**1. Introduction**

The design work on aircraft constructions, with applicable standards and requirements imposed by regulations applicable to aircraft design taken into account, represents a discipline significantly different than other fields of modern engineering. In fact, as opposed to rules commonly applicable to design of technical structures, in view of the need to limit the mass, in the case of airframe structures there is a necessity to allow phenomena involving loss of stability with respect to some of their components under the in-flight conditions.

From the historical point of view, the issue of the loss of stability was a factor significantly slowing down the progress in aviation at early stage of its development. In aspiration to ensure safety, a large group of designers adhered to the lattice structure concepts for many years. The first attempts to develop some more advanced solutions were based on the use of corrugated sheetmetalas theskinmaterialforwingsandfuselages.Suchsolutionwasadoptedinnumerous constructions manufactured on a mass scale, e.g. Ford Trimotor or Junkers 52 (Fig. 1).

With increasing availability of more and more reliable and powerful aircraft engines and the related improvement of aircraft performance parameters, it became necessary to use smooth skin materials constituting integral components of semi-monocoque and monocoque struc‐ tures. On the other hand, in striving after development of optimum constructions with the mass criterion met at the same time, it became impossible to use the sheet metal with thickness allowing to achieve critical loads with values exceeding the allowable loads.

Similarly as for other types of constructions, the principal rule involved preventing bar systems, such as stringers, frame components, or spar flanges, from buckling. In the case of the loss of stability, such elements of the structure were considered damaged.

© 2014 Kopecki; licensee InTech. This is a paper 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.

**Figure 1.** Airplanes with the skin made of corrugated sheet metal: Ford Trimotor (left) and Junkers Ju-52 (right)

At the same time it has been found that the skin stability loss is not dangerous for the whole of the structure provided its character is local, i.e. it occurs within the area of skin segments limited by components of the skeleton, such as frames and stringers, and represents an elastic phenomenon.

In such situation, it has become a standard that some local buckling of airframe skin is admitted in the in-flight conditions. In the current state of the art, the rule applies mainly to isotropic materials, e.g. metals. It should be emphasized that in case when the post-buckling deforma‐ tion field encompasses also components of the framing, such as bars constituting frame components or stringers, the skin buckling is considered global, and the structure is assumed to be destroyed. It can be therefore concluded that a loss of stability is of a local type when it encompasses skin segments limited with components of the framing.

nonlinear analysis still causes numerous problems and requires application of additional tools

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The essence of FEM-based nonlinear numerical analysis comes down to determination of a relationship between a set of parameters determining the state of the structure, known as state parameters, and a set of control parameters related to the load. The latter can be, in general, related to and expressed by means of a single control parameter. On the other hand, the number of state parameters corresponds to the total number of degrees of freedom of the analyzed system. Such relationship is known as the equilibrium path and for a system with an arbitrary number of degrees of freedom can be interpreted as a hypersurface in the state hyperspace where the dimension of the hyperspace corresponds to the number of degrees of freedom [1-4]. The equilibrium path fulfils the matrix equation of residual forces which, for a single control

> **ru 0** ( , , l

where **u** is the state vector containing components of displacements of nodes of the structure corresponding to its current geometrical configuration, *λ* is the control parameter representing a function of the load, and **r** is the residual vector containing non-balanced force components

Procedures employed in modern numerical software packages, apart from the prognostic phase allowing to determine the next point on the equilibrium path, comprise also a correction phase offering the possibility to compensate divergences between the actual equilibrium path

) = (1)

to verify the results.

parameter, has the following form:

**Figure 2.** Damaged Aloha Airlines' Boeing 737

related to the current system deformation state.

Further experience collected in operation of airframes constructed with the use of the abovementioned standard revealed another issue connected with limited operating durability of such structures. In fact, cyclic nature of loads which a plane is subjected to in the course of flight induces occurrence of fatigue phenomena which, when undiagnosed or underestimated, may lead to destruction of the structure. Examples include such aviation accidents as e.g. disasters of De Havilland Comet airplanes in the years 1953–54 or the accident occurring in the course of flight of Aloha Airlines' Boeing 737 when a fatigue-induced gap developed in the skin resulted in explosive decompression of the plane's fuselage and breaking off a large fragment of the fuselage skin (Fig. 2).

Nature and intensity of fatigue-induced changes in a structure is related to the stress distri‐ bution which, assuming admissibility of skin stability loss, means that it is necessary to carry out detailed analyses of post-buckling deformation states.

The tool that become used commonly for this purpose is the nonlinear numerical analysis based on the finite element method (FEM), allowing to represent actual deformations of thinwalled structures and the related stress distributions. However, in so far as application of FEM in linear problems became a routine in the engineering practice, and results being obtained with the use of commercial software packages are, on the whole, correct and reliable, the Numerical Simulations of Post-Critical Behaviour of Thin-Walled Load-Bearing Structures Applied in Aviation http://dx.doi.org/10.5772/57218 141

**Figure 2.** Damaged Aloha Airlines' Boeing 737

At the same time it has been found that the skin stability loss is not dangerous for the whole of the structure provided its character is local, i.e. it occurs within the area of skin segments limited by components of the skeleton, such as frames and stringers, and represents an elastic

**Figure 1.** Airplanes with the skin made of corrugated sheet metal: Ford Trimotor (left) and Junkers Ju-52 (right)

In such situation, it has become a standard that some local buckling of airframe skin is admitted in the in-flight conditions. In the current state of the art, the rule applies mainly to isotropic materials, e.g. metals. It should be emphasized that in case when the post-buckling deforma‐ tion field encompasses also components of the framing, such as bars constituting frame components or stringers, the skin buckling is considered global, and the structure is assumed to be destroyed. It can be therefore concluded that a loss of stability is of a local type when it

Further experience collected in operation of airframes constructed with the use of the abovementioned standard revealed another issue connected with limited operating durability of such structures. In fact, cyclic nature of loads which a plane is subjected to in the course of flight induces occurrence of fatigue phenomena which, when undiagnosed or underestimated, may lead to destruction of the structure. Examples include such aviation accidents as e.g. disasters of De Havilland Comet airplanes in the years 1953–54 or the accident occurring in the course of flight of Aloha Airlines' Boeing 737 when a fatigue-induced gap developed in the skin resulted in explosive decompression of the plane's fuselage and breaking off a large

Nature and intensity of fatigue-induced changes in a structure is related to the stress distri‐ bution which, assuming admissibility of skin stability loss, means that it is necessary to carry

The tool that become used commonly for this purpose is the nonlinear numerical analysis based on the finite element method (FEM), allowing to represent actual deformations of thinwalled structures and the related stress distributions. However, in so far as application of FEM in linear problems became a routine in the engineering practice, and results being obtained with the use of commercial software packages are, on the whole, correct and reliable, the

encompasses skin segments limited with components of the framing.

fragment of the fuselage skin (Fig. 2).

out detailed analyses of post-buckling deformation states.

phenomenon.

140 Computational and Numerical Simulations

nonlinear analysis still causes numerous problems and requires application of additional tools to verify the results.

The essence of FEM-based nonlinear numerical analysis comes down to determination of a relationship between a set of parameters determining the state of the structure, known as state parameters, and a set of control parameters related to the load. The latter can be, in general, related to and expressed by means of a single control parameter. On the other hand, the number of state parameters corresponds to the total number of degrees of freedom of the analyzed system. Such relationship is known as the equilibrium path and for a system with an arbitrary number of degrees of freedom can be interpreted as a hypersurface in the state hyperspace where the dimension of the hyperspace corresponds to the number of degrees of freedom [1-4]. The equilibrium path fulfils the matrix equation of residual forces which, for a single control parameter, has the following form:

$$\mathbf{r}\left(\mathbf{u}, \mathbb{X}\right) \; = \mathbf{0},\tag{1}$$

where **u** is the state vector containing components of displacements of nodes of the structure corresponding to its current geometrical configuration, *λ* is the control parameter representing a function of the load, and **r** is the residual vector containing non-balanced force components related to the current system deformation state.

Procedures employed in modern numerical software packages, apart from the prognostic phase allowing to determine the next point on the equilibrium path, comprise also a correction phase offering the possibility to compensate divergences between the actual equilibrium path 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 as the increment control equation or the equation of constraints,

$$c\left(\Delta \mathbf{u}\_{n'} \,\,\Delta \mathbb{A}\_n\right) = \,\, 0,\tag{2}$$

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].

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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

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

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).

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

out additional experiments.

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

package.

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 state *n* + 1.

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

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 of other factors.

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 relatively expensive and frequently difficult to execute.

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