MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2018/5260639 5260639 Research Article Analytical Solution of the Stability Problem for the Truncated Hemispherical Shell under Tensile Loading http://orcid.org/0000-0002-9964-5882 Kalamkarov Alexander L. 1 Andrianov Igor I. 2 Campos-Canton Eric 1 Department of Mechanical Engineering Dalhousie University Halifax Nova Scotia Canada B3H 4R2 dal.ca 2 Institut für Bauforschung RWTH Aachen University Schinkelstr. 3 52062 Aachen Germany rwth-aachen.de 2018 12112018 2018 02 10 2018 01 11 2018 12112018 2018 Copyright © 2018 Alexander L. Kalamkarov and Igor I. Andrianov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Natural Sciences and Engineering Research Council of Canada
1. Introduction

The tensile instability of a truncated hemispherical shell was analyzed for the first time by Yao (1963) . It was assumed that the equatorial edge of the shell was clamped and the other edge was subjected to the constant tensile loads. After buckling occurred a large number of buckling waves were formed along the circumferential coordinate. The prebuckling state was assumed as linear membrane, and the bifurcation equations were based on the theory of shallow shells, for The bifurcation equations were solved by Yao  using the Bubnov-Galerkin approach. An empirical formula for the critical load was proposed on the basis of the numerical results. Test data for five samples of various geometry provided by Douglas Aircraft Company were also reported in .

This problem was revisited by Wu and Cheng (1970) . They used Sanders’ shell theory and assumed that prebuckling state was nonlinearly axisymmetric. A finite difference numerical approach was used for solving the bifurcation problem. The numerical results obtained in [1, 2] were in a good agreement.

Bushnell (1967)  (see also [4, 5] for more details) performed the FEM numerical simulations. He showed that for a shallow spherical cap the linear membrane state assumption was not justifiable, and it could lead to the critical load levels different from those predicted by the nonlinear solution. At the same time, for close to hemisphere truncated spherical cap influence of edge effect was negligible. Application of Reissner’s simplified equations did not lead to significant differences in the values of bifurcation load.

Grigolyuk and Lipovtsev (1970)  (see also ) applied finite difference approach for analysis of buckling of shells of revolution under tensile loading. They used theory of shallow shells and bifurcation theory. They took into account the momentness of the prebuckling state. Numerical results obtained in  for(1)α>10231-ν2hRpractically coincide with results obtained by Yao . The angle α is defined in Figure 1.

Analysis carried out in Bushnell (1967) [3, 4] allows obtaining the following estimate of applicability of the linear theory:(2)π2-α>20231-ν2hR.

Radhamohan and Prasad (1974)  used Sanders’ equations and the numerical method to prove that the different boundary conditions did not affect the buckling phenomena of hemispherical shells even when the prebuckling state was represented by a general nonlinear bending state.

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., [3, 9].

2. Analytical Solution

The truncated shell of revolution under study is shown in Figure 1 .

The determining system of partial differential equations in the spherical coordinate system can be written as follows:(3)1121-ν2hR24w-2F-p2wϕ2-sec2ϕ2wθ2+tanϕwϕsec2ϕ=0;4F+2w=0,where(4)4=22;2=2ϕ2+sec2ϕ2θ2-tanϕϕ;w=WR;F=ΦEhR2;p=PEh;W is the radial displacement; Φ is the stress function; E is Young’s modulus; ν is Poisson’s ratio; P is the linear tensile force applied to the upper edge of the shell.

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 [24, 8], where more accurate Sanders’ and Reissner’s theories of shells were used. The second assumption is justified under the conditions given by (1) and (2). These conditions are assumed in the sequel.

System of (3) must be supplemented by boundary conditions; for example,(5)w=wϕ=F=Fϕ=0atϕ=0,α.

Boundary value problem defined by (3) and (5) describes bifurcation of the shell of revolution under study under the assumption of the membrane prebuckling state. Note that as shown in  the buckling conditions have a small effect on the buckling load.

Considering that the loss of stability occurs with the formation of a large number of half-waves in the annular direction, (1) can be simplified as follows:(6)1121-ν2hR2sec4ϕ4wθ4-sec2ϕ2Fθ2+psec4ϕ2wθ2=0;sec2ϕ4Fθ4+2wθ2=0.

Equations (6) can be reduced to the following single equation:(7)1121-ν2hR24wθ4+cosec4ϕw+p2wθ2=0.

Circumferential prebuckling stress is given by the formula(8)Tθpr=-Psec2ϕ

It is logical to fix the value of the variable ϕ in (7) at which the prebuckling stress Tθpr reaches its maximum value. It follows from (8) that this value is ϕ=α. However, due to the presence of the edge effect, this value will be shifted by an amount of the order of h/R characterizing the extent of the edge effect.

Using asymptotic estimates from  and numerical data from [3, 4], it can be assumed that the maximum value of Tθpr is reached at ϕ=β=α-4h/R. Consequently, ϕ=β is assumed in (7).

Accordingly, ϕ=α is substituted in (6):(9)1121-ν2hR24wθ4+cosec4βw+p2wθ2=0.

Equation (9) describes the bifurcation loss of stability of a circular ring with an elastic filler of the Winkler type.

Writing down its solution in the form w=sinnθ and minimizing by n yields the following formula for the bifurcation tension load and number of circumferential waves:(10)p=h31-ν2Rcos2β0.577h1-ν2Rcos2β,(11)n=121-ν24Rhcosβ.

3. Comparison with the Empirical Formulas, Numerical and Experimental Results

Yao  and Bushnell [3, 4] derived simple empirical formulae for the bifurcation tension load using numerical data. For a truncated hemisphere, they can be written as follows:

according to Yao , (12)p=1.57cos2αhR1.12;

according to Bushnell [3, 4],(13)p=0.622h1-ν2Rcos2α-3.1hR;(14)n=1.84Rhcosα-4.2hR.The structure of formulae (12) and (14) coincides with (10) derived using the asymptotic method. The comparison of numerical results obtained using the presently derived formula (10) and the formula (12)  for some values of parameters is given in Table 1.

Comparison of numerical, asymptotic, and experimental results.

 α 0.1287 0.1306 0.1889 R / h 1600 480 757 1 0 4 p ; p calculated from (10) 3.78 12.87 7.48 1 0 4 p ; p calculated from (12)  3.81 13.49 7.05 1 0 4 p e x ; pex experimental data from  1.47 6.15 4.72

The accuracy of the asymptotic solution, as one would expect, increases with the ratio R/h and is quite satisfactory.

Formula (13) [3, 4] yields values of the bifurcation load by 8-10% higher in the considered range of parameters than formula (10).

Regarding comparison with the experimental data, the test results have been reported in  for 5 samples. The theoretical values are found to be greater than the experimental ones. The explanation of this common mismatch in the theory of elastic stability of thin-walled systems is given in [9, 11]. It is related to a very high sensitivity to the initial geometrical imperfections and residual stresses.

4. Some Generalizations

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:(15)D2B1R24wθ4+cosec4ϕw+p2wθ2=0.

Here

w = W / R ; F = Φ / B 1 R 2 ; p = P / B 1 ; D2 is the bending stiffness in circumferential direction; B1 is the membrane stiffness in meridional direction.

Note that in deriving (15), it was assumed that the orthotropy of the shell was not very strong, i.e., the flexural (membrane) stiffness in the circumferential and meridional directions is of the same order: D1~D2, B1~B2.

The following expressions are obtained for the bifurcation tension load and number of circumferential waves:(16)p=D2B1R2cos2β;n=B1R2D24cosβ;β=α-2B1R2D24.

Consider now the general case of shell of revolution with radii of curvature R1,R2. The original equation can be represented as(17)1121-ν2hR224wθ4+cosec4ϕR2R12w+pR2R12wθ2=0.

In deriving (17), it was assumed that the variation of the radii of curvature in spatial coordinates is small: Ri/θ~Ri;Ri/ϕ~Ri,i=1,2.

The following expressions are obtained in this case for the bifurcation tension load and number of circumferential waves:(18)p=h31-ν2R2βcos2β0.577h1-ν2R2βcos2β;n=121-ν24R2βhcosβ.

5. Conclusions

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 [10, 1214] for the original boundary value problem, as it is done in the present paper. In this way, after substantial simplifications, a simple analytical formula (10) for the bifurcation tension load is derived.

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 [1, 3, 4].

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

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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

Yao J. C. Buckling of a truncated hemisphere under axial tension AIAA Journal 1963 1 10 2316 2319 2-s2.0-85003368799 10.2514/3.2059 Wu M. T. Cheng S. Nonlinear asymmetric buckling of truncated spherical shells Journal of Applied Mechanics 1970 37 3 651 660 2-s2.0-85003330843 10.1115/1.3408594 Zbl0201.27103 Bushnell D. Bifurcation phenomena in spherical shells under concentrated and ring loads AIAA Journal 1967 5 11 2034 2040 2-s2.0-79954556684 10.2514/3.4358 Bushnell D. Computerized Buckling Analysis of Shells 1989 Dordrecht, The Netherlands Kluwer AP Shilkrut D. Stability of Nonlinear Shells on the Example of Spherical Shells 2002 Amsterdam, The Netherlands Elsevier 10.1016/S0922-5382(02)80029-0 Grigolyuk E. I. Lipovtsev Yu. V. To the solution of one class of eigenvalue problem for thin shells of revolution. Problemy Mekhaniki Tverdogo Deformiruemogo Tela (Problems of Mechanics of Solids) 1970 Leningrad, Russia Sudostroyenie Grigoljuk I. Kabanov V. V. Stability of Shells 1978 Nauka, Moscow Radhamohan S. K. Prasad B. Asymmetric buckling of toroidal shells under axial tension AIAA Journal 1974 12 4 511 515 2-s2.0-0016050045 10.2514/3.49277 Zbl0281.73029 Pedersen P. T. Jensen J. J. Buckling of spherical cargo tanks for liquid natural gas Transactions of the Royal Institution of Naval Architects 1976 118 193 205 Goldenveizer A. L. Theory of elastic thin shells 1961 Oxford Pergamon Press MR0135763 Hutchinson J. W. Initial post-buckling behavior of toroidal shell segments International Journal of Solids and Structures 1967 3 1 97 115 2-s2.0-0346869553 10.1016/0020-7683(67)90046-7 Andrianov I. V. Awrejcewicz J. Danishevs'kyy V. V. Ivankov A. Asymptotic Methods in the Theory of Plates with Mixed Boundary Conditions 2014 Chichester, UK Wiley 2-s2.0-84921979770 Kalamkarov A. L. Asymptotic homogenization method and micromechanical models for composite materials and thin-walled composite structures Mathematical methods and models in composites 2014 5 Imp. Coll. Press, London 1 60 Comput. Exp. Methods Struct. 10.1142/9781848167858_0001 MR3289880 Zbl1302.74136 Andrianov I. I. Analytical investigation of buckling of a cylindrical shell subjected to nonuniform external pressure Mathematics and Mechanics of Solids 2018 108128651875617 10.1177/1081286518756179