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This and the next 2 images are from:
Francois Claude Bardi, “Plastic buckling and collapse of circular cylinders under axial compression”, Ph.D. dissertation, The University of Texas at Auston, August 2006
ABSTRACT: This study is concerned with the plastic buckling of relatively thick tubes and the ensuing succession of instabilities leading to their failure. The first instability is uniform axisymmetric wrinkling that is treated as a plastic bifurcation. As the wrinkles grow, the axial rigidity of the shell is gradually reduced. This eventually leads to a limit load instability beyond which failure in the form of localized deformation takes place. The problem is studied using experiments and analyses. Stainless steel specimens with D/t of 23-52 were custom-designed to avoid stress concentrations and reproduce long uniform pipe conditions. The specimens were compressed to failure under displacement control. In all cases, a second bifurcation involving non-axisymmetric mode of deformation preceded the limit load. The bifurcation into axisymmetric wrinkling was determined by monitoring the development of wrinkles on the surface of the tubes. This critical state was successfully predicted using an anisotropic deformation theory of plasticity. The anisotropy of the material was established experimentally and modeled using Hill's quadratic anisotropic yield criterion. The problem was first modeled as uniform axisymmetric wrinkling. The model uses Sanders’ shell kinematics assuming small strains and moderately small rotations and includes a modified flow theory of plasticity to accommodate the anisotropy observed in the tubes. Small axisymmetric imperfections based on the critical half-wavelength were integrated into the model. The problem was formulated through the principle of virtual work and solved using Newton’s method. The solution correctly simulates the growth of wrinkles resulting in a limit load instability. The model included second bifurcation calculations from axisymmetric to non-axisymmetric configuration. Second bifurcation instabilities were found to occur before the limit load developed. For this reason, a second model was developed in which non-axisymmetric deformation of the shell was simulated by introducing both axisymmetric and non-axisymmetric imperfections. Non-axisymmetric responses were found to be highly sensitive to the imperfections. Each experiment was first reproduced accurately by choosing the right combination of imperfections. However, to achieve a satisfactory prediction of the limit state over the whole range of D/t, a thorough parametric study of the imperfection sensitivity was performed. The relative amplitude of the axisymmetric imperfection to the non-axisymmetric imperfection was found to define whether the shell deforms axisymmetrically or not. Furthermore, if one of the imperfections governs the deformation configuration, then the effect of the second onto the response is negligible. Thus, a constant axisymmetric imperfection of 0.05% of the pipe wall thickness and a non-axisymmetric imperfection proportional to (D/t)2 / m3 yielded accurate predictions of both mode of deformation and limit load.
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