7. Conclusions

In conclusion let us sum up the problem of stability of the discussed elastica in case when w>0.
As follows from the above discussion, in this case the rectilinear form of equilibrium will always be stable, while we consider infinitesimal deviations from the point of balance. It can be seen clearly that with the increasing value p the local maximum of energy occurs at a value approaching closer and closer to a=0, but not reaching it. This only proves the fact that for large values of p it is easier to unbalance the system when it is in a stable, rectilinear state of equilibrium, causing some finite displacement to it. In order, however, for the system to assume a new, curvilinear form of stable equilibrium, it is necessary to pass the maximum of potential energy corresponding to the unstable form of equilibrium (Figure 5). The greater the force p is, the less displacement is needed for the system to assume a new form of equilibrium.
The force p_k as defined by the formula (37) can be called the critical force, above which (except for the rectilinear form of stability) there is also a curvilinear form of stable equilibrium of the system.
Due to the assumption of inextensibility of the elastica there arose the limitation of the value of the shape parameter a, which has to be less than . Thus, in turn, some limitations for the value w and axial force p followed in the determination of the curvilinear stable solutions as in the limits of admissible values a, which are illustrated in Figure 6.

References

1. Dym C L 1974 Stability Theory and Its Applications to Structural Mechanics (Leyden: Noordhoff Int. Publishing)
2. Naleszkiewicz J 1958 Problems of Elastic Stability (Warsaw: PWN)
3. Forray M J 1968 Variational Calculus in Science and Engineering (New York: McGraw-Hill)
4. Timoshenko S P and Gere J M 1961 Theory of Elastic Stability (New York: McGraw-Hill)

About the author:

PhD Piotr Szablewski is a lecturer at Technical University of Lodz, Department of Technical Mechanics and Informatics, Lodz, Poland. He has over 18 years experience in the teaching of mechanics and numerical methods in engineering. Particularly, he is interested in theoretical methods and numerical simulation.

He is interested in theoretical mechanics and its applications in woven and non-woven textiles (particularly - textile composites). He tries to develop textile mechanics on Department of Technical Mechanics and Informatics - Technical University of Lodz. During the last three years he has worked on applications of numerical method for building geometrical models for textile composites and predicting mechanical parameters. Currently he’s working on simulation and optimization geometrical parameters of composites, to obtain higher quality for structural strength of textile composites.

He has also experience in working with international student groups in the field of mechanics.

PhD Piotr Szablewski
Department of Technical Mechanics and Informatics
Technical University of Lodz
Zeromskiego 116
90-924 Lodz
Poland
e-mail: piotr.szablewski@p.lodz.pl

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