EC
15,5
552
An energy-conserving
co-rotational procedure
for the dynamics of shell
structures
H.G. Zhong and M.A. Crisfield
Department of Aeronautics, Imperial College,
London, UK
1. Introduction
The non-linear analysis of shell structures is still a challenging problem.
Considerable progress can be made by adopting the simplest possible shell
formulation which involves the combination of Morley’s triangular bending
element and the constant strain triangle. For non-linear static analysis, this
element was embedded within a co-rotational framework by Peng and
Crisfield[1] (see also [2-4]). In the present paper, we will extend the concepts to
dynamics.
However, before moving to dynamics, we re-visit the static formulation with
a view to the derivation of a more straightforward formulation. This is achieved
by borrowing ideas from Nour-Omid and Rankin[5] and Rankin and Brogan[6]
who introduced a matrix connecting the spin of the element frame to the change
in nodal variables. These concepts were applied to continua by Moita and
Crisfield[7] (see also [4]).
Having presented a neat static formulation for the shell, the paper moves on
to consider non-linear dynamics. The initial formulation uses the Newmark
trapezoidal rule[8] and modifications involving the HHT-
α
scheme[9] and the
generalized CH-
α
scheme[10]. It has been shown by Simo and co-workers[11-13]
and by the second author and co-workers[14-17] that the standard Newmark
trapezoidal rule can introduce severe numerical instabilities in the presence of
large deformations (in particular, rotations). The numerical damping,
introduced by the
α
-schemes, is intended to overcome these difficulties.
However, the methods have only been shown to lead to dissipation for linear
systems. Preliminary results by the authors and co-workers[16,17] have
indicated that this energy dissipation does not always extend to the non-linear
regime. These issues will be explored further in the current paper.
An alternative remedy is to develop various forms of energy-conserving
algorithms. Such concepts were originally introduced by Haug et al.[18] and by
Hughes et al.[19] and involved enforcing the energy-conservation as a
constraint with the aid of a Lagrangian multiplier. This approach has recently
Engineering Computations,
Vol. 15 No. 5, 1998, pp. 552-576,
© MCB University Press, 0264-4401
Received January 1997
Accepted November 1997