**6. Conclusion**

The first part of this work presents the results of a combined experimental/numerical effort that aims to assess the performance of different plastic stress flow in predicting the burst of welded steel tubes loaded under internal pressure. The prediction of the stress-strain characteristic with the anisotropic effect of tubular material has been proposed. Once the expanding diameter, the internal pressure and the wall thickness were obtained from the results of the bulge tests, the effective stress and effective strain could be calculated. The bulge tests carried out until bursting showed that all the fissures are initiated in the central area of the expanding zone not far from the weld zone.

Using the Nelder-Mead (NM) simplex search method, a flow stress curve (Swift's model) that best fits the stress-strain of the used anisotropic material could be determined with consideration global response (force/displacement). The local behaviour (stress/strain) of the welded joints and the HAZ is identified numerically using ABAQUS solver from global results (force/depth) of nanoindentation tests. The identified hardening coefficients are introduced by Swift model. From the simulations carried out, it is clear the influence of the plastic flow behaviour of the WT in the final results (thickness distribution, stress instability, tube circularity and critical thinning and rupture).

It is also clear that to predict with more accuracy the results, the model used for simulation has to be as realistic as possible. Therefore, future work in this area will include the experimental identification approach of the hardening model coupled with damage. Indeed, we think that measurements of displacements and strains without contact can improve results quality. The suggested model coupled with ductile damage can contribute to the deduction of forming limit diagrams.

The plastic deformation of a circular sheet hydraulically expanded into a complex female die was explored using experimental procedure and numerical method using ABAQUS/EXPLICIT code©. As future work, one can study others optimization techniques without using derivatives to make a numerical comparison between these different techniques and integration of adaptive remeshing procedure of sheet forming processes.

In the second part of this work, an efficient method was proposed to optimize the THP with taking into account the uncertainties that can affect the process. The optimization process consists to minimize an objective function based on the wrinkling tendency of the tube under probabilistic constraints that ensure to decrease the risk of potential failure as necking and severe thinning. This method can ensure a stable process by determining a load path that can be insensitive to the variations that can affect input parameters. Construction of the objective function and reliability analysis was done based on the response surface method (RSM). The study shows that the RSM is an effective way to reduce the number of simulations and keep a good accuracy for the optimization.

Probabilistic approach revealed several advantages and promoter way than conventional deterministic methodologies, however, probabilistic approach need precise information on the probability distributions of the uncertainty and is sometimes scarce or even absent. Moreover, some uncertainties are not random in nature and cannot be defined in a probabilistic framework.
