Keywords: finite elements, curved, elastic, shells, positions, large deformation.

The good representation of shells exhibiting large displacement, rotation and strain is a continuous activity of a large number of research works. Some recent works may be cited as for example: Petchsasithon & Gosling [1] Pimenta et al. [2], Campello et al. [3], Sze et al. [4], Hong et al. [5], Sansour & Kollmann [6] and Bischof & Ramm [7] to confirm the current importance of the subject.

The work of Bischoff & Ramm [7] about the physical meaning of higher order kinematics and static variables for three-dimensional shell formulation clarifies all aspects of shell modeling and provide more physical insight to the shell behaviour. These aspects are crucial for proposing coherent models for finite element shell analysis.

This work presents an alternative shell formulation based on a non-conventional nodal parameters. This formulation has its origin in the so-called positional formulation proposed in the works of Coda & Greco [8] and Greco & Coda [9] for two dimensional frame analysis. The novelty of the proposed technique is the parameters considered, i.e., nodal positions (not displacements) and generalized vector components that comprises both direction cossines and shell thickness at the same time.

The objective of this work is not to achieve a better model than the ones proposed by Bischoff & Ramm [7], Pimenta et al. [2] or Sansour & Kollman [6] for example, but to offer an alternative way to achieve the numerical procedure for geometrically non-linear analysis of shell using curved triangular (or any geometry) elements without the use of 'cumbersome notations' [7]. To facilitate even more the understanding of the positional process, non-linear engineering strain is adopted, taking advantage of simple linear engineering stress-strain relations.

The resulting formulation presents six degrees of freedom by node and considers linear thickness variation during the deformation process. Curved triangular elements with cubic approximation for both positions and generalized vector components are adopted. The modified positional mapping, as presented, fulfils a three-dimensional representation and requires a three-dimensional relaxed constitutive relation to avoid Poisson thickness locking. Several numerical simulations are presented in order to illustrate and confirm the accuracy and applicability of the formulation.

1

Petchsasithon, A. and Gosling, P.D. "A locking-free hexahedral element for the geometrically non-linear analysis of arbitrary shells", Comput Mech (35) 94-114, 2005. doi:10.1007/s00466-004-0604-y

2

Pimenta, P.M., Campello, E.M.B, Wrigers, P., "A fully nonlinear multi-parameter shell model with thickness variation and a triangular shell finite element", Computational Mechanics (34), 181-193, 2004. doi:10.1007/s00466-004-0564-2

3

Campelo, E.M.B, Pimeta, P.M., Wrigers, P., "A triangular finite shell element based on a fully nonlinear shell formulation", Computational Mechanics (31), 505-517, 2003. doi:10.1007/s00466-003-0458-8

4

Sze K.Y., Chan W.K., Pian T.H.H. "An 8-node hybrid-stress solid-shell element for geometric nonlinear analysis of elastic shells", Int. J. Numer. Meth. Engrg. 55: 853-878, (2002) doi:10.1002/nme.535

5

Hong W.I., Kim J.H., Kim Y.H., Lee, S.W., "An assumed strain triangular curved solid shell element formulation for analysis of plates and shells undergoing finite rotations", Int. J. Numerical Methods Engng 52, 747-761, 2001. doi:10.1002/nme.234

6

Sansour C., Kollmann F.G., "Families of 4-node and 9-node .nite elements for a .nite deformation shell theory. Anassessment of hybrid stress, hybrid strain and enhancedstrain elements", Comput. Mech. 24: 435-447, 2000. doi:10.1007/s004660050003

7

Bischoff M., Ramm E. "On the physical significance of higher order kinematic and static variables in a three-dimensional shell formulation", International Journal of Solids and Structures (46-47): 6933-6960 Nov, 2000. doi:10.1016/S0020-7683(99)00321-2

8

Coda H.B., Greco M. "A simple FEM formulation for large deflection 2D frame analysis based on position description", Comput. Meth. Appl. Mech. Eng. 193 (33-35): 3541-3557, 2004. doi:10.1016/j.cma.2004.01.005

9

Greco M., Coda H.B. "Positional FEM formulation for flexible multi-body dynamic analysis Journal of Sound and Vibration" 290 (3-5): 1141-1174, 2006. doi:10.1016/j.jsv.2005.05.018

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