11EXEEBIMEKΕAL RESULTS "SET A"
4.2 NUMERICAL ANALYSIS ,,SET A"
The first numerical analysis involves those specimens having a .030" wall thickness.
Uniaxial properties for "set A" are obtained as discussed in Chapter 2. Loading for this set consists of applying a prescribed tensile stress to the specimen before the pressure is numerically increased in incremental steps. After the test shell buckles in the numerical analysis (i.e., stiffness determinant changes sign), the buckling calculations can be
restarted at a lower load with smaller load increments to obtain a more accurate bifurcation point. Bifurcation points are obtained with a numerical accuracy of five pounds per square inch.
For example, if a shell was determined to buckle between 900 and 950 psi, the analysis was restarted with 5 psi increments from 900 psi. The final buckling load was determined to be halfway between the last numerically calculated loads.
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4.2.1 CONSTANT TENSION "SET A”
Figure 4.4 shows the results obtained from both numericaΓ(BOSOR5) and
experimental work. The analysis allows for both plastic models to be used to calculate the bifurcation point. Either incremental theory G (G bar) or deformation theory can be specified for the plastic analysis. In-plane warping at the ends is restrained for the results presented in Figure 4.4. Numerical results using the physically less acceptable
deformation theory correspond better to the experimental results than those obtained with the more physically sound incremental theory. Deformation theory seems to perform better although it overpredicts or underpredicts depending upon the level of axial tension to which the shell is subjected. Underprediction of actual experimental values is not common in buckling theory and is one of the reasons why caution should be exercised when
seeking justification for deformation theory results. Especially, when a numerically perfect shell analysis yields a lower buckling load than the observed experimental buckling load of a physically imperfect specimen, there is reason to question the analysis and plasticity model. For proportional loading (external pressure only), both theories predict
approximately the same result and within 5% of the observed buckling pressure. As long as load-paths do not diverge too much from this proportional path, both theories do relatively well. This does not contradict theory, since deformation and incremental theory are the same for proportional loading. However, as paths tend towards more
nonproportional loading, significant differences between the two theories start to emerge.
Some empirical justification for the inclusion of the "less respected" deformation theory in a numerical analysis such as the BOSOR5 code can be found here. Deformation theory, although path-independent, captures the weakening behavior of the material with increasing axial load as observed in the experiment. Incremental theory, in contrast, still displays the stiffening character as observed for axially stiffened elastic shells (axial tension on elastic shells increases resistance to buckling because of external pressure).
Deformation theory predicts buckling to occur in five circumferential waves, whereas
experimentally, a change in buckling waveform (five to four waves) is observed as the axial load-increases. This change of circumferential buckling wave number is predicted by the J2 incremental theory. In the case of analysis with incremental theory, the buckling curve consists of two curved segments, each representing a fixed number of
circumferential waves. Above approximately 12,000 psi tensile stress, incremental theory shows a significant stiffening trend, which is not observed experimentally. Overall the
"sense" of the incremental data is incorrect as can be observed from Fig 4.4. The apparent discrepancies between incremental and deformation theory buckling loads for proportional loading are due to end effects. When the ends of the short shell are unrestrained (i.e., endplugs are neglected), both theories predict identical buckling pressures and adhere to the generally known result that deformation theory and incremental theory are the same for proportional loading.
A closer investigation of incremental theory as implemented by Bushnell produces the result that the shear response during buckling has been modified to correspond to the shearing response predicted by a deformation theory model. There is no shear present in the prebuckling analysis because of the nature of the problem, but when the shell deforms nonaxisymmetrically during buckling, shearing is introduced and needs to be accounted for in the numerical analysis. The shear modification will be addressed in Chapter 5, but to understand the actual response of pure J2 incremental model, the shear term was changed to an elastic response as predicted by J2 incremental theory when no shear was present in the prebuckling analysis. Results using this shear response are also shown in Figure 4.4 and are indicated by "incremental theory (G)," whereas results using the modified BOSOR5 shear response are denoted by "incremental theory (G) (G bar)," It is clear that pure incremental theory results in even larger disagreement with experimental results.
Bushnell's use of a modified shearing response is justified from a standpoint that
predictive capability is slightly improved, but results are still unsatisfactory. Modification of the shear response reduces the error but does not physically improve the "sense" of the
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results.
Resistance to buckling and the number of circumferential waves that form during buckling is also influenced by the amount of in-plane warping allowed at the ends [Ref.
34]. In Figures 4.5a and b the effects of end warping on the buckling load are presented.
When in-plane warping is unrestrained instead of restrained (in-plane axial displacements at the ends of the shell are free to assume any shape necessary to compensate for the formation of the out-of-plane buckling wave-form) during the buckling analysis, buckling resistance of the test shell is considerably reduced. The actual boundary condition during buckling is most likely between these two extremes and within the shaded areas as shown in Figures 4.5 a and b. The shaded regions in Figure 4.5b become more narrow as the axial stress component increases. This indicates that sensitivity to the type of end
condition reduces with increasing axial stress, which seems physically correct. Including these effects in the analysis does not explain the discrepancy between the experimental results and the numerical analysis.
Wall thickness of the model-shell in the BOSOR5 analysis has been taken to be the average thickness measured in the actual experiments. However, it seems that taking the lower bound of the shell thickness data may be more appropriate, since buckling of the shell starts at the thinnest section in the shell wall. This will reduce the predicted buckling pressure by approximately 5-8% according to some sample runs when no axial load was applied. Although a thinner wall thickness reduces the gap between the numerical data and the experimental data, the "sense" of BOSOR5 data is not expected to change with this parameter.
4,2,2 CONSTANT PRESSURE "SET A"
Analysis of the reversed loading problem is unsuccessful, using incremental theory.
BOSOR5 predicts failure in tension because of the accumulation of excessively large axial strains, and bifurcation as observed in the experiments is not predicted in the analysis.
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.