**2. Objective and the point of the research**

Thin-walled aircraft load-carrying structures, in view of requirements applicable to them, are typically characterized with significant sophistication of the applied solutions. However, despite their advanced degree of complexity, they still have a number of characteristic features following from assumptions on which their operation is based. Thus, the loss of stability of thin-walled shell segments used in such structures is a result, in general, of a distribution of tangential stresses interpreted as a field of tensions. It can be therefore stated that post-buckling deformation patterns of a skin segments limited by components of the framing depend on factors decisive for stress distributions, i.e. proportions between dimensions of skin segments (rectangular in general), curvature radii related to these dimensions, and the load intensity [10].

Occurrence of post-buckling deformations corresponding to rapid changes in combinations of state parameters, known as bifurcation, in the case of the load only slightly exceeding the critical value, depends in great measure on geometrical imperfections and, to some extent, can exhibit random nature [11, 12]. In most cases, however, the ultimate pattern of post-buckling deformations corresponding to the maximum load is the same in each of load cycles. It is therefore possible to distinct the so-called nature of post-buckling deformations in case of structure designs characterized with specific, fixed geometrical features [13,14]. With knowl‐ edge of this nature, i.e. availability of data concerning post-buckling deformation patterns for a sufficiently broad spectrum of variants of the structure, it seems to be possible to use them as a tool for verification of results of nonlinear numerical analyses without necessity to carry out additional experiments.

In the present study, an attempt was made to determine the nature of post-buckling defor‐ mation of a characteristic fragment of the typical semi-monocoque aircraft structure by means of carrying out a series of relevant model experiments and confronting the results with the outcome of nonlinear numerical analyses performed with the use of commercial software package.

The subject of the research was a closed, semi-monocoque thin-walled cylindrical shell structure which corresponded to a fragment of an aircraft fuselage tail section (Fig. 3).

**Figure 3.** Examined part of the aircraft structure

and the solution determined in the prognostic phase, or the so-called "drift error". The correction phase consists in the use of an additional equation to be met by the system, known

where increments Δ**u***n* = **u***n*+1 – **u***<sup>n</sup>* and Δ*λ<sup>n</sup>* = *λ<sup>n</sup>*+1 – *λn* correspond to transition from state *n* to

In view of the large number of degrees of freedom and state parameters related to them, deformation processes are represented in practice by means of a relationship between a control parameter related to the load and a selected geometrical quantity linked to deformation of the

As was already mentioned above, results of FEM-based nonlinear numerical analyses require verification. Relying unquestioningly on such results alone can lead to significant errors in design processes through adopting incorrect solutions as a base for construction design assumptions. The problem consisting in arriving at incorrect deformation patterns as a result of numerical calculations is a consequence of the fact that numerical procedures employed in commercial software packets contain a large number of algorithms the course of which depends on choice of certain control parameters. These in turn follow from the applied boundary conditions, selection of prognostic procedures, correction strategies, and a number

In view of practical impossibility to obtain appropriate solutions for complex thin-walled structures in a purely analytical way, the basic tool that can be used for verification of results nonlinear numerical analyses is the experiment, by its nature representing an undertaking

In case of systems characterized with high degree of complexity or having geometrical singularities of any kind (e.g. cut-outs), execution of an appropriate experiments is absolutely necessary. It should be however emphasized that semi-monocoque aircraft structures include, in many cases, some typical components with characteristic, repeatable geometrical features. In such cases, it seems to be purposeful to create a base of standard solutions, containing result of experiments aimed at determination of deformation patterns of the analyzed structure for a given range of post-buckling loads justified by actual in-flight conditions. Such data could

Thin-walled aircraft load-carrying structures, in view of requirements applicable to them, are typically characterized with significant sophistication of the applied solutions. However, despite their advanced degree of complexity, they still have a number of characteristic features

constitute a base sufficient to verify results of nonlinear numerical analyses.

(2)

( , 0, ) *n n c* DD = **u** l

system. The relationship is called the representative equilibrium path [5-9].

relatively expensive and frequently difficult to execute.

**2. Objective and the point of the research**

as the increment control equation or the equation of constraints,

state *n* + 1.

142 Computational and Numerical Simulations

of other factors.

In the in-flight conditions, the structure can be subjected to bending and twisting, as a result of aerodynamic forces exerted on tail control surfaces. The structure components responsible

In all the cases it has been assumed that cross-sections of stringers applied in similar structures have geometrical characteristics preventing them from buckling in conditions of actual operating loads. For this reasons, their dimensions were intentionally exaggerated in model

Numerical Simulations of Post-Critical Behaviour of Thin-Walled Load-Bearing Structures Applied in Aviation

http://dx.doi.org/10.5772/57218

145

All experimental models were made of polycarbonate for which the following material constants have been determined: *E* = 3000 MPa, *v* = 0.36. Selection of the material was dictated by its isotropic properties and low Young modulus which allowed to limit the applied loads

The models were subjected to constrained torsion with the use of experimental set-up allowing to apply loads gravitationally (Fig. 5). As the representative equilibrium path, the relationship between the total angle of torsion of the structure and the torsional moment was selected. In view of the lack of possibility to register instantaneous changes of the load related to bifurcation changes of combinations of the parameters of state occurring in the structure, the presented equilibrium paths were determined for steady-state conditions as a result of which they have

As expected, in the case of the first variant of the examined structure, occurrence of postbuckling deformation had a violent nature. Magnitude of post-buckling deformations and the resulting significant value of the total angle of torsion make application of similar solution in actual aircraft impracticable. Despite the fact that the loss of stability had a local nature,

The deformation pattern as such was characterized with occurrence of folds observed in all four skin segments (Fig. 6). In the course of experiment, Atos optical scanner of GOM Optical Measuring Techniques brand was used to register the geometry of the deformed skin.

deformations occurring in this case would mean loss of rigidity of the fuselage.

experiments.

to relatively low values.

"smooth" courses.

**Figure 5.** The experimental set-up

**Figure 4.** Geometry of the examined structure

for transfer of bending loads are the stringers, cross-sections of which are selected based on the rod stability conditions, with the safety factor provided by aircraft construction regulations taken into account. However, appropriate torsional strength of the structure and its required torsional rigidity must be ensured by the skin in which a distribution of tangential stresses is developed creating conditions favorable for the loss of stability. The distributions, as was already mentioned, depend on a number of factors of geometrical nature, related to the number of frames and stringers determining size and shape of skin segments.

To determine characteristic features of deformation of the tested structure type and examine in detail the nature of the involved phenomena, it is necessary to analyze different geometrical variants. In order to be able to use the obtained results as a universal tool supporting the design process, it would be advisable to examine as broad spectrum of such variants as possible. This study presents only a few examples of such analyses, while the fundamental objective of the work consisted in development of a methodology for creation of sets of results obtained from appropriate model experiments and their numerical representations. Comparison of repre‐ sentative equilibrium paths of the examined systems and convergence of deformation patterns was aimed at determination of recommendations applicable to modeling structures of that type and carrying out nonlinear FEM analyses.
