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 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.

Solutions of a singular integro-differential equation related to the adhesive contact problems of elasticity theory

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nugzar Shavlakadze ◽  
Otar Jokhadze
Keyword(s):  
Differential Equation ◽  
Analytic Functions ◽  
Contact Problems ◽  
Approximate Solutions ◽  
Integral Transforms ◽  
Algebraic Equations ◽  
Adhesive Interaction ◽  
Normal Contact ◽  
Integro Differential Equation ◽  
Linear Algebraic Equations

Abstract Exact and approximate solutions of a some type singular integro-differential equation related to problems of adhesive interaction between elastic thin half-infinite or finite homogeneous patch and elastic plate are investigated. For the patch loaded with vertical forces, there holds a standard model in which vertical elastic displacements are assumed to be constant. Using the theory of analytic functions, integral transforms and orthogonal polynomials, the singular integro-differential equation is reduced to a different boundary value problem of the theory of analytic functions or to an infinite system of linear algebraic equations. Exact or approximate solutions of such problems and asymptotic estimates of normal contact stresses are obtained.

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.

scholarly journals Mathematical Modeling of the Contact Interaction of Two Elastic Bodies Using the Mortar Method

2018 ◽  
pp. 26-44
Author(s):  
I. V. Stankevich ◽  
P. S. Aronov
Keyword(s):  
Finite Elements ◽  
Contact Line ◽  
Relaxation Method ◽  
Contact Problems ◽  
Algebraic Equations ◽  
Mortar Method ◽  
Classical Formulation ◽  
Linear Algebraic Equations ◽  
Different Types ◽  
Lagrange Functional

The article discusses the development of an algorithm for solving contact problems of elasticity theory. Solving such problems is often associated with necessity of using mismatched grids. Their joining can be carried out both with the help of iterative procedures that form the so-called Schwarz alternating methods, and with the help of the Lagrange multipliers method or the penalty method. The algorithm constructed in the article uses the mortar method for matching the finite elements on the contact line. All these methods of joining the grids make it possible to ensure continuity of displacements and stresses near the contact line. However, one of the main advantages of the mortar method is the possibility of independent choice of different types of finite elements and form functions both on both boundaries of two bodies on the contact line, and when integrating along it. The application of this method in conjunction with the classical formulation of the finite element method based on the minimization of the Lagrange functional leads to a system of linear algebraic equations with a saddle point. The article discusses in detail its numerical solution based on the modified symmetric successive upper relaxation method.The results of the constructed algorithm are demonstrated on three test contact problems. They analyze the stress-strain state of differently loaded contacting two-dimensional plates. The examples considered show that continuity of the displacements of displacements and stresses is preserved near the contact line. The versatility of the developed algorithm leaves the possibility of further analysis of the effectiveness of the mortar method using different types of finite elements and form functions.

scholarly journals Contact interaction of a predeformed plate which lies without friction on rigid base with a parabolic indenter

2021 ◽  
Vol 102 (2) ◽  
pp. 87-95
Author(s):  
Hryhorii Habrusiev ◽  
Iryna Habrusieva
Keyword(s):  
Contact Interaction ◽  
Rigid Base ◽  
Algebraic Equations ◽  
Dual Integral Equations ◽  
Linear Algebraic Equations ◽  
Vertical Displacements ◽  
Smooth Contact ◽  
Incompressible Solids ◽  
Initial Strains

Within the framework of linearized formulation of a problem of the elasticity theory, the stress-strain state of a predeformed plate, which is modeled by a prestressed layer, is analyzed in the case of its smooth contact interaction with a rigid axisymmetric parabolic indenter. The dual integral equations of the problem are solved by representing the quested-for functions in the form of a partial series sum by the Bessel functions with unknown coefficients. Finite systems of linear algebraic equations are obtained for determination of these coefficients. The influence of the initial strains on the magnitude and features of the contact stresses and vertical displacements on the surface of the plate is analyzed for the case of compressible and incompressible solids. In order to illustrate the results, the cases of the Bartenev – Khazanovich and the harmonic-type potentials are addressed.

The Analysis of Contact Stresses in Rolling Element Bearings

1979 ◽  
Vol 101 (1) ◽  
pp. 105-109 ◽  
Author(s):  
M. J. Hartnett
Keyword(s):  
Three Dimensional ◽  
Contact Problems ◽  
Rolling Element Bearing ◽  
Rolling Element Bearings ◽  
Algebraic Equations ◽  
Linear Elastic ◽  
Linear Algebraic Equations ◽  
Rolling Element ◽  
Contact Symmetry ◽  
Force Displacement

A numerical solution is presented which can be used to analyze the complete range of frictionless contact problems found in rolling element bearings. A three dimensional, linear elastic solution to the problem is developed by combining the Boussinesq force-displacement relationships for a half-space with a modified flexibility method. In this manner a stable system of linear algebraic equations in terms of the unknown surface pressures is formed, with no restrictions placed upon either contact symmetry or material connectivity. Several numerical examples of common but hitherto unsolved contact problems prevalent in rolling element bearing applications are also presented.

Asymptotic Solutions for Contact Problems: The Effect of an Internal Discontinuity in Surface Profile

2006 ◽  
Vol 220 (4) ◽  
pp. 387-391 ◽  
Author(s):  
C M Churchman ◽  
A Sackfield ◽  
D A Hills
Keyword(s):  
Contact Problem ◽  
Contact Pressure ◽  
Half Plane ◽  
Surface Profile ◽  
Contact Problems ◽  
Asymptotic Solutions ◽  
Plane Contact ◽  
Plane Contact Problem ◽  
Full Solution

The contact pressure adjacent to the apex of a tilted punch is studied and used to form a refined, two-term asymptote for the contact pressure at a point of discontinuous gradient interior to a half-plane contact problem. The asymptote is compared with the full solution for an example problem, the wheel with a flat.

The Contact Problem of a Plate Pressed Between Two Spheres

1964 ◽  
Vol 31 (4) ◽  
pp. 659-666 ◽  
Author(s):  
Yih-O Tu ◽  
D. C. Gazis
Keyword(s):  
Contact Problem ◽  
Legendre Polynomials ◽  
Plate Thickness ◽  
Elastic Contact ◽  
Algebraic Equations ◽  
Total Load ◽  
Linear Algebraic Equations ◽  
Even Order ◽  
Axisymmetric Bodies ◽  
Maximum Contact

The elastic contact of a plate with two axisymmetric bodies, generally of dissimilar elastic properties and curvatures, is considered. The solution of the resulting two integral equations is given as a truncated series of Legendre polynomials of even order whose coefficients may be determined from a system of linear algebraic equations. The relationships between the total load, radii of curvature of the bodies, contact radii, approach, radial displacements, maximum contact stress, and the plate thickness can then be computed in terms of these coefficients. Numerical computations have been carried out for the contact of a plate with two identical spheres.

Hertz Contact Problem between Wheel and Rail

2013 ◽  
Vol 837 ◽  
pp. 733-738 ◽  
Author(s):  
Tiberiu Axinte
Keyword(s):  
Plastic Deformation ◽  
Finite Element ◽  
Contact Problem ◽  
Contact Pressure ◽  
Three Dimensional ◽  
Contact Problems ◽  
Rolling Contact ◽  
Hertz Contact ◽  
Shakedown Limit ◽  
Plastic Shakedown

Rail-wheel contact problems have been analyzed by the use of the three-dimensional finite element models. Based on these models, the paper presents a study regarding the applicability of the Hertz contact to rail-wheel contact problems. Beside a standard rail, the study also considers a crane rail and a switching component. The bodies of the contact problem are the standard rail UIC60 and the standard wheel UICORE. The maximum contact pressure which the material can support in the elastic range in steady state conditions is known as the shakedown limit. With an operating contact pressure below the shakedown limit the rail would be expected to remain elastic a long period of its lifecycle. However, examination of rail cross-sections shows severe plastic deformation in a sub-surface layer of a few tens of microns thickness; the contact patch size is in tens of millimeters. Three-dimensional elastic-plastic rolling contact stress analysis was conducted incorporating elastic and plastic shakedown concepts. The Hertzian distribution was assumed for the normal surface contact load over a circular contact area. The tangential forces in both the rolling and lateral directions were considered and were assumed to be proportional to the Hertzian pressure. The elastic and plastic shakedown limits obtained for the three-dimensional contact problem revealed the role of both longitudinal and lateral shear traction on the shakedown results. An advanced cyclic plasticity model was implemented into a finite element code via the material subroutine. Finite element simulations were conducted in order to study the influences of the tangential surface forces in the two shear directions on residual stresses and residual strains. The Hertz theory is restricted to frictionless surfaces and perfectly elastic solids, but it is the best method for determining deformations and stress from pitch of contact. Form change due to wear and plastic deformation of a rail can reduce the service life of a track. The purpose of this investigation was to study the development of these damage mechanisms on new and three years old rails in a commuter track over a period of two years.

scholarly journals РЕШЕНИЕ ПРОСТРАНСТВЕННОЙ КОНТАКТНОЙ ЗАДАЧИ ШАРНИРНЫХ УЗЛОВ ОПИРАНИЯ БАЛКИ НА УПРУГИЕ ЧЕТВЕРТЬПРОСТРАНСТВО И ОДНУ ВОСЬМУЮ ПРОСТРАНСТВА

2020 ◽  
Author(s):  
S. Bosakov ◽  
P. Skachok
Keyword(s):  
Contact Problem ◽  
Iterative Process ◽  
Beam Width ◽  
Contact Stresses ◽  
Elastic Space ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Quarter Space ◽  
Left And Right ◽  
Flexibility Index

The article discusses the solution of the spatial contact problem arising when calculating a reinforced concrete rafter beam pivotally supported by concrete walls. The walls are modeled by the elastic quarter-space on the left and by one-eighth of the elastic space on the right. This contact problem is solved using the numerical method - the Zhemochkin method. For this purpose, the contact area is divided into fragments. Rigid one-way ties are set in the center of each fragment to implement contact between the beam and the wall. It is assumed that the forces arising in these ties provide uniform distribution of reactive pressures in the appropriate fragment. Then, the system of linear algebraic equations for the mixed method of structural mechanics shall be prepared and solved. Different Green functions are assumed for the left and right wall. The problem under consideration is nonlinear, and it requires an iterative process to calculate the effective area of contact and the values of the related reactive pressures. The iterative process shall be finished when contact stresses at the boundary of separation of the structure from the walls are identically equal to zero, or when there are no stretched Zhemochkin ties. Isolines of contact stresses and vertical displacements of the contact areas of the walls are plotted for the flexibility index corresponding to the real ratio of rigidity of supported structures and the flexibility index corresponding to the support of the absolutely rigid beam. The function is found, describing the torque arising in the beam versus the distance from the edge of one eighth of the elastic space. A beam can be considered as supported on the left and right by the elastic quarter-space when the distance from the beam axis and the edge of one-eighth of the space exceeds the twofold beam width. В статье рассматривается решение пространственной контактной задачи, возникающей при расчете железобетонной стропильной балки, шарнирно опираемой на бетонные стены. Стены моделируются слева упругим четвертьпространством и справа -одной восьмой пространства. Данная контактная задача решается с использованием численного метода -метода Б. Н. Жемочкина. Для этого область контакта разбивается на участки. В центрах каждого участка устанавливаются жесткие односторонние связи, через которые осуществляется контакт балки со стеной. При этом предполагается, что усилия, возникающие в установленных связях, вызывают равномерное распределение реактивных давлений в соответствующем участке. Далее составляется и решается система линейных алгебраических уравнений смешанного метода строительной механики. Для левой и правой стен принимаются различные функции Грина. Рассматриваемая задача является нелинейной и требует итерационного процесса для определения фактической области контакта с величинами соответствующих реактивных давлений. Моментом окончания итерационного процесса служит тождественное равенство нулю контактных напряжений на границе отрыва конструкции от стен либо отсутствие растянутых связей Б. Н. Жемочкина. Построены изолинии контактных напряжений и вертикальных перемещений контактных областей стен при показателе гибкости, соответствующем реальному соотношению жесткостей опираемых конструкций, и показателе гибкости, соответствующем опиранию абсолютно жесткой балки. Установлена зависимость возникающего крутящего момента в балке от расстояния до края одной восьмой упругого пространства. Балку можно считать как опираемую слева и справа на упругое четвертьпространство, когда расстояние от оси балки и края одной восьмой пространства превышает двойную ширину балки.

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