Stability of Complete Circular and Non-Circular Toroidal Shells

Abstract

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.

Simplified theoretical solution of circular toroidal shell with ribs under uniform external pressure

“…w k n 1 12 while the deformation and main stress at the middle of curvebeam can also be calculated by formula (14) and (15) w k n pa Et 14 Then from Eqs. (9) and (10), the normal displacement w and internal displacement u can be obtained too, so the stresses at the key position of curve beam may be solved totally from Eqs.…”

“…(9) and (10), the normal displacement w and internal displacement u can be obtained too, so the stresses at the key position of curve beam may be solved totally from Eqs. (12)- (15).…”

“…The buckling of the segments of toroidal shells had been solved by Stein and Mcelman [13] and Hutchinson [14]. Galletly [15,16] had studied the stability of closed toroidal shells with circular or non-circular cross-section. Blachut [17] had obtained three toroidal models experiments and revealed the collapse pressure by different material, geometry and manufactory, compared and confirmed with numerical results.…”

Buckling analysis of an orthotropic thin shell of revolution using differential quadrature

“…The buckling of complete isotropic toroidal shells under uniform external pressure has been considered by a number of authors [6][7][8][9][10][11]. Sobel and Flügge [6] presented a theoretical series solution to equations corresponding to the Sanders shell theory.…”

“…Wang and Zhang [9] derived asymptotic stability equations from the non-linear Sanders equations of equilibrium. Galletly and Blachut [10] calculated results for toroidal shells of circular and elliptical cross-section using a finite difference program. Blachut and Jaiswal [11] conducted a detailed numerical study in which extensive results were given for toroidal shells of circular and elliptical cross-sections.…”