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Branch Identification in Elastic Stability Analysis

Magnusson, Anders LU (2000)
Abstract (Swedish)
Popular Abstract in Swedish

Avhandlingen behandlar metoder för att kunna beräkna den last en struktur kan bära. En struktur kan i detta sammanhang vara allt från en bro till en mjölkkartong. För att kunna göra en korrekt bedömning av den lastbärande förmågan måste hänsyn tas till olinjära effekter som härrör från stora deformationer. Sådana olinjära effekter kan orsaka instabilt beteende hos en struktur, vilket i sin tur kan leda till kollaps vid betydligt lägre belastning än vad som hade förutspåtts av en analys där olinjära effekter inte beaktats.



I avhandlingen presenteras olika metoder för att analysera beteendet hos en struktur, efter det att den har passerat kritiska punkter. Tyngdpunkten ligger i... (More)
Popular Abstract in Swedish

Avhandlingen behandlar metoder för att kunna beräkna den last en struktur kan bära. En struktur kan i detta sammanhang vara allt från en bro till en mjölkkartong. För att kunna göra en korrekt bedömning av den lastbärande förmågan måste hänsyn tas till olinjära effekter som härrör från stora deformationer. Sådana olinjära effekter kan orsaka instabilt beteende hos en struktur, vilket i sin tur kan leda till kollaps vid betydligt lägre belastning än vad som hade förutspåtts av en analys där olinjära effekter inte beaktats.



I avhandlingen presenteras olika metoder för att analysera beteendet hos en struktur, efter det att den har passerat kritiska punkter. Tyngdpunkten ligger i utveckling av numeriska metoder som kan användas vid finitaelementberäkningar. Genom att beräkna jämviktskurvor, d.v.s. kurvor som beskriver jämviktslägen för olika laster, för en struktur utan störningar kan slutsatser dras om hur en verklig struktur, med störningar, skulle bete sig.



Speciellt intresse riktas mot bifurkationspunkter, vilket är en punkt där jämviktskurvan förgrenar sig. Ett stört system tenderar att följa någon av de bifurkerade grenarna, som alltså har stor betydelse för en korrekt bedömning av den lastbärande förmågan hos strukturen. I avhandlingen ges flera exempel på olika typer av bifurkationer och en metod att identifiera alla utgående grenar från dessa har tagits fram. (Less)
Abstract
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... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
opponent
  • Prof Mikkola, Martti, Helsinki University of Technology
organization
publishing date
type
Thesis
publication status
published
subject
keywords
vacuum technology, hydraulics, Mechanical engineering, Produktionsteknik, Production technology, FEM, computer simulation, bifurcation, stability, vibration and acoustic engineering, Maskinteknik, hydraulik, vakuumteknik, vibrationer, akustik
pages
135 pages
publisher
Division of Solid Mechanics, Box 118, 221 00 Lund, Sweden,
defense location
sal M:B, M-huset, LTH, Ole Römers väg 1
defense date
2000-05-19 10:15
external identifiers
  • Other:ISRN: LUTFD2/TFHF--00/1021--SE (1-135)
ISBN
91-7874-073-8
language
English
LU publication?
yes
id
89cf45ce-4dc1-4b86-9770-4b1eccbe10cd (old id 40527)
date added to LUP
2007-07-31 16:07:31
date last changed
2016-09-19 08:45:09
@misc{89cf45ce-4dc1-4b86-9770-4b1eccbe10cd,
  abstract     = {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.<br/><br>
<br/><br>
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.<br/><br>
<br/><br>
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.<br/><br>
<br/><br>
The methods is applied mainly on truss-bar structures, which exhibit many different types of singularities, and yet are computationally cheap.<br/><br>
<br/><br>
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.},
  author       = {Magnusson, Anders},
  isbn         = {91-7874-073-8},
  keyword      = {vacuum technology,hydraulics,Mechanical engineering,Produktionsteknik,Production technology,FEM,computer simulation,bifurcation,stability,vibration and acoustic engineering,Maskinteknik,hydraulik,vakuumteknik,vibrationer,akustik},
  language     = {eng},
  pages        = {135},
  publisher    = {ARRAY(0xa4a10e0)},
  title        = {Branch Identification in Elastic Stability Analysis},
  year         = {2000},
}