Generally, the discrete model of arbitrary shells was treated by formulations possessing 6 degrees of freedom at each node (6 DOF/node) in finite element analysis. If any region of the shell structure is flat or very shallow, singularity of system equations will appear. Many methods were suggested to overcome this difficulty. The most important work was the efficient formulation proposed by Simo and his co-workers, which offers an optimal parameterization of the rotation field with 5 or 6 DOF/node. It is suitable for a smooth shell and also for intersections of shell surfaces, especially in non-linear applications. Following this formulation, a four-node shell element is developed. A singularity-free description of the rotation tensor in original state is suggested in this paper. The global updating formulae for finite rotation are described. Both 5 and 6 DOF formulations are numerically investigated for linear and non-linear calculations. The behaviour of numerical directors is discussed. The results demonstrate that an 'average director' is not suitable for the shell intersection with a big angle in non-linear applications.

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