Keywords: bifurcation buckling, collapse, non-uniform length, cylindrical shell, non-uniform compression, contact.

In many real life situations one needs to connect several cylindrical segments to form a prime load bearing structure. When loading is an axial compression - the interaction between two neighbouring cylindrical segments becomes critical. Possibility of buckling of either one, or both segments further complicates segment-to-segment interaction. This is not a new problem but it is far from being satisfactorily understood or solved. The type of instability induced by cylinder-cylinder interaction is relevant to many industries, i.e. aerospace, civil, and mechanical.

The prime objective of this paper is to examine buckling which can occur due to uneven contact between two cylindrical segments when subjected to axial compression. This is an entirely numerical study. Contact stresses at segment-to-segment interface are limited to one end of a single cylindrical shell. It is assumed that a single cylinder has a non-uniform and 'wavy' axial length at one end. The assembly stresses are induced by clamping the cylinder at one end, and by compressing it by a rigid plate, moving axially, at the other end. The non-uniform axial length has a sinusoidal shape along the compressed edge. The number of waves, , is varied between two and six. Also, the amplitude, , is taken as a variable. The ratio of axial amplitude, , to the wall thickness, , i.e. is varied between . The length-to-shell radius, , is kept constant throughout and equal to whilst the radius-to-thickness ratio, , varies from 165 to 1000. Results are provided for mild steel cylinders with typical material properties (i.e. with the Young's modulus, GPa, Poisson's ratio, , and with the yield point MPa). The initial contact between the rigid surface and deformable cylinder occurs at -points. As the rigid plate moves further down the contact area between the plate and the cylinder changes. As a result of axial compression the load carrying capacity of the cylinder can be limited either by collapse or by bifurcation buckling. A set of four different boundary conditions applied at 'the sinusoidal edge' of axially compressed cylinder has been used to examine the load carrying capacity. Modelling of contact problem is based on a standard 'slave-master' algorithm with no friction between the rigid surface and deformable cylinder.

This numerical study of static stability of cylinders with variable axial length and subjected to axial compression by a rigid plate has revealed a complicated nature of contact between the rigid plate and deformable shell.

For thicker shells, i.e. with , the magnitude of bifurcation buckling load and its position on the load deflection curve depends on the -ratio. For smaller values of the -ratio the failure is controlled by collapse, and bifurcation mode occurs on the-post-collapse-path, and hence it remains irrelevant from a practical point of view. For larger magnitudes of the -ratio, on the other hand, bifurcation can appear well below the collapse load on the initial, primary path of the load deflection curve. This can be dangerous if not accounted for during the design stage. Equally interesting is the size of contact between imperfect cylinder and rigid plate which triggers buckling. Generally larger the amplitude less contact is needed to trigger bifurcation buckling.

For thinner shells, e.g., , asymmetric bifurcation buckling ends the load carrying capacity but only for smaller values of the -ratio. For larger -ratios collapse remains the controlling mode of failure.

Within the investigated range of , i.e. , and the buckling load is being grossly reduced for all cylinders with sinusoidal imperfection of axial length. The thinner shells suffer the largest reduction of the load carrying capacity especially for small magnitudes of the imperfection. Some background information on buckling and/or collapse of cylindrical shell with non-uniform axial length can be found in references [1,2,3,4].

1

Albus, J., Gomez-Garcia, J., Ory, H., "Control of assembly induced stresses and deformations due to waviness of the interface flanges of the ESC-A upper stage", in "Proceedings of 52nd Intl Astronautical Congress", Toulouse, 1-9, 2001.

2

Guggenberger, W., Greiner, R., Rotter, J.M., "The behaviour of locally-supported cylindrical shells: unstiffened shells", J. Constructional Steel, 56, 175-197, 2000. doi:10.1016/S0143-974X(99)00102-9

3

Galletly, G.D., Blachut, J., "Axially compressed cylindrical shells - a comparison of experiment and theory", in "Inelastic Solids and Structures", Kleiber, M., and König, J.A., (Editors), Pineridge Press, Swansea, 257-276, ISBN 0-906674-67-0, 1990.

4

Blachut, J., "Pressure vessel components: some recent developments in strength and buckling", Progress in Structural Engineering and Materials, 1, 415-421, 1998. doi:10.1002/pse.2260010410

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