246 A. Bagheri et al. Fig. 30.3 The value of as a function of the bridge’s span a and width b for the plate’s thickness of 0.6 m and: (a) the skew angle of 0ı and (b) the skew angle of 45ı structure which causes to increase the bridge’s modal frequency, and this effect increases the value of which can be seen by comparing Fig. 30.2a, b. In order to see the effect of the plate’s thickness on , Fig. 30.3 shows the results of for the plate’s thickness of 0.6 m. By comparing Figs. 30.2 and 30.3, it can be found that the value of decreases by increasing the plate’s thickness at short spans and widths. The changing plate’s thickness affects both the mass and stiffness of bridge, and also changes the contribution of the parapet’s stiffness on the global stiffness of the structure. Therefore, it can be concluded that there is a complex relation between and the plate’s thickness. 30.2.3 Mapping Process An artificial neural network (ANN) was employed to predict the relationship between the coefficient of and the bridge’s parameters, namely the bridge’s spana,widthb, skew angle , and thickness t. Neural networks are one of the most effective soft computing algorithms for data fitting and classification. They mimic a human brain by implementing the interconnection of artificial neurons. In this study, a feed-forward back-propagation ANN with three layers was used. The input layer received the data vector containing the four parameters of bridge. The hidden layer processed the data by multiplying the input vectors by weights and adding biases. The results constituted the argument of a transfer function that squashed the output values into a certain range. For the hidden layer, ten neurons were used by means of a trial and error method, and the hyperbolic tangent sigmoid transfer function was employed. For the input and output layers, a liner transfer function was used. The output layer had one node that provided the value of . To train the network, the Levenberg-Marquardt algorithm was used because of its high performance and speed [8]. To study the ability of the network to estimate the value of , we used all 14,256 data for the training. Figure 30.4 shows the ANN estimated versus the actual , and it can be seen that there is no difference between them, because there is a linear function between the estimated and target values. This trained ANN will be used to estimate in identifying process of the structural stiffness of a tested slab bridge.
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