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This is FIg. 5.10 in the dissertation.
From:
Luc Wullschleger, “Numerical investigation of the buckling behavior of axially compressed circular cylinders having parametric initial dimple imperfections”, PhD dissertation, Swiss Federal Institute of Technology, Zurich, 2006
(Abstract continued from the previous slide) Several parameter studies were conducted for a number of cylinders having dimples of different initial amplitude but fixed initial circumferential width and axial height. In addition, for a few cylinders and for some predefined initial amplitudes the initial axial height and circumferential width to the dimple was searched which reduce the axial cylinder stability the most. Finally the influence of the relative position of a second identical dent to the load carrying capacity was investigated. These series of analyses aimed at investigating whether there are single, localized initial dimples which reduce the nominal axial buckling load of an unstiffened circular cylinder more than imperfections derived from classical buckling patterns of ideal shells, and whether there is a worst geometry of such a single dimple imperfection. Further: is the instability behaviour the same for isotropic shells as for laminated composite shells having such a localized dimple imperfection? And, is there an important interaction between two initial dimples?
The dimple-parameter studies required a large amount of static and transient dynamic FE analyses. Most of the calculations performed were nonlinear buckling analyses, i.e. nonlinear static stress analyses under consideration of large displacements and rotations using Updated Lagrangian formulations with additional linear eigenvalue calculations, conducted after a selected number of small loading steps to determine the stability of pre-buckling states of stress and deformation. To manage the large number of shells with different bucking loads and behaviour considered, the nonlinear buckling analysis was adapted for an adaptive load step control which utilizes the intermediately extracted eigenvalues. For a selection of cylinders and dimples additional nonlinear transient dynamic analyses were conducted in order to research into the particular deformation processes of such shells under axial loading. Because of the relatively slow compression velocities assumed the implicit "single-step Houbolt” method for time integration was preferred to the more common explicit operators. To reduce the number of time increments needed for stepwise convergence significant inertia damping was introduced. (Ab
In a classical analysis, for ideal, thin-walled unstiffened isotropic circular cylinders of medium length under pure axial compression the load-carrying capacity can be predicted analytically by means of simple equations. These equations follow from solving the coupled partial differential equations for equilibrium and compatibility in simply-supported cylinder with harmonic functions. There are also closed-form solutions of the Donnell-type shell equations available for thin-walled orthotropic composite cylinders. Such a classical analysis, however, is applicable exclusively for perfect cylinder geometry.
For the imperfection shapes and sizes considered no test results were available against which the FE analysis results could have been benchmarked. Instead, they are supported by convincing results of such calculations for similar cylinders with perfect geometry and for laminated CFRP cylinders with their measured imperfections included. The results of the ideal cylinders could be compared with values achieved with classical analyses, whereas for the analysis results of the CFRP shells with measured imperfections test data was available for comparisons.
(Abstract continued on next slide)
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