3.1. Force Prediction in Single Connections
The first task was to identify the forces in the bolts in a single connection. Each contained a screw, which was not equipped with a force sensor, but the elastic wave signals were recorded there during a static tensile test. With reference to this particular screw, the axial forces were predicted.
In previous works [
11,
17], it was assumed that data from all bolts (2, 4, 5) were to be used for the purpose of ANN training, assuming a constant distribution of patterns for testing and validation. The new approach proposed herein involves the learning, testing, and validation patterns being separated into individual bolts from a given connection. Among the possible combinations, the following division of patterns was adopted (short names are provided in brackets; they are also used in the data descriptions in the charts and tables):
This is to achieve the repeatability of force identification for various bolts in the connection and then possibly extend the database to include bolts from the other connections.
After the first series of simulations, it turned out that none of the signal parameter sets (PCA, TA, encoder) allowed us to obtain a satisfactory level of identification accuracy for the axial forces in the screws. This was due to the fact that there were relatively significant differences between the signals measured in the individual bolts. Of course, this can be influenced not only by the excitation pattern and its parameters (especially the frequency), but also by the likely differences in the attachment of the screws (some of them may touch the edge of the hole, while others may not). However, these are limitations arising both in the measurement capabilities and factors that occur in real constructions. For this reason, the obtained pattern database derived from laboratory experiments of connection models with real dimensions is a good test of the proper operation of the developed diagnostic system.
At the current stage of research, it was decided to extend the database by the elongation measured on one of the screws. This measurement was made using inductive sensors applied to both ends of the S1 screw. In this way, the input data gained an additional physical sense, which seems to be the key issue in this case.
The input vectors consisted of the elongation of S1 and:
Twelve principal components (PCA);
Six amplitudes of the response signals and theirs six arrival times (TA);
Twelve parameters obtained from the encoder.
The obtained values of the identification errors for two exemplary connections (P1) with respect to the input data used are shown in
Table 2. It can be seen that the smallest learning errors were obtained for the PCA, but this also led to the largest validation errors. Time and amplitude (TA) had the lowest validation error, but reasonably good accuracy was obtained in the case of the encoder. A graphic comparison of the results for the encoder and PCA is shown in
Figure 8. Therein, the F2 and F5 forces measured in the S2 and S5 bolts, as well as the respective force values obtained from the ANN (test and valid) are shown. It can be seen there that they are in good agreement, especially the input vector taken from the encoder. In addition, the predicted values of the axial forces in bolt S3 (predict) show the nature of the changes to be very similar to the other screws.
If we look at the fragment of results that are enlarged (
Figure 8c,d), we see that training the ANN using the input data taken from the encoder led to the averaging of the identified quantities (
Figure 8c), which is very interesting. However, in the case of the principal components (
Figure 8d), the results obtained followed the measured values, which in the case of the measurement errors may lead to them becoming stronger.
The same approach was also repeated for the other connections. For each case, it was possible to identify the axial forces in the bolts at a similar level of accuracy. As an example, the list of errors obtained for the P3 connection can be analyzed (
Table 3). This time, the best validation results were obtained for the encoder data. Although they were very similar to the TA results, the errors for learning and testing were at a lower level. Therefore, in other cases presented in the paper, encoder data were used as the input to the ANN.
3.2. Force Prediction in Sets of Connections
In the next task, an attempt was made to combine the data from two and more connections. The idea was to check whether it was possible to train the diagnostic system on one connection and use it to assess the forces in the bolts of the other connections. Therefore, connections with the same bolt lengths were grouped, and the results of the simulations carried out are presented in the following sections.
3.2.1. P2P3
The first set of connections consisted of two connections (P2P3). They corresponded in terms of the dimensions of the end plates and the class of screws. In this case, two sets of input vectors were also analyzed:
The ANN used data from the P2 = {S2, S4, S5} connection to learn, while data from P3 = {S4, S5} were used for testing and P3 = {S2} for validation;
The ANN used data from the P3 = {S2, S4, S5} connection to learn, while data from P2 = {S4, S5} were used for testing and P2 = {S2} for validation.
In both cases, the prediction was made with respect to the S3 screws from both connections (P2, P3). The obtained results of testing, validation, and prediction are shown in
Figure 9. There was a good agreement between the measured axial forces (F2, F5) and the values obtained from the ANNs.
It can be seen in
Table 4 that lower statistical parameters were observed when data from P3 were used for learning and P2 for training and validation.
3.2.2. Set of P2P3P4 Connections
The promising results obtained for the first set prompted us to extend it with data related to the P4 connection, which differed not only in the dimensions of the front plate, but also in the class of screws. Training patterns in this case were separated as follows: learning was performed on patterns from P2 and P3 connections (i.e., P2 = {S2, S4, S5}, P3 = {S4, S5}), while testing and validation involved a set of patterns related to P3 and P4 connections (i.e., [P3 = {S2}, P4 = {S2}], and P4 = {S4, S5}, respectively). An example of the results obtained from testing and prediction is shown in
Figure 10. As before, there is good agreement between the measured axial forces F2 and the values estimated by the ANN (
Figure 10a). The predicted values of the forces in the S3 screws (which were not equipped with force sensors) for all three connections are shown in
Figure 10b. The nature of their changes is similar to the other bolts in the analyzed connections.
3.2.3. Set of P2P3P4P6 Connections
The dataset used to train the ANN was expanded to include results from the next connection (P6). The patterns in this process were divided according to the following scheme: the learning was carried out on data from bolts P2 = {S2, S4, S5}, P3 = {S4, S5}, P4 = {S4, S5}; the testing involved data related to [P3 = {S2}, P4 = {S2}, P6 = {S2}]; while the validation was performed using data from P6 = {S4, S5}. The obtained results from the testing and prediction of the axial forces in the S2 bolts in the analyzed connections are shown in
Figure 11.
The characteristic pattern in this case was the noticeable smoothing of the test results. The determined elongation values indicated that in a certain phase of the tensile test, the length of the S1 screw was temporarily shortened. The real reason, however, may be the fact that the end plates of the P6 connection were deformed and the screws were bent (see
Figure 2). Thus, the changes registered by the sensors were the result of the bolt end rotating and not its actual shortening.
Summarizing the results obtained so far, it can be stated that adding information related to the elongation of one of the bolts allows for good estimation of the identified values of the axial forces in the bolts of flange connections. However, taking into account that the application of this approach in practice is very difficult or even impossible, an attempt was also made to replace the bolt elongation with its increments. This idea and preliminary results are briefly described in the next section.
3.3. Force Prediction Using Load/Elongation Increments
The concept of identifying forces in bolts using load or elongation increments consists of the fact that the load change in the tested connection is introduced. This is accompanied by the simultaneous measurement of elastic wave propagation and the elongation in one of the screws. Then, the obtained elongation increment is added to the other parameters provided in the input vector to the ANN.
To show the potential of the proposed approach, preliminary simulations were performed on the data related to the P3 connection. For this purpose, the database was divided into relatively small increments. It turned out that better identification results were obtained when these increments were not uniform, but variable with a random Gaussian distribution. Thus, the database used for ANN training consisted of elongation increments of the S1 bolt and the parameters of signals (taken from the encoder) measured at the following bolts: S4 for learning; S2 for testing; S5 for validation; and S3 for prediction.
The results of the preliminary tests are presented in
Figure 12a together with the results corresponding to the ANN trained directly using the elongation of the S1 screw (
Figure 12b). It can be seen that there were some differences in the accuracy of testing, validation, and prediction of the axial forces. The error values calculated on this basis are collected in
Table 5. Despite the clearly superior results in the case of training the ANN with the direct elongation of the S1 screw, the results of both approaches remained in good agreement with the results of the experimental measurements of the F2 and F5 axial forces in the S2 and S5 screws. Therefore, in future studies, this approach will also be extended to other connections.
As a summary of the approaches analyzed so far, the results of axial force prediction in the S3 bolt of the P3 connection (which was not equipped with a force sensor) are summarized in
Figure 13a. There, we can see the results obtained for the ANNs trained on the basis of data from a single connection (P3 with elongation, P3inwith elongation increments), two connections (P2P3), and three connections (P2P3P4). The results related to P3 and P2P3P4 are similar to each other (which is clearly seen in
Figure 13b), while P3in and P2P3 slightly differ from them.