Finite Element Modeling of Crushing Behaviour of Thin Tubes with ...

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10/19 Paper: ASAT-13-ST-34 initial inward folding of the pentagonal tube was appeared in only two non-successive sidewalls. After complete constituting of the initial folds, similar sequential folds were regularly created and the crushing was propagated along the sidewalls of all polygonal tubes. The folds generated in all tube models were in contact and adjoining each other except for that ones of the triangular tube. This may be attributed to the acute inner angles between the sidewalls which increase the strength of the triangular tube and allow the rigid body behaviour to appear early before occurring a complete adjoining of folds. On the other hand, the crushing behaviour of the elliptical tube was quite complex and totally different from the deformation characteristics of the other tube models. The upper cross-section of the elliptical tube was initially stretched in the direction of maximum diameter as shown in Fig. (6). Afterwards, the crushing was propagated suddenly allover a region bounded by the upper stretched cross-section and another similar one located nearly at the middle height of the tube. Progressive compression generated similar crushing characteristics on the rest of the elliptical tube model. When the walls of the tubes stopped folding and behaved as a rigid body, the number of folds was observed maximum with fourteen folds in the circular tube and minimum with three folds in the triangular tube. Load- and Energy-Displacement Characteristics Figures (5-10) indicate the initiation and end of the typical axial load- and the energydisplacement curves of various tube models. The absorbed energy was obtained by integrating the area under the axial load-displacement curve until the folding of the walls ended. Afterwards, the mean crushing load was calculated by dividing the absorbed energy by the total displacement. Maintaining all parameters of the finite element model, a series of the load-displacement curves were obtained from different shape tube models. The general loaddisplacement characteristics of all tubes can be described by a rapid rise of the force initiated due to the elastic compression of the tubes. After that a rapid drop of the force was observed when one end of the tube models started folding. A steep rising of the force was again noticed after the first folding of walls ended and the second folding of the adjacent walls started generation. A drop of the force occurred again when a new folding started, and the force rose if the walls came into contact. This response was repeated until the folding ended and the models behaved as a rigid body. A uniform response was noted in all tube curves except for the elliptical tube, the response was irregular. It is worth mentioning that all the numerical results of the energy-displacement relations were nearly linear all over the crushing intervals. Comparing the force-displacement curves seen in Figs. (5-10) can help analyze the behaviours of different crushing mechanisms. It was clearly seen in Fig. (5) that the higher number of local peaks along the force-displacement curve reflected the great number of folds created in the circular tube. The difference between the upper and lower bounds of the load values was significantly large in the square and triangular tube curves and quite narrow in the hexagonal tube curve. For the pentagonal tube, the difference between upper and lower bounds of the load values started large then damped gradually with the compression progress until reaching a stable load as shown in Fig.(8). On the other side, the general crushing response of the elliptical tube was totally different where the values of the load and consequently the absorbed energy were pronouncedly high and the load peaks were scattered in a distinct manner all over the curve. This may be attributed to the disturbance occurred during the folding process of this tube shape. Figure (11) shows that the absorbed energy of the finite element tube model decreases with an increase in the tube side breadth-to-thickness ratio up to the tube of four sides. Then, the absorbed energy starts increasing at the triangular tube. The relationship between the absorbed energy and the side breadth-to-thickness ratio of the tubes can be empirically expressed as:
11/19 Paper: ASAT-13-ST-34 E 5.19 1.48cos[(0.05 Sb/ t) 0.6] (1) where; E = Absorbed energy Sb = Side breadth, T = Tube thickness Initial Peak Load and Mean Crushing Force Figure (12) depicts the nonlinear relation between the increasing of the number of sides and the decreasing of the side breadth of a polygonal tube of 300mm perimeter. Variations of the initial peak load and mean crushing force with different number of sides (3-6) and various side breadths (50-100 mm) are shown in Figs. (13,14) for the polygonal tubes. Generally, it was observed that the initial peak load highly increased with the increasing of the number of sides and greatly decreased with the increasing of the side breadth. Moreover, the initial peak loads of the triangular and square tubes had closed magnitudes while values of the initial peak loads of the pentagonal and hexagonal tubes were sharply far off. On the other side, the mean crushing load was changed in a different manner with the number of sides and the side breadth of the tubes. The mean crushing load greatly decreased with the increasing of both the number of sides and the side breadth until reaching a minimum value at the square tube. Subsequently, the mean crushing load began to increase with the increasing of the number of sides and the side breadth of the tubes. It is worth mentioning that the initial peak load and mean crushing force of the round (circular and elliptical) tubes were greatly higher than that obtained from the polygonal tubes as compared in Table (2). The Fold Depth and the Side Breadth of Tubes The fold depth was investigated through a longitudinal path of the tube. This path was located at the middle of one arbitrary face for all the sided tubes. The fold depth of the sided tubes was compared with that obtained from the circular tube overall the tube span as shown in Fig. (15). The bottom end of all tubes was located at the origin of the horizontal axis, however the inward and/or outward fold profiles were recorded on the vertical axis of Fig. (15). It can be noticed that the folds of the circular tube have the lowest depth (approximately 8 mm) and bulged mainly outward. The profiles of all the folds are not so different except for the pentagonal tube where its fold profile was not symmetric about the middle path. On the other side and apart from the triangular tube, the fold depth was seen maximum in the square tube and it generally decreased with increasing the number of sides of the polygonal tubes. In other words, starting from a polygonal tube with four sides, the initiated fold depth is decreased with the decreasing of the tube side breadth as depicted in Fig. (16). The relationship between the fold depth and the side breadth-to-thickness ratio of the tubes can be empirically expressed as: where; Fd = Fold depth, Sb = Side breadth, t = Tube thickness Fd 8.302exp[0.021( Sb / t)] (2)
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Finite Element Modeling of Crushing Behaviour of Thin Tubes with ...

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