1 Introduction

SHM is a technology to automate the inspection process in order to assess and evaluate the health condition of structures in real-time or at specified time intervals. SHM limits the number of collapsed structures, gives an opportunity to repair them, extends their lives and therefore it avoids demolishing them and constructing new structures. Thus, money will be saved, and the environment will be protected as well.

There are four different sequential levels: detection, identification, quantification and prediction. The higher levels of SHM are, the more complicated SHM technology is. SHM technology can be divided into two main categories, namely physical model–based and statistical/data model–based approaches. Physical model–based concentrates on the changes in vibration parameters such as frequency, mode shape and damping by using Finite Element Analysis (FEA) with experiment data. The change of mode shape combined with Mac or Comac function are used to detect damage in [1, 2]. The frequency of a bridge can be extracted indirectly from a passing vehicle and then use for SHM [3, 4]. Statistical model-based methodology only considers the differences between two group of responses: responses under the intact condition and the responses under the damaged condition [5]. [6,7,8] success in using statistical analysis response measurement to detect damage in structure.

AANN is a statistical technique, which has been widely used in SHM during last decades. They proved that AANN is successfully used to detect damage [9,10,11]. In this paper, transmissibility is used incorporated with AANN to detect damage. The transmissibility in a single-degree-of-freedom system is defined as the ratio between the amplitude of the response displacement and the amplitude of the motion. The idea of transmissibility can be extended to a system with N degree-of-freedom. Transmissibility use output only response measurement, defined on output to output relationship. Transmissibility makes possible to detect damage without any assumption about the nature of excitations even though different loading conditions are applied during the experiments. In this paper, an indicator based on transmissibility function is used to get the input data for AANN.

2 Theoretical Derivation

2.1 Transmissibility of Motion

Firstly, we consider the relationship between responses and forces in term of reacceptance. If one has a vector FA of magnitudes of applied forces at coordinates A, a vector XU of unknown response amplitudes at coordinates U and vector XK of known response amplitudes at coordinates K. The reader may find more information about transmissibility in reference [12].

One may establish the following relationship:

$$ X_{U} = \, H_{UA} F_{A} $$
(1)
$$ X_{K} = \, H_{KA} F_{A} $$
(2)

Where HUA and HKA are the reacceptance frequency response matrices related coordinates U and A, and K and A, respectively. Eliminating FA in two equation, we have:

$$ X_{U} = H_{UA} H_{KA}^{ + } X_{K} $$
(3)

Or

$$ X_{U} = T_{UK} X_{K} $$
(4)

Thus, the transmissibility matrix is defined as:

$$ T_{UK} = H_{UA} H_{KA}^{ + } $$
(5)

The transmissibility matrix could be evaluated directedly from the measurement of responses. From Eq. (4), we also have:

$$ T_{UK} = X_{U} X_{K}^{ - 1} $$
(6)

If we consider a simply supported beam, the dynamic equilibrium equation can be written as:

$$ M\ddot{x}\left( t \right) + Cx \left( t \right) + K\dot{x}\left( t \right) = f\left( t \right) $$
(7)

Where M, C and K are the mass, damping and stiffness matrices of the system, respectively.

f(t) is the input vector and x(t) contains the responses of each DOF of the system.

Solving this equation, we can calculate the displacement, velocity and the acceleration.

Herein, for a harmonic applied force at given coordinate, the transmissibility between point i and a reference to point j can be defined as:

$$ T_{{\left( {i,j} \right)}} \left( \omega \right) = X_{i} \left( \omega \right)X_{j}^{ - 1} \left( \omega \right) $$
(8)

Where Xi and Xj are the complex amplitudes of the system response xi(t) and xj(t), respectively.

2.2 Auto-Associative Neural Network

The human brain has about 1012 neurons and about 1014 neural connections between them. This complex pairing system gives us the ability to analyze, process information, emotions, etc. The brain is able to organize and control its basic elements (single neurons) to perform tasks such as identification, control and analysis in much more effective way than the current computer. For example, the human brain is able to identify familiar faces in the crowd, estimating the distance of the observed object’s moving velocities in the period from 100 ms to 200 ms, the speed at which the computer and computational software are still not available today. The ability to analyze handwriting, audio, and sound analysis, understand foreign languages and dialects is also a difficult task to simulate.

Artificial Neural Network (ANN) attempt to bring computer a little closer to brain’s capacity by imitating certain aspects of information processing in the brain, in a highly simplified way. ANN investigates the capabilities of the human brain and reproduces those capabilities on machines, equipment or software. Once trained ANN is able to recognize similarities presented with a new input pattern, resulting in a predicted output pattern [13]. ANN has two basic processes: the learning process and the testing process. Learning process is the process of creating knowledge from existing information. This process is done for a set of sample data called metrics. Learning consists of three tasks; (a) compute output, (b) compare output with desired target then adjust weight and (c) repeat the process. The quality of the learning process is expressed by a target function or a function error. The learning process will reduce the error of this function. The learning process is usually learned on learning algorithms to optimize target functions. To evaluate the effectiveness of the learning process, one needs to use an additional testing process. A trained network is tested with a new set of data, called the test data set. The test variance will show the operability of ANN with new data, which has not appeared during the learning process. The low-test error corresponds to the ability to handle new good cases. AANN Application process consists of:

  1. 1.

    Collect data

  2. 2.

    Separate in to training and test Sets

  3. 3.

    Defined a network structure

  4. 4.

    Select a learning algorithm

  5. 5.

    Set parameters, values, initialize weights

  6. 6.

    Transform data to network Input

  7. 7.

    Start training, and determine and revise weights

  8. 8.

    Stop and test

  9. 9.

    Implementation: Use the network with new case

3 Numerical Example

A FE model of an experimentally tested simply supported beam, shown in Fig. 1, is divided in 18 elements as shown in Fig. 2. The beam is made of steel and has I100 cross-section. Young’s modulus is 190.98 GPa, the density is 7800 kg/m3, and the length is 3 m. The beam is fitted with acceleration sensors in fixed positions. Based on the data of the accelerometers, we can calculate the experimental fundamental frequency as 34.26 Hz. Then, using the SAP2000 software to update the FE model, of the numerical fundamental frequency is determined as 34.26 Hz. The dynamic force is applied to the beam at node 3 and, in turn, the accelerations are obtained at the other nodes. To investigate the damage in the beam, we reduce the stiffness of each element in the beam. For each element, a stiffness reduction from 0% to 50% with an interval of 1% is recorded. For each single damage, there will be 51 scenarios, i.e. from D1 to D50 and D0 is for an intact case.

Fig. 1.
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Experimental beam

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Fig. 2.
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FE model beam

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3.1 Input for AANN

As discussed above, the transmissibility matrix T34, T35, T36, T37 could be evaluated directly from the measurement of the responses at nodes 3, 4, 5, 6 and 7 using Eq. (8). Zhou et al. [9], the indicator takes the sum of transmissibility along the specific frequency range that can be described as:

$$ TI_{i} = \int_{{f_{min} }}^{{f_{max} }} {T_{i,j} df} $$
(9)

Where fmin and fmax are the low and high boundaries for the integration area as seen in Fig. 3.

Fig. 3.
figure 3

A description of the frequency band

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The choose of fmin and fmax greatly influences results and this is usually done by experience. With 4 functions T34, T35, T36, T37 we have 4 indicators TI1, TI2, TI3, TI4 can be used as input for the network. The intact beam natural frequency of mode 1 is 34.26 Hz. Survey Fig. 6, we saw the first peak of transmissibility function near the mode 1 most, then we used this peak to calculate.

As we know in advance, natural frequencies are important factors in assessing structural failure. There are also number of studies that make use of frequencies as damage indicators. However, in many cases, they are changing very slightly and then we don’t have enough evidence to conclude that whether structure is damaged or not [14]. The natural frequencies of a simple beam can be determined using acceleration sensors located on the beam. In this paper, the natural frequencies are used as an input for the network.

3.2 Target for AANN

The first two levels of SHM are detection and localization of damage. This paper concentrates on those two indexes as the target for the network. The severity of damage is shown by the percentage decrease of the stiffness of damage section. Damages are introduced in four elements 4, 5, 6, 7. For each damage cases, we have 50 scenarios as explained earlier. All input parameters for the network are calculated based on the damaged position and the severity of the damage, respectively. The network diagram is shown in Fig. 4.

Fig. 4.
figure 4

The schematic structure of multi-layer perceptron neural networks model

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3.3 Damage Detection Procedure

Step 1: Transmissibility is estimated from Eq. (4) for all measurements.

Step 2: The TI index is calculated based on the Eq. (9) and used as input data for AANN. This process is performed repeatedly with multiple defect locations and varying severity of damage (the stiffness reduces from 0% to 50%) as described above. Damage locations and deterioration levels are retained as a target for AANN.

Step 3: The AANN working as described in Sect. 2.2.

Step 4: Results analysis: Based on the objective function that evaluates whether the network is performing well or not. If the operation is not good, we can change the Norton number or change the frequency domain. If the result is satisfactory, the calculation is finished, and the network can be used.

3.4 Result Discussion

3.4.1 Intact Beam

We apply different forces to a beam position and measure the transmissibility function in the existing beam model. Because of symmetric, the T34 is almost the same with T36 and T37 is a horizontal line (Fig. 5). From the results shown in Figs. 5 and 6, we observe that although the applied force is different, the transmissibility function remains unchanged. This emphases that the transfer function can be determined from the response only, which can be characterized by the dynamic effect of the system and independent of the force. This makes it easier to apply it in practice cases because sometimes it is extremely difficult to measure the applied force on a structural in-service.

Fig. 5.
figure 5

The transmissibility with input 1

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Fig. 6.
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The transmissibility T34 with difference inputs

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3.4.2 Damaged Beam

The transmissibility’s of damaged beam (the location of damage section is node 6) are shown in Figs. 7 and 8. It is easy to see that even when the damage is very small, the transmission function changes immediately. As the stiffness decreases, the peak of the transfer function moves slightly to the left Fig. 9. This suggests that the transfer function is more sensitive to the detection of damage than to the severity of the damage.

Fig. 7.
figure 7

The D1 transmissibility function damage in node 6

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Fig. 8.
figure 8

The D50 transmissibility function damage in node 6

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Fig. 9.
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Compared the first peak of the transmissibility with difference scenarios damage in node 6

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As discussed above, beams will be damaged at different levels by changing the stiffness of each element. This is done at positions close to points 4, 5, 6, 7. TI indicator is calculated and used as input data for AANN network. 70% sample of the data is used to train the network, which is adjusted according to its error. 15% sample is used to measure the network generalization, and to halt training when generalization stops improving. 15% sample is used to test the network, which has no effect on training and so provides an independent measure of network performance during and after training.

Figure 10 shows the network outputs with respect to targets for training, validation and test sets. The results are perfectly fit, the data falls along 45-degree line, where network outputs are equal to the targets and with R value in each case of slightly above 0.99.

Fig. 10.
figure 10

Correlation between actual and predicted values in training phase, validation phase and test phase

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Figure 11 shows the predicted results of AANN function with 4 damage positions and 50 levels. The vertical axis indicates the location of the damage section (at 4 nodes 4, 5, 6, 7) and the horizontal axis is 50° of damage respectively (from D1 to D50). The graph shows that if the stiffness decreased by less than 5%, the results are less accurate. However, with more severity of damage, the network’s predicted results are relatively accurate. This is understandable because the structure needs enough changes to be identifiable and distinguishable. With the above results, we can confirm that after training the AANN network, we can completely detect the degree of damage and its location.

Fig. 11.
figure 11

Comparison between actual damage location and AANN prediction

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4 Conclusion

This study proved that using transmissibility together with AANN could be used to detect and localize damage. It used response measurements only, and the use of AANN make it possible to detect damage once the base-line was defined.

The object of this study was simply supported beams that were simulated with various degrees of failure, i.e. damage positions and severities. Research has shown that, with the data collected, the network after learning was completely capable to identify damage. It should be noted, however, that the selection of input parameters greatly influences the results. Using this method requires large number of data sets to train and test the network, which is a drawback. In this paper, we analysis each of the single damage cases, however many cases may contain multiple damages. To solve this problem, we need a very large data sets. However, today with the development of structural analysis software, the structure can be modeled using FEA and then get data to train the network, thus opening a new direction for SHM technology.