A CFRP Passenger Floor Stanchion Underwent Dynamic Buckling Structural Testing

The work focuses on the study of the structural behaviour of a composite �oor beam in the cargo area of a commercial aircraft subjected to static and dynamic loads (typical of hard or crash landing). Experimental tests have been performed in the laboratories of the Dept. of Industrial Engineering (UniNA) jointly with the development of numerical models suitable to correctly simulate the phenomenon through the LS-DYNA software. The de�nition of a robust numerical model allowed to evaluate the possibility of buckling triggering. The test article was equipped with potting supports on both ends of the tested beam, �lling the pots with epoxy resin toughened with glass �ber nanoparticles. This allowed to uniformly load the beam ends in compression and to carry out the tests loading the specimen statically and dynamically, to observe the differences in the behaviour of the beam under two different types of applied load. The result obtained through the comparison between the numerical model and the experimental test was that the dynamic buckling was triggered by a quantitatively smaller load than in the static case. Furthermore, it has been observed that the experimental compressive displacement to trigger the dynamic buckling instability was greater than the displacement observed in the static case.


Introduction
Buckling is an instability phenomenon that is common in "thin" structures (those with at least one relatively tiny dimension in comparison to the others).Buckling was formerly thought to be a totally static occurrence.The classic example is the Euler beam, in which a beam clamped at one end is axially compressed at the free end.
Depending on the applied load, the beam can return to the initial equilibrium con guration (stable equilibrium, below the critical compressive load), move to a new equilibrium condition different from the initial one but with certain constraints (indifferent equilibrium, at the critical compressive load), or move away from the initial equilibrium con guration inde nitely (unstable equilibrium, above the critical compressive load).The buckling load is the smallest load for which equilibrium is indifferent.
Buckling can, however, be produced by time dependent loads.Many writers [1][2][3] have explored the application of a time-dependent axial stress to a beam, which causes lateral vibrations and can eventually lead to instability.Dynamic buckling is a relatively new phenomenon.One of the rst researcherinvestigating dynamic buckling is Zizicas [4], who proposed a theoretical solution for the solution of a simply supported rectangular plate exposed to variable oor loads over time.A criterion that connected dynamic buckling to load duration was developed by Budiansky [5], Lindberg and Florence [6] where the effect of a high intensity, short duration load was investigated.
In [7] results of dynamic buckling due to impulsive loads applied to thin-walled carbon-ber reinforced plastic shell structures subjected to compressive axial loads are reported.The approach taken was based on the equations of motion, numerically solved using an explicit nite element code.It was shown that also geometric imperfections as well as the duration of the loading period had a great in uence on the dynamic buckling of the shells.
In [8][9][10] the relationship between dynamic and static buckling load has been pointed out, depending on several factors such as the duration of impact (i.e., duration of load application), panel geometry ( at or curved panel) and also the presence of initial imperfections in the geometry.Particularly for short duration loads, the dynamic buckling load increases relative to the equivalent static buckling load.With increasing load duration, on the other hand, the dynamic critical load becomes much lower than the static one.Regarding the geometry, the dynamic buckling load increases relative to the static one as the curvature increases.
Other parameters in uencing this phenomenon are the boundary conditions and the strain rate effects, investigated in [11] and [12], respectively.
In recent years, the dynamic buckling analysis has been extended also to stiffened panels.In [13] the dynamic buckling of simply supported stiffened panels under in-plane impact load was studied using the theory of large de ection.The in uence of imperfection on dynamic responses of stiffened panels under in-plane impact was investigated in [14], nding the dynamic buckling to be sensitive to local imperfection, but not to overall ones.Also, in the case of stiffened panels, the boundary conditions play a fundamental role, as expected from the previous results.In detail, the effect of the constrained edge rotation on dynamic buckling of stiffened panels under uniaxial in-plane impact is investigated in [15].Then, Yang et al. [16] showed that the in uence of the rotation constraint around the unloaded edges on dynamic buckling was more signi cant than that around the loaded edges.
The de nition of dynamic buckling is arbitrary, and consequently there is still no unambiguous criterion for its determination, nor are there guidelines for the design of structures resistant to it.Budiansky and Hutchinson proposed a criterion that leads to a rational de nition of dynamic instability [17].A widely accepted de nition of dynamic instability of imperfect structures states that instability occurs when a small increase in load intensity results in unlimited growth of de ections [18].
In this study, the structural behaviour of a composite oor beam subjected to low-frequency cyclic load conditions has been explored.Three distinct loads-below, equal, and over the static critical buckling load -have been considered to determine the structural response.
The aim is to conduct an experimental numerical investigation of the dynamic buckling phenomenology on a composite material beam.It was important to modify the test object to make sure the experimental test was carried out correctly.In particular, both the plate ends were equipped with pottings, made out of steel pots lled with epoxy resin toughened by glass ber nanoparticles to guarantee that the compressive force was applied symmetrically and uniformly over the entire beam cross-section.The resin was cured in the pots, at 60°C temperature and 10% humidity.
The department laboratory's test equipment was used to conduct the experimental tests.LS-DYNA software was used to mathematically recreate the treated phenomena.The Matlab working environment was used to process the results.

Case study description
The structure examined during this study was a beam that is part of the lower section of a fuselage trunk of a commercial aircraft, as showed in Fig. 1.
For the sake of completeness, the total length of the beam was 380 mm, also accounting for the connection areas with the frame and the oor truss of the considered fuselage section.The beam has been cut orthogonally with respect to the loading direction, in order to consider only its free length, the one in which the dynamic buckling phenomena can happen.
Based on that, the test article has a length of 315 mm and a cross-sectional area of 343.6 mm 2 .Such a length of the beam also accounts of the terminal areas, covered by two rectangular pottings with a length of 25.40 mm.The pottings were needed to adapt the testing machine to the actual application, as previously described.
The test article and the numerical model, discretized in the Finite Element (FE) environment, are reported in Fig. 2.
The mechanical properties reported in Table 1, referred to the single lamina of the composite material adopted for the beam manufacturing.Concerning the numerical model, to nd the optimal balance between computational costs and the correctness of the ndings in terms of expected stiffness, a preliminary mesh convergence study has been performed.As a result, several static linear studies with varied in-plane con gurations and throughthe-thickness element sizes have been carried out.Particularly, three distinct mesh element sizes-coarser (8 mm), moderate (4 mm), and ner (2 mm) were taken into consideration and three through-thethickness con gurations were accounted for.Thus, nine distinct model in total have been examined.
Concerning the in-plane con gurations, the stanchion model was realized by considering shell elements in the middle plane of the cross-sectional area, then extruding them along the longitudinal direction.The discriminating parameter between the three adopted con gurations is the number of integration points, which was set to be 16 for the rst con guration.The second and the third con gurations have been obtained by considering respectively 4 and 2 integration points in through-the-thickness direction of shell elements.
Noteworthy is that in the rst con guration, given the number of integration points, it has been possible to reproduce exactly the stanchion material stacking sequence.On the other hand, for the second con guration, the cross-sectional area was ideally divided into 3 areas, and to each of them the properties of the equivalent laminate thus obtained were given.Following the same approach, also the third con guration has been realized, in this case the whole cross-sectional area is considered.
The adopted material card was the *MAT_LAMINATED_COMPOSITE_FABRIC, ful lled with the parameters reported in Table 1.This is clearly valid for the rst con guration.For the second one, given the assumed division in 3 areas, such parameters were valid only for the middle area, since comprehending all the plies 0-degrees oriented of the stacking sequence (as a matter of fact the only parameter changing is the thickness).For the other two areas the material properties for the equivalent sub-laminate were theoretically computed.The same approach has been adopted for the third con guration; in that case the whole equivalent laminate is considered.The material properties for the equivalent sub-laminate and laminate are reported respectively in Tables 2 and 3.The best ratio between results effectiveness and computational results was provided by the model characterized by a moderate size mesh, with 16 integration points for the shell elements.
The model boundary conditions have been set compatibly with the experimental test conditions.The beam has one base xed (the top surface), while the other is free to move along the longitudinal axis.Constraints must then be added to simulate the mechanical behaviour of potting.Speci cally, potting was introduced to impose compression-only stresses at the end of the beam to prevent it from buckling in these areas.Constraints were, therefore, applied to lock the degrees of freedom related to the three rotations and in-plane translations.Concerning the loading conditions, it is realized by a displacement curve imposed on the rigid frame, modeled through solid elements with the same mesh size as the beam.This choice is justi ed by the aim of obtaining a stable contact between the frame itself and the bottom surface of the beam, through which the load is transferred.

Results
The numerical model has been veri ed by comparison with the stiffness and failure ndings of an experimental test program, the compressive experimental test up to ultimate failure and reported in the last work [19] underlined the results: According to these experimental results, the failure load was 103 KN, and up to the failure load, there were no buckling events, both experimentally and numerically.
The Table 4 reports the failure displacements and loads, there is good agreement between the solutions of both formulations.Then, a compressive experimental test designed to measure the structure's stiffness was repeated without taking the failure into account to con rm the linear deformation on the 315 mm long stanchion made of composite material that combines carbon bres with a highly toughened epoxy matrix and a 15ply lamination sequence.
Figure 4 shows the experimental test set-up, resembling what gured out previously.The department laboratory's test equipment was used to conduct the experimental tests.In particular it is characterized by two rigid steel blocks, the top and the bottom ones, both connected to hydraulic pistons.The top plate is locked after the sample positioning, while the bottom one is free to move along the longitudinal direction to apply the desired load, compatibly with the assigned displacement curve.
Figure 5 shows the good agreement of the stiffness and failure load between the numerically predicted solutions and the outcomes of the experimental campaign.Additionally, the Hashin's failure criteria were used to compute the test article's failure mode, which was then predicted with a high degree of accuracy.
Once the numerical model was validated by static test, the dynamic experimental test is executed applying a load increase and constant speed of 100 mm/s.The explicit numerical investigations were carried out to investigate how the dynamic buckling phenomenon would have occurred, as reported in the paper [19], where it was evaluated the variation of stiffness and displacements out of the plane, and it was seen that numerically the instability is due to dynamic load where a decrease in stiffness was evaluated as a warning of approaching buckling.Every explicit analysis has been run with a structural dampening of 2%.
In Fig. 6 the force versus time is plotted; from the analysis of the following graphs, the phenomenon of dynamic buckling instability can be observed, which occurs after 0.18 ms.Close to this time value, a buckling load equal to 89 kN and a displacement value equal to 0.063 mm are noted.
As opposed to the static gure, which shows that the critical load has a displacement of 0.03 mm and is about 105.6 kN.The structure collapses completely at values greater than this load.

Conclusions
The work focuses on the study of the structural behaviour of a composite oor beam in the cargo area of a commercial aircraft subjected to static and dynamic loads (typical of hard or crash landing).The research state of the art on the relevant topic of dynamic buckling is provided.The performed activity is fully detailed, both from experimental and numerical point of view.In detail, once the numerical model has been validated statically the dynamic bucking phenomena is numerically investigated.It is clear from an examination of the ndings that for composite material structures, the buckling load in a dynamic compression test assumes a lower value than the buckling load in a quasi-static compression test.
Additionally, in a dynamic compression test, a larger displacement respect to the static case is required to cause the buckling phenomena.Finally, it can be shown that during the dynamic compression test, after buckling occurred, the structure does not collapse, but rather the beam may continue to operate in post buckling.
The Fig. 7 reports the differences between the numerical simulation about the post buckling static and dynamic cases, when the dynamic buckling load value is exceeded.The post-buckling displacements are transmitted by the lower plate throughout the beam, in the quasi-static compression test, the structure collapses and the displacements are concentrated near the load area.In the case of the dynamic compression test, it is observed that the curve force vs time exhibits a plateau close to the buckling load, so a different transmission of stress is observed in the longitudinal element; in fact, a greater quantity of surface is called upon to collaborate, making it a more e cient.This is because some of the mechanical energy resulting from the application of the load is lost due to the deformation that took place after the beam protruded from its axis.

Declarations Figures
Page 11/ Time history of the force in the experimental test.

Figure 1 Section
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Figure 2 Left
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