**3. Conclusion**

216 Nonlinearity, Bifurcation and Chaos – Theory and Applications

long "horizontal section".

The best result was obtained only after the fundamental change of the concept of FEM model, when the different kind of finite elements was applied as a representation of stringers (thick shell element was used instead of a recommended beam element). However this solution, from the point of view of mathematical description is much less correct, it turned out much more effective in case of relatively low values of the total torsion angle of the structure.

The results of analysis of this FEM model version, obtained using secant prediction method

**Figure 20.** The deformation distribution (left) and reduced stress distribution acc. to Huber-Mises hypothesis (right) for 100% of the maximum load (stringers modelled with thick shell bilinear element)

The relation between the representative equilibrium paths is presented in Figure 21.

**Figure 21.** The presentation of the representative equilibrium paths

The strain-correction strategy turned out most effective in case of significant, violent change of the form of deformations, when the representative equilibrium path contains relatively

and strain correction strategy (Figure1), are presented in Figure 20.

The presented examples of load-bearing structures represent only some of those many used in the modern aviation technology. But the criterion applied while selecting them as objects of experimental and numerical analyses was its representativeness for the most commonly met elements of constructions, in case of which the occurrence of a local stability loss is acceptable in the conditions of service load.

The fundamental conclusion that can be drawn from the presented results of the research is the absolute need for using experimental verifications with regard to FEM nonlinear numerical analyses of this type of structures. The more so that even in the cases in which the correctness of the results obtained seems unquestionable, they may be in fact burdened with errors resulting from the very limited reliability of the numerical procedures used in commercial programs.

Based on the nonlinear numerical analyses, related to the presented structures, frequently repeated many times, a general recommendation may also be formulated for the maximum possible limitation of the size of a task. Striving for increasing the accuracy of the calculations by increasing the density of finite elements mesh, applied successfully in linear analyses, may turn out ineffective in case of a nonlinear analysis and may lead to incorrect results or the lack of convergence of calculations.

The numerical representation of bifurcation, by virtue of the mere idea of the discrete representation of continuous systems, must be simplified in case of the finite elements method. In such a situation, based on the quoted examples, the need must be emphasised for obtaining the indispensable convergence of the experimental and obtained numerically relations between a selected geometric parameter characterising the essence of a structure's deformation and a selected value relating to the load, recognised as representative equilibrium paths. This convergence, in combination with the accepted as sufficient similarity of post-critical deformation forms, constitutes the grounds for accepting the reliability of stress distributions determined by means of numerical methods.
