1. Introduction
Buckling is an instability phenomenon typical of “thin” structures (characterized by at least one very small dimension compared to the others). Usually, buckling has been considered a purely static phenomenon. The classic example is that of the Euler beam in which a beam stuck at one end is loaded from the tip to the other, with a compressive load lower than the limit of elasticity of the material. Theoretically, if the force is perfectly centered and the beam is free of imperfections, the latter would remain in equilibrium under the action of any load. In order for the instability to occur, it is mandatory to destabilize the beam by means of an external action, immediately removed. After the perturbation, three cases can occur, depending on the applied load: The beam returns to the initial equilibrium configuration (stable equilibrium); the beam moves to a new equilibrium condition different from the initial one (indifferent equilibrium); the beam moves away indefinitely from the initial equilibrium configuration (unstable equilibrium). The so-called buckling load is the lowest of the loads for which equilibrium is indifferent. However, buckling can also be caused by loads that vary over time [
1,
2,
3]. The application of a time-dependent axial load to a beam, which then induces lateral vibrations and can eventually lead to instability, is something that has been investigated by many authors [
4,
5,
6,
7,
8,
9,
10].
Generally, dynamic buckling depends on prescribed dynamic loads, and is distinguished between buckling from oscillatory loads and buckling from transient loads, so two forms of dynamic buckling are classified: Vibration and pulse buckling.
Vibration buckling is characterized by oscillating loads producing amplification of the vibrations up to unacceptable values, and, in the absence of damping or nonlinear effects, a phenomenon similar to vibration resonance is predicted. In the vibration buckling, the force producing the instability is considered as a parameter multiplying the displacement, while in the vibration resonance, the load exciting the motion is evaluated as an applied force.
In pulse buckling, the presence of applied transient loads generates much larger and unacceptable amplitudes than the static buckling load of the bar as a result of the plastic response or large-deformation post-buckled state. The pulse buckling is characterized by buckling increasing rather than oscillating, and the modes produce wavelengths larger than the Euler wavelength for a given load.
The dynamic buckling of the column in an aircraft landing stanchion is a great deal and, in this work, the structural behavior of a composite floor beam subjected to low-frequency cyclic load conditions has been investigated. Indeed, the structural limits of beam-like structures is driven by their stability, rather than by their strength. The loss of stability induced by static loads is a well-known phenomenon, extensively investigated [
11]. However, the loads that actually act on the structures are characterized by dynamic behavior, which can result in dynamic instability [
12,
13,
14,
15]. For these reasons, more detailed studies on the dynamic buckling phenomenon became mandatory in order to accurately predict the structural limits of the components subjected to realistic loads [
16].
Dynamic buckling has a relatively recent history. One of the first research studies on dynamic buckling can be found in [
17], where a theoretical solution considering a simply supported rectangular plate subjected to varying floor loads over time has been developed. In [
18], a criterion that relates dynamic buckling to the duration of the load has been introduced. The effects of high-intensity and short-duration loads have been studied in [
19]. According to the investigation, long-lasting critical dynamic buckling loads may be of less intensity than the corresponding static buckling loads.
In [
20], a numerical and experimental investigation was performed on composite laminated shells by correlating their natural vibration frequencies and mode shapes with the buckling loads and modes. Moreover, parametric studies on the dynamic instability of carbon-fiber-reinforced plastic (CFRP) cylindrical shells were introduced in [
21]. Different parameters were considered, such as the shape and the duration of pulse loading, and the initial geometric imperfection of the structure. According to the study, the dynamic load factor (DLF), defined as the ratio between the dynamic and the static buckling loads, is higher than that if short-pulse-duration loadings are used. Then, as the pulse duration increases, the DLF decreases, being lower than that in the vicinity of the natural frequency of the structure.
Other works have focused on the investigation of dynamic buckling on damaged composite structures. In [
22], the nonlinear dynamic pulse buckling of a plate with embedded delamination was numerically investigated, and the DLF was computed. Different types of pulse loadings (sinusoidal, exponential, and rectangular) and different delamination sizes and depths were considered. Moreover, in [
23], the dynamic instability of damaged composite components was studied, and the effects of the damage and its position were assessed. The damage was numerically simulated by reducing the stiffness of the components.
Different studies on the dynamic buckling induced by impact loading can be found in the literature [
24,
25,
26]. In [
27], the dynamic buckling of an elastic cylindrical shell subjected to axial impact was presented, and the influence of the boundary condition on the critical velocity of the impactor was assessed. In [
28], the dynamic pulse buckling of composite laminated beams subjected to impact in the axial direction was investigated. In particular, the influence of different beams parameters (transverse and axial inertias, shear deformation, cross-section’s rotational inertia, and axial shortening) on the beams’ pulse buckling response was assessed.
Interesting insight on the dynamic instability can be found in [
29]. A very detailed study was presented by the authors, which categorized the phenomenon according to the nature of the external load, whose direction can vary depending on the deformation of the structure, classifying the dynamic buckling problems as flutter or pulse buckling.
In this work, detailed numerical analyses have been presented, aimed to investigate the dynamic buckling response of the composite stanchion, in the finite element code Abaqus environment (release 2019, Dassault Systèmes Simulia Corp: Providence, Rhode Island, RI, SUA). Preliminary finite element models have been presented and compared, considering different in-plane and through-the-thickness element dimensions, different element types, and different material formulations, in order to find the model that more faithfully reproduces the investigated phenomenon. The latter has been used for the subsequent numerical investigation: The static stability limit of the composite stanchion (identified by its static critical buckling load) has been found; then, three different low-frequency cyclic load conditions have been considered (below, close, and above the static critical buckling load). In
Section 2, the test cases are described, while in
Section 3, the numerical analyses are introduced, including preliminary analyses aimed to determine the most suitable numerical model able to describe the dynamic phenomenon, and the results are critically discussed.
2. Test Case Description
The investigated test case, which has been previously validated in [
30,
31,
32], is representative of the stanchion of the cargo floor of a regional aircraft of general aviation [
33]. Images of the stanchions and their location in a typical fuselage section are reported in
Figure 1.
Two different test articles, with different geometrical characteristics, have been considered. In particular, Test Article 1 is 380 mm long, while the length of Test Article 2 is 315 mm. More details on the test cases’ geometry and boundary conditions can be found in
Figure 2. According to
Figure 2, both sides of the specimens have been potted. A compressive displacement has been applied on one side, while the opposite side has been fixed.
The stanchion material is a composite made of carbon fibers and epoxy matrix at high polymerization temperature. The mechanical properties of the composite lamina are reported in
Table 1, where ρ is the nominal density, th is the lamina thickness, E is Young’s modulus, G is the shear modulus, υ is Poisson’s ratio, X
t and X
c are respectively the longitudinal tensile and compressive strengths, Y
t and Y
c are respectively the transversal tensile and compressive strengths, and S
c is the shear strength. The stacking sequence of the laminate is [45; −45; 90; −45; 45; 0; 0; 0; 0; 0; 45; −45; 90; −45; 45].
4. Conclusions
In this work, the structural behavior of a composite floor beam subjected to cyclical low-frequency compressive loads has been presented. Sensitivity analyses have been carried out in order to find the most suitable numerical model able to simulate the dynamic analyses, considering different mesh discretization, element, and section formulations. The numerical model has been validated with respect to experimental data obtained under quasi-static loading conditions, thus adopting the implicit formulation. In order to investigate the influence of the solution scheme on the numerical results, an explicit–implicit comparison has been performed. The results highlighted that the adopted technique (smoothed step) allows us to minimize the inertial effects and, thus, to have a good agreement between the two solver schemes. Therefore, the explicit model can also be considered indirectly validated. Further, considering that the applied loading velocity, in dynamic cyclic loading conditions, is very small (lower than 0.1 m/s), the previous consideration can be extended to the final model.
Explicit non-linear analyses have been performed, applying dynamic 10 Hz frequency compressive loads. Load cycles with different amplitudes have been considered: 1 mm (about 70% of the static buckling value), 1.53 mm (close to the static buckling value), and finally 2 mm (above the static buckling value).
The results showed that low-frequency cyclic loads (compared to the test article’s own frequencies, that are in the order of hundreds Hz) with peak values lower than the static buckling load value are not capable of triggering significant instability. This behavior is confirmed by comparing the stiffness of the models as a function of time for the different load levels applied at the same 10 Hz frequency.
In particular, all reported results highlight that the applied load velocity (obtained from the load frequency) is small enough to not generate any resonance issue; therefore, the deformation elastic wave is able to cover the entire specimen length in a time smaller than that related to a single load cycle. Thus, considering the applied load frequency, no significant differences were observed with respect to the static case. All of this will be confirmed by future experimental activities in which the frequency-damping dependency will also be investigated (also beyond the applied frequency). This parameter, assumed constant in this work, could generate a slowdown in the propagation of the elastic wave and, thus, anticipate resonance effects.