scholarly journals Relaxation methods applied to engineering problems. VIII. Plane-potential problems involving specified normal gradients

1943 ◽  
Vol 182 (989) ◽  
pp. 129-151 ◽  
Keyword(s):  
Harmonic Function ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Boundary Values ◽  
Approximate Computation ◽  
Relaxation Methods ◽  
Potential Problems ◽  
Engineering Problems ◽  
Direct Attack

Since every plane-harmonic function is associated with a conjugate, problems in which normal gradients are specified on the boundary can be transformed into problems in which boundary values are specified. There then remains, however, the problem of deducing a function ψ from its conjugate ϕ, and this, when the conjugate has been determined only approximately, entails uncertainties which were exemplified in Part V. To minimize the errors of approximate computation ψ and ϕ should be determined severally and independently, consequently a method of direct attack is still needed on problems in which normal gradients are specified. Recent applications have, moreover, presented cases in which the boundary conditions are ‘mixed’, i.e. values are specified at some parts of the boundary, gradients at others. Here, two methods are propounded for the satisfaction of mixed boundary conditions, the first applicable also to cases in which normal gradients alone are specified. Test examples indicate that the wanted extension of method is now available.

scholarly journals Mixed boundary conditions in the relaxational treatment of biharmonic problems (plane strain or stress)

1947 ◽  
Vol 189 (1019) ◽  
pp. 535-543 ◽  
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Plane Strain ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Elastic Plates ◽  
Relaxation Methods ◽  
Valuable Alternative ◽  
Engineering Problems ◽  
Biharmonic Problems

Relaxation methods have already been applied to the solution of four problems of (i) extension and (ii) flexure of flat elastic plates, in which ( a ) displacement or ( b ) traction is specified at the boundary. Here the method is adapted to the case in which the two types of boundary condition are mixed, where photo-elastic methods are difficult to apply. Two examples are treated by relaxation methods, and the results obtained indicate that this method may be a valuable alternative in engineering problems.

scholarly journals Relaxation methods applied to engineering problems - Biharmonic analysis as applied to the flexure and extension of flat elastic plates

1945 ◽  
Vol 239 (810) ◽  
pp. 419-460 ◽  
Keyword(s):  
Boundary Conditions ◽  
Elastic Plate ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Relaxation Method ◽  
Elastic Plates ◽  
Relaxation Methods ◽  
Engineering Problems ◽  
Edge Conditions

By extension of technique described in earlier papers, biharmonic analysis and the solution of the equation V%> = W are brought within range of the relaxation method. Special attention is given to the problem of a fiat elastic plate which is either bent or stretched (the second case being that to which photo-elastic methods are commonly applied). In all, four cases are presented, since the edge conditions may specify either tractions or displacements both in the flexural and in the extensional problem: one example of each is treated. Mixed boundary conditions (tractions specified at some points, displacements at others) are not considered in this paper. It would seem that only slight modifications of method will be required to deal with acolo-tropic plates (which present much greater difficulties in an orthodox analysis).

scholarly journals Benchmark Studies for the Generalized Streamwise Periodic Heat Transfer Problem

2008 ◽  
Vol 130 (11) ◽  
Author(s):  
Steven B. Beale
Keyword(s):  
Heat Transfer ◽  
Boundary Conditions ◽  
Transfer Problem ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Wall Boundary Conditions ◽  
Engineering Problems ◽  
Complex Engineering ◽  
The Common ◽  
Periodic Heat

This is a comparison of calculations performed with a scheme for handling streamwise-periodic boundary conditions with known solutions to the common problem of fully developed heat transfer in a plane duct. Constant value, constant flux, mixed boundary conditions, and linear wall flux (conjugate heat transfer) are all considered. Agreement is, in every case, near exact showing that the methodology may be applied with confidence to complex engineering problems with a variety of thermal wall boundary conditions.

Relaxation Methods Applied to Two Problems of Two-Dimensional Stress Distribution Involving Mixed Boundary Conditions

1948 ◽  
Vol 1 (2) ◽  
pp. 135
Author(s):  
WH Wittrick ◽  
W Howard
Keyword(s):  
Stress Distribution ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Two Dimensional ◽  
Stress Distributions ◽  
Relaxation Methods

Relaxation methods have been used to determine the stress distributions in both a rectangular and a highly tapered plate under tension when the load is applied through absolutely rigid clamps. Both problems require the treatment of boundary conditions involving the values of both stresses and displacements. The solutions were obtained in terms of displacements and the stresses were subsequently determined from them.

scholarly journals Solution of a Problem Linear Plane Elasticity with Mixed Boundary Conditions by the Method of Boundary Integrals

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Nahed S. Hussein
Keyword(s):  
Boundary Conditions ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Stress Component ◽  
Mixed Boundary Conditions ◽  
Plane Elasticity ◽  
Boundary Integral ◽  
Unknown Boundary ◽  
Algebraic Equations ◽  
Boundary Values

A numerical boundary integral scheme is proposed for the solution to the system of…eld equations of plane. The stresses are prescribed on one-half of the circle, while the displacements are given. The considered problem with mixed boundary conditions in the circle is replaced by two problems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way, and the problem at this stage is reduced to the solution to a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution and the unknown boundary values of stresses or displacements on proper parts of the boundary. On the basis of the obtained results, it is inferred that a stress component has a singularity at each of the two separation points, thought to be of logarithmic type. The results are discussed and boundary plots are given. We have also calculated the unknown functions in the bulk directly from the given boundary conditions using the boundary collocation method. The obtained results in the bulk are discussed and three-dimensional plots are given. A tentative form for the singular solution is proposed and the corresponding singular stresses and displacements are plotted in the bulk. The form of the singular tangential stress is seen to be compatible with the boundary values obtained earlier. The efficiency of the used numerical schemes is discussed.

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