Nonshallow Spherical Shells Research Articles

Large Deflections and Stability of Spherical Segment under Thermal and Force Loading

The effect of cryogenic and high temperatures on the nonlinear straining processes of the non-shallow spherical shells under external axisymmetric and non-axisymmetric pressure has been numerically investigated. Along with this, the limit loads buckling have been determined. Cases of substantial differences between reached and initial temperatures are considered. According to the calculations, a diversity of the buckling forms in the analyzed problems has been shown. It has been found that the largest critical loads for the deep spherical shell are observed at cryogenic temperatures for the analyzed thermal and force loading and its values decrease considerably at higher temperatures. The calculations have shown a significant dependence of the stress-strain state of the thermo-stressed segments on the accounting completeness of the temperature dependence of the material characteristics.

Nonlinear straining of shallow and non-shallow spherical shells under thermal and force loadings

Numerical results of the study of nonlinear bending and the stability loss of spherical segments with a rigidly embedded edge are obtained under the influence of external normal pressure and temperature changing in the range from cryogenic to increased. The dependence of the material characteristics on the level of the operating temperature is taken into account. A significant dependence of the stress-strain state and the stability of the segment on the nature of the interaction of external normal pressure and temperature is shown. With axisymmetric temperature and force loading, it is established that the critical loads for a shallow segment are greatest in the vicinity of the initial temperature at low temperature stresses, while for the deep spherical shell, the greatest critical loads are observed at cryogenic temperatures with a significant decrease in their values at elevated temperatures.

Numerical analysis of nonlinear straining of non-shallow spherical shell subjected to thermal and force loadings

The influence of low and high temperatures on the process of nonlinear straining and the critical load buckling of the elastic-plastic deep spherical shells under external axisymmetric and non-axisymmetric loading has been investigated in this paper. We have studied the cases where reached temperature is substantially different from the initial one. For the analyzed thermal and force loading it has been found that the highest critical loads for the deep spherical shell are observed at cryogenic temperatures, and they considerably decrease at higher temperatures.

Nonshallow spherical shells

In this paper, spherical bodies of shell type are discussed, where the displacement vector is independent of the thickness coordinate. Their internal geometry is alterable towards the thickness (nonshallow spherical shells). A planar deformation model for spherical bodies of shell type is obtained. General representations of the system of equilibrium equations are expressed with the help of three holomorphic functions for the spherical shells. The components of the stresses and displacements and the boundary conditions for the components of the stresses and displacements are also expressed with the help of three holomorphic functions.

Solution of two-dimensional boundary-value problems of the theory of thin composite shells

Discrete analogues of the boundary-value problems of a two-dimensional refined theory of anisotropic shells taking into account the transverse shear deformation are presented. The systems of resolving equations in the general form are obtained for arbitrary nonshallow shells of variable curvature whose coordinate lines of the reduction surface may not coincide with the lines of principal curvatures. The algebraic problems of determining the stress-strain state in shells made of composite materials with stress concentrators under various kinds of loads are obtained as particular cases of the schemes presented. The results of calculating the stress concentration near a nonsmall circular hole in a transversely isotropic nonshallow spherical shell under internal pressure are presented. The dependences of stress concentration factors on the hole dimension and on a change in the shear stiffness of the shells are studied. A comparison between the calculation results obtained within the framework of the theories of shallow and nonshallow shells is given.

AN APPROXIMATION TO THE VIBRATIONS OF OBLATE SPHEROIDAL SHELLS

This paper presents the results of investigating the axi-symmetric free vibrations of an isotropic thin oblate spheroidal shell. An oblate spheroid is considered as a continuous system constructed from two spherical shell caps by matching the continuous boundary conditions. This approximation technique is used to express the results in a continuous function analytical formation. The radii and opening angles of the spherical elements are chosen according to the eccentricity of the oblate spheroid.It is shown that for an eccentricity: smaller than 0·6 the Rayleigh method reasonably estimates natural frequencies; larger than 0·95 both of the shallow and non-shallow spherical shell theories predict natural frequencies in close agreement; and equal to zero an exact thin sphere solution emerges.The method presented herein is clearly an engineering approximation for special purpose shells and does not require the computer storage of numerical methods. Also, it may be quite useful when utilised judiciously in many applications, such as in force response analysis. Comparison is made between the present technique, the Rayleigh method, and experimental data for two laboratory models. The formulation can be extended to many types of oblate spheroids with different support conditions and still retain the functional representation.

Classical Versus Improved Thin Shell Theories: A Theoretical Argument or a Design Concern?

Differences in the response of thin nonshallow spherical shells resulting from the choice of the adopted shell theory (classical or improved) are addressed analytically and through a series of representative shell problems. The analytical approach utilized to study the variation between the two theoretical models is based on the response resulting from Singular loads. The differences are quantified in a set of problems that reflect on the assumptions used in formulating the analytical description of the two theories in question. The broad scope of this paper is to examine the impact of shear deformability, introduced by the improved theory on the stress field when amplified under specific loading and geometric conditions, when those are of primary concern to the engineers. Such cases associated with stress concentration around cutouts, interaction of shells with nozzles, stress field in the vicinity of concentrated surface loads, etc. The mathematical formulation is based on the derivation of appropriate Green functions and the computational scheme is formed upon a special type of boundary integral equation. Comparison solutions for stress concentration around circular cutouts of twisted and sheared shells, stress amplification in the junction of shell with nozzles, and local stress field induced by concentrated loads are presented for the two theories.

An Indirect Boundary Integral Equation Applied to Nonshallow Spherical Shell Problems With Arbitrary Boundary Constraints

The fundamental singular solutions of a complete elastic spherical shell are utilized and, via the superposition principle, an indirect boundary integral equation is formulated. The singular solutions correspond to the action of normal point loads, concentrated tangential loads, and surface moments which apply in a self-equilibrating fashion over the spherical surface. With singular solutions of such property in hand, an arbitrary spherical shell with surface loading and any set of consistent boundary constraints is embedded onto the complete sphere. A set of fictitious load vectors is introduced along the boundary line which, together with the prescribed surface traction, is required to satisfy the constraints at the boundary. The idea of an auxiliary boundary is also introduced and the solution to a number of representative shell problems is shown as compared to the available analytical and finite element method results.

Buckling of Spherical Viscoelastic Shells

The creep buckling of thin nonshallow spherical shells is treated, the material of which is assumed to be isotropic and linearly viscoelastic with special temperature dependence of its mechanical properties so as to fit the thermorheologically simple body model. The shell is subjected to uniform external pressure smaller than the classical buckling pressure for corresponding elastic shells. The prebuckling stress state is assumed to be the membrane state, and total deflections are less than or equal to the shell thickness. The classical stability criteria, uniqueness of equilibrium, is used in formulating the problem, defined by a linear integrodifferential equation in terms of displacement field W, and describing the adjacent equilibrium configuration, and it is assumed to be a function of coordinates and time. A first-step approximation to the governing equation is derived and used to determine the external critical pressure loading as a function of temperature, time, and shell geometry. For a shell made of a three-element model material a “safe load limit” is established.

Analytical solutions for open nonshallow spherical shell vibrations

This paper is concerned with axisymmetric as well as nonsymmetric vibrations of open nonshallow thin elastic spherical shells. Without employing the usual auxiliary variables for reduction of shell motion equations introduced by Van der Neut and Berry, independent analytical solutions for middle‐surface displacements are obtained and explicitly expressed in terms of associated Legendre functions. In order to gain physical insights into the free‐vibration characteristics of an open nonshallow shell, theoretical calculations together with asymptotic descriptions are made of natural frequencies and mode shapes of a hemispherical shell with a free edge. The numerical predictions obtained herein compare excellently against the experimental results obtained previously by Hwang and recently in the David W. Taylor Naval Ship R & D Center, Bethesda, Maryland. Essential features of shell dynamics are ultimately displayed by normalized frequency (Ω) and nondimensional shell thickness parameter (β). Five families of natural frequencies, i.e., low Rayleigh bending, mixed bending‐membrane, torsional, bending, and membrane frequencies, are found. The corresponding mode shapes exhibit distinctive displacement patterns. [Work supported by Naval Sea Systems Command.]

On the asymptotic value of the upper critical pressure of nonshallow spherical shells: PMM vol. 38, n≗4, 1974, pp. 760–765

The upper critical pressures of nonshallow spherical shells under uniform external pressure are determined by an asymptotic method [1, 2] as a function of the apperture angle and the mode of support. The case is considered when the parameter of relative thin-walledness is sufficiently small. The values found agree well with the results of computations by direct numerical methods and permit their continuation into the domain of arbitrarily thin shells where a machine computation becomes of very low efficiency.

Equations of the axisymmetric state of stress and strain of a nonshallow spherical shell of nonlinear elastic material under large deformations: PMM vol. 37, n≗5, 1973, pp. 934–939

A sequential derivation is given of the fundamental axisymmetric strain equations of a spherical shell made from nonlinear elastic material under the assumption of smallness of the relative elongations as compared with unity and under arbitrary angles of rotation. The most widely used versions of the boundary value problems for the investigation of nonshallow shells under large deformations were proposed in [1–5], The general formulation of boundary value problems for the investigation of nonshallow shells under large deformations is found in [6, 7]. Only the geometric nonlinearity is taken into account in all these papers. Equations and boundary conditions for the axisymmetric deformation of a nonshallow spherical shell under finite displacements, taking account of a physical nonlinearity, are presented herein.

Thermally induced vibrations of thin nonshallow spherical shells.

Thermally induced vibrations of thin nonshallow spherical shells.

On Vibrations of Spherical Shells

By the use of suitable auxiliary variables, the equations of motion of nonshallow spherical shells are reduced to a system of two uncoupled and one partly coupled equations. The solution of these equations is given in terms of associated Legendre functions.

偏平球殼理論とその応用(その3)

It is known that the general solutions for elastic shallow spherical tnin shells without surface load can be expressed by means of the Kelvin function in terms of φ, stress function, and w, displacement normal to the snell surface. In this paper some impotant points which are deficient in the existing theories are revised: physical meanigs of mathematical solutions are considered and the terms giving multi-valued displacements are excluded, membrane stress and rigid-body displacement terms in the antisymmetrical case (n=1 whenw=w_n cos nθ) which can not be derived from the mathematical solutions, are added. Formulas for dending and membrane stresses and displacements are presented as complete solutions, and particular solutions for various loads and temperature distributions are tabulated. As specific applications, shallow sphericall shells for roof denies and large tracking antennae are discussed. In these examples equations for deciding the unknown constants are derived from boundary conditions in the form of matrix. They can readily be used in the design of the similar structures. In paragraphs 1 and 2, the fundamental equations of bending problem of shallow spherical shells without surface load (equilibrium and compatibility equations) are established. From paragraph 3 to 6, the general solutions are derived by means of the Kelvin function (ber, bei, ker, kei), and the fundamental character of this function is discussed. Furtner, cosidering on the nature of the stresses and displacements resulted from the solution as well as on the rigid body displacements, complete form of the solutions is accompished. Particular solutions for various loads and temparature conditions are taculated in paragraph 7 and complete solutions for shells without surface load are formulated in paragraph 8, both for stresses and displacements. In and after paragraph 9 are the applications of the solutions to the actual structures. Paragraph 9 is the example of a shallow spherical shell stbjected to exially symmetrical edge moment, and ressults from the shallow shell theory are compared with those from Hetenyi's approximate theory for non-shallow spherical shells. In peragraph 10 and 11, the reinforced shallow spherical shell roof subjected to various loads and a temperature distribution under several kinds of supporting conditions is discussed. Paragraph 12 is the example of a large antenna, the outer edge free and the inner edge clamped, under the follewing lord conditions: a. own weight in two standing states, vertical and horizontal to the ground. b. various wind loads. c. axially symmetrical and antisymmetrical distributions of temperature increment.

2156 球殼非対称曲げ実用近似解の精度(構造,昭和38年度(仙台)大会学術講演要旨集)

2156 球殼非対称曲げ実用近似解の精度(構造,昭和38年度(仙台)大会学術講演要旨集)