Drilling rotation constraint for shell elements in implicit and explicit analyses

A subordinate but interesting detail in theory and application of shell elements is investigated in this study, namely the drilling rotation constraint approach. Standard shell elements exhibit 3 translational and 3 rotational degrees-of-freedom at each node. While two nodal rotations are directly associated with bending and twisting modes, the third rotation about the shell normal (also known as the drilling rotation) does not provide any resisting force or stiffness by itself. This fact leads to zero valued components in the stiffness matrix for implicit analyses, which in turn results in a system of equations that cannot be solved. Therefore a small amount of stiffness in form of a torsional spring is artificially added just to remedy the singularity but not to affect the solution too much. This is absolutely necessary to deal with implicit analysis, otherwise no results could be obtained. On the other hand, it might be helpful to have this option also available in explicit analyses to improve results in special situations. It is the intention of this paper to present the theoretical background of this phenomenon and to illustrate the influence of the constraint method in several numerical examples.