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This paper presents a solution for the problem of three cylindrical shells, one of which is slotted lengthwise. With slot length as one of the parameters, the solution includes the solid three-shell problem as a particular case. The results are applicable to the stress problem of a closed-end vessel supported either by a slotted or an ordinary skirt, or one that is mounted on top of another vessel. It is shown that slotting substantially reduces the critical stress at the junction.

Thermal stresses in cylinders, perfectly insulated on the outside and subjected to internal heat generation due to γ-ray radiation, are considered. Numerical values for the governing stress are presented up to q = 7 and η = 100. The asymptotic values are indicated, which permits the treatment of any problem of practical interest. Three simplified solutions, containing only elementary functions, are evaluated. These approximations are found to be surprisingly accurate, at least one of them being of sufficient engineering accuracy over the entire range of variables.

The stress distribution in circular bends formed from straight pipe and subjected to internal pressure is presented. The analysis determines first the thickness distribution after forming, followed by an examination of the validity of membrane solutions for a complete torus. The predicted thickness distribution is in excellent agreement with experimental results as well as with three previous empirical formulations. The analysis shows that stresses in curved bends will always exceed the governing hoop stress in a comparable straight pipe, the deviation becoming greater with increased corrosion allowance. The authors recommend that, for purposes of acceptance by the ASA B31 Code, an excess stress level of the order of 5 per cent be allowed to develop in pipe bends. The resulting bend radius to pipe diameter limitations are listed in Table 3 and shown in Fig. 5. The last section of the paper deals with the major assumptions made in the derivation, and finds all of them to be acceptable for purposes of this analysis.

Instability (maximum pressure) criteria for materials obeying a parabolic stress-strain law are investigated. Two special solutions are formulated, accompanied by the instability condition obtained from the author’s approximate theory. Bursting pressure is most accurately predicted by the approximate theory. In turn, the polar deflection and strain at instability are more closely predicted by special solution II. It is shown that neither the “pureness” nor the annealed condition of a metal gives adequate assurance of reliability in frangible disk service. The way to insure this requirement is to confirm experimentally that, in the uniaxial test, fracture will not occur before the logarithmic strain exceeds roughly (0.36 + 0.73 m), or that in a bulge test the applied hydrostatic pressure passes through a maximum prior to bursting.

Factors determining the performance of welded skirt supports for vertical process vessels are discussed. The importance of thermal effects is emphasized. A comprehensive analysis of local stresses with closed-form solutions is presented, permitting assessment of expected performance in service on a fatigue basis. The expected shape of skirt distortion is briefly mentioned. Suggestions and recommendations are made for design practice, control of thermal stresses, weld and fabrication detail for optimum practical quality, as dictated by service conditions. A numerical example is presented to illustrate the application of the analysis.

Solutions are presented for the bursting strength of thin-walled cylinders of finite length with various end restraints, including (a) two rigid end plates, (b) two hemispherical caps, and (c) one rigid end plate and one hemispherical cap. The deformation theory of plasticity is used, together with the Mises yield criterion, for materials obeying a linear strain-hardening law. Comparisons of the results are made to determine the influence of end restraints on burst pressure with numerical results presented for cylinder lengths from 0 (sphere) to ∞, and for a range of hardening constants. The results show that the effect of shortness is small (less than 8 percent), and the influence of end conditions negligible, in the range l > 3. For shorter cylinders, the finite length of the shell serves to increase burst pressures, to values nearly double those for the infinite cylinder, as l → 0. Also, the effect of end restraints becomes very pronounced, the cylinder with the two rigid end plates being the strongest, and that with two hemispherical caps the weakest; all other types of end closures are assumed to have a strengthening effect intermediate to these two cases.

An elastic analysis of U-shaped expansion joints under axial loads and internal or external pressure is presented. The analysis employs the energy method for the toroidal sections, and the theory of symmetrical bending of circular plates augmented by thickwalled cylinder analysis for the annular plate connecting the two toroidal sections. The general solution permits the investigation of any U-shaped expansion joint falling in the range of thin shells, for any arbitrary combination of axial force and pressure loading. Different forming radii may be assigned to the inner and outer toroidal sections, and it is permissible to vary the average thicknesses of the two toroidal sections and the interconnecting annular plate independently. Expressions are given for the load-extension and load-stress characteristics of U-shaped expansion joints, and a numerical example is presented comparing the present solution with results of existing approximate analyses. The method presented here lends itself readily to programming on an electronic computer.

A solution is obtained by means of the Ritz method for the “large-deflection” problem of a clamped elliptical plate of constant thickness, subjected to a uniformly distributed load. Two shapes of elliptical plate are treated, in addition to the limiting cases of the circular plate and infinite strip, for which the exact solutions are known. Center deflections as well as total stresses at the center and edge decrease as one proceeds from the infinite strip through the elliptical shapes to the circular plate, holding the width of the plates constant. The relation between edge-stress at the semiminor axis (maximum stress in the plate) and center deflection is found to be practically independent of the proportions of the elliptical plate. Hence the governing stress may be determined from a single curve for a given load on an elliptical plate of arbitrary dimensions, if the center deflection is known.

An investigation is presented for the plastic behavior of clamped circular membranes subjected to hydrostatic pressure, based on the Hencky-Mises theory. New strain-displacement relations, valid for finite deformations in the plastic domain, are introduced. The solution is presented in terms of two simultaneous implicit integral equations, which can be solved by numerical methods. The solution also permits the formulation of the instability condition. Experimental work consisted of bulge tests carried out on annealed copper plates of two thicknesses, whose τ-γ curve was previously determined from uniaxial tests. A good correlation is obtained between theoretical predictions and experimental results.