step must now be implemented. In this step, the model-updating process using the objective function defined by (7) is used to determine damage extents with the aid of two optimization algorithms of EO and PSO in order to compare the performance of EO with PSO. DD results associated with this second step are shown in Table 8.

As can be seen from the tables, both of the algorithms have almost accurately estimated the extent of damage.

For the sake of comparison, convergence rates of both optimizers, EO and PSO, have been shown for both cases (using 1 mode and 5 modes) in Figure 11.

Obviously, both algorithms have converged almost before 60 iterations in both cases with EO more quickly obtaining final results for the objective function.

According to what the author has stated in reference [50], his proposed method is capable of identifying the locations of damaged elements using modal information from the first 3 modes, while 5 modes are needed to determine damage severities alongside damaged locations. However, the results of this comparative study have shown that, thanks to the powerful objective function proposed in the present paper, modal information from only 1 mode is enough for the method to predict both damage locations and severities. This is indicative of the novelty and efficiency of the suggested damage index and the objective function in this paper.

As a result of this comparative study, it can also be concluded that the objective function proposed by this paper can adapt to different optimization algorithms as the results obtained by the PSO algorithm have been completely satisfactory.

4.1. Experimental Verification Study

The effectiveness of the proposed method was clearly demonstrated through the previous numerical examples while also comparing the suggested method with a previous study to show the accuracy and the efficacy of the presented DD technique. In this section, the suggested method is tested on an experimental study as well to show the performance of the method in the real world. The experimental setup selected for this section is an eight-element simply supported steel beam tested by Moradi Pour in 2020 [51]. The tested beam is shown in Figure 12, and the properties of the beam are listed in Table 9. The beam was tested at Banyo Pilot Plant Precinct of Queensland University of Technology.

In this experimental study, element 7 is 25% damaged. It is attempted here to detect the damaged element by means of the two-step method of this study with the aid of EO. DD results obtained from the first step for this scenario are shown in Figure 13. It must be mentioned that DD has been implemented in this example using modal information from the first mode, the first three modes, and the first five modes of the structure to compare the results.

According to the figure, element 7 has been correctly detected as damaged and will be considered in the second step to determine the severity of the existing damage. The results estimated by the model-updating process in the second step are shown in Figure 14.

As it is clear from Figure 14, the method has accurately and successfully estimated the extent of damage in the detected damaged element. This outstanding result indicates that the suggested method works well under real conditions, which shows the practicality of the DD technique. The convergence rate of the EO algorithm in this example is shown in Figure 15.

As was mentioned earlier, the damaged element in this structure has been detected using modal information from the first mode, the first three modes, and the first five modes of the structure. As can be seen from Figure 15, the algorithm has been able to converge to the desired value with a small number of iterations, which shows that the method can successfully perform using different numbers of modal information.

5. Conclusions

A two-step damage detection approach was proposed in this paper to identify damaged elements in the structure while also determining the extent of damage in these structural members. Although model updating-based damage detection techniques are quite efficient and accurate for damage detection purposes, they are incapable of solving large-scale problems associated with real-world structures. Therefore, the two-step method in this paper was suggested to reduce the size of the problem in the model-updating process. In this way, both the speed and the accuracy of the method significantly increased, leading to promising outcomes. In this respect, the first step of the process was aimed at identifying damaged elements using a static strain-based damage index, called SSEBI in this paper. Therefore, the only variables introduced to the second step–the model-updating procedure–were the suspicious elements introduced by the previous stage. Accordingly, the extents of damage in these elements were then identified over the course of the model-updating step with great accuracy. As the size of the problem in the second step had been significantly reduced, the convergence rate of the optimizer as well as the quality of the results had improved considerably. In the model-updating step, a novel objective function was defined in this paper which was completely capable of detecting damage while being rather insensitive to measurement noise. In fact, the suggested objective function was defined using a correlative measure to scale the correlation between numerical and experimental parameters. The damage-sensitive parameter used for the objective function was static strain energy. The suggested technique was tested on three numerical examples to validate its effectiveness. The effects of the perturbing factor of measurement noise were also included in the examples to make the results more comparable to real situations. Apart from this, information from only a limited number of modes was used to show the effectiveness of the suggested approach in case of incomplete modal information. Besides, a comparative example was also tested which made a comparison between the damage index introduced by this paper with that of another research. The results of the numerical study and the comparative example clearly demonstrate the efficacy and accuracy of the method in detecting damage in different structures, even when measurement noise is present. Another result worth mentioning was that the suggested method was completely in compliance with different optimization algorithms such as PSO. Finally, an experimental verification study was also carried out to demonstrate the capability of the method in detecting damaged elements in real structures. The experimental study was also successful, and the method could detect the damaged element accurately, which is indicative of the practicality of the suggested technique in real-world applications.

The suggested method performs better in structures where there are more degrees of freedom (DOFs) at each of the nodes.

It would be a good research idea for future studies to estimate the remaining lives of the structures using the suggested method of this paper.

Data Availability

There are no available data for this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.