ANNs have emerged as the powerful tools for damage quantification in different structures, including beams [21, 22], bridges [23, 24, 25] and plates [26, 27]. ANN is formed by interconnected neurons consisting of input, hidden, and output layers. The quantity of neurons in each layer may differ. All of the neurons in the layers are connected to each other. The feedforward neural network with backpropagation (BP) algorithm constructed in MATLAB R2020a is shown in Fig. 8 along with the structure of ANN used in the work. The network is trained using the Levenberg-Marquardt BP method, which is a feature of NN tools. A log-sigmoidal transfer function is shared by the 25 neurons in each of the two hidden layers that make up the NN.
Out of total 100 elements, 64 elements (excluding the 36 elements which are next to the clamped support) are damaged due to reduction in ME ranging from 5–30% with an increment of 5%. This has resulted in study of 6 damage cases. The GFEQDR is obtained for 100 locations, which means that, 600 DI’s (6x100) are obtained for each of the damaged elements out of 6 cases. Therefore, 38400 DI’s are obtained (64x600) for 64 damaged elements. Using 38400 DIs as input, 26880 data are used for network training, 5760 data are used for validation, and the remaining 5760 data are used for testing. More data are used to train the network in order to achieve higher output accuracy.
The effectiveness of the trained NN can be estimated by utilizing regression analysis and mse. Figure 9 demonstrates the regression plots of the trained neural network, along with the regression value (R) indicating the relationship between the targets and outputs. A relationship with a value of '0' indicates randomness, while one with a value near '1' indicates proximity. R = 0.99994, denotes a network that has been properly trained. For the network that was trained (Y = T), the best fit line and the perfect fit line (in Fig. 8) overlapped. This demonstrates that, a close correspondence between predicted and desired results is achieved.
When the mse of the validation process reaches the target (10− 06), the optimal ANN is obtained at 39 epochs, as shown in Fig. 10. The difference between the expected ANN results (target) and the validation test results is used to compute the mse. The mse settles down to 0.0000075183 after 39 epochs. Figure 11 depicts the performance of gradient, mu, and validation check steps during the training process of the model. The first two plots show the attained values of gradient, and mu at every epoch. Hence, the optimal values of gradient and mu are achieved at 45 epochs. The validation plot explains that, after 39 epochs, the system fails six times, thereby the training process stops. Figure 12 shows the variation between the DI predicted by ANN, and the DI (βn) computed from Eq. 4 for all the damaged elements. The R2 values, for the elements damaged are very near to 1. Moreover, the error histogram in Fig. 13, refers the difference of error between targets and outputs, for the training and testing stages of NN. These errors indicate as to how much predicted values differ from the target values. The total error of ANN ranges from − 0.00905 to 0.01714. Out of 384 target values, 240 values got 0.000603 error,125 values got − 0.00078 error,19 values got larger error, which means 95% data samples have errors that are close to zero. Once we train the network, the same network can be adopted to estimate the DI’s for the unknown damaged cases also.
Several more testing samples are used to evaluate the trained ANN. Table 3, refers to the comparison between ANN output and the DI’s (βn) computed from Eq. 4. The results indicate the maximum percentage of difference which is 0.30, between ANN output and the expected outcome. Table 4 depicts the quantification equation, for each of the damaged elements, by drawing the linear trendline in the graph in excel. These equations relate, percentage reduction in ME (when percentage reduction in effected) with the β. The DI’s can be easily computed for the given damage severity, which is effected by the percentage reduction of ME, without using any code/programme. This is a non-destructive method of assessing the DIs and is highly helpful in the health monitoring of LCP’s.
Table 3
Comparison of ANN output with expected output
Element Number | % reduction in ME | Expected output | ANN output | % Difference |
5 | 6 | 0.0453 | 0.0454 | 0.22% |
23 | 17 | 0.1339 | 0.1335 | 0.30% |
31 | 12 | 0.0708 | 0.0710 | 0.28% |
32 | 24 | 0.3208 | 0.3209 | 0.03% |
71 | 26 | 0.3742 | 0.3746 | 0.11% |
Table 4
Damage quantification equation for different elements
Element number | Damage quantification equation | R2 | Element number | Damage quantification equation | R2 |
5 | 0.0097x − 0.0183 | 0.995 | 55 | 0.016x − 0.0323 | 0.995 |
8 | 0.0014x − 0.001 | 0.999 | 56 | 0.0161x − 0.0333 | 0.994 |
9 | 0.0124x − 0.0243 | 0.995 | 57 | 0.0052x − 0.0109 | 0.994 |
12 | 0.0069x − 0.0102 | 0.997 | 58 | 0.0097x − 0.0183 | 0.995 |
13 | 0.0052x − 0.0109 | 0.994 | 61 | 0.0124x − 0.0243 | 0.995 |
14 | 0.0136x − 0.0267 | 0.995 | 62 | 0.012x − 0.0253 | 0.994 |
17 | 0.009x − 0.0139 | 0.997 | 63 | 0.0166x − 0.0343 | 0.994 |
18 | 0.0143x − 0.0289 | 0.995 | 64 | 0.016x − 0.0323 | 0.995 |
19 | 0.012x − 0.0253 | 0.994 | 65 | 0.0159x − 0.0321 | 0.995 |
20 | 0.0141x − 0.0277 | 0.995 | 66 | 0.0143x − 0.0289 | 0.995 |
23 | 0.009x − 0.0139 | 0.997 | 67 | 0.0014x − 0.001 | 0.999 |
24 | 0.0148x − 0.0293 | 0.995 | 70 | 0.0136x − 0.0267 | 0.995 |
25 | 0.0161x − 0.0333 | 0.994 | 71 | 0.0157x − 0.0328 | 0.994 |
26 | 0.0157x − 0.0328 | 0.994 | 72 | 0.0166x − 0.0343 | 0.994 |
27 | 0.0141x − 0.0277 | 0.995 | 73 | 0.0159x − 0.0321 | 0.995 |
31 | 0.0069x − 0.0102 | 0.997 | 74 | 0.0148x − 0.0293 | 0.995 |
32 | 0.0148x − 0.0293 | 0.995 | 75 | 0.0069x − 0.0102 | 0.997 |
33 | 0.0159x − 0.0321 | 0.995 | 78 | 0.0141x − 0.0277 | 0.995 |
34 | 0.0166x − 0.0343 | 0.994 | 79 | 0.0157x − 0.0328 | 0.994 |
35 | 0.0157x − 0.0328 | 0.994 | 80 | 0.0161x − 0.0333 | 0.994 |
36 | 0.0136x − 0.0267 | 0.995 | 81 | 0.0148x − 0.0293 | 0.995 |
41 | 0.0014x − 0.001 | 0.999 | 82 | 0.009x − 0.0139 | 0.997 |
42 | 0.0143x − 0.0289 | 0.995 | 85 | 0.0141x − 0.0277 | 0.995 |
43 | 0.0159x − 0.0321 | 0.995 | 86 | 0.012x − 0.0253 | 0.994 |
44 | 0.016x − 0.0323 | 0.995 | 87 | 0.0143x − 0.0289 | 0.995 |
45 | 0.0166x − 0.0343 | 0.994 | 88 | 0.009x − 0.0139 | 0.997 |
46 | 0.012x − 0.0253 | 0.994 | 91 | 0.0136x − 0.0267 | 0.995 |
47 | 0.0124x − 0.0243 | 0.995 | 92 | 0.0052x − 0.0109 | 0.994 |
51 | 0.0097x − 0.0183 | 0.995 | 93 | 0.0069x − 0.0102 | 0.997 |
52 | 0.0052x − 0.0109 | 0.994 | 97 | 0.0124x − 0.0243 | 0.995 |
53 | 0.0161x − 0.0333 | 0.994 | 98 | 0.0014x − 0.001 | 0.999 |
54 | 0.016x − 0.0323 | 0.995 | 100 | 0.0097x − 0.0183 | 0.995 |
x denotes the damage severity |