A comparative study on the elastic-plastic collapse strength of initially imperfect deep spherical shells

Abstract

Unfortunately, earlier tests conducted in Ref.[2] provided only one-fourth of the collapse strength predicted by the classical buckling theory. The huge discrepancy existed between the theory and the test is traceable. The test specimens used in Ref. [2] were formed from flat plates, which inevitable introduced significant departures from sphericity as well as variations in thickness and residual stresses. Since the initial imperfection among these adverse factors introduced has been assumed to be the primary source that affects the collapse strength of shell structures [3][4][5][6], the discrepancy is consequential and their comparison is inappropriate.To eliminate or at least partially reduce the adverse effect from flat plates, a series of nearly perfect machined shells were made in Refs.[7-9].The test results showed their collapse -2-strength was nearly 90% of the classical buckling pressure.These tests not only provide a strong support to the classical small deflection buckling theory of initially perfect spherical shells, but indicate that the initial imperfection does play a significant role in reducing shell load-carrying capacity.In view of the practicality, it is very difficult, if not impossible, to manufacture or measure most spherical shells with sufficient accuracy to justify the use of classical shell buckling formulas in design. It thus becomes evident that we should consider the unevenness factors in the shell buckling analysis. Since most contributions to the unevenness factors, such as variation in thickness, residual stress, boundary conditions, etc. may be, at least on occasions, are fairly well controlable, the effect of initial departures from sphericity appears most worthy of investigation.In connection with this investigation, the large deflection elastic buckling analysis was performed in Ref.[5] for complete spherical shells with a dimple type of initial imperfections.Focused only on shallow spherical portions of these complete shells, numerical solutions of these modified shell structures[6] compare quite satisfactorily with those of [5].The comparison by itself prompts a basic assumption that the collapse of initially imperfect shell structures is primarily a local phenomenon and therefore critically dependent on local geometry.For the purpose of the same investigation with an extension -3- As mentioned in the previous Section, the large deformation elastic-plastic thin shallow spherical shell theory [4] is utilized in this paper to predict the collapse strength of initially imperfect deep spherical shells. A shell is called "thin" if the ratio of its thickness to the radius of curvature of its middle surface is much less than unity; and a spherical shell is called "shallow" if its rise at the center is less than, say, one-eight of its base diameter.The geometry of a clamped spherical cap is shown in Fig. l(a), in which H is the central height, R the shell radius to the midsurface of the shell, a the base radius, and h the shell thickness; W(r) and U(r) are displacement components al...