Edge Influence Coefficients for Toroidal Shells of Positive Gaussian Curvature

1960 ◽  
Vol 82 (1) ◽  
pp. 60-68 ◽  
Author(s):  
G. D. Galletly
Keyword(s):  
Gaussian Curvature ◽  
Constant Thickness ◽  
Toroidal Shell ◽  
Uniform Pressure ◽  
Dimensionless Form ◽  
Positive Gaussian Curvature ◽  
Influence Coefficients ◽  
Bending Loads ◽  
Edge Influence ◽  
Toroidal Shells

Tables are given for the edge deformations of constant-thickness toroidal shells subject to uniform pressure and edge bending loads. Over one hundred different shell geometries were investigated and the results are presented in dimensionless form. Possession of these coefficients, which were obtained on a digital computer, means that a rapid and accurate formulation of the compatibility equations at toroidal shell junctions is now possible.

Edge Influence Coefficients for Toroidal Shells of Negative Gaussian Curvature

1960 ◽  
Vol 82 (1) ◽  
pp. 69-75 ◽  
Author(s):  
G. D. Galletly
Keyword(s):  
Gaussian Curvature ◽  
Constant Thickness ◽  
Toroidal Shell ◽  
Uniform Pressure ◽  
Simple Manner ◽  
Negative Gaussian Curvature ◽  
Influence Coefficients ◽  
Bending Loads ◽  
Edge Influence ◽  
Toroidal Shells

Continuing the work presented in reference [1], the present paper gives additional tables for the edge deformations of constant-thickness toroidal shells subject to edge bending loads and uniform pressure. The two papers together thus cover a wide variety of toroidal shell geometries and enable a designer to calculate in a simple manner the edge moments and shears at toroidal shell junctions.

On the Accuracy of Some Shell Solutions

1959 ◽  
Vol 26 (4) ◽  
pp. 577-583
Author(s):  
G. D. Galletly ◽  
J. R. M. Radok
Keyword(s):  
Negative Curvature ◽  
Numerical Solutions ◽  
Asymptotic Solutions ◽  
Thin Shells ◽  
Shells Of Revolution ◽  
Equilibrium Equations ◽  
Influence Coefficients ◽  
Bending Loads ◽  
Edge Influence

Abstract R. B. Dingle’s method [1] for finding asymptotic solutions of ordinary differential equations of a type such as occur in the bending theory of thin shells of revolution is presented in outline. This method leads to the same results as R. E. Langer’s method [2], recently used for problems of this kind, and permits a simple analytical and less formal interpretation of the asymptotic treatment of such equations. A comparison is given of edge influence coefficients due to bending loads, obtained by use of these asymptotic solutions and numerical integration of the equilibrium equations, respectively. The particular shells investigated are of the open-crown, ellipsoidal, and negative-curvature toroidal types. The results indicate that the agreement between these solutions is satisfactory. In the presence of uniform pressure, the use of the membrane solutions for the determination of the particular integrals appears to lead to acceptable results in the case of ellipsoidal shells. However, in the case of toroidal shells, the membrane and the numerical solutions disagree significantly.

On the Linear Bending Theory of a Pressurized Torus

1968 ◽  
Vol 35 (1) ◽  
pp. 160-162 ◽  
Author(s):  
Jaan Kiusalaas ◽  
K. S. Dhanju
Keyword(s):  
The Other ◽  
Toroidal Shell ◽  
Uniform Pressure ◽  
Other Hand ◽  
Manufacturing Errors ◽  
Shell Equations ◽  
Toroidal Shells

The intent of this note is to point out the role of initial displacements (due to manufacturing errors) in the analysis of toroidal shells by linear bending theory. It is shown that the stresses and displacement are very sensitive to small variations of the meridional curvature in the case of uniform pressure. On the other hand, the analysis is comparatively immune to these imperfections when the loading consists of forces or moments applied to the edges of the shell. These conclusions are drawn from the toroidal shell equations of Novozhilov and Zienova, which have been modified such as to include highest-order terms in initial displacements. No solution of the equations is attempted.

Nonlinear Corrections for Edge Bending of Shells

1980 ◽  
Vol 47 (4) ◽  
pp. 861-865 ◽  
Author(s):  
G. V. Ranjan ◽  
C. R. Steele
Keyword(s):  
Differential Equations ◽  
Spherical Shell ◽  
Asymptotic Expansions ◽  
Axial Displacement ◽  
Exponential Functions ◽  
Spherical Cap ◽  
Influence Coefficients ◽  
Edge Loading ◽  
General Geometry ◽  
Edge Influence

Asymptotic expansions for self-equilibrating edge loading are derived in terms of exponential functions, from which formulas for the stiffness and flexibility edge influence coefficients are obtained, which include the quadratic nonlinear terms. The flexibility coefficients agree with those previously obtained by Van Dyke for the pressurized spherical shell and provide the generalization to general geometry and loading. In addition, the axial displacement is obtained. The nonlinear terms in the differential equations can be identified as “prestress” and “quadratic rotation.” To assess the importance of the latter, the problem of a pressurized spherical cap with roller supported edges is considered. Results show that whether the rotation at the edge is constrained or not, the quadratic rotation terms do not have a large effect on the axial displacement. The effect will be large for problems with small membrane stresses.

Influence Coefficients for Hemispherical Shells With Small Openings at the Vertex

1955 ◽  
Vol 22 (1) ◽  
pp. 20-24
Author(s):  
G. D. Galletly
Keyword(s):  
Constant Thickness ◽  
Influence Coefficient ◽  
Circular Opening ◽  
Central Angle ◽  
Calculation Time ◽  
Numerical Example ◽  
Hemispherical Shell ◽  
Influence Coefficients ◽  
Hemispherical Shells

Abstract Three methods of obtaining the influence coefficients for a thin, constant-thickness, hemispherical shell with a circular opening at the vertex were investigated and utilized in a numerical example. Bearing in mind both accuracy and calculation time, it was concluded that when the total central angle subtended by the opening is less than approximately 30 deg, good results for the influence coefficient calculation will be obtained by using Method II in the text of the paper.

scholarly journals Umbilical points on surfaces in RN

1985 ◽  
Vol 100 ◽  
pp. 135-143 ◽  
Author(s):  
Kazuyuki Enomoto
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Normal Connection ◽  
Round Sphere ◽  
Positive Gaussian Curvature ◽  
Connected Surface ◽  
Umbilical Points ◽  
The Mean

Let ϕ: M → RN be an isometric imbedding of a compact, connected surface M into a Euclidean space RN. ψ is said to be umbilical at a point p of M if all principal curvatures are equal for any normal direction. It is known that if the Euler characteristic of M is not zero and N = 3, then ψ is umbilical at some point on M. In this paper we study umbilical points of surfaces of higher codimension. In Theorem 1, we show that if M is homeomorphic to either a 2-sphere or a 2-dimensional projective space and if the normal connection of ψ is flat, then ψ is umbilical at some point on M. In Section 2, we consider a surface M whose Gaussian curvature is positive constant. If the surface is compact and N = 3, Liebmann’s theorem says that it must be a round sphere. However, if N ≥ 4, the surface is not rigid: For any isometric imbedding Φ of R3 into R4 Φ(S2(r)) is a compact surface of constant positive Gaussian curvature 1/r2. We use Theorem 1 to show that if the normal connection of ψ is flat and the length of the mean curvature vector of ψ is constant, then ψ(M) is a round sphere in some R3 ⊂ RN. When N = 4, our conditions on ψ is satisfied if the mean curvature vector is parallel with respect to the normal connection. Our theorem fails if the surface is not compact, while the corresponding theorem holds locally for a surface with parallel mean curvature vector (See Remark (i) in Section 3).

scholarly journals Curved Bistable Composite Slit Tubes with Positive Gaussian Curvature

AIAA Journal ◽  
2018 ◽  
Vol 56 (4) ◽  
pp. 1679-1688 ◽  
Author(s):  
Geoffrey Knott ◽  
Andrew Viquerat
Keyword(s):  
Gaussian Curvature ◽  
Positive Gaussian Curvature
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