Membrane Theory Of Shells Research Articles

Effect of bending on the axisymmetric vibrations of a spheroidal model of the head

Abstract The head is modelled as an elastic prolate spheroidal shell filled with an inviscid, compressible fluid. Bending effects are included, and the free vibration frequency spectrum obtained is compared with that of an earlier spheroidal model using extensional (membrane) shell theory and with a spherical model including bending. The differences between the present results and those reported previously are significant.

Coupled-Circuit Representation of the Vibrations of a Thin Ferroelectric Cylindrical Shell

This paper describes a lumped-parameter equivalent circuit that represents the radially symmetric vibratory motion of a finite-length cylindrical shell, which is composed of a radially poled ferroceramic material of the barium titanate type. The equivalent circuit consists of two branches that are coupled together. One branch describes the axial motion of the cylindrical shell and the other branch the radial motion. Such a two-branch equivalent circuit is necessary, since the radially symmetric vibration of a finite-length cylindrical shell is a two-dimensional problem, owing to the fact that the radial and axial motions are closely coupled due to the finiteness of the Poisson's ratio of the shell material. The equivalent circuit is derived using membrane shell theory and is thus able to describe the motion of the cylindrical shell at frequencies up to and including the lowest radial and axial resonances. By means of this circuit, it is possible to analyze the effect of axial boundary conditions when designing transducers that use the radial motion of thin cylindrical shells in the generation and reception of underwater sound.

Free vibration of an inflated oblate spheroidal shell

An approximate analytical solution of the subject problem is developed. The vibration of the oblate spheroid is assumed to be governed by the membrane theory of shells. It is also assumed that the square of eccentricity of the oblate spheroid is small and constant during inflation and that the spheroid is composed of a neo-Hookean material. The first step in the solution process concerns an exact solution of the free vibration problem of an inflated spherical shell. The free vibration of the oblate spheroid is then obtained by Galerkin's method with the modal solutions of the inflated spherical shell being used. The frequencies and mode shapes of both types of shell are given and compared to similar linear solutions. Comparisons of the behavior of oblate spheroidal and spherical shells are given. An interesting instability phenomenon of the vibration of inflated spherical shells is discussed and the constant eccentricity assumption is justified by comparisons with test results.

Longitudinal impact on a circular cylindrical tube

The longitudinal strain histories produced by impact in a long circular cylindrical tube are determined using membrane shell theory. Solutions are generated by a method which reformulates the boundary value problem in terms of Volterra integral equations which are then numerically resolved. The predicted histories are compared with strain histories measured in a corresponding experiment, and the correlation is found to be better than that based on histories predicted from elementary rod theory corrected to account for lateral inertia.

Hemodynamic Flow in Anisotropic, Viscoelastic Thick-Wall Vessels

Hemodynamic flow in anisotropic, viscoelastic thick-wall vessels is analyzed. For small amplitude harmonic waves, the wavelengths of which are large compared to the vessel radius, the fluid motion is governed by the linearized form of the Navier-Stokes equations and the vessel motion is described by the classical Navier equations for solids. Compressible and incompressible transversely isotropic vessels with frequency dependent material moduli are considered. Longitudinal constraints are also incorporated. The quantities calculated are: the velocities of propagation, transmission per wavelength, fluid impedance, and the displacements and stresses at the fluid-vessel interface. The results indicate that the earlier anisotropic analyses, which incorporate membrane or thick shell theories, agree qualitatively with the present thick-wall analysis. However, quantitative differences among the analyses do occur, especially as the compressibility of the vessel material decreases. For the thick-wall analysis, the effects of anisotropy on the various quantities of interest are established. In particular, it is shown that the fluid impedance is relatively insensitive to the degree of anisotropy, but the resistance and inductance are very sensitive to anisotropy at the high and low values of frequency respectively. In general, anisotropy significantly affects the first mode transmission per wavelength and second mode phase velocity. The displacements and stresses at the fluid-vessel interface are also altered by anisotropy. In addition, the arterial system was found to be essentially independent of moderate variations in the axial shear modulus of the material. Furthermore, it is demonstrated that the degree of viscoelasticity associated with the vessel material substantially alters the transmission per wavelength but affects the phase velocity to a much lesser degree. For the suggested value of the spring parameter associated with the longitudinal constraint, it is found that the additional mass and the viscous parameters have little effect on the system behavior. The overall effect of the longitudinal constraint is to completely subdue the second mode of transmission, increase the first mode phase velocity, decrease the first mode transmission per wavelength and axial displacement, and to increase the modulus and decrease the phase of the first mode fluid impedance. In addition to confirming the importance of anisotropy, viscoelasticity, and the longitudinal constraint, the investigation demonstrates that an appropriate thick-wall analysis is required in order to adequately describe the fluid-vessel behavior.

On the membrane stress function for shells

In this work the system of three equations of equilibrium of membrane theory of shells is reduced to a single partial differential equation of second order through a transformation involving the first order partial derivatives of a stress function. This type of transformation is possible only if the parameters of the shell surface satisfy certain conditions. As particular cases the shells of revolution, the shells in the form of Dupin's cyclide of 4th order and of 3rd order are considered.

Centrifugal Effects in Inflated, Rotating Bias-Ply Tires

An equation is derived which governs the dynamic equilibrium contour taken by an inflated and rotating, but otherwise unloaded, bias-ply tire. This equation, which is in the form of a hyperelliptic integral, is based on the membrane theory of shells and the netting analysis of composite materials. Results are obtained for the meridional geom etry of a typical two-ply automobile tire, which are in reasonable agreement with ex perimental measurements. This integral, which describes the dynamic equilibrium contour of the tire, is then used to obtain an algebraic cord load formula. The formula shows how the cord tension at any location is dependent on mass distribution, angular velocity, inflation pressure, deformed tire geometry, and number of cords in the tire. Centrifugal effects, as they influence cord tension, are more pronounced in the bead region than in the crown region. Cord forces are experimentally measured up to 1600 rpm, using the tirecord tension transducer which is a strain-gage instrumented tension link embedded in the tire and placed in series with the cord. Agreement between the theoretically predicted and ex perimentally measured cord forces is good, especially in the range of angular velocities and inflation pressures used in service.

Frequency Analysis of Complete Spherical Shells

The results of the transverse frequency analysis of complete spherical shells by elastic theory and classical, improved, and membrane shell theories are compared. Fundamental frequencies less than the lowest pure shear frequency of a solid sphere are shown to be accurately predicted by classical shell theory. For shells of vanishing thickness, the equations of elasticity are shown to reduce exactly to the membrane shell equations. Evaluation of the distortional energy associated with each transverse frequency allows a classification of the three frequencies of transverse vibration in each mode shape. This classification is based on predictions by improved shell theory as to the dominant character of the behavior associated with extensional, bending or shear distortion.

Reduction of the equations of the membrane theory of shells.

Reduction of the equations of the membrane theory of shells.

On the correctness of certain problems of the membrane theory of shells of negative curvature: PMM vol. 33, n≗4, 1969, pp. 676–687

On the correctness of certain problems of the membrane theory of shells of negative curvature: PMM vol. 33, n≗4, 1969, pp. 676–687

Notes on Nonlinear Shell Theory

Exact tensor equations of equilibrium are derived for nonlinear membrane shell theory and small perturbations of pressurized membrane shells. Approximate equations (for sufficiently small initial strains and rotations) for small perturbations of pressurized membrane shells are also given. Exact equations for the general nonlinear shell theory (including bending) are discussed, and approximate equations (again for small initial strains and rotations) are derived for the small perturbations, buckling, and vibration of stressed shells. These last equations are given in an Appendix in lines-of-curvature coordinates in classical notation.

Four-Pole Parameters for Impedance Analyses of Conical and Cylindrical Shells under Axial Excitations

A combined analytical and experimental study is presented to demonstrate that the four-pole parameters of a truncated, thin conical shell, under axial excitations, may be accurately obtained by applying membrane shell theory. A general calculation procedure is described, and the numerical results are applied to determine the mechanical impedances of shell-mass systems, and then compared with test data for three shell models with semivertex angles 0°, 15°, and 30° over a frequency range from 20 to 600 cps. The exce lent agreement indicates that the four-pole parameters calculated through the present analysis are adequate for vibration analyses, if the input excitation frequency is appreciably lower than a theoretical singularity, on the frequency spectrum inherent to the membrane shell theory. If the excitation frequency is near or above this singularity, a more accurate bending shell theory must be used.

DOMES

Summary The basic structural action of domes is first explained, and the reasons for the economy of material which is possible with this structural form are indicated. An outline is then given of the membrane theory of shells in the form of a surface of revolution, such as spherical domes, and of other forms such as elliptic paraboloids which are also used in the design of roofs in and other some brane stresses are given. The effect of the boundary conditions (edges) of the shell in modifying the membrane stresses is then discussed by reference to the approximate theory for spherical domes supported continuously around the periphery. Domes supported at discrete points present more difficult problems which are indicated in relation to the Academy of Sciences building, the dome is referred to. Other questions dealt with include the effect of Prestressing the rim of a dome and of shrinkage in concrete. An account is also given of the stability of domes, the danger of failure by buckling, and means taken to prevent this.

Vibrations of a Thick-Walled Cylindrical Shell—Comparison of the Exact Theory with Approximate Theories

The results for vibrations of an elastic cylinder as predicted by a number of the approximate shell theories are compared with the results of the exact theory. It is found that the membrane theory of shells is accurate for predicting frequencies and displacement ratios of cylinders with appreciable thickness. Furthermore, the theories which include bending effects as well as membrane effects are good at even the shorter wavelengths; and the theories which include rotatory inertia and shear are accurate over most of the wavelength spectrum of the lowest branch of rather thick shells. However, for the very thick shell (with a ratio of inside radius to outside radius less than 0.5), only the exact theory shows the full characteristics of the displacement distribution.

Three-Dimensional and Shell-Theory Analysis of Axially Symmetric Motions of Cylinders

Abstract The frequency (or phase velocity) of axially symmetric free vibrations in an elastic, isotropic, circular cylinder of medium thickness is studied on the basis of the three-dimensional linear theory of elasticity and several different shell theories. To be in good agreement with the solution of the three-dimensional equations for short wave lengths, an approximate theory has to include the influence of rotatory inertia and transverse shear deformation, for example, in a manner similar to Mindlin’s plate theory. A shell theory of this (Timoshenko) type is deduced from the three-dimensional elasticity theory. From a comparison of phase velocities it appears that, to a good approximation, membrane and curvature effects on one hand, and on the other hand, flexural, rotatory-inertia, and shear-deformation effects are mutually exclusive in two ranges of wave lengths, separated by a “transition” wave length. Thus, in the full range of wave lengths, the associated lowest phase velocities may be determined on the basis of the membrane shell theory (for wave lengths larger than the transition wave length) and on the basis of Mindlin’s plate theory (for wave lengths smaller than the transition wave length).

The Effect of a Surrounding Fluid on Pressure Waves in a Fluid-Filled Elastic Tube

Abstract The analysis of the transmission of pressure waves in a fluid-filled elastic tube has been extended to the case where the tube is surrounded by a fluid medium. The sound pressure inside the tube is the resultant of a number of modes, some of which are nonpropagating, while others propagate at their own characteristic phase velocities. Neglecting end effects, and for continuously generated waves, it is found that only the modes whose velocity is larger than the sound velocity of the surrounding medium radiate sound energy radially outward. These modes will be damped out by radiation losses, while modes having a phase velocity smaller than this sound velocity are propagated without attenuation (if viscous and heat-transfer losses are neglected). Consequently, if the fluid is the same in the surrounding medium and in the tube only the lowest mode, which resembles a plane wave, propagates unattenuated. In any case, the mass-loading of the surrounding fluid lowers the phase velocities of the propagating modes, particularly at intermediate frequencies. It is shown that in this application the membrane theory of shells will lead to incorrect results, even in thin-walled tubes. This is illustrated by comparison with experimental data.

The Propagation of Elastic Waves in Thin-Walled Cylindrical Shells

This paper presents a study of the propagation of elastic waves in the walls of cylindrical shells. Two modes of propagation in the axial direction are found to exist; at high frequencies, the one corresponds to flexural waves, the other to longitudinal waves. The dispersion curves are in good agreement with published experimental data. Hitherto available analyses are based on the membrane theory of shells, which disregards bending stresses; these analyses lead to partly different conclusions in that they predict a “dead” frequency region where no propagation occurs, and in that they do not account for the high-frequency flexural mode. The discrepancy carries over into the evaluation of the natural frequencies of the tube, and of the dispersion of pressure waves propagating in liquid-filled tubes. Published experimental data on the latter process at ultrasonic frequencies also support the present analysis. Results of this analysis have been applied to the investigation of the dynamics of barium titanate transducers in the form of cylincrical shells. Some important concepts of sound insulation by lightweight partitions, such as the “coincidence dip,” depend essentially on the ratio of the sound velocity in air to the phase velocity of elastic waves in the partition; the present analysis might therefore also find a practical application in the adaptation of these concepts to cylindrical surfaces, such as the skin of aircraft fuselages or architectural shells.

On the Membrane Theory of Shell in Form of a Surface of Revolution

In the membrane theory of a shell in form of a surface of revolution, using a relation between the fundamental magnitudes of a surface and by the selection of some suitable variable, it is shown that in general the equ. of stress resultant, the equ. of deformation and the equ. of neutral surface will be reducible to a differential equ. of second order with an invariant R23/R1. Particurally, when the invariant is equal to a constant, the form of shell is a quadrics of revolution, and by this result the general solution of stress and deformation of shell in the form of a quadrics of revtion under an arbitary distributed load is obtained. Furthermore the dispersion of stress under olua concentrated load is analyxed by the method of complex variable.

On the reliability of the membrane theory of shells of revolution

On the reliability of the membrane theory of shells of revolution

Note on the Membrane Theory of Shell of Revolution

Note on the Membrane Theory of Shell of Revolution