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.

A unified treatment is presented of two well-known problems which have until now been considered separately. The two problems are: (a) the linear problem of pure bending of curved tubes, and (b) the nonlinear problem of pure bending of straight tubes. In both problems the effect of uniform internal pressure is included. The essential step in the present analysis is the treatment of the flattening of the cross sections of the tube by means of a theory of finite bending of circular rings. The general results of the paper are used to obtain improved values for the stability parameters in the problem of flattening instability of originally straight tubes acted upon by end bending moments, and also to obtain results on the effect of slight original curvature of the beam axis in the problem of flattening instability.

An exact solution is obtained of the nonlinear equations for finite deflections of thin elastic plates, for a rectangular plate acted upon by twisting moments T and bending moments M simultaneously applied to two opposite edges of the plate. The principal results of the paper are relations between the moments T and M , on the one hand, and the angle of twist per unit of length θ , and the curvature k of the plate, on the other hand. These relations which are of the form T = T ( θ , k ) and M = M ( θ , k ) generalize previously known relations for the special cases for which either k = 0 or θ = 0. A numerical evaluation of the functions T ( θ , k ) and M ( θ , k ) reveals the existence of a characteristic jump phenomenon.

Known solutions for transverse bending and plane stress of flat circular ring plates are generalized so as to apply to shallow helicoidal shells. The pitch of the middle surface of the shell is responsible for a coupling of what would be separate problems of plane stress and transverse bending for flat plates. Explicit results are obtained for an important sample problem. A general property of helicoidal cantilever shells is stated. Criteria are obtained indicating, in terms of the parameters of the shell ( a ) the range of applicability of the results obtained, and ( b ) the range in which the shell behaves like a flat plate.

In a recent paper (1) the author considered anew the problem of determining corrections to the Saint Venant theory of torsion, induced by restraint against warping at the end section of the beam. The present note is intended to furnish some further discussion of these earlier results.

A system of equations is developed for the theory of bending of thin elastic plates which takes into account the transverse shear deformability of the plate. This system of equations is of such nature that three boundary conditions can and must be prescribed along the edge of the plate. The general solution of the system of equations is obtained in terms of two plane harmonic functions and one function which is the general solution of the equation Δ ψ − (10/ h 2 ) ψ = 0. The general results of the paper are applied ( a ) to the problem of torsion of a rectangular plate, ( b ) to the problems of plain bending and pure twisting of an infinite plate with a circular hole. In these two problems important differences are noted between the results of the present theory and the results obtained by means of the classical plate theory. It is indicated that the present theory may be applied to other problems where the deviations from the results of classical plate theory are of interest. Among these other problems is the determination of the reactions along the edges of a simply supported rectangular plate, where the classical theory leads to concentrated reactions at the corners of the plate. These concentrated reactions will not occur in the solution of the foregoing problem by means of the theory given in the present paper.

This paper deals with the problem of the distribution of stress in cantilever beams of narrow rectangular flanged cross section with one end of the beam rigidly built-in. Since an exact solution of this plane-stress problem appears difficult to obtain, an approximate solution is derived by applying the principle of least work. Instead of the linear normal stress distribution of the elementary beam theory, a third-degree polynomial is assumed, and the spanwise variation of this stress curve is determined by means of the calculus of variations. Numerical results are obtained with regard to the stresses at the built-in end of the beam, in their dependence upon ( a ) the span-height ratio of the beam, ( b ) the flange area-web area ratio of the beam, ( c ) Poisson’s ratio of the material, and ( d ) the distribution of load along the span. It is found that the deviations from the results of the elementary theory may be appreciable when the distance of the center of gravity of the load curve from the built-in end of the beam is less than twice the height of the cross section of the beam.

We depart from a three-dimensional statement of the problem of small bending of elastic plates, for a survey of approximate two-dimensional theories, beginning with Kirchhoff’s fourth-order formulation. After discussing various variational statements of the three-dimensional problem, we describe the development of two-dimensional sixth-order theories by Bolle´, Hencky, Mindlin, and Reissner which take account of the effect of transverse shear deformation. Additionally, we report on an early analysis by Le´vy, on a direct two-dimensional formulation of sixth-order theory, on constitutive coupling of bending and stretching of laminated plates, on higher than sixth-order theories, and on an asymptotic analysis of sixth-order theory which leads to a fourth-order interior solution contribution with first-order transverse shear deformation effects included, as well as to a sequentially determined second-order edge zone solution contribution.

Recent simplifications of linear shell theory through consideration of transverse shear deformations and stress moments with axes normal to the shell middle surface suggest analogous approaches to the corresponding problem of nonlinear theory. As a first step in this direction consideration is given here to the classical subject of finite symmetrical deformations of shells of revolution. The principal new results of the present analysis concern the form of strain-displacement and compatibility differential equations.

The paper presents an asymptotic expansion procedure (with reference to decreasing values of wall-thickness shell-radius ratio) for the solutions of the linear theory of the three simultaneous equations for displacements in isotropic circular cylindrical shells. The resultant expansions are applied to a number of fundamental boundary-value problems for a semi-infinite shell. The analysis shows the importance of the joint consideration of differential equations and boundary conditions for a clarification of the asymptotic nature of the solution to be determined.

In the first part of the paper the basic equations of three-dimensional elasticity are formulated as a system of four equilibrium equations and seven stress displacement relations, together with a variational problem which has these eleven equations as Euler equations. In the second part of the paper, the new variational problem is used for a derivation of shell theory which accounts particularly simply for the differences between the resultants N 12 and N 21 and the couples M 12 and M 21 . In the third part of the paper a solution is given of the torsion problem for circumferentially nonhomogeneous circular cylindrical shells, as an explicit demonstration of the fact that certain terms in the shell equations which are often of negligible influence sometimes are of considerable influence.

The stress-strain relations of Flu¨gge and Byrne for thin elastic shells are inverted to express strain quantities, and therewith the strain energy, in terms of stress resultants and couples. In this form, and upon omission of terms which are small of order h 2 /R 2 , the stress-strain relations and the strain-energy expression are shown to be simply related to corresponding results of Trefftz. The strain-energy formula of Trefftz is generalized to arbitrary orthogonal middle surface co-ordinates.