Insight into finite element shell discretizations by use of the “basic shell mathematical model”

The objective of this paper is to gain insight into finite element discretizations of shells using the basic shell mathematical model and, in particular, regarding the sources of “locking”. We briefly review the “basic shell mathematical model” and present a formulation of shell finite elements based on this model. These shell finite elements are equivalent to the widely-used continuum mechanics based shell finite elements. We consider a free hyperboloid shell problem, which is known to be difficult to solve accurately. Using a fine mesh of MITC9 elements based on the basic shell mathematical model, a detailed analysis is performed giving the distributions of all strain terms. A similar analysis using the MITC6 shell element shows why this element locks when the shell thickness is very small.