Keywords: elastic stability, buckling, post-buckling, complementary energy principles, dual stability criterion, geometrically exact beams, three-dimensional frames, finite element analysis.

Stability analyses of framed structures under the action of conservative loads have been carried out by many researchers. From a variational point of view, these studies have almost invariably been based on the principle of minimum total potential energy. Only a few works using complementary-energy based principles to address the stability analysis of structural systems have been proposed in the literature. Almost all of these studies are, however, limited to the cases of bifurcation instability.

The only work found in the literature focusing on a general stability criterion, relying on complementary-energy based principles and applicable to both buckling and post-buckling analyses of elastic structural systems, is due to Valid [1]. Based on this approach, a complementary-energy based stability criterion (dual criterion) will be presented for the geometrically exact quasi-static analysis of elastic three-dimensional framed structures. The framed systems are modelled using a Reissner-Simo beam theory, which is valid for arbitrarily large displacements, rotations and strains, and which takes into account the first-order shear deformations of the cross-sections of the beams [2,3,4].

The criterion is derived in two fundamental steps: the first consists in introducing the compatibility equations, by means of the so-called Lagrangian multiplier method, into the variational framework of the principle of stationary total potential energy; the second consists in using the well known (primal) stability criterion given by the condition of minimum total potential energy. This criterion naturally incorporates shear deformation effects, which is a feature neglected in most of the available buckling (and also post-buckling) analyses.

Both buckling and post-buckling numerical analyses of several benchmark problems will be presented. The analyses will be carried out using the equilibrium-based finite element model introduced in [5], which relies on a modified principle of complementary energy, in conjunction with the present dual stability criterion. In addition, the analyses will be compared to those rendered by the standard two-node displacement/rotation-based finite element model and its associated (primal) stability criterion.

1

R. Valid, "The Structural Stability Criterion for Mixed Principles", in "Hybrid and mixed finite element methods", 289-323, Wiley, New York, 1983.

2

E. Reissner, "On One-Dimensional Large-Displacement Finite-Strain Beam Theory", Studies in Applied Mathematics, 11, 87-95, 1973.

3

J. Simo, "A Finite Strain Beam Formulation. The Three-Dimensional Dynamic Problem. Part I", Computer Methods in Applied Mechanics and Engineering, 49, 55-70, 1985. doi:10.1016/0045-7825(85)90050-7

4

J. Simo, L. Vu-Quoc, "A Three-Dimensional Finite-Strain Rod Model. Part II: Computational Aspects", Computer Methods in Applied Mechanics and Engineering, 58, 79-116, 1986. doi:10.1016/0045-7825(86)90079-4

5

H.A.F.A. Santos, P.M. Pimenta, J.P. Moitinho de Almeida, "An equilibrium-based finite element formulation for the geometrically exact analysis of three-dimensional framed structures", Computer Methods in Applied Mechanics and Engineering, 2009, (Submitted).

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