**1. Introduction**

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, within the range of admissible loads.

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 corresponding to actual deformations of the analysed structure.

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

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

Therefore, a prerequisite condition for obtaining a proper form of the numerical model deformation is to retain the conformity between numerical bifurcations and bifurcations in the actual structure. In order to determine such conformity it is required to verify the results obtained by an appropriate model experiment or by using the data obtained during the tests of the actual object. It is often troublesome to obtain reliable results of nonlinear numerical analyses and it requires an appropriate choice of numerical methods dependent upon the type of the analysed structure and precise determination of parameters controlling the course of procedures.

Due to the number of state parameters, the full equilibrium path should be interpreted as hyper-surface in state hyperspace, satisfying the matrix equation for residual forces:

$$\mathbf{r}(\mathbf{u}, \Lambda) = \mathbf{0},\tag{1}$$

Numerical Reproducing of a Bifurcation in the Stress Distribution Obtaining Process in Post-Critical Deformation States of Aircraft Load-Bearing Structures 203

structure's deformation, an increment of which corresponds to a change of the value of all or

**Figure 1.** Graphic presentation of various correction strategies for the representative system of one

In case of a large number of state parameters it is not possible at all to represent the character of bifurcation by applying a representative equilibrium path. Sometimes, changes of state parameters resulting from local bifurcation may show the lack of perceptible influence on the representative value, which results in non-occurrence of any characteristic points on the representative path. In general, however, these changes cause a temporary

**Figure 2.** Bifurcation points on a representative equilibrium path (u - representative geometric value,

some state parameters [5-9].

degree of freedom

drop in the control parameter value (Figure 2).

λ – control parameter related to load).

where **u** is the state vector containing structure nodes' displacement components corresponding to current geometrical configuration, is a matrix composed of control parameters corresponding to current load state, and **r** is the residual vector containing uncompensated components of forces related to current system deformation state. The set of control parameters may be expressed by a single parameter that is a function of the load. Equation (1) takes then the following form:

$$\mathbf{r}(\mathbf{u}, \lambda) = \mathbf{0},\tag{2}$$

called a monoparametric equation of residual forces.

The prediction-correction methods of determining the consecutive points of the equilibrium path used in modern programs contain also a correction phase based on the satisfaction of an additional equation by the system, called an increment control equation or constraints equation:

$$\mathbf{c}(\Delta \mathbf{u}\_{n'}, \Delta \mathcal{J}\_n) = \mathbf{0},\tag{3}$$

where the increments:

$$
\Delta \mathbf{u}\_n = \mathbf{u}\_{n+1} - \mathbf{u}\_n \text{ and } \Delta \mathcal{k}\_n = \mathcal{k}\_{n+1} - \mathcal{k}\_n \tag{4}
$$

correspond to the transition from *n*-th state to *n+1*-th state*.* 

The graphic interpretation of the increment control equation is presented in Figure 1.

In order to find out whether there is full conformity between the character of actual deformations and their numerical representation it would be required to compare the combinations of the relevant state parameters in all the phases of the course of the phenomenon considered herein. Because of the complication of such a comparative system, the deformation processes are represented in practice by applying substitute characteristics called representative equilibrium paths. They define the relations between a control parameter related to load and a selected, characteristic geometric value related to a structure's deformation, an increment of which corresponds to a change of the value of all or some state parameters [5-9].

202 Nonlinearity, Bifurcation and Chaos – Theory and Applications

Equation (1) takes then the following form:

called a monoparametric equation of residual forces.

( , ) 0, *n n c* **u**

correspond to the transition from *n*-th state to *n+1*-th state*.* 

course of procedures.

equation:

where the increments:

Therefore, a prerequisite condition for obtaining a proper form of the numerical model deformation is to retain the conformity between numerical bifurcations and bifurcations in the actual structure. In order to determine such conformity it is required to verify the results obtained by an appropriate model experiment or by using the data obtained during the tests of the actual object. It is often troublesome to obtain reliable results of nonlinear numerical analyses and it requires an appropriate choice of numerical methods dependent upon the type of the analysed structure and precise determination of parameters controlling the

Due to the number of state parameters, the full equilibrium path should be interpreted as

where **u** is the state vector containing structure nodes' displacement components corresponding to current geometrical configuration, is a matrix composed of control parameters corresponding to current load state, and **r** is the residual vector containing uncompensated components of forces related to current system deformation state. The set of control parameters may be expressed by a single parameter that is a function of the load.

The prediction-correction methods of determining the consecutive points of the equilibrium path used in modern programs contain also a correction phase based on the satisfaction of an additional equation by the system, called an increment control equation or constraints

*n n n nn n* 1 1 *and* 

In order to find out whether there is full conformity between the character of actual deformations and their numerical representation it would be required to compare the combinations of the relevant state parameters in all the phases of the course of the phenomenon considered herein. Because of the complication of such a comparative system, the deformation processes are represented in practice by applying substitute characteristics called representative equilibrium paths. They define the relations between a control parameter related to load and a selected, characteristic geometric value related to a

The graphic interpretation of the increment control equation is presented in Figure 1.

 **uu u** (4)

**ru 0** (,) , (1)

**ru 0** (,) , *λ* (2)

(3)

hyper-surface in state hyperspace, satisfying the matrix equation for residual forces:

**Figure 1.** Graphic presentation of various correction strategies for the representative system of one degree of freedom

In case of a large number of state parameters it is not possible at all to represent the character of bifurcation by applying a representative equilibrium path. Sometimes, changes of state parameters resulting from local bifurcation may show the lack of perceptible influence on the representative value, which results in non-occurrence of any characteristic points on the representative path. In general, however, these changes cause a temporary drop in the control parameter value (Figure 2).

**Figure 2.** Bifurcation points on a representative equilibrium path (u - representative geometric value, λ – control parameter related to load).

So both the experiment itself and nonlinear numerical analyses may result only in a representative equilibrium path. In that case, the problem of the numerical representation of bifurcation comes down to the preservation of conformity of the representative equilibrium path obtained by a numerical method with the one obtained experimentally, where a sine qua non for the application of the solution uniqueness rule is to recognise the similarity of the post-critical deformation forms of the experimental and numerical models as sufficient [10,11].

Numerical Reproducing of a Bifurcation in the Stress Distribution Obtaining Process in Post-Critical Deformation States of Aircraft Load-Bearing Structures 205

numerical representative equilibrium path of the same level of simplification as in the case

The comparative analysis of such representative equilibrium paths is not, however, a method that allows a complete enough verification of the reliability of the results of numerical calculations. An example of a problem in which the calculated results have been deemed incorrect despite the seeming full conformity of the representative equilibrium paths is a thin-shell open cylindrical structure with edges strengthened by stringers, working in the conditions of constrained torsion (Figure 4). This type of systems is quite often used in aviation structures. They form areas of cockpits and large cut-outs, e.g. in cargo airplanes and they are usually adjacent to much stiffer fragments of the structures

**Figure 4.** A schematic view of the tested structure and comparison of the deformation forms obtained

The area adjacent directly to the closing frame turned out to be crucial in the problem under consideration. The stringer strengthening the edge of the structure was buckled, and the experimental model sustained plastic deformation. The relation between the total torsion angle of the examined structure and the torque moment constituting the load was adopted

of the experiment.

[12,13].

experimentally and numerically

**2. Analyses of example structures** 

An additional problem occurred during the experimental determination of the equilibrium path, resulting from the lack of abilities of recording the said temporary, little drops in load, arising from local bifurcations, causing changes of the values of some state parameters. In the majority of experiments, the load of the tested model is achieved by force control, e.g. using a gravitational system, or displacement, by means of various types of load-applying devices (Figure 3).

**Figure 3.** A stand for testing thin-shell structures subject to torsion: left – a version with a loading system controlling the displacement (turnbuckle), right – a version with a system controlling by force (gravitational).

However, even in case of devices with high level of technical advancement, in general it is not possible to register precisely short-lasting force changes, occurring from the beginning of a bifurcation phenomenon to the moment of reaching the consecutive deformation form by the model. Therefore, the representative equilibrium path obtained as a result of the experiment is of smooth characteristics, and its formation is based on measuring points corresponding to the consecutive deformation states determined.

In case of nonlinear numerical analysis in the finite element approach, the accuracy of the obtaining of the representative equilibrium path may be much more accurate. The existing commercial programs usually offer the results of all the increment steps, followed during the calculation process, and thus they also allow observing slight fluctuations of the control parameter. The only limitation here is exclusively the value of the incremental step itself. In spite of this, due to the lack of possibilities of relating the results obtained to the relevant detailed changes of the experimental characteristics, it seems appropriate to determine the numerical representative equilibrium path of the same level of simplification as in the case of the experiment.
