Branch Identification in Elastic Stability Analysis
Research output: Thesis › Doctoral Thesis (compilation)
Abstract
In this thesis, methods to determine the static postbuckling behaviour of elastic structures undergoing large deformations, are considered and developed. Since the governing nonlinear 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 arclength 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 LyapunovSchmidt 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 trussbar 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 closedform 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.
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 arclength 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 LyapunovSchmidt 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 trussbar 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 closedform 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.
Details
Authors  

Organisations  
Research areas and keywords  Subject classification (UKÄ) – MANDATORY
Keywords

Original language  English 

Qualification  Doctor 
Awarding Institution  
Supervisors/Assistant supervisor 

Award date  2000 May 19 
Publisher 

Print ISBNs  9178740738 
Publication status  Published  2000 
Publication category  Research 
Bibliographic note
Defence details
Date: 20000519
Time: 10:15
Place: sal M:B, Mhuset, LTH, Ole Römers väg 1
External reviewer(s)
Name: Mikkola, Martti
Title: Prof
Affiliation: Helsinki University of Technology
