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This and the next several slides are from the 2009 GENOPT paper, "Minimum weight design of imperfect isogrid-stiffened ellipsoidal shells under uniform external pressure" by David Bushnell, presented at the AIAA 50th Structures, Structural Dynamics, and Materials Conference, Palm Springs, California, 2009, AIAA Paper 2009-2702, called "2009 GENOPT paper" in the next several slides.
ABSTRACT from the 2009 GENOPT paper:
GENOPT, a program that can be used to optimize anything, and BIGBOSOR4, a program for stress, buckling, and vibration analysis of segmented, branched, stiffened, elastic shells of revolution, are combined to create a capability to optimize a specific kind of shell of revolution: an internally isogrid-stiffened elastic ellipsoidal shell subjected to uniform external pressure.
Optimum designs are obtained for isogrid-stiffened and unstiffened axisymmetrically imperfect and perfect titanium 2:1 ellipsoidal shells.
The decision variables are the shell skin thickness at several user-selected meridional stations, the height of the isogrid stiffeners at the same meridional stations, the spacing of the isogrid stiffeners (constant over the entire shell), and the thickness of the isogrid stiffeners (also constant over the entire shell).
The design constraints involve maximum stress in the isogrid stiffeners, maximum stress in the shell skin, local buckling of an isogrid stiffener, local buckling of the shell skin between isogrid stiffeners, general nonlinear bifurcation buckling, nonlinear axisymmetric collapse, and maximum normal displacement at the apex of the dome.
Optimum designs first obtained by GENOPT/BIGBOSOR4 are subsequently evaluated by the use of STAGS, a general-purpose finite element computer program.
It is found that in order to obtain reasonably good agreement between predictions from BIGBOSOR4 and STAGS it is necessary to model the ellipsoidal shell as an "equivalent" ellipsoidal shell consisting of a spherical cap and a series of toroidal shell segments that closely approximates the true ellipsoidal meridional shape.
The equivalent ellipsoidal shell is optimized with up to four axisymmetric buckling modal imperfections, each imperfection shape assumed to be present by itself. Computations include both plus and minus axisymmetric buckling modal imperfection shapes.
At each design cycle and for the plus and minus version of each axisymmetric imperfection shape the following analyses are conducted: 1. linear general axisymmetric bifurcation buckling analysis (in order to obtain the axisymmetric linear buckling modal imperfection shapes), 2. nonlinear axisymmetric stress analysis at the design pressure, 3. nonlinear axisymmetric collapse analysis, and 4. nonlinear non-axisymmetric bifurcation buckling analysis.
For each axisymmetric imperfection shape the design margins include an axisymmetric collapse margin, a general buckling margin, a margin involving the normal displacement of the apex of the shell, and local skin and stiffener stress margins and local skin and stiffener buckling margins within two approximately equal meridional regions of the equivalent ellipsoidal shell.
There is generally good agreement of the predictions from STAGS and from BIGBOSOR4 for the elastic behavior of the perfect stiffened and unstiffened optimized shells and for the behavior of the imperfect stiffened optimized shells with axisymmetric buckling modal imperfections.
Optimization with the use of only axisymmetric buckling modal imperfections has a disadvantage in the case of the unstiffened imperfect shell under certain conditions: the optimum design of the axisymmetrically imperfect unstiffened shell evolves in such a way that, according to predictions from STAGS, a non-axisymmetric buckling modal imperfection with the same amplitude as an axisymmetric buckling modal imperfection causes collapse of the shell at an external pressure far below the design pressure. This disadvantage is easily overcome if, during optimization cycles, the unstiffened shell wall in the neighborhood of the apex is forced to remain thick enough so that local axisymmetric buckling does not occur primarily at and near the apex but instead occurs primarily in the remainder of the shell.
An extensive study of some of the previously optimized elastic shells is conducted with STAGS including elastic-plastic material properties. The effect on collapse pressure of initial imperfections in the form of off-center residual dents produced by load cycles applied before application of the uniform external pressure is determined and compared with the effect on collapse pressure of imperfections in the form of non-axisymmetric and axisymmetric linear buckling modes, especially the non-axisymmetric linear buckling modal imperfection with n=1 circumferential wave, which seems to be the most harmful imperfection shape for optimized externally pressurized ellipsoidal shells.
For the optimized unstiffened shell it is found that a residual dent that locally resembles the n=1 linear buckling modal imperfection shape is just as harmful as the entire n=1 linear buckling modal imperfection shape.
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