The Accuracy of Donnell’s Equations

1955 ◽  
Vol 22 (3) ◽  
pp. 329-334
Author(s):  
N. J. Hoff
Keyword(s):  
Cylindrical Shell ◽  
Circular Cylindrical Shell ◽  
Basic Parameters ◽  
Thin Walled ◽  
Small Deformations

Abstract Solutions of Donnell’s equations of the small deformations of the perfectly elastic thin-walled circular cylindrical shell are compared with those obtainable from Flügge’s equations. The range of the basic parameters is found within which the two solutions are approximately equal.

On Radial Deflections of a Cylinder Subjected to Equal and Opposite Concentrated Radial Loads: Infinitely Long Cylinder and Finite-Length Cylinder With Simply Supported Ends

1957 ◽  
Vol 24 (2) ◽  
pp. 278-282
Author(s):  
S. W. Yuan ◽  
L. Ting
Keyword(s):  
Mathematical Method ◽  
Cylindrical Shell ◽  
Finite Length ◽  
Circular Cylindrical Shell ◽  
Short Length ◽  
Simply Supported ◽  
Thin Walled ◽  
Long Cylinder

Abstract The radial deformations of a thin-walled circular cylindrical shell subjected to a pair of equal and opposite concentrated radial loads were obtained in (1) for the cases of infinitely long cylinders and cylinders of finite length simply supported at the ends. Based on the mathematical method given in (1) this problem is reexamined in the present paper by using Flügge’s equations (2, 3). It is found that the results obtained in (1) are quite satisfactory for short-length cylinders (L/α ≤ 10) with simply supported ends but not satisfactory for infinitely long cylinders.

Buckling of a Thin-Walled Circular Cylindrical Shell Heated Along an Axial Strip

1964 ◽  
Vol 31 (2) ◽  
pp. 253-258 ◽  
Author(s):  
Nicholas J. Hoff ◽  
Chi-Chang Chao ◽  
Wayne A. Madsen
Keyword(s):  
Cylindrical Shell ◽  
Critical Stress ◽  
Circular Cylindrical Shell ◽  
Axial Stress ◽  
Elastic Stability ◽  
Nonuniform Heating ◽  
Small Deflection ◽  
Thin Walled ◽  
Axial Compressive Stress ◽  
Deflection Theory

The elastic stability of a thin-walled circular cylindrical shell is investigated by means of the small-deflection theory when the shell is subjected to such nonuniform heating as causes a uniform axial compressive stress to arise in a band of width 2b while the rest of the shell is free of stress. The critical value of the compressive axial stress is found to be equal to the critical stress of the same circular cylindrical shell when subjected to uniform axial compression provided the band is not extremely narrow. In the latter case the critical stress of the band is higher than that of the uniformly compressed shell.

Thermoelastic Response of Thin-Walled Cylindrical Shell Under the Effect of Moving Point Load and Heat Source

2011 ◽  
Author(s):  
Erno Keskinen ◽  
Timo Karvinen ◽  
Vladimir Dospel ◽  
Michel Cotsaftis
Keyword(s):  
Cylindrical Shell ◽  
Temperature Distribution ◽  
Thermal Effects ◽  
Circular Cylindrical Shell ◽  
Thermoelastic Problem ◽  
Work Piece ◽  
Thermal Deformations ◽  
Thin Walled ◽  
Eigenfunction Basis ◽  
Modal Coordinates

Cylinder grinding has been the subject of an intensive research, because delay-type resonances, commonly known as chatter-vibrations, have been reason for serious surface quality problems in industry [1]. As a result of this activity it has been developed a simulation platform, on which the complete grinding process including delay-resonances can be driven [2]. This platform consists of models for the grinder, for the cylindrical work piece and for the stone-cylinder grinding contact. The elastic cylinder model is based on analytical eigenfunctions in bending vibrations, which basis has been used to present the rotordynamic equations of cylinder in modal coordinates. Stone-cylinder interaction mechanism has been derived by combining the rules of mass and momentum transfer in the material removal process. The contribution of this paper is to update the platform to include the thermal effects of the work body undergoing shell deformations. Following the method to use the eigenfunctions of a thin-walled circular cylindrical shell to describe the rotordynamic motion of the work body, a promising method could be to use in a similar way the eigenfunctions of a thermally isolated cylinder to solve the temperature distribution of the cylinder. The temperature distribution and terms related to the non-homogeneous boundary conditions will then be the input to the thermoelastic problem. It can be shown that the eigenfunction basis consists of trigonometric functions in axial and circumferential directions while the radial eigenfunctions are Bessel functions. The stone-cylinder interface has to be updated also to include thermal effects. A portion of the mechanical power is transferred to the work piece. The rest goes to the stone, to the material, which is removed and to the cutting coolant. On the other hand, thermal deformations modify the grinding forces, which are loading the work piece. The solution of the coupled thermal and thermoelastic problem will be done in terms of modal coordinates corresponding to the eigenfunction basis. This leads to numerical time integration of two groups of differential equations, the solution of which can be used to perform the temperature distributions and the corresponding thermal deformations.

Strongly Nonlinear Vibrations of a Hyperelastic Thin-walled Cylindrical Shell Based on the Modified Lindstedt–Poincaré Method

2019 ◽  
Vol 19 (12) ◽  
pp. 1950160 ◽  
Author(s):  
Jing Zhang ◽  
Jie Xu ◽  
Xuegang Yuan ◽  
Wenzheng Zhang ◽  
Datian Niu
Keyword(s):  
Cylindrical Shell ◽  
Nonlinear System ◽  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Harmonic Excitation ◽  
System Of Differential Equations ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Hardening Behavior ◽  
Thin Walled

Some significant behaviors on strongly nonlinear vibrations are examined for a thin-walled cylindrical shell composed of the classical incompressible Mooney–Rivlin material and subjected to a single radial harmonic excitation at the inner surface. First, with the aid of Donnell’s nonlinear shallow-shell theory, Lagrange’s equations and the assumption of small strains, a nonlinear system of differential equations for the large deflection vibration of a thin-walled shell is obtained. Second, based on the condensation method, the nonlinear system of differential equations is reduced to a strongly nonlinear Duffing equation with a large parameter. Finally, by the appropriate parameter transformation and modified Lindstedt–Poincar[Formula: see text] method, the response curves for the amplitude-frequency and phase-frequency relations are presented. Numerical results demonstrate that the geometrically nonlinear characteristic of the shell undergoing large vibrations shows a hardening behavior, while the nonlinearity of the hyperelastic material should weak the hardening behavior to some extent.

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