The Problem of an Elastic Stiffener Bonded to a Half Plane

1971 ◽  
Vol 38 (4) ◽  
pp. 937-941 ◽  
Author(s):  
F. Erdogan ◽  
G. D. Gupta
Keyword(s):  
Mechanical Properties ◽  
Integral Equation ◽  
Contact Problem ◽  
Half Plane ◽  
Elastic Constants ◽  
Infinite System ◽  
Stress Singularity ◽  
Algebraic Equations ◽  
Numerical Example ◽  
Linear Algebraic Equations

The contact problem of an elastic stiffener bonded to an elastic half plane with different mechanical properties is considered. The governing integral equation is reduced to an infinite system of linear algebraic equations. It is shown that, depending on the value of a parameter which is a function of the elastic constants and the thickness of the stiffener, the system is either regular or quasi-regular. A complete numerical example is given for which the strength of the stress singularity and the contact stresses are tabulated.

Simultaneous Wiener–Hopf equations

1980 ◽  
Vol 58 (3) ◽  
pp. 420-428 ◽  
Author(s):  
A. D. Rawlins
Keyword(s):  
Integral Equation ◽  
Half Plane ◽  
Fredholm Integral Equation ◽  
Coupled System ◽  
Algebraic Equations ◽  
Branch Points ◽  
Linear Algebraic Equations ◽  
Upper Half Plane ◽  
Infinite Set ◽  
Special Case

In Noble's book The Wiener Hopf Technique, Pergamon, 1958, he considers the coupled system of Wiener–Hopf equations (§4.4, pp. 153–154)[Formula: see text]He shows that provided the functions L(α), M(α), Q(α), and R(α) have only simple pole singularities the solution can be reduced to two sets of infinite simultaneous linear algebraic equations. In this article a different approach is used which gives the solution in the form of a Fredholm integral equation of the second kind. This Fredholm integral equation can be reduced to infinite sets of simultaneous linear algebraic equations under the less restrictive conditions that either (i) L(α)/M(α) has no branch points in the lower α-half plane: Im(α) < τ+; or (ii) Q(α)/R(α) has no branch points in the upper α-half plane: Im(α) > τ−. In the special case considered by Noble if L(α)/M(α) (or Q(α)/R(α)) only have simple poles in the lower (upper) half plane then the Fredholm integral equation reduces to one infinite set of simultaneous equations. This extends the Wiener–Hopf technique to yet a larger class of boundary value problems, and simplifies the numerical computations.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

A Diagonally Dominant Solution for the Cylinder End Problem

1981 ◽  
Vol 48 (4) ◽  
pp. 876-880 ◽  
Author(s):  
T. D. Gerhardt ◽  
Shun Cheng
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Infinite Cylinder ◽  
Algebraic Equations ◽  
Precise Calculation ◽  
Linear Algebraic Equations ◽  
Present Solution ◽  
Diagonally Dominant ◽  
Expansion Coefficients

An improved elasticity solution for the cylinder problem with axisymmetric torsionless end loading is presented. Consideration is given to the specification of arbitrary stresses on the end of a semi-infinite cylinder with a stress-free lateral surface. As is known from the literature, the solution to this problem is obtained in the form of a nonorthogonal eigenfunction expansion. Previous solutions have utilized functions biorthogonal to the eigenfunctions to generate an infinite system of linear algebraic equations for determination of the unknown expansion coefficients. However, this system of linear equations has matrices which are not diagonally dominant. Consequently, numerical instability of the calculated eigenfunction coefficients is observed when the number of equations kept before truncation is varied. This instability has an adverse effect on the convergence of the calculated end stresses. In the current paper, a new Galerkin formulation is presented which makes this system of equations diagonally dominant. This results in the precise calculation of the eigenfunction coefficients, regardless of how many equations are kept before truncation. By consideration of a numerical example, the present solution is shown to yield an accurate calculation of cylinder stresses and displacements.

scholarly journals A convergence theorem for singular integral equations

1981 ◽  
Vol 22 (4) ◽  
pp. 539-552 ◽  
Author(s):  
David Elliott
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Convergence Theory ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

AbstractThe principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

scholarly journals Two Contact Problems for Annular Rigid Stamp on an Elastic Half Space

2018 ◽  
Vol 17 (6) ◽  
pp. 458-464
Author(s):  
S. V. Bosakov
Keyword(s):  
Functional Equation ◽  
Half Space ◽  
Bessel Functions ◽  
Infinite System ◽  
Annular Plate ◽  
Contact Problems ◽  
Elastic Half Space ◽  
Contact Stresses ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

The paper presents solutions of two contact problems for the annular plate die on an elastic half-space under the action of axisymmetrically applied force and moment. Such problems usually arise in the calculation of rigid foundations with the sole of the annular shape in chimneys, cooling towers, water towers and other high-rise buildings on the wind load and the load from its own weight. Both problems are formulated in the form of triple integral equations, which are reduced to one integral equation by the method of substitution. In the case of the axisymmetric problem, the kernel of the integral equation depends on the product of three Bessel functions. Using the formula to represent two Bessel functions in the form of a double row on the works of hypergeometric functions Bessel function, the problem reduces to a functional equation that connects the movement of the stamp with the unknown coefficients of the distribution of contact stresses. The resulting functional equation is reduced to an infinite system of linear algebraic equations, which is solved by truncation. Under the action of a moment on the annular plate  die, the distribution of contact stresses is searched as a series by the products of the Legendre attached functions with a weight corresponding to the features in the contact stresses at the die edges. Using the spectral G. Ya. Popov ratio for the ring plate, the problem is again reduced to an infinite system of linear algebraic equations, which is also solved by the truncation method. Two examples of calculations for an annular plate die on an elastic half-space on the action of axisymmetrically applied force and moment are given. A comparison of the results of calculations on the proposed approach with the results for the round stamp and for the annular  stamp with the solutions of other authors is made.

scholarly journals Diffraction of the field of vertical electric dipole on the spiral conductive sphere in the presence of a cone

2018 ◽  
Keyword(s):  
Hilbert Space ◽  
Electric Dipole ◽  
Infinite System ◽  
Matrix Operator ◽  
Algebraic Equations ◽  
Problem Statement ◽  
Vertical Electric Dipole ◽  
Linear Algebraic Equations ◽  
Method Of Regularization ◽  
The Matrix

The problem of diffraction of a vertical electric dipole field on a spiral conductive sphere and a cone has been solved. By the method of regularization of the matrix operator of the problem, an infinite system of linear algebraic equations of the second kind with a compact matrix operator in Hilbert space $\ell_2$ is obtained. Some limiting variants of the problem statement are considered.

scholarly journals STADY OF CONTACT INTERACTION OF ELEMENT OF BREAST-WALL MADE OF UTILIZED TYRES WITH RETAINING GROUND

2016 ◽  
Vol 3 (2) ◽  
pp. 55-57
Author(s):  
Фахраддин Габибов ◽  
Fakhraddin Gabibov ◽  
Намик Халафов ◽  
Namik Khalafov
Keyword(s):  
Contact Problem ◽  
Contact Pressure ◽  
Contact Interaction ◽  
Contact Problems ◽  
Algebraic Equations ◽  
Combined Model ◽  
Internal Surfaces ◽  
Linear Algebraic Equations ◽  
Breast Wall

They work out various kinds of breast-walls in which tyres are threaded on the columns deepened into the ground. The space between the internal surfaces of tyres and external surfaces of columns is filled with compressed ground or cement-ground. Using Vlasov–Leontyev combined model they consider the problem of contact interaction of round elements of mentioned constructions with elastic padding at the contact. Basing on the method of Furie rows for solution of contact problems they get endless system of linear algebraic equations of Furie coefficients. As a result of solution of contact problem they find wanted values of deformation and contact pressure

scholarly journals Numerical solution of a weakly singular integral equation by the method of orthogonal polynomials

2021 ◽  
pp. 98-103
Author(s):  
Sergei M. Sheshko
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Orthogonal Polynomials ◽  
Numerical Solution ◽  
Singular Integral ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Expansion Coefficients ◽  
Analytical Expressions

A scheme is constructed for the numerical solution of a singular integral equation with a logarithmic kernel by the method of orthogonal polynomials. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

scholarly journals On the stationary vibrations of a rectangular plate subjected to stress prescribed partially at the circumference

1991 ◽  
Vol 14 (3) ◽  
pp. 581-586
Author(s):  
M. G. El Sheikh ◽  
A. H. Khater ◽  
D. K. Callebaut
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Rectangular Plate ◽  
Singular Integral ◽  
Fourier Transformation ◽  
Algebraic System ◽  
Infinite System ◽  
Algebraic Equations ◽  
Periodical Problem

The stationary periodical problem of a vibrating rectangular plate, stressed at a segment while fixed elsewhere at one of its edges, is considered. Using the finite Fourier transformation, the problem is converted to a singular integral equation that in turn can be reduced to an infinite system of algebraic equations. The truncation of the algebraic system is justified.

scholarly journals Nonsteady frictional heating on a sliding contact

2018 ◽  
Vol 226 ◽  
pp. 03030
Author(s):  
Vladimir B. Zelentsov ◽  
Boris I. Mitrin
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Contact Problem ◽  
Half Plane ◽  
Frictional Heating ◽  
Contact Stresses ◽  
Sliding Contact ◽  
Singular Equation ◽  
Static Contact ◽  
Class Of Functions

We consider quasi-static contact problem on frictional heating on a sliding contact of a rotating rigid cylinder and a half-plane. The cylinder is pressed towards the half-plane material. The problem is reduced to solution of a singular integral equation with respect to contact stresses. Solution of the singular equation is looked for in a class of functions limited on the edge, with two additional conditions to determine timedependent boundaries of the contact area. Temperature at the contact and inside the half-plane is determined in terms of contact stresses.

Sign in / Sign up

Export Citation Format

Share Document