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By means of a simple example, a stretched string under transverse load, finite element and finite difference methods which are so widely used in engineering are illustrated. The finite element method is shown to be an essentially modified Raleigh-Ritz procedure. The finite difference technique is applied directly to the string differential equation; an energy related approach is also discussed. The manner in which a combination finite element/finite difference solution can be effected for the same physical problem is treated. Application of both the finite element and finite difference methods to more complex problems as well as selected programs and depositories are mentioned.

The title problem is solved by estimating the maximum strain rate field in a circular membrane associated with a modal velocity condition when one-half the kinetic energy in the system has been dissipated. The radial stress may only vary in the radial direction, but is taken as a constant with time at any given location. Internal plastic energy absorbed during the large deformation is found by integrating while accounting for rate sensitivity effects; this absorbed energy is equated to the initial kinetic energy in the system to obtain the final deformed configuration. Correlation of analytically determined results with a series of experiments reported elsewhere is very good. The procedure described here is potentially extendable to any planform membrane shape as well as to axisymmetric shells. The method is applicable to plate problems when membrane effects dominate over bending, that is for deformations of the order of two thicknesses or more. A simple general formula for final plate deformation is devised and also agrees well with experimental results.

A string supported impulsively loaded mass is used to study large deformation rate sensitivity effects where membrane action is dominant. It is found that an overall correction factor can be devised using physical properties associated with the average strain rate. Maximum strain rate occurs with a velocity field corresponding to the deformation state wherein half the initial kinetic energy has been dissipated. (If V0 is initial velocity, V0/2 is associated with maximum strain rate.) Exact and approximate solutions for a broad range of parameters serve to illustrate and verify the procedure. A discussion is presented to show how the same methodology could also be applied via a modal approach to an arbitrary three-dimensional structure undergoing large deformations, if the primary mechanism for energy absorption is from membrane action.

The nonlinear relaxation method, an iterative approach used in conjunction with finite-difference approximations, is illustrated via the solution to a very simple problem. Subsequently, the method is used to solve three geometrically nonlinear problems in mechanics: finite bending of a circular thin walled tube, the large deflection membrane response of a spherical cap, and finite deformations of a uniformly loaded circular membrane. Formulations for the three problems are quite different but this difference does not inhibit the use of the nonlinear relaxation technique. Solutions were obtained in approximately one man day per problem including the total time devoted to examining, planning, programming, debugging, etc. Solutions compare very favorably with results found elsewhere in the literature. The essential and important advantages of the nonlinear relaxation technique are (a) versatility and ease of application, (b) efficiency with respect to people and computer time utilized, (c) insensitivity to starting values as far as convergence is concerned, and (d) simplicity of logic that makes it a trivial task to learn how to employ it.

The title problem has been solved using the nonlinear relaxation technique to treat the finite-difference representation of the differential equations governing cap response. For clamped spherical caps designated by λ = 6, buckling loads are somewhat lower for the same imperfection magnitudes than those obtained previously for asymmetric and axisymmetric buckling. The three differential equations of the Marguerre type are set up in terms of three displacement quantities as the dependent variables. The equations are formulated so as to allow arbitrary initial imperfections, boundary conditions, and loading. The computer program setup to solve these equations was checked by successfully solving simpler problems, the solutions of which are in the literature. Although the buckling loads for spherical caps with asymmetric initial imperfections appear reasonable, it is not possible to ascertain at this point whether the cap results are indicative of what the full shell might do.

A numerical method of solution for the nonlinear deflection of thin flat membranes subjected to normal forces as well as forces in the plane of the membrane has been given by Hencky. The equations he solved were first derived by Föppl, and they also follow directly from the von Kármán nonlinear plate equations on formally making the plate stiffness zero. Föppl’s equations are two in number, one being of fourth order and the other second. The unknown quantities are a stress function and the normal displacement. Hencky’s method of solution does not seem capable of easy generalization. The same class of membrane problems is reconsidered here. By casting the problem entirely in terms of displacement components three simultaneous nonlinear second-order partial differential equations are obtained, and a technique is here devised by means of which these equations can be solved without difficulty using finite-difference approximations in conjunction with a relaxation-iteration procedure. Various simple preliminary examples are discussed, after which an example involving a rectangular region is considered in complete detail. The technique devised can be used for a region of any shape, including the case where the boundary is curved, and also for any arbitrary given load system.