Rectangular Plates on Unilateral Edge Supports: Part 1—Theory and Numerical Analysis

1986 ◽  
Vol 53 (1) ◽  
pp. 146-150 ◽  
Author(s):  
J. P. Dempsey ◽  
Hui Li
Keyword(s):  
Singular Integral Equations ◽  
Integral Transforms ◽  
Rectangular Plates ◽  
Centrally Symmetric ◽  
Solution Methods ◽  
Dual Series Equations ◽  
Finite Integral ◽  
Loss Of Contact ◽  
Laterally Loaded ◽  
Numerical Approaches

The corners of a simply supported, laterally loaded rectangular plate must be anchored to prevent them from lifting off the supports. If no such anchors are provided, and the supports are unilateral or capable of exerting forces in one direction only, parts of the plate will bend away from the supports upon loading. The loss of contact when uplift of laterally loaded rectangular plates is not prevented is examined in this paper. Arbitrary centrally symmetric loading is considered. Finite integral transforms convert the coupled dual-series equations that result from the Levy-Nadai approach to two coupled singular integral equations. Different solution methods are applicable for sagged and unsagged supports; these two numerical approaches are discussed in detail.

Rectangular Plates on Unilateral Edge Supports: Part 2—Implementation; Concentrated and Uniform Loading

1986 ◽  
Vol 53 (1) ◽  
pp. 151-156 ◽  
Author(s):  
J. P. Dempsey ◽  
Hui Li
Keyword(s):  
Unilateral Contact ◽  
Rectangular Plates ◽  
Uniform Pressure ◽  
Computer Implementation ◽  
Concentrated Loads ◽  
Uniform Loading ◽  
Loss Of Contact ◽  
Implementation Aspects ◽  
Laterally Loaded ◽  
Support Reactions

Rectangular plates in unilateral contact with sagged and unsagged supports laterally loaded by centrally concentrated loads and uniform pressure are examined. The loss of contact and the redistribution of deflections, moments, and support reactions are presented. Computer implementation aspects are discussed.

Loss of Contact in the Vicinity of a Right-Angle Corner for a Simply Supported, Laterally Loaded Plate

1981 ◽  
Vol 48 (3) ◽  
pp. 597-600 ◽  
Author(s):  
L. M. Keer ◽  
A. F. Mak
Keyword(s):  
Integral Transform ◽  
Space Region ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Concentrated Force ◽  
Simply Supported ◽  
Quarter Space ◽  
Loss Of Contact ◽  
Restraining Force ◽  
Laterally Loaded

The solutions to problems of laterally loaded, simply supported rectangular plates are classical ones that can be found in standard textbooks. It is found that forces directed downward must be present to prevent the corners of the plate from rising up during bending. The objective of the present analysis is to determine the extent to which such a plate will rise if the corner force is not present and the plate is unilaterally constrained. Rather than determine the solution for a rectangular plate, we consider a laterally loaded, simply supported plate which occupies a quarter space region. The plate is unilaterally constrained and may rise at the corner due to an absence of restraining force there. Using integral transform techniques appropriate to the quarter space for elastic plates, the region of lost contact is determined for a general loading. The special loading due to a concentrated force is given as an example.

Contact Between Plates and Unilateral Supports

1984 ◽  
Vol 51 (2) ◽  
pp. 324-328 ◽  
Author(s):  
J. P. Dempsey ◽  
L. M. Keer ◽  
N. B. Patel ◽  
M. L. Glasser
Keyword(s):  
Integral Transform ◽  
Plate Theory ◽  
Classical Plate Theory ◽  
Dual Series Equations ◽  
Receding Contact ◽  
Finite Integral ◽  
Simple Supports ◽  
Square Plate ◽  
Laterally Loaded ◽  
Support Reactions

The tendency of a laterally loaded, unilaterally constrained, rectangular plate to separate from its simple supports motivates one to consider the actual extent of contact. In the case of a square plate, an appropriately chosen finite integral transform converts the dual series equations that result from the Levy-Nadai approach to one singular integral equation which can be solved by standard methods. Being a receding contact problem, the extent of contact depends on the geometry and elastic properties of the plate only. The support reactions are integrated to confirm that total equilibrium is obtained using classical plate theory.

Nusselt Numbers in Rectangular Ducts With Laminar Viscous Dissipation

1999 ◽  
Vol 121 (4) ◽  
pp. 1083-1087 ◽  
Author(s):  
G. L. Morini ◽  
M. Spiga
Keyword(s):  
Boundary Condition ◽  
Temperature Distribution ◽  
Viscous Dissipation ◽  
Integral Transforms ◽  
Navier Stokes ◽  
Brinkman Number ◽  
Balance Equations ◽  
Nusselt Numbers ◽  
Finite Integral ◽  
Axial Heat

In this paper, the steady temperature distribution and the Nusselt numbers are analytically determined for a Newtonian incompressible fluid in a rectangular duct, in fully developed laminar flow with viscous dissipation, for any combination of heated and adiabatic sides of the duct, in H1 boundary condition, and neglecting the axial heat conduction in the fluid. The Navier-Stokes and the energy balance equations are solved using the technique of the finite integral transforms. For a duct with four uniformly heated sides (4 version), the temperature distribution and the Nusselt numbers are obtained as a function of the aspect ratio and of the Brinkman number and presented in graphs and tables. Finally it is proved that the temperature field in a fully developed T boundary condition can be obtained as a particular case of the H1 problem and that the corresponding Nusselt numbers do not depend on the Brinkman number.

scholarly journals Dual-series equations formulation for static deformation of plates with a partial internal line support

2007 ◽  
Vol 34 (3) ◽  
pp. 221-248 ◽  
Author(s):  
Yos Sompornjaroensuk ◽  
Kraiwood Kiattikomol
Keyword(s):  
Integral Equation ◽  
Fredholm Integral Equation ◽  
Hankel Transform ◽  
Internal Line ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Distributed Load ◽  
Dual Series Equations ◽  
Finite Hankel Transform ◽  
Standard Techniques

The paper deals with the application of dual-series equations to the problem of rectangular plates having at least two parallel simply supported edges and a partial internal line support located at the centre where the length of internal line support can be varied symmetrically, loaded with a uniformly distributed load. By choosing the proper finite Hankel transform, the dual-series equations can be reduced to the form of a Fredholm integral equation which can be solved conveniently by using standard techniques. The solutions of integral equation and the deformations for each case of the plates are given and discussed in details.

Practical numerical considerations for the Alekseev–Mikhailenko method

1990 ◽  
Vol 27 (8) ◽  
pp. 1023-1030 ◽  
Author(s):  
P. F. Daley ◽  
F. Hron
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Integral Transforms ◽  
Hankel Transform ◽  
Radial Symmetry ◽  
Hyperbolic Partial Differential Equations ◽  
Finite Integral ◽  
Finite Hankel Transform ◽  
Difference Methods ◽  
New Generation

Programs that utilize the Alekseev–Mikhailenko method are becoming viable seismic interpretation aids because of the availability of a new generation of supercomputers. This method is highly numerically accurate, employing a combination of finite integral transforms and finite difference methods, for the solution of hyperbolic partial differential equations, to yield the total seismic wave field.In this paper two questions of a numerical nature are addressed. For coupled P–Sv wave propagation with radial symmetry, Hankel transforms of order 0 and 1 are required to cast the problem in a form suitable for solution by finite difference methods. The inverse series summations would normally require that the two sets of roots of the transcendental equations be employed, corresponding to the zeroes of the Bessel functions of order 0 and 1. This matter is clarified, and it is shown that both inverse series summations may be performed by considering only one set of roots.The second topic involves providing practical means of determining the lower and upper bounds of a truncated series that suitably approximates the infinite inverse series summation of the finite Hankel transform. It is shown that the number of terms in the truncated series generally decreases with increasing duration of the source pulse and that the truncated series may be further reduced if near-vertical-incidence seismic traces are avoided.

Plastic Reserve Strength of Rectangular Plates

1984 ◽  
Vol 28 (03) ◽  
pp. 173-185
Author(s):  
Ching-Yuan Chu ◽  
William S. Vorus
Keyword(s):  
Upper Bound ◽  
Load Capacity ◽  
Programming Algorithm ◽  
Rectangular Plates ◽  
Unknown Parameters ◽  
Various Boundary Conditions ◽  
Elastic Load ◽  
Minimization Procedure ◽  
Laterally Loaded ◽  
Reserve Strength

An upper-bound functional which attains its minima at collapse of thin laterally loaded plates is formulated. A general collapse pattern for various thin plate problems is constructed in terms of a set of unknown parameters. An appropriate mathematical programming algorithm is applied to search for the correct parameters in the minimization procedure. Rectangular plates under uniform load are treated specifically. The least upper-bound solutions for rectangular plates with various boundary conditions are calculated. The corresponding elastic solutions are also calculated. From these elastic and plastic solutions the reserve strength of the plate beyond its elastic load capacity is predicted.

scholarly journals An analytical and numerical approach to a bilateral contact problem with nonmonotone friction

2013 ◽  
Vol 23 (2) ◽  
pp. 263-276 ◽  
Author(s):  
Mikaël Barboteu ◽  
Krzysztof Bartosz ◽  
Piotr Kalita
Keyword(s):  
Contact Problem ◽  
Frictional Contact ◽  
Numerical Approach ◽  
Bundle Method ◽  
Nonconvex Problem ◽  
Proximal Bundle Method ◽  
Loss Of Contact ◽  
The Galerkin Method ◽  
Bilateral Contact ◽  
Numerical Approaches

We consider a mathematical model which describes the contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is bilateral, i.e., there is no loss of contact. The friction is modeled with a nonmotonone law. The purpose of this work is to provide an error estimate for the Galerkin method as well as to present and compare two numerical methods for solving the resulting nonsmooth and nonconvex frictional contact problem. The first approach is based on the nonconvex proximal bundle method, whereas the second one deals with the approximation of a nonconvex problem by a sequence of nonsmooth convex programming problems. Some numerical experiments are realized to compare the two numerical approaches.

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