SUMMARYThis paper should not be only regarded as a presentation of new shell elements but rather as a methodology which can be applied to most classical shell elements and has two aims: Achieving the same results in bending cases while breaking from plane stress state hypothesis and adding a normal stress component for process simulations such as hydro-forming, hemming, sheet metal forming with bottoming, flanging, incremental forming and so on. Owing to the non-linear applications quoted before, only shell elements with one integration point on the mid-plane are selected: Triangles that are naturally constant strain elements and reduced integration quadrilaterals. The method mainly consists of adding a central node at the center (of gravity for a triangle) with two degrees of freedom: Two translations normal to the mid-surface for which one corresponds to the bottom surface ('lower skin' of the shell) and the other to the top surface ('upper skin' of the shell). Then a full 3D constitutive strain-stress behavior can be used. For triangles in bending state-either based on Kirchhoff's or on Mindlin's assumptions-, it is shown that the results are exactly the same as those given by the initial formulation of these elements using a plane stress hypothesis. For quadrilaterals, the results are slightly different but many numerical examples-including non-linear computations-prove that those differences are not significant.

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