Buckling Of Shallow Spherical Shells Research Articles

Snap-through Buckling of Shallow Spherical Shells under Ring Loads

Snap-through buckling behaviour of rigid vinyl polyethylene shallow spherical shells, undergoing large displacements, subjected to static ring loads is investigated by using finite difference and the Newton-Raphson Method. The load-deflection diagrams corresponding to various values of thickness, depth and ring diameter of the shell with simply supported and clamped edges are sketched and compared. The accuracy of the used algorithm and the prepared computer program are tested by comparing some of the numerical results obtained in this study with those obtained by an experimental study, available in the relevant literature. Further steps on the concerning subject are achieved.

ПОТЕРЯ УСТОЙЧИВОСТИ ОСЕСИММЕТРИЧНЫХ ФОРМ РАВНОВЕСИЯ ПОЛОГИХ СФЕРИЧЕСКИХ ОБОЛОЧЕК ПОД ДЕЙСТВИЕМ ВНУТРЕННЕГО ДАВЛЕНИЯ

ПОТЕРЯ УСТОЙЧИВОСТИ ОСЕСИММЕТРИЧНЫХ ФОРМ РАВНОВЕСИЯ ПОЛОГИХ СФЕРИЧЕСКИХ ОБОЛОЧЕК ПОД ДЕЙСТВИЕМ ВНУТРЕННЕГО ДАВЛЕНИЯ

Buckling of Externally Pressurized Shallow Spherical Caps from Composites

Buckling of shallow spherical shells subjected to external pressure has been exploited, for example, in pressure safety devices. Shallow spherical caps have also been used as mirrors and more recently they have been considered as energy concentrators in space applications. Hence there is a renewed interest in their static stability performance. This paper addresses two issues: (1) buckling performance of caps, with shallowness parameter, lambda, varying from 3.5 to 7.5, and (2) the effect of boundary conditions on the buckling strength. In the first case, for elastic buckling of caps, there is a sudden drop in the buckling strength for small magnitude of cap’s geometrical parameter lambda, i.e., for lambda ≈ 4.0. This has recently been re-confirmed for steel caps. However, it is not clear whether this drop can be eliminated or reduced in caps made from composites mainly through the lamination sequence and also by appropriate supporting of the caps perimeter—as typically used in inflatable space antenna. The current paper examines spherical caps made from composites subjected to quasi-static external pressure. The effect of different lamination sequences on the buckling strength is examined for a range of the shallowness parameter. In the second part, it is proposed to use a closed toroidal shell as means of supporting the cap’s perimeter. The supporting toroidal shells can be of different cross-section, i.e., circular or elliptical. The toroids can also be internally pressurized, differently laminated, etc. Additionally the point of attachment can be chosen in such a way that buckling load is as large as possible. The effect of some of these parameters on the buckling pressures is quantified. This is an entirely numerical study.

Snap-Through Buckling Analyses of Composite Shallow Spherical Shells

The purpose of this work is to present some numerical results on nonlinear, snap-through buckling of shallow spherical shells made of laminated composite materials under general static loading. Isotropic shells are covered as a special case of the general problem. A special-purpose finite-difference computer program has been developed. Moreover, the general-purpose computer program ANSYS has also been used. Effects of various material properties on the nonlinear behavior of the shells are investigated. One of the conclusions of the study is that ANSYS may yield erroneous results in some cases. Another conclusion is that the buckling curves of composite shallow spherical shells have a shape similar to that of the corresponding isotropic shell. This finding implies that the geometry of a shell may be an important factor in giving a characteristic shape to the buckling curve, at least for the shell parameters considered here.

Nonlinear analysis of axisymmetric deformation of shallow spherical shells by least squares collocation method

A nonlinear analysis for deformation and buckling of shallow spherical shells with a central circular opening is presented. The shell is under a vertical ring load at the opening and/or a uniformly distributed surface load. The method of least square collocation is used in the analysis. Examples for the degenerated case of annular plates and shallow spherical shells with a central opening under different loading conditions are presented. The results agree well with existing solutions, and the method used is effective and accurate resulting in steady convergent solutions according to the numerical examples. While the buckling load for clamped shallow spherical shells subjected to uniformly distributed surface load increases with the increase of the rise-to-thickness ratio, the buckling load decreases when the sehll is under a uniformly distributed ring load at the central opening.

Nonlinear dynamic response and dynamic buckling of shallow spherical shells with circular hole

In this paper, the nonlinear equations of motion for shallow spherical shells with axisymmetric deformation including transverse shear are derived. The nonlinear static and dynamic response and dynamic buckling of shallow spherical shells with circular hole on elastically restrained edge are investigated. By using the orthogonal point collocation method for space and Newmark-β scheme for time, the displacement functions are separated and the nonlinear differential equations are replaced by linear algebraic equations to seek solutions. The numerical results are presented for different cases and compared with available data.

Non-axisymmetric buckling of shallow spherical shells

The buckling of elastic shallow orthotropic spherical shells subjected to a transverse load is investigated on the basis of geometrically non-linear equilibrium equations in a non-axisymmetric formulation. By using the method of finite differences and a continuation procedure in the prameters in combination with a Newton operator method an algorithm is constructed to determine the state of shell stress and strain in the pre- and post-critical stages. The upper critical loads (CL) of spherical shells are determined for different external pressure distribution laws taking perturbing factors such as initial harmonic and azimuthal imperfection directions in the shape of the shell middle surface and analogous load deviations from a uniformly distributed load into account. Under the imperfections mentioned, good agreement is obtained with the results for the upper CL found by the theory of buckling and the initial post-critical behaviour /1–4/. Special attention is paid to an investigation of the non-axisymmetrical buckling of an isotropic spherical shell closed at the apex. It is shown that the presence of small initial imperfections is the reason for a substantial reduction in the upper critical load and, moreover, its values can be determined by the formula for unimodal buckling not at the least bifurcation point but at the one following if the initial damage component proportional to the harmonic natural mode of this second bifurcation point is predominant.

Effects of Initial Imperfections and Boundary Conditions on Buckling of Shallow Spherical Shells

Effects of Initial Imperfections and Boundary Conditions on Buckling of Shallow Spherical Shells

Experimental investigation of the buckling of shallow spherical shells

In the present paper, the buckling behavior of clamped thin shallow spherical shells under external pressure is studied. Seventy-nine plastic shells formed by thermovacuum process were tested. The distributions of initial geometrical imperfections and vertical displacements were minutely measured with a differential transformer. It was possible to control the symmetrical initial geometrical imperfection of each specimen. Results indicate that the buckling phenomena of shallow spherical shells vary greatly with the symmetrical initial imperfection parameter η. In the case of the geometrical parameter λ larger than 5.5, the amplitude of the asymmetrical displacement component with the bifurcation buckling wave calculated by Huang becomes large immediately before buckling. The validity of Huang's theory for an initially perfect shell is experimentally demonstrated.

Effects of Boundary Conditions on Buckling of Shallow Spherical Shells

外圧を受ける部分球かくの座屈に及ぼす軸対称な初期不整と境界条件の不完全性の影響を有限要素法により解析した.これにより初期不整および境界条件の不完全性が座屈に及ぼす作用は球かくの形状パラメータあるいは変形モードによって全く異なることを明らかにした.この事実により従来の実験データに見られる全く異なった座屈圧力の分布に統一的な説明を与えることができるようになった.

Axisymmetrie Dynamic Buckling of Shallow Spherical Shells

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und MechanikVolume 60, Issue 2 p. 118-120 Kleine Mitteilungen Axisymmetrie Dynamic Buckling of Shallow Spherical Shells Prof. R. S. Alwar, Prof. R. S. Alwar Department of Applied Mechanics, I. I. T., Madras 600036, IndiaSearch for more papers by this authorYogendra Nath, Yogendra Nath Department of Applied Mechanics, I. I. T., Madras 600036, IndiaSearch for more papers by this authorB. Sekhar Reddi, B. Sekhar Reddi Department of Applied Mechanics, I. I. T., Madras 600036, IndiaSearch for more papers by this author Prof. R. S. Alwar, Prof. R. S. Alwar Department of Applied Mechanics, I. I. T., Madras 600036, IndiaSearch for more papers by this authorYogendra Nath, Yogendra Nath Department of Applied Mechanics, I. I. T., Madras 600036, IndiaSearch for more papers by this authorB. Sekhar Reddi, B. Sekhar Reddi Department of Applied Mechanics, I. I. T., Madras 600036, IndiaSearch for more papers by this author First published: 1980 https://doi.org/10.1002/zamm.19800600214Citations: 12AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume60, Issue21980Pages 118-120 RelatedInformation

Closure to “Discussion of ‘Dynamic Buckling of Shallow Spherical Shells’” (1974, ASME J. Appl. Mech., 41, p. 834)

Closure to “Discussion of ‘Dynamic Buckling of Shallow Spherical Shells’” (1974, ASME J. Appl. Mech., 41, p. 834)

On the Dynamic Buckling of Shallow Spherical Shells

On the Dynamic Buckling of Shallow Spherical Shells

Application of Finite Element Method to Axisymmetric Buckling of Shallow Spherical Shells under External Pressure

The finite element method is applied to the axisymmetric buckling of shallow spherical shells under uniform external pressure. In order to analyze the nonlinear behavior of shells with sufficient accuracy and efficiency, the simple load incremental method is modified by introducing the mid-increment stiffness matrix and residual load correction. The analysis is performed on a fairly wide range of the geometric parameter α to obtain pressure-deflection curves in both pre-and post-buckling states. The results appear generally satisfactory, at least up to the early stage of post-buckling. In particular, the upper and lower buckling pressures are obtained in excellent agreement with known analytical solutions.

Dynamic Buckling of Shallow Spherical Shells

The dynamic behavior of clamped shallow spherical shells subjected to axisymmetric and nearly axisymmetric step-pressure loads is examined using a digital computer program for the geometrically nonlinear static and dynamic analysis of arbitrarily loaded shells of revolution. A criterion for dynamic buckling under the nearly axisymmetric load is proposed and critical buckling pressures are determined for a large range of shell sizes.

Inelastic Buckling of Shallow Spherical Shells Under External Pressure

In this paper the inelastic effect on the axisymmetric buckling load of a clamped shallow spherical shell subjected to an external pressure is investigated. A variational principle derived from a set of rate equilibrium equations and expressed in terms of Lagrangian variables is introduced. Then, the variational principle is employed with a step-by-step numerical process based on a Rayleigh-Ritz technique to determine the initial pressure. The spherical shell is idealized as a structure composed of four thin load-carrying layers of equal thickness and made of a work hardening material. The numerical results of the present analysis compare favorably with those of available experimental work.

Application of Finite Element Method to Axisymmetric Buckling of Shallow Spherical Shells under External Pressure

The finite element method is applied to the axisymmetric buckling of shallow spherical shells under uniform external pressure. In order to analyze the nonlinear behavior of shells with sufficient accuracy and efficiency, the simple load incremental method is modified by introducing the mid-increment stiffness matrix and residual load correction. The analysis is performed on a fairly wide range of the geometric parameter α to obtain pressure-deflection curves in both pre-and post-buckling states. The results appear generally satisfactory, at least up to the early stage of post-buckling. In particular, the upper and lower buckling pressures are obtained in excellent agreement with known analytical solutions.

Limit surfaces for fibrous composite plates

Deformations of Thin Spherical Shells, AIAA Journal, Vol. 3, No. 9, Sept. 1965, pp. 1716-1725. 3 Bushnell, D., Behavior of Shells of Revolution, AIAA Journal, Vol. 5, No. 3, March 1967, pp. 433-439. 4 Huang, N. C., Axisymmetric Snap-through of Elastic Clamped Shallow Spherical Shells, AIAA Journal, Vol. 7, No. 2, Feb. 1969, pp. 215-220. 5 Fitch, J. R. and Budiansky, B., Buckling and Post Buckling Behavior of Spherical Caps under Load, AIAA Journal, Vol. 8, No. 4, April 1970, pp. 686-693. 6 Weinitschke, H. J., On Buckling of Shallow Spherical Shells, Journal of Mathematics and Physics, Vol. 44, No. 2, June 1965, pp. 141-163. 7 Mescall, J., Large Nonsymmetric Deflections of Thin Shells of Revolution, Proceedings of the Army Symposium on Solid Mechanics, Army Materials and Mechanics Research Center, Watertown, Mass., Sept. 1968. 8 Leicester, R. H., Finite Deformations of Shallow Shells, Journal of the Engineering Mechanics Division of ASCE, Vol. 94, No. EM 6, Dec. 1968, pp. 1409-1423. 9 Ball, R. E., A Geometrically Nonlinear Analysis of Arbitrarily Loaded Shells of Revolution, CR-909, Jan. 1968, NASA. 10 Famili, J. and Archer, R. R., Finite Deformation of Shallow Spherical Shells, AIAA Journal, Vol. 3, No. 3, March 1965, pp. 506-510. 11 Stricklin, J. A., Haisler, W. E., MacDougall, H. R., and Stebbins, F. J., Analysis of Shells of Revolution by the Matrix Displacement Method, AIAA Journal, Vol. 6, No. 12, Dec. 1968, pp. 2306-2312. 12 Stricklin, J. A. and Martinez, J. E., Dynamic Buckling of Clamped Spherical Caps under Step Pressure Loadings, AIAA Journal, Vol. 7, No. 6, June 1969, pp. 1212-1213. 13 Liepins, A. A., Asymmetric Nonlinear Response and Buckling of Shallow Spherical Shells, CR-1376, June 1969, NASA. 14 Perrone, N. and Kao, R., A General Relaxation Technique for Solving Nonlinear Problems, Presented at the 12th International Congress of Applied Mechanics, Stanford, Calif., Sept. 1968. 15 Southwell, R. V., Relaxation Methods in Theoretical Physics, Vol. II, The Clarendon Press, Oxford, 1956, pp. 432-494. 16 Shaw, F. S. and Perrone, N., A Numerical Solution for the Nonlinear Deflection of Membranes, Journal of Applied Mechanics, Vol. 76, June 1954 pp. 117-128. 17 Famili, J., On the Vibration, Buckling, and Post Buckling Behavior of Shells of Revolution under Symmetrical and Unsymmetrical Load, Doctoral thesis, 1965, Engineering Div., Case Institute of Technology, Cleveland, Ohio.

Two-dimensional finite-difference equations for shallow spherical shells.

Linear static deformation and buckling of shallow spherical shells under asymmetric load by solving two dimensional finite difference equations

On Asymmetric Buckling of Shallow Spherical Shells

On Asymmetric Buckling of Shallow Spherical Shells