Propagation of Axisymmetric Waves in an Unlimited Elastic Shell

1960 ◽  
Vol 27 (4) ◽  
pp. 690-695 ◽  
Author(s):  
A. Kalnins ◽  
P. M. Naghdi
Keyword(s):  
Spherical Shell ◽  
Entire Range ◽  
Energy Input ◽  
Elastic Shell ◽  
Mechanical Impedance ◽  
Stress Waves ◽  
Axisymmetric Stress ◽  
Spherical Shells ◽  
Concentrated Load ◽  
Shallow Shells

This investigation is concerned with the propagation of axisymmetric stress waves in unlimited thin shallow elastic spherical shells. In particular, a solution is obtained for an unlimited shallow spherical shell subjected to a harmonically oscillating concentrated load at the apex. This solution, exact within the scope of the linear theory of shallow shells, has an outward propagating wave character in the entire range of forcing frequency. Appropriate expressions for the mechanical impedance and the energy input are derived, and numerical results are obtained for the axial displacement corresponding to various forcing frequencies.

Optimal Forms of Shallow Shells With Circular Boundary, Part 2: Maximum Buckling Load

1984 ◽  
Vol 51 (3) ◽  
pp. 531-535 ◽  
Author(s):  
R. H. Plaut ◽  
L. W. Johnson
Keyword(s):  
Critical Load ◽  
Elastic Shell ◽  
Material Surface ◽  
Fundamental Vibration ◽  
Uniform Loading ◽  
Concentrated Load ◽  
Shallow Shells ◽  
Simply Supported ◽  
Circular Boundary ◽  
Boundary Part

In Part 1, optimal forms were determined for maximizing the fundamental vibration frequency of a thin, shallow, axisymmetric, elastic shell with given circular boundary. Our objective in this part is to maximize the critical load for buckling under a uniformly distributed load or a concentrated load at the center. Again, the shell form is varied and the material, surface area, and uniform thickness of the shell are specified. Both clamped and simply supported boundary conditions are considered for the case of uniform loading, while one example is presented involving a concentrated load acting on a clamped shell. The optimality condition leads to forms that have zero slope at the boundary if it is clamped. The maximum critical load is sometimes associated with a limit point and sometimes with a bifurcation point. It is often substantially higher than the critical load for the corresponding spherical shell.

Elastic-Stress Waves Produced by Pressure Loads on a Spherical Shell

1955 ◽  
Vol 22 (4) ◽  
pp. 473-478
Author(s):  
J. H. Huth ◽  
J. D. Cole
Keyword(s):  
Spherical Shell ◽  
Pressure Wave ◽  
Elastic Stress ◽  
Elastic Shell ◽  
Stress Waves ◽  
Blast Waves ◽  
Constant Speed ◽  
Spherical Elastic Shell ◽  
Technical Interest

Abstract The paper treats the problem of stresses in a spherical elastic shell subjected to a plane pressure wave traveling across it with constant speed, a case of technical interest when considering the effect of blast waves on the structure of a missile in flight.

Experimental Buckling Modes of Clamped Shallow Shells Under Concentrated Load

1966 ◽  
Vol 33 (2) ◽  
pp. 297-304 ◽  
Author(s):  
F. A. Penning
Keyword(s):  
Elastic Stability ◽  
Plastic Behavior ◽  
Spherical Shells ◽  
Finite Area ◽  
Concentrated Load ◽  
Buckling Loads ◽  
Buckling Modes ◽  
Shallow Shells

Elastic stability of clamped shallow spherical shells subjected to a small finite-area load at the apex is studied experimentally. Upper and lower critical buckling loads and prebuckled and postbuckled deflections are found for λ between 10 and 17. Buckling modes have been observed and defined. Nonaxisymmetric deformations are noted. Shells for λ less than 8 1/2 exhibit plastic behavior and do not buckle.

Deformation of Open Spherical Shells Under Arbitrarily Located Concentrated Loads

1966 ◽  
Vol 33 (2) ◽  
pp. 305-312 ◽  
Author(s):  
J. P. Wilkinson ◽  
A. Kalnins
Keyword(s):  
Spherical Shell ◽  
Green’S Function ◽  
Green's Function ◽  
Driving Frequency ◽  
Legendre Functions ◽  
Spherical Shells ◽  
Concentrated Load ◽  
Associated Legendre Functions ◽  
Rotationally Symmetric ◽  
In Series

An exact solution is derived for the Green’s function of an open rotationally symmetric spherical shell subjected to any consistent boundary conditions. The fundamental singularity of the Green’s function is expanded in series according to the addition theorems of Legendre functions, and the solution for a spherical shell subjected to an arbitrarily situated, harmonically oscillating, normal, concentrated load is obtained explicitly in terms of associated Legendre functions. The corresponding static Green’s function is obtained simply by setting the driving frequency equal to zero. Numerical results for the displacements and stress resultants of an example are presented in detail.

A Comparison of Theoretical and Experimental Results From Spherical Shells With a Single Radially Attached Nozzle

1967 ◽  
Vol 89 (3) ◽  
pp. 333-338 ◽  
Author(s):  
F. J. Witt ◽  
R. C. Gwaltney ◽  
R. L. Maxwell ◽  
R. W. Holland
Keyword(s):  
Spherical Shell ◽  
Internal Pressure ◽  
Experimental Results ◽  
Strain Gages ◽  
Spherical Shells ◽  
Axial Thrust ◽  
Outside Diameter ◽  
Theoretical Results

A series of steel models having single nozzles radially and nonradially attached to a spherical shell is presently being examined by means of strain gages. Parameters being studied are nozzle dimensions, length of internal nozzle protrusions, and angles of attachment. The loads are internal pressure and axial thrust and moment loadings on the nozzle. This paper presents both experimental and theoretical results from six of the configurations having radially attached nozzles for which the sphere dimensions are equal and the outside diameter of the attached nozzle is constant. In some instances the nozzle protrudes through the vessel.

A fast approach for acoustic source localization on a thin spherical shell

2021 ◽  
pp. 147592172110419
Author(s):  
Zixian Zhou ◽  
Zhiwen Cui ◽  
Tribikram Kundu
Keyword(s):  
Velocity Profile ◽  
Spherical Shell ◽  
Source Localization ◽  
Pressure Vessels ◽  
Solution Technique ◽  
Acoustic Source ◽  
Spherical Shells ◽  
Fast Approach ◽  
Acoustic Source Localization ◽  
Thin Spherical Shell

Thin spherical shell structures are wildly used as pressure vessels in the industry because of their property of having equal in-plane normal stresses in all directions. Since very large pressure difference between the inside and outside of the wall exists, any formation of defects in the pressure vessel wall has a huge safety risk. Therefore, it is necessary to quickly locate the area where the defect maybe located in the early stage of defect formation and make repair on time. The conventional acoustic source localization techniques for spherical shells require either direction-dependent velocity profile knowledge or a large number of sensors to form an array. In this study, we propose a fast approach for acoustic source localization on thin isotropic and anisotropic spherical shells. A solution technique based on the time difference of arrival on a thin spherical shell without the prior knowledge of direction-dependent velocity profile is provided. With the help of “L”-shaped sensor clusters, only 6 sensors are required to quickly predict the acoustic source location for anisotropic spherical shells. For isotropic spherical shells, only 4 sensors are required. Simulation and experimental results show that this technique works well for both isotropic and anisotropic spherical shells.

scholarly journals Axially Symmetric Vibrations of Composite Poroelastic Spherical Shell

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Rajitha Gurijala ◽  
Malla Reddy Perati
Keyword(s):  
Wave Propagation ◽  
Phase Velocity ◽  
Spherical Shell ◽  
Wave Number ◽  
Biot’S Theory ◽  
Impervious Surfaces ◽  
Axially Symmetric ◽  
Spherical Shells ◽  
Biot's Theory

This paper deals with axially symmetric vibrations of composite poroelastic spherical shell consisting of two spherical shells (inner one and outer one), each of which retains its own distinctive properties. The frequency equations for pervious and impervious surfaces are obtained within the framework of Biot’s theory of wave propagation in poroelastic solids. Nondimensional frequency against the ratio of outer and inner radii is computed for two types of sandstone spherical shells and the results are presented graphically. From the graphs, nondimensional frequency values are periodic in nature, but in the case of ring modes, frequency values increase with the increase of the ratio. The nondimensional phase velocity as a function of wave number is also computed for two types of sandstone spherical shells and for the spherical bone implanted with titanium. In the case of sandstone shells, the trend is periodic and distinct from the case of bone. In the case of bone, when the wave number lies between 2 and 3, the phase velocity values are periodic, and when the wave number lies between 0.1 and 1, the phase velocity values decrease.

Small-Scale Yielding at the Tip of a Through-Crack in a Shell

1980 ◽  
Vol 102 (2) ◽  
pp. 167-173 ◽  
Author(s):  
J. D. Achenbach ◽  
T. Bubenik
Keyword(s):  
Nonlinear Behavior ◽  
Crack Edge ◽  
Small Scale ◽  
Small Scale Yielding ◽  
Spherical Shells ◽  
Shallow Shells ◽  
Scale Yielding ◽  
Hardening Curve ◽  
Thickness Variations ◽  
Independent Integral

Deformation theory is used to model plastic deformation at the tip of a through-crack in a thin shell. In the vicinity of the crack the shell is subjected to both stretching and bending, but stretching is assumed to dominate. Thus the stresses are tensile, but with a nonuniform distribution through the thickness, which depends on the material properties as well as on the geometry. The nonlinear near-tip fields (which are singular) have been analyzed asymptotically. Cracks in shallow shells and spherical shells have been investigated in some detail. It is shown that the angular variations are the same as for generalized plane-stress plate problems. Assuming small-scale yielding a path-independent integral, which is valid in a region close to the crack edge, is used to connect the nonlinear near-tip fields with the corresponding singular parts of the linear fields. It is shown that the nonlinear behavior significantly affects the through-the-thickness variations of the near-tip fields. The singular parts of the membrane stresses tend to become more uniform through the thickness of the shell with a flatter strain-hardening curve.

Stresses in a Spherical Shell With a Circular Elastic Inclusion

1974 ◽  
Vol 96 (3) ◽  
pp. 228-233
Author(s):  
P. Prakash ◽  
K. P. Rao
Keyword(s):  
Differential Equations ◽  
Spherical Shell ◽  
Elastic Inclusion ◽  
Circular Inclusion ◽  
Shallow Spherical Shell ◽  
Spherical Shells ◽  
Hankel Functions ◽  
Circular Elastic Inclusion ◽  
Rigid Circular Inclusion

The problem of a circular elastic inclusion in a thin pressurized spherical shell is considered. Using Reissner’s differential equations governing the behavior of a thin shallow spherical shell, the solutions for the two regions are obtained in terms of Bessel and Hankel functions. Particular cases of a rigid circular inclusion free to move with the shell and a clamped rigid circular inclusion are also considered. Results are presented in nondimensional form which will greatly facilitate their use in the design of spherical shells containing a rigid or an elastic inclusion.

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