48 H. M. Imounga et al. Fig. 6.3 Flowchart of the proposed methodology Table 6.3 Deflection and tensile stress means Parameters No evidence With evidence Experimental data w(mm) 0.53 0.82 0.6 σVM (MPa) 51.12 61.62 – The methodology used to update the deflection and the Von Mises stress requires an iteration of mechanical (finite element code) and probabilistic analyses (Monte Carlo simulation, Bayesian network), see Fig. 6.3. Monte Carlo simulations of the input random variables of the mechanical model (E and ft) and of the load (F), carried out in the MATLAB software, are introduced into the Castem code to calculate 1000 values of the output parameters of the model (w, σVM). These values allow building the conditional probability tables of the nodes of the Bayesian network. This step allows propagating the uncertainties of the input parameters to the outputs. The data from the experimental tests observed on the mid-span deflection are then introduced into the Bayesian network as evidence in order to update the probabilities of all the nodes. The deflection measurements can also be used to indirectly identify the mean values, standard deviation, and type of distribution of other mechanical parameters. The Bayesian network outputs are the a posteriori probabilities of each of the nodes, from which the laws of the nodes and their parameters (mean, standard deviation) could be determined after updating. 6.4 Results and Discussions The values of the maximum mid-span deflection before and after the update with the Bayesian network are presented in Table 6.3. The mean deflection value for 10,000 cycles to the unload is also presented in Table 6.3 [5]. We can note that the mean values of w obtained from the Bayesian network are higher than the experimental value; and that obtained without the addition of evidence is closer to experimental value. In addition, the value of the maximum tension stress increases with the introduction of the evidence of the deflection. This difference shows on the one hand that the uncertainties linked to the parameters of the concrete propagate in the mechanical model before the update and on the other hand that they are linked to the experimental evidence propagated through the Bayesian network. Uncertainties on experimental measurements are more important than the uncertainties linked to the parameters. These preliminary results can be improved by adjusting the limit values, the number of states, or even the number of Monte Carlo simulations, which was fixed at 1000 simulations because of the computation time. Figure 6.4 shows the deflection histograms with and without evidence. We can see that the a posteriori histogram of w adjusts with the introduction of evidence from the trials. This adjustment is due to the fact to had integrated additional information on the deflection. The proposed methodology therefore makes it possible to update the output parameters of the mechanical model by integrating experimental measurements. 6.5 Conclusion This chapter proposes a methodology which combines a mechanical model and probabilistic approaches. The methodology makes it possible to account for the uncertainties linked to the material, loading, and experimental measurements. It is also useful to integrate the observations of the tests and to update the other parent nodes by using Bayesian networks. The results show that the uncertainties related to the experimental measurements are greater than those linked to the characteristics of the material in view of the difference between the numerical values of the deflection before and after the introduction of the evidence. Although the number of simulations is small, the numerical values of the deflection are not very far from those of the experimental test. Future work will focus on the analysis of cracking under cyclic load, taking into account the phenomena of opening and closing of the crack. Another aspect of the improvement of the methodology will be the
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