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.

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.

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.

Cylindrical Shells Under Line Load

1957 ◽  
Vol 24 (4) ◽  
pp. 553-558
Author(s):  
R. M. Cooper
Keyword(s):  
Cylindrical Shell ◽  
Radial Displacement ◽  
Circular Cylindrical Shell ◽  
Transverse Shear Deformation ◽  
System Of Equations ◽  
Principle Of Superposition ◽  
Line Load ◽  
Simply Supported ◽  
Shell Equations ◽  
Shell Geometry

Abstract The problem of a line load along a segment of a generator of a simply supported circular cylindrical shell is treated using shallow cylindrical shell equations which include the effect of transverse-shear deformation. The line load is first treated as a sinusoidally-varying edge load over the length of the shell, with boundary conditions prescribed along the loaded generator such that the continuity of the shell is maintained. The solution for the problem of a uniform line load over a segment of a generator is obtained from the preceding solution, using the principle of superposition. By means of a numerical example it is shown that the results predicted by the Donnell equations for the stresses are in excellent agreement with those obtained from the system of equations employed here. However, the radial displacement predicted by the Donnell equations is in error by as much as 20 per cent in the range of shell geometry considered.

Effet des imperfections géométriques sur la stabilité des coques élastiques cylindriques

2004 ◽  
Vol 31 (1) ◽  
pp. 27-32 ◽  
Author(s):  
Abdellatif Khamlichi ◽  
Mohammed Bezzazi ◽  
Larbi Elbakkali ◽  
Ali Limam
Keyword(s):  
Critical Load ◽  
Axial Symmetry ◽  
A Priori ◽  
Severe Case ◽  
Thin Shells ◽  
Geometrical Imperfections ◽  
Localized Defects ◽  
Shell Equations ◽  
Selection Of ◽  
Numerical Approaches

The effects of geometrical imperfections on the critical load of elastic cylindrical shells when subjected to axial compression are studied through analytical modelling. In addition to distributed defects of both axisymmetric or asymmetric forms, emphasis is put on the more severe case of localized defects satisfying the axial symmetry. The Von Kármán – Donnell shell equations were used. The obtained results show that shell strength at buckling varies very much with the defect amplitude. These variations are not monotonic in general. They indicate however a clear reduction of the shell critical load for some defects revealed as the most dangerous ones. The proposed method does not consider the complete coupled situation that may arise from interactions between several localized defects. It facilitates nevertheless straightforward initializing of closer analyses if such couplings are to be taken into account by means of special numerical approaches, because it enables fast a priori selection of the most hazardous isolated defects.Key words: stability, buckling, imperfections, thin shells, silos, localized defects.

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 On shallow shell equations

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