Previous investigations have raised some doubts about the accuracy of flow theory predictions for a few plate and shell plastic buckling problems. The present series of buckling experiments on machined, mild steel, cylindrical shell models under non-proportional biaxial loading (axial tension plus external pressure) was designed to provide additional data for the evaluation of the J2 plasticity theories. Numerical calculations were carried out with the BOSOR 5 shell buckling program, using the J2 deformation and flow theories, and these were compared with the test results. Neither theory can be said to predict plastic buckling accurately. However, deformation theory predicted the bifurcation buckling loads reasonably well, whereas flow theory was often incorrect.

In the first part of the paper a comparison of various theoretical methods for predicting the elastic buckling pressures of externally pressurized complete circular toroidal (that is doughnut shaped) shells is given. Calculations were also carried out using the BOSOR 5 shell buckling program. The accuracy of the latter was checked using two independent programs. The available experimental results on toroidal shells, from the United States and Russia, are summarized and their buckling pressures calculated using BOSOR 5. A simple equation for predicting the buckling pressures of these shells, due to Jordan, is also briefly reviewed. The buckling of externally pressurized complete elliptical toroidal shells is then discussed. Both buckling pressures and buckling modes for selected geometries are given. Depending on the major/minor axis ratio, the buckling pressures of the elliptical toroids can be larger or smaller than those of the corresponding circular toroids. Some part-cylindrical, part-spherical cross-sections were also explored. Buckling due to internal pressure is considered in the third part of the paper. It is predicted to occur for several complete toroids of elliptical cross-section. It is believed that this is a new result. It still has to be verified experimentally.

Plastic buckling/collapse pressures for externally pressurized imperfect hemispherical shells were calculated for several values of' the yield point (nyp), the radius-thickness ratio (R/t) and the amplitude of the initial imperfection at Ehe pole (ao). The well-known elasticplastic shell buckling program BOSOR 5 was used in the calculations and two axisynzmetric initid imperfection shapes were studied, viz. a localized increased-radius type and a Legendre polynomial.The 8numerical collapse pressures (p,) for both types of imperfection were normalized and plotted versus , i (g parameter proportional to J{(R/tXa JE)}). Approximate algebraic equations were then derived which give pc/pyp as a.function o f 2 and dolt. The values of p, given by tiese equations agree well with the computer results. Using the maximum values of the geometric shape deviations allowed by some national Codes, the corresponding theoretical buckling strengths were calculated. These were then compared with an approximate lower bound of test results obtained on externally pressurized spherical shells. The agreement between the two curves was not very goodfor BS 5500 but was fair,for the DnV rules. The agreement with BS 5500 can be improved b y increasing simp, the arc length over which the initial imperfections are measured. The foregoing lower bound of test results on externally pressurized spherical shells can also be obtained, approximately, using increased-radius and Legendre polynomial imperfections in which the ratio Rimp/R is not restricted. The magnitude of the initial imperfection required for approximate agreement between the experimental and theoretical results was 6,/t x 0.5. This seems a reasonable value. However, more study of this aspect of the problem is required in both the elastic and plastic buckling regions. The limitution of Rimp/R < 1.3 imposed by some Codes should also be reviewed, particularly in the plastic regime. NOTATIONdiameter of cylindrical shell modulus of elasticity degree of Legendre polynomial [equation number of waves in the circumferential direction collapse pressure of imperfect hemispherical shell elastic buckling pressure of perfect spherical shell from linear theory = 1.21E(t/R)' for v = 0.3 elastic, or plastic, bifurcation buckling pressure of perfect clamped hemisphere (from BOSOR 5) yield pressure of spherical shell = 2(t/R)ayp Legendre polynomial of degree n radius of perfect hemispherical shell (also radial coordinate on Figs 8 and 9) radius of imperfect hemispherical shell arc length over which radial imperfections, or radii of curvature, are measured thickness of shell axial coordinate on Figs 8 and 9 semi-angle of imperfection (see Fig. 2) amplitude (inwards herein) of radial imperfection at the pole (rp = 0) {12(1 -V~)}"~IX,/(R/~) % 1.82a.,,/(Rt) for v = 0.3

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In the diameter-to-thickness range 250 < D/t < 1000, internally pressurized torispherical shells can fail either by plastic buckling or by axisymmetric yielding. However, the present Code rules cater only for the axisymmetric yielding mode and they also restrict the D/t ratios to being less than 500. The rules are based on limit analysis results and these can be conservative for this problem. With regard to internal pressure buckling, there are as yet no design rules in either the American or the British pressure vessel Codes to prevent its occurrence. To provide guidance for a more accurate formulation of design rules for both of these failure modes over the range 300 < D/t < 1500, the authors have made a series of calculations to determine the values of Pcr (the internal buckling pressure) and pc (the axisymmetric yielding pressure) for perfect torispherical shells. The availability of these results, obtained with a finite-deflection shell theory, enables curves to be drawn showing when buckling is the controlling failure mode and when axisymmetric yield controls. A comparison is also made, for D/t < 600, between the controlling failure pressures mentioned above and the Drucker-Shield limit pressures. The ratio between the former and the latter varied between 1.2 and 1.8, depending on the geometry of the shell and the magnitude of the yield point, σyp. Considerable economies in the designs of many torispherical shells could, therefore, be achieved if the relevant sections of the Codes were to be modified to take advantage of the foregoing results. The controlling failure pressure curves also indicate how Code rules to prevent plastic buckling for D/t > 500 might be formulated. For the benefit of designers, the numerical values of pcr and pc were transformed, using curve-fitting techniques, into simple approximate equations. Although these equations are for perfect torispherical shells, they should be very beneficial when analysing the related problems of fabricated torispheres in practice.