Shells Of Elliptical Cross-section Research Articles

Stress Distribution Near a Circular Hole in a Flexible Orthotropic Cylindrical Shell of Elliptical Cross-Section

A numerical technique for solving geometrically nonlinear problems for an orthotropic cylindrical shell of elliptical cross-section weakened by a circular hole is developed. The system of governing equations is derived using the Kirchhoff–Love theory of deep shells and Hooke’s law for orthotropic materials. The technique employs the procedure of incremental loading, as well as the modified Newton–Kantorovich and finite-element methods. The effect of finite deflections and mechanical and geometrical parameters on the stress–strain state of the shell near the circular hole acted upon by axial tensile forces is analyzed.

Casimir energy of a cylindrical shell of elliptical cross section

We calculate the increase in the number of modes (the Kac number) per unit length and the change in the zero-point energy (the Casimir energy) of the electromagnetic field resulting from the introduction of a thin perfectly conducting cylindrical shell of elliptical cross-section. Along the way we give a novel route to the calculation of these physical quantities. The Casimir energy is found to be attractive with the circular case corresponding to the energy maximum and the large eccentricity limit being the divergent energy minimum. As a result, with only Casimir stresses present, a fixed area shell is unstable with respect to collapse onto itself. This instability is argued to persist at arbitrary temperature.

Analysis of Stresses and Displacements in Orthotropic Noncircular Cylindrical Shells Under Centrifugal Loads

This paper addresses the class of stress–strain problems for thin orthotropic cylindrical shells of arbitrary cross section under centrifugal loads. Separating out the variables for a simply supported shell yields a system of ordinary differential equations for which the boundary-value problem is solved by a stable numerical method. Study is made of the distribution of displacements in shells of elliptical cross section versus the ratio of ellipse exes and the eccentricity of the axis of revolution relative to the geometrical axis of symmetry

On the stress-strain state of toroidal shells of elliptical cross section formed from nonlinear elastic orthotropic materials

Problems of the statics of toroidal shells with elliptical cross sections, which are formed from nonlinear elastic orthotropic materials, were examined. Investigations were carried out in accordance with a procedure based on the method of successive approximations, the variational-difference method, and the method of Lagrangian multipliers. Calculations were performed by varying the ellipticities of the cross section of the torus and the radii of the circular axis of the torus over a broad range. Numerical and analytical results are presented in tables and graphs. Attention was focused on the accuracy of the computational results obtained, and agreement between the internal and external forces in the cross section was adopted as an integral criterion.

Buckling of imperfect elastic elliptic toroidal shells subjected to uniform external pressure

The effects of axisymmetric initial geometric imperfections on the buckling pressures of externally pressurized steel elastic toroidal shells of elliptical cross-section are studied in this paper. The shapes of the axisymmetric localized imperfections are increased-radius circular ‘flat spots’ and smooth dimples. In most cases, the locations of the imperfections are at the north (Φ = 180°) and south (Φ = 0°) poles of the cross-section. The main shells investigated are prolate ellipses having k(≡ a/ b) = 1.5, R/ b = 4 and 10 and b/ t = 50, 100 and 500. Some studies of shells with k = 2.0 and 2.5 and k = 0.833 (an oblate ellipse) are also included.

Fluid column resonances of water-filled cylindrical shells of elliptical cross section

The acoustic resonances of a submerged fluid-filled cylindrical shell are analyzed as the shell cross section is deformed from circular to elliptical geometry. A schlieren visualization system is used to image standing wave fields within insonified water-filled shells of different eccentricities, and to locate the resonance frequencies of the fluid column. The acoustic behavior of elliptical cavities with infinite and finite surface impedances are modeled and the theory used to predict the resonance frequencies of the fluid column and calculate the pressure distribution in the acoustic field. As the symmetry of the circular shell is broken by deforming it to the more general ellipse the resonance spectrum changes; mode splittings and level crossings are observed as the eccentricity increases. The experimental observations of resonance patterns and frequency variation are in good agreement with the theoretical predictions.

Shock-resistant flextensional transducer

An underwater sonar projection includes a one piece hollow shell of elliptical cross section having a number of transducer stacks positioned within the shell and compressively prestressed by the shell to vibrate along the major axis thereof which causes the broad surfaces between the small radius ends to vibrate in and out. To resist large external shocks the shell is made somewhat heavier than usual, a center support member or beam is placed along the minor axis of the shell and an adjustable support rod is carried in the beam which is adjusted to a length just sufficient to clear the vibrating surfaces of the projector in normal operation but which functions to limit the inward movement of the shell if exposed to an explosion. The transducer stacks include a number of columns of piezoelectric disks on each side of the support beam which are bonded together, each stack having a tail member fastened to said center support beam and a head member between the disks and the end of the shell. To minimize damage to the disks, the stacks are wrapped with glass fiber cloth. Two or more such projector units may be fastened together and the ends closed by endplates.

Buckling and Initial Postbuckling Behavior of Oval Cylindrical Shells Under Axial Compression

Buckling and initial postbuckling behavior is determined for thin, elastic cylindrical shells of elliptical cross section. This study complements the buckling and advanced postbuckling calculations reported by Kempner and Chen on a similar class of shells. The initial postbuckling analysis indicates that, like compressed circular cylinders, the oval cylinders will be highly sensitive to small geometrical imperfections and may buckle at loads well below the predictions for the perfect shell. On the other hand, buckling will not necessarily result in complete collapse. A series of simple tests has been performed which provide qualitative verification of the major features of the theory.

Discussion: “Stresses and Deformations of Toroidal Shells of Elliptical Cross Section: With Applications to the Problems of Bending of Curved Tubes and of the Bourdon Gage” (Clark, R. A., Gilroy, T. I., and Reissner, E., 1952, ASME J. Appl. Mech., 19, pp. 37–48)

Discussion: “Stresses and Deformations of Toroidal Shells of Elliptical Cross Section: With Applications to the Problems of Bending of Curved Tubes and of the Bourdon Gage” (Clark, R. A., Gilroy, T. I., and Reissner, E., 1952, ASME J. Appl. Mech., 19, pp. 37–48)

Closure to “Discussion of ‘Stresses and Deformations of Toroidal Shells of Elliptical Cross Section: With Applications to the Problems of Bending of Curved Tubes and of the Bourdon Gage’” (1952, ASME J. Appl. Mech., 19, pp. 565–566)

Closure to “Discussion of ‘Stresses and Deformations of Toroidal Shells of Elliptical Cross Section: With Applications to the Problems of Bending of Curved Tubes and of the Bourdon Gage’” (1952, ASME J. Appl. Mech., 19, pp. 565–566)

Stresses and Deformations of Toroidal Shells of Elliptical Cross Section: With Applications to the Problems of Bending of Curved Tubes and of the Bourdon Gage

Abstract This paper is concerned with the application of the theory of thin shells to several problems for toroidal shells with elliptical cross section. These problems are as follows: (a) Closed shell subjected to uniform normal wall pressure. (b) Open shell subjected to end bending moments. (c) Combination of the results for the first and second problems in such a way as to obtain results for the stresses and deformations in Bourdon tubes. In all three problems the distribution of stresses is axially symmetric but only in the first problem are the displacements axially symmetric. The magnitude of stresses and deformations for given loads depends in all three problems on the magnitude of the two parameters bc/ah and b/c where b and c are the semiaxes of the elliptical section, a is the distance of the center of the section from the axis of revolution, and h is the thickness of the wall of the shell. For sufficiently small values of bc/ah trigonometric series solutions are obtained. For sufficiently large values of bc/ah asymptotic solutions are obtained. Numerical results are given for various quantities of practical interest as a function of bc/ah for the values 2, 1, 1/2, 1/4 of the semiaxes ratio b/c. It is suggested that the analysis be extended to still smaller values of b/c and to cross sections other than elliptical.