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.

scholarly journals A Green’s Function Approach for Dynamic Stress Analysis of Spherical Shell under the Isotropic Impact Load

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Jincheng Lv ◽  
Shike Zhang ◽  
Xinsheng Yuan
Keyword(s):  
Stress Distribution ◽  
Spherical Shell ◽  
Green’S Function ◽  
Green's Function ◽  
Dynamic Stress ◽  
Initial Conditions ◽  
Impact Load ◽  
Static Solution ◽  
Function Approach ◽  
Dynamic Stress Distribution

A Green’s function approach is developed for the analytic solution of thick-walled spherical shell under an isotropic impact load, which involves building Green’s function of this problem by using the appropriate boundary conditions of thick-walled spherical shell. This method can be used to analyze displacement distribution and dynamic stress distribution of the thick-walled spherical shell. The advantages of this method are able(1)to avoid the superposition process of quasi-static solution and free vibration solution during decomposition of dynamic general solution of dynamics,(2)to well adapt for various initial conditions, and(3)to conveniently analyze the dynamic stress distribution using numerical calculation. Finally, a special case is performed to verify that the proposed Green’s function method is able to accurately analyze the dynamic stress distribution of thick-walled spherical shell under an isotropic impact load.

XIII.—Some Integrals, with respect to their Degrees, of Associated Legendre Functions

1935 ◽  
Vol 54 ◽  
pp. 135-144 ◽  
Author(s):  
T. M. MacRobert
Keyword(s):  
Type A ◽  
Fourier Integral ◽  
Contour Integration ◽  
Legendre Functions ◽  
Integral Type ◽  
Integral Theorem ◽  
Associated Legendre Functions ◽  
In Series ◽  
Special Case

Little is known regarding the integration of Legendre Functions with respect to their degrees. In this paper several such integrals are evaluated, three different methods being employed. In § 2 proofs are given of a number of formulae which are required later. In § 3 an example is given of the evaluation of an integral by contour integration. The following section contains the proof of a formula of the Fourier Integral type, a special case of which was given in a previous paper (Proc. Roy. Soc. Edin., vol. li, 1931, p. 123). In § 5 an integral is evaluated by employing Fourier's Integral Theorem; while in § 6 other integrals are evaluated by. means of expansions in series.

scholarly journals Some applications of Legendre numbers

1988 ◽  
Vol 11 (2) ◽  
pp. 405-412 ◽  
Author(s):  
Paul W. Haggard
Keyword(s):  
Legendre Polynomials ◽  
Legendre Functions ◽  
Linear Combinations ◽  
Associated Legendre Functions ◽  
In Series ◽  
Derivatives Of

The associated Legendre functions are defined using the Legendre numbers. From these the associated Legendre polynomials are obtained and the derivatives of these polynomials atx=0are derived by using properties of the Legendre numbers. These derivatives are then used to expand the associated Legendre polynomials andxnin series of Legendre polynomials. Other applications include evaluating certain integrals, expressing polynomials as linear combinations of Legendre polynomials, and expressing linear combinations of Legendre polynomials as polynomials. A connection between Legendre and Pascal numbers is also given.

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.

The complex wavenumber eigenvalues of Laplace’s tidal equations for oceans bounded by meridians

1995 ◽  
Vol 449 (1935) ◽  
pp. 51-64 ◽  
Keyword(s):  
Gravity Waves ◽  
Homogeneous System ◽  
Legendre Functions ◽  
Constant Depth ◽  
Eigenvalue Spectrum ◽  
Associated Legendre Functions ◽  
Complex Eigenvalues ◽  
Basin Scale ◽  
Tide Height ◽  
In Series

The complex wavenumber eigenvalues of Laplace’s tidal equations are determined for an ocean of constant depth bounded by meridians. A Galerkin method is used to expand the tide height and velocities in series of associated Legendre functions. A homogeneous system of equations results from the continuity and momentum equations. The frequency and depth are fixed, so that the meridional wavenumbers are the eigenvalues. This gives rise to a generalized eigenvalue problem that must be solved numerically by iteration. The eigenvalues are not integers and represent inertia-gravity wave solutions at the specified tidal forcing frequency that can be excited by the presence of meridional boundaries. Those complex eigenvalues represent solutions that decay away from meridional boundaries. The eigenvalue spectrum is investigated for the semi-diurnal, fortnightly, and monthly tides. One complex wavenumber for the semi-diurnal tide explains the amphidromic systems within 20° of a north-south coastline. The fortnightly and monthly tides have only real wavenumber eigenvalues. The basin scale deviation of these tides from equilibrium is attributed to low wavenumber divergent inertia-gravity waves.

Application of the R-Function Theory for the Bending Problem of Shallow Spherical Shells with a Dodecagon Domain

2012 ◽  
Vol 468-471 ◽  
pp. 8-12 ◽  
Author(s):  
Shan Qing Li ◽  
Hong Yuan
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Function Theory ◽  
Homogeneous Boundary Condition ◽  
Fredholm Integral Equations ◽  
Winkler Foundation ◽  
Boundary Equation ◽  
Spherical Shells ◽  
Suitable Form ◽  
Good Agreement

The R-function theory is applied to describe the dodecagon domain of shallow spherical shells on Winkler foundation, and it is also used to construct a quasi-Green’s function. The quasi-Green’s function satisfies the homogeneous boundary condition of the problem. Then the differential equation of the problem is reduced to two simultaneous Fredholm integral equations of the second kind by the Green’s formula. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. A comparison with the ANSYS finite element solution shows a good agreement, and it demonstrates the feasibility and efficiency of the present method.

On Fundamental Solutions and Green’s Functions in the Theory of Elastic Plates

1966 ◽  
Vol 33 (1) ◽  
pp. 31-38 ◽  
Author(s):  
A. Kalnins
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Bessel Functions ◽  
Arbitrary Point ◽  
Fundamental Solutions ◽  
Addition Theorem ◽  
Reciprocal Theorem ◽  
Elastic Plates ◽  
Concentrated Load ◽  
The Reciprocal Theorem

This paper is concerned with fundamental solutions of static and dynamic linear inextensional theories of thin elastic plates. It is shown that the appropriate conditions which a fundamental singularity must satisfy at the pole follow from the requirement that the reciprocal theorem is satisfied everywhere in the region occupied by the plate. Furthermore, dynamic Green’s function for a plate bounded by two concentric circular boundaries is derived by means of the addition theorem of Bessel functions. The derived Green’s function represents the response of the plate to a harmonically oscillating normal concentrated load situated at an arbitrary point on the plate.

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