Alexander Kerzhaev

• PhD
• Senior Researcher at Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences

A method is proposed for constructing exact solutions to boundary value problems of the theory of elasticity in a rectangle with stiffeners located inside the region (an inhomogeneous problem). The solutions are represented by series in Papkovich–Fadle eigenfunctions, the coefficients of which are determined explicitly. The method is based on the P...

The paper deals with a periodic boundary value problem of the theory of elasticity for a half-strip with mixed boundary conditions at its end. The boundary conditions on the long sides correspond to the periodic continuation of the solution into a half-plane, i.e. the solution is represented in the form of trigonometric Fourier series. The construc...

The paper presents some basic points of the theory of expansions in Papkovich–Fadle eigenfunctions in the polar coordinate system. Formulas for the Papkovich–Fadle eigenfunctions corresponding to the boundary value problem of the theory of elasticity for a truncated wedge with free long sides are given. Equations for determining the functions biort...

We construct exact solutions of three boundary value problems in the theory of elasticity for an infinite strip with a central transverse cut on which constant normal stresses are specified (even‐symmetric deformation). We consider three variants of homogeneous boundary conditions on the strip sides: (1) free sides, (2) firmly clamped sides, and (3...

The paper presents examples of exact solutions to boundary value problems of the theory of elasticity for a truncated wedge. The straight sides of the wedge are free, and normal stresses are given on its circular “end” (the tangential stresses are considered equal to zero). The solutions are sought in the form of expansions in Papkovich–Fadle eigen...

The paper presents formulas describing an exact solution to the boundary value problem of the theory of elasticity in a rectangle in which the horizontal sides are rigidly clamped, and normal and tangential stresses are given on the vertical ones. Only an odd-symmetric deformation of the rectangle with respect to the horizontal axis of symmetry and...

We present the formulas that describe the exact solutions of the boundary value problems in the theory of elasticity for a half-strip and a rectangle in which the horizontal sides are firmly clamped, while normal and tangential stresses are specified on the vertical ones. We consider only an even-symmetric deformation of the half-strip and the rect...

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infin...

The article deals with a boundary value problem for a rectangle whose horizontal sides are rigidly clamped, and the ends are free. In the centre of the rectangle, a vertical cut is made on which a discontinuity of the longitudinal displacements is given. An exact solution to the problem is constructed in the form of series in Papkovich–Fadle eigenf...

We construct examples of exact solutions of the temperature problem for a square: the sides of the square are (i) free and (ii) firmly clamped. Initially, we solve the inhomogeneous problem for an infinite plane. The known exact solutions for a square, with which the boundary conditions on the sides of the square are satisfied, are added to this so...

In this paper, in the form of expansions in Papkovich–Fadle eigenfunctions, an exact solution to the boundary value problem of the theory of elasticity is constructed for a rectangle whose horizontal sides are rigidly clamped. The expansion coefficients are determined with the help of biorthogonal functions by simple formulas.

The paper presents a method for determining thermal stresses in an elastic free square plate (the plane problem). First, we construct the solution to the nonhomogeneous temperature problem for an infinite plane. Then, we add to this solution the solution for a square, with the help of which the required boundary conditions on its sides are satisfie...

This paper presents a method for determining thermal stresses in an elastic clamped square with a given temperature distribution (the plane problem). First, the solution to the temperature problem for an infinite plane is constructed. Then, the solution for a square is added to this solution, with the help of which the boundary conditions on its si...

This paper provides an exact solution to the boundary value problem of elasticity theory for a square loaded with tangential stresses along all its sides (even-symmetric deformation with respect to the central axes). The solution is represented as series in Papkovich–Fadle eigenfunctions. The coefficients of the series are determined by simple clos...

We have constructed an exact solution to the boundary value problem in the theory of elasticity for a half‐strip with identical stiffeners along its long sides and a load acting at its end (even‐symmetric deformation). The solution is represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from closed formulas. The...

The paper deals with the problem of relieving residual stresses in an elastic domain of rectangular shape with free sides as a result of the formation of a discontinuity of particular shape. First, we construct the solution to the problem of residual stresses in an infinite strip with free sides and with a central transverse cut on which a disconti...

An exact solution is constructed for the boundary value problem of the theory of elasticity in a rectangle, in which two opposite (horizontal) sides are free, and normal stresses are given on the other two (the ends of the rectangle) (even-symmetric deformation relative to the central axes). The solution is constructed in the form of expansions in...

This paper presents an exact solution to the nonhomogeneous boundary value problem of the theory of elasticity for a clamped rectangle. First, a solution to the nonhomogeneous problem for an infinite strip with clamped sides is constructed. Then, a solution for the rectangle is added to this solution, with the help of which the boundary conditions...

We derive the formulas that describe the exact solution of the boundary value problem in the theory of elasticity for a rectangle in which two opposite (horizontal) sides are free and stresses are specified (all cases of symmetry relative to the central axes) on the other two sides (rectangle ends). The formulas for a half-strip are also given. The...

In the paper, we give examples of exact solutions to two boundary value problems for plates stiffened by ribs in the form of explicit expansions in Papkovich–Fadle eigenfunctions. It is assumed that the stiffeners work only in tension-compression, and their flexural rigidity is equal to zero. The unknown expansion coefficients are determined in sim...

In the paper, on the basis of the solution of the biharmonic problem for a smooth semi-strip, we first give an exact solution to a boundary value problem for a semi-strip whose longitudinal sides are supported by stiffeners in bending and tension-compression. The solution is given in the form of explicit expansions in Papkovich-Fadle eigenfunctions...

In the paper, for the first time we give exact solutions to two nonhomogeneous boundary value problems of the theory of elasticity for a rectangle with free long sides. Inside the rectangle there are applied two equal concentrated forces directed oppositely along the horizontal axis (even-symmetric deformation). The method of solution is based on t...

In the paper, we construct an exact solution to a boundary value problem of the theory of elasticity for a rectangle in which the longitudinal sides are free, while normal and tangential stresses are given at the ends (even-symmetric deformation with respect to the central axes). The solution is represented in the form of series in Papkovich–Fadle...

In the 1950s, the method of initial functions (MIF) was developed in the Soviet Union. It rapidly became popular among research scientists, civil engineers, and, later, among strength engineers engaged in the aerospace industry. Since the MIF is known comparatively little to the Western reader, a brief overview of the publications devoted to the fo...

Based on the theoretical research results about the instability mechanism and online monitoring in more than ten years, as well as the problems about tailings dam instability, in-situ online monitoring and the failure forecast warning are researched deeply and systematically. In order to reveal the particularity of tailings dam structure, the compl...

In solving boundary value problems of elasticity theory in a half-strip in the form of series in Papkovich–Fadle eigenfunctions, there are always two representations for these functions. In the paper we consider Lagrange expansions in these representations. Lagrange expansions are the expansions of only one function in a series in any single system...

Using the boundary value problem on the bending of a thin elastic semi-infinite plate in which the long sides are free, while a self-balanced bending moment and a generalized shearing force are specified at its end as an example, we consider the main steps in constructing exact solutions to the boundary value problems of bending of thin elastic rec...

In this paper for the first time we have constructed the exact solutions of two boundary value problems of the theory of elasticity for an infinite strip with a central transverse crack on which a constant normal stress is given (even-symmetric deformation). In the first problem the sides of the strip are free, while in the second they are rigidly...

We propose a method of solving mixed boundary value problems in the theory of elasticity in an infinite horizontal strip that has two points of change in the type of boundary conditions located on the upper and lower sides of the strip and lying at the ends of a vertical segment—the line of the joint between the left and right half-strips. The solu...

We have constructed the solution of the first basic odd-symmetric boundary value problem in the theory of elasticity in a half-strip with free longitudinal sides. The solution is represented as series in Papkovich–Fadle eigenfunctions whose coefficients are found in an explicit form by using functions biorthogonal to the Papkovich–Fadle eigenfuncti...

The paper is dedicated to the classical problem of elasticity theory – to the solution of the biharmonic equation in a semi-strip (rectangle) and to some conclusions that follow from the analysis of exact solutions of the biharmonic problem. The formulas describing the distribution of stresses and displacements in the semi-strip (rectangle) with fr...