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Analytical solution of the problem of buckling of truncated hemispherical shell of revolution, subjected to tension loading, is obtained. Assumption of membrane prebuckling state is applied, and the range of applicability of this assumption is estimated. The developed algorithm is based on the asymptotic simplification procedure of bifurcation equations. The formula for the bifurcation tension load is derived and compared with the earlier published empirical and numerical results. It is shown that it is sufficiently accurate and can be used in engineering practice.

The tensile instability of a truncated hemispherical shell was analyzed for the first time by Yao (1963) [

This problem was revisited by Wu and Cheng (1970) [

Bushnell (1967) [

Grigolyuk and Lipovtsev (1970) [

Truncated hemispherical shell under tensile loading.

Analysis carried out in Bushnell (1967) [

Radhamohan and Prasad (1974) [

In the present paper a simple formula for the bifurcation load is derived for a truncated hemispherical shell under the tensile loading.

Note that the problem under study has both the fundamental analytical importance and the practical applied significance. Very large spherical tanks for transporting liquid natural gas are supported by the relatively short cylindrical shells. When the tank is filled by less than half its weight creates axial tension that might cause buckling; see, e.g., [

The truncated shell of revolution under study is shown in Figure

The determining system of partial differential equations in the spherical coordinate system can be written as follows:

The shallow shell theory will be used and the membrane prebuckling state will be assumed.

The validity of the first assumption follows from the results of [

System of (

Boundary value problem defined by (

Considering that the loss of stability occurs with the formation of a large number of half-waves in the annular direction, (

Equations (

Circumferential prebuckling stress is given by the formula

It is logical to fix the value of the variable

Using asymptotic estimates from [

Accordingly,

Equation (

Writing down its solution in the form

Yao [

according to Yao [

according to Bushnell [

Comparison of numerical, asymptotic, and experimental results.

| 0.1287 | 0.1306 | 0.1889 |

| |||

| 1600 | 480 | 757 |

| |||

| 3.78 | 12.87 | 7.48 |

| |||

| 3.81 | 13.49 | 7.05 |

| |||

| 1.47 | 6.15 | 4.72 |

The accuracy of the asymptotic solution, as one would expect, increases with the ratio

Formula (

Regarding comparison with the experimental data, the test results have been reported in [

One of the advantages of an asymptotic solution is that it allows a simple generalization to other thin-walled shells. As the first example of generalization, consider an orthotropic hemispherical shell. The simplified stability equation in this case can be written as follows:

Here

Note that in deriving (

The following expressions are obtained for the bifurcation tension load and number of circumferential waves:

Consider now the general case of shell of revolution with radii of curvature

In deriving (

The following expressions are obtained in this case for the bifurcation tension load and number of circumferential waves:

The problem of buckling of truncated hemispherical shell subjected to tension loading is solved analytically using the asymptotic simplification procedure for the bifurcation equations. The membrane prebuckling state is assumed, and the range of applicability of this assumption is estimated. The formula for the bifurcation tension load is derived and compared with the earlier published empirical and numerical results. It is shown that it is sufficiently accurate and can be used in engineering practice.

The results of the present paper demonstrate that it is possible to obtain a simple and accurate engineering formula in two ways.

The first one is to apply the asymptotic method [

The second approach involves obtaining a large array of numerical data, applying standard codes and consequently reprocessing information using the empirical formulae. This approach was used in [

It is worth noting that, for the problem under consideration, both approaches led to approximately similar results, thereby confirming their reliability.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This work has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).