An effective method for damage assessment based on limited measured locations in skeletal structures

Abstract

This article proposes a new damage detection method using Modal Test Analysis Model and artificial neural networks. A challenge in damage detection problems is lack of measured degrees of freedom, as well as limitations of attached sensors. Modal Test Analysis Model has been used in order to estimate unmeasured degrees of freedom. An experimental cantilever beam was used to show Modal Test Analysis Model’s efficiency in estimation of unmeasured mode shapes. To solve the inverse problem of damage detection, mode shapes estimated by Modal Test Analysis Model were used as inputs, and characteristics of the damage served as outputs of the artificial neural network. The sensitivity analysis carried out for each example showing the performance of artificial neural network after mode shape expansion was efficiently improved. Three numerical examples for plane and space truss structures are considered, in order to verify effectiveness of the proposed method. Results demonstrate a high accuracy of Modal Test Analysis Model and artificial neural network for structural damage detection.

Keywords

artificial neural networks damage detection experimental cantilever beam Modal Test Analysis Model plane truss space truss

Introduction

Safety is an important issue for all civil structures such as bridges, buildings, and offshore jacket platforms. Structural damages as a result of environmental factors, repetitive and excessive loadings are inevitable (Nguyen et al., 2016). Vibration-based damage identification techniques are established upon the idea that any damage exerted to the structural elements changes the stiffness, thereby affecting the dynamic characteristics such as natural frequencies and mode shapes (Prawin et al., 2020).

Natural frequency–based methods are capable of revealing global changes, and mode shape–based methods are sensitive to local changes (Hosseini et al., 2017). Ghannadi and Kourehli (2019b) introduced a model updating method based on natural frequencies and natural frequency vector assurance criterion. Their method was applied to laboratory scale structures in order to detect and quantify the damages. Khatir et al. (2018b) conducted experimental tests for crack identification in beam-like structures using natural frequencies. Some studies have also used the data from the frequencies in order to detect damages in composite elements (Khatir et al., 2015, 2018a). Zenzen et al. (2018) presented an approach to identify the severity and location of damage in beam and truss structures using frequency response function data. In terms of application of natural frequencies for crack identification in plates, Khatir et al. (2020) employed a combination of artificial neural network (ANN) and Jaya optimization algorithm to develop a robust and accurate method. Some methods have been developed using mode shapes. To address the optimization-based damage detection problem, an objective function is utilized consisting of modal assurance criterion (MAC) flexibility and natural frequency differences formulated by Ghannadi and Kourehli (2019c). Khatir et al. (2019) proposed the Cornwell indicator for damage quantification in laminated composite plates. A modified Cornwell indicator was used to perform damage detection in skeletal structures (Tiachacht et al., 2018). Ghannadi and Kourehli (2020) have made a comparison between two objective functions. The first one is based on the MAC and the second one on modified total modal assurance criterion (MTMAC). Their research shows that MTMAC provides more accurate results in structural damage identification.

Structures modeled as a discrete system have numerous degrees of freedom (DOFs). Therefore, a large number of frequencies and mode shapes are obtained. There is no economic justification for measurement of mode shapes and frequencies at all DOFs (Rezaiee et al., 2020). Furthermore, measurement of mode shapes is even more difficult than frequencies (Humar et al., 2006). There are two common solutions to address the incompleteness of measurement. First, the finite element model (FEM) was reduced to fit the size of the measured DOFs. Ghannadi and Kourehli (2018) investigated different FEM reduction techniques and accuracy studies on each reduction technique such as Guyan, improved reduced system (IRS), iterated improved reduced system (IIRS), and system equivalent reduction expansion process (SEREP).

Second, unmeasured DOFs were estimated by expansion methods. Guyan, IRS, and SEREP are also used as expansion techniques. Guyan is a static method and may give accurate mode shape expansion estimations only when there are enough DOFs to represent the mass inertia of the actual tested dynamic system (Chen, 2010). Au et al. (2003) presented an application of SEREP expansion and genetic algorithm for damage assessment in beam-like structures. Ghannadi et al. (2020) proposed a methodology of damage detection based on SEREP expansion and grey wolf optimizer. Abdollahi and Tavakkoli (2019) employed two mode shape expansion techniques namely SEREP, and dynamic expansion to estimate the unmeasured modal data in solid structures. In this article, damage detection problems are solved as an optimization scheme through minimizing an objective function consisting of a mode shape term. Rezaiee et al. (2020) presented a sensitivity-based FEM updating strategy. Their method includes an SEREP expansion in order to address deficiency of mode shapes. Lale Arefi et al. (2020) used the IRS method to estimate the slave DOF in a planar frame and a spatial truss. They have then formulated the modified modal strain energy-based index to detect damaged elements when a limited number of sensors are installed.

Li et al. (2008) applied different methods of model reduction and expansion for offshore jacket structure. Their results showed that model reduction is always better than model expansion. Liu and Li (2012) developed a rapid direct-mode shape expansion method and applied it for offshore jacket structures. Their proposed method includes a hybrid vector which was assembled through measurements at master DOFs and constant values at slave DOFs. In a two-step method, Kourehli (2018) predicted unmeasured mode shapes by least squares support vector machine. He has then determined the location and severity of the damages using predicted mode shapes obtained from the first step. Ghannadi and Kourehli (2019a) used the ANN and a non-smooth version of SEREP expansion for damage detection of discrete structures. The efficiency of this method was validated by the spring-mass system.

The opening of this article is an introduction to vibration-based damage detection technique, using natural frequencies, mode shapes, and other dynamic characteristics, alongside a review on FEM reduction and expansion techniques. Measuring all DOFs is impossible due to lack of attached sensors. This shortage of measured DOFs can be solved in two ways, either by reducing the FEM to fit the size of experimental one or by expanding the experimental data to obtain unmeasured DOFs in the FEM.

For the first time, this article presents a damage detection and severity identification approach based on a mode shape expansion method called Modal Test Analysis Model (Modal TAM) and ANN. To training the ANN, expanded mode shape vectors considered as inputs, whereas damage location and severity vectors are targets. When tested against new data, the ANN-learnt model receives an expanded mode shape vector as input, and damage location and severity are outputs. Therefore, ANN immediately reacts to the new test data after the first learning.

Theoretical background

It is assumed that damage reduces stiffness of the elements, and the mass is constant during the life-cycles (Ghahremani et al., 2020). Stiffness of damaged structures with nel number of elements and damage parameter d_i (i = 1, 2, …, nel) are formulated by equation (1)

K_{d} = \sum_{i = 1}^{nel} (1 - d_{i}) k_{i}^{e}

(1)

where K_d is the stiffness of damaged system and $k_{i}^{e}$ represents the stiffness matrix of the ith element. d_i parameter may have values between 0 and 1 indicating intact and fully damaged structures, respectively.

TAM

The fundamental idea behind the TAM is to generate a transform between the full FEM and the experimental modal model (EMM). This can be formulated as follows

{\begin{matrix} x_{a} \\ x_{o} \end{matrix}}_{m \times 1} = {[T_{TAM}]}_{m \times a} {x_{a}}_{a \times 1}

(2)

where [T] is a TAM transformation matrix which transforms the EMM to the full FEM, and subscripts a and o are active (a-set) and omitted (o-set) DOFs, respectively. m dimension of the full FEM DOFs and ${\begin{matrix} x_{a} \\ x_{o} \end{matrix}}$ is the displacement vector at full FEM DOFs (Mains, 2013).

Modal TAM

The Modal TAM is similar to SEREP, but a-set modes are unmodified. This is achieved by realizing that there is a direct representation of {X_a}

{x_{a}} = [I] {x_{a}}

(3)

Transformation matrix of Modal TAM is as follows

[T_{TAM}] = {[\begin{matrix} [I] \\ [Φ_{o}] {[Φ_{a}]}^{+} \end{matrix}]}_{m \times a}

(4)

where $[Φ_{o}]$ and $[Φ_{a}]$ are the o-set and a-set partition of the matrix of analytically obtained mode shapes, respectively. Also, “+” represents pseudo-inverse of a matrix. Modal TAM has a very high accuracy since the mode shapes of the FEM are used to formulate the transformation matrix. It has been successfully implemented on several large structures (Mains, 2013; Qu, 2004).

Performance of Modal TAM

To evaluate applicability of Modal TAM for mode shape expansion, an experimental cantilever beam is employed (see Figure 1). As illustrated in Figure 1, six accelerometers (B&K8340-type uni-axial) are located on this cantilever beam, and measurements are performed in vertical direction. In this experiment, the cantilever beam was excited by ambient vibration. B&K3560 data acquisition system with 17 channels is also used. Experimental dynamic characteristics for healthy state and damaged scenarios are extracted. Details can be found in Altunışık et al., (2017). As can be seen in Figures 2 and 3, damage scenarios are considered as single or multiple cracks at the locations of A-A, B-B, and C-C. In order to investigate the performance of Modal TAM for mode shape expansion, three different patterns of measurements for both undamaged and damaged beams are assumed:

Pattern #1: Active DOFs ={1, 3, 4}, Omitted DOFs ={2, 5, 6}.

Pattern #2: Active DOFs ={2, 5, 6}, Omitted DOFs ={1, 3, 4}.

Pattern #3: Active DOFs ={3, 4, 5}, Omitted DOFs ={1, 2, 6}.

For instance, only three DOFs 1, 3, and 4 are considered as measured (active) in pattern #1. The rest are unmeasured (omitted), estimated by Modal TAM. Table 1 shows MAC values between experimental and expanded mode shape vectors. All MAC values are very close to 1, which represents that expanded mode shapes have a good agreement with experimental ones. In conclusion, Modal TAM provides an accurate estimation for unmeasured mode shapes.

Figure 1.

Experimental cantilever beam and accelerometer locations (Altunışık et al., 2017).

Figure 2.

Damage locations on cantilever beam.

Figure 3.

Different damage scenarios for cantilever beam (Altunışık et al., 2019).

Table 1.

MAC values between experimental and expanded mode shapes.

Scenarios	Pattern #1		Pattern #2		Pattern #3
Scenarios	Mode	MAC	Mode	MAC	Mode	MAC
Undamaged	1	1.0000	1	0.9997	1	0.9999
	2	0.9889	2	0.9954	2	0.9965
	3	0.9964	3	0.9925	3	0.9963
Damaged—1	1	0.9998	1	0.9999	1	0.9998
	2	0.9876	2	0.9968	2	0.9988
	3	0.9982	3	0.9926	3	0.9963
Damaged—2	1	0.9999	1	0.9996	1	0.9998
	2	0.9872	2	0.9966	2	0.9981
	3	0.9890	3	0.9767	3	0.9876
Damaged—3	1	0.9999	1	0.9998	1	0.9998
	2	0.9895	2	0.9969	2	0.9983
	3	0.9944	3	0.9853	3	0.9957
Damaged—4	1	0.9996	1	0.9997	1	0.9996
	2	0.9943	2	0.9949	2	0.9970
	3	0.9640	3	0.9251	3	0.9959
Damaged—5	1	0.9997	1	0.9999	1	0.9997
	2	0.9960	2	0.9981	2	0.9986
	3	0.9728	3	0.9699	3	0.9919
Damaged—6	1	0.9998	1	0.9999	1	0.9997
	2	0.9997	2	0.9991	2	0.9999
	3	0.9957	3	0.9882	3	0.9948

MAC: modal assurance criterion.

MAC describes the coherence between the mode shape vectors, defined as follows

MA C_{i} = \frac{{({Φ_{i}^{x}}^{T} {Φ_{i}^{y}})}^{2}}{({Φ_{i}^{y}}^{T} {Φ_{i}^{y}}) ({Φ_{i}^{x}}^{T} {Φ_{i}^{x}})}

(5)

where ${Φ_{i}^{x}}$ and ${Φ_{i}^{y}}$ represent the vectors corresponding to the ith mode shape (Kaveh and Dadras Eslamlou, 2019).

Selecting training algorithm

ANN can be trained using different algorithms such as Levenberg–Marquardt (LM) or Bayesian Regulation. The study of LynDee et al. (2011) comprises different ANN training algorithms for vibration-based damage identification problems, where the results show the LM algorithm provides the best generalization performance. But that requires high computer memory and longer training time. Kourehli (2015) have obtained best performances using LM for damage detection in structures with a low number of elements (10 or less). In general, LM is suitable for training small and medium-sized problems (Beale et al., 2010; Yu and Wilamowski, 2011). Conjugate gradient algorithm with Polak and Ribière updates (CGP) requires less memory and shorter time for training process. See more details on CGP algorithm in Hagan et al. (2014). In this article, the ANN was trained using CGP to solve inverse problems of damage detection. The performance of the training algorithm has been measured by mean squared error (MSE) and R value. The mathematical relations of these parameters are given in equations (6) and (7)

MSE = \frac{1}{N} \sum_{i = 1}^{N} {(P_{i} - O_{i})}^{2}

(6)

R = \frac{\sum_{i = 1}^{N} (P_{i} - P_{m}) (O_{i} - O_{m})}{\sqrt{{(P_{i} - P_{m})}^{2}} \sqrt{{(O_{i} - O_{m})}^{2}}}

(7)

where O_i is the measured value, P_i stands for prediction values, N is the number of data points, O_m is the mean observation value, and P_m is the mean prediction value. In fact, the R parameter was chosen to determine the correlation between predicted and measured values. If the R value is more than 0.8, it means that there is a strong correlation between measured and predicted values. In summary, the MSE is closer to zero and R is closer to one, which would be a good performance for the ANN (Kaveh et al., 2018).

Numerical examples

Numerical examples of plane truss and space truss are used in this section to specify the location and severity of damages. To utilize machine learning in damage detection or any similar problems, datasets shall be generated for training processes. The architectural design of the ANN consisting of the number of input samples and the number of neurons in the hidden layer is achieved through trial and error. The main focus of this article is to overcome measurement limitations. Therefore, the performance of ANN before and after mode shapes expansion is considered to serve as sensitivity analysis.

Experimental measurements may be erroneous due to noises coming from environmental conditions or measurement errors. The equation below pollutes the mode shapes obtained from FEM with a certain percentage of noise to expose them to real measurement conditions

\overset{*}{D} = Data \times (1 + β λ)

(8)

where Data consists of any component of the eigenvectors matrix, β is the noise level, λ is a random value uniformly distributed in [−1, 1] interval, and $\overset{*}{D}$ are noise-polluted components (Kaveh and Dadras Eslamlou, 2018).

All of the numerical examples and ANN configuration were performed using MATLAB software. The flowchart in Figure 4 shows damage detection steps. It can be seen from Figure 4 that the efficiency of the proposed method is also examined with the effect of noise in the input samples.

Figure 4.

The flowchart of damage detection steps.

Thirty-one-bar plane truss

Plane truss shown in Figure 5 consists of 31 elements and 14 nodes (Law and Lin, 2014). Each node has two translational DOFs (X and Y directions). The length of horizontal and vertical members was considered to be 1.52 m, while diagonal members’ length was 2.1496 m. Material properties used for this example are given as follows: cross-section A= 0.0025 m², mass density $ρ$ = 2770 kg/m³, and Young’s modulus E = 70 GPa. Many researchers including Kim et al. (2019), Wei et al. (2018), Seyedpoor et al. (2019), Kim et al. (2018), Ghiasi et al. (2018), and Kim et al. (2016) have attempted to address this problem.

Figure 5.

Plane truss with 31 elements (Law and Lin, 2014).

To generate the machine learning datasets, damage parameters d_i = 0 and d_i = 0.1–0.5 were considered for elements 1 through 31. A total of 6000 random samples were then selected from all possible permutations, 4200 random samples were used for ANN training and the remaining 1800 were utilized for test and validation. All generated datasets are assumed to be measured by sensors set at limited nodes of the plane truss. Three different patterns of measurements are used to investigate the impact of sensor arrangement on the performance of ANN:

Pattern #1: Active nodes = {2, 3}, Omitted nodes = {4 up to 14}

Pattern #2: Active nodes = {2, 3, 4}, Omitted nodes = {5 up to 14}

Pattern #3: Active nodes = {2, 3, 4, 5}, Omitted nodes = {6 up to 14}

In pattern #1, only two vertical DOFs are measured at nodes 2 and 3. The rest of the DOFs are estimated using Modal TAM. Each input sample contains the first two mode shape vectors of the structure.

In pattern #2, only three vertical DOFs are measured at nodes 2, 3, and 4. Similar to the previous pattern, the rest of the DOFs are estimated using Modal TAM. For this pattern, each input sample includes the first three mode shape vectors.

In pattern #3, four vertical DOFs are assumed to be measured, and there are more measured nodes than pattern #2 and pattern #1. For this pattern, each input sample is constructed by the first four mode shape vectors.

The architecture of ANN consisted of an input layer, a hidden layer with 200 neurons, and an output layer. The performance of ANN before and after mode shape expansion method considering the three patterns is listed in Table 2. The results show that after mode shape expansion method is applied, R values are modified as closer to one. Some smaller values of MSE are also achieved. Therefore, the performance of ANN is significantly improved.

Table 2.

Performance report for 31-bar plane truss.

	Pattern #1		Pattern #2		Pattern #3
	R	MSE	R	MSE	R	MSE
Before expansion
Training	0.89990	2.12E–03	0.94447	1.203E–03	0.99434	1.25E–04
Test	0.89855	2.17E–03	0.94383	1.222E–03	0.99339	1.48E–04
Validation	0.89721	2.15E–03	0.94467	1.215E–03	0.99362	1.43E–04
After expansion
Training	0.99463	1.19E–04	0.99630	8.20E–05	0.99799	4.5E–05
Test	0.99450	1.24E–04	0.99606	8.70E–05	0.99770	5.1E–05
Validation	0.99443	1.23E–04	0.99606	8.80E–05	0.99763	5.4E–05

MSE: mean squared error.

After applying the mode shape expansion method, pattern #3 has the lowest MSE and highest R value compared to pattern #1 and pattern #2. Therefore, pattern #3 is adopted to report damage detection results.

Table 3 shows the damage scenarios for this example. The damage identification results of two damage scenarios in comparison with the actual severity and location of damages are shown in Figures 6 and 7. It can be found that for both scenarios, the presented method based on expanded mode shapes and ANN can accurately detect all damaged elements in the 31-bar plane truss. Moreover, the severity and location of damaged elements are still accurately detected for both scenarios, even under the influence of different noise levels.

Table 3.

Damage scenarios for the 31-bar plane truss.

Scenarios	Elements	Damage severities
1	20	0.3
2	4	0.14
	11	0.13
	20	0.26

Figure 6.

Damage detection results for 31-bar plane truss: scenario 1.

Figure 7.

Damage detection results for 31-bar plane truss: scenario 2.

Fifty-two-bar space truss

The space truss shown in Figure 8 consists of 52 elements and 21 nodes (Jalili et al., 2014). Each node has three translational DOFs (X, Y, and Z directions). The mass of nodes in FEM is neglected. Material properties utilized for this example are given as follows: cross-section A= 2 × 10⁻⁴ m², mass density $ρ$ = 7800 kg/m³, and Young’s modulus E = 210 GPa. Some studies on this matter have been previously conducted by several researchers, such as Seyedpoor and Montazer (2016), Truong et al. (2020), and Nobahari et al. (2019).

Figure 8.

Space truss with 52 elements: (a) plan view and (b) elevation view (Jalili et al., 2014).

Similar to the former example, in order to generate machine learning datasets, damage parameters d_i = 0 and d_i = 0.1–0.5 were considered for elements 1 through 52. A total of 8000 random samples were then selected from all possible permutations, 5600 random samples were used for ANN training and the remaining 2400 samples were used for test and validation. It should be noted that only limited sensors are assumed to be placed at nodes in order to generate the datasets.

To demonstrate the sensitivity of measured points on the performance of ANN, three different patterns of active nodes are proposed as follows:

Pattern #1: Active nodes = {5, 6}, Omitted nodes = {1 up to 4, 7 up to 13}

Pattern #2: Active nodes = {5, 6, 7}, Omitted nodes = {1 up to 4, 8 up to 13}

Pattern #3: Active nodes = {5, 6, 7, 8}, Omitted nodes = {1 up to 4, 9 up to 13}

In pattern #1, only two sensors in vertical direction were placed at nodes 5 and 6. In pattern #2, three vertical DOFs of nodes 5, 6, and 7 are measured. Similarly, four vertical DOFs of nodes 5, 6, 7, and 8 are measured in pattern #3. Finally, Modal TAM is employed to determine unmeasured DOFs. First two, first three, and first four of mode shape vectors are used as input samples in pattern #1, pattern #2, and pattern #3, respectively. The architecture of ANN consisted of an input layer, a hidden layer with 350 neurons, and an output layer.

Table 4 presents the performance of ANN for three measured patterns, before and after applying the mode shape expansion method. It can be witnessed that for all patterns, Modal TAM can effectively improve the performance of ANN performing as the mode shape expansion method.

Table 4.

Performance report for 52-bar space truss.

	Pattern #1		Pattern #2		Pattern #3
	R	MSE	R	MSE	R	MSE
Before expansion
Training	0.66711	6.33E–03	0.81780	3.78E–03	0.90268	2.12E–03
Test	0.65915	6.52E–03	0.81180	3.94E–03	0.89026	2.38E–03
Validation	0.66340	6.46E–03	0.80712	3.97E–03	0.88745	2.42E–03
After expansion
Training	0.98841	2.63E–04	0.99355	1.47E–04	0.99260	1.68E–04
Test	0.98702	2.94E–04	0.99028	2.21E–04	0.98986	2.32E–04
Validation	0.98756	2.85E–04	0.99003	2.25E–04	0.98971	2.35E–04

MSE: mean squared error.

From the results shown in Table 4, it can be found that when using pattern #2 to train the ANN, smaller values of MSE are achieved. In addition, R values are very close to 1 and represent a good agreement between predicted and actual outputs. Therefore, after applying the mode shape expansion method, pattern #2 is selected to report the damage identification results.

Table 5 shows the damage scenarios for this example. Figures 9 and 10 demonstrate the efficiency and reliability of the proposed approach with both the noisy and noise-free input samples. It can be observed that for both damage scenarios in 52-bar space truss, the location of damages is detected with high accuracy. However, there are some false damaged elements that appear in noisy conditions.

Table 5.

Damage scenarios for 52-bar space truss.

Scenarios	Elements	Damage severities
1	18	0.33
2	27	0.37
	43	0.48

Figure 9.

Damage detection results for 52-bar space truss: scenario 1.

Figure 10.

Damage detection results for 52-bar space truss: scenario 2.

Seventy-two-bar space truss

Figure 11 shows a 72-bar space truss and its node and element numbering schemes (Dizangian and Ghasemi, 2015). This example consists of 20 nodes, and each node has three translational DOFs (X, Y, and Z directions). The mass of nodes is neglected in FEM. Material properties utilized for this example are given as follows: cross-section A = 0.0025 m², mass density $ρ$ = 2770 kg/m³, and Young’s modulus E = 69.8 GPa. Many researchers including Kaveh et al. (2019), Bureerat and Pholdee (2018), Nguyen-Thoi et al. (2018), Shahrouzi and Sabzi (2018), Kaveh and Zolghadr (2015), Pholdee and Bureeratand (2016), and Kaveh et al. (2016) have attempted to address this problem.

Figure 11.

Space truss with 72 elements (Dizangian and Ghasemi, 2015).

Damage parameters d_i = 0 and d_i = 0.1–0.5 were considered for elements 1 through 72. A total of 16,000 random samples were then selected from all possible permutations to generate machine learning datasets, 11,200 random samples were used for ANN training and the remaining 4800 samples were utilized for test and validation. Similar to the previous examples, mode shapes of incomplete DOFs are available, and machine learning datasets are collected from limited sensors. The performance of ANN has assessments through three patterns of measured locations:

Pattern #1: Active nodes = {9, 10}, Omitted nodes = {5 up to 8, 11 up to 20}

Pattern #2: Active nodes = {9, 10, 11}, Omitted nodes = {5 up to 8, 12 up to 20}

Pattern #3: Active nodes = {9, 10, 11, 12}, Omitted nodes = {5 up to 8, 13 up to 20}

Pattern #1 covers two vertical measured DOFs at nodes 9 and 10. Pattern #2 includes three measured DOFs of nodes 9, 10, and 11. In pattern #3, four sensors were installed at nodes 9, 10, 11, and 12 only to measure the vertical responses. Meanwhile, the rest of the unmeasured DOFs are determined by Modal TAM. For pattern #1, first two mode shape vectors are selected as input samples. The first three and first four mode shapes are also considered as input samples in pattern #2 and pattern #3, respectively.

The architecture of ANN included an input layer, a hidden layer with 400 neurons, and an output layer. The sensitivity analysis carried out on the performance of ANN for different patterns of measurements is shown in Table 6. The results in Table 6 show that after applying the mode shape expansion method in pattern #1, a very small modification is observed for the performance of ANN. While after applying the mode shape expansion method in pattern #2 and pattern #3, the performance of ANN was significantly improved. After applying the mode shape expansion method in pattern #3, very small values of MSE are calculated. Besides, compared to pattern #1 and pattern #2, R values are very close to 1 and represent an excellent correlation between predicted and actual values. Hence, the results of damage detection are based on pattern #3. Table 7 shows the damage scenarios for this example. Figures 12 and 13 show the identification results of 72-bar space truss. The identified damage severities are close to the actual values, and there is no confusion in localization. When applying a certain percentage of noise, some false identification is observed. As expected, the damage identification results demonstrate that the increment of noise level in the input samples has a minor effect on the precision of the presented approach. The results of this numerical example again confirm the reliability and efficiency of the ANN combined with Modal TAM in identifying the severity and location of damage.

Table 6.

Performance report for 72-bar space truss.

	Pattern #1		Pattern #2		Pattern #3
	R	MSE	R	MSE	R	MSE
Before expansion
Training	0.84083	1.96E–03	0.85284	1.83E–03	0.92473	9.76E–04
Test	0.84013	2.00E–03	0.85203	1.84E–03	0.92354	9.83E–04
Validation	0.84004	1.97E–03	0.84970	1.87E–03	0.92193	1.01E–03
After expansion
Training	0.84373	1.94E–03	0.92857	9.24E–04	0.99796	2.70E–05
Test	0.84075	1.96E–03	0.92739	9.39E–04	0.99778	3.00E–05
Validation	0.83614	2.01E–03	0.92706	9.49E–04	0.99710	3.90E–05

MSE: mean squared error.

Table 7.

Damage scenarios for 72-bar space truss.

Scenarios	Elements	Damage severities
1	26	0.28
	64	0.45
2	26	0.45
	51	0.4
	59	0.24

Figure 12.

Damage detection results for 72-bar space truss: scenario 1.

Figure 13.

Damage detection results for 72-bar space truss: scenario 2.

Conclusion

This study proposed a new inverse method properly determining damage locations and severities in three numerical examples including plane and space truss structures based on the ANN and Modal TAM. A challenge in damage detection problems is lack of measured DOFs and limitations of sensors installed on the structures. Therefore, the mode shape expansion method called Modal TAM was applied to estimate unmeasured DOFs. Applicability of Modal TAM was validated using an experimental cantilever beam in a healthy state and different damage scenarios. MAC values between experimental and expanded beams are very close to 1, which shows that expanded mode shapes have a good agreement with experimental ones. To solve the inverse problem of damage detection, mode shapes estimated by Modal TAM were used as inputs, whereas location and severity of damage were considered to be outputs of the ANN. Sensitivity analyses were performed for numerical examples considering different arrangements of measured points, where the performance of ANN (R value and MSE) is efficiently improved after mode shape expansion. Numerical results show that the combination of the Modal TAM and ANN is more efficient for damage detection and identification of its severity.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Parsa Ghannadi

Seyed Sina Kourehli

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