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Abstract
The structural stability of a simple benchmark problem – the snap through of a shallow cylindrical roof – is revisited. The problem is analysed in a robust manner using a technique known as generalised pathfollowing, which combines the mathematical domains of finite element analysis and numerical continuation.
Using this technique, the wellknown arclength method for tracing equilibrium paths in loaddisplacement space is extended to explore other interesting paths on a twodimensional solution manifold in threedimensional loaddisplacementparameter space. These paths include unconventional equilibrium paths traced by varying a model parameter, e.g. shell thickness, as well as critical paths that describe points where the tangential stiffness matrix is singular – that is where buckling, snapthrough or other instabilities occur. For the chosen benchmark problem, a localised region of stable equilibrium exists on one of the unstable equilibrium branches. The evolution of this stable region with variations in shell thickness is explored by tracing the locus of critical
limit points that separate the stable and unstable portions of the equilibrium curve. Hence, without resorting to computationally expensive parametric studies, we establish a twodimensional manifold of stable equilibria in displacementloadthickness space, which can be graphically interpreted as an island of stability.
Using this technique, the wellknown arclength method for tracing equilibrium paths in loaddisplacement space is extended to explore other interesting paths on a twodimensional solution manifold in threedimensional loaddisplacementparameter space. These paths include unconventional equilibrium paths traced by varying a model parameter, e.g. shell thickness, as well as critical paths that describe points where the tangential stiffness matrix is singular – that is where buckling, snapthrough or other instabilities occur. For the chosen benchmark problem, a localised region of stable equilibrium exists on one of the unstable equilibrium branches. The evolution of this stable region with variations in shell thickness is explored by tracing the locus of critical
limit points that separate the stable and unstable portions of the equilibrium curve. Hence, without resorting to computationally expensive parametric studies, we establish a twodimensional manifold of stable equilibria in displacementloadthickness space, which can be graphically interpreted as an island of stability.
Original language  English 

Title of host publication  Shell Structures: Theory and Applications Volume 4 
Subtitle of host publication  Proceedings of the 11th International Conference "Shell Structures: Theory and Applications, (SSTA 2017), October 1113, 2017, Gdansk, Poland 
Editors  Wojciech Pietraszkiewicz, Wojciech Witkowski 
Publisher  Routledge 
Pages  223226 
Number of pages  4 
ISBN (Print)  9781138050457 
Publication status  Published  25 Sep 2017 
Event  11th International Conference "Shell Structures: Theory and Applications"  Gdansk, Poland Duration: 11 Oct 2017 → 13 Oct 2017 
Conference
Conference  11th International Conference "Shell Structures: Theory and Applications" 

Country  Poland 
City  Gdansk 
Period  11/10/17 → 13/10/17 
Structured keywords
 ACCIS Core Team
Keywords
 Generalised pathfollowing
 Bifurcation
 Buckling
 Postbuckling
 Nonlinearity
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Projects
 1 Finished

Structural Efficiency and Multifunctionality of WellBehaved Nonlinear Composite Structures
1/04/15 → 31/08/20
Project: Research