Bending of a circular plate with a radial crack whose flanks are in contact

Bending of the circle isotropic plate with radial crack with contact shores is investigated. For want of solving of a problem was considered, that the shores of a crack come in contact on the upper basis on all it’s length. A solving of a problem constructed with use of methods of the theory of functions of a complex variable and complex potentials. The system of singular integral equations was untied numerically with the help of method of mechanical quadratures. The numerical analysis of a problem is conducted, because of which constructed graphic dependence of contact pressure and moments intensity factor and strain intensity factor

Bilateral bending of the plate with a circular hole and a disk in presence of a full smooth linear contact of components

У роботі досліджено задачу про двобічний згин ізотропної пластини розподіленими згинальними моментами на нескінченності за існування у пластині кругового отвору, в який вставлена без натягу кругова шайба із матеріалу пластини. Під дією зовнішнього навантаження береги шайби і отвору приходять у гладкий контакт уздовж колової лінії на одній з основ пластини. Розв’язок задачі подано у вигляді суперпозиції двох розв’язків: плоскої задачі теорії пружності і задачі згину пластини з використанням класичної теорії. За допомогою методів теорії функцій комплексної змінної та комплексних потенціалів задачу зведено до низки задач лінійного спряження, на основі яких отримано явні вирази для комплексних потенціалів, контактного зусилля між берегами шайби й отвору. З’ясовано межі зміни відношення згинальних моментів на нескінченності, коли існує розв’язок задачі у такій постановці. Подано графічні залежності для контактного зусилля між берегами кругового отвору і шайби, на основі яких можна виявити, де і за яких умов береги шайби і кругового отвору пластини відставатимуть.This paper concerns with the problem of the bilateral bending of an isotropic plate with the bending moments distributed on infinity at presence of a circular hole in a plate into which the circular disk from a plate material is inserted without a tension. Under the influence of external loading, the edges of a disk and a hole come into a smooth contact along a circular curve on one of plate bases. The problem solution is presented in the form of superposition of two solutions: a plane problem of elasticity and a problem of bending of a plate with the use of a classical theory. Using the complex variable method and complex potentials the problem is reduced to a number of problems of linear conjugation, based on which the explicit expressions for complex potentials and contact force between the disk and the hole edge are received. The limits of the bending moments ratio on infinity when there exists the problem solution in such formulation are determined. The graphic dependences for contact force between the edge of a circular hole and a disk are presented. Based on them it is possible to detect where and under what conditions the edges of a disk and a circular hole in a plate will lag behind

Bending of a plate with two equal symmetric cracks along a circular arch with allowance for the contact of their flanks

In this paper the problem of one-sided bend of a plate with two symmetric cracks along the arc of circle by bend moments on infinity including a contact of their sides in the presence of geometric and physical symmetry of a sum has been investigated. By virtue of contact of crack’s sides a task solution is given in the form of solution of two tasks: plane problem and bend (classical theory). Using complex potentials and methods of the theory complex variable quantity function, task solution has been reduced to linear conjugation, on the base of which the equation for finding the contact effort between crack’s sides is got. Evident forms for complex potentials, contact efforts between crack’s sides, coefficients of intensity efforts, moments are got for carrying out the numerical analysis, results of which are shown graphically

Reissner’s plate bilateral bending containing coaxial through-the-thickness slit and crack taking into account contact zone width of its faces

Досліджено напружено-деформований стан ізотропної пластини з прямолінійними співвісними наскрізними щілиною та тріщиною за двобічного згину розподіленими моментами на нескінченності. Береги дефектів до прикладання зовнішнього навантаження були вільними від нього, а під дією згинальних моментів на нескінченності береги тріщини прийшли у гладкий контакт уздовж області сталої ширини поблизу однієї з основ пластини. На основі методів теорії функцій комплексної змінної і комплексних потенціалів плоскої задачі теорії пружності та теорії згину пластин за Рейсснером розв’язок задачі зведено до системи сингулярних інтегральних рівнянь як на тріщині, так і на щілині, яку розв’язано числово за допомогою методу механічних квадратур. Побудовано графічні залежності для контактного зусилля між берегами тріщини, коефіцієнтів інтенсивності зусиль і моментів за різних параметрів задачі.Construction of the cracked plate bending problems solutions forms an important and actual in terms of theory development and practical applications scientific direction. They make it possible to determine stress and displacement distribution near the defects tips, as well as make recommendations for selecting optimal geometrical, physical and mechanical characteristics of plates depending on operating conditions in order to prevent destruction of engineering constructions. In this paper the stress-strain state of boundless isotropic plate with coaxial through-the-thickness slit and crack, the faces of which are free from the external loading is investigated. The plate is under the action of the uniformly distributed in a remote part bending moments, which vectors are parallel and perpendicular to the axe of the defects. It is assumed that under external loading the crack faces come in a smooth contact on all crack length along the two-dimensional area of constant width near the upper plate basis. As a result of the crack faces contact the solution of problem is presented in the form of two related problems solutions: the theory of elasticity plane problem and the problem of plates bending based on the equations of Reissner theory. On the basis of complex variable function theory methods and complex potentials the system of singular integral equations is obtained which is reduced by the mechanical quadratures method to the infinite system of linear algebraic equations. This system is solved numerically by the method of reduction using Gauss with a choice of main entry. The numerical analysis of problem at some parameters values is carried out and graphic dependences for contact force between the faces of crack, force and moment intensity factors are constructed. In particular cases known in the scientific literature results for Reissner’s plate bending problems with one crack considering the contact zone width of its faces, with two coaxial slits as well as for appropriate problems solved using equations of the classical theory of plates bending are obtained

Bilateral flexure of the plate with two symmetrical through the thickness circular arc cracks with account of their edges contact

В роботі досліджено задачу про двосторонній згин пластини з двома симетричними тріщинами по дузі кола розподіленими згинальними моментами на нескінченності з урахуванням контакту їх берегів при наявній геометричній і фізичній симетрії задачі. В силу контакту берегів тріщини розв’язок задачі подано у вигляді розв’язку двох задач: плоскої задачі та згину (класична теорія). Використовуючи комплексні потенціали і методи теорії функцій комплексної змінної, розв’язок задачі зведений до задач лінійного спряження, на основі якого отримано рівняння для знаходження контактного зусилля між берегами тріщин. Записані явні вирази для комплексних потенціалів, контактних зусиль між берегами тріщин, коефіцієнтів інтенсивності зусиль і моментів та проведено їх числовий аналіз, результати якого подано графічно.On the ground of the two-dimensional problem of the theory of elasticity and of the classical theory of plate flexure bilateral flexure of the arc cracks with extrinsic symmetric load causing smooth contact of its edges along the whole length of one of its surfaces by moments applied at infinity has been investigated. With the use of the theory of the complex variable functions methods the solution of the problem has been brought to the problems of linear conjugation and their analytic solution has been built. Numerical analysis of contact pressure and moments-intensive factors are presented in the form of diagrams

Combined Bending with Tension of Isotropic Plate with Crack Considering Crack Banks Contact and Plastic Zones at its Tops

Stress-strain state of isotropic plate with rectilinear through-crack at combined action of bending and tension, realized by applying distributed forces and bending moments at infinity, the vectors of which are parallel and perpendicular to the crack, is investigated. Under the influence of the internal stress the crack faces contacts on area of constant width near the upper base of plate, and plastic zones forms in its tips. Using methods of the theory of complex variables, complex potentials plane problem of elasticity theory and the classical theory of plates bending, solving of the problem is reduced to the set of linear conjugation problems and their analytical solution is built in a class of functions of limited plastic zones in the crack tips. The conditions of existence of the solution of the problem in these terms are determined. Using Treska plasticity conditions in the form of surface layer or the plastic hinge, the length of plastic zone and crack opening displacement are found analytically. Their numerical analysis for various parameters of the problem is conducted

Biaxial loading of a plate containing a hole and two co-axial through cracks

The paper presents the solution linear elasticity problem for an isotropic plate weakened by a hole and two co-axial cracks. The plate is exerted by uniform traction at infinity. The corresponding 2D problem is solved by the method of Kolosova-Muskhelishvili complex potentials. The method implies reduction of the problem to simultaneous singular integral equations (SIE) for the functions defined the region of the cracks and hole. For particular case the solution the SIE is obtained analytically in a closed form. A thorough analysis of the stress intensity factors (SIF) is carried out for various cases of the hole shape: penny-shaped, elliptical and rectangular

Pure bending of strip (beam) with the arbitrarily oriented cross-cutting crack

The problem of pure bending of strip (beam) with transverse rectilinear crack, edges of which are free from acuter load, is investigated in this paper. Under bending moment its edges may not contact or smoothly contact throughout its area length or part. Dependently on where it is located.Using methods of theory of functions of complex variable and complex potentials the problem at issue has been reduced to the problems of linear conjugation, their analytical solution is found. Explicit expressions on complex potentials is written. Based on the energy criterion of destruction stress intensity factors are determined. Limit value of moment when the crack begins to propagate is found. For the case when crack edges partially contact, area length of contact of her edges is determined. Numerical analysis of critical moment of failure of strip (beams) is performed under various parameters of the problem, which are related to the mechanical state of crack. The corresponding graphic dependencies are constructed.Key words: crack, strip (beam), complex potentials, bending moment, stress intensity factors.Pages of the article in the issue: 142-145Language of the article: Ukrainia

Combined bending with tension of isotropic plate with crack considering crack banks contact and plastic zones at its tops

Stress-strain state of isotropic plate with rectilinear through-crack at combined action of bending and tension, realized by applying distributed forces and bending moments at infinity, the vectors of which are parallel and perpendicular to the crack, is investigated. Under the influence of the internal stress the crack faces contacts on area of constant width near the upper base of plate, and plastic zones forms in its tips. Using methods of the theory of complex variables, complex potentials plane problem of elasticity theory and the classical theory of plates bending, solving of the problem is reduced to the set of linear conjugation problems and their analytical solution is built in a class of functions of limited plastic zones in the crack tips. The conditions of existence of the solution of the problem in these terms are determined. Using Treska plasticity conditions in the form of surface layer or the plastic hinge, the length of plastic zone and crack opening displacement are found analytically. Their numerical analysis for various parameters of the problem is conducted