Keywords: collapse, axial loading, cylindrical tube, FEM, quasi-static, chamfer.

This paper presents a FEM based method for solving axisymmetric progressive buckling of cylindrical tubes under quasi-static loading between two rigid plates. As a result of symmetrical geometry, loading and boundary conditions axisymmetric elements are used. Some experimental tests have been simulated by this method. The model is found to reproduce the crushing response to a significant degree of accuracy. Mean and maximum collapse loads together with final shape of tube are greatly in accordance with experimental results. The most important item of the model is the chamfer which is suggested as an applicable way to achieve a desired collapse mode.

A circular section metal tube under axial compressive loading collapses in one of the following modes: Euler, diamond, concertina, mixture of concertina and diamond or barreling. When loading is quasi-static, collapse mode depends on material behavior, diameter to thickness ratio (), length to diameter ratio () and boundary conditions.

In 1983 quasi-static collapse mode of about 70 aluminum tubes, having a variety of and ratios were investigated [1]. The results of this experimental study were presented in a useful graph where boundaries of various buckling modes were determined approximately. In 1992 a new model of progressive crushing of tubes was developed, in which an active crushing zone contained two semi-circle folds [2]. In 1998 thin-walled aluminum and steel cylindrical tubes under dynamic and quasi-static loading were studied experimentally. The effect of aluminum and steel tubes dimension and annealing of steel tubes on collapse mode were investigated [3].

A numerical analysis of axisymmetric buckling of thin-walled cylindrical tubes based on elastic-viscoplastic material model and small strain assumptions was presented in 1999 and estimation of fold's length was in a good agreement with Alexander's, Abramowitz's and Singace's theoretical models [4]. Aluminum cylindrical tubes were tested under quasi-static compressive loading in 2001 and in this experiments range of was wider than previous works [5]. These researchers developed some experimental relations for mean and first maximum crushing loads in concertina and diamond buckling modes. In 2003 an experimental study of aluminum cylindrical shells under quasi-static loading was developed [6]. In this research geometrical characteristics of tubes ( and ) were selected so that predicted buckling modes were concertina, diamond or mixed mode.

The objective of this study is to model axisymmetric progressive collapse of cylindrical tubes under axial quasi-static loading between two rigid grips of a tension-compression machine. The symmetry of geometry and loading with respect to the mid-length plane of tube imposes a mid-length symmetry of model's collapse which is different from experimental observations. It occurs because the model is perfect but the real tube has some imperfections. In this study the proposed solution includes inserting an imperfection to the model which is a chamfer made at one end of the tube. It was observed that the onset of wrinkling occurred at imperfect end.

The authors have tested some aluminum tubes with different length to diameter and diameter to thickness ratios such that the tubes often collapsed in concertina, diamond or mixed mode [6]. It was found that progressive buckling started in concertina mode and then changed to diamond mode. To verify the accuracy of FEM model, the concertina part of buckling of this study and another experimental research [5] is modeled and the results are compared directly with the experiments.

1

Andrews, K.R.F., England, G.L., Ghani, E., "Classification of Axial Collapse of Cylindrical Tubes Under Quasi-Static Loading", Int. J. Mech. Sci., Vol. 25, No. 9-10, p687, 1983. doi:10.1016/0020-7403(83)90076-0

2

Wierzbicki, T.S., Bhat, W., Abramowitz, D., "Revised-A Two Folding Elements Model of Progressive Crushing of Tubes", Int. J. Solids Structures, Vol. 29, No. 24, p3269, 1992. doi:10.1016/0020-7683(92)90040-Z

3

Gupta, N.K., "Some Aspects of Axial Collapse of Cylindrical Thin-Walled Tubes", Thin-Walled Structures, 32, p111, 1998. doi:10.1016/S0263-8231(98)00029-9

4

Mikkelsen, L.P., "A Numerical Axisymmetric Collapse of Viscoplastic Cylindrical Shells under Compression", Int. J. Solids and Structures, 36, p643, 1999. doi:10.1016/S0020-7683(98)00037-7

5

Guillow, S.R., Grzebieta, G. Lu R.H., "Quasi-Static Axial Compression of Thin- Walled Circular Aluminum Tubes", Int. J. Mech. Sci., 43, p2103, 2001. doi:10.1016/S0020-7403(01)00031-5

6

Shakeri, M., Beiglou, A.A., Mirzaee, M., "Axial collapse of cylindrical shells under quasi-static loading", in " 11th annual conference of mechanical engineering, ISME 2003", article No.:S-STR-20.

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