Keywords

1 Introduction

Structural damage detection is an essential approach to prevent sudden collapse of structures and avoid casualties and heavy economic losses. A series of damage detection methods have been proposed [1, 6, 11, 18, 25], the principle of vibration-based methods is that the modal parameters (e.g., natural frequencies and modal shapes) of a structure vary with the changes of the structural physical parameters such as the stiffness and mass. By collecting the vibration excitation and response data of a structure, the modal parameters are obtained, and then the potential damage of the structure is detected by analyzing the change of its modal parameters. Frequency-based structural damage detection method has been used in damage detection of composite structures [2, 3]. Modal-based methods show that local damage causes irregularity of mode shapes [21] which is evident for relatively large damage [17]. Nevertheless, the changes of natural frequencies and mode shapes are unable to detect small damage [30]. As the modal strain energy (MSE) is related to the second-order derivatives of mode shapes for beam-like structures, it is much more sensitive to the damage than natural frequencies and mode shapes. The MSE has been used to successfully locate structural damage and quantify the damage level [4, 10, 16, 20, 22] for simple structures. Structural damage can cause changes in many mechanical parameters, normally, a single damage index is generally impossible to reflect all types of damage of the real structures. Thus it is essential to develop a comprehensive damage detection method, such as artificial neural networks (ANNs) [8], which is able to integrate multiple damage features into a detection method. An ANN is similar to a human brain and has excellent non-linear learning ability. Combination of ANNs and traditional damage indicators for obtaining structural damage information may advance the damage detection technology.

It has been demonstrated that an ANN is able to locate and quantify structural damage owing to its powerful data fitting as well as pattern recognition capability [7, 9], It has achieved promising results [19, 24]. The traditional ANN (e.g., BP neural network) has its inherent shortcomings, such as low convergence rate, time-consuming, and over-fitting of data [27, 28], etc. When the traditional ANN is used to detect damage location, its training time is very long for high-dimension input data. To overcome the limitations of ANNs, the convolutional neural network (CNN), with convolution layers and pooling layers, has been developed to extract the features of the image [29] and been proved successful. It has more powerful feature learning ability and feature expression ability [12] than traditional ANNs. At present, the CNN has been widely used in license plate detection, face recognition and other fields [13, 23, 26]. The CNN has also been applied to SHM, such as crack detection [5, 14] and damage feature extraction from low-order vibration signals [15]. The application of CNNs to SHM provides a new intelligent method for structural damage detection. It is expected that CNNs can be applied to predict the locations and levels of damage in a structure.

In this paper, a CNN is proposed to detect damage of a simply supported beam; the modal strain energy is used as the input data of the CNN as it can reflect the damage information of structures, and comparisons are made between the performance of the CNN and traditional BP neural networks.

2 Methods

The CNN was trained with the modal strain energy of various damage scenarios and then used to predict new damage scenarios. In this paper, the collected data were arranged into a two-dimensional matrix as the CNN input data. The abnormality of the modal strain energy was extracted by the CNN to predict the damage location and damage level.

2.1 Numerical Calculation and Sample

The beam model used in this paper had a length of 9 m and a rectangular cross section of 0.3 m × 0.2 m. The structure was divided into 36 elements as numbered in Fig. 1.

Fig. 1
figure 1

Simply supported beam model with 36 elements

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The Young’s modulus, Poisson’s ratio and density of the steel beam model were 211 GPa, 0.288, and 7800 kg/m3 respectively. Structural damage was simulated by reducing the Young’s modulus of the concerned element. Totally 18 damage levels for each element were simulated, which were from 5% to 90% with an increment of 5%. In-house python scripts were used to prepare the training samples for the CNN.

Firstly, the detection of damage location is studied. The validation set was based on the cases with 45% damage in an element and the intact case (totally 37 samples), and the testing sets included the cases with either 10% or 60% damage in an element (72 samples).The training set included the following five groups, each group included 37 samples, i.e., 36 scenarios with damage in only one element plus the intact case.

  • Dataset (A): only 15% damage in one element for each damage scenario plus the intact case;

  • Dataset (B): only 30% damage in one element for each damage scenario plus the intact case;

  • Dataset (C): only 75% damage in one element for each damage scenario plus the intact case;

  • Dataset (D): only 90% damage in one element for each damage scenario plus the intact case.

Then, the detection of damage level is studied. The validation set was based on the data of 45% damage in an element and the intact case (37 samples), the samples for 60% damage in an element (36 samples) were used to test the CNN fitting effect, the training set included the following five groups.

  • Dataset (A): Every element was simulated with 15% damage levels, thus there were 36 damage scenarios, plus 1 intact case, and thus there were totally 37 samples.

  • Dataset (B): Every element was simulated with 75% damage levels, thus there were 36 damage scenarios, plus 1 intact case, and thus there were totally 37 samples.

  • Dataset (C): Every element was simulated with 15%, 30%, 75% and 90% damage levels, thus there were 144 damage scenarios, plus 1 intact case, and thus there were totally 145 samples.

  • Dataset (D): Every element was simulated with 16 damage levels from 5% to 90% with an increment of 5%, thus there were 36 × 16 = 576 damage scenarios, plus 1 intact case, and thus there were totally 577 samples.

2.2 Convolutional Neural Network

The CNN was designed and trained using the Deep Learning Toolbox in MATLAB (MathWorks Inc, Natick, MA, US). The network included an input layer, 2 convolution layers, 1 pooling layer, an activation layer, a fully connected layer and output layer (classification layer or regression layer); for classification problems, a softmax layer was added after the fully connected layer. The explanations for the activation function and convolution and pooling processes were seen in the appendix A. In this paper, a CNN was designed for damage detection. The network architecture and structural parameters of the CNN were shown in Fig. 2 and Table 1.

Fig. 2
figure 2

CNN framework. C1, C2: Convolution layers; P: Pooling layer; FC: Fully Connected layer

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Table 1 Structural parameters of the CNN
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2.3 Input and Output of Network

The CNN input data, i.e., the modal strain energy of each element, was collected for each damage scenario, and a matrix of 6 × 6 was constructed as the input. This paper used a classification method to detect the damage location, which was set to different categories, i.e. the intact condition was set to 0, the damage on the element 1 set to 1, the damage on the element 2 set to 2, and so on.

For detection damage level, by replacing the softmax layer and classification layer with a regression layer of the CNN, the classification problem was transformed into a regression problem. The network output was set as a vector consisting of 36 elements, i.e. the 15% damage in Element 1 was set as [0.15, 0, 0… 0, 0, 0], 30% damage of Element 1 was set to [0.3, 0, 0, 0… 0, 0, 0], and so on; the intact case was defined to be a vector of 36 zeros.

3 Results

3.1 Detection of Single Damage Location

The CNN was trained with the 4 training sets described in Sect. 2.1 separately. The detection results were shown in Table 2.

Table 2 Prediction results by CNN
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According to the detection effect of the four different Datasets, no matter which dataset was used for training, the prediction accuracy was 100% for the testing set with the damage level of 60%. For the testing set with the damage level of 10%, the prediction accuracy reached 100% only for the CNN trained by Dataset (A).

In order to compare the detection effect between the CNN and a traditional BP neural network, Dataset (A) was inputted into BP neural networks, and the prediction results were shown in Table 3.

Table 3 Prediction results by BP neural networks
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Table 3 showed that the detection effect of the BP neural networks was worse than that of the CNN, and the detection effect of BP neural networks did not change significantly with the node number of the hidden layer. The best detection effect was 77.8%.

3.2 Detection of Damage Level

The testing sets were inputted into the trained network as described in Sect. 2.1. The detection results were shown in Table 4, the relative error decreases with the increase of training samples, and the minimum relative error of the damage level was about 5%.

Table 4 Prediction result of different Datasets
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The relative error was used to evaluate the detection effect of damage level, and the formula was as follows:

$$R_{Error} = \frac{{\left| {y_{p} - y_{t} } \right|}}{{y_{t} }} \times 100\%$$
(1)

where \(y_{p}\) and \(y_{t}\) were the predicted value and target value of a testing sample. The relative error for n testing samples was:

$$R\_Error = \frac{1}{n}\sum\nolimits_{1}^{n} {\left( {\frac{{\left| {y_{p} - y_{t} } \right|}}{{y_{t} }}} \right) \times 100\% }$$
(2)

The Dataset (D) were inputted into the BP neural networks. After training, the detection effect was obtained and shown in Table 5.

Table 5 Detection effect of the BP neural networks
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As shown in Table 5, with the increase of the nodes of the hidden layer, the relative error of the BP neural networks decreased gradually, and the error was lower than that of the CNN, but the training time increased significantly. The fitting effect of the BP neural networks for damage level was better than that of the CNN. Table 6 showed the uptimes of the CNN and the BP neural networks which had the most comparative relative errors to that of the CNN.

Table 6 Comparisons of uptime between CNN and BP neural networks
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When the relative errors were similar, the uptime of the CNN was about 5%–40% that of the BP neural networks, the iteration time of the CNN was only about 1% that of the BP neural networks. For the smallest relative error (1%) of the BP neural network in Table 5, the relative error of CNN was only 5%, but its uptime was only 2% that of the BP neural network.

4 Discussions

The comparisons of damage locations detected between the CNNs and BP neural networks showed that the CNNs had better classification ability than BP neural networks to extract damage location features. The CNNs used partially connected network structure to extract the main damage features, ignoring trivial information. While the BP neural networks adopted a fully connected network structure, each element in the raw data had an impact on the results.

It can be seen from Sect. 3.2 that the detection effect (damage level) of the BP neural networks was slightly better than that of the CNN. When the BP neural network had 36 and 42 nodes in the hidden layer, the relative errors were 6% and 2%, respectively, and the CNN were 5%. The uptime was 10 min 24 s, 78 min 58 s, 4 min 21 s, respectively. Thus, when the relative error was similar, the uptime of CNN was 5%–40% that of the BP neural networks.

In summary, the BP neural networks had excellent regression fitting effects but consumed a lot of computing resources. Because of its partial connection, the CNN lost some information and its fitting effect was sacrificed in the training process, but the calculation speed was faster than that of the BP neural network.

5 Conclusion

Based on the above discussions, the following conclusions can be drawn:

  1. 1)

    The CNN achieved excellent detection results for structural damage location and level.

  2. 2)

    The CNN had more advantages than the BP neural networks in detecting damage location.

  3. 3)

    The CNN was more economical in computational costs than a BP neural network.

According to the results of this paper, the combination of the CNN and the modal strain energy as a new damage detection method has great potential in structural damage detection.