On the Accuracy of Some Shell Solutions

1959 ◽  
Vol 26 (4) ◽  
pp. 577-583
Author(s):  
G. D. Galletly ◽  
J. R. M. Radok
Keyword(s):  
Negative Curvature ◽  
Numerical Solutions ◽  
Asymptotic Solutions ◽  
Thin Shells ◽  
Shells Of Revolution ◽  
Equilibrium Equations ◽  
Influence Coefficients ◽  
Bending Loads ◽  
Edge Influence

Abstract R. B. Dingle’s method [1] for finding asymptotic solutions of ordinary differential equations of a type such as occur in the bending theory of thin shells of revolution is presented in outline. This method leads to the same results as R. E. Langer’s method [2], recently used for problems of this kind, and permits a simple analytical and less formal interpretation of the asymptotic treatment of such equations. A comparison is given of edge influence coefficients due to bending loads, obtained by use of these asymptotic solutions and numerical integration of the equilibrium equations, respectively. The particular shells investigated are of the open-crown, ellipsoidal, and negative-curvature toroidal types. The results indicate that the agreement between these solutions is satisfactory. In the presence of uniform pressure, the use of the membrane solutions for the determination of the particular integrals appears to lead to acceptable results in the case of ellipsoidal shells. However, in the case of toroidal shells, the membrane and the numerical solutions disagree significantly.

Influence Coefficients for Thin Smooth Shells of Revolution Subjected to Symmetric Loads

1962 ◽  
Vol 29 (2) ◽  
pp. 335-339 ◽  
Author(s):  
B. R. Baker ◽  
G. B. Cline
Keyword(s):  
Differential Equations ◽  
Asymptotic Integration ◽  
Asymptotic Solutions ◽  
Thin Shells ◽  
Uniform Thickness ◽  
Shells Of Revolution ◽  
Influence Coefficients ◽  
Independent Variable

The differential equations governing the deformation of shells of revolution of uniform thickness subjected to axisymmetric self-equilibrating edge loads are transformed into a form suitable for asymptotic integration. Asymptotic solutions are obtained for all sufficiently thin shells that possess a smooth meridian curve and that are spherical in the neighborhood of the apex. For design use, influence coefficients are derived and presented graphically as functions of the transformed independent variable ξ. The variation of ξ with the meridional tangent angle φ is given analytically and graphically for several common meridian curves—the parabola, the ellipse, and the sphere.

scholarly journals Calculation of thin isotropic shells beyond the elastic limit by the method of elastic solutions

2018 ◽  
Vol 196 ◽  
pp. 01014 ◽  
Author(s):  
Avgustina Astakhova
Keyword(s):  
Elastic Limit ◽  
Physical Nonlinearity ◽  
Elasticity Tensor ◽  
Thin Shells ◽  
Equilibrium Equations ◽  
Spherical Shells ◽  
Plastic Deformations ◽  
Elastic Plastic ◽  
Internal Forces

The paper focuses on the model of calculation of thin isotropic shells beyond the elastic limit. The determination of the stress-strain state of thin shells is based on the small elastic-plastic deformations theory and the elastic solutions method. In the present work the building of the solution based on the equilibrium equations and geometric relations of linear theory of thin shells in curved coordinate system α and β, and the relations between deformations and forces based on the Hirchhoff-Lave hypothesis and the small elastic-plastic deformations theory are presented. Internal forces tensor is presented in the form of its expansion to the elasticity tensor and the additional terms tensor expressed the physical nonlinearity of the problem. The functions expressed the physical nonlinearity of the material are determined. The relations that allow to determine the range of elastic-plastic deformations on the surface of the present shell and their changing in shell thickness are presented. The examples of the calculation demonstrate the convergence of elastic-plastic deformations method and the range of elastic-plastic deformations in thickness in the spherical shell. Spherical shells with the angle of half-life regarding 90 degree vertical symmetry axis under the action of equally distributed ring loads are observed.

Edge Influence Coefficients for Toroidal Shells of Negative Gaussian Curvature

1960 ◽  
Vol 82 (1) ◽  
pp. 69-75 ◽  
Author(s):  
G. D. Galletly
Keyword(s):  
Gaussian Curvature ◽  
Constant Thickness ◽  
Toroidal Shell ◽  
Uniform Pressure ◽  
Simple Manner ◽  
Negative Gaussian Curvature ◽  
Influence Coefficients ◽  
Bending Loads ◽  
Edge Influence ◽  
Toroidal Shells

Continuing the work presented in reference [1], the present paper gives additional tables for the edge deformations of constant-thickness toroidal shells subject to edge bending loads and uniform pressure. The two papers together thus cover a wide variety of toroidal shell geometries and enable a designer to calculate in a simple manner the edge moments and shears at toroidal shell junctions.

scholarly journals Метод точного расчёта длины цилиндрической оболочки

2020 ◽  
Author(s):  
A.A. Karpachev
Keyword(s):  
Critical Pressure ◽  
Thin Shells ◽  
System Of Equations ◽  
Linear Algebraic System ◽  
Shells Of Revolution ◽  
Equilibrium Equations ◽  
Eighth Order ◽  
Bodies Of Revolution ◽  
Axisymmetric Bodies ◽  
Additional Assumptions

Как известно, для решения конкретных задач расчета прочности и устойчивости оболочек вращения используется теория расчетов осесимметричных тел вращения произвольной формы, основанная на гипотезах Кирхгофа и предположениях об однородности и изотропности материалов изготовления. В общей теории тонких оболочек данная задача сводится к решению системы уравнений равновесия в частных производных восьмого порядка. Для цилиндрических оболочек ввиду принятых допущений система уравнений равновесия в перемещениях преобразуется в линейную алгебраическую систему. Из данной системы на основе дополнительных допущений получают простое уравнение, из которого и определяется величина критического давления устойчивости по заданной длине оболочки. Однако из основной системы уравнений возможно решение обратной задачи: по заданной величине критического давления определять точное значение длины цилиндрической оболочки. При этом задача имеет точное решение без каких либо дополнительных допущений и упрощений системы уравнений.As is known, to solve specific problems of calculating the strength and stability of shells of revolution, the theory of calculations of axisymmetric bodies of revolution of arbitrary shape is used, based on Kirchhoff hypotheses and assumptions about the homogeneity and isotropy of manufacturing materials. In the general theory of thin shells, this problem reduces to solving a system of eighth-order partial differential equilibrium equations. For cylindrical shells, in view of the accepted assumptions, the system of equations of equilibrium in displacements is transformed into a linear algebraic system. From this system, on the basis of additional assumptions, a simple equation is obtained, from which the critical pressure of stability for a given shell length is determined. However, it is possible to solve the inverse problem from the main system of equations: determine the exact value of the length of a cylindrical shell for a given critical pressure. Moreover, the problem has an exact solution without any additional assumptions and simplifications of the system of equations.

scholarly journals The thickness functions of the shells of revolution subjected to axisymmetrical loads

2005 ◽  
Vol 27 (2) ◽  
pp. 66-73
Author(s):  
Ngo Huong Nhu ◽  
Pham Hong Nga
Keyword(s):  
Differential Equations ◽  
Thin Shell ◽  
Numerical Solutions ◽  
External Pressure ◽  
Shells Of Revolution ◽  
Varying Thickness ◽  
Shell Design ◽  
Mathematical Difficulties ◽  
Thickness Function

The inverse problems for determining the meridian shape or varying thickness function of momentless shells of revolution under given loads were concerned in many works [2, 3, 4]. However, for the complexity of loads or configuration of a shell these problems haven' t bee.n solved perfectly because of its mathematical difficulties. In this paper, the problem for determining the thickness function of shells of revolution such as a parabola, sphere arc! under axisymmetrical loads is considered. The general integro-differential equations for determination of the meridian form and shell thickness are obtained. A solution of differential equations by semi-analytical and numerical methods for the thickness is presented. The numerical solutions are given for the parabola under external pressure, the sphere immerged in the fluid and the sphere arc. Obtained results may be used in the thin shell design.

Edge Influence Coefficients for Toroidal Shells of Positive Gaussian Curvature

1960 ◽  
Vol 82 (1) ◽  
pp. 60-68 ◽  
Author(s):  
G. D. Galletly
Keyword(s):  
Gaussian Curvature ◽  
Constant Thickness ◽  
Toroidal Shell ◽  
Uniform Pressure ◽  
Dimensionless Form ◽  
Positive Gaussian Curvature ◽  
Influence Coefficients ◽  
Bending Loads ◽  
Edge Influence ◽  
Toroidal Shells

Tables are given for the edge deformations of constant-thickness toroidal shells subject to uniform pressure and edge bending loads. Over one hundred different shell geometries were investigated and the results are presented in dimensionless form. Possession of these coefficients, which were obtained on a digital computer, means that a rapid and accurate formulation of the compatibility equations at toroidal shell junctions is now possible.

On Influence Coefficients and Nonlinearity for Thin Shells of Revolution

1959 ◽  
Vol 26 (1) ◽  
pp. 69-72
Author(s):  
Eric Reissner
Keyword(s):  
Linear Theory ◽  
Thin Shells ◽  
Shells Of Revolution ◽  
Nonlinear Formulation ◽  
Elastic Shells ◽  
Edge Zone ◽  
Influence Coefficients ◽  
Rotationally Symmetric

Abstract The paper is concerned with a nonlinear formulation of the problem of rotationally symmetric deformations of thin elastic shells of revolution, which are acted upon by edge forces and moments. Determined are, in particular, nonlinear corrections to the known results of the linear theory, for edge displacements and rotations. The calculations are for cases for which thickness and curvature of the shell are such as to insure that stresses and deformations are effectively contained within a narrow edge zone of the shell.

An Approach to Optimum Shape Determination for a Class of Thin Shells of Revolution

1968 ◽  
Vol 35 (3) ◽  
pp. 524-529 ◽  
Author(s):  
Han-Chung Wang ◽  
Will J. Worley
Keyword(s):  
Thin Shell ◽  
Failure Criteria ◽  
Optimum Shape ◽  
Thin Shells ◽  
Shells Of Revolution ◽  
Numerical Iteration ◽  
Shape Determination ◽  
Selection Of ◽  
Convex Shell

A method is presented for the determination of an optimum shape of a convex shell of revolution with respect to volume and weight. The technique depends on selecting a multiparameter equation and varying the parameters to achieve a near optimum shape for prescribed failure criteria. As an illustration of the method, the first quadrant of the meridian (x/a)α + (y/b)β = 1 is selected. Here a, b, α, and β are positive constants not necessarily integers, with α and β equal to or greater than unity. Variations in shape are expressed in terms of the parameters b/a, α and β. The procedure is applied to the selection of a thin shell which will fit within the space defined by a circular cylinder of radius b and length 2a. The shell is optimized, in terms of α and β, with respect to volume and weight. The numerical iteration was performed by means of a digital computer.

Maysel's formula generalized for piezoelectric vibrations - Application to thin shells of revolution

1996 ◽  
Author(s):  
Hans Irschik ◽  
Franz Ziegler ◽  
Hans Irschik ◽  
Franz Ziegler
Keyword(s):  
Thin Shells ◽  
Shells Of Revolution
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