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. This contribution deals with shell element formulations involving three rotational degrees of freedom. Its purpose is twofold. Firstly, a review of relevant non-linear shell theories as well as finite element implementations is given and corresponding defects are addressed. Secondly, a geometrically linear shell element formulation is presented which is free of any of these defects. Conventional shell theories involve two rotational degrees of freedom, the drilling rotation being excluded. Three standard methods of incorporating the drilling degree of freedom are considered: (a) The application of a proper rotation constraint condition, (b) shell theories intrinsically involving three rotational degrees of freedom (here called ‘micropolar theories’) and (c) introducing the drilling degree of freedom at the level of the finite element discretization (Allman-type shape functions). Here, reference implementations relying on approaches (a) to (c) as well as their reasonable combinations are considered. It is demonstrated that each of these implementations reveals at least one questionable feature: Formulations based on a micropolar approach involve ad hoc assumptions related to the constitutive law. Formulations involving a rotation constraint imply the application of a problem-dependent penalty or regularization parameter. Finally, Allman-type shell elements suffer from a lower convergence rate for bending dominated problems compared to isoparametric ones. The proposed shell element relies on a conventional (nonpolar) shell theory and applies a drill rotation constraint via a penalty formulation. A crucial point is the application of properly designed enhanced strain fields to avoid in-plane locking phenomena. It is demonstrated numerically that the resulting implementation overperforms existing shell elements. Most importantly, the way of incorporating the rotation constraint proves to constitute a proper penalty formulation in the sense that for sufficiently large values of the penalty parameter the results are independent of the latter. Moreover, this threshold is independent of the problem. The relation to existing implementations of the drill rotation constraint, all of them requiring the application of problem dependent penalty parameters, is discussed. Concluding, it is explicated how the present formulation derives from a micropolar approach.

A NEW SHELL FINITE ELEMENT WITH DRILLING DEGREES OF FREEDOM AND ITS RELATION TO EXISTING FORMULATIONS