**3. Experimental research**

The subject of the research were three variants of a thin-walled cylindrical structure, geomet‐ rical details of which are presented in Fig. 4. The first variant was a design solution representing a limiting case with particularly small number of framing components. Employing, in the next variant, an additional frame situated half length of the skin segment, and further increasing the number of stringers, corresponded to modifications possible to be employed in practice, as a result of which the size and geometrical relationships characterizing skin segments change significantly.

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 experiments.

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 to relatively low values.

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 "smooth" courses.

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

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

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

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

The subject of the research were three variants of a thin-walled cylindrical structure, geomet‐ rical details of which are presented in Fig. 4. The first variant was a design solution representing a limiting case with particularly small number of framing components. Employing, in the next variant, an additional frame situated half length of the skin segment, and further increasing the number of stringers, corresponded to modifications possible to be employed in practice, as a result of which the size and geometrical relationships characterizing skin segments change

of frames and stringers determining size and shape of skin segments.

type and carrying out nonlinear FEM analyses.

**3. Experimental research**

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

144 Computational and Numerical Simulations

significantly.

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.

equilibrium paths proves, the change of skin segment sizes and proportions as well as changed ratio of the segment dimensions and the curvature radius have brought the effect in the form of further increase of torsional rigidity of the examined structure with simultaneous small

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

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

147

The pattern of post-buckling deformations has significantly changed, taking the form of single shallow folds (Fig. 8). The course of the phenomenon as such was more gentle in this case than

Therefore, in this case the skin stability loss resulted in development of small geometrical defects of the fuselage inducing a local drag coefficient increase; however, higher rigidity

**Figure 8.** Advanced post-buckling deformation of the examined structure (left) and distribution of contour lines re‐ flecting magnitude of deformations obtained by means of the projection moiré technique (right) — variant 3

**Figure 9.** Comparison of representative equilibrium paths determined in the course of experimental research

guarantees that basic aerodynamic properties of the aircraft are maintained.

reduction of the critical load [10,12 ].

in the variants examined earlier.

**Figure 6.** Advanced post-buckling deformation of the examined structure (left) and distribution of contour lines re‐ flecting magnitude of deformations obtained by means of the projection moiré technique (right) — variant 1

In connection with a violent development of deformation, the representative equilibrium path contains a characteristic horizontal segment corresponding to a large change of state parameter combinations at virtually fixed load value (Fig. 9).

The next stage consisted in examination of the second variant of the structure in which an additional frame was employed (Fig. 4, variant 2). The change in proportions of basic skin segment dimensions resulted in occurrence of a post-buckling deformation pattern signifi‐ cantly different than this observed in the first case (Fig. 7).

**Figure 7.** Advanced post-buckling deformation of the examined structure (left) and distribution of contour lines re‐ flecting magnitude of deformations obtained by means of the projection moiré technique (right) — variant 2

Occurrence of a double fold resulted in more gentle course of the phenomenon, which manifested itself in absence of any large jump on the equilibrium path (Fig. 9). An increase of the load critical value and torsional rigidity of the system with respect to the first variant was also observed.

The third variant of the structure was a modification of the second variant reinforced with four evenly distributed additional stringers (Fig. 4, variant 3). As comparison of representative equilibrium paths proves, the change of skin segment sizes and proportions as well as changed ratio of the segment dimensions and the curvature radius have brought the effect in the form of further increase of torsional rigidity of the examined structure with simultaneous small reduction of the critical load [10,12 ].

The pattern of post-buckling deformations has significantly changed, taking the form of single shallow folds (Fig. 8). The course of the phenomenon as such was more gentle in this case than in the variants examined earlier.

Therefore, in this case the skin stability loss resulted in development of small geometrical defects of the fuselage inducing a local drag coefficient increase; however, higher rigidity guarantees that basic aerodynamic properties of the aircraft are maintained.

**Figure 6.** Advanced post-buckling deformation of the examined structure (left) and distribution of contour lines re‐ flecting magnitude of deformations obtained by means of the projection moiré technique (right) — variant 1

In connection with a violent development of deformation, the representative equilibrium path contains a characteristic horizontal segment corresponding to a large change of state parameter

The next stage consisted in examination of the second variant of the structure in which an additional frame was employed (Fig. 4, variant 2). The change in proportions of basic skin segment dimensions resulted in occurrence of a post-buckling deformation pattern signifi‐

**Figure 7.** Advanced post-buckling deformation of the examined structure (left) and distribution of contour lines re‐ flecting magnitude of deformations obtained by means of the projection moiré technique (right) — variant 2

Occurrence of a double fold resulted in more gentle course of the phenomenon, which manifested itself in absence of any large jump on the equilibrium path (Fig. 9). An increase of the load critical value and torsional rigidity of the system with respect to the first variant was

The third variant of the structure was a modification of the second variant reinforced with four evenly distributed additional stringers (Fig. 4, variant 3). As comparison of representative

combinations at virtually fixed load value (Fig. 9).

146 Computational and Numerical Simulations

also observed.

cantly different than this observed in the first case (Fig. 7).

**Figure 8.** Advanced post-buckling deformation of the examined structure (left) and distribution of contour lines re‐ flecting magnitude of deformations obtained by means of the projection moiré technique (right) — variant 3

**Figure 9.** Comparison of representative equilibrium paths determined in the course of experimental research
