Numerical Analysis of Thin Shell Problems With Hierarchic High Order Finite Elements
by
Terenzio Scapolla
Dip. Matematica, Università di Pavia, Italia
Coauthors: Lucia Della Croce (università di Pavia)

As in the case of the plate bending model, even a shell model can be derived according to the different physical assumptions. When the fibers are supposed to keep normal to the surface after deformation, i.e., the Kirchhoff hypothesis are assumed, we arrive at the Koiter's model. When the normals to the undeformed middle surface remain straight but not necessarily normal to the deformed middle surface, i.e., the Reissner Mindlin hypothesis are assumed, we arrive at the Naghdi's model.

It is known that despite its simple approach the discretization of the Reissner-Mindlin model is not straightforward both in plates and shells frames. The inclusion of transverse shear strain effect in the finite element models introduce an undesirable numerical effect, the so called shear locking phenomenon. Consequently, as the thickness of the plate and shell becomes extremely thin, the shear strain energy predicted by the finite element analysis can be magnified unreasonably, even though the average value of the shear strain over the area tends to zero.

Finite element schemes for shells problem also suffer of the so-called membrane locking, i.e., the finite element approximation of the membrane component of the energy is unstable with respect to the thickness of the shell. Several solutions to avoid the numerical locking have been proposed. Mixed formulations, reduced integration, and its offspring, selective reduced integration, have been often used to mitigate the effects of shear and membrane locking.

In this paper we consider the shell model arising from the Naghdi formulation. A displacement finite element scheme has been developed using finite element of hierarchic type of degree ranging from one to four. In order to analyze the behavior of our finite elements respect to the membrane and shear locking, we have dealt with two test problems often used to assess the performance of numerical formulations based on the degenerated solid approach. The two tests are representative of extremely discriminating situations.

Date received: June 15, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadk-07.