**4. FEM-based numerical analysis**

Examining the nature of post-buckling deformations occurring in thin-walled structures with the intent to use the obtained results as a tool useful in aircraft design processes, together with carrying out appropriate experiments, it seems to be purposeful to develop recommendations concerning methods of numerical modeling of considered structures and selection of most effective numerical methods. It should be emphasized that the practice of developing dedi‐ cated software based on the finite elements method by entities dealing with aircraft structures design is a relatively rare phenomenon. As a rule, different types of routines are used available on the commercial software market.

Analyses discussed in the present study were carried out on the grounds of MSC MARC program. In all cases, the mounting of the model was reproduced by locking all degrees of freedom of selected points on the upper frame (corresponding to location of bolt joints in the experimental model) and a pair of forces was applied at appropriate points of the lower frame (corresponding to locations where the ties were attached). To represent the skin, four-node shell elements of the thin-shell type were used, with six degrees of freedom at each of the nodes and bilinear shape functions. The frames have been modeled with the use of thick-shell elements with similar properties. In case of stringers, according to software authors' recom‐ mendations, beam-type elements were employed, based on the Euler-Bernoulli model [14].

recognized satisfactorily convergent with experimental characteristics. In line with the rule of uniqueness of solutions, according to which one and only one distribution of the reduced stress corresponds to each deformation state, the obtained reduced stress distribution can be

**Figure 10.** Geometrical model of the structure developed in MSC Patran environment with boundary conditions and

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**Figure 11.** Displacement distribution (left) and the reduced stress according to Huber-Mises hypothesis (right) for

When numerical models for the remaining variants of the structure were developed, the same concept as for application of constraints, loads, and additional forces initiating post-buckling

Application of numerical methods identical to those used previously allowed to obtain postbuckling deformation distributions and the corresponding reduced stress distributions

100% of the maximum load (stringers modeled by means of bilinear thick-walled shell elements)

satisfactorily consistent with results of experiments (Figs. 14 and 15).

deformation was adopted as in the first variant (Fig. 13).

therefore also considered reliable (Fig. 11).

load

The first variant of the examined structure, in view of the observed post-buckling deformation scale and violent course of the related phenomena, turned out to be very troublesome from the point of view of FEM-based nonlinear numerical simulation. In fact, lack of ability to represent symmetry or antisymmetry of post-buckling deformation states is a characteristic feature of algorithms employed in majority of commercial software packages. In absence of geometrical imperfections of the structure, the state parameter combination change occurred only in one segment of the structure. Errors of that type follow, in general, from unreliability of algorithm used to select an appropriate equilibrium path variant after reaching a bifurcation point [15].

In order to initiate deformation patterns corresponding to actual ones, additional loads in the form of forces with small values normal to the skin applied at central points of skin segments have been introduced to the numerical model (Fig. 10). However, despite the obtained repeatability of deformation in individual segments, faulty results were obtained for a number of consecutive numerical models or solutions in the full load range were impossible to obtain.

A significant improvement of effectiveness of the analysis process and quality of the obtained results was achieved by changing the concept used to model the stringers. In successive versions of numerical models, thick-walled shell elements were used to represent these components of the structure. From among numerical models available in the software package, after a series of tests, a combination of the prognostic secant method and the strain correction method was adopted [16]. The deformation distribution obtained numerically was found satisfactory from the point of view of both qualitative and quantitative similarity to deforma‐ tion patterns obtained experimentally. Also representative equilibrium paths (Fig. 12) were

**4. FEM-based numerical analysis**

148 Computational and Numerical Simulations

on the commercial software market.

point [15].

Examining the nature of post-buckling deformations occurring in thin-walled structures with the intent to use the obtained results as a tool useful in aircraft design processes, together with carrying out appropriate experiments, it seems to be purposeful to develop recommendations concerning methods of numerical modeling of considered structures and selection of most effective numerical methods. It should be emphasized that the practice of developing dedi‐ cated software based on the finite elements method by entities dealing with aircraft structures design is a relatively rare phenomenon. As a rule, different types of routines are used available

Analyses discussed in the present study were carried out on the grounds of MSC MARC program. In all cases, the mounting of the model was reproduced by locking all degrees of freedom of selected points on the upper frame (corresponding to location of bolt joints in the experimental model) and a pair of forces was applied at appropriate points of the lower frame (corresponding to locations where the ties were attached). To represent the skin, four-node shell elements of the thin-shell type were used, with six degrees of freedom at each of the nodes and bilinear shape functions. The frames have been modeled with the use of thick-shell elements with similar properties. In case of stringers, according to software authors' recom‐ mendations, beam-type elements were employed, based on the Euler-Bernoulli model [14].

The first variant of the examined structure, in view of the observed post-buckling deformation scale and violent course of the related phenomena, turned out to be very troublesome from the point of view of FEM-based nonlinear numerical simulation. In fact, lack of ability to represent symmetry or antisymmetry of post-buckling deformation states is a characteristic feature of algorithms employed in majority of commercial software packages. In absence of geometrical imperfections of the structure, the state parameter combination change occurred only in one segment of the structure. Errors of that type follow, in general, from unreliability of algorithm used to select an appropriate equilibrium path variant after reaching a bifurcation

In order to initiate deformation patterns corresponding to actual ones, additional loads in the form of forces with small values normal to the skin applied at central points of skin segments have been introduced to the numerical model (Fig. 10). However, despite the obtained repeatability of deformation in individual segments, faulty results were obtained for a number of consecutive numerical models or solutions in the full load range were impossible to obtain.

A significant improvement of effectiveness of the analysis process and quality of the obtained results was achieved by changing the concept used to model the stringers. In successive versions of numerical models, thick-walled shell elements were used to represent these components of the structure. From among numerical models available in the software package, after a series of tests, a combination of the prognostic secant method and the strain correction method was adopted [16]. The deformation distribution obtained numerically was found satisfactory from the point of view of both qualitative and quantitative similarity to deforma‐ tion patterns obtained experimentally. Also representative equilibrium paths (Fig. 12) were

**Figure 10.** Geometrical model of the structure developed in MSC Patran environment with boundary conditions and load

recognized satisfactorily convergent with experimental characteristics. In line with the rule of uniqueness of solutions, according to which one and only one distribution of the reduced stress corresponds to each deformation state, the obtained reduced stress distribution can be therefore also considered reliable (Fig. 11).

**Figure 11.** Displacement distribution (left) and the reduced stress according to Huber-Mises hypothesis (right) for 100% of the maximum load (stringers modeled by means of bilinear thick-walled shell elements)

When numerical models for the remaining variants of the structure were developed, the same concept as for application of constraints, loads, and additional forces initiating post-buckling deformation was adopted as in the first variant (Fig. 13).

Application of numerical methods identical to those used previously allowed to obtain postbuckling deformation distributions and the corresponding reduced stress distributions satisfactorily consistent with results of experiments (Figs. 14 and 15).

Plots of representative equilibrium paths for variants 2 and 3 (Figs. 16, 17) are characterized with significantly better conformity with experimental characteristics then in the case of variant 1. In the pre-buckling range, the consistence is almost perfect. On the other hand, for maximum loads, the error of representation of the total angle of torsion does not exceed 10%. It allows to conclude that the adopted modeling method and selection of numerical procedures turned out to be satisfactory.

**Figure 14.** Distribution of the displacement (left) and the reduced stress according to Huber-Mises hypothesis (right)

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**Figure 15.** Distribution of the displacement (left) and the reduced stress according to Huber-Mises hypothesis (right)

for 100% of the maximum load — variant 2

for 100% of the maximum load — variant 3

**Figure 16.** Comparison of representative equilibrium paths — variant 2

**Figure 12.** Comparison of representative equilibrium paths — variant 1

**Figure 13.** Geometrical models of variant 2 (left) and variant 3 (right) of the examined structure developed in MSC Patran environment, with boundary conditions and loads

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Plots of representative equilibrium paths for variants 2 and 3 (Figs. 16, 17) are characterized with significantly better conformity with experimental characteristics then in the case of variant 1. In the pre-buckling range, the consistence is almost perfect. On the other hand, for maximum loads, the error of representation of the total angle of torsion does not exceed 10%. It allows to conclude that the adopted modeling method and selection of numerical procedures

**Figure 13.** Geometrical models of variant 2 (left) and variant 3 (right) of the examined structure developed in MSC

turned out to be satisfactory.

150 Computational and Numerical Simulations

**Figure 12.** Comparison of representative equilibrium paths — variant 1

Patran environment, with boundary conditions and loads

**Figure 14.** Distribution of the displacement (left) and the reduced stress according to Huber-Mises hypothesis (right) for 100% of the maximum load — variant 2

**Figure 15.** Distribution of the displacement (left) and the reduced stress according to Huber-Mises hypothesis (right) for 100% of the maximum load — variant 3

**Figure 16.** Comparison of representative equilibrium paths — variant 2

izing skin segments on one hand and location of folds and the related deformations on the other which in turn determine magnitude of the structure's total angle of torsion. Realization of such research program would require application of the above-described experimental procedure to consecutive versions of the model with fixed curvature radius and different cylinder lengths, and then to another series of models with a fixed length and different diameters. This would allow to determine characteristic combinations of geometrical param‐ eters which are connected to fundamental changes in post-buckling deformation patterns

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As was already noted earlier, results of experiments allow to conclude that, in general, the structure rigidity increases with increasing number of components of the structure framing. It should be however borne in mind that in the case of aircraft structures, there is an absolute necessity to strive after minimization of the mass which limits the possibility to increase the number of frames and stringers. It seems therefore to be possible to determine a limiting number of framing components above which further increase of the weight is no more justified

With a sufficiently broad range of test results being available, it would be possible to use them as a base of standards for verification of results of nonlinear numerical analyses, as the nature of post-buckling deformations, with geometrical proportions and rigidity relationships between elements of the structure maintained, is not subject to any major changes when other isotropic materials are used or other load values are applied. This was confirmed by numerical

The main conclusion following from the presented numerical calculation cases is the necessity to strive to reduce the size of the task and avoid any numerical singularities which, in the case of nonlinear analysis, may result from using different finite element types in the model.

Despite difficulties related frequently to carrying out nonlinear numerical analyses of FEM models of thin-walled structures subjected to advanced deformation states, commercial FEM programs represent a tool allowing for effective determination of stress distributions in such states. However, one should always bear in mind the absolute necessity to verify results obtained this way, either by making use of the above-discussed base of standard solutions, or

\* Faculty of Mechanical Engineering and Aeronautics, Rzeszów University of Technology,

resulting in increase of torsional rigidity of the structure.

tests performed with the use of models presented here.

by performing an appropriate experiment.

Address all correspondence to: t\_kopecki@poczta.wp.pl

**Author details**

Tomasz Kopecki\*

Rzeszów, Poland

by measurable improvement of strength and rigidity of the structure.

**Figure 17.** Comparison of representative equilibrium paths — variant 3

### **5. Summary and conclusions**

As it was emphasized in the introduction, the research results presented in this study represent a fragment of the cycle of experiments that should be executed in order to test the whole of the physical phenomena involved in the loss of stability of the examined structure. It can be stated on the grounds of the executed experiments that construction solutions of structures of the type analyzed in this study that comprise too small quantity of framing components, are characterized with deformations far too large to be used in actual aircraft constructions. Considering the three presented variants of the thin-walled cylindrical structure, it seems that the last of them could be used in practical applications.

The fundamental observation that can be made on the grounds of relatively small number of the cases examined here is an increase of torsional rigidity of the structure with increasing number of components of the framing. The increase is caused partly by rigidity of stringers alone, however another reason consists in the change of relationship between the skin segment surface areas and their linear dimensions on one hand and the skin curvature radius on the other. Increasing the number of frames and stringers results in a decrease of average size of skin segments which, at fixed curvature radius value, is the cause of relative "flattening" of skin components. Reduction of the value following from the above-mentioned relationship limits, in a natural way, the depth of folds developed as a result of the loss of stability, and therefore also the scale of deformation. The deformation pattern, and thus also the number and relative position of the folds occurring in individual skin segments, depends also on the ratio of their linear dimensions which is decisive for the nature of the field of tensions devel‐ oping in the segments [10]. To be able to call a post-buckling deformation research program the completed task, it seems to be necessary to perform a series of experiments aimed at determination of detailed relationships between ratios of geometrical parameters character‐ izing skin segments on one hand and location of folds and the related deformations on the other which in turn determine magnitude of the structure's total angle of torsion. Realization of such research program would require application of the above-described experimental procedure to consecutive versions of the model with fixed curvature radius and different cylinder lengths, and then to another series of models with a fixed length and different diameters. This would allow to determine characteristic combinations of geometrical param‐ eters which are connected to fundamental changes in post-buckling deformation patterns resulting in increase of torsional rigidity of the structure.

As was already noted earlier, results of experiments allow to conclude that, in general, the structure rigidity increases with increasing number of components of the structure framing. It should be however borne in mind that in the case of aircraft structures, there is an absolute necessity to strive after minimization of the mass which limits the possibility to increase the number of frames and stringers. It seems therefore to be possible to determine a limiting number of framing components above which further increase of the weight is no more justified by measurable improvement of strength and rigidity of the structure.

With a sufficiently broad range of test results being available, it would be possible to use them as a base of standards for verification of results of nonlinear numerical analyses, as the nature of post-buckling deformations, with geometrical proportions and rigidity relationships between elements of the structure maintained, is not subject to any major changes when other isotropic materials are used or other load values are applied. This was confirmed by numerical tests performed with the use of models presented here.

The main conclusion following from the presented numerical calculation cases is the necessity to strive to reduce the size of the task and avoid any numerical singularities which, in the case of nonlinear analysis, may result from using different finite element types in the model.

Despite difficulties related frequently to carrying out nonlinear numerical analyses of FEM models of thin-walled structures subjected to advanced deformation states, commercial FEM programs represent a tool allowing for effective determination of stress distributions in such states. However, one should always bear in mind the absolute necessity to verify results obtained this way, either by making use of the above-discussed base of standard solutions, or by performing an appropriate experiment.
