Branch Identification in Elastic Stability Analysis

Sammanfattning: In this thesis, methods to determine the static postbuckling behaviour of elastic structures undergoing large deformations, are considered and developed. Since the governing non-linear equations usually becomes too complex to be handled analytically, the main focus has been on developing methods that can be incorporated into a numerical solution scheme, such as the finite element method. First methods to numerically track equilibrium curves and to calculate singular points along the equilibrium path will be discussed. The path following technique adopted is the well stablished arc-length method. To calculate the singular points along the equilibrium path, an extended system of equations is used which directly calculates the location of the singular point. The main interest in this thesis is the treatment of the singular points along the equilibrium path, especially for bifurcation points. For bifurcation points an asymptotic expansion method is developed, which combines a Lyapunov-Schmidt decomposition of the solution space with asymptotic expansions of both the displacements and the load, as well as of the equilibrium equations. This method can accurately predict the postbuckling behaviour on the secondary branches, at least in the vicinity of the bifurcation, for both asymmetric and symmetric single and multiple bifurcations. Special care is taken for symmetric multiple bifurcations, where higher order expansions have to be used to obtain correct results. The inclusion of higher order terms in the expansion allows for correct treatment of certain bifurcation points where the number of secondary paths emerging are larger than usually assumed. The methods is applied mainly on truss-bar structures, which exhibit many different types of singularities, and yet are computationally cheap. Finally, a classic stability problem is examined, namely the elastica. Contrary to the classical elastica problem the beam axis is here allowed to extend. This leads to a formulation where a closed-form solution can be obtained in terms of elliptical integrals. The considered form of the elastica shows some interesting stability phenomena compared to the classical inextensible case, e.g. the buckling load and the number of bifurcation points depend on the slenderness of the beam, and for certain values of the slenderness the load is initially decreasing on a postbuckling branch. The developed numerical methods are then applied to the elastica problem, where it is found that the properties predicted from the analytical treatment are in close agreement with the finite element results.