On the Derivation of the Differential Equations of Linear Shallow Shell Theory

1984 ◽  
pp. 12-21 ◽  
Author(s):  
E. Reissner
Keyword(s):  
Differential Equations ◽  
Shallow Shell ◽  
Shell Theory

Comparison of Beam and Shell Theories for the Vibrations of Thin Turbomachinery Blades

1983 ◽  
Vol 105 (2) ◽  
pp. 383-392 ◽  
Author(s):  
A. W. Leissa ◽  
M. S. Ewing
Keyword(s):  
Shallow Shell ◽  
Shell Theory ◽  
Free Vibrations ◽  
Two Dimensional ◽  
Analysis Method ◽  
One Dimensional ◽  
Low Aspect Ratio ◽  
Turbomachinery Blades ◽  
Dimensional Analysis Method ◽  
Beam Theories

A great deal of published literature exists which analyzes the free vibrations of turbomachinery blades by means of one-dimensional beam theories. Recently, a more accurate, two-dimensional analysis method has been developed based upon shallow shell theory. The present paper summarizes the two types of theories and makes quantitative comparisons of frequencies obtained by them. Numerical results are presented for cambered and/or twisted blades of uniform thickness. Significant differences between the theories are found to occur, especially for low aspect ratio blades. The causes of these differences are discussed.

An Integral Equation Solution for Limit Loads Applied to Lugs on Cylindrical Shells

1991 ◽  
Vol 113 (2) ◽  
pp. 308-313 ◽  
Author(s):  
G. N. Brooks
Keyword(s):  
Lower Bound ◽  
Load Distribution ◽  
Shallow Shell ◽  
Shell Theory ◽  
Normal Force ◽  
Cylindrical Shells ◽  
Yield Condition ◽  
Convolution Integral ◽  
Concentrated Load ◽  
Lower Bound Limit Analysis

A lower-bound limit analysis of loaded integral lugs on cylindrical shells is presented. Normal force and circumferential and longitudinal moment loadings on the lug are considered. The equilibrium solution, necessary for a lower bound, is obtained as a convolution integral of the concentrated load solutions of linear shallow shell theory. The load distribution is chosen to satisfy the yield condition everywhere, while maximizing the load. A simplified yield condition in terms of the shell stress resultants is used. Failure is assumed to occur in the shell, not the lug. Encouraging comparisons with available experimental results for moment-loaded rectangular lugs on pipes are presented. The use of shallow shell theory enables the problem geometry to be described by one less parameter than complete shell theory.

FREE VIBRATION RESPONSE OF CYLINDRICAL PANELS WITH HIGHER ORDER SHEAR DEFORMATION THEORY

2005 ◽  
Vol 05 (03) ◽  
pp. 409-434 ◽  
Author(s):  
HUMAYUN R. H. KABIR ◽  
HASAN ASKAR
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Shear Deformation ◽  
Shell Theory ◽  
Fourier Series Expansion ◽  
Partial Differential ◽  
Cylindrical Panels

Presented here is an analytical solution to the free vibration problem of an isotropic cylindrical panel with SS2-type simply supported boundary conditions based on Reddy's third order shear deformation shell theory. Using the principle of virtual work, the Reddy's shell theory generates five highly coupled partial differential equations in terms of three unknown displacements and two unknown rotations. The partial differential equations in conjunction with the prescribed boundary conditions are solved using displacement functions expressed in terms of double Fourier series expansion. Cylindrical panels with various aspect and thickness ratios are considered in the study of convergence behavior and parametric variation of the eigenvalues. The eigenvalues and mode shapes obtained in this study are compared with those obtained from the finite element software package ANSYS. The hitherto unavailable analytical solutions can be used as benchmarks for checking the accuracy of various approximate methods such as the Rayleigh–Ritz, finite element and finite difference methods.

Vibrations of Twisted Rotating Blades

1984 ◽  
Vol 106 (2) ◽  
pp. 251-257 ◽  
Author(s):  
A. W. Leissa ◽  
J. K. Lee ◽  
A. J. Wang
Keyword(s):  
Shallow Shell ◽  
Shell Theory ◽  
Analytical Procedure ◽  
Ritz Method ◽  
Two Dimensional ◽  
Algebraic Polynomials ◽  
Shape Information ◽  
Rotating Blades ◽  
Turbomachinery Blades ◽  
Twisted Blade

The literature dealing with vibrations of turbomachinery blades is voluminous, but the vast majority of it treats the blades as beams. In a previous paper a two-dimensional analytical procedure was developed and demonstrated on simple models of blades having camber. The procedure utilizes shallow shell theory along with the classical Ritz method for solving the vibration problem. Displacement functions are taken as algebraic polynomials. In the present paper the method is demonstrated on blade models having camber. Comparisons are first made with results in the literature for nonrotating twisted plates and various disagreements between results are pointed out. A method for depicting mode shape information is demonstrated, permitting one to examine all three components of displacement. Finally, the analytical procedure is demonstrated on rotating twisted blade modes, both without and with camber.

On Finite Symmetrical Deflections of Thin Shells of Revolution

1969 ◽  
Vol 36 (2) ◽  
pp. 267-270 ◽  
Author(s):  
Eric Reissner
Keyword(s):  
Differential Equations ◽  
Nonlinear Theory ◽  
Transverse Shear ◽  
Shell Theory ◽  
Middle Surface ◽  
Thin Shells ◽  
Shells Of Revolution ◽  
Shear Deformations ◽  
Linear Shell Theory ◽  
Classical Subject

Recent simplifications of linear shell theory through consideration of transverse shear deformations and stress moments with axes normal to the shell middle surface suggest analogous approaches to the corresponding problem of nonlinear theory. As a first step in this direction consideration is given here to the classical subject of finite symmetrical deformations of shells of revolution. The principal new results of the present analysis concern the form of strain-displacement and compatibility differential equations.

On the Equations of Linear Shallow Shell Theory

1969 ◽  
Vol 48 (2) ◽  
pp. 133-145 ◽  
Author(s):  
E. Reissner ◽  
F. Y. M. Wan
Keyword(s):  
Shallow Shell ◽  
Shell Theory

On the Initial Response of a Spherical Shell to a Concentrated Force

1962 ◽  
Vol 29 (4) ◽  
pp. 689-695 ◽  
Author(s):  
M. A. Medick
Keyword(s):  
Spherical Shell ◽  
Shallow Shell ◽  
Shell Theory ◽  
Initial Response ◽  
Shallow Spherical Shell ◽  
Restricted Class ◽  
Concentrated Force ◽  
Elastic Shells

This paper is concerned with the initial response of a restricted class of thin elastic shells to localized transient loadings. Attention is restricted to those shells which are essentially spherical and shallow in a neighborhood of the loading. The initial response within this neighborhood can be approximated by the response of a (shallow) spherical shell segment to a concentrated force within the framework of a modified shallow-shell theory.

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