The paper is concerned with a nonlinear formulation of the problem of rotationally symmetric deformations of thin elastic shells of revolution, which are acted upon by edge forces and moments. Determined are, in particular, nonlinear corrections to the known results of the linear theory, for edge displacements and rotations. The calculations are for cases for which thickness and curvature of the shell are such as to insure that stresses and deformations are effectively contained within a narrow edge zone of the shell.

The yield surface for a thin cylindrical shell is shown to be a very good approximation to the yield surface for any symmetrically loaded thin shell of revolution. Hexagonal prism approximations to this yield surface, appropriate for pressure vessel analysis, are described and discussed in terms of limit analysis. Procedures suitable for finding upper and lower bounds on the limit pressure for the complete vessel are developed and evaluated. They are applied for illustration to a portion of a toroidal zone or knuckle held rigidly at the two bounding planes. The combined end force and moment which can be carried by an unflanged cylinder also is discussed.

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.

A strain-energy expression is derived for thin isotropic elastic shells in terms of the displacements of the middle surface of the shell. This expression is confined to small-deflection theory, and the condition of plane stress previously used in the theory of thin shells is retained. A simplified expression is also obtained by the introduction of the Kirchhoff-Love hypothesis, and the relative merits of these two expressions are discussed. The strain-energy expression is applied to the thin circular cylinder, and the result is compared with various strain-energy expressions developed by previous authors.

The differential-pressure test is an original, nondestructive, experimental technique for determining incipient buckling pressures of thin shells subjected to external pressure. Extensions of the basic test permit study of many buckling parameters as well as other mechanical factors in thin-shell design and evaluation. The salient feature of the technique is the filling of the internal volume of the shell with a compressible fluid, such as water, to control the magnitude and rate of shell deformation. The incipient buckling pressure is detected by noting the point at which the difference in internal and external pressure becomes constant. Experimental verification of the technique and its nondestructive aspect is presented. Applications and limitations of the test are discussed.

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.

In the general theory of shells expressions are obtained for the changes of curvature and the twist, and revisions are introduced in the equations of equilibrium.

In applying the elastic shell models to monolayer or few-layer two-dimensional (2D) materials, an effective thickness has to be defined to capture their tensile and out-of-plane mechanical behaviors. This thin-shell thickness differs from the interlayer distance of their layer-by-layer assembly in the bulk and is directly related to the Föppl–von Karman number that characterizes the mechanism of nonlinear structural deformation. In this work, we assess such a definition for a wide spectrum of 2D crystals of current interest. Based on first-principles calculations, we report that the discrepancy between the thin-shell thickness and interlayer distance is weakened for 2D materials with lower tensile stiffness, higher bending stiffness, or more number of atomic layers. For multilayer assembly of 2D materials, the tensile and bending stiffness have different scaling relations with the number of layers, and the thin-shell thickness per layer approaches the interlayer distance as the number of layers increases. These findings lay the ground for constructing continuum models of 2D materials with both tensile and bending deformation.

In this paper, we combine recent developments in modeling of fatigue-damage, isogeometric analysis (IGA) of thin-shell structures, and structural health monitoring (SHM) to develop a computational steering framework for fatigue-damage prediction in full-scale laminated composite structures. The main constituents of the proposed framework are described in detail, and the framework is deployed in the context of an actual fatigue test of a full-scale wind-turbine blade structure. The results indicate that using an advanced computational model informed by in situ SHM data leads to accurate prediction of the damage zone formation, damage progression, and eventual failure of the structure. Although the blade fatigue simulation was driven by test data obtained prior to the computation, the proposed computational steering framework may be deployed concurrently with structures undergoing fatigue loading.

The equilibrium equations and boundary conditions in terms of the second Piola–Kirchhoff membrane stress and moment are given in this note, which are necessary for the finite deformation analysis of shells.

This paper provides the theoretical collapse loads of thick, long cylindrical shells subject to pressure and axial forces. Tubes are made of isotropic, perfectly plastic von Mises’ material. Axial strains are assumed to be constant but possibly different from zero, so that elongation is permitted. This assumption, together with axial symmetry and the isochoric nature of plastic flow, unambiguously defines the set of possible collapse mechanisms, and collapse loads are computed on this basis. Results are contrasted to those presently available, based on thin-shell assumptions. Comparison shows that differences are of engineering significance, well worth considering for thick tubes, such as those envisaged in some nuclear power plant applications.

This paper presents an exact solution and an approximate solution, using the expansion of transient wave functions in a series of eigenfunctions, for the transient response of an infinitely long and multilayered circular cylinder subjected to uniformly distributed dynamic pressures at the boundaries. Numerical results are given to illustrate the effects of the layer properties on the interfacial stresses and the spatial and temporal variations of the displacement and stresses. In particular, the exact solution is used to examine the applicability of the thin shell theories to the transient response of multilayered cylinders.

In the spirit of Mindlin and others who have used series expansions to express transverse dependences in thin bodies, the present work uses Ritz expansions in a variational formulation for cylindrical shell vibrations. By expanding displacements in spatial coordinates, integral expressions for strain and kinetic energy are converted to quadratic sums involving time-dependent generalized coordinates. Hamilton’s principle provides ordinary differential equations for these coordinates. This view-point yields physical insight into the mechanisms of energy storage and avoids the geometrically thin assumption inherent to many formulations. A set of Legendre polynomials multiplied by a radial factor represent the radial dependences of displacement components, while circumferential variations are represented by sinusoidal functions. Excellent agreement in natural frequencies is found between this approach and analytical solutions over the entire range of shell thicknesses, including the limiting case of a solid cylinder. Comparisons to several thin shell theories are given, leading to conclusions about the range of validity of these theories.

Nonlinear bending deformations of generally laminated circular cylindrical shells subjected to an axisymmetric radial pressure, axial loading, and torsion are considered. Deformations of such shells are axisymmetric, i.e., while cross-sections can rotate about the shell axis, they remain circular and all derivatives with respect to the circumferential coordinate are equal to zero. In this case a nonlinear term is included into the axial strain only. Excluding this strain from the equilibrium equations, one can reduce the analysis to integration of a system of linear differential equations. In the first part of the paper the solution based on Love’s first approximation and Donnell’s theories is shown for thin shells. While out-of-surface boundary conditions can be arbitrary, several combinations of in-surface conditions are discussed. The second part of the paper outlines the approach to the solution for shear-deformable geometrically nonlinear shells. First-order shear-deformation theory is used in the analysis. The approach is based on exclusion of the nonlinear term (axial strain) and subsequent solution of linear differential equations of equilibrium by the operational method. This approach can be also used for an analysis of thin, generally laminated cylindrical shells.

Work–conjugate boundary conditions for a class of nonlinear theories of thin shells formulated in terms of displacements of the reference surface are discussed. Applying theorems of the theory of differential forms it is shown that many of the sets of static boundary conditions which have been proposed in the literature do not possess work–conjugate geometric counterparts. The general form of four geometric boundary conditions and their work–conjugate static boundary conditions is constructed and three particular cases are analyzed. The boundary conditions given here are valid for unrestricted displacements, rotations, strains and/or changes of curvatures of the reference surface.

The finite axisymmetric deformation of a thin shell of revolution is treated in this analysis. The governing differential equations are given for a hyperelastic shell material with the Mooney-Rivlin strain-energy-density function. These equations are solved numerically using a 4th-order Runge-Kutta integration method. A generalized Newton-Raphson iteration procedure is used to systematically improve trial solutions of the differential equations. The governing differential equations are differentiated with respect to a set of generalized coordinates to derive associated rate equations. The rate equations are solved numerically to generate the tangent stiffness matrix which is used to determine the load deformation history of the shell with incremental loading. Numerical examples are presented to illustrate the major characteristics of nonlinear shell behavior.

The problem of a circumferential through-crack at the fixed end of a cylinder in flexure is considered herein. An expression for the energy-release rate, in closed form, is obtained according to linear fracture mechanics and thin shell theory. The expression consists of two terms, one of which is the energy-release due to shear and the other is the energy-release due to bending. The former is relatively negligible except for very short cylinders. A formula for the flexibility factor is also derived.

Plasticity in shells is often contained near the ends of a segment where the bending stresses are significant. Outside of this local neighborhood the behavior is elastic. Thus, an axisymmetric shell can be divided along its axis into a purely elastic region away from an end and the local region where plasticity is present. The moment-curvature relation in the elastic-plastic region is calculated using the Tresca yield condition. Use of the Tresca yield condition greatly simplifies this derivation because the principal directions are known. This moment-curvature relationship is “exact” in the sense that only the standard assumptions of thin shell theory are made. The solutions of the elastic and plastic regions are matched at their intersection for an efficient numerical solution. The technique is used here to study the semi-infinite clamped cylindrical shell with an internal pressure loading.