COLLAPSE BEHAVIOR OF DYNAMICALLY CRASHED HEXAGONAL THIN WALLED STRUCTURES WITH MULTI-CELL CROSS-SECTION GEOMETRY

In previous studies, the cross-section geometries were either single cell (circular, prismatic) or multi-cell structures with identical cell properties. In the present research the collapse characterization of multi-cell structures consisting of three different types of cell formation defined by either two wide or two narrow angles separating the ladders is conducted. The shape characteristics of interest sum up the layout of the interior walls and their constraints over the outer structure walls. To study the control of cross-section geometry over the crashing mechanism, local or global progressive buckling response, energy absorption and crash load for the structures in discussion FEA simulations of the impact were performed.


INTRODUCTION
The progressive buckling of metallic thin-walled structures has been known as one of the best impact absorbing mechanism [1].Today, elements based upon the progressive bucking response are used in high volume production systems such as vehicle or train manufacturing industry and other fields.The main purpose of these elements is to absorb the energy released during a crash instance in a controlled manor [2].Research of crashing energy absorption is mandatory in the field of safety design regarding passenger transportation.Shell structures are probably the most utilized products in many engineering applications because of their crashworthiness compared to material volume [1].Various types of structures are utilized in the construction of controlled body crashing zones.These elements have different cross-sections like circular, elliptical, square, top-hat, double-hat, polygonal shapes.
The top-hat and double-hat structures are mainly studied due to their facile possibility of application in car body construction.Regardless the cross-section shape, one of the most crashworthy parameter is the total energy that a given structure can ultimately absorb [3][4][5].To absorb and dissipate the crash energy effectively, a tubular structure is required to point out a vast stress range.So, by increasing the range of the unavoidable strain the improvement of the dissipated energy can be performed.The upmost engineering field of application regarding the crashworthy structures is the crash protection [6].
Nowadays, one important issue in car manufacturing is the increased desirability in using new materials.It is possible that well known materials like deep-drawing steels might be discarded in favor of UHSLA steels, aluminum or magnesium alloys and various grades of polymeric materials or composites.These changes are undertaken for reasons like: the total weight reduction to allow for more safety components, the strengthening of the structures and cost reductions [7,8].
Improvement of nonlinear FEA codes such as LS_Dyna V971 made possible the analyses of the crash phenomenon as a difficult nonlinear matter using the explicit time integration technique.This code is purposely designed and applicable for the transitory dynamic analysis regarding highly nonlinear problems such as impact situations [9].
One possible solution of controlling the crash area and deformation of tubular structures is by designing multicell cross section thin-walled structures.Product manufacturing using extrusion processes provide facile production of various prismatic multi-cell components.Previous studies pointed out the fact that multi-cell structures with an increased amount of corners could increase the capacity of energy absorption regarding thinwalled structures [10].
The impact conditions were carefully reproducing the real impact tests which were examined in previous studies.So, the moving mass was described as a rigid body which presents only one degree of freedom along the z axis (translation) with a specific velocity and a certain weight.
In previous studies, the cross-section geometries were either single cell (circular, prismatic) or multi-cell structures with identical cell properties.In the present research the collapse characterization of multi-cell structures consisting of three different types of cell formation defined by either two wide or two narrow angles separating the ladders is conducted.The shape characteristics of interest sum up the layout of the interior walls and their constraints over the outer structure walls.Furthermore, the FEA simulations of the impact were performed in order to research the control of cross-section geometry over the crashing mechanism, local or global progressive buckling response, energy absorption and crash load for the structures in discussion.

Structural components
There were used four types of cross-section geometries as presented in Figure 1.The side of the enclosed cross sections was 46.67 mm for all types of structures.The reason for having these dimensions of the sides is for having 280 mm for both crash box length and cross section perimeter respectively.One structure, named in this research, model 1 has no ribs of any kind.However, all other three structures, second, third and fourth model respectively present different types of internal ribs.Detailed description of the structures cross-sections is illustrated in Figure 1.

Materials characterization
The material attributed for these thin-walled structures is a FeP-05-MB steel grade used in car body manufacturing.The sheet metal thickness for this research was 0.82 mm.In order to find the mechanical characteristics and response, tensile tests were conducted.The most important data is presented in Table 1.1)) the forming limit curve (Figure 2) of the studied material was obtained.In the forming limit diagram, the material presents lower formability characteristics on the biaxial strains (right side of the chart).This behaviour is strongly influenced by the low yield strength value of the material.

= 1+ +
(1 where: is the strain rate, C and P are the Cowper-Symonds parameters regarding the strain rate, K is the resistance coefficient, e represents the elastic deformation, p ef is the real (effective) plastic deformation.The strain hardening coefficient is represented by n [9].Fig. 2. Forming limit curve characteristic for FeP-05-MB steel used in this research.

Numerical setup
The numerical analysis was conducted through the explicit non-linear finite element platform LS-Dyna V971 R5.0.This programme was used in conjunction with the pre-and post-processor LSPrePost V4.0.The strain rate effect was accounted by using the Cowper and Symonds model that scales the flow stress with the factor: 1+ ( /C) 1/P .
All the structures from this study were modelled using Belytschko-Tsay shell elements having five points of integration through the thickness and one point of integration in the element plane.This element formulation gives greater computational efficiency compared with other existent shell element formulation [9].Equal characteristics of the material were attributed to each structure no matter the cross-section shape.*Mat_Piecewise_Linear_Plasticity material model was used to characterise the steel sheets for the thin-walled structures and rigid elements were used to characterise the crashing tool.The size of mesh is 2.5 mm for the steel sheets used in the thin-walled structures and 20 mm for the tools.
The velocity boundary conditions were applied on the top rigid plane in the axial direction while the bottom was kept into place.According to the real gravitational crash tests, the impact velocity was established to be equal to 6.5 m/s.This velocity is similar to the one of a 125 kg metal bulk dropped from a height of 2.2 m.Clamped boundary conditions were applied at the bottom of the structure.Furthermore, two types of contacts were prescribed for the simulation analysis.Between the column and the rigid plane an automatic node to surface contact was prescribed.For the contact between the lobes during deformation, an automatic single surface contact was adopted.

RESULTS AND DISCUSSION
In the case of the collapse mode the folds formation and shape can be clearly characterised in all four types of structures.Simulated impact collapse mode in different stages and the load/energy graphs in conjunction with displacement are all depicted in Figures 3-6.Analysing all the crashed structures, it can be stated that all structures formed an asymmetric progressive buckling mode.This type of collapse mode is also called the diamond mode [1].This behaviour is related with the fact that all the structures have corners and the width of the 0.0 0.2 0.4 0.6 0.8 -0.30 -0.20 -0.10 0.00 0.10 major strain minor strain side is large enough to generate a successive inward and outward fold while the corners of the structure are still resisting to the crashing load delaying the collapse.For all cross-sections shapes, the folds created due to impact deformation are in contact with each other, which determine by consequence the symmetrical inward and outward formation on opposite walls of these folds.The structure which shows the highest number of folds has the cross-section as model 3 and a number of 8 fully formed lobs.The structure with the lowest number of the folds has cross-section as model 1 and a number of 4 fully formed lobs.As the figures illustrate, model 1 and model 2 structures show unstable collapse mode toward the end of the crashing stage.The collapsing axis of these structures is slightly changing toward the end of the crash causing critical buckling.This is an effect which it must be avoided in the design of safety structures.
On the other hand, structures with the cross-section as third and fourth model respectively, show consistent diamond collapse mode on the entire crashing sequence.Also, the collapsing axis is kept collinear to the crashing load direction.In the presented diagrams, the absorbed energy curve is obtained by surface integration of the space below the load/displacement curve after the termination of the progressive buckling.After the first fold is created and the peak load appears, a load reversal cycle during progressive buckling follows.This phenomenon occurs until the end of the collapse sequence.
The greatest difference between the load peaks and the load bottoms is present at the structures with the crosssection as model 2, which indicates great inertial behaviour.Inertial behaviour is unwanted in safety structures [6].Opposite to this, the structures with the cross-section as model 3 present the smallest difference between the load peaks and load bottoms respectively, which represents consistent collapse behaviour.
Analysing the graphs above it was found that the mean crash load value increases as the number of folds increases.This happens due to the dynamic inertia of the material mostly determined by the resistance coefficient and the strain hardening index.Furthermore, an increasing trend is recorded between the number of folds and the number of facets.Thus, if the number of facets increases the number of folds increases also.As a consequence, the mean load could have a higher value if the structure presents a higher number of facets.Data supporting the statement above is presented extensively in

CONCLUSIONS
Analyzing the presented data it was found that if the structure has the cross-section as model 2, the crash load can increase up to 0.3 % and up to 118 % for the absorbed energy.If the structure has the cross-section as model 3, the crash load can increase up to 37 % and it can be reported an increase of the absorbed energy up to 186 %.
If the structure has the cross-section as model 4, the crash load can increase up to 33 % and it can be reported an increase of the absorbed energy up to 170 %.
So, it can be concluded that for steel hexagonal thin-walled structures with multi-cell cross-section geometry, the crashworthiness can be improved by connecting the internal corners of the structure to the external corners of a hexagonal cell paced inside the parent structure.Due to asymmetric progressive buckling the folding process is maintained consistent regardless the cell number or the position of the middle hexagonal cell of the structures in discussion.

Fig. 1 .
Fig. 1.Cross-section geometries of the thin-walled structures studied in the present research.

Fig. 3 .Fig. 4 .
Fig. 3. Cross-section of model 1, collapse mode depiction and the graph displaying the relation between the load/energy and displacement.

Fig. 5 .
Fig. 5. Cross-section of the third model, collapse mode depiction and the graph displaying the relation between the load/energy and displacement.

Fig. 6 .
Fig. 6.Cross-section of the fourth model, collapse mode depiction and the graph displaying the relation between the load/energy and displacement.

Table 1 .
Mechanical properties of the material.Formability characteristics of the sheet metal were obtained by Marciniak formability test.Using the data from the tests and the modified Hollomon's law of plasticity (equation (

Table 2 .
Table 2 displayed below.Correlative data regarding the crashing behaviour of thin-walled structures.