Mathematical Model Development on the Deformation Behaviour of Symmetric Hexagonal of Various Angles and Square Tubes under Lateral Loading

The purpose of this research is to develop a mathematical model of the collapse behaviour of symmetric hexagonal tubes. For that, a finite element analysis procedure was conducted using ABAQUS software to determine the lateral collapse behaviour of symmetric hexagonal of angles, 30°, 45° and 60° and square tubes to compare the results with the cylindrical tube. Then, a new predictive mathematical model of the lateral collapse behaviour for the generalized symmetrical geometric tubes is developed based on rigid, perfectly plastic model and the energy balance method. The newly mathematical model was validated with the simulation method results. It was discovered that symmetric hexagonal and square tubes performed different deformation behaviour than the cylindrical tube. Square and symmetric hexagonal with 15° tubes performed Type II deformation behaviour. Symmetric hexagonal tubes with 30°, 45° and 60° performed Type I with the perfectly plastic collapse behaviour whereas cylindrical tube performed Type I with strain hardening deformation behaviour. The mathematical prediction model had managed to model the deformation behaviour of symmetric hexagonal tubes with 30°, 45° and 60° but failed to model the square and symmetric hexagonal with 15° tubes because it was the perfectly plastic model which suitable for Type I with perfectly plastic deformation behaviour.


Introduction
The crashworthiness or the response quality of a vehicle during collision or impact has become one of the important engineering studies in the designing of vehicles.This study is important in order to improve on the crashworthy properties of a vehicle so that the fatalities rate on the occupants involved in the crash could be reduced (Johnson, 1990).When a collision occur, the energy absorption system should react by absorbing and dissipating the kinetic energy into an irreversible or inelastic energy such as plastic deformation, viscous energy and friction or fracture energy (Olabi, Morris, & Hashmi, 2007).One of the popular energy absorption systems is the lateral compression of cylindrical tube.
Several experimental and analytical studies which basically focused on the cylindrical tube had been performed with regards to the lateral collapse of tubes compressed between two flat rigid plates.The first mathematical model was developed based on rigid perfectly plastic material model where the load-deformation prediction was based on the energy balance method and it took into account the geometrical components of stiffening phenomenon (De Runtz & Hodge, 1963).The model included four stationary plastic hinges where the plastic deformation occurred to replicate the experimental quasi-static lateral collapse of mild steel cylindrical tube which was compressed between two flat rigid plates.Another separate mathematical model assumed that there were six plastic hinges (5)  Figures 9 (c) -(e) show that the symmetric hexagonal with 30°, 45° and 60° tubes had a flat or nearly flat force and ended with high rising force.Figures 9 (a) -(e) show that under the mathematical model methods, all the diagrams had a flat force and ended with high rising force.

Discussion
The comparison of force vs deformation/total height relationship under the simulation method for cylindrical, square and symmetric hexagonal of the various angles i.e.
15°, 30°, 45° and 60° tubes had shown that the square and symmetric hexagonal tubes was different than the cylindrical tube regarding the deformation behviour pattern at the plastic collapse zone.All the tube structures had an early and immediately increasing force which was the elastic collapse zone.Then, there was the plastic collapse zone where all of the three types of structures had shown the different behaviours.The cylindrical tube had a steadily increasing curve force, the square and symmetric hexagonal with 15° tubes had an initial peak, and the rest of symmetric hexagonal tubes i.e. with 30°, 45° and 60° tubes had a long flat or nearly flat force.For all of the tube structures in this study, the force ended with a high rising force which was the densification zone for all the tubes.
The initial peak phenomena for square and symmetric hexagonal with 15° tubes was in accordance with the Type II deformation behaviour (Calladine & English, 1984) since a high early force was needed against the elastic deformation of the vertical sides of square tube and the oblique sides with acute angle of symmetric hexagonal with 15° tube before plastic deformation could take place.The cylindrical and symmetric hexagonal with 30°, 45° and 60° tubes closely resembled the Type I deformation behaviour with the long flat or nearly flat or the steady increasing force (Calladine & English, 1984) since no high force was needed to collapse their oblique sides.For this type of collapse behaviour, the early linear increasing force is the elastic deformation zone, the following long flat force or plateau is the plastic deformation zone and the last high rising force is the densification zone (Gibson & Ashby, 1997).
The plastic collapse for the cylindrical tube was due to the four stationary hinges (De Runtz & Hodge, 1963) or six hinges around the perimeter of the cylindrical tube (Burton & Craig, 1963).Based on the mathematical model prediction and simulation performed, the plastic collapse for symmetric hexagonal tubes was due to the stationary plastic hinge at all the six vertices which made the four oblique sides at the left and right of the symmetric hexagonal tubes to collapse.The steadily increasing force of the cylindrical tube resembled the strain hardening deformation whereas the flat force of the symmetric hexagonal tubes resembled the perfectly plastic deformation behaviour (Li et al., 2006).
For square and symmetric hexagonal tubes, the yield stress was reduced when the angle was increased.For square tube, 120kN, for hexagon tubes, when 15°, 40kN, when 30°, 20kN, when 45°, 10kN and when 60°, 5kN.This shows that the increased of has reduced the elastic deformation and hence reducing the possibility of injury and damage due to the effect of repetitive acceleration and deceleration under the elastic deformation zone (Johnson & Reid, 1978).
In contrast, the long flat force or the stroke length became longer when the angle was increased.For square tube the percentage of stroke length to the tube's size was 44%.The stroke length percentage for hexagon tubes was 60% when 15°, 64% when 30°, and 72% when 45°, but slightly reduced to 68% when 60°.The longer stroke will produce a longer plastic displacement, increasing the rate of hindering after an impact and hence will reduce the possibility of severe injury and damage (Lu & Yu, 2003).Thus, this shows that the increased of has improved the energy-absorption mechanism of symmetric hexagonal tubes.
The validation of the mathematical model with the simulation method showed a big gap for square and symmetric hexagonal tubes with a small angle, .The gap was reduced when the angle was increased and when 60° the result was almost close.The gap was due to the yield stress, for square and symmetric hexagonal tubes with small were very high.This was because the deformation behaviour for square tube and the symmetric hexagonal tube with small were in accordance with the Type II and the deformation behaviour gradually change to the Type I behaviour as the angle for symmetric hexagonal tube was increased.Since, the mathematical model was based on the rigid, perfectly plastic model which resembled Type I deformation behaviour, it showed contrasting results to the Type II deformation behaviour.However, the mathematical model had managed to show the general deformation pattern of the symmetric hexagonal tubes where there is an existence of a long flat force before densification occurred at around 80% of the collapse.

Conclusion
The validation of the mathematical model with the simulation method has shown that for square and symmetric hexagonal tubes with 15°, both methods did not have good agreement since these tubes had Type II pattern with an initial peak but the mathematic model was based on the rigid, perfectly plastic model which resembled Type I deformation behaviour without the initial peak.When the angle for symmetric hexagonal tubes was increased, the deformation behaviour of the hexagonal tubes had gradually changed to the Type I behaviour which showed the agreement of both methods was getting better.The symmetric hexagonal tubes with 60° showed very close results between both methods.
For symmetric hexagonal tubes, the increased of the angle had reduced the yield stress but increased the plastic deformation thus improved the energy-absorption system by reducing the possibility of damage and injury during the collision. Fig Figure 4 Figure 8