Singular Solutions to the Shallow Shell Equations

1970 ◽  
Vol 37 (2) ◽  
pp. 361-366 ◽  
Author(s):  
J. Lyell Sanders
Keyword(s):  
Closed Form ◽  
Theoretical Investigation ◽  
Shallow Shell ◽  
Fourier Transforms ◽  
Complex Form ◽  
Middle Surface ◽  
Singular Solutions ◽  
Concentrated Loads ◽  
Shell Equations ◽  
Complete Solutions

The paper contains a theoretical investigation of those multivalued singular solutions to the shallow shell equations which correspond physically to concentrated loads and dislocations. Use of the shell equations in complex form permits a unified treatment of the load and dislocation problems. The analysis is limited to the case of shells with a quadratic middle surface, and Fourier transforms of the solutions are obtained. Complete solutions in closed form in the case of a shallow sphere are given in the Appendix, including some results not previously published.

Point Load on a Shallow Elliptic Paraboloid

1966 ◽  
Vol 33 (3) ◽  
pp. 575-585 ◽  
Author(s):  
Kevin Forsberg ◽  
Wilhelm Flu¨gge
Keyword(s):  
Power Series ◽  
Axial Symmetry ◽  
Shallow Shell ◽  
Point Load ◽  
Singular Solutions ◽  
Elliptic Paraboloid ◽  
Concentrated Loading ◽  
Shell Equations ◽  
Shell Geometry ◽  
Thin Shallow Shell

The present work is a study of a thin shallow shell having a specific type of deviation from axial symmetry, i.e., the portion of an elliptic paraboloid near its vertex. The singular solutions to the homogeneous shallow-shell equations are expressed as power series in terms of a parameter γ, which is a measure of the deviation of the shell geometry from axial symmetry. These singular solutions can be directly related to concentrated loading at the vertex of the shell. The solution converges in the range γ = 0 (sphere) to γ = 1/2 (cylinder). Detailed graphical results are presented for the stress resultants and radial deflection of a shell subjected to a point load at its vertex.

Admissible Singular Solutions and the Hypercircle Method in Shallow Shell Theory

1977 ◽  
Vol 44 (1) ◽  
pp. 117-122 ◽  
Author(s):  
H. Antes
Keyword(s):  
Exact Solution ◽  
Singular Point ◽  
Iterative Procedure ◽  
Shallow Shell ◽  
Shell Theory ◽  
Plate Theory ◽  
Singular Solutions ◽  
Admissible Solutions ◽  
Shell Equations

The object of this study is the construction of geometrically and statically admissible solutions of the basic shallow shell equations in the case of singular loads, especially for the use in the hypersphere theorems. An iterative procedure extends known solutions of plate theory to the classical and an improved shallow shell theory. The results contain all important terms of the exact solution near the singular point.

scholarly journals Closed-Form Expressions for the Analysis of Wave Propagation in Overhead Distribution Lines

Energies ◽  
2020 ◽  
Vol 13 (17) ◽  
pp. 4519
Author(s):  
Theofilos A. Papadopoulos ◽  
Andreas I. Chrysochos ◽  
Christos K. Traianos ◽  
Grigoris Papagiannis
Keyword(s):  
Wave Propagation ◽  
Closed Form ◽  
System Analysis ◽  
Complex Form ◽  
Displacement Current ◽  
Decomposition Algorithms ◽  
Overhead Lines ◽  
Power System Analysis ◽  
Electromagnetic Transient ◽  
Time Domain Electromagnetic

The calculation of the influence of the imperfect earth on overhead conductors is an important issue in power system analysis. Rigorous solutions contain infinite integrals; thus, due to their complex form, different simplified closed-form expressions have been proposed in the literature. This paper presents a detailed analysis of the effect of different closed-form expressions on the investigation of the wave propagation of distribution overhead lines (OHLs). A sensitivity analysis is applied to determine the most important properties influencing the calculation of the OHL parameters. The accuracy of several closed-form earth impedance models is evaluated as well as the influence of the displacement current and imperfect earth on the shunt admittance, which are further employed in the calculation of the propagation characteristics of OHLs. The frequency-dependence of the soil electrical properties, as well as the application of different modal decomposition algorithms, are also investigated. Finally, results on the basis of frequency-domain signal scans and time-domain electromagnetic transient responses are also discussed.

Pressurized Cylindrical Shell With a Rigid Circular Inclusion

1978 ◽  
Vol 100 (2) ◽  
pp. 158-163 ◽  
Author(s):  
D. H. Bonde ◽  
K. P. Rao
Keyword(s):  
Differential Equation ◽  
Cylindrical Shell ◽  
Shallow Shell ◽  
Circular Inclusion ◽  
Hankel Functions ◽  
Curvature Parameter ◽  
Shell Equations ◽  
Limiting Case ◽  
Rigid Circular Inclusion ◽  
Good Agreement

The effect of a rigid circular inclusion on stresses in a cylindrical shell subjected to internal pressure has been studied. The two linear shallow shell equations governing the behavior of a cylindrical shell are converted into a single differential equation involving a curvature parameter and a potential function in nondimensionalized form. The solution in terms of Hankel functions is used to find membrane and bending stressses. Boundary conditions at the inclusion shell junction are expressed in a simple form involving the in-plane strains and change of curvature. Good agreement has been obtained for the limiting case of a flat plate. The shell results are plotted in nondimensional form for ready use.

scholarly journals Solutions of Unified Fractional Schrödinger Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
V. B. L. Chaurasia ◽  
Devendra Kumar
Keyword(s):  
Schrödinger Equation ◽  
Closed Form ◽  
Numerical Computation ◽  
Schrodinger Equation ◽  
Fourier Transforms ◽  
Schrödinger Equations ◽  
Laplace And Fourier Transforms ◽  
Fractional Schrödinger Equation ◽  
Fractional Schrödinger Equations ◽  
Fractional Schrodinger Equation

We obtain the solution of a unified fractional Schrödinger equation. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the Mittag-Leffler function. The result obtained here is quite general in nature and capable of yielding a very large number of results (new and known) hitherto scattered in the literature. Most of results obtained are in a form suitable for numerical computation.

scholarly journals On shallow shell equations

2009 ◽  
Vol 2 (3) ◽  
pp. 697-722 ◽  
Author(s):  
Peng-Fei Yao ◽  
Keyword(s):  
Shallow Shell ◽  
Shell Equations

Projected Measures: A Simple Way to Characterize Fractal Structures and Interfaces

Fractals ◽  
1997 ◽  
Vol 05 (02) ◽  
pp. 295-308
Author(s):  
Massimiliano Giona ◽  
Manuela Giustiniani ◽  
Oreste Patierno
Keyword(s):  
Closed Form ◽  
Fourier Transforms ◽  
Fractal Structures ◽  
Multifractal Spectra ◽  
Self Similar ◽  
The Moment ◽  
Dynamic Phenomena

The properties of projected measures of fractal objects are investigated in detail. In general, projected measures display multifractal features which play a role in the evolution of dynamic phenomena on/through fractal structures. Closed-form results are obtained for the moment hierarchy of model fractal interfaces. The distinction between self-similar and self-affine interfaces is discussed by considering the properties of multifractal spectra, the orientational effects in the behavior of the moment hierarchies, and the scaling of the corresponding Fourier transforms. The implications of the properties of projected measures in the characterization of transfer phenomena across fractal interfaces are briefly analyzed.

Analysis of a Semi-Infinite Strip With Constant Stress Flanges Under Concentrated Loads

1966 ◽  
Vol 33 (3) ◽  
pp. 571-574 ◽  
Author(s):  
H. R. Meck
Keyword(s):  
Stress Distribution ◽  
Exact Solutions ◽  
Closed Form ◽  
Complex Variable ◽  
Constant Stress ◽  
Infinite Strip ◽  
Concentrated Loads ◽  
Simple Complex ◽  
Variable Analysis

An analysis is presented for a semi-infinite strip reinforced by flanges and subjected to concentrated in-plane loads at the ends of the flanges. The taper which results in constant flange stress is determined, and the stress distribution in the sheet is also found. A simple complex variable analysis is used which leads to exact solutions in closed form.

Self-Equilibrated Singular Solutions of a Complete Spherical Shell: Classical Theory Approach

1989 ◽  
Vol 56 (1) ◽  
pp. 105-112 ◽  
Author(s):  
Nikolaos Simos ◽  
Ali M. Sadegh
Keyword(s):  
Spherical Shell ◽  
Closed Form ◽  
Mathematical Analysis ◽  
Stress Function ◽  
Elastic Response ◽  
Singular Solutions ◽  
Transverse Displacement ◽  
System Of Equations ◽  
Theory Approach ◽  
Elementary Functions

The elastic response of a complete spherical shell under the influence of concentrated loads (normal point loads, concentrated tangential loads, and concentrated surface moments) which apply in a self-equilibrating fashion is obtained. The mathematical analysis incorporates the classical uncoupled system of equations for the transverse displacement W and a stress function F. The solution formulae for all three types of singular loading are in closed form and they are expressed in terms of complex Legendre and other elementary functions. The two latter portions of the analysis are associated with a multivalued stress function F which leads to a single-valued stress and displacement formulae. The intricacies of the solutions and their singular character are also discussed. Lastly, some representative shell problems are evaluated.

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