11EXEEBIMEKΕAL RESULTS "SET A"
4.3 NUMERICAL ANALYSIS "SET B"
Buckling was definitely observed in the experiments and should be predicted by the bifurcation analysis. The path independent deformation theory predicts buckling at the same locations in stress space (as in the constant tension numerical analysis), except that these points are approached along another stress path.
From an engineering standpoint this seems not to be too much of a problem, since experimentally the shell under reversed loading buckled very close to the bifurcation points for the constant tension experiments. Analysis with deformation theory is the only
analysis that predicts bifurcation for both loading conditions and therefore adds to the paradox that the seemingly (physically) incorrect deformation theory performs (again) better than the more physically sound incremental theory. It is important to realize that underprediction as observed for part of the loading problem remains a definite indication that deformation theory is also incorrect. Comer theory as discussed in the last section of this chapter experiences the same problem as the J2 incremental theory for this reversed load-path. Comer theory does not predict bifurcation for this reversed load-path and eventually predicts failure that is due to excessive accumulation of axial strains.
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In Figures 4.6 and 4.7 results are shown using the original deformation and
incremental theories G (G bar) as they are available in the BOSOR5 analysis. Figure 4.6 shows the result using the axial material curve in the BOSOR5 analysis. For this set circumferential properties were obtained in addition to the axial properties, and the analysis is repeated using an average circumferential stress-strain curve. These results will show the effect of material definition on the numerical analysis. Results are shown in Figure 4.7.
Since more waves form around the circumference of the shell, it is likely that circumferential material behavior may be of greater significance in the buckling analysis than the axial behavior. Variation between axial and circumferential material behavior seems to be small enough to warrant the use of isotropic assumptions in the case of "set A." Circumferential properties do show an earlier and more gradual transition from linear to nonlinear material behavior although the difference is small. Results are shown in Figure 4.7. It is important to realize that circumferential properties are obtained under uniaxial "hoop" tension that is due to internal pressurization of the test shell. In actuality, the hoop stress in the experiment is compressive and material properties may vary slightly using a compressive test. It is much more difficult to obtain compressive properties of the specimen, since the shell will buckle before much of the plastic behavior can be observed, but ideally one should use an external pressure test.
Results using either material curve and J2 incremental theory display a stiffening behavior with increasing axial stress, which is not observed experimentally. For low axial loads, bifurcation prediction using either material curve is approximately the same. This may be a result of the thin (0.028") shell wall, resulting in a buckling stress, which is very near the elastic region and therefore is not very much affected by the nonlinear part of the material curve. However when axial loads become large, the effective stress state is much deeper into the plastic range, where the difference between the axial and circumferential material behavior is more pronounced. Comparing both figures and assuming that
compressive behavior is identical to tensile behavior, it can be said that the proper definition of the uniaxial stress-strain curve affects the analysis, but not to such a degree that the "sense" of the results is changed.
It is interesting to note that results obtained with circumferential properties and deformation theory are located entirely outside of the experimental results when circumferential properties are used. The underprediction that was distressing in the previous analysis does not occur here. Deformation theory captures the reduction in buckling resistance with increasing axial load and is also able to predict bifurcation for the reversed load-path.
4,3.2 CONSTANT PRESSURE "SET B"
Numerical analysis using BOSOR5 is unsuccessful since failure is predicted to occur because of large strains at load values well beyond the observed buckling loads.
Deformation theory does predict buckling independent of the load path as in "set A." A complete analysis using this reversed load-path was not performed, since path
independence of the deformation theory is expected to produce similar results as those shown in Figures 4.6 and 4.7. Using incremental theory (G) (G bar), the numerical shells do not bifurcate but buckling is again observed in the experiments using this
reversed load-path. An improved model that obtains better results for the constant tension tests will, it is hoped also increase the predictive capability for bifurcation under increasing axial tension and constant pressure. It seems logical that improvement should first be sought in the constant tension experiment with varying external pressure, since it is the external pressure that causes bifurcation and collapse of the shell. Reversed loading changes material stiffness until the shell can no longer resist buckling (due to the external pressure), and bifurcation in this case is an entirely different phenomenon. Correct modeling of this behavior seems to be even more difficult since axial tension indirectly induces buckling (external pressure is primarily responsible). It appears that the analysis
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using incremental properties is unable to recognize the "softening" influence of the axial load on the stiffness of the shell. The shell still fails because of accumulation of excessive axial strains, as if the presence of external pressure is not recognized.