PSI - Issue 68

Morteza Khomami Abadi et al. / Procedia Structural Integrity 68 (2025) 1312–1318 Author name / Structural Integrity Procedia 00 (2025) 000–000

1316

5

three levels is used. The five variables and 72 tests are set to evaluate the stress intensity factors in the GMDH algorithm. Five input variables are ! ( = 1 . . . 5) , are defined as the height (ℎ) in beam or width ( ) in plate, the ratio of the length to the height ( /ℎ) in beam or ( / ) in plate, the ratio of the crack location to the length ( " / ) , and the ratio of the crack depth to the height of the beam ( /ℎ) or ( / ) in plate, and the last input variable ( # ) is identified as the loading conditions, and ( $ ) is specified as the output of the problem (SIFs). Different data sets between the training and the testing of the GMDH neural network are presented in [13, 14]. In this research, a trial-and-error procedure is applied to select a reliable data set in the determination of desired results. This technique resulted in the selection of 65% of the data for training. The numerical ranges of the variables used in the GMDH algorithm are defined in Table 1. Table 1. Range of the input variables. Input variables Minimum and Maximum limits Normalization ! =ℎ , ( ) 0, 0.5 -1, 1 " = /ℎ , / 0, 10 -1, 1 # = $ / 0, 0.75 -1, 1 % = /ℎ , / 0-0.75 -1, 1 & = ( . ) 0, 500000 1.5, 15 ' = ( / " ) 0, 1000 5, 12 Table 2 is designed to represent the equations of the stress intensity factors based on the Eq. 8. The GMDH results for arbitrary beam dimensions, different boundary conditions, and other arbitrary parameters (considered in the identified range of variables) can be determined. Table 2. Constants of the stress intensity factor equation.

Input of Neuron Output of Neuron ( ! " # % & ! , % ! 0.0442 0.1860 0.2572 0.0442 −0.0496 −0.0415 # , & " 0.1198 0.0579 0.8060 −0.1903 −0.0604 −0.0208 " , & # 0.0222 0.3364 1.0336 0.0222 −0.1722 −0.0581 ! , " % −0.0406 0.6125 0.9436 −0.3972 0.0932 0.1342 # , ! & 0.0033 0.9942 0.1531 −0.0072 0.0033 −0.0052 % , & ' −0.0056 0.1511 0.8875 −0.2214 −0.1522 0.4335

Similarly, the above algorithm was derived for cracked plates. The coefficients of the Table 2 were determined and placed in Eq. 8, and finally the stress intensity factors were extracted for single edge and center cracked plates. 4. Research results and discussion In this section, the accuracy of GMDH neural network for determining stress intensity factors is validated with references. Then, the results of stress intensity factors are presented in some new cases that did not have a solution until now. 4.1. Validation of the stress intensity factor The comparison of GMDH neural network results with Kumar results is carried out for the validation of the extracted stress intensity factor. The validity of the results is shown in Fig. 3. The results are evaluated by a cracked F-F beam under bending according to the range in Table 1. The beam is shown in Fig. 1. The results demonstrates that the GMDH algorithm can determine the stress intensity factor with an error less than 7.04%. The sources of errors of the GMDH method are the number of initial test based on the DOE, The wide range of variables, the number of hidden layer and the level of the factorial design.