1 Introduction

Structural health monitoring (SHM) is crucial not only for maintaining civil engineering infrastructure but also for other engineering disciplines, ensuring reliability and safety, detecting damage evolution, and predicting performance deterioration [1, 2]. Furthermore, SHM allows real-time or near-real-time monitoring of structural performance, ensuring safety, reliability, cost reduction, and extended service life [3, 4]. Non-destructive testing (NDT) is one of the popular methods that allow engineers to evaluate the structural integrity of materials and components without inflicting any harm [5]. Data obtained during SHM includes static or dynamic properties, such as static deformation, vibration intensity, strain responses, and modal parameters [6, 7]. Based on the ability, damage detection is divided into four levels: [8].

  • 1. First level: To determine if damage exists in a structure (detection),

  • 2. Second level: To predict the location of damaged parts or members (localization),

  • 3. Third level: To calculate the amount of damage (assessment),

  • 4. Fourth level: To determine the remaining life time of the damaged parts.

(residual life time)

Many structural damage detection methods have been proposed, including the DLV, mode shape curvature method, and modal strain energy [9,10,11,12]. The DLV method is commonly used for truss structures, which are in the second level of damage detection methods, and this DLV method may inaccurately predict damaged member locations. In the previous a few decades, many innovative optimization techniques have emerged. These methods have been employed to deal with structural damage identification problems by employing an inverse solution moving from classic optimisation techniques to more advanced and reliable optimisation techniques [13] and had been applied to several types of structures such as plane truss, space truss, plane frame, plate like structures and even for structural connections [14,15,16]. This is in line with extensive research undertaken by [17, 18], where optimization techniques are integrated with dynamic/vibration-based properties of structures to accurately detect structural damage.

This study introduces a modified method that rectifies inaccurate predictions and assesses the loss of strength in the members using DLV, by extending the DLV method into the third level of damage detection. The proposed method utilises DLV, MAC, and optimisation algorithms, specifically FA, to identify the location of damaged members and accurately quantify the reduction in stiffness of these members. Previously, this method has been successfully implemented in plane truss structures [19]. Thus, in this study, the proposed method will be applied to space truss structures to show its ability to predict the location and the stiffness losses of damaged members. There are two types of space truss structures configuration with various damage scenario considered.

1.1 Damage locating vector (DLV)

The DLV method was primarily proposed by Bernal [9], focusing on changes in the flexibility matrix between healthy and damaged conditions. Furthermore, the method uses static loading techniques for damage detection and is categorized as the second level of damage detection, determining the damaged part locations, and this DLV method includes three steps [9].

  • 1. Obtain the change of the flexibility matrix as:

    $$\Delta F={F}_{U}-{F}_{D}$$
    (1)

where \(\Delta F\) is the change of the flexibility matrix, \({F}_{U}\) represents healthy condition flexibility matrix, \({F}_{D}\) represents damaged condition flexibility matrix.

  • 2. Find the singular value decomposition of \(\Delta F\).

$$\Delta F=U\left[\begin{array}{cc}{s}_{1}& 0\\ 0& {s}_{2}\end{array}\right]{V}^{T}$$
(2)

where \({s}_{2}\) represents the independent values that are 'small'. Under optimal circumstances, \({s}_{2}\) consists entirely of zeros and the DLV vectors correspond directly to the columns of V that correspond to the null space. To identify DLVs in a noisy environment, perform these steps:

  1. a.

    Calculate each member’s stresses using V as a series of vector loads.

  2. b.

    Assemble a single characterizing stress for each individual members

  3. c.

    Calculate the svn value for each member corresponding to series of vector load

    $${svn}_{j}=\sqrt{\frac{{s}_{j}{s}_{j}^{2}}{{s}_{q}{c}_{q}^{2}}}$$
    (3)

where,

$${s}_{q}{c}_{q}^{2}=max({s}_{j}{c}_{j}^{2}) \,for \,j=1:m$$
(4)

  • 3. The vectors with \(svn\) ≤ 0.20 can be classified as DLVs, and once the set of DLV vectors has been obtained, the damage detection of the part is performed as follows:

    $${nsi}_{i}=\frac{{\sigma }_{i}}{{\sigma }_{max}}$$
    (5)

where, \({nsi}_{i}\) = normalized stress value for member-i, \({\sigma }_{i}\) = stress at member-i, \({\sigma }_{max}\) = maximum stress.

When the normalized stress index (nsi) of a part approaches zero, it shows the fraction is potentially damaged. However, internal forces, such as axial forces, bending moments, or shear forces, can replace the stress criterion. This substitution is practicable because stress values are a function of internal forces for constant cross-sectional properties. Replacing the nsi value with internal forces significantly reduces computational time in which Equation (5) can be modified accordingly.

$${normalized \,axial \,forces}_{i}=\frac{{AF}_{i}}{{AF}_{max}}$$
(6)

where, \({AF}_{i}\) represent an axial force for member-i and \({AF}_{max}\) is the maximum axial force.

1.2 Firefly algorithm (FA)

In civil engineering, optimization techniques are frequently utilized to produce the best possible outcomes in study or design for numerous structural design [20]. Furthermore, an optimisation method can be used to identify structural degradation. Many studies have been conducted in order to find out structural degradation utilising optimisation algorithms [21,22,23]. In this study, the FA, initially proposed by Yang [24] based on firefly performance, includes two crucial factors which are attractiveness and the variation of light intensity. The objective function is matched by attractiveness, as measured by the firefly’s brightness.

The fundamental equation in the FA calculates the movement of a firefly i attracted to others, while Equation (7) is used to determine the more attractive (brighter) location of firefly j.

$${x}_{i}^{t+1}={x}_{i}^{t}+{\beta }_{0}{e}^{-\gamma {r}_{ij}^{-}}({x}_{i}^{t}+{x}_{i}^{t})+{\alpha \varepsilon }_{i}^{t}$$
(7)

The second term, β0, represents attractiveness at zero distance (r = 0), while \({e}^{-\gamma {r}_{ij}^{-}}\) is for attractiveness. The third term, \({x}_{i}^{t}\), serves as the randomization parameter (mutation parameter), and \({\varepsilon }_{i}^{t}\) is a random vector number based on Gaussian or uniform distribution at time t.

Generally, FA has six steps as follows:

  1. 1.

    Initiating an initial population of n fireflies \({x}_{i}^{t}\) (i = 1, 2, 3, 4, …, n).

  2. 2.

    Determining the light intensity Ii at xi according to the objective function, \(f({x}_{i}^{t})\).

  3. 3.

    Defining light absorption coefficient, \(\gamma \).

  4. 4.

    For i = 1:n (all fireflies)

    For j = 1:n (all fireflies)

    When Ii<Ij

    move the firefly from i to j using equation (7)

    end when

    evaluate new solutions and update the light intensity

    end for loop j

    end for loop i

  5. 5.

    Sort the fireflies and determine the current global best g*.

  6. 6.

    Interpretation the result

FA is a distinctive algorithm that combines the strengths of three different optimization methods which include Differential Evolution (DE), Simulated Annealing (SA), and Accelerated Particle Swarm Optimization (APSO) as expressed in Eq. (7). When \(\alpha =0\) and \(\beta =0\), FA degenerates into DE, when β0 = 0, FA transforms into SA, and substituting \({x}_{i}^{t}\) with g* causes FA to deteriorate into APSO. Consequently, FA possesses the advantages of all three algorithms, rendering it more adaptable as well as efficient and this adaptability allows FA to surpass other algorithms such as Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) [25].

When the normalized stress index (nsi) of a part approaches zero, it shows the fraction is potentially damaged. However, internal forces, such as axial forces, bending moments, or shear forces, can replace the stress criterion. This substitution is practicable because stress values are a function of internal forces for constant cross-sectional properties. Replacing the nsi value with internal forces significantly reduces computational time in which equation (5) can be modified accordingly.

1.3 Modal assurance criterion (MAC)

MAC is a method used to assess the similarity between two mode shapes which evaluates the consistency between two vectors namely, the estimated modal vector and the original modal vector originated from experimental tests [26]. The MAC value ranges from 0 to 1, showing inconsistency when the MAC value is close to zero and consistency when the MAC value is close to 1. The criteria have been frequently employed in cases of damage detection [27, 28].

The MAC can be defined in terms of the mode shape of the damaged structure \((\mathrm{\phi d})\) and the mode shape of the predicted structural model \((\mathrm{\phi a})\), as follows.

$${MAC}_{j}=\frac{{\left|{\{\phi a\}}_{j}^{T}{\{\phi d\}}_{j}\right|}^{2}}{\left({\{\phi a\}}_{j}^{T}{\{\phi d\}}_{j}\right)\left({\{\phi a\}}_{j}^{T}{\{\phi d\}}_{j}\right)}$$
(8)

where \({\{\phi a\}}_{j}\) is the mode shape vector of the predicted structure for j-mode, and \({\{\phi d\}}_{j}\) is the mode shape vector of the damaged structure for j-mode. When the MAC value is 1, the mode shape is identical (similar), however, when the MAC value is zero, there is no connection between two modal vectors.

2 Methodology

2.1 Combination of DLV method and MAC-FA

This study introduced a third-level damage detection method by combining the DLV method, MAC, and FA. This method addressed the DLV limitation method in accurately predicting damaged parts in a structure. The method used static and dynamic properties of structures, with DLV functioning as the static method for damage detection, while MAC-FA was based on the mode shapes of the structures. This analysis extended previous inquiries to assess the strength of the method in predicting either the location or the stiffness losses of damaged members [12, 19, 29, 30]. Figure 1 showed the flowchart of the proposed method that began with the flexibility matrix of the healthy and damaged condition, the mode shape of the damaged condition, and the predicted mode shape of the structures as input variables. The DLV method was then used to determine parts that might have inaccurately predicted damaged parts at this stage. Subsequently, the number of fireflies was introduced based on the number of damaged units, with each representing stiffness losses. The FA with MAC criterion was then applied to obtain the stiffness loss of the damaged members. The proposed method showed its advantage by modifying inaccuracies from the DLV method, as the stiffness losses of the damaged part were close to 0 when the DLV method failed to accurately predict the parts, and all calculations as well as coding were developed using MATLAB [30, 31]

Fig. 1
figure 1

Flowchart of DLV, MAC, and FA combination

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2.2 Fitness and penalty function

The fitness function in this algorithm was the addition of MAC values for n-modes to be considered. As the FA used a minimization function and the MAC value represented the correlation of two different mode shapes (ϕa and ϕd), where the MAC increased with a higher correlation between two mode shapes, the fitness function was modified accordingly.

$$F=\frac{1}{\sum_{j=1}^{n}{MAC}_{j}}$$
(9)

where F was the fitness value, MAC represented the modal assurance condition of two-mode shapes (\(\phi a\) and \(\phi d\)), and n was the number of mode shapes to be considered.

The firefly location showed the remaining stiffness of the damaged part, and once the firefly location was outside the value of 0 and 1, the penalty function would be automatically activated. Consequently, the fitness function was set to the maximum value the program could read.

3 Result and discussion

3.1 Application 1: MAC-FA was the third level of damage detection for obtaining the loss of stiffness of the space truss structure

Considering a 25-bar space truss structure as shown in Figure 2, this structure had been used in many optimization problem [32,33,34] and referenced for this optimization problem. The cross-section and modulus of elasticity were 1354.8 mm2 and 69.95 MPa for all parts. Four damaged scenarios were considered as shown in Table 1, where the damaged part was represented by loss of stiffness, and in this case, the part was also represented as the reduction of modulus of elasticity.

Fig. 2
figure 2

25-bar space truss

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Table 1 Damaged scenario
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To analyze the damage, the location of the damaged part was determined using the DLV method that was applied to the structure for all damaged scenarios. The DLV approach is initiated by examining differences in the flexibility matrix between damaged and undamaged structures. Mathematically, equation (1) can be used. Once the flexibility matrix has been obtained, the subsequent procedure involves determining the DLV matrix by applying equations (2) to (4). This DLV matrix is then utilized as a load vector on undamaged structures to calculate the axial forces of every member. The axial forces are then normalized using equation (5). Elements with normalized axial forces approaching zero are identified as being damaged. Figures 2 and 3 showed the normalized axial forces for all members of each damaged scenario using the DLV method. In this case, the DLV accurately predicted the damaged part for all scenarios. For example, in the fourth scenario (Fig. 4b), the normalized axial forces for parts 4, 8, 11, and 17 were zero. Those parts could be considered damaged but with no information regarding the loss of stiffness.

Fig. 3
figure 3

Normal axial forces for (a) first damaged scenario, (b) second scenario

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Fig. 4
figure 4

Normal axial forces for (a) third scenario, (b) fourth damaged scenario

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After determining the location of damaged parts using DLV, MAC-FA was applied, and the FA parameters were detailed in Table 2. Considering all degrees of freedom in this application, the minimum fitness value achievable based on Equation (8) was 0.055555556. This minimum fitness value showed that all predicted mode shapes matched the mode shape of the structure, with a fitness value of 0.0555556. Figure 5 showed the fitness reduction for the duplication across all damaged scenarios. It was seen from Fig. 5 that convergence was achieved in fewer than 10 iterations.

Table 2 FA parameters
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Fig. 5
figure 5

Fitness reduction for all damaged scenarios

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Tables 3, 4, 5 and 6 represent the results for all damage scenarios acquired using the proposed method. It was noticed that the proposed method accurately predicted the decrease in stiffness of the damaged part for all damage scenarios.

Table 3 Test model specifications and test conditions
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Table 4 The second damaged scenario result for the first application
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Table 5 The third damaged scenario result for the first application
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Table 6 The fourth damaged scenario result for the first application
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3.2 Application 2: MAC-FA as improvement technique for damage locating vector method of space truss structure

In the first application, DLV precisely determined the location of the damaged part, while MAC-FA served as an algorithm to calculate the stiffness losses for damaged segments. In the second application, MAC-FA was used to correct the location of parts predicted by the DLV method. In this instance, the DLV method predicted more damaged fractions than there were. Besides improving the accuracy in predicting the damage location, the MAC-FA method also accurately determined the loss of stiffness in damaged portions.

A pyramid truss structure with 32 bars was considered (Figure 6), as discussed in [35, 36], and the base length in the x- and y-directions was 2000 mm, while the height of the structure was 707 mm. DLV method was performed to obtain the location of the damaged members.

Fig. 6
figure 6

Pyramid module of space truss

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Four damaged scenarios were considered:

  1. 1.

    The first damaged scenario was a single damage, in which part 11 had a stiffness reduction of 50% from the initial stiffness.

  2. 2.

    The second scenario was double damage, in which members 16 and 29 had 50% and 75% stiffness reduction from initial stiffness, respectively.

  3. 3.

    The third scenario was triple damage, in which parts 8, 18, and 31 had stiffness reduction of 10%, 60%, and 38%, respectively.

  4. 4.

    The fourth scenario was multiple damage, in which six members, i.e., parts 2, 9, 16, 23, 28, and 32 were simulated to have stiffness reduction of 35%, 10%, 60%, 28%, 70%, and 38%, respectively.

In Figures 7 and 8, DLV method accurately detected damaged parts in the space truss structure for the first and second scenarios. However, the method faced challenges in predicting damaged members for the third and fourth scenarios. In the third scenario, DLV mistakenly labeled member-7 as damaged, along with members 8, 18, and 31. Figure 8a showed that the normalized axial force for member 7 was nearly zero, leading to an incorrect prediction by the DLV method.

Fig. 7
figure 7

Normal Axial Force for (a) first damaged scenario, (b) second damaged scenario

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Fig. 8
figure 8

Normal Axial Force for (a) third scenario, (b) fourth damaged scenario

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A similar situation was recorded in the fourth scenario where DLV identified member 1 as damaged. However, Figure 8b showed that the normalized axial force for member 1 was close to zero, indicating it was essentially a healthy part. Despite accurately identifying damaged parts in the first two scenarios, DLV could not predict stiffness reduction in these parts and this limitation could be tackled by using MAC-FA.

For this second application, MAC-FA was used four times to ensure the convergence, maintaining consistent lower and upper bound values as shown in Table 7. Due to space constraints, only the results for the first scenario were presented, meanwhile, Figure 9 showed the fitness reduction for each iteration with different lower and upper bound values. Similarly with the first application, the result was obtained less than 10 iterations. Considering 5 mode shapes, the minimum fitness achievable based on equation (8) was 0.2, and convergence was achieved in fewer than 10 repetitions, with a minimum fitness value of 0.200009.

Table 7 Lower and upper bound for each run
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Fig. 9
figure 9

Fitness reduction with various lower and upper bound values

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In order to demonstrate the speed at which FA converges, Genetic Algorithm (GA) is employed to achieve a reduction in the stiffness of the damaged component. This algorithm has been used in previous study [37]. Figure 10 shows the results achieved through the utilization of GA. It is noticeable that a significant number of generations, specifically around 120, are required to find a solution. In contrast, employing FA leads in the attainment of a solution in fewer than 10 iterations. This demonstrates that the technique of FA possesses the capacity to achieve convergent outcomes at a faster speed when compared to GA.

Fig. 10
figure 10

Fitness reduction using GA

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Tables 8, 9, 10 and 11 showed the results obtained by DLV-MAC-FA in the first and second scenarios, where DLV accurately predicted the damaged members (Tables 8 and 9), while MAC-FA determined stiffness losses. Consequently, there was no difference between the remaining stiffness values obtained from the proposed method and the defined scenarios.

Table 8 The first damaged scenario result for the second application
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Table 9 The second damaged scenario result for the second application
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Table 10 The third damaged scenario result for the second application
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Table 11 The fourth damaged scenario result for the second application
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Table 10 particularly showed MAC-FA results for the third damaged scenario. By comparing these results with Figure 8a from the DLV method, MAC-FA accurately identified member 7 as a healthy member with a remaining stiffness of 0.9994. Therefore, member 7 was categorized as healthy, while members 8, 18, and 31 were identified as damaged with stiffness reductions as shown in Table 10.

Table 11 showed results for the fourth scenario obtained from MAC-FA. Although DLV identified member 1 as damaged, MAC-FA showed a remaining stiffness of 0.9834, categorizing member 1 as healthy. Other members were successfully predicted, as shown in Table 11.

4 Conclusion

In conclusion, this study addressed the improvement of DLV, with a dual focus on accurately predicting damaged locations and quantifying the stiffness loss in affected structural parts. The precision of DLV in identifying damaged locations in structures was significantly improved by combining MAC and FA. Furthermore, the combined effectiveness of these methods was shown through evaluations of two distinct space truss structures which before applied on plane truss structures [37]. The first application included a space truss structure with 25 bars, while the second analyzed a pyramid space truss with 32 bars. In both applications, DLV-MAC-FA successfully predicted the damaged location and the loss of fitness of the damaged parts. To achieve the desired result, DLV was used for predicting damaged parts, and MAC-FA combination was used to evaluate stiffness loss and refine the damaged location. Therefore, the proposed method could be categorized as a level three detection method, capable of evaluating both the loss of stiffness and the remaining strength of structural members. The stability of this method was shown in its convergence to near-optimal results, even when using different upper and lower bounds for the loss of stiffness.