Effects of the constraint’s curvature on structural instability: tensile buckling and multiple bifurcations

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 fibre-reinforced 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

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
http://poliba.academia.edu/micheleciavarella

bigoni is right, from Son and followers there is a lot

http://hal.archives-ouvertes.fr/hal-00105477/

http://paloma.eng.tau.ac.il/~herzl/1998/THE%20POST-BUCKLING%20RESPONSE%20OF%20A%20BI-LATERALLY.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. 2629-2650
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 35400-000 Ouro
Preto, MG, Brazil
b. Department of Civil Engineering, PUC-Rio, Rua Marquês de São Vicente, 225, Gávea 22453-900 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 uni-dimensional 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, beam-columns 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 S0020-7683(07)00517-3

Publication Type:
Article

ISSN:
0020-7683

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Or this one on "constrained Euler Buckling"

Computer Methods in Applied Mechanics and Engineering

Elsevier Science

  info contratto
Vol: 170, Issue: 3-4

March 12, 1999 pp. 175-207

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, H-1521 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 path-following 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

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