Large Deflection of Plates Under Hydrostatic Pressure with Special Reference to Liquid Containers

A large-deflection stress analysis is made for square plates under hydrostatic pressure with several flexural and membrane boundary conditions, and with special reference to conditions related to flat-plate components in liquid containers and partitions. The analysis is based on von Karman's nonlinear plate equations for elastic isotropic plates using graded-mesh finite-difference approximations together with an iterative procedure. The influence on plate behavior of membrane and flexural boundary conditions is discussed. It is concluded that in thin-plated containers membrane stresses of the same order of magnitude as bending stresses develop. Solutions are offered nondimensionally in a tabular form for a number of more frequent membrane and bending boundary conditions suitable for design purposes. The application of the solutions is illustrated through numerical examples.

A Numerical Solution for the Nonlinear Deflection of Membranes

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.