More than a hundred years after the publication of Love's seminal paper on the vibrations and deformation of a thin elastic shell, engineers, scientists, and mathematicians continue to struggle with some of the more complicated issues relating to such structures, despite the advent in recent years of extremely powerful computers and appropriate software. In particular the buckling of cylindrical shell structures under axial loading has received a great deal of attention since the turn of the century. This case has been addressed from a number of different points of view. However, none of these have yielded wholly satisfactory solutions to the stability problems faced.
In this thesis a new approach is proposed for analysing some of the more challenging examples relating to shell buckling. It is suggested that a careful examination of a heavily deformed shell structure in the post-buckling regime and the processes leading up to this state of deformation will establish some of the principles involved in its loss of stability. The complex deformation patterns observed experimentally after buckling may be reduced to a much more elementary description consisting of inextensional and transitional regions. Hence in this dissertation two relatively simple shell inversion problems exhibiting corresponding representative features are studied in detail.
Experiments investigating the inversion of a cylindrical shell under a radial point load are outlined. Due to a free boundary large elastic inextensional deformations may occur. Some straightforward empirical formulae are found that well describe the characteristics of the deformed geometry; and the applied load is established as a function of the deformation.
The inversion of a spherical shell under a radial point load is considered as a means of examining some of the basic features observed in the experiments performed, whilst eliminating the complexities arising from the lack of symmetry in the cylindrical case. A 'simple model' is detailed which clearly highlights the major components of the total strain energy of the deformed surface and their origin. A number of formulae are fond that relate the loading and peak stresses to the radial displacement under the load. The results from this model are confirmed using a finite-element analysis, which also provides a more comprehensive description of the overall stress state of the shell.
The finite-element method is likewise applied to the case of the cylindrical shell as an extension of the experiments. Findings from the latter are employed to confirm the validity of the numerical analysis. A detailed account is given of the distribution of stresses and strains over the deformed surface; and the similarities with the case of the spherical shell are outlined.
Finally, an analytical model is established for the case of the cylindrical shell using the empirical formulae. The derivation of a condensed expression (Equation 6.61) for the total strain energy is presented. It is shown that the model represents well the relevant data obtained numerically. Notably, it is discovered that the two terms of Equation 6.61 correspond to the two most prominent regions of the deformed shell surface.
[Cambridge University | CUED | Structures Group | Geotechnical Group]
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