A Numerical Solution for the Nonlinear Deflection of Membranes

1954 ◽  
Vol 21 (2) ◽  
pp. 117-128
Author(s):  
F. S. Shaw ◽  
N. Perrone
Keyword(s):  
Normal Displacement ◽  
Normal Forces ◽  
Finite Difference Approximations ◽  
Method Of Solution ◽  
Plate Equations ◽  
Complete Detail ◽  
Flat Membranes ◽  
Relaxation Iteration ◽  
Nonlinear Plate Equations ◽  
Numerical Method Of Solution

Abstract 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.

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

1972 ◽  
Vol 16 (04) ◽  
pp. 261-270
Author(s):  
B. Aalami
Keyword(s):  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Special Reference ◽  
Large Deflection ◽  
Finite Difference Approximations ◽  
Isotropic Plates ◽  
Plate Equations ◽  
Bending Stresses ◽  
Order Of Magnitude ◽  
Nonlinear Plate Equations

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.

Numerical Methods for Solving Physically Nonlinear Problems for Inhomogeneous Thick-Walled Shells

2017 ◽  
Vol 865 ◽  
pp. 325-330 ◽  
Author(s):  
Vladimir I. Andreev ◽  
Lyudmila S. Polyakova
Keyword(s):  
Numerical Method ◽  
Nonlinear Problems ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Deformation Diagram ◽  
Method Of Solution ◽  
Properties Of Materials ◽  
Physical Fields ◽  
Cylinders And Spheres ◽  
Numerical Method Of Solution

The paper proposes the numerical method of solution the problems of calculation the stress state in thick-walled cylinders and spheres from physically nonlinear inhomogeneous material. The urgency of solved problem due to the change of mechanical properties of materials under the influence of different physical fields (temperature, humidity, radiation, etc.). The deformation diagram describes the three-parameter formula. The numerical method used the method of successive approximations. The results of numerical calculation are compared with the test analytical solutions obtaining the authors with some restrictions on diagram parameters. The obtained results can be considered quite satisfactory.

Structural acoustic problem and dynamic nonlinear plate equations

2020 ◽  
pp. 1-21
Author(s):  
Jaouad Oudaani ◽  
Mustapha Raïssouli ◽  
Abdelkrim El Mouatasim
Keyword(s):  
Nonlinear Plate ◽  
Acoustic Problem ◽  
Plate Equations ◽  
Nonlinear Plate Equations

Analysis of Shells of Revolution Subjected to Symmetrical and Nonsymmetrical Loads

1964 ◽  
Vol 31 (3) ◽  
pp. 467-476 ◽  
Author(s):  
A. Kalnins
Keyword(s):  
Direct Integration ◽  
Shells Of Revolution ◽  
Natural Boundary ◽  
First Order ◽  
Rotationally Symmetric ◽  
Dependent Variables ◽  
Method Of Solution ◽  
Natural Boundary Conditions ◽  
Numerical Method Of Solution ◽  
Theory Of Shells

The boundary-value problem of deformation of a rotationally symmetric shell is stated in terms of a new system of first-order ordinary differential equations which can be derived for any consistent linear bending theory of shells. The dependent variables contained in this system of equations are those quantities which appear in the natural boundary conditions on a rotationally symmetric edge of a shell of revolution. A numerical method of solution which combines the advantages of both the direct integration and the finite-difference approach is developed for the analysis of rotationally symmetric shells. This method eliminates the loss of accuracy encountered in the usual application of the direct integration approach to the analysis of shells. For the purpose of illustration, stresses and displacements of a pressurized torus are calculated and detailed numerical results are presented.

ON FLAME PROPAGATION IN EXPLOSIVE MIXTURES OF GASES: I. GENERAL THEORY

1956 ◽  
Vol 34 (3) ◽  
pp. 313-323 ◽  
Author(s):  
Robert Sandri
Keyword(s):  
Temperature Distribution ◽  
Flame Propagation ◽  
Energy Equation ◽  
Approximation Formula ◽  
Flame Velocity ◽  
System Of Differential Equations ◽  
Flame Zone ◽  
Method Of Solution ◽  
Set Up ◽  
Numerical Method Of Solution

The system of differential equations of flame propagation is set up and discussed. It is shown that, without any major influences being neglected, the energy equation can be reduced to the form[Formula: see text]with the boundary conditions[Formula: see text]Some qualities of the solutions of this equation are discussed and a simple numerical method of solution is described. The flame velocity V0 is found as an eigenvalue of the energy equation. The temperature distribution in the flame zone can then be found by an ordinary quadrature. Further, an approximation formula for finding V0 directly is derived[Formula: see text]where F(η) is proportional to [Formula: see text]and has a maximum for η = ηm.

The numerical solution of certain integral equations with non-integrable kernels arising in the theory of crack propagation and elastic wave diffraction

1969 ◽  
Vol 265 (1163) ◽  
pp. 353-381 ◽  
Keyword(s):  
Integral Equation ◽  
Elastic Wave ◽  
Numerical Solution ◽  
Wave Diffraction ◽  
Crack Surface ◽  
Relative Displacement ◽  
Method Of Solution ◽  
Set Up ◽  
Prestressed State ◽  
Numerical Method Of Solution

An infinite elastic medium is initially at rest in a prestressed state of plane- or anti-plane strain. At time t = 0 a plane crack comes into existence which occupies a strip parallel to the y axis and whose width varies in time. Assuming that the components of the traction are known on the crack surface it is possible to set up an integral equation on the area of the crack for the relative displacement across the crack. Although the kernel of this integral equation is non-integrable a method is found for discretizing it and a numerical method of solution is carried out. The results, which in some cases are the solutions of diffraction problems, are presented graphically.

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