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Normal stress has some role in the deformation analysis of hydroforming processes. In this study, analytical modeling is pursued to evaluate the effect of normal stress on the hydro-mechanical deep drawing (HDD) process. Analyses are carried out for axisymmetric elements of the formed cup-shaped part for increments of the punch travel. The formulations are obtained using mechanical and geometrical relations and the finite difference method, thereby being solved by proper numerical algorithms. Furthermore, in the present work, part thickness is variable, the loading and straining are non-proportional, and bending/unbending effects over the part curvature are considered. The results show that...