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Effects of the constraint’s curvature on structural instability: tensile buckling and multiple bifurcations
Thu, 20120322 05:00  Davide Bigoni
Can a singledegreeoffreedom elastic structure have two buckling loads?
We provide a positive answer to this question, see http://www.ing.unitn.it/~bigoni/multiple_bifurcations.html
More information about my research activity can be found in http://www.ing.unitn.it/~bigoni/
More information about our experiments can be found in http://ssmg.unitn.it/
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Davide, since you are the world leader on buckling, a question..
Suppose I have the problem in the figure.
This is remotely the problem I am studying in railways subject to thermal buckling.The beam represents the rail, and the restraints, unilateral ones, the ballast (there would be only one of course).
Consider the problem of a nominally straight beam, flexural rigidity EI
and length L. It is pressed between two rigid half planes by a pressure
p. For example, the lower half plane is fixed, the beam rests on it and
the second half plane is now pressed down from above with a force pL.
Whilst the assembly is in this state, the beam is now compressed by an axial force F.
The question is "If F is gradually increased, will the beam ever buckle, and if so at what value of F?"
With Jim Barber, we developed some simple Rayleigh Ritz solution for this.
The example I sent is to develop some thinking methods. In particular,
as stated, it seems that the system is always stable if the load is big
enough to flatten the initial perturbation. However, no bodies are
completely rigid (meaning a stiff Winkler foundation might be more
exact, but if the materials of the beam and the foundation are the same,
this would probably give a very high buckling load). Also, if the
surfaces are rough, we know that no amount of compression is enough to
flatten the roughness, so there is a dimension to do with roughness
here, seen as part of the initial perturbation question.
Even more challenging is that fact that if by some accident we can get a
slight separation, the system will go from stable to unstable. In other
words, the potential barrier that ensures stability under the linear
theory gets smaller and smaller as the axial load P increases. Common
sense suggests that there is some limiting value of this barrier below
which the linear theory is meaningless. In fact, it was an attempt to
quantify this that led me to think of there being an initial
perturbation, but as you see, it doesn't work. Even with the
perturbation, the problem becomes stable in the linear limit.
I think there is a great deal more to it than this. Firstly, the track
problem is different on the two sides  upward motion is less restrained
than downwards, and all sorts of coupled modes might be important.
But is there literature on this problem?
Also, I am wondering, would any fiber reinforced material be a possible
model, provided we neglect adhesion of the fiber (delaminated fiber),
and provided the compression is mainly effective on the fiber? There
must be tons of papers on that.
But for fibrereinforced composites, the fibre is usually stiffer than the
matrix, so it is like a rather soft elastic foundation. The cases I am
looking at are at the opposite extreme. For example, if the material of
the foundation has the same modulus as the beam, elastic deformation of
the foundation would be as if everything was one solid block and hence
stable. Thus any instablity must come from separation of the block as a
result of initial perturbation somehow.
Michele Ciavarella, Politecnico di BARI  Italy, Rector's delegate.
http://poliba.academia.edu/micheleciavarella
Dear Mike
Buckling with unilateral constraints is cool!
I am not an expert of this, but there is a lot of work. I would like to invite you to consider the work done by Professor Nguyen Quoc Son and coworkers. You will find a lot of references!
Best wishes,
Davide
Davide, I can see you have not much time to help !!! ;)
Michele Ciavarella, Politecnico di BARI  Italy, Rector's delegate.
http://poliba.academia.edu/micheleciavarella
... I'm doing my best ...
but these are though days ... too much politics for scientists in the Italian Universities!
However I did find some references following your suggestion
bigoni is right, from Son and followers there is a lot
http://hal.archivesouvertes.fr/hal00105477/
http://paloma.eng.tau.ac.il/~herzl/1998/THE%20POSTBUCKLING%20RESPONSE%20OF%20A%20BILATERALLY.pdf
http://buildtech.aalto.fi/fi/ajankohtaista/tapahtumat/udine02.pdf
which one do you suggest?
Perhaps even better
Nonlinear analysis of structural elements under unilateral contact constraints by a Ritz type approach
Journal: International Journal of Solids and Structures
Author:
Silveira Ricardo A.M.,
Pereira Wellington L.A.,
Gonçalves Paulo B.,
Volume: 45, Issue: 9, May 1, 2008, pp. 26292650
Score: 3.081263
Bibliographic Page
 Article Full Text PDF (1059.36 KB)
Title:
Nonlinear analysis of structural elements under unilateral contact constraints by a Ritz type approach
Authors:
Silveira, Ricardo A.M.a; Pereira, Wellington L.A.a; Gonçalves, Paulo B.b
Affiliations:
a.
Department of Civil Engineering, School of Mines, Federal University of
Ouro Preto, Campus Universitário, Morro do Cruzeiro 35400000 Ouro
Preto, MG, Brazil
b. Department of Civil Engineering, PUCRio, Rua Marquês de São Vicente, 225, Gávea 22453900 Rio de Janeiro, RJ, Brazil
Keywords:
Unilateral contact; Stability; Ritz method; Tensionless foundation; Nonlinear structural analysis
Abstract (English):
A
nonlinear modal solution methodology capable of solving equilibrium and
stability problems of unidimensional structural elements (beams,
columns and arches) with unilateral contact constraints is presented in
this work. The contact constraints are imposed by an elastic foundation
of the Winkler type, where special attention is given to the case in
which the foundation reacts in compression only, characterizing the
contact as unilateral. A Ritz type approach with moveable boundaries,
where the coordinates defining the limits of the contact regions are
considered as additional variables of the problem, is proposed to solve
this class of unilateral contact problems. The methodology is
illustrated by particular problems involving beams, beamcolumns and
arches, and the results are compared with available results obtained by
finite element and mathematical programming techniques. It is concluded
that the Ritz type approach proposed is particularly suited for the
analysis of structural problems where the number, but not the length, of
the contact regions between the bodies are known a priori. Therefore,
it can substitute in these cases finite element applications and be used
as a benchmark for more general and complex formulations as well.
Publisher:
Elsevier Science
Language of Publication:
English
Item Identifiers:
6183 10.1016/j.ijsolstr.2007.12.012 S00207683(07)005173
Publication Type:
Article
ISSN:
00207683
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info contratto
Vol: 170, Issue: 34
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Bibliographic Page
Article Full Text PDF (1.58 MB)
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Title:
Constrained Euler buckling: an interplay of computation and analysis
Authors:
Holmes, Philipa; Domokos, Gáborb; Schmitt, Johnc; Szeberényi, Imre
Affiliations:
a. Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
b. Department of Strength of Materials, Technical University of Budapest, H1521 Budapest, Hungary
c. Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Abstract (English):
We
consider elastic buckling of an inextensible beam with hinged ends and
fixed end displacements, confined to the plane, and in the presence of
rigid, frictionless sidewalls which constrain overall lateral
displacements. We formulate the geometrically nonlinear (Euler) problem
and develop global search and pathfollowing algorithms to find
equilibria in various classes satisfying different contact patterns and
hence boundary conditions. We derive complete analytical results for
the case of line contacts with the sidewalls, and partial results for
point contact and mixed cases. The analysis is essential to
understanding the numerical results, for in contrast to the
unconstrained problem, we find a very rich bifurcation structure, with
the cardinality of branches growing exponentially with mode number.
Michele Ciavarella, Politecnico di BARI  Italy, Rector's delegate.
http://poliba.academia.edu/micheleciavarella
Very Interesting Work!
Hi Dr. Bigoni,
Thanks for sharing this nice work. The video, in particular, is very clear and informative on how (especially nonlinear) boundary conditions can make for very interesting phenomena.
Matt Pharr
Nice to meeting you!
Hi Matt, nice of meeting you! Best wishes, Davide