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

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8/19 Paper: ASAT-13-ST-34 hole at mid height of the tube on the energy absorption characteristics and collapse mode transitions was examined. Results revealed that a diamond deformation mode was generated in all the tube specimens. A great number of studies have been carried out on the axial crushing of thin-walled tubes. Wierzbicki and Abramowicz [5] developed a simplified model based on rigid plastic assumptions and obtained the mean crushing force for square tubes. Abramowicz and Jones [6,7] conducted a series of experiments, and modified the collapse models by taking account of strain-rate sensitivity and effective crushing distance. Grzebieta [8] proposed a method for determining the load history between a peak and a minimum during an oscillation of the load-compression curve of round tubes. Gupta and Velmurugan [9] studied experimentally the internal/external folding of round tubes. The folding parameter (ratio of the inside to the total fold length) was determined experimentally and employed in a proposed analysis. Wierzbicki et al. [10] studied the axi-symmetric mode of deformation of round tubes by considering partly internal and partly external folding. It was shown that the load compression curve is dependent on the folding parameter, while the mean collapse load and the folding length are independent of it. Closed form solutions were obtained for the instantaneous and mean crushing force, the effective crush distance and the length of the local folding wave. Singace et al. [11-13] gave an analysis of the axi-symmetric and multi-lobe or diamond mode of deformation to determine the eccentricity factor and crushing load. It was concluded that measured values of the eccentricity factor and the critical folding angles obtained for tubes of different materials and geometric ratios are independent of the tubes material and geometric ratios. There is a considerable amount of published data on the response of composite tubes to axial crushing [14]. Many of these studies utilize circular cross-section tubular specimens to determine the energy absorption capability of the material. Farley [15] studied the effect of specimen geometry on the energy absorption capability of composite materials. He found that, changes in section lay-up that lead to an increase in modulus lead to higher crush strengths and energy absorption. Mamalis et al. [16-18] analyzed the collapse behavior and deformation mechanism of thin-walled non-circular composite tube, thin-walled composite conical shell, and braced elliptical tube. They contributed with a valuable data of crashworthiness of composite structures. The aims of this study are to obtain numerical data on the crushing of different cross-section tubes, and to develop simple empirical expressions for predicting the absorbed energy and the fold formation of steel tubes. A series of finite element calculations was carried out on six tube models crushed axially in a quasi-static condition by using ABAQUS/Explicit. The cross-section shapes of the six tubes were circle, ellipse, triangle, square, pentagon and hexagon. Effect of the side breadth and the generated fold depth on the initiated peak load and the mean crushing force of different tubes were carefully examined. Finite Element Model The finite element simulation of the case described herein was carried out using the commercial code ABAQUS/Explicit version 6.4. Six tubes with different geometrical crosssections were modeled in the present study. All the tube models were developed with same lengths and equal cross-section perimeters of 300 mm each. A uniform thickness of 1 mm is considered for all the wall tubes. The difference between the whole tubes was only the shape of the cross-sections. The latter were selected from uniform geometries like a circle, an ellipse, a triangle, a square, a pentagon and a hexagon. Detailed description of the tube crosssections is illustrated in Fig. (1). For comparing reasons; the material used in the simulation was provided from the actual material properties of steel experimentally measured by Paik et al. [1]. The mechanical properties of the steel material were assigned into the ABAQUS input
9/19 Paper: ASAT-13-ST-34 file. A list of numerical values of the basic material properties required for the tube models is given in Fig. (2). All the tube models used in the present finite element simulation were generated by using the element S4R. This element is a three-dimensional doubly curved fournode shell element. Each node has three-displacement and three-rotation degrees of freedom. Moreover, this element is considered a general-purpose shell element where it allows for large strains as load increases. In order to predict the overall response accurately, the mesh of finite elements was fine and uniform with equal number of elements along the length of the tube and through its circumferential direction. On the other side, all the tube models were crushed axially between upper and lower rigid parallel plates. These two parallel plates were simulated by using three-dimensional four-node rigid elements R3D4. The tubes were crushed by pushing down the upper rigid plate. The latter was fixed in all degrees of freedom except the vertical downward displacement where the loading was applied. However, the lower rigid plate was stationary by constraining its whole degrees of freedom. All tube models rested free on the lower rigid plate however the applied force was attained as a reaction created from the pushing of the upper plate. The quasi-static loading condition was achieved by moving the upper plate slowly downward over a sufficiently long time. It is noted that the tubes were not completely free due to the effect of friction between the tube models and the crushing parallel plates. The progressive deformation shape was continuously monitored for each tube and the corresponding force-axial displacement curve was depicted. Experimental Validation The validation of the finite element tube model was made by direct comparison with the experimental results and the collapse observations of the square tube US-2 subjected to quasistatic axial compression and examined by Paik et al. [1]. The structural geometry of the specimen and the properties of the used material are shown in Fig. (2). All other testing details used in the experimental works were necessarily simulated in finite element model. The progressive collapse of the finite element tube model and the examined specimen at various stages of the compression are compared in Fig. (3). Visual examination of the pictures showed good agreements between the numerical simulations and the experimental outcomes. Comparing the predicted load-displacement behaviour with the experimental results in [1] and the obtained calculations in [2] revealed good correlation between curves at the same displacement values as shown in Fig. (4). Table (1) lists the comparison of results between the previously published work and the present finite element analysis. Investigation of Crushing Behaviour The present numerical simulation provides an opportunity to predict the fold formation of various tubes. Figures (5-10) display the collapse mechanisms accompanied by the corresponding axial- and energy-displacement curves of different tube models. Folding modes were symmetric about the principal axes of all the tubes cross-sections. Moreover, an axisymmetric mode was noticed during the deformation of the circular tube. The entire tube models initiated folding from the top end in contact with the movable upper plate. However, the circular tube began to fold from the bottom end in contact with the fixed lower plate. It is worth mentioning that, the folding process of polygonal tubes was initiated by random interior and exterior rotating of the upper edges of sidewalls. Tubes of square and hexagonal crosssections created symmetric inward and outward folds on each two opposite sidewalls. Folds of the triangular tube initiated by curving its upper edges completely inward. However, the
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Finite Element Modeling of Crushing Behaviour of Thin Tubes with ...

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