A simple indicator and branch switching technique for hidden unstable equilibrium paths

The unfolding, via imperfections, on an asymmetric bifurcation may result in both stable and unstable equilibrium paths. In practive it it quite possible that an analysis will be performed on the stable path only without the analyst knowing of the adjacent unstable path. This process could lead to a disastrous design. In this paper we propose a simple indicator based on the local minimum of the smallest pivot of the tangent stuffness matrix, which can be used to warn the analyst of the danger. In addition if he wishes, he can employ a simple eigenmode injection technique to analyse the unstable path. Numerical examples are presented which prove that both the indicator and branching technique are efficient and reliable.     

An energy conserving co-rotational procedure for non-linear dynamics with finite elements

A new procedure is proposed for implicit dynamic analysis using the finite element method. The main aim is to give stable solutions with large time-steps in the presence of significant rigid body motions, in particular rotations. In contrast to most conventional approaches, the time integration strategy is closely linked to the element technology with the latter involving a form of co-rotational procedure. For the undamped situation, the solution procedure leads to an algorithm that exactly conserves energy when constant external forces are applied (i.e. with gravity loading).     

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