**8. References**


Goege, D., Fuellekrug, U., Sinapius, M., Link, M., & Gaul, L. (2005, Vol. 46, No. 5). INTL - A Strategy for the Identification and Characterization of Non-Linearities within Modal Survey Testing. *AIAA Journal*, pp. 974-986.

198 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**Author details** 

Ulrich Fuellekrug

**8. References** 

Verlag.

*Processing*, pp. 269-290.

*Chaos Solitons & Fractals*, pp. 687-708.

England: Research Studies Press Ltd.

*Sound and Vibration*, pp. 14-18.

(pp. 123-155). Wiesloch (Germany).

DLR\_FB\_2004-36.

applying Phase Separation Techniques. Thus, the experimental effort of applying the Phase

Al\_Hadid, M. A., & Wright, J. R. (1989, Vol. 3, No. 3). Developments in the Force-State Mapping Technique for Non-Linear Systems and the Extension to the Localisation of Non-Linear Elements in a Lumped Parameter System. *Mechanical Systems and Signal* 

Awrejcewicz, J., & Krysko, V. A. (2008). *Chaos in Structural Mechanics.* Berlin: Springer-

Awrejcewicz, J., Krysko, V. A., Papkova, I. V., & Krysko, A. V. (2012, 45). Routes to chaos in continuous mechanical systems. Part 1: Mathematical models and solution methods.

Awrejcewicz, J., Krysko, V. A., Papkova, I. V., & Krysko, A. V. (2012, 25). Routes to chaos in continuous mechanical systems. Part 3: The Lyapunov exponents, hyper, hyper-hyper

Crawley, E. F., & Aubert, A. C. (1986, Vol. 24, No. 1). Identification o f Nonlinear Structural

Ewins, D. J. (2000). *Modal Testing: Theory, Practice and Application.* Baldock, Hertfordshire,

Fuellekrug, U. (1988, Vol. 27-1). Survey of Parameter Estimation Methods in Experimental

Gloth, G., & Goege, D. (2004). Handling of Non-Linear Structural Characteristics in Ground Vibration Testing. *Proceedings of the International Conference on Noise and Vibration* 

Gloth, G., Degener, M., Fuellekrug, U., Gschwilm, J., Sinapius, M., Fargette, P., & Levadoux, B. (2001, Vol. 35, No. 11). New Ground Vibration Testing Techniques for Large Aircraft.

Goege, D. (2004). *Schnelle Identifikation und Charaktersisierung von Linearitaetsabweichungen in der experimentellen Modalanalyse grosser Luft- und Raumfahrtstrukturen.* Forschungsbericht

Goege, D., & Fuellekrug, U. (2004). Analyse von nichtlinearem Schwingungsverhalten bei großen Luft- und Raumfahrtstrukturen. *VDI Schwingungstagung, VDI-Berichte Nr. 1825*,

Modal Analysis. *Journal of the Society of Environmental Engineers*, pp. 33-44.

and spatial-temporal chaos. *Chaos Solitons & Fractals*, pp. 721-736.

Elements by Force-State Mapping. *AIAA Journal*, pp. 155-162.

*Engineering (ISMA)*, (pp. 2129-2143). Leuven (Belgium).

*Institute of Aeroelasticity, Deutsches Zentrum fuer Luft- und Raumfahrt (DLR), Germany* 

Resonance Method could be avoided leading to a reduced test duration.


Wrigth, J., Platten, M., Cooper, J., & Sarmast, M. (2003). Experimental Identification of Continuous Non-Linear Systems Using an Extension of Force Appropriation. *Proceedings of the 21st International Modal Analysis Conference (IMAC-XXI).* Kissimmee, FL (USA).

**Chapter 8** 

© 2012 Kopecki, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

**Numerical Reproducing of a Bifurcation in** 

**the Stress Distribution Obtaining Process in** 

**Post-Critical Deformation States of Aircraft** 

Modern aviation structures are characterised by widespread application of thin-shell loadbearing systems. The strict requirements with regard to the levels of transferred loads and the need to minimise a structure mass often become causes for accepting physical phenomena that in case of other structures are considered as inadmissible. An example of such a phenomenon is the loss of stability of shells that are parts of load-bearing structures,

Thus, an important stage in design work on an aircraft load-bearing structure is to determine stress distribution in the post-critical deformation state. One of the tools used to achieve this aim is nonlinear finite elements method analysis. The assessment of the reliability of the results thus obtained is based on the solution uniqueness rule, according to which a specific deformation form can correspond to one and only one stress state. In order to apply this rule it is required to obtain numerical model's displacements distribution fully

An element deciding about a structure's deformation state is the effect of a rapid change of the structure's shape occurring when the critical load levels are crossed. From the numerical point of view, this phenomenon is interpreted as a change of the relation between state parameters corresponding to particular degrees of freedom of the system and the control parameter related to the load. This relation, defined as the equilibrium path, in case of an occurrence of mentioned phenomenon, has an alternative character, defined as bifurcation. Therefore, the fact of taking a new deformation form by the structure corresponds to a

**Load-Bearing Structures** 

Additional information is available at the end of the chapter

Tomasz Kopecki

**1. Introduction** 

http://dx.doi.org/10.5772/48069

within the range of admissible loads.

corresponding to actual deformations of the analysed structure.

sudden change to the alternative branch of the equilibrium path [1-4].
