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.

A Mixed Boundary-Value Problem of Elasticity With Parabolic Boundary

1957 ◽  
Vol 24 (1) ◽  
pp. 122-124
Author(s):  
Gunadhar Paria
Keyword(s):  
Stress Distribution ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Complex Variable ◽  
Hilbert Problem ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Two Dimensional ◽  
Mixed Boundary Value Problem ◽  
Parabolic Boundary

Abstract The problem of finding the stress distribution in a two-dimensional elastic body with parabolic boundary, subject to mixed boundary conditions, has been reduced to the solution of the nonhomogeneous Hilbert problem following the method of complex variable. The result has been compared with that for a straight boundary.

Modeling of Mixed Convection Between Vertical Parallel Plates in Electric Equipment Immersed in High-Pr Liquids

2019 ◽  
Vol 11 (5) ◽  
Author(s):  
Thomas B. Gradinger ◽  
T. Laneryd
Keyword(s):  
Heat Flux ◽  
Reynolds Number ◽  
Natural Convection ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Two Dimensional ◽  
Parallel Plates ◽  
One Dimensional ◽  
Electric Equipment

Natural-convection cooling with oil or other fluids of high Prandtl number plays an important role in many technical applications such as transformers or other electric equipment. For design and optimization, one-dimensional (1D) flow models are of great value. A standard configuration in such models is flow between vertical parallel plates. Accurate modeling of heat transfer, buoyancy, and pressure drop for this configuration is therefore of high importance but gets challenging as the influence of buoyancy rises. For increasing ratio of Grashof to Reynolds number, the accuracy of one-dimensional models based on the locally forced-flow assumption drops. In the present work, buoyancy corrections for use in one-dimensional models are developed and verified. Based on two-dimensional (2D) simulations of buoyant flow using finite-element solver COMSOL Multiphysics, corrections are derived for the local Nusselt number, the local friction coefficient, and a parameter relating velocity-weighted and volumetric mean temperature. The corrections are expressed in terms of the ratio of local Grashof to Reynolds number and a normalized distance from the channel inlet, both readily available in a one-dimensional model. The corrections universally apply to constant wall temperature, constant wall heat flux, and mixed boundary conditions. The developed correlations are tested against two-dimensional simulations for a case of mixed boundary conditions and are found to yield high accuracy in temperature, wall heat flux, and wall shear stress. An application example of a natural-convection loop with two finned heat exchangers shows the influence on mass-flow rate and top-to-bottom temperature difference.

Static Stability Behaviour of Plate Elements with Non-Uniform, in-Plane Stress Distribution

1979 ◽  
Vol 21 (5) ◽  
pp. 363-365
Author(s):  
P. K. Datta
Keyword(s):  
Stress Distribution ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Plane Stress ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Static Stability ◽  
Buckling Loads ◽  
Plate Elements ◽  
Edge Loading

The results of analytically and experimentally determined buckling loads of a rectangular plate, subjected to partial edge loading and mixed boundary conditions, are presented.

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 Vectorized Matlab Codes for Linear Two-Dimensional Elasticity

2007 ◽  
Vol 15 (3) ◽  
pp. 157-172 ◽  
Author(s):  
Jonas Koko
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Linear Elasticity ◽  
Numerical Experiments ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Two Dimensional ◽  
Linear Finite Element ◽  
Matlab Implementation

A vectorized Matlab implementation for the linear finite element is provided for the two-dimensional linear elasticity with mixed boundary conditions. Vectorization means that there is no loop over triangles. Numerical experiments show that our implementation is more efficient than the standard implementation with a loop over all triangles.

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