Neural component N 1

To create this component the parametric analysis has to be performed. The set of discrete results should be stored in a knowledge-base. In this case the analysis has been performed in accordance to the Polish Code for the bridge loads (PN-85/S-10030) for the single load class (local lines and provisional bridges, where load class coefficient α = 1.0). In this analysis span length Lt and cross section location ξ are the parameters taken into the consideration and the maximal bending moment M’p,max caused by the mentioned load is the single result. The results of these simulations have been presented in Fig. 8-5. Theoretical span length Lt and location of cross section ξ are its inputs. The maximal bending moment M’p,max is the output of this component.

Fig. 8-5 Results of the static analysis: envelopes of bending moment caused by moving load (for local lines and provisional bridges, α = 1.00) for various length of spans

The following neural network architecture has been proposed:

• input layer related to the parameters of analysis with the following input parameters:

theoretical span length Lt and cross section location ξ,

• hidden layer(s),

• output layer with one output signal of the network M’p,max.

Before training and testing processes each envelope of bending moments (Fig. 8-6, line 2) needs to be transformed because of the problems with approximation of boundary conditions.

The symmetrical form of the envelopes allows for analysing of their halves only (Fig. 8-6, line 1). The mentioned graphs have been extended by their mirror reflection on the left (Fig.

8-6, line 3) and by horizontal line on the right (Fig. 8-6, line 4). The mentioned problems moved to the new boundaries but fortunately outside of the analysed domain.

ξ (x/L) M (MNm)

L (m) 10.0

9.0 8.0 7.0 6.0 5.0

4.0 3.0

2.0 1.0 0.0

0.0 0.5

1.0

19.0 15.0 12.0 10.0 7.0

5.0 3.0

Fig. 8-6 Example of bending moment envelope and its transformation for application in NNT

Three neural networks of various architectures (Fig. 8-7, Fig. 8-8 and Fig. 8-9) have been analysed and compared during creation of N1 component.

Fig. 8-7 Component N1: network architecture “2-10-10-1”

The first network (Fig. 8-7) consists of four layers (input, two hidden and output layer). The input layer corresponds to the parameters of the analysis and consists of two neurons (Lt and ξ). Each of the hidden layers consists of the ten neurons. The output layer is composed of the single neuron (M’p,max). Architecture of the network determines its name: “2-10-10-1”.

The next neural network example (see Fig. 8-8) consists of three layers (input, one hidden and output layer). The input layer consists of two neurons (Lt and ξ). The hidden layer is

Theoretical span length Lt

(2) (3)

(4) (1)

Bending momentM

2-10-10-1

M’p,max

Lt

ξ

x0=1

composed of 20 neurons. The output layer is composed of the single neuron (M’p,max).

Architecture of the network determines the network name: “2-20-1”.

Fig. 8-8 Component N1: network architecture “2-20-1”

The third neural network example has been presented in Fig. 8-9. The network is composed of four layers (input, two hidden and output layer). The input layer consists of two neurons (Lt

and ξ). Each of the hidden layers consists of ten neurons.

Fig. 8-9 Component N1: network architecture “2-5-5-1“

The output layer is composed of the single neuron (M’p,max). Architecture of the network determines the network name: “2-5-5-1”.

To analyse efficiency of created neural networks a comparison of accuracy achieved by three various architectures of designed neural networks have been presented in Table 8-1.

2-5-5-1

Lt

ξ x0 = 1

M’p,max 2-20-1

Lt

ξ x0 = 1

M’p,max

Table 8-1 Component N1: training and testing errors of various network architectures Training error Testing error

Network architecture

Avg RMS Avg RMS 2-5-5-1 0.0075 0.0091 0.0104 0.0117 2-10-10-1 0.0061 0.0057 0.0095 0.0092

2-20-1 0.0057 0.0054 0.0067 0.0064

This table presents two error groups (training and testing). The efficiency of the trained neural networks is presented also graphically, i.e. response of these networks against expected and analytically obtained values of envelopes of bending moments for various span lengths (Fig.

8-10 to Fig. 8-15). In this way the response of the neural networks can be easily compared and discussed in order to select the best architecture for further application in the expert tool.

Fig. 8-10 Component N1: the network answers and expected values for Lt = 3.0 m

The presented results cover span lengths between 3.0 and 19.0 m and the worst behaviour of all networks can be observed for the shortest span length, Lt = 3.0 m (Fig. 8-10).

During the creation and training of neural networks some problems with the approximation of boundary conditions occurred. To solve this problem the analysed set of data has been extended by points related to the shorter spans (Lt = 3.0 m) and though the range of application of this methodology starts from 6.0 m, for the training purpose these spans have been considered. For the spans of length Lt = 3.7 m (Fig. 8-11) the accuracy of this analysis leaves a lot to be desired, but for longer spans the results are much better (Fig. 8-12).

pattern 2-5-5-1 2-10-10-1 2-20-1 Legend

0.0 0.2 0.4 0.6 M [MNm]

0.0 0.1 0.2 0.3 0.4 0.5

Fig. 8-11 Component N1: network answers and expected values for Lt = 3.7 m

Fig. 8-12 Component N1: network answers and expected values for Lt = 5.5 m

In the presented figures various behaviours of the created networks can be observed. The network of the architecture “2-5-5-1” represents the best performance.

0.0 0.1 0.2 0.3 0.4 0.5

ξ [-]

M [MNm]

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

0.0

pattern 2-5-5-1 2-10-10-1 2-20-1 Legend 0.0

0.2 0.4 0.6 0.8 1.0

M [MNm]

ξ [-]

pattern 2-5-5-1 2-10-10-1 2-20-1 Legend

0.0 0.1 0.2 0.3 0.4 0.5

Fig. 8-13 Component N1: network answers and expected values for Lt = 10.0 m

Fig. 8-14 Component N1: network answers and expected values for Lt = 15.0 m

The simulations for 10.0, 15.0 and 19.0 meters (Fig. 8-13 to Fig. 8-15) represent the highest accuracy of obtained results. The best accuracy has been achieved for the architecture called

“2-5-5-1” and this composition has been typed for component N1 of the expert tool.

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 M

0.0 0.1 0.2 0.3 0.4 0.5

ξ [-]

pattern 2-5-5-1 2-10-10-1 2-20-1 Legend 0.0

1.0 2.0 3.0 4.0

M [MNm]

0.0 0.1 0.2 0.3 0.5

ξ [-]

pattern 2-5-5-1 2-10-10-1 2-20-1 Legend

Fig. 8-15 Component N1: network answers and expected values for Lt = 19.0 m