Keywords: post-collapse behaviour, sequential limit analysis, shells.

The prediction of the post-collapse behaviour of structures, in particular shells, has been for a long time a challenging problem, solved by modern solution methods at the price of significant computational cost. In the last decades the ever increasing requirements in terms of safety stimulated the research in the field. In several situations, the designer has to deal with the definition of proper zones, or whole structures, capable of absorbing possible impacts in order to reduce damage to devices and/or people. These shock absorbers dissipate the impact energy by developing plastic strains and large deformations, so as to make acceptable the consequent acceleration. As an example, this approach has been applied to the design of rail vehicles and automotives.

Different methods can be employed to predict the post-collapse response. The most complete, although computationally demanding, simulations are provided by large displacement, incremental elastic-plastic analyses. Simplified methods, such as mechanism analysis are no doubt faster, but their applicability is limited to specific situations. A good balance between computational efficiency and accuracy of results is provided by sequential limit analysis. As its name suggests, the method is based on a sequence of limit analyses performed on meshes updated on the basis of the collapse mechanism provided by the previous step. Since elastic strains are neglected, only the rigid-plastic response is tracked, which however provides a meaningful piece of information on the energy that the structure can dissipate and on its deformation capabilities.

In this paper, the limit analysis problem is solved by means of the procedure suggested in [1] and applied sussessfully in different cases. In contrast to alternative approaches, mostly based on regularization techniques, this procedure deals with the non-smooth nature of the function to be minimized by detecting and eliminating from the problem the finite elements that do not undergo plastic flow in the collapse mechanism, which are considered as rigid. To enforce this condition, it is essential that the finite element be formulated on the basis of the natural approach introduced by Argyris [2], permitting the separation of deformation modes from rigid body motions: in fact, in a rigid element the first vanish, but the latter must survive.

Recently, a simple triangular shell element, named TRIC, has been developed by Argyris and co-workers [3], and its formulation rests on the natural approach, making it a spontaneous candidate for the procedure. The element is simple but sophisticated (using authors' terminology), non-conforming but converging, accounts for transverse shears without locking at the thin shell limit and several elastic computations assess its robustness and accuracy also in situations involving large displacements and rotations.

In this study, a rigid-plastic version of the TRIC element is proposed, which appears suited for sequential limit analysis. The different context demands some modifications, which were reduced as much as possible, so as to maintain the quality of the element performances. The computational experience gained so far (some examples are presented) assess that the procedure is computationally effective, robust and numerically stable both for increasing (stable) and decreasing (unstable) post-collapse responses and seems sufficient to establish that the marriage between TRIC and sequential limit analysis is a successful one.

1

Capsoni, L. Corradi, "A finite element formulation for the rigid-plastic limit analysis problem", Int. J. Num. Meth. Eng., 40, 2063-2086, 1997. doi:10.1002/(SICI)1097-0207(19970615)40:11<2063::AID-NME159>3.0.CO;2-#

2

J. H. Argyris, H. Balmer, J. St. Doltsinis, P. C. Dunne, M. Haase, M. Muller, D. W. Scharpf, "Finite element method: The natural approach", Comp. Meth. Appl. Mech. and Eng., 17/18, 1-106, 1979. doi:10.1016/0045-7825(79)90083-5

3

J. H. Argyris, L. Tenek, L. Olofsson, "TRIC: a simple but sophisticated 3-node triangular element based on six rigid-body and 12 straining modes for fast computational simulations of arbitrary isotropic and laminated composite shells", Comp. Meth. Appl. Mech. and Eng., 145, 11-85, 1997. doi:10.1016/S0045-7825(96)01233-9

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